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Rotor module for SDT, User Manual

1. Velocity computations are currently incorrect with u4 ignored in the rotation effect So that viscous damping loads can be added 1FCalr Rra K 2n o tala 5 11 FCB g K tale tB a For a linearization around a given state needed for frequency domain computa tions or building a sensor observation matrix qac Rar QAL 525 In global basis stiffness matrix of a celas link is given by R u E 5 13 which leads to the following stiffness matrix H 0 k 1 I 0p k I RE 5 14 0 I I I 0 I Rar I l where qa DOFs are in the local basis motion relative to the shaft in its initial position and qg are in the global frame data describing this link is stored in model stack as a p spring pro entry Stiff ness and damping are stored respectively as 3rd and 5th column of the data il field standard linear spring see sdtveb p spring NDdata fields Non linearities listi25 type sel JCoef drot lab string RotCenter a FindElt command to find celas of RotCenter type this field should not be used JCoef field should be used instead and has priority Stiffness used for Jacobian computation Damping is not taken in account in Jacobian in this case coefficient of celas stiffness and damping for jacobian computation Default is 1 the rotation DOF label nl rotCenter This non linearity can be used to connect 2 points A and B where t
2. 4 External Load Load fe_cyclicb DiskEngineLoad r 201 er gt cf def Load fecom showdefarrow OBSOLETE DiskFRF D MS rest This command allows to build the Frequency Response Functions of a disk model either full or reduced A load and a set of observation DOF have to be defined and added to the model with fe casel The frequency range is stored in the stack as a info Freq entry The general call is xFefe cyclicb DiskFRFD disk lossfac xF fe cyclicb DiskFRFD rest disk lossfac cf sel xF fe_cyclicb DiskFRFMS disk def damp xFefe cyclicb DiskFRFMS rest disk def damp cf sel The command FRFD assembles the matrices of the model then uses them to compute the response An optional loss factor can be specified The command FRFMS synthetizes the response from a set of modeshapes A damping ratio for all modes can be specified The option rest recovers xxxEB recovers expands interpolates the re sponse computed on the reduced model to a given selection of physical DOF With out selection the response is expanded to the whole physical DOF set This option must be disabled when dealing with a full disk model gsobsolete The following example builds both direct and sythetized responses of a reduced disk model to a 2EO excitation cf demo_cyclic testload disk 5 nor reload model 4 the call to fe cyclicb basis is already done fe cyclicb reduce 1 1 int cf disk
3. param kO param k1 param c1 data Sens 2 1 03 translation sensor defining nl maxwell inputs Non linearities listii9 4 define associated property ri p spring default ri feutil rmfield r1 name ri NLdata data r1 il 3 param k0 ri il 1 2100 mdlestack set mdl pro zener r1 define option for time integration optenl spring TimeOpt opt NeedUVA 1 1 1 opt Follow 1 opt RelTol 1e 5 opt Opt 7 1 factor type sparse opt Opt 4 param dt opt Opt 5 param N NSteps opt IterInit model FNLO model FNL 0 needed for internal state xxx opt IterEnd eval opt Residual 7 to compute real FNL for current stat 4 Initial state ri data SE K 3 1 0 ri ri 1 initial displacement for 1N load mdlestack set mdl curve q0 struct def r1 D0F 1 03 4 Time computation ci iiplot 3 def0 fe time opt md1 compute 4 The same but NL as a model SE2 data SE SE2 Elt end 1 end42 1 6 Inf abs massi 1 param m Ol SE2 fe caseg assemble secdof matdes 2 3 1 reset SE2 ri SE2 K 3 1 0 rier1 1 SE2 stack_set SE2 curve q0O struct def r1 DOF SE2 D F def20 fe_time opt SE2 compute F20 SE2 K 2 def20 v SE2 K 3 def20 def F20 F20 1 NL20 struct X def20 data 1 LIN F 2 Lit Gel Lin aQel L lini 1 Y F20 fe c def20 D0F 3 03 def20 def fe_c def20 DO0F 2 NL20 name NLfromLI
4. 2 3 1 cyclic structure basicsl 2 3 2 Fourier transform for shaft computations 2 9 8 Solutions in periodic media 8 CHAPTER 2 THEORETICAL REMINDERS The following sections give a number of theoretical reminders on things used for the toolbox THIS IS VERY INCOMPLETE AND NOT VERY ORDERED 2 1 Rotating bodies 2 1 1 Problem definition in a rotating frame The developments of this section are derived from internal work on the SDT Rotor module which is currently only distributed to SNECMA The results shown here can be seen as a summary of those found in Ref which treats the problem with a strong emphasis on the theoretical formalism Other classical references that treat of the problem of rotating bodies are 2 3 4 Particles located in point p of the body fixed frame are at location x at time t One defines the displacement u by x p t p u p t 2 1 At time t a reference point of the rotating body is assumed to have a rigid rotation speed w with respect to the reference frame in the present study this speed is related to a global rotation around a fixed axis characterized by angle 0 The velocity is thus given by sa p t ME o 2 0 2 2 This expression can easily be derived by decomposing the position in body fixed coor dinates x x and noting that the derivatives the base vectors ey Ot Q t epi In implementations one repl
5. An unbalanced mass is represented by a rotating load in the global fixed frame sin Qt cos Mt Fo 4 10 0 0 4 1 RIGID DISK EXAMPLE 47 4 1 3 Validation with 3D model disk This validation example considers a disk meshed with hexa8 volume elements Local frame matrices are computed and projected on the geometrical rigid body modes x and z translations and corresponding rotation One checks matching of e matching of theoretical and numerical matrices e response amplitudes of the disk with unbalanced mass in global and local frame e massi and beami elements gives the same gyroscopic matrices in the global frame mattype 70 4 model of disk R1i 01 R2 15 h 0 03 rho 7800 md d_rotor sprintf testvoldisk dim 15g 2 15g 15g 5 24 R1 R2 h md DOF md fe_caseg assemble se matdes 2 1 7 8 not md disk assumed to be rigid rb feutil geomrb md cf feplot md cf def rb md K feutil tkt rb def 1 3 4 6 md K md DOF 1 01 1 03 1 04 1 06 nd K 2 0 md K 2 7 rigid disk md K 2 1 1 5e5 md K 2 2 2 5e5 bearing md K 2 3 3 1e5 md K 2 4 4 1e5 bearing rot XXX Campbell in local rotating frame fe rotor campbell critical md linspace 0 9000 31 4 Unbalanced mass w 1 t 12 wrange logspace 0 3 100 Q for j1 1 length wrange w wrange j1 4 rotation matrix R cos w t sin w t sin w t cos w t R R zeros 2 zeros 2 R Rp w sin w t cos w t cos
6. Ke Q 1 Xo 0 Examples can be found in section 3 4 xxx Whirldir Internal command to display the direction of the modes on the Campbell diagrams xxx details Assemble model fe_rotor Assemble model zCoef r1 This command is used to assem ble mass stiffness and damping matrices taking in account all the rotor matrices gyroscopic coupling centrifugal softening etc Default zCoef can be obtain using model fe def zCoef default model r1 fe rotore7 is a data structure whose fields are parameters used in zCoef In particular one should specify the rotation speed ri 0mega rad s Command option cell can be used to return only matrices in a cell array RotatingLoad This command builds a rotating DofLoad on a model node for frequency analysis model fe rotor RotatingLoad NodeId fO theta0 vexponent model Nodeld is the id of the node where load will be applied 0 is the amplitude of the load theta0 is the angle formed by the load initial direction and the first global direction in the plane orthogonal to rotation axis x for y and z rotation and y for x rotation The amplitude of the load can depend on a power of pulsation defined with wexponent Resulting complex load is of the form m 1 T in the plane orthogonal to the rotation axis sign of 2 depends on rotation axis Forced response to this load can be computed using ForcedResponse ForcedResponse This co
7. RunOpt FileTeig 1 end 4 if norm def_int data def_int data 1 gt 1 1 def_ext data def_ext data 1 gt 1 Zerror ShaftTEig internal and external prestress produce different results Note that to build a disk model from a sector you should use modelefe eyclicb DiskFromS 1 model sector xxx Arnaud need to check adaptation for multi disk models 3 7 Time domain analysis Once rotor model is created one can perform time computation nl spring 3 7 TIME DOMAIN ANALYSIS 39 function is very useful in that case Gyroscopic effects can be treated as non linear load depending on instant value of a model DOF observation of the rotation speed for example Links between fixed and rotating parts can also be modelized as non linear loads 3 7 1 Simple example Following example simply computes the response to an impact on the disk assuming that rotation speed is 1000 RPM 1 rotor TestShaftDiskMdl build simple 1D rotor 4 Define Time option and other parameters opt fe_time TimeOpt Newmark 25 5 1e 4 1e4 7 TimeOpt Range struct Omega 1000 60 2 pi modelestack set model info Range Range opt matdes 2 1 70 model stack set model info Time pt opt model fe_rotor TimeOptAssemble model initial impact model fe case model DofLoad In struct def l1 DUF 2 02 gt curve sprintf TestStep 15g opt Opt 4 b
8. The alternate command model fe_cyclic build nsec epsl len intersect model I is much faster but does not implement strict node tolerancing and may thus need an adjustement of eps1 to higher values Command options are 62 fe cyclicos nsec is the optional number of sectors An automatic determination of the number of angular sectors is implemented from the angle between the left and right interface nodes with the minimum radius This guess may fail in some situtations so that the argument may be necessary nsec 1 is used for periodic structures and you should then provide the trans lation step For periodic solutions modelefe eyclic build 1 z ty tz epsl Len intersect model LeftNodeSelect specifies 3 components for the spatial periodicity Fix will adjust node positions to make the left and right nodes sets match exactly epsllen gives the tolerance for edge node matching equal can be used to build a simple periodicity condition for use outside of fe cyclic This option is not relevant for cyclic symmetry y ByMat is used to allow matching by Matld which allows for proper matching of coincident nodes model demosdt demo sector 5 cf model fe_cyclic build epsl 1e 6 model LoadCentrifugal The command is used to build centrifugal loads based on an info Omega stack entry in the form data struct data 0 1000 unit RPM model stack_set model info
9. c ci cos k Az fer Im U 0 2 18 and one solves for U amp knowning the transform of the applied loads For a discrete load on the first slice n the fourier coefficients of the load is F x f 0 One can usefully note that the wave length L covers the full interval of positive lengths that U x U and that the half spectrum corresponds to a wavelength of Ax Using the property of symmetry allows the use of computations for wavelengths larger than Ax only In the fe cyclic eig calls one specifies the period as a number of steps pAz The phase angle is thus amp Az 4 spectrum is symmetric for 2 moi femesh testhexa8b divide 2 1 1 molefe eyclic build 1 1 0 mo1 4 symmetry along right edge r2 fe cyclic build 1 1 0 mo1 r2 fe_case r2 getdata Symmetry moi fe_case mol FixDof y_symmetry r2 IntNodes 1 02 moi stack_set mo1 info EigOpt 2 2 1e31 range 1 linspace 01 99 21 def fe_cyclic teig all sprintf g range mo1 figure 1 plot 1 def data 2 def data 1 x 14 CHAPTER 2 THEORETICAL REMINDERS feplot mo1 def r1 ill unique def data 2 first Ek fe_caseg enerm bygroup mol fe_def subdef def i1 if norm Ek Y 4 1 gt sqrt eps error Inconsistent energy end if 1 2 7 manual verification n k mdof fe mknl moi assemble NoT
10. k 0 For an unbalanced mass mb at a position of L 3 on the shaft and at a distance of db of the rotation axis F m db s n where m sin mb cos Qt 3 For a simple asynchronous load of amplitude f0 rotating at sQ F f0 sin sQt Frequency analysis is performed Harmonic solutions are computed under the form R Qoexp ssQt 1 for unbal anced mass and s lt gt 1 for asynchronous load Implementation of this case can be found in d_rotor Lalanne2DOF One can 4 2 SIMPLE 2DOF MODEL OF SHAFT WITH DISK 51 define values of parameters kxx czx m a mb f0 s etc as fields of a data structure r1 given as argument d_rotor Lalanne2DOF r1 A campbell and fre quency forced response are then computed at the rotation speeds defined in RPM in the field om of the parameter data structure One can ask only for the model using model d rotor Lalanne2DOFMdl r1 If this first case no bearing is considered Koearing 0 and Chearing 0 beta 0 ri struct om linspace 0 9e3 101 s 1 kxx 0 kzz 0 kKxa 0 cxx 1e2xbeta czz Be2xbeta cxz 0 czx 0 Ried rotor TestLalanne2DOF ri 70 pp b eo eo ap b pb popper L 09905 ppbb g Frequency Hz m o 4 1 4 4 4 4 4 gt o 1 4 444 14444 4444 74444 2 3 4 5 6 7 8 9 Rotation speed 1000 RPM o Fig
11. 1 solve Mode mode1 def0 nl_solve Mode stat fullDOF model defT nl_solve Mode time model hist nl solve Mode time data nly model histC nl_solve Mode cpx time dataOnly model defCenl solve Mode cpx time alpha real MyBas ful1D F model defienl solve Mode skip fullDUF model SineSveep xxx nl mesh Purpose Description Integrated mesh modifications for non linear applications Some non linearities require surface volume remeshing e g definition of con forming interfaces for contact or adpations generation of thin interface layers This function regroups such functionalities Mesh generation are performed by fe gnsh interface to gmsh and fe tetgen interface to tetgen see help fe tetgen Conform The Conform call is an integrated call to generate comforming meshes between two facing interfaces T he command generates a conforming surface mesh of the face to replace merges it with the conform mesh of the second interface replaces the model face mesh and remeshes the model volume to yield a new equivalent volume with a conform face mesh moi nl mesh conform eltsel FindElt model sel sel eltSelToReplace eltSelForReplacement model is a standard SDT model sel is a cell array containing in each line two FindElt commands specifying the element selection face to remesh and the element selection face to use for the conforming interface for r
12. 4 1 1 Matrices in rotating frame 5 4 1 3 Validation with 3D model disk 4 2 Simple 2DOF model of shaft with disk 43 1D modelsil 4 3 1 1D example in a fixed frame 4 3 2 1D models in a rotating body fixed framel AA D TOTOT Lice Rcge oa m wee SOE H R e mox ow 25 5 Function reference 22777 25 75 21777 00 7751777 2277 75575 CONTENTS CONTENTS Creating a new non linearity nl_fun m sp mex utils 098 Bibliography 130 132 135 142 144 145 146 CONTENTS Installation dE Contents 6 CHAPTER 1 INSTALLATION The SDT Rotor toolbox is intalled as a patch on an SDT installation You can download the toolbox from http www sdtools com distrib beta rotor patch dis p Then within Matlab 1 cd to location where you saved the patch MUST differ from SDT path cd tempdir 2 check that you will overwrite the expected SDT installation path which feplot 3 install the patch you must have write permission on the directory rotor patch dis Theoretical reminders p uui Contents 2 1 Rotating bodies 2 1 1 Problem definition in a rotating frame 2 2 Problem definition in a fixed framel 2 3 Fourier analysis of structuresi
13. Syntax Obsolete functionality model fe cyclicb Basis norm all int rb model orders omegas def fe_cyclicb DiskEig DISK def fe_cyclicb DiskEngineLoad EO model def fe_cyclicb DiskFRFD rest DISK lossfac cf sel def fe_cyclicb DiskFRFMS restl DISK def damp cf sel K fe cyclicb DiskMatrices mk DISK Eltselection fe cyclicb Fourier MODENUM phys rotor disk red test T L fig model fe cyclicb PolyAssemble NoT model params DISK fe cyclicb Reduce NODEIDO ELTIDO intl sector def model fe cyclicb Renumber emodel def fe_cyclicb Display cf def def fe_cyclicb ShaftEngineLoad E0 model def fe_cyclicb ShaftFRFD L restl model lossfac cf sel def fe_cyclicb ShaftFRFMS rest model def damp cf sel fe_cyclic ShaftPrep model def mdl fe cyclicC ShaftRing round N1 N2 N3 autoclose NA4 rimi rim2 fe cyclicb C ShaftLoad mode1 def md1 fe_cyclicb ShaftTEig ORDERS model OBSOLETE Basis all norm int rb This command allows to build a set of modes e with the harmonics specified in an array 1 r 2 for the rotation vectors specified in a cell uis Wiz war Way W2z The general call is model def fe_cyclicb basis all int norm rb sector orders omegas The output is a struct array containing the modeshapes If only one output is required the bas
14. cylinder is linked to the block by 3 celas preserving the pivot link moi demosdt demoConnection vol meshes models moi fe_case mo1 fixdof base z 1 clamps the cylinder base ri struct Origin 0 5 0 5 0 5 axis 0 11 radius 1 rt91 01 Iength 1 Np IProId 111 planes 4 Cylinder side nl springios ri nl spring ConnectionCyl r1 defines planes r3 r r3 ProId 1 Z Block side linke connection linki eteuet C1 r3 O2 1 link celas 0 12345 12345 1000 1e9 Defines c md1 idof nl_spring ConnectionBuild mo1 link builds connection cf feplot mdl displays in feplot fecom promodelviewon fecom curtab Cases link1 2 def fe eig mdl 5 20 1e31 computes the first 20 modes if length find def data lt le 3 gt 1 sdtw _err connection failed enc cf def def fecom ColorDataAll 7 displays modes See also t_nlspring 2beam example ConnectionCyl Define a cylinder connection data nl_spring ConnectionCyl data data is a standard connection screw data structure see fe_caseg ConnectionScrew with following additional fields rtol radius tolerance for cylinder selection length length of the cylinder Npt number of planes equally distributed on the whole length If Npt 0 ends of the cylinder are included in the connection points Prold Prold of the e
15. fe cyclicb MeshRimLine2Patch cf model Elt fecom ShowPatch save cf Stack info ViewMesh 4 Once the geometry generated typical calls are fesuper Sebuildsel initrot cf cf Stack info ViewMesh fe_cyclicb Displaysel 2 cf def cf Stack info ViewMesh 3 2 Bearing and support representations 3 2 BEARING AND SUPPORT REPRESENTATIONS 27 3 2 1 Linear bearing One can simply represent a support or a bearing by a linear spring that is to say a celas element for more details see sdtweb celas A celas element is announced in the model element matrix by the header Inf abs celas and format is as follow NodeId1 NodeId2 DofID1 DofID2 ProID E1tID Kv Mv Cv Nodeld1 and NodeId2 define between which nodes celas is connected if NodeId2 0 celas is grounded DofID1 and DofID2 which dof are connected and Kv Mv and Cv give respectively stiffness mass and damping A typical bearing for a rotor turning around x will then be defined by v Lx Figure 3 5 Simple disk on long rod model The following generates a spring bearing connected to node n1 stiffness is 1e9 and damping 1e2 in the xy plane The use of a spring property is necessary for non linear time domain applications model d rotor TestDiskLongBeam ni feutil findnode 2 71 amp x 0 amp y 0 model model feutil AddEl1t model celas n1 12 1001 model
16. generalized load with respect to the set of generalized coordinates option red has to be specified Energy based computations are not available for generalized loads fe cyclicb Mesh Purpose MeshAddRim MeshAddRim deals with already existing inter disk volumic rings It renumbers their model to be integrated with the already defined disk super elements Note that adjusting the geometric tolerance with option eps1 val can be important cf demo_cyclic buildstep0 See sdtweb demo_cyclic m Step0 Extract rim model into mol would come from other reading cf mdl Elt moi Elt feutil RemoveElt EltName SE cf mdl mo1 Node cf mdl Node moi Node feutil GetnodeGroupAll mo1 cf mdl Node 4 Combine moi rim and cf mdl disks fe cyclicb MeshAddRim epsl 1e 3 cf mo1 cf sel EltName SE fecom showpatch Other commands documented in the tutorial are MeshRim described below and illustrated in section 3 1 6 is an automated procedure to mesh rims form a selection of nodes on two disks e MeshRimLine2Patch described in section is used to build view meshes of large models MeshRim This command is used to mesh rims as volumes or penalty springs e kp val is used to specify the penalty stiffness e epsl val specifies length tolerance for node matching e masterdisk val specifies the disk s on whose surface the slave nodes will be matched e masterdisk val specifies the slave disk s whose
17. mega data model fe cyclic LoadCentrifugal mode1 Fig def fe_cyclic eig nd am model Eig pt computes ndiam diameter modes us ing the cyclic symmetry assumption For nd am 0 these modes are complex to account for the inter sector phase shifts EigOpt are standard options passed to fe eig This example computes the two diameter modes of a three bladed disk also used in the d cms2 demo cfe cyclic model demosdt demo sector model fe_cyclic build 3 model groupal1 fe case model info def fe cyclic eig 2 model 16 20 111 fe_cyclic display 3 model def The basic functionality of this command is significantly extended in fe_cyclicb ShaftEig that is part of the SDT Rotor toolbox See also Section fe rotor Purpose The fe_rotor function implements classical solutions for rotor dynamics appli cations AddMass mdl fe rotor AddMass mdl RO0 This command can be used to add a local mass on a 3d rotor mesh The mass is added and linked to existing node using a MPC connection RO is a data structure with fields e mxyz matrix whose first 3 columns define x y and z coordinates of added mass and the 4th defines the mass e ProId the Prold of the rotor where mass is added Command options are e DofLoad defines a 6 direction load on added masses usefull for reduction purpose e mpcmaster modifies MPC connection so that mass DOF are master rat
18. modes then the modeshapes are recovered on the blades only Model Initialization cf demo_cyclic testload disk 5 reset reset file rather than load Mode Computations def fe eig cf mdl 5 20 1e3 11 1e 8 4 Restitution cf def fesuper SeDef cf def obsolete o OBSOLETE DiskSeLoad This commands transforms an external load expressed on the physical DOF set into a generalized load expressed on the generalized DOF set It is necessary to initialize the restitution of the underlying disk model since this transformation is the inverse operation Model Initialization cf demo_cyclic testload disk 5 nor 4xxx obsolete Model Reduction fe eyclicb reduce 1 1 int cf 4 External Load Load fe cyclicb DiskEngineLoad 1 cf fe cyclicb DiskSeDeflnit cf Rload fe_cyclicb DiskSeLoad cf Load Reduce int The Reduce command is used to generate a disk model from a set of cyclic modes associated with multiple harmonics The general call is disk SEsector fe cyclicb reduce Nodeld0 Eltld0 int model fe cyclicb reduce NodeIdO EltIdO int cf The second call uses directly a global model stored in the variable cf and stores the reduced model as an entry info diskmodel in cf Stack If required new starting points for numbering the generalized DOF and the associated elements are defined in the command string through the two integers Nodel1d0 a
19. om 60 2 pi Range model stack_set model info Range r1 R1i fe_rotor forcedresponse model compute forced response iiplot R1 plot response In global rotating frame load is static w 0 Following example deals with a rigid disk in global rotating frame Ri 01 R2 15 h 0 03 md d_rotor sprintf C TestVolDisk dim 15g 15g 2 15g 2 16 R1 R2 h md DOF md fe_caseg assemble se matdes 2 1 7 8 md disk assumed to be rigid rb feutil geomrb md cf feplot md cf def rb md K feutil tkt rb def 1 3 4 61 md K md DOF 1 01 1 03 1 04 1 061 md K 2 0 md K 2 rigid disk md K 2 1 1 5e5 md K 2 2 2 5e5 bearing md K 2 3 3 1e5 md K 2 4 4 1e5 bearing rot Unbalanced mass or asynchrone load mb 1e 4 db 0 15 7 mass distance to axis s 0 om sort 12789 2750 10 2820 11760 10 11840 linspace 0 20000 101 RPM md fe_rotor sprintf RotatingLoad 1 15g 2 mb db md ri struct Omega om 60x2xpi w s om 60 2 pi 7 Range md stack_set md info Range r1 Ri fe rotor forcedresponse md compute forced response iiplot R1 plot response SEBuild This command can be used to build a superelement of rotor from a 3d rotor model It includes Craig Bampton reduction bearing rings adding fe rotores SE fe rotor SEBuild model RO model is a 3d mesh of rotor RO is a data structure with following
20. pass is performed with gmsh to provide an SDT contour model This is not supported for other file types model nl_mesh contourFrom file stp not specifying the type Call Contour generates an SDT face mesh from an SDT beamifl or beamiB con tour 1383n mesh model nl mesh contour model model is an SDT beam model defining a closed contour e Command option lcval allows specifying a characteristic length for gmsh e Command option 1cminval allows specifying a minimal characteristic length for gmsh e Command option quad allows generating quadratic meshes e Command option keepNode asks to keep the original contour NodeId for the contour command e Command option diag asks to output the gmsh log file for diagnostic problems e Command option single tells n1 mesh that a single coutour is defined This is usefull when several closed contours are defined since it is impossible to automatically decide whether each contour is independant or if they define a single complex contour e Command option groupval is used in combination to the single command option This allows specifying which contour group will be meshed while other possible contours will define holes e Command option algo val allows specifying which algorithm gmsh must use this depends on the gmsh version report to the gmsh documentation for more details e Command option AllowContourMod allows gmsh adding nodes on the conto
21. sel displays the outer enveloppe of the selected disks in sel with the inter sector common surfaces removed sefe cyclicb cfedemo cyclic buildstepO0 fe_cyclicb DisplaySkin cf disk1 fecom showpatch set cf o 1 FaceAlpha 33 fe cyclicb DisplayInterDisk cf nodes displays the two ring surfaces used to define the inter stage volumic interface cf demo_cyclic buildstep0 4 rim nodes ni feutil getnode Nodeld cf mdl 112 18 24 1127 1133 11391 fe eyclicb DisplaylnterDisk cf n1 fe cyclicb DisplaySymmetrySurface cf displays the nodes in a eyclic symmetric condition as colored surfaces Mono harmonic solutions These commands apply to sector models used to compute mono harmonic eigenso lutions ShaftTEig ShaftEig ShaftSolve These commands compute mono harmonic solutions with specified Fourier harmon ics classical cyclic solution for single stage models For a tutorial on generating the proper models see section For the associated theory please refer to 9 The calling format is def fe_cyclicb shaft Teig delta list model ShatTeig accepts multiple diameters in the delta 1251 and packages individual calls that to ShatTeig For a disk example section 3 5 2 The main variants in the call are as follows e Diameter 1 is used to ask for the computation of fixed edge modes e The eigenvalue options info Eig pt should be set i
22. tempdir donnees secteur dat cf feplot Fix sector edges fe cyclic buildepsl 1 fix cf mdl fecom curtabCases Symmetry fecom view3 proViewOn Figure 3 2 Left and right sides of the sector The left and right edges of sectors should be conform Practically you need to mesh the two surfaces first possibly generate the second by rotation of the first Then mesh the interior You will note that this typically requires at least two layers of element for tetral0 meshing 3 1 ROTOR MESHING 23 3 1 5 Utilities for handling slanted blades Typically blades have an angle with respect to the e e plane In a number of cases removing this angle makes node and element selection easier With two nodes the FixTheta command modifies node positions by removing the angle The transformation is saved in info FixTheta stack entry and the back transformation is obtained using a FixTheta call with no argument cf feplot 2 cf d rotor TestSector sdtuveb d rotor mitTestSector ni cf mdl Node save nodes cg feplot 5 4 place rectified model in figure 5 cg mdl fe cyclicb MeshFixTheta 10061 10086 offset 80 cf mdl fecom cg view2 verify that the back step works cf mdl fe_cyclicb MeshFixTheta cg mdl fecom cf viev1 if norm cf mdl Node n1 51e 10 error Back transform failed end 3 1 6 Disk connections in multi stage cyclic symmetry In multi stage cyclic symmetr
23. u 0 v 0 09 u n v n It follows that Y k for all k Z 1 The intersector continuity condition says that the nodes that are common to two sectors have the same motion By convention the left nodes in SDT rotor are those with the lowest clockwise polar angle the left nodes of sector have thus equal motion than the right nodes or sector N 1 Thus cz ceg Y k e 79 0 Which leads to the constraint ci cos ko c sin k a cr Rela 1 sin a erl ci cos 6 cr oe Im q xl 2 13 12 CHAPTER 2 THEORETICAL REMINDERS The Fourier transform being a linear relation one can actually rewrite the rela tion as a y E Y For N even Yo i ws 2 0 001 B Re Y ie 11 2cos akn 2sin akn 1 Im Yy 2 14 YN 1 i Note that ET E is a diagonal matrix with N 2N 2N N on the diagonal Posing Inq is the identity matrix whose size is the number of physical DOF of the sector and amp is the Kronecker product the fourier DOFs Q sorted as in 2 14 and physical DOFs on the whole disk are related by 4 E amp Ing Qx 2 15 When performing harmonic computations one typically uses the fact that the model matrices are identical for each sector As a result the transformed matrices are block diagonal which is the basis for the ability to compute full responses based on independent computation of each Fourier solution Q When adding a ma
24. 1 0 01 for displacement Dof 0 1 0 for velocity and 0 0 1 acceleration exponent Exponent of the DOF def data structure with fields def which defines vector V and DOF whi defines corresponding DOF nlspring nlspring defines a non linear effort from rheological information stop tabulated damping or stiffness laws etc between 2 DOF To define a non linear spring one has to add a classic celas element linear spring Non linearities listi21 between only 2 DOF The non linear aspect is described by associated properties as a pro entry in the model Stack For the moment the 2 DOF of the non linear spring have to be given in the dof1 and dof2 fields of this entry One can describe non linearity by a formal rheological description using one or more of following fields in the pro Stack entry e But dumax KO cO dumin ki c1 For du from dumin to dumax f 0 For du gt dumax KO stiffness is applied to du dumax and for du lt dumin k1 stiffness is applied to du dumin Damping is not taken in account at this time due to tabulated law strategy e Fsec fsec cpenal For dv lt fsec cpenal or for dv gt fsec cpenal f fsec is applied For fsec cpenal lt dv lt fsec cpenal f cpenal dv is applied If omitted cpenal 1e5 e K e This information will be converted in tabulated laws Fu and Fv using low level call that should be automatically called at the beginning of time computation One can also descri
25. 4 1 w 0 1 v 0 Ow 4 1 RIGID DISK EXAMPLE 45 Rotation matrix 2 3 along Y axis is given by 0 0 1 0 ek 1 0 4 2 0 Using matrix expression given in section in body fixed local rotating frame and proyecting the integrals by assuming rigid disk motion m 0 0 0 0 m O0 0 M 4 M 00h 0 4 3 0 0 with I Iw imm m 0 1 0 0 D 2m9 27757 4 4 4m h2 00 0 5 0 5 0 1 0 0 0 0 1 0 MEM Kc mQ 0 0 i 0 4 5 h2 000 5 M dr D L Kciqr Fr 4 6 An unbalanced mass is represented by a static load in the local rotating frame Fr 4 7 O o 4 1 2 Matrices in global fixed frame Local frame indexed with z is rotating in global fixed frame indexed with according to rotation matrix 46 CHAPTER 4 VALIDATION cos Qt sin Qt 4 8 Ri sin Qt cos Qt so that we have ar Ri ac r Ri c Pi qe Gn Ri c 2 dc Pi qe and equation can thus be rewritten as RT MR faq ka 2RT MR je RTDR l a RT MR RT DR RT K R qe Ma da Da dc Kea qa Fe 4 9 1 0 One has RTR ELLE 0 1 0 the translation terms disappear and one has thus in the gyroscopic effect 000 0 000 0 Del 20 000 00 with I imm 1 2mh 12 and the xxx UNEXPECTED xxx centrifugal softening in the global frame 00 0 0 0100 0 0 oNu ars og 00 0 4I v
26. K in this order u and v variables of caller workspace can be needed e ener to compute for each def stored in model d1 def structure that is typi cally computed modes some associated energies i08nl spring freq frequency in Hz damping damping ratio 97 C 65 20 enerK total strain energy K 16 enerC 7 K NLlink enerK strain energy for each NL link T LENL nkl i NL nk enerK for each NL link di Cn unk SetPro mdl nl_spring SetPro Prold 4 ParamNamei Value1 mdl This command is used to change some nl_spring properties parameters 2 is the Prold of corresponding p spring property ParamName the name of parameter to change k for il 3 c for il 5 or the fieldname in NLdata and Value the value to assign It is possible to define a new property by specifying an NLdata structure in third argument mdl nl_spring SetPro Prold 4 mdl NLdata If the property already exists it is overwritten by the NLdata structure provided Standard NLdata structures depend on the non linear function see for more details They can be obtained through the nl function command db see n1 fun for more details GetPro pro nl spring GetPro model This command is used to get non linear properties in the model stack Command option ID allows getting a specific non linear property by specifying its Prold Command option type nl_fun allows getting the non linear propert
27. Model Initialization cfedemo cyclic testdisk 5 blade noK nor cf 2 sector fe_cyclicb polyassemble noT cf Stack disk1 10 500 10001 Case sector D Flefe mknl init sector K sector K 1 feutilb sumkcoef fsector K 2 4 1 25072 250741 Mode Computations def fe_eig K 1 K 2 Case T sector DOF 5 10 111 cf model sector cf def def OBSOLETE DiskSeDefInit This command initializes the restitution of the generalized modes computed with the DiskEig command on the physical DOF set The restitution bases are stored in the Stack as a info SeRestit entry which contains the following fields e a field DOF that contains the physical DOF set e a field adof that contains the generalized DOF set e a cell Restit where the first column gives the indices in the physical DOF vector associated with each sector the second column gives the indices in the generalized DOF vector associated with each sector and the third column con tains the transformation matrix from physical to generalized DOF coordinates e a cell cGL that contains one matrix per sector which is a local to global frame transformation matrix The restitution is performed with the command SeDef of fesuper In this com mand a patch can be defined as a selection of elements so that the modeshapes are expanded only to the physical DOF of their nodes The following example builds a reduced model computes a set of generalized
28. TimeOpt can be set using nl spring Time pt The folloving steps are required for a time simulation e Definition of non linear properties T hese are stored as pro entries of the model stack The associated property function must handle non linearities which is currently only the case for p spring and p contact A non linearity is always associated with elements or superelements typically a celas element A given group of elements can only be associated with a single non linearity type The information needed to describe the non linearity is stored in a NLdata field e Model initialization using the an fe case assemble call in is followed by the building of a model NL stack that describes all non linearities of the model in a format that is suitable for efficient time domain integration This translation is performed by the n1 spring NL command e Jacobian computation see nl spring NLJacobianUpdate e Residual computations are performed through mk1 utils The nominal resid ual call is r fc mkl utils residual r model u v a opt Case 103 104nl spring Supported non linearities Seefnllist for supported non linearities and n1 fun to add your own non linearities ConnectionBuild One can define a set of non linear links between 2 parts of a model using a call of the form model idoflenl1 spring ConnectionBuild model data idof is a second optionnal out argument It returns the list of DOF conc
29. below this is the reason for fixing the axial motion of point 136 in disk1 and using the FixTan which adds a tangential translation constraint on the first disk if 1 1 7 One disk case cf demo cyclic testrotor 7 blade cf 2 Sel disk1 groupall else Two disk case cf demo_cyclic testrotor 7 10 blade cf 2 reset Sel EltName SE diski 1 2 xroupall disk2 1 5 groupal1 end cf Stack disk1 fe case cf Stacki disk1 FixDof Axial 136 03 4 Linear static response cf Stack info Omega struct data 250 unit RPM d0 fe_cyclicb shaftsolvestatic FixTan cf mdl fe eyclicb displaySel cf d0 Sel fecom ColorDataEvalRadZ Non linear static response cf Stack info Omega struct data 250 unit RPM d0 fe_cyclicb shaftsolvestatic FixTan nlstep 1e 10 cf mdl 34 CHAPTER 3 TOOLBOX TUTORIAL fe_cyclicb displaySel cf d0 Sel fecom ColorDataEvalRadZ XXX example with thermal loading xxx Example with temperature dependent properties 3 5 2 Single stage mode computations As a first example one will consider computations of a single disk using mono harmonic solutions Call shaftTeig in fe cyclicbhllows to compute the specified mono harmonic normal modes Mono harmonic normal modes are recovered to the rotor with the help from command shaf
30. cf Stack diskmodel 4 External Load Load fe_cyclicb DiskEngineLoad 2 cf fe cyclicb DiskSeDeflnit cf Rload fe_cyclicb DiskSeLoad cf Load disk fe case disk dofload Blade load Rload freq 11500 5 30001 disk stack_set disk info Freq freq 4 Restitution to Blade Dofs tips feutil FindNode r gt 201 cf mdl disk fe_case disk SensDof Sensors disk DOF Transfert Functions Direct xFd fe_cyclicb disk frfd rest disk 002 cf tips Transfert Functions Modal Synthesis disk stack_set disk info EigOpt 5 50 1e3 11 1e 81 def disk fe_cyclicb diskeig disk xFms fe cyclicb disk frfms rest disk def 001 cf tips Response Plots i liplot XF iicom curveXF XF Blade resp D struct w xFd data xf xFd def dof xFd D F XF Blade resp MS struct w xFms data xf xFms def dof xFms DOF iicom subMagPha iicom ci IIxOnly Blade resp D Blade resp MS OBSOLETE DiskPlot This commands provides a graphical representation of the generalied quantities modeshapes or load on a patch whose nodes represent the generalized DOFs obsoleteos The following example deals with such representation for both the modes and a 3EO excitation Model Initialization cf demo_cyclic testload disk 5 nor reset Model Redu
31. data structre see sdtweb def with an additional ID field The data field is equivalent to the ones of complex modes see f e It is a matrix of two columns respectively giving the frequency and the target damping ratio for each mode Non linearities listi29 Since modal damping implies a modal sensor the features performs both by default It is however possible to simplify it as a pure modal sensor The theory around modal sensing damping can be found in 12 The pro NLdata structure has fields type string nl modaldmp e Curveld the curve ID stacked in model which provides the shapes and their damping ratios e SensorOnly to use the feature only as a modal sensor in a def data structure The NLdata structure generation can be integrated using an nl_modaldmp db call See sdtweb nl springisetpro for this integration This is used in transient simulations and in complex mode computations see nl contact See Creating a new non linearity nl fun m Purpose The structure of allows creating any new non linearity through the use of a dedicated function named nl fun m This function which non linearity name will be fun will be automatically called by for classical operations The function structure has been designed to comply with specific needs Stan dard calls have been defined which are detailed below e Residue computation called by nk1 utils sdtwebmkl utils must ouput the entry for
32. defefeutilb placeindof mdof def feutilb dtkt real def def m feutilb dtkt imag def def m end Toolbox tutorial 0050 Contents 3 1 Rotor meshing 17 3 1 1 Meshing utilities 17 S x r E Mae EM ND tes 19 17777 20 3 1 4 From sector to shaft in the case of cyclic symmetry 22 23 3 1 6 Disk connections in multi stage cyclic symmetry 23 3 1 7 View meshes for cyclic synmetry 25 177v7v7v7v 26 3 2 1 Linear bearing 27 127777 28 5 e fe AA 28 3 3 1 Fixed frame models 28 Gs Sn Gn E eh 29 3 4 Frequency domain analysis full model 29 3 4 1 Campbell diagrams full model 30 3 4 2 Blade with centrifugal stiffening 30 m mol TT 31 3 4 4 Forced frequency response to unbalanced load 32 3 5 Solvers for models with cyclic symmetry 33 210 Ade Org y oie 33 255 34 5577 35 5500 Bese ae ae Hees cea SO a i eA 36 de A Sy Bee De ee 37 3 5 6 Forced frequency response to unbalanced load 37 T 37 15 MT ecco ME 3057707 re Masas ka lak To EF 3 1 ROTOR MESHING 17 fe rotorlmodule can use 2 different frames to describe rotating effects rotating frame and fixed frame Rotors can also be describes at 3 levels of modelization 1d 2d o
33. defining the property for each rivet separately e Command option MatIdval allows setting the modified mesh MatlHd to val e Command option Proldval allows setting the modified mesh Prold to val e Command option fill outputs in second argument a compatible mesh of the rivet bores Following example meshes a rectangular contour with a few rivet drillings inside Generate a global contour model struct Node 1000 0 00 200010 00 300010 20 2000 6 2901 ED model Eltefeutil biectBeamLine 120230340 4 1 model model feutil refinebeam 2 model Zfeplot model uon mesh define rivet positions eventually planes RO struct Orig 3 1 0 6 1 0 9 1 0 wadHole 27 95 9 2 radWash 8 982 81 1 3 model nl_mesh Rivet model RO cf feplot model GmshVol This call integrates the generation of a volumic mesh from a face mesh with gmsh model nl mesh GMSHvol model model is a standard SDT face mesh model e Command option setmat allows specifying a specific MatId to the output mesh e Command option setpro allows specifying a specific ProId to the output mesh e Command option keepFaces asks to keep orignal NodeId of the nodes located on the face mesh e Command option 1c specifies a characteristic length for gmsh e Command option clmin specifies a minimal mesh length for gmsh e Command option clmax specifies a maximal mesh length for gmsh ExtrudeLayer This c
34. edge nodes will be con nected to the master disk surface 79 safe cyclicb Mesh MeshSecAddNode MeshSecAddNode epsl espi is used to place sensors relatively to a disk rotor model It uses a cell array to define sensors see sdtweb sensor scell It first matches each sensor to a surface in a sector of a disk with an optional tolerance epsi that allows to tune the research It then adds these surface elements to the selection The latter serves as the input of the subsequent SensMatch call that 1 automatically deals with the specified direction of measure column DirSpec by detecting the normals computing vectors from the latter etc 2 builds the observation matrix at each sensor cta entries that relates the DOF of the matched surface to the observation at this sensor along the measurement direction Load 360 reduced shaft model sdtweb demo cyclic mitStep2 cfedemo cyclic buildstep2 sensors cf Stack Test tdof disp sensors display sensor properties 4 match sensors sens sel fe_cyclicb MeshSecAddNode epsl 20 cf sensors cf mdl fe case cf mdl SensDof Test sens 4 display sensors fecom showpatch fecom cf curtab Cases Test The model to which the sensors are matched is made of super elements already reduced call to MeshSecAddNode with the supplementary input of a generalized quantity with a list of generalized coordinates stored in DOF automat
35. frame That is the Eulerian frame one has u 0 and u r 9 2 u p 2 cos 0 One now considers a disk that has a steady state deformation in the rotating frame 0 and 2 1 eg The displacement of a particle located at at time t is given by ug pa t 1 cos 9 0 eon Rod u p its velocity is 2 3 Fourier analysis of structures For more details you can refer to 5 that is available on line 2 3 FOURIER ANALYSIS OF STRUCTURES 11 2 3 1 cyclic structure basics For cyclic system with N sectors angle a 27 N a point load placed at angle na is associated with the harmonic load Y k L y n e ien 2 9 n e this can be used to simply compute the k diameter response Using the symmetry of the spectrum one can recover the full spatial response by inverse fourier transform For N odd N 1 2 y n Y 0 M 2Re Y k e 2 10 kel for N even N 2 1 y n Y 0 1 Y N 2 5 2Re Y k e 2 11 ki The displacements y n of each sector are however expressed in local coordinates when applying conditions a transformation 0 to global coordinates is thus needed q n cosa sina q n q n y sino cosa 0 q n y 2 12 qin Glob 9 1 qin Loc For a point on the axis the in plane response is given by two components u 0 v 0 the response at other points of the series is given by but the point being coincident on has
36. il p_spring model il dbval 100 1e10 1e2 fe simul mode model ri struct Origin 20 10 0 axis 0 11 radius 3 planes 11 5 1 111 1 3 1 5 0 1 112 1 44 28 CHAPTER 3 TOOLBOX TUTORIAL MasterCelas 10 123456 123456 10 1e141 NevNode 0 links connection bearing Ci 1 02 313 link celas 0 23 0 103 0 1 b 0 11 NLdata struct type RotCenter lab Bearing 8617 Prold 12 6069 1 drot n17 06221 model Run Opt idof nl spring ConnectionBuild model links cf mdl RunOpt idof nl_spring ConnectionBuild cf mdl links For 3d rotors if there is no node on the axis of the shaft where one want to model the bearing one can add a node whose displacement depends on displacements of nodes on a ring of the shaft using see sdtweb fe_case connectionscrew Then one can connect a grounded celas to this node xxx a full example is needed XXX 3 2 2 Non linear bearings in the time domain Documentation of nl spring capabilities should go here 3 3 Gyroscopic effects SDT supports gyroscopic matrices in both rotating body fixed and fixed frames Eulerian representation of an axisymmetric structure When performing assembly the matrices in the local rotating frame are obtained with MatType 7 gyroscopic and MatType 8 centrifugal softening MatType 70 corresponds to the gyrosco
37. init model cf feplot cf model rotor2d buildFrom2D mo1 SE cf Stack disk1 enforce boundary cond on sector and assemble SE fe_case SE FixDof Base z 1 01 SE fe_cyclic assemble se SE cf Stack disk1 SE fecom viev1 4 automated building of Campbell diagram XXX NEED REVISION Run ptestruct targ 1 i Range linspace 0 1 30 cf def rotor2d teig cf Run0pt d1 fe_def subdof cf def feutil findnode r 0 SE 3 4 2 Blade with centrifugal stiffening One considers stiffness matrices that are dependent on the rotation speed As suming that a second order polynomial representation is sufficient one can build a 3 4 FREQUENCY DOMAIN ANALYSIS FULL MODEL 31 vector of weigthing coefficients 1 1 1 1 1 ag Q 25 03 ag 3 1 a3 02 Q3 Such that the stiffness at rotation speed Q is approximated by K 0 04 3 2 Qi The zCoef uses velocity Omega in rad s This example now treats computation at variable rotation speeds Model Initialization modeledemo cyclic testblade cf feplot model Compute matrix coefficients for a multi stage rotor range struct data 0 0 1 0 0 8e3 0 0 16e3 unit RPM 4 Assembling in the feplot figure allows memory offload fe cyclicb polyassemble noT cf range X struct data linspace 0 16e3 10 unit RPM fe rotor Campbell cf
38. mold 11 9 01 4 now view as 3D model mo2d rotorid iTo2d 1 5e 2 mold cf feplot cf model rotor2d buildFrom2D nsec16 mo2d You can also add beam through rotorid Addbeam command moid rotorid AddBeam x1 x2 0 7 r2 0 77 MatID 1 1 Z add rod moid rotorid AddBeam xi 0 5 x2 0 8 r1 0 77 r2 0 90 MatID 1 moid Zadd tube mo2d rotorid iTo2d lc 5e 2 moid cf feplot cf model mo2d fecom colordatapro 2D For 2D rotor representations the SkyLineTo2d command eases the generation of simple rotors Note how in the following example Nan separators are used to generate a rotor in multiple parts center shaft first then after the separator various disks xy 0 0 0 0 00945 0 0088 0 00945 0 0088 0 0057 042 0 0057 0 042 0 00938 0 057 0 00938 057 0 008 0 0637 0 008 0 0637 0 00595 0919 0 00595 0 0919 0 00925 0 0979 0 00925 0979 0 006 0 121 0 006 0 121 0 008 127 0 008 0 127 0 0095 0 142 0 0095 142 0 006 0 175 0 00565 0 226 0 00565 0 226 0 0088 232 0 0088 NaN 0 0057 Disk 0 00952 0 0057 0 00952 0 008 0 0209 0 008 0 0209 0 0369 0 0242 0 0369 0 0242 0 0572 0 0266 0 0572 0 0266 0 0369 0 0299 0 0369 0 0299 0 008 0 0413 0 008 NaN 00565 Separator giving non zero internal diameter 00565 0 177 01 0 0057 0 177 01 0 02 0 191 0 02 0 191 0 0057 NaN 00565 Separator giving non zero internal diameter 00565 Oo XO O O O O O 3 1 ROTOR MESHING 19 0 197 005 0 0374 0 215 005 0 0374 7 I
39. of the InitV process diverge and last dt that leads to convergence is kept Then Np is increased by 100 steps until the deformation is converged on the stabilization periods that is to say that a criteria taking in account standard deviation mean of the deformation and the ratio of the last Np 10 steps upon previous Np 10 steps on each Np period is less than a tolerance 2 0 See 2beam example XXX NL model nl spring NL model This command is used to build NL field data for time integration from NLdata field in NL p spring property entries in the input model Stack The command option storefnl can be used to specify the way of computing and storing a non linear effort associated to NL for those which support it NLJacobianUpdate opt nl spring JacobianCall returns the callback used to update or initial ize the jacobian ki used in iterative methods The said jacobian must take non linearities into account For the case of a non linear spring the most important gradient of the tabulated law Fu is added as stiffness between the 2 DOF to the stiffness matrix and the most important gradient of Fv to the damping matrix For non linear iterations in a Newmark scheme the Jacobian is given by ki model K 3 kj opt 2 opt 1 dt model K 2 cj 1 opt 1 dt 2 mo Accepted command options associated to variants of the call are TangentMdl to return tangent model It is assumed that model K 1 3 cor respond to M C and
40. option but no harmonic are given it computes only fixed interface modes The six rigid body modes of a cyclic symmetric structure are mono harmonic with harmonic 0 1T along and 1R around the axis of symmetry and harmonic 1 2T along and 2R around the other axes Thus the rb option is used to compute two more flexible modes with harmonic 0 and four more flexible modes with harmonic 1 OBSOLETE DiskEig ord This command allows to compute the approximate modes of a reduced disk model built with the command Reduce of fe_cyclicb Command fails and is no longer maintained zzz see with arnaud cf demo_cyclic testload disk 5 nor reload model fe_cyclicb reduce 1 1 int cf disk cf Stack diskmodel Mode computations disk stack_set disk info EigOpt 5 30 1e3 11 1e 81 def disk fe_cyclicb diskeig ord disk obsolete disp def data cf def fe_cyclicb Display cf def The ord option is used to identify the Fourier harmonic coefficient associated with each mode when dealing with the reduced model of a tuned disk When dealing with a mistuned disk whose modes are multi harmonic this returns the coefficient whose harmonic is the greatest DiskEngineLoad EO sel This commands builds a physical load spatially mono harmonic on a specified set of nodes If no selection is present all nodes are used Model Initialization cf demo_cyclic testdisk 7 blade cf 2
41. processed with a Singular Value Decomposition See Ref 5 for details handle option controls whether the resulting bases are stored in memory or on the disk norestit suppresses the explicit construction of the Restit variable normally stored in cf mdl Stack Once the sector superelements have been generated the disk model is assembled using the subsequent fesuper fassemble cf call which generates the reduced disk model in cf Stacki SE MVR A compact example is provided below A fully developed example can be found in demo cyclic ShaftMulti cf defledemo cyclic buildstep1 fe_cyclicb ShaftPrep svdtruncate mseqi norestit cf def fesuper fassemble cf ShaftLoad ShaftSELoad fe cyclicb ShaftLoadMulti cf data generates a reduced mono harmonic load for each disk specified in data It is a cell array whose a typical line is diski def delta 2 where for each disk 7 the shape def and harmonic coefficient delta 4 are specified fe cyclicb ShaftSELoad cf def generates a reduced load from its phys ical counterpart In practice it is used for very specific loading cases e g single DOF load or random load Developed examples are presented inldemo eyclic ShaftMulti 4 define model cfedemo cyclic buildstep4 build the excitation on first all sectors with specified diameters 3 on diski and 4 on disk 2 data disk1 cf Stack disk1 Stac
42. traj for mode computations based on states provided as an additional argu ment xxx e skip skips fe timekimulations and use the data provided in model to com pute the modes For a static computation the model stack entry curve q0 will be used If empty a zero displacement static state will be used For a transient mode computation the model stack entry curve TSIM will be used if empty an error will occur For options skip and or traj a deformation is can be provided either the time deformation or a static deformation in the model stack This deformation is stacked in the working model under the name curve DEF TRAJ and can be accessed during all initialization phases Accepted command options to control the output format are e dataOnly to save only the frequency damping data does not store the de formation field The output is then under a frequency tracking curve in the iiplotformat e fullDOF to output the deformation fields restituted on the unconstrained DOF e alpha requires cpx to also output the real mode participation to the com plex modes This is in fact the projection of the complex modes on the real mode basis 134nl solve e real ModeBas requires cpx to specify a particular real mode basis on which the complex modes will be computed The real mode basis is supposed to be stored in the model stack entry curve ModeBas Sample calls to extract tangent modes are given below 20
43. 00 101 7 RPM model fe rotor sprintf rotatingload 180 15g 2 f0 model ri struct Omega om 60 2 pi w s om 60 2 pi Range modelestack set model info Range r1 Ri fe_rotor forcedresponse model compute forced response iiplot R1 4 plot response 4 5 DATA STRUCTURE REFERENCE 59 4 Post radial deformation Q abs R1 Y 1 2 unbalanced along x figure semilogy om Q xlabel Rotation speed RPM ylabel radial def amplitude m if s 0 title Unbalanced mass else title sprintf Asynchronous load 15g Omega s end Unbalanced mass T 10 500 M 1 400 19 10 2 L 4142 3 300 Bw m g 2001 13 400 1307 8 or n 9 n L El 10 B 4 0 2 4 6 8 10 12 14 16 18 0 0 5 1 15 2 Rotation speed 1000 RPM Rotation speed RPM x 10 Figure 4 12 Left Campbell diagram Right Response to unbalanced mass Unbalanced mass excites the forward whirl modes Maximum response is found at critical speeds rotation speeds that induce a complex mode of 0 Hz frequency in the rotating frame Campbell critical speed 2789 RPM matches computed frequency response 4 5 Data structure reference XXX 60 CHAPTER 4 VALIDATION Function reference 25555 Contents 77 62 2775 65 250203 71 17 74 57777 5 27777 79 227777 83 257 95 27755 103 ee 11
44. 2 P uda da d dre 114 E 25757 132 220035 135 15 142 57 7 144 fe eyclic Purpose Support for eyclic symmetry computations model fe cyclic build NSEC model LeftNodeSelect Syntax def fe cyclic eig NDIAM model Eig pt Description groups all commands needed to compute responses assuming cyclic symmetry For more details on the associated theory you can refer to 8 Assemble struct This command supports the computations linked to the assembly of gyroscopic cou pling gyroscopic stiffness and tangent stiffness in geometrically non linear elasticity The input arguments are the model and the rotation vector in rad s model demosdt demo sector all K model Case fe_case assemble matdes 2 1 NoT cell model SE fe_cyclic assemble struct model 10 10001 7 def fe_eig K 1 2 Case T model DOF 6 20 0 Non rotating modes def2 fe_eig K 1 SE K 4 Case T model DOF 6 20 01 Rotating mode shapes def data def2 data Note that the rotation speed can also be specified using a stack entry model stack set model info Omega 10 10001 Build model fe cyclic build nsec epsl len model LeftNodeSelect addsa cyclic symmetry entry in the model case It automatically rotates the nodes selected with by 27 nsec and finds the corresponding nodes on the other sector face The default for LeftNodeSelect is GroupA11 which selects all nodes
45. 4 Compute response def fe time model iiplot def 3 7 2 Gyroscopic effects Gyroscopic coupling is represented by a load Q t D Q 1 V where Q is the rotation speed and D the gyroscopic coupling matrix That can be applied at each step of time using the DofKuva n1 spring non linearity see sdtweb nl_spring An unbalanced mass is a load proportional to rotation speed In the local rotating frame it can be described using n1 spring DofV non linearity Following example deals with the simple 1D rotor and performs a time inte gration in fixed global frame taking in account the gyroscopic effect for an initial impact on the disk Note that the non linear holding of the gyroscopic effect is not necessary here since the global frame is considered and rotation speed is assumed 40 CHAPTER 3 TOOLBOX TUTORIAL to be nt 1000 7 The gyroscopic effects in green in figure below are oupli e y and z x 107 1s hy MN Sem AMI gunu RA 7 e 3 7T y top and z bottom deflection with green and without blue gyro effect model d rotor TestShaftDiskMdl model fe case model FixDof Ends 3 7 TIME DOMAIN ANALYSIS 41 11 01 1 02 1 03 4 01 4 02 4 03 0 011 TimeOpt opt nl_spring TimeOpt gamma 501 opt NeedUVA 1 1 1 opt IterEnd if ite gt 90 evalin caller assignin base def opt Follow 1 opt RelTol 1e 5 opt Opt 7 1 factor t
46. Data info Eig pt 3 01 fe_cyclicb Display cf def fecom ColorDataEvalTanZ ColorBarTitle TanZ fe eyclicb Purpose Description Support for advanced cyclic symmetry computations fe cyclicb groups advanced commands used to build and manipulate reduced order models of single symmetric structures and their assemblies For more details on the associated theory you can refer to 8 Rotor Construction DiskFromSector This command builds a disk rotor model from a physical sector model s Shafts can be generated by combining multiple calls to disk from sector Once disks are combine their connection trough rim models is described in section Command DiskFromSector also handles the construction of the cyclic sector models Cyclic symmetry information can be already given in the sector model calls to e cyclic C build done beforehand or done in the command In the later case an optional 1 tol can be declared so that it is propagated to the subsequent call to fe cyclic build epsl tol where tol is the desired tolerance for left right interface node matching The example below demonstrates the capability of the function for two disks with 7 and 10 blades respectively cfedemo eyclic buildstep0 sectori cf Stack diski sector2 cf Stack disk2 4 build diski from sectori model fe cyclicb DiskFromSector epsl 1e 6 sector1 4 build disk2 from sector2 a
47. J P Lain Dynamique des rotors Cours de l cole Centrale de Lyon 2005 A Sternchtiss Multi level parametric reduced models of rotating bladed disk assemblies PhD thesis Ecole Centrale de Paris 2009 M Lalanne and G Ferraris Rotordynamics prediction in Engineering Wiley 1998 G Lallement C Berriet R and S Updating finite element models using static deformations International Modal Analysis Conference pp 494 499 1992 R G and V C Calcul modal par sous structuration classique et cyclique Code_Aster Version 5 0 R4 06 02 B pp 1 34 1998 Sternchtiss A and Balmes E and Jean P and Lombard JP Reduction of Multistage disk models application to an industrial rotor in 012502 2008 Paper Number GT2008 012502 E Balmes Orthogonal maximum sequence sensor placements algorithms for modal tests expansion and visibility January 2005 145 146 BIBLIOGRAPHY 11 A Sternch ss and E Balmes On the reduction of quasi cyclic disks with variable rotation speeds in Proceedings of the International Conference on Advanced Acoustics and Vibration Engineering ISMA pp 3925 3939 2006 12 G Vermot Des Roches Frequency and time simulation of squeal instabilities Application to the design of industrial automotive brakes PhD thesis Ecole Centrale Paris CIFRE SD Tools 2010
48. N iiplot NL20 iicom IIxOnly NL 1 NLfromLIN if norm ci Stack NL 1 Y 2 end NL20 Y 2 end norm NL20 Y 2 end sdtw err something has changed end DofKuva DofKuva defines a non linear load of the form 120Non linearities list factor Do ferrent K V type DofKuva lab Label of the non linearity Dof Dof of Case DOF Dofuva 1 for displacement Dof 1 01 for velocity and 0 1 for acceleration MatTyp Type of the matrix K see Mat Type Desired matrix is automatically assembled before time computation factor Scalar factor exponent Exponent of the DOF uva Type of vector V 1 O 0 for displacement 0 1 0 for velocity and 0 O 1 for acceleration For example one can take in account gyroscopic effect in a time computation with a NL of the form model stack set model pro DofKuva1005 2 gyroscopic effects struct il 1005 fe mat p spring SI 1 0 type p spring NLdata struct type DofKuva lab gyroscopic effect Bof 1 060 D fuva O 1 01 7HatTyp 7 gt factor 1 exponent 1 uva 0 1 01 DofV DofV defines a non linear effort of the following form product of a fixed vector and a dof Do ferronent V type Dof V lab Label of the non linearity Dof Dof of Case DOF Dof 1 f loci 111 NDdata fields Dofuva
49. Rotor Module for the Structural Dynamics Toolbox For Use with MATLAB R VERY EXPERIMENTAL NOT ALL FUNCTIONALITY IS DOCUMENTED Some functionality in the manual may not be distributed User s Guide Etienne Balmes Jean Philippe Bianchi Version 1 0 Arnaud Sternch ss Hovv to Contact SD Tools 33 1 44 24 6371 Phone 33 6 77 17 29 99 Fax SDTools Mail 44 rue Vergniaud 75013 Paris France www sdtools com Web support sdtools com Support web site support sdtools com gf project openfem The FEM engine developped by SD I info sdtools com Sales pricing and general information SDT Rotor Module User s Guide on October 1 2012 Copyright 1991 2012 by SDTools The software described in this document is furnished under a license agreement The software may be used or copied only under the terms of the license agreement No part of this manual in its paper PDF and HTML versions may be copied printed photocopied or reproduced in any form without prior written consent from SDTools Structural Dynamics Toolbox is a registered trademark of SDTools OpenFEM is a registered trademark of INRIA and SDTools MATLAB is a registered trademark of The MathWorks Inc Other products or brand names are trademarks or registered trademarks of their respective holders Contents 1 Installation 2 3 Theoretical reminders 2 1 Rotating bodies 2 1 1 Problem definition in a rotating framel 2 2 Pr
50. US ERG The maximum loss factor is n 5 6 m spa 2 JR R and obtained for pulsation Ki Ko m x 5 7 di Ci K pen Ko is the static stiffness of the model Typically K and C is defined so that the damping is maximal for the frequency of interest sNon linearities list Following example considers Ko 1000N m K 500N m and C1 1 4N s m These parameters lead to a maximum loss factor of 20 1496 for a frequency of 46 41Hz The module and the loss factor are represented in figure 5 3 1600 T 1400 Yo x 2 1200 4 1000 10 10 10 10 10 10 freq Hz 0 4 eu 0 3 4 0 2 Pan 4 01 4 0 1 1 m m 10 10 10 10 10 10 freq Hz Figure 5 3 Module and loss factor Following example consists in a mass of 1e 2kg linked to the ground by the zener model Initial displacement corresponding to a 1N load on the mass is imposed and then a time simulation is performed parameters param m 1e 2 param dt 1e 4 param N 1e3 param k0 1e3 param ki param k0 2 param ci 1 4 7 zener parameters define model mdl struct Node 1 01 Elt Inf abs C massi 0 1 param m 01 define nl maxvell data data nl maxvell sprintf db zener kO 15g k1 2 15g ci 15g
51. a discretized vectors DOFs are placed either sequentially x y z at all nodes of the element or separated x at all nodes while the operations BE 11 Nzyz imply the use of vectors This 2 dimensional product notation however directly reflects the numerical implementation as is thus deemed preferable In the applications considered in this study one will use a fixed axis of rotation Q w t e The matrices and loads are thus proportional to the scalars w w and w One will thus simply use Dg w w Dg 1 which results in significant computational cost savings since the matrix only needs to be computed for a single velocity One proceeds similarly for the other matrices and loads 10 CHAPTER 2 THEORETICAL REMINDERS 2 2 Problem definition in a fixed frame Fully axisymmetric rotors can be modeled in a fixed frame using an Eulerian repre sentation where particles are moving under a deformed mesh Particles located at point p po t po u po t in the deformed Eulerian frame have a velocity given by xxx tos ocu PU CP t t z 2 6 The body fixed frame verifies 0 4 Ot which can be written as pg r 0 z Ro p The matching of displacements in both frames is given by tu p t ue pa t 2 7 The velocity of a particle in the disk is given by ead I0 pa I C26 2 8 Validation example One first considers a disk that has a steady state de formation in the global
52. aces the vector product w t by the product by the corresponding skew symmetric matrix 0 wz t w t Gi wlt 0 wz t 2 3 wy t welt 0 The acceleration derived from the velocity expression is given by a 2 66 s 2 2 a 07 x 9 0 p u 2 0 a 27 o u 2 1 ROTATING BODIES 9 where three contributions rows of the equation are typically considered the ac celeration in the rotating frame the Coriolis acceleration and the centrifugal accel eration The virtual work of acceleration quantities is thus typically expressed as id MJ 45 Dg at Ka Ke a fe fa 2 5 with the following element level expressions The displacement within an element is given by the position and the element shape functions N in the three directions LYZ x p AD Nus 4 The matrices and loads are integrated over the volume So in the reference con figuration and are given by M Po Newall 11 Nzyz dSo the mass in the rotating frame Dy Js 200 Naye Nzyz dSo the gyroscopic coupling Ke po Nes 1951 Niyz dSo the centrifugal softening stiffening e Ka s 00 RET Nryz 486 the centrifugal acceleration f s Po Noyz 1971 p dSo the centrifugal load s 00 Neel d ip dSo the Coriolis load It is acknowledged that the notations used can be somewhat confusing Indeed in
53. and allows to build the Frequency Response Functions of a rotor model either full or reduced A load and a set of observation DOF have to be defined and added to the model with e case The frequency range is stored in the stack as a info Freq entry The general call is xF fe_cyclicb ShaftFRFD disk lossfac xF fe_cyclicb ShaftFRFD rest disk lossfac cf sel xF fe_cyclicb ShaftFRFMS disk def damp xF fe_cyclicb ShaftFRFMS rest disk def damp cf sel o fe cyclicb The command FRFD assembles the matrices of the model then uses them to compute the response An optional loss factor can be specified The command FRFMS synthetizes the response from a set of modeshapes A damping ratio for all modes can be specified The option rest restores the response computed on the reduced model to a given selection of physical DOF Without selection the response is restored to the whole physical DOF set This option must be disabled when dealing with a full rotor model The example developed in demo cyclic ShaftMulti builds the sythetized re sponse of a reduced rotor model to a random excitation 4 define model cf load def demo_cyclic buildstep5 Compute the response to the random excitation mdl fe case cf Stack mvr dofload Load load mdlestack set mdl info Freq linspace 900 2000 2201 xFefe cyclicb shaft frfms mdl def 001 select where to resto
54. ass Asynchronous load 0 52 10 v r 7 10 E E TEN 3 10 dE 3 10 B H 3 10 8 107 6 E Theoretical 8 SDT q l a ee ne 10 i 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Rotation speed RPM Rotation speed RPM Figure 4 7 Response to an unbalanced load left and asynchronous load right Asymmetric rotor with z bearing stiffness Now there are 2 maximums of response that match critical rotation speeds for 4 2 SIMPLE 2DOF MODEL OF SHAFT WITH DISK 53 both forward and backward modes This is the same for the asynchronous load Bearing stiffness along z is taken in account so that rotor is not symmetric kzz 5e5 kez leb and kzz kzx 0 Damping in the bearing is also consid ered with crx 1006 czz 5000 and ezz czz 0 beta 15 ri struct om linspace 0 20e3 201 s 1 kxx 165 kez Beb kxz 0 7kzx 0 cxx 1e2xbeta czz 5e2 beta cxz 0 czx 0 d_rotor TestLalanne2D0F r1 Frequency Hz Frequency Hz Frequency Hz Unbalanced mass 100 A t z 10 80 pe UN poe gt pb pee H OF per s 10 444444 m 40 5444444 5 7499 444 4 4 4 hip 10 L 1 n L n L L L L 205 5 10 15 20 02 04 06 08 1 12 14 16 18 2 Rotation speed 1000 RPM Rotation speed RPM
55. ating part An example can be found in t nlspring 2beam XXX this must be generalized to make it separated from psa08 Temporary strategy was to define prior a RotCenter link then convert it to the nl rotcenter link using mdlepsa08 n1 rotcenterFix cf mdl R0 command This should no longer be used XXX Default uses the damping and stiffness defined in the il field of the p spring pro entry to model a linear spring damper between the 2 parts stiffness il 3 and damping il 5 Defining a xb parameter the Excite NONL law will be applied instead of the spring damper Parameter that are to be defined are xb Radial clearance kb Stiffness at radial clearance cb Damping at radial clearance Stiffness and damping at initial position are given in corresponding p spring prop erties il 3 and il 5 For example cf mdl nl spring setpro Prold 103 k 371 c 2000e 3 xb 0 03 kb 37100 cb 5 scr mal One can also use a squeeze film type law defining a muRL3 field with following parameters XXX this need to be reactivated and updated muRL3 oil viscosity bearing radius bearing length boundary Boundary condition 0 Sommerfeld 1 Gumbel p 0 N_theta Number of integration point Defining a RelF field one can define a sub call to another non linearity to compute the force resulting from the excentricity computed by nl_rotcenter In Non linearities listi27 that case the RelF field contains the NLdata field of the s
56. b NL unl The matrix is expected to be in transposed form NL b v handle mklst b These conventions allow reuse of a c matrix for command Note that the sign conven tions when using unl to return a non linear force are opposite to what is done when the result is added to fc see sdtweb nl fun to compare conventions Non linearities list Purpose List of supported non linearities t is possible to create new ones sdtweb nl fun nl inout nl inout is the more general non linearity using observation and command matrix fnt b x f C u Cw The pro NLdata structure has fields 114 Non linearities listi15 type lab iB 6 nl inout Label of the non linearity Command and observation definition If the NLdata DOF field is defined placement of the observation matrix in done during the model assembly phase Sens Load lternate form for b c xxx Sens ell array of the form Load Fu JElt unlO SensType SensData where SensType is a string defining the sensor type and SensData a matrix with the sensor data see sdtveb sensor data structure defining the command as a load with DOF and def fields defines the function of C u and C v defining the load Multiple forms are possible Currently a cell array of tables for Fu a nu meric curve ID for a curve in the stack 2 predefined laws ex ist Friction f f cO cO and BumpStop dumax 0 02 kO 500 cO dumin 0 02 ki 500 ci O a bumpstop
57. be non linearity with a tabulated effort relative displacement and effort relative velocity law between the DOF dof2 dof1 respectively in the Fu and Fv fields of the pro Stack entry First column of Fu resp Fv gives the relative displacements resp velocities and second column gives the efforts One can give a coefficient av factor of Fv depending on relative displacement as a third column of Fu It can be useful to describe a non linearity depending on relative displacement and relative velocity Force applied is F av du Fv dv It is used in particular to describe damping in a stop But NL Following example performs a non linear time computation on a simple 2 node model model Node 1 000 000 2 000 0011 model Elt Inf abs celas 0 0 1233 3 Q L0 10 4 linear celas 12 3 3 100 2 00 Associated to nl HH 122Non linearities list model Stack pro celasi struct il L100 fe mat p spring SI 1 163 1 0 type p spring NLdata struct type nlspring daft 1 03 d012 42 09 99 but 1e 3 163 4e 4 1e3 01 model fe_case model FixDof base 1 model fe_case model DofLoad in struct DOF 2 03 def 1 model fe_curve model set input TestStep 0 02 model fe_case model setcurve in input opt nl spring Time pt opt Opt 4 5 1e 3 1e4 opt NeedUVA 1 1 0 def f
58. beami gyroscopic matrix 70 Omega 0 1 0 axis of the disk mdbeami struct Node 1 h 2 0mega 20 2 EG Int abs beami 0 1211 0 0 012 mdbeami il p beam converttoi p beam sprintf dbval 1 TUBE 2 15g 15g R2 R1 mdbeam1 pl m_elastic dbval 1 steel mdbeami fe_case mdbeam1 fixdof fixi 1 find Omega 0 3 100 2 find Omega 0 3 100 rb feutil geomrb mdbeam1 mdbeami fe_case mdbeam1 DofSet fix2 rb rb fe_simul static mdbeam1 mdbeami fe_case mdbeam1 remove fix2 mdbeam1 DOF mdbeam1 fe_caseg assemble se matdes 2 1 70 mdbeam1 mdbeami DOF fe case gettdof mdbeam1 rb feutil placeindof mdbeam1 D F rb mdbeam1 Kefeutil TKT rb def setdiff 1 6 find Omega 0 3 mdbeami K if norm w mdbeam1 K 3 DGth norm DGth 1e 8 sdtw err beaml gyro 70 unmatch 4 1 RIGID DISK EXAMPLE 49 end The response to the unbalanced mass is the same in both of the 2 frames The maximum of response matches rotation speed found as critical speed for forward whirl in local and global Campbell diagram Note that we can compute Campbell in local frame from Campbell in global frame with E Oli for forward modes and ye z for backward modes The 2 Campbell diagrams have been computed using matrices in corresponding frame and we check that we can pass from one to other u
59. ce minus the non linear force computed The call performed is nl fun r2 fc model u v a opt Case This call is low level and must modify fc using sp util setinput as fc fnl where fnl is the non linear force computed Note that this is the only possible call for nargin 8 Note that allows a formalism with precomputed observations using fields unl e Jacobian computation must output the tangent stiffness and tangent damp ing matrices associated to the non linearity The call performed is kj2 cj2 nl_fun NL model u v opt Case RunOpt This call must output either empty matrices if no tangent nor Jacobian matrix is associated to the non linearity or matrices expressed on the DOF vector of Case DOF The first matrix is the tangent stiffness matrix the second on is the tangent damping matrix e Initializations for must initialize the model non linearity for non linear forces computation The call must generate the non linearity stored in model NL it can optionally generate non linear DOF and labels The call performed is of the type NL nl_fun init data mo1 NL is a struct containing at least the field type with the n1 fun handle e g NL type Onl fun data contains the Stack pro entry and mo1 is the model named mo1 where the call is performed ParamEdit returns the ParamEdit string allowing integrated parameters in terpretation for internal SDT use The call performed is of the type 130 Crea
60. ction fe_cyclicb reduce 1 1 int cf disk cf Stack diskmodel Mode Computations disk stack_set disk info EigOpt 5 50 1e3 11 1e 8 def disk fe_cyclicb diskeig disk fe_cyclicb diskplot 3 cf def 4 External Load Load fe_cyclicb DiskEngineLoad 3 cf fe eyclicb DiskSeDeflnit cf Rload fe_cyclicb DiskSeLoad cf Load fe_cyclicb diskplot 4 cf Rload OBSOLETE Disk internal commands DiskRest computes the Fourier Recovery Matrix associated with the specified har monic This function is used internally by fe cyclicb DiskMatrices returns the reduced matrices assembled from the sector superele ment matrices When the option mk is specified it returns the mass and stiffnes matrices only cf demo_cyclic testload disk 5 nor reset reload model fe cyclicb reduce 1 1 int cf disk cf Stack diskmodel KrOefe cyclicb diskmatrices disk disk il 1001 1 1 25072 250741 Kr fe_cyclicb diskmatrices mk disk PolyAssemble noT supports the computations of the coefficients of the ma trix polynomial from the assembly of the stiffness matrices including gyroscopic and nonlinear tangent stiffnesses at the given values of the parameters Three values are required The following example assembles the mass and full stiffness matrices at 0 500 and 1000 rps then computes the modes of the free sector at 250 rps 1000bsolete
61. d default material pro 4 Assemble nominal matrices md13d fe_caseg assemble reset secdof matdes 2 1 7 8 md13d One can also find an example of a 3D rotor ind rotor TestVolShaftDiskMd1 data Rs 1 shaft radius Ra 0 15 gt disk radius Le 0 4 shaft length La 0 03 disk thickness yd D A S disk position on the shaft yb 0 4 2 3 bearing stifness position on the shaft 0 Tol ie model d_rotor TestVolShaftDiskMdl data rotor2d BuildFrom2D convert 2d model to 3d model using cyclic symetry For example to convert previous 2D model mdl12d rotorid 1To2d quad 1c0 02 mo1d md13d rotor2d buildFrom2D close nsec3 div8 md12d 05 elt length Tol 22 CHAPTER 3 TOOLBOX TUTORIAL 3 1 4 From sector to shaft in the case of cyclic symmetry Closing a disk The SDT function only handles single sector models using symmetry conditions SDT Rotor extends the capabilities by dealing with a full possibly multi stage then called shaft rotor model Building a disk shaft model is done in two steps For each sector one de fines matching edges using a fe cyclic Build command the generates a disk shaft model using fe cyclicb DiskFromSector demosdt download back http support sdtools com gf douvnload forummessage 4 11 donnees secteur dat 4 removeface removes skin elements samcef read removeface fullfile sdtdef
62. dd a rotor bearing as a celas element molderotorid AddBearing mold pos moid is a 1d rotor model pos is the x y z position of the bearing Command options are e keep if not present remove existing celas at given position e DOF DOF that defines the DOF concerned by bearing link e k k the spring stiffness c c the spring damping T1 vrrotorid ProlD ProlD the spring ProlD For example moid rotorid AddBearing DOF 123 k 1e4 keep moid pos AddBeam moid rotorid AddBeam moid This command adds a beam on the axis of the rotor accordin to following argument specified as command option x1 z1 beginning of the beam in the axis direction x2 z2 end of the beam in the axis direction ri ri inner radius of the beam By default 0 r2 r2 outer radius of the beam MatID MatID MatID If not given new MatID is used ProID ProlD Note that the associated element property contains the section information according to r1 and r2 so that existing ProID will be lost If not given new ProID is used refine Refine existing beams to ensure that beams have coincident nodes For example moid rotorid AddBeam xi 0 5 x2 0 7 ri 0 11 r2 0 77 MatID 1 mold AddNodeRefine lmold indlerotorid AddNodeRefine mold xyz This low level call command adds nodes at given positions xyz one row per node 3 columns x y and z or one column to define distance along the rotor axis refinin
63. delize link between rotating and non rotating parts In the fixed basis one can use a non linear spring to model the bearing The nl maxwell non linearity describes a set of generalized maxwell rheological models that can be used as non linear bearing in the global frame As an illustration following example replaces the linear bearing of the 1D rotor by a zener model of the link model d rotor TestShaftDiskMdl Build model model fe_case model FixDof Ends 1 01 1 02 1 03 4 01 4 02 4 03 0 01 Clamp ends TimeOpt opt nl spring TimeOpt opt Follow 1 opt RelTol 1e 5 opt Opt 7 1 factor type sparse opt Opt 4 1e 4 opt Opt 5 0 6e4 NSteps opt IterInit model FNLO model FNL 0 Initial impact model fe_case model DofLoad In struct det te DOF 202 lt lt curve sprintf TestStepA 15g opt Opt 4 5 NL bearings define nl maxvell data model il ismember model il 1 100 101 remove properties k0 500000 ki k0 2 c1 600 20 41 for f 54 15 Hz data nl maxwell sprintf db zener KO 15g ki 2 15g c1 15g kO k1 c1 data Sens 2 3 02 translation sensor defining nl maxwell inputs ri p spring default ri feutil rmfield ri name ri NLdata data ri1 i1 3 k0 translation sensor defining nl maxvell inputs ri il 1 100 modelestack set model pro bearingy r1 ri NLdata Sens trans 3 03 translation sensor defining n
64. e 300r T vb N Frequency Hz a a N Q o o o N o o o o 4 F o al aa cee m z gt A 5 b ob n l PPE Pe DP bb bb b p pb oL UP bob Pyb bib bb bu b pb 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Rotation speed 1000 RPM Rotation speed 1000 RPM 080500 Frequency Hz 0 10 20 30 40 50 60 70 80 90 Rotation speed 1000 RPM Figure 4 9 Campbell diagrams Left 1D rotor up projected on 2 DOF down full model right Lalanne 2 DOF rotor 4 8 1D MODELS 55 Following example performs preceding comparison LALANNE 2DOF model2DOF m 14 29 k 1 195e6 a 2 871 kxx 0 kzz 0 kxz 0 kzx 0 bearing 4 define model matrices model2DOF K m eye 2 k eye 2 sin pi 2 3 2 kxx kxz kzx kzz a O 1 1 model2D0F DOF 1 01 03 model2D0F Klab m k Dg model2D0F Opt 1 0 2 1 70 rotate model so that x rotating axis model2DOF K feutil tkt 1 0 1 model2DOF K model2DOF K feutil tkt O 1 0 1 0 1 model12DOF K for ji 1 3 model2DOF K j1 model2D0F K j1 2 31 2 31 end model2D0F DOF 1 02 1 03 model2D0F stack_set model2D0F info Omega 1 01 SHAFTDISK beamitmassi gyro 70 model1 TRl d rotor TestshaftdiskMdl TRLalanne model Elt feutil removeelt eltname celas model project matrices on 2 sin shape functions equivalent to lalanne 2DOF m mod
65. e fe_cyclicb fourier 1 50 mono egyfrac cf fe_def subdef def def data 2 1 sdemo cyclic xxxNeed display curve fe cyclicb fourier mono egyfrac cf 3 Curve ri sortrows def data if any r1 1 6 1 gt r1 7 1 1e5 error Missing rigid modes end Model Reduction fe_cyclicb ShaftPrep svdtruncate mseqi handle norestit cf def fesuper fassemble cf Mode Computations defr fe_eig cf Stack MVR 5 50 1e3 11 1e 81 Sel disk2 innode r gt 150 disk21 innode r gt 150 disk1 groupall diski1 groupall1 withnode x gt 0 fesuper sebuildsel initrot cf Sel cf def feutilb placeindof cf sel cna 1 adof defr fecom colordataevala Post processing spatial spectra ef sel reset cf def fe_cyclicb fourier 7 25 red egyfrac cf 4 cf defr fe_cyclicb fourier 7 25 red egyfrac sortbyd cf 5 cf defr focus on mode 7 fe_cyclicb fourier 7 red egyfrac cf 6 cf defr Now one wants to treat the case of a forced response for loads defined on the disks only The command ShaftLoadMulti allows to build one mono harmonic excitation per disk For each disk 2 the shape def 2 and harmonic coefficient delta 4 are specified in a cell array whose a typical line is diski def 2 delta i Call fourier provides a means to checkout the spatial harmonic content of the
66. e filled by user in order to define what matrices will be assembled Default is 2 1 7 for mass stiffness and gyroscopic coupling matrix see sdtweb MatType for available matrices for example 8 for centrifugal softening Before time integration desired matrices will be assembled taking in account the info Omega model stack entry that defines rotation vector norm should be equal to one so that matrices can be assembled once for different computation with different rotation speed Rotation speed is taken as first parameter on the info Range model stack entry in that case rotation vector norm must be one If there is no Range entry rotation is taken equal to the norm of rotation vector Corresponding M C K matrices are then built fe rotor Assemble call and will be used for Newmark time integration rotorld Purpose This fo agneral details about 14 rptors and section B12 fer pb full example of 1d meshing 1To2D From a 1d model with beami elements masses are not converted for the moment builds a 2d rotor LC l defines the max length of elements 1To3D From a 1d model with beami elements masses are not converted for the moment builds a 3d rotor one can see section for a tutorial e LC 1 defines the max length of elements e div n defines the number of sector of 3d mesh The grounded celas are assumed to be bearings RBE3 rings are defined at each bearing AddBearing This command a
67. e_time opt mode1 4 XXX need to update timeoptexplicit if 1 2 opt nl spring Time ptExplicit opt Opt 4 5 1e 3 1e3 opt NeedUVA 1 1 d2 fe time opt model figure 10 subplot 211 plot def data def def xlabel Time s title displacement subplot 212 plot def data def v xlabel Time s title velocity subplot 211 hold on plot d2 data d2 def r xlabel Time s title displacement hold off subplot 212 hold on plot d2 data d2 v r xlabel Time s title velocity legend Implicit Explicit hold off end Following example deals with a clamped free beam with a bilateral bump stop at the free end XXX image 4 define model L 1 b 1e 2 h 2e 2 e 1e 3 dimensions model model Node 1 000 000 2 000 model Elt Inf abs celas 0 0 20 20 100 1 110 0 4 linear celas Non linearities listi23 Inf abs beami 0 iz 31 oO 1D 0 a model feutil sprintf RefineBeam 15g L 20 model model fe_case model FixDof base 1 7 clamps 1st end model fe_case model FixDof 2D 0 03 0 04 0 05 7 2D motion model properties model pl m_elastic dbval 1 steel model il p_beam sprintf dbval 1 BOX 15g 2 15g 4 15g 15g b h e e Bump stop NL model stack_set model pro celas1 etruct 711 100 fTe mat puspring BI 1 16 9 0 0 0 l TYpe p Spr
68. ec number of sectors By default one considers 10 sectors for a different value use buildFrom2D nsec M e div l number of divisions in each sector The sector is modeled as a superelement called disk1 Model Initialization model2D rotor2d test simpledisk back cf feplot rotor2d buildFrom2D model2D SE cf Stack disk1 enforce boundary cond on sector and assemble SE fe_case SE FixDof Base z 1 01 SE stack_set SE info Omega 0 0 0 10001 define speed range SE fe_cyclic assemble se SE cf Stack disk1 SE fecom view1 cf Stack info EigOpt 5 20 0 define eigenvalue options RunOpt struct targ 1 define target diameter Range linspace 0 1 30 define speed points relative to range cf def hist rotor2d teig cf RunOpt figure 1 rotor2d plot hist set gca ylim 0 2501 74 demo cyclic Purpose Combines examples of the use of fe cyclicb commands Syntax demo eyclic testdisk nsec demo eyclic testrotor nsec1 nsec2 Disk Moved to section 3 5 2 ShaftMono Moved to section 3 5 3 Variable speed One considers stiffness matrices that are dependent on the rotation speep Assuming that a second order polynomial representation is sufficient one can build a vector of weigthing coefficients 1 a L uw ti 1 02 Q 02 03 ag 5 1 023 02 Q2 Q2 023 S
69. ed mass or asynchronous load mb 1e 4 40 0 15 7 mass distance to axis sel f0 1 7 s 1 unbalanced load s lt gt 1 asynchronous load om sort 3296 3200 10 3500 linspace 0 20000 101 RPM RotatingLoad Nodeld 20 thetaO exponent define complex rotating load if s 1 unbalanced mass model fe rotor sprintf rotatingload 2 15g 90 2 mb db mode1 else asynchronousload model fe rotor sprintf rotatingload 2 4 15g 90 0 f0 mode1 end ri struct Omega om 60 2 pi v skom 60x2xpi Range model stack_set model info Range r1 Ri fe rotor forcedresponse model compute forced response iiplot R1 plot response 4 Post radial deformation Q max abs R1 Y 11 2 figure semilogy om Q xlabel Rotation speed RPM ylabel radial def amplitude m if s 1 title Unbalanced mass else title sprintf Asynchronous load 15g Omega s end 4 4 3D ROTOR 57 N o o Frequency Hz a o o o 2 3 4 Rotation speed 1000 RPM Unbalanced mass Asynchronous load 19 o o o o radial def amplitude m i radial def amplitude m o 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Rotation speed RPM Rotation speed RPM o Figure 4 10 Top Campbell diagrams Bottom Responses to unbalanced mass and asynchronous load FigH 10 shows the radial deformation response f
70. eel shaft material model il p_beam dbval 1 circle 1e 2 shaft r 1e 2 model il p spring model il dbval 100 5e5 dbval 101 5e5 bearings ends boundaries model fe case model fixdof Ends 1 0121 02 1 03 4 01 4 02 4 0312 4 Assemble nominal matrices model fe caseg assemble reset secdof matdes 2 1 70 model cf feplot model 4 For solution see sdtveb freqstud Note that at this time only fixed frame representation is available for such 1d rotors beami and massi elements Gyroscopic coupling is then computed under MatType 70 The formula for gyroscopic coupling can be found in 6 The nom inal representation for these models is then the Eulerian point of view where the displacement of the rotor is expressed in a non rotating frame For the 1D represen tation the model nodes are always placed on the nominal rotation axis Thermal and pre stress effects are not accounted for 3 1 3 Meshing 3D rotor from 1D and 2D models The rotor module supports all 3D elements of SDT 2D models are only consid ered through an extrusion and 3D cyclic symmetry One can import a volume model or mesh it using feutil meshing commands This section describes procedures to mesh volumes from 1D and 2D models From 1D rotor model meshed using beam1 elements one can create a 2D model using rotorid 1To3D command Note that massi elements can not be converted for the moment For example with sect
71. ef 1 model fe_curve model set input TestStep 0 02 model fe_case model setcurve in input Define the TimeOpt and compute the solution optenl spring Time pt opt Opt 4 5 1e 3 1e4 opt NeedUVA 1 1 01 def fe_time opt model1 nl solve Purpose Description Integrated non linear simulations The simulation of non linearities require special handling in SDT which is pack aged in the non linear toolbox This function aims at performing classical studies such as done by fe simullor classical SDT models with this special handling See nllist for the list of supported non linearities Static To compute the static state of a model with non linearities q0 nl_solve static model It is possible to use custom fe timelsimulation properties using the model stack entry info TimeOptStat See nl_ spring TimeOpt for fields and defaults It is possible to use as command option any field from the usual static simulation option see sdtweb nl springfTime pt to have more details E g To redefine on the fly the maximum number of iteration one can enter q0 opt nl_solve static maxiter 100 model By default the StaticNewton algorithm implemented in fe time is called An Uzawa algorithm is also implemented in n1 solve under the method static nl solve uzawa This algorithm is very different from the StaticNewton one since here the solution is not increm
72. el fe caseg assemble secdof matdes 2 1 70 3 model TRefeutilb placeindof model DOF TR model K feutil TKT TR def model K model DOF 2 02 2 03 4 compare with reference matrices from Lalanne for 1 1 3 Compare matrices model K j1 model2DOF K ji if normest model K j1 model2D0F K j1 normest model K j1 gt 0 05 sdtw _err 2DOF and shaftdisk unmatch for mat s model Klab j1 end end Campbell with SDT matrices reduced on Lalanne shapes ri struct data linspace 0 9e3 unit RPM fe rotor campbell nodir crit cf1 model r1 Nov compare full beam model and 2D shapes model DOF model K if ishandle 1 close 1 end fe rotor campbell nodir crit cf1 model2D F r1 set findall 1 type line color k linewidth 2 linestyle hold on fe_rotor campbell nodir crit cf1 model r1 56 CHAPTER 4 VALIDATION hold off set gca ylim 10 1601 h findall 1 type line legend h end 0 3 2 DOF beam Following example computes frequency response to unbalanced or asynchronous load model d rotor TestShaftDiskMd1 2 Model Initialization 4 Assemble nominal matrices model fe caseg assemble reset secdof matdes 2 1 70 model Campbell diagram and critical speeds fe rotor campbell full critical model linspace 0 20000 30 set gca ylim 10 3501 Unbalanc
73. ented but fully re computed at each iiteration This is useful when some non linear forces do not derive from potentials Com mand StaticUzawa can be used in n1 solve to access it qoenl solve static Uzawa mode1 Mode The definition of modes for non linear models is not straight forward This com mand aims at computing tangent modes as function of a non linear model current state The resolution thus concerns a linear model with tangent stiffness damping matrices corresponding to the model current diplacement velocity accleration state The eigenvalue solvers used are then fe eigfor real modes and fe_ceigfor complex modes 132 nl solveus3 By default modes tangent to a static state are computed A static simulation is performed to produce a model state from which tangent matrices are computed It is also possible to compute tangent modes at specific instants during a transient simulation at SaveTimes instant and to store frequency damping data and defor mations Command options a A set of command options allows precising the mode computation wanted and the output Accepted command options to control the model computation itself are e cpx for complex mode computation default is real mode computation e stat for mode computation based on a static state typically after an fe_timebtat simulation e time for mode computations during a transient simulation exclusive with the default stat option e
74. eplacement Command option eltsel allows specifying in a string a FindElt command restraining the working area in the original model Command option smartSize allows generating a conforming mesh with a co herent mesh characteristic length Command option gmsh allows using gmsh to mesh the final volume Command option tetgen allows using tetgen to mesh the final volume by default Command option output asks to output the generated mesh in a mat file Command option OrigContour asks to keep original positions of mid nodes of the quadratic faces delimiting the volume to remesh This may however yield mesh wraping problems when the face to remesh is much coarser than the mesh trace to place for conformity 135 senl mesh Command option mergeTo allows specifying a FindElt selection command in a string to replace the mesh on another model selection than the one used to generate the conforming interface which uses eltsel e It is also possible to provide additional arguments which will be passed the the nl meshcover call performed in the procedure Limitations The Conform call only supports generation of conforming inter faces when one interface contour fully contains the other interface contour Handling of more complex contour configurations has not been implemented Besides this function has been designed to handle planar surfaces Additional operation to work on non planar surfaces are left to the user e g pre post
75. eristic length to Val to the interstice mesher Command option algo name allows specifying the meshing algorithm name to the gmsh mesher See the gmsh documentation for more information Replace The call Replace is designed to replace parts of a model face mesh with new given face meshes It is performed as model nl_mesh replace model nodesToReplace NewModel nodeIDtoKeep model is a standard SDT face shells model nodesToReplace is a cell array containing vectors of Nodeld specifying the areas to be replaced NevModel is a cell array containing the new face models which will be merged to the mesh skin in co herence with the removed elements specified by nodesToReplace nodeIDtoKeep is an optionnal argument specifying NodeId of the original model for nodes whose NodeId must not change in the transformation e Command option setMat allows defining a specific MatId to the output mesh e Command option setPro allows defining a specific ProId to the output mesh e Command option keepNoCheck in combination with the use of a third argu ment nodeIDtoKeep assumes the nodes numbering is correct and forces the nodes original numerotaion without check Contour Call ContourFrom generates SDT beamipr beam3 contour models for CAD defini tions All formats readble by gmsh can theoretically be used Only the geo stp and igs are tested Since geo files can contain geometric yet undiscretized objects a 1D meshing
76. erned by links it can be useful in order to reduce super elements keeping idof as interfaces DOF for instance data contains all the information needed to define links It is a 3 column stack like cell array First column contains the string connection the second the name of the non linear link described in the third column that contains a data structure with following fields C define nodes to connect in first C1 and second component C2 It can be a vector of Nodeld or a screw data structure slave nodes of the model nodes via RBE3 links see see sdtweb fe casettconnectionscrew link defines how to link component 1 to component 2 It is a 1x2 cell array First cell defines the type of link Equa1Dof or Celas and the second gives information about the link For celas link it is a standard element matrix row with replacing Nodeld 0 DoflIdi DofId2 ProId EltId Kv Mv Cv Bv NLdata optional defines non linearity associated to celas link See the list in list of supported NL If this field is not present or empty only linear link is considered PID optional is a 1x2 line vector that defines PID second column of Node matrix see sdtveb node of connected node 1rst column for 1rst component DID optional is the same as above defining DID third column of Node matrix see sdtveb node of connected nodes Following example defines model with a cylinder and a hole in a block The
77. esponses xF cf Stack xF mvr ci iiplot 3 iicom ci curvereset XF iicom ci curveXF XF Test vexF data XF Test xf sens ctaxxF def XF Test lab sens lab iicom SubMagPha iicom ci IIx nly Test animate response at sensors afe cyclicb Mesh cf sel 1 groupall cf sel 2 Test cf def struct def sens cta xF def DUE ef OStack i T st tdof i l o data xF data cf 0 1 sel 2 def 1 ch 1 ty8 scc10 edgecolor r linewidth 2 MeshSensMatch supports topology correlation for a SensDof entry defined on a multi stage shaft model MeshSurfSet generates standard surface sets for a sector MeshCylSurf model fe_cyclicb MeshCylSurf epsl val model NodeList makes a surface perfectly cylindrical The node list of nodes to be used on the left edge may be omitted it will be found automatically MeshFixTheta provides handling utilities for slanted sectors see section MeshFixRadial Generates MPC constraints for using DOFs given in radial coordinates 01 is radial 02 tangential 03 axial The following example illustrates a case where tangential motion of nodes 5 and 6 is set cf demo_cyclic sprintf testrotor i cf 5 5 cf Stack disk1 fe_cyclicb MeshFixRadial cf BStack diski l radial 5 02 5 02 def fe_cyclicb shaftteig 0 1 stack set cf mdl Get
78. example can be found in sdtweb t_nlspring BumpStop THIS IS NOT FULLY GENERAL In particular c and b must have the same size for the moment XXX need to be detailed XXX not available yet Optional Define an offset in observation before applying Fu as unl cxu unl0 It can be a vector giving direct offset as many lines as c It can also be a string defining what offset to apply the only strategy implemented is un10 q0 to remove observation of the static from observation at each time step It can also be the name of a curve stored in model stack No jacobian is computed for this non linearity Maybe it will be needed in some Cases nl maxwell 1 11 describes rheological models using stiffness and damping 116Non linearities list type 201 maxwell lab Label of the non linearity Sens Observation definition Cell array of the form SensType SensData where SensType is a string defining the sensor type and SensData a matrix with the sensor data see sdtweb sensor Load data structure defining the command as a load with DOF and def fields SE superelement that defines the rheological model Only matrices are used K field Mass matrix is ignored The DOF field is unused and first DOF are assumed to be the observations defined and following correspond to internal states Jacobian is computed using a Guyan condensation keeping only the observation internal states are condensed
79. example model stack_set model info Omega 0 1 01 will define a rotation axis along Y Note that the norm of this vector is assumed to be the rotation speed in rad s The norm of the rotation vector should be 1 so that matrices are assembled for a rotation speed of 1 rad s Indeed it is assumed that gyroscopic and centrifugal softening matrices have been assembled for a rotation speed of 1 rad s in many functions such as fe rotor etc 3 4 Frequency domain analysis full model Supported computations in fe_rotor are e Campbell diagram building 30 CHAPTER 3 TOOLBOX TUTORIAL e Direct computation of critical speeds e Frequency domain response to unbalance and harmonic loads on bearings e Time domain simulation accounting for non linear bearings is under develop ment these are performed with the shaft in a rotating frame and the stator fixed 3 4 1 Campbell diagrams full model Campbell diagrams are implemented in the fe rotor Campbell command model d rotor TestShaftDiskMdl fe rotor Campbell crit cf100 model linspace 0 8000 100 figure 100 set gca ylim 10 2001 Command options are cf2 to define figure where to plot crit to compute critical speeds stability to display a stability diagram damping rotation speed The following is an example of a simple disk rotating at the end of a long rod For other details see Model Initialization moi rotor2d test simpledisk back
80. fields xbea vector defining position of bearings according to rotor axis bea data structure of bearing infos bea length is the length of the bearings bea Npt the number of points on one bearing bea Prold is the Prold of the journal mxyz is optional Defines the mass position reduction is performed so that DOF are created at these positions EigOpt eig option for mode computation for Craig Bampton reduction One can use following options noreduce no reduction is performed rigid use rigid ring for bearings instead of rbe3 rings Lalanne 3D rotor model d_rotor TestVolShaftDiskMdl Build model 4 remove bearing and boundary conitions model Elt feutil RemoveElt proidi1000 1001 model remove linbearings model fe case model Remove Ends Build SE RO struct mxyz 0 0894892917244 0 118333333333 0 0516666666667 Pc EigO0pt 5 20 1e31 eig option for CraigBampton RO xbea 0 0 4 RO bea length 0 01 RO bea Npt 1 RO bea ProId 1 proid of the bearing SE fe_rotor SEBuild rigid model RO i Following is now in SEbuild con TimeOpt model fe rotor TimeOpt model Defines a default TimeOpt in the info TimeOpt model stack entry for time integration with rotor models Following command options are available zofe rotor e Assemble Defines AssembleCall field in order to take in account gyroscopic effects in time computation TimeOpt field matdes should b
81. g beams of the model mot1d if needed Command options are e epsl epsi Tolerance for new node adding For example moid ind rotorid AddNodeRefine moid 0 2 rotorld 3 Skyline mdlerotorid Skyline xy This command can be used to generate rotor model from skyline description xy is a 2 column matrix whose 1st column defines position according to rotation axis and 2nd column defines radius at corresponding positions One can define multiple parts rotor using separator NaN in the 1st column For 2D rotor meshing use NaN r to define internal radius r of the corresponding part e SkylineToBeam builds 1d beam rotor model See example in 1D subsection of section 3 1 1 e SkylineTo2D a 2D rotor model that can be used to generate 3D model by revolution or using cyclic symetry see rotor2d BuildFrom2D see example in 2D subsection of section 3 1 1 Following command options are accepted Lc 1 specify maximum length 1 of elements rotor2d Purpose The rotor2d function lets you build a superelement representation for rotor applications starting from a 2D model A tutorial is given in section 77 teig Needs description test Simpledisk BuildFrom2D model rotor2d buildFrom2D mode12D builds a 3D sector model by revolution of a 2D section In the rotor module the symmetry axis is always z so that if the 2D mesh is given in zy coordinates a permutation is performed e ns
82. gh 1 m is added to the residual e 1 12 can be a data structure describing modal damping with following fields def M vectors as columns data c modal damping coefficients as a vector cj 201 K optional additional damping matrix This matrix must be in a mkl transposed v handle format use v handle mklst K to convert a matlab matrix to this format Note that model K 2 K is taken in ac count for the jacobian computation whereas modal damping is not Corresponding additional residue term is T M 9j x cj di M T v e model NL can be a stack of non linearities Column 3 provides a structure with the following standard fields type a MATLAB function handle which is then called through NL type NL fc model u v Also some older NL string giving the type name 112 mkl utils s unl vnl observation matrix for non linear displacements and velocities stored in row format obtained with NL cev handle mklst sparse NL c pre allocated memory for the result of NL c u Must be consistent with the number of rows in NL c The computation is handled by mk1 utils if exists pre allocated memory for the result of NL c v Must be consis tent with the number of rows in NL c The computation is handled by mkl utils command matrix for non linear loads At the end of the NL type call it is expected that NL un1 contains the non linear component loads such that the residual becomes r r NL
83. he motion of A is defined in a rotating frame associated with angle 04 and large angle rotation Rrc 6A More generally A and B are no real nodes but defined implicitely as ob servation matrices nl rotcenter is an extension of documented above using observation matrices which is more general type sel JCoef drot lab Weights Stack string nl rotcenter a FindElt command to find elemnts asosciated to the NL link XXX really used 7 coefficient of celas stiffness and damping for jacobian computation Default is 1 the rotation DOF label optional Weight of the stiffness in a pivot link in fact computed force is multiplied by the weight factors before being applied so that the sum of weight coef divided by number of points by pivot should be equal to 1 Stack of cta coupling Of the form cta name ri r2 where cta is a constant string defining the type of the link name a string containing the name of corresponding links r1 is the observation in the first rotating part It is a data structure with fields Node defining the nodes involved 6eNon linearities list cta defining the observation matrix DOF defining corresponding DOF as many columns as in cta and SeName defining as a string the name of the superelement where cta is defined XXX if omitted it is assumed that DOF and cta are defined on the model DOF no superelement r2 is the same for the non rot
84. herthan slave Campbell fe rotor Campbell model RunOpt computes the Campbell diagram and dis plays it with fe_rotor plotcampbell out RunOpt if no output argument is requested RunOpt can be a vector of rotation speeds RPM or a data structure with at least field Omega containing rotation speeds and other fields giving other variables used in zCoef for example par field Accepted command options are 65 oofe rotor e cf i can be used to force display of the diagram in a specific figure e nodir avoids the call to determine the rotation direction e full forces the use of full matrices Without the argument full matrix eigenvalue calls are only performed with less than 100 DOFs e crit overlays the critical speed computation e stability displays stability diagram damping Omega Examples can be found in section Critical Computation of critical speeds It can be called when computed Campbell using command option crit see Campbe11 Critical speed are computed assuming deformation is a complex mode at the same frequency as the rotation speed w 0 in the global fixed frame or equal to 0 w 0 in the local rotating frame In the global fixed frame M X C QD Q 1 X K Q Ke Q 1 X 0 With X 02 M 4D Q 1 10 K Xo In the local rotating frame M X C QD Q 1 X K 92 Ke O 1 X 0 With X Xo constant K
85. his func tion automatically removes the segments that connect nodes of two different disks e an array of elements connecting nodes of the blades of each disk returned by a previous fe fmesh 3dLineInitAddInfo Quad4 The selection is stored in the cf mdl Stack info VievMesh If the reset 26 CHAPTER 3 TOOLBOX TUTORIAL option is not specified the current selection is appended to the existing one For each project you should typically edit a script similar to the following cfedemo cyclic BuildStepO0 Viewing mesh step 1 disk elements fe_cyclicb DisplayAllEdges cf Zfe fmesh 3dLinelnit cf 4 right click Type or Done L 1 3 5 15 26 0 1121 1123 1125 1135 1146 0 13 15 18 1133 1135 1138 fe cyclicb MeshRimLine2Patch reset cf L 4 Viewing mesh step 2 blade elements 4 diski fe_cyclicb DisplayFirst cf disk1 Zfe fmesh 3dLineInitAddInfo Quad4 cf pick four nodes to form a quad 4 use right click Type or Done to display Elt Inf abs quad4 154 156 152 148 1 1 148 146 150 154 1 1 146 142 144 150 1 1 142 111 83 144 1 1 fe_cyclicb MeshRimLine2Patch cf Elt disk2 fe_cyclicb DisplayFirst cf disk2 fe fmesh 3dLineInitAddInfo Quad4 cf pick four nodes to form a quad model Elt Inf abs quad4 1274 1276 1272 1270 1 131270 1272 1268 1266 1 1 1266 1268 1264 1262 1 1 1262 1231 1203 1264 1 1
86. i cf mdl X set gca ylim O 30001 Now a more standard non linear static computation for a range or rotation speeds modeledemo cyclic testblade cf feplot model r2 linspace 1e3 15e3 20 range struct data r2 0 1 unit RPM model stack_set model info Omega range model fe_cyclic LoadCentrifugal model opt fe simul NLStaticOpt opt MaxIter 100 opt JacobianUpdate 1 def fe_time opt model def data 1 range data 3 3 4 8 Complex modes To compute complex modes at a given rotation speed one can use Campbell command with option cmode Complexe modes are the computed using a 32 CHAPTER 3 TOOLBOX TUTORIAL real mode basis projection Mode options should be stored in the info Eig pt model stack entry The complexe modes are return for the first rotation speed given as input argument Once modes are computed you can display them in feplot The fecom ShowTraj command displays trajectories For example model d rotor TestShaftDiskMd1 model fe caseg assemble SE secdof matdes 2 1 707 model assemble model def fe rotor Campbell cmode cf100 model1 100 7 C modes at 100 RPM cf feplot model def fecom viev3 iimouse resetvie fecom cf ShowTraj 2277 Hz 0 00 Figure 3 6 Complex mode and trajectory 3 4 4 Forced frequency response to unbalanced load One can compute the frequency response to an unbalanced
87. ically builds the observation matrix that expands the generalized modeshapes to the sensors along the measurement direction A first application of this is to animate generalized modeshapes on the experi mental wire frame as proposed in the example below cfedemo cyclic buildstep4 def cf Stack def mvr fe cyclicb Meshs match sensors sensors stack_get cf mdl info Test GetData r1 sel sens fe_cyclicb MeshSecAddNode epsl 20 cf sensors tdof def 4 add wireframe ri Elt r1 Elt zeros size ri Elt 1 1 cf Stack WireFrame r1 Elt feutil setgroupall egid 1 r1 cf mdl fe_case cf mdl SensDof Test r1 animate generalized modes at sensors cf sel 1 groupall cf sel 2 Test cf def struct def sens cta def def D F e1 COStack Test tdof 1 data def dasta cf 0 1 sel 2 def 1 ch 1 ty8 scc10 edgecolor r linewidth 2 A second application is to display and animate generalized response or transfer functions cf demo_cyclic buildstep6 xF cf Stack xF_mvr match sensors sensors stack get cf mdl info Test GetData r1 sel sens fe_cyclicb MeshSecAddNode epsl 20 cf sensors tdof xF 4 add wireframe ri Eltelr1 Elt zeros size ri Flt 1 1 cf Stack WireFrame ri Elt feutil setgroupall egid 1 r1 cf mdl fe_case cf mdl SensDof Test r1 display r
88. ies of a specific type See n1list for more details on types of non linearities nl springio TimeOpt This command returns usual default TimeOpt for non linear simulations By default the output is the same as the TimeOptNLNewmark presented below See also fe time for TimeOpt definition details Supported TimeOpt commands are TimeOptNLNewmark or Time pt to obtain the TimeOpt for NLNevmark simu lations Use TimeOpt gamma 51 to introduce numerical damping by directly giving gamma e TineOptStat to perform static simulations see also e time n1 so1ve e TimeOptTheta to perform time simulations with the 0 method see Numerical damping can be introduced using TimeOptTheta alpha 05 the specified a value will be added to 6 so that the coefficient used in the simulations will be 01 0 a Time ptExplicit to perform time simulations with the explicit Newmark scheme From standard fe_timefimulations the following TimeOpt fields are added or modified e Jacobian field is modified to take into account non linearities see NLJacobianUpda e Residual field is modified to take into account non linearities and to use mkl utils to improve computation times see sdtweb mkl utils e AssembleCall field is modified to perform non linearities initialization af ter assembly AssembleCall is the string passed to generated by nl_spring AssembleCall DutputInit field is modified to also check non linearitie
89. igenvectors with multiple diameter in a single run For large models this can take many hours so that intermediate file saves are used to allow restarts One then typically expects to have set a cf mdl Dbfile file DB mat to allow memory off loading during the computation cf demo_cyclic testrotor 7 blade cf 2 rootefullfile sdtdef tempdir Disk 7 Batch Asetpref SdtRuntime ExecLocal 1 May be needed 4 cf mdl Dbfile root DB mat 7 out of core matrices fe cyclicb shaft Teig 1 5 batch root mat multiple Disk 7 batch diam mat files are generated Now reload pointers to selected solutions RO struct Fmax 8000 diams 0 1 51 di fe_cyclicb DefList root RO ssfe_cyclicb ShaftSeAssemble fe_cyclicb ShaftSeAssemble force cf mdl fname is used to assemble su perelement matrices of each of the disk superelements If a curve StaticState is defined in the model stack assemble is used otherwise SE fe mkn1 SE NoT is called If the force option is omitted an attempt to reload the variable Stack SE disk from the file is first made and assembly is only performed if that variable does not contain the matrices If a md1 Dbfile field is defined the argument fname may be omitted reset Xxx Deflist Def Cyclic symmetry results can be stored in three main forms e the basic form valid for real vectors only stores real and imaginary compone
90. ina NLdata struct C type nlspring dofl1 2 02 do0f12 0 gt but 10 02 5e2 0 02 5e2 0 gap k umin 2 if 1 model fe_case model DofLoad in struct DOF 2 02 def 50 model fe_curve model set input TestStep 0 02 else f linspace 12 18 3 model fe_case model DofLoad in struct DOF 2 02 def 1 model fe_curve model set input sprintf Testeval cos 15g t f 1 3 end model fe_case model setcurve in input 4 Time computation opt nl_spring TimeOpt opt Opt 4 5 1e 3 1e4 opt NeedUVA 1 1 0 def fe time opt model RotCenter The Rotcenter joint is used to introduce a penalized translation link between two nodes A and B rotation DOFs of NL entry are ignored where the motion of A is defined in a rotating frame associated with angle 04 and large angle rotation Rrce 6A The indices G and L are used to indicate vectors in global and local coordinates respectively 124Non linearities list The positions of nodes are given by wale Rer ra 4 pB lunio which leads to expressions of the loads as Fa Ruel K esla tralo K za a xB q l To account for viscous damping loads in the joints one must also compute ve locities Using 2 2 one obtains Gale Rer ia W0 pa 5 10 12
91. ion rotor model xy lU OF 0 2141 gist 1 1 1 17171 18 211 molderotorid skyline ToBeam xy Mesh as beams Add bearings as spring elements 3 1 ROTOR MESHING 21 moid rotorid AddBearing DOF 123 k 1e4 keep mold 0 1 OO 4 moid rotorid AddBearing DOF 123 k 1e4 keep moid 1 9 0 2 mdl3d rotorid i1To3d quad 1c0 02 div24 moid build 3d rotor cf feplot md13d Command option quad force the use of quadratic elements hexa20 instead of linear elements hexa8 Shaft is meshed using hexa8 degenerated elements The edges on the axis of the shaft are using the same nodes as the beam nodes so that bearings described by celas in 1D intial rotor can remain the same RBE3 rings are created at each bear ing celas Concentrated masses no inertia on the axis are also left THIS IS NO LONGER TRUE BUT SHOULD BE REACTIVATED XXX massi elements with inertia that represents disks are meshed as volume disks of arbitrary thickness dt 0 005 with radius R computed so that inertia along rotation axis is the same Ir 0 5m R3 R2 and then density is computed to match the mass m m z x R2 R x dt Ru is shaft radius Young modulus of disk is taken at 100 steel modulus Gyroscopic coupling and centrifugal stifness for 3D elements are only described in the rotating frame that is to say under MatType 7 and MatType 8 mdl3d fe mat defaultil md134 7 default element pro md13d fe_mat defaultpl md13
92. ion and rewrite Model Initialization model demo_cyclic testrotor 7 10 blade cf feplot model Compute matrix coefficients for a multi stage rotor range struct data 0 1 0 800 0 1600 unit RPM Assembling in the feplot figure allows memory offload model fe_cyclicb shaftRimAsSe model Needed for PolyAssemble 3 6 FULL ROTOR MODEL FROM CYCLIC COMPUTATION 37 fe_cyclicb polyassemble noT cf range Now run a mono harmonic multi speed computation cf Stack info Omega struct data range data 1 unit RPM cf Stack info EigOpt 5 20 1e3 define eigenvalue options MVR fe_cyclicb shafteig 1 ReAssemble 2 NoN buildMVR cf 7MVR Stack end end 2 0 4 17 0 0 2 skip problem with geomet rc struct data linspace 1 8000 50 unit RPM hist fe_rotor Campbell MVR rc fe_rotor plotCampbell hist struct fig 100 axProp ylim 0 3e3 X Sel disk1 groupall disk2 groupall Zfe eyclicb Displaysel cf def Sel 3 5 5 Complex modes Need documentation here 3 5 6 Forced frequency response to unbalanced load Need documentation here 3 6 Full rotor model from cyclic computation 3 6 1 Single stage full rotor example Starting from the mono harmonic computation in section One builds a full shaft model that will allow predict
93. ion of all the modes H5 close fclose all may be needed for overwrite cf demo cyclic testrotor 7 blade cf 2 cf Stack info Omega struct data 250 unit RPM 38 CHAPTER 3 TOOLBOX TUTORIAL dO fe eyclicb shaftsolvestatic 0 cf mdl auto display with no arg cf Stack curve StaticState d0 Run pt Root fullfile sdtdef tempdir Disk 7 Run pt FileSeefullfile sdtdef tempdir Disk 7 SE cf mdl fe cyclicb shaftSeAssemble reset cf mdl Run pt FileSe Run pt FileTeig fullfile sdtdef tempdir Disk 7 TEIG mat 4 See sdtweb fe_cyclicb ShaftEig for batch option def fe cyclicb shaft Teig 1 5 cf mdl 4 Now build a multi harmonic model fe cyclicb C ShaftPrep handle cf def fesuper fassemble cf Mode Computations defr fe_eig cf Stack MVR 5 50 1e3 11 1e 8 cf sel reset cf def fesuper sedef cf defr def_ext fe_cyclicb DefList RunOpt FileTeig 1 end 4 4 the same results should be achived by assembling the prestress inside shaftTeig xxx update matrices only once not for each diameter cf demo_cyclic testrotor 7 blade cf 2 cf Stack info Omega struct data 250 unit RPM cf mdl il 1001 fe_mat p_super SI 1 1 0 1 fe_cyclicb shaft Teig 0 1 5 batch reassemble cf mdl Run pt FileTeig def int fe cyclicb DefList
94. is is added to the model as a TR field The field data refers to the harmonic in column 2 and the rotation speed in column 3 The number of computed modes is controlled by the field info Eig pt in the stack cf demo cyclic testload disk 5 nor reset reset file rather than 1 95 960bsolete 4 Set of Cyclic Modes Fixed Interface Modes sector stack set cf Stack disk1 info Eig pt 5 4 11 1e 81 sectorestack set sector info Eig ptFixlnt I5 4 11 1e 81 sector fe cyclicb basis all int norm sector 0 3 110 0 0 0 0 2501 cf model sector cf def sector TR The a11 option is used to get both modeshapes associated with a double eigen value in the case where harmonics are not or half the number of sectors when applicable For more information on cyclic symmetry please refer to The norm option ensures that modes are orthonormalized in mass and stiffness because of convergence problems caused by the presence of double eigenvalues This option is not required when the eigenvalue problem is solved with Nastran fe_eig method 50 xxx discuss with EB When the int option is added modes of the initial sector with its left and right interfaces fixed clamped are also computed and added at the beginnning of the output These modes have a 1 in the field data The computation parameters are specified in the info EigOptFixInt stack entry When this
95. k Enforce mode 7 3 disk2 cf Stack disk2 Stack Enforce mode 7 4 diski17 struct def l DOPF cf Stack diskil E1 2 0 9 01 3 disk217 struct def 1 DOP cf Stack disk21 L EIE UA 104 01 4 def34 fe_cyclicb ShaftLoadMulti cf data def34 def sum def34 def 2 one column in def34 def per row in data fe_cyclicbs1 def34 data 0 0 fe_cyclicb fourier 1 red cf 1 cf def34 checkout shape 4 now keep the same shapes but force delta 0 on both disks data 3 0 def00 fe_cyclicb ShaftLoadMulti cf data def00 def sum def00 def 2 one column in def00 def per row in data def00 data 0 0 fe_cyclicb fourier 1 red cf 1 cf def00 checkout shape now try a random load ri fesuper fnode cf md1 ri rii 1 4 01 v1 5 1 9 09 ris 10 0S1 defrnd struct DU F ri def ll defrnd def randn length defrnd DOF 1 defrnd fe cyclicb ShaftSELoad cf defrnd defrnd data O 0 fe eyclicb fourier 1 red cf 1 cf defrnd 7 checkout shape 4 now consider a physical load on first sector of disk 1 ri fesuper fnode cf md1 rie ril 1 4 019T1 1 9 02 0 12 021 defsp struct CDOF r1 def defsp def fe_c defsp DOF 1 01 02 03 randn 3 1 defsp fe cyclicb ShaftSELoad cf defsp defsp data 0 0 fe eyclicb fourier 1 red cf 1 cf defsp retrieve a Dirac s comb ShaftFRF D MS restl This comm
96. l maxwell inputs ri il 1 101 model stack set model pro bearingz r1 ci iiplot 3 4 results will be store there def0 fe time opt model compute Validation 7 Contents 257577 44 su ss m d den ee 44 25 77777 CO 45 4 1 3 Validation with 3D model disk 47 4 2 Simple 2DOF model of shaft with disk 50 220077 aE 54 4 3 1 1D example in a fixed frame 54 4 3 2 10 models in a rotating body fixed framel 57 5675577 57 4 5 Data structure reference 59 44 CHAPTER 4 VALIDATION 4 1 Rigid disk example Y v Figure 4 1 Simple rigid disk We consider a simple rigid disk of axis Y thickness h radius R mass m This simple example is very useful because we can easily compute matrices in both of global and local frame for a simple description of motion with 4 DOF Besides we can compute in SDT equivalent matrices for a mass1 rigid disk for a disk described by a beam1 element and for a volume disk in hexa8 elements Then we can compare gyroscopic matrices and make sure that their implementation for each element is correct 4 1 1 Matrices in rotating frame The motion of the disk is described by 2 translations u and we and 2 rotations DOF 6 and 0 Disk is assumed to be rigid so displacements at each point of the disk are given by the shape functions u 10 0 u a 0 w u 0 Nays fal
97. lds DOF and def that defined 0 reference for Fu and Fv tab laws tab model nl_spring tab model This command is used to convert formal rheological description data stored in model Stack to a tabulated law description The format is likely to change due to optimization of the compiled functionality in mk1 utils see mkl_utils nl springiii BlockSave BlockLoad Undocumented intermediate save of a time block for long simulations that do not fit in memory mkl utils Purpose For detailed callback information see sdtweb nlspring_timeopt section 77 Residual command is used to compute standard residue mkl utils residual r model u v a opt Case call modifies variable r in mem ory according to following standard residue computation r model K i a model K 2 v model K 3 u fnl fc Typically in fe timel computations one has opt Residual r full fc mkl_utils residual r model u v a opt Case with fc the time load resulting from DofLoad entries in model Case and fnl is the sum of the non linear efforts if any computed directly by mk1 utils rotcenter mocirc2 in the non linear functions see sdtweb nl fun orinnl spring mkl utils then calls the adequate n1 fun function n1 spring by default automat ically Model information specifically supported by the residual command are opt Rayleigh if the field exists defines a global Rayleigh damping and opt Raylei
98. le cf 4 Force single harmonic 4 xxx Rewrite needed here ShaftMulti The second example is a non monoharmonic shaft computation The following example builds a reduced order model from a set of mono harmonic modeshapes whose Fourier harmonics are 0 1 and 2 and sector modes with fixed interfaces The latter are computed within the framework of mono harmonic com putations they are called with 1 in shaftTeig No confusion is possible since the demo cyclic true mono harmonic solutions with 1 are solutions with 1 However the restitution of fixed interface solutions with Display Se1 has no sense Call shaftprep aims to build the reduced kinematic subspaces of the sector super elements from the specified target solutions Prior to that it separates the sector mesh into two the slice with the left interface nodes the inter sector super element and the remaining elements the sector super element It projects the matrices of the sector super elements onto their individual subspaces and the ma trices of the inter sector slices onto the subspaces of its two neighbouring sector super elements Command fassemble in first projects the finite elements matrices of the inter disk volumic interface onto the subspaces of its neighbouring disks It then assembles the reduced matrices of the sector super elements and inter sector slices to form the reduced matrices of the disks Finally it assembles these reduced
99. lements containing nodes to connect 10601 spring radius rtol Figure 5 1 ConnectionCyl InitV 0 1 spring InitV model d0 RO0 InitV computes the initial static displacement and velocity associated to a DOF initial position and velocity dO is a data structure with field DOF containing the DofId where initial value is applyed and def containing initial displacement and velocity at this DOF RO is a optional input argument data structure with following fields that define dt time step for time integration dq increment for initial vel computation Nv number of time steps to reach d0 def 1 displacement is imposed as a 0 5 1 cos time function on these time steps Np number of steps to stabilize at d0 def 1 and d0 def 1 dq If input argument RO omitted options are get from info initvopt Stack entry If there is no such entry InitV parameters are computed using optim process see below Displacement at q0 and q0 dq is obtained meaning the last Np 10 steps of each stabilization period and initial velocity is computed from those 2 displacements to match d0 def 2 at dO DOF nl springior q0 RO 2n1 spring InitV optim model d0 can be used to find input pa rameters RO Optimization of dt and Np is performed from given or default values Parameters dq and Nv are kept at given or default value First dt is optimized dt is increased multiplied by 4 until time integration
100. load or to an asyn chronous load using and command see sdtweb fe rotor for more details Definition of the load is different in local ro tating frame or global fixed frame One can see the validation part of this document for various examples Following example computes the response to an unbalanced mass on the 1D rotor model mb 1e 4 db 0 15 mass distance to axis om linspace 0 6000 201 RPM model fe_rotor sprintf RotatingLoad 2 2 15g 90 2 mbxdb model unbalance ri struct mega om 60 2 pi w om 60 2 pi Range modelestack set model info Range r1 Riefe rotor ForcedResponse model compute forced response iiplot R1 4 plot response 3 5 SOLVERS FOR MODELS WITH CYCLIC SYMMETRY 33 3 5 Solvers for models with cyclic symmetry The SDT Rotor module contains e classical cyclic symmetry solvers where one assumes the solution to be asso ciated with a specific number of diameters spatial harmonic associated to the Fourier transform of a periodic geometry see 5 for more details e full rotor reduced models where cyclic symmetry solutions are used to build a reduced model for various stages The associated solvers are discussed in section 3 5 1 Static response Resolution of static responses is performed using fe_cyclicb ShaftSolvel You should be aware that non linear static iterations may fail to converge if you have rigid body modes in your system In the example
101. matrices and that of the volumic interfaces to form the reduced matrices of the whole rotor Here too a selection can be specified so that the generalized modeshapes can be recovered to a subset of physical DOF relying on the true mesh or a viewing mesh The selections are defined like in DisplaySel however both the sector and inter sector super elements have to be considered so that the recovery concerns the whole underlying bladed sector The Fourier harmonic contents of the generalized modeshapes can be obtained without recovery with the help from command fourier of When spec ified egyfrac returns the fraction of strain energy per harmonic per disk so that energy localization within a disk can be achieved with the supplementary informa tion of which harmonics are involved in the response The graph displays the disks from top to bottom and for each disk the possible harmonics between 0 and N 2 if applicable Another way to display the same information is to group the harmonics first and then the disks This is done with the sortbyd option It adds the proper amount of zeros for harmonics that are not present on a given disk mono mat to name a model that contains the mono harmonic description Model Initialization cfedemo cyclic testrotor 7 10 blade cf 2 Mono harmonic Solutions model stack_set cf mdl info EigOpt 5 10 1e3 11 1e 81 def sectors fe_cyclicb shaftteig 1 0 1 2 model batch Curv
102. mmand can be used to compute forced response to a frequency load for example a rotating load built using RotatinglLoad Riefe rotor ForcedResponse model Complex load must be prior defined as a In entry in the case of the model Observation can be defined as output SensDof entry If not only dof correspond ing to the In load will be returned Range of computation must be defined in the info Range Stack entry of the model as a structure of data with fields Omega defining rotation speeds and w defining pulsations of the load at corresponding rotation speeds These fields must be of the same length If necessary one can add other fields for variables used in a defined zCoef If w is equal to zero frequency dependence of load is assumed to concern the vector it is usefull to described unbalanced mass in the local rotating frame since load amplitude depends on 02 but load is static in this frame so that w 0 esfe rotor Following example defines an unbalanced load on the simple 1d ShaftDisk model defined in d rotor and computes the forced response local frame model d rotor TestShaftDiskMdl 2 build simple model model fe caseg assemble se matdes 2 1 70 model assemble model mb 1e 4 db 0 15 mass distance to axis om linspace 0 6000 201 RPM model fe rotor sprintf RotatingLoad 2 2 15g 90 2 mb db model unbalance ri struct Omega om 60x2xpi v
103. mpeller 0 215 005 0 006 003 0 222 002 0 006 003 between sub disks 0 222 002 0 0775 0 2259 0 0775 14 3 0 mo2d rotorid skyline To2d 1c 005 xy x axis mo2d pl 1 fe mat m elastic 1 1 200000000000 0 29 7800 2 fe mat m elastic 1 1 7 17e10 0 33 2830 742830 3 fe mat m elastic 1 1 3 07e12 0 3 78 7800 Ji mo2d Elt feutil findelt matid 2 3 4 5 mo2d 5 6 2 mo2d Elt feutil findelt innode x gt 15 amp y lt 0057 mo2d 5 6 3 mo2d Elt feutil findelt innode x gt 22 amp y lt 01 mo2d 5 6 3 feplot mo2d fecom colordatamat cf feplot cf model rotor2d buildFrom2D nsec16 mo2d cf sel disk1 ColorDataMat fecom view1 Figure 3 1 Example rotor generated with meshing utilities 3 1 2 Basic 1D rotor example Following example builds by hand a simple 1D rotor with one shaft one disk and 2 bearing stifnesses It is almost the same as one accessible through d_rotor TestShaftDiskMdl 4 define mesh model struct Node 1000 00 0 20 00 0 4 3 0 0 20 CHAPTER 3 TOOLBOX TUTORIAL 3000 0 4x2 300 4000 0 4 0 01 model Elt feutil ObjectBeamLine 1 4 define shaft model Elt feutil ObjectMass model 2 16 5 16 5 16 5 0 18608 0 093 0 093 add disk model feutil AddElt model celas 3 0 20 100 0 0 3 0 3 0 101 0 01 add bearings y and z stifness define properties model pl m_elastic dbval 1 st
104. n the model stack e ReAssemble forces reassembly rather than reuse of disk matrices that may have been precomputed and saved in the sector superelements e thermal takes thermal loading into account Thermal state should be stored as a case entry called DofSet ThermalState See example in section B 5 1 fe cyclicbs for computations at prestressed state curve StaticState should be defined all xxx FixTan is used to enforce no tangential motion of one interface node This is used for static analysis of freely rotating rotors example in section 3 5 1 NoN stores the imaginary component of the eigenvector using a DOF shift by 50 thus 51 is the imaginary z translation This is necessary if further computations require complex fields by considering different components there is no difficulty storing the spatial Fourier transform of a complex field model Dbfile when the field is defined to a proper file name intermediate matrices above the preference getpref SDT OutOfCoreBufferSize 100 in MB are stored in the database file which uses the standard HDF5 based mat format MATLAB j 7 3 nlstep tol is used to compute the large non linear large transformation problem with a tolerance q q tol See an example in section 3 5 1 For static computations the centrifugal load is rebuilt at each step using model fe cyclic loadcentrifugal model1 for the rim and sectors batch option is used to compute e
105. nd EltIdO The initial set of modes has to be given as a field TR of the model This operation can be performed directly by using the command Basis of fe cyclicb as shown in the following script Model Initialization cf demo_cyclic testload disk 5 nor cf reset fe cyclicb reduce 1 1 int cf disk cf Stack diskmodel 1020bsolete The reduction basis is built by separating right left and interior motion from the cyclic modeshapes When the int option is invoked fixed interface modes of the sector are added to the set of interior modes For more information about this procedure refer to 11 nl spring Purpose Syntax Description Non linear links force modelization for time simulation model nl spring tab mode1 nl spring command Uptenl spring TimeOpt nl spring supports non linear connections and loads for transient analysis Non linear springs between 2 DOF seelnlspring loads which depend on DOF values see DofKuval DofV springs between 2 nodes in different bases see RotCenterl etc A full list of non linearities is given below Standard non linear simulations are handled byinl solvel Below is a description of the inner mechanisms of a non linear simulation with the non linear toolbox After the non linearity definition a proper TimeOpt is required to set the good fe time calls to perform a non linear Newmark time integration A default
106. nd append to diski model fe cyclicb DiskFromSector epsl 1e 6 model sector2 fe_cyclicb DisplayFirst model Avoids full display for large models In cases when cf already contains one sector per disk the shaft model can be cre ated in a single operation with the command fe_cyclicb diskfromsector cf s where sel disk selects the sector model of disk i The example below illustrate this by putting the two sector models into a single one prior to the rotor assembly cfedemo eyclic buildstep0 See sdtweb demo eyclic Step0 83 safe cyclicb sectors cf Stack diski Build a model with two sectors sectors feutil addtest sectors cf Stack disk2 sectors Stack cf model sectors build rotor from sectors and auto display fe_cyclicb DiskFromSector epsl 1e 3 cf group1 2 group3 4 During the build process sectors are automatically renumbered so that node numbers are left interface interior right interface in order matching that of the left interface The renumbering can be forced with the renumber option This allows to have nodal overlap between the superelements of two adjacent sectors The command then adds a mpc disk2 end multiple point constraint to account for the fact that the disk is closed circumferentially Mesh Meshing utilities See fe cyclicb Mesh ConnectionRing ConnectionRing builds a ring connection where the structure is fi
107. neStep Number of step to refine the optimal parameter pair found in the first step Command option refine must be added to perform the refine step The last argument ofact 1 1u is needed in order to call directly spfmex utils Available command options are setopt use default RO refine performs refine step for optimal search fact to benchmark factorization step solve to benchmark resolution step plot to plot history in iiplot xxx this need to be generalised call to gvdr utils plot xxx Following example optimize only solving ki rand 20 RO struct nCompt 100 number of computation for result averaging 142 spfmex utilsi maxDomain 2 4 7 parameter 1 maxZeros logspace 3 1 5 parameter 2 refine8tep 3 refine results to most relevant parameters spfmex utils ofactoptim solve refine ki RO ofact 1 lu method sol extrotor Purpose External rtfiqrpfyossvvr sdtools com help mpc html 144 Bibliography 1 10 C Desceliers Dynamique non lin aire en d placements finis des structures tridi mensionnelles visco lastiques en rotation PhD thesis Ecole Centrale de Paris 2001 M G radin and D Rixen Mechanical Vibrations Theory and Application to Structural Dynamics John Wiley amp Wiley and Sons 1994 also in French Masson Paris 1993 J Batoz and G Dhatt Mod lisation des Structures par El ments Finis Herm s Paris 1990
108. ns A first super element is attached to the elements with the left interface nodes it is called the inter sector super element A second one is attached to the remaining elements to form the sector super element A reduced kinematic subspace of the sector super element with the definition above is built from def disk by disk Vectors are first sorted with respect to their contribution to the considered disk if the svdtruncate option is used Then they are sorted according to their contribution to subsets of physical DOF of the initial sector If one specifies mseq 0 default call these subsets are 1 DOFs within either the inter sector interface right interface or the inter stage interface s 2 remaining DOFs left interface and interior DOF If mseq 1 is enforced these subsets are 1 DOFs common to the inter stage and inter sector interface s 2 DOFs within the inter sector interface right interface 3 DOFs within the inter stage interfaces 4 remaining DOFs left interface and interior DOFs Both these sortings make the subsets of vectors linearly independent from each other They require that fixed edge solutions are stored at the beginning in def The following step is to make the vectors linearly independent within each set Vectors in sets 1 2 and 4 when applicable are processed with an Iterative oofe cyclicb Maximum Sequence Algorithm 10 Vectors in set 3 when applicable are
109. nts for the spatial fourier transform as real and imaginary components at a single DOF e the long vector form uses additionnal DOFs shifted by 50 Starting from the basic form one would have d1 DOF d1 DOF d1 DOF 5 di defelreal d1 def imag The test for usage of this format is that the last dof is above 05 rem d2 DOF end 1 gt 5 e the double vector form uses the nominal DOFs of the first sector but store the real and imaginary parts as consecutive vectors di def real di def 1 imag di def 1 d1 data di data 1 NaN NaN DefDouble DefLong DefBasic commands allow transformations between for mats while handling out of core files properly When reading results def fe_cyclicb DefList root reads all root mat files and combines the vectors into a single deformation set in the double vector for mat Selection of diameters and frequency range during the read process is peformed using RO struct Fmax 8000 diams 0 1 5 181 di fe_cyclicb DefList root RO fe cyclicbso Specific cases require to sort the output vectors according to the list of diameters specified in the diams field especially when one wants to put fixed interfaces solu tions first for reduction purposes The command to use becomes DefListSortDiam Full rotor SE model ShaftPrep fe eyclicb ShaftPrep cf def generates reduced sector super elements Each bladed sector is divided into two regio
110. oblem definition in a fixed framel 2 3 Fourier analysis of structures 2 3 1 cyclic structure basics 2 9 2 Fourier transform for shaft computations 2 3 3 Solutions in periodic medial Toolbox tutorial 3 1 Rotor meshing 2557 3 1 3 Meshing 3D rotor from 1D and 2D models 3 1 4 From sector to shaft in the case of cyclic symmetry 3 1 5 Utilities for handling slanted blades 3 1 6 Disk connections in multi stage cyclic symmetry 3 1 7 View meshes for cyclic symmetry 3 2 Bearing and support representations 3 2 44 Linear bearing 3 2 2 Non linear bearings in the time domain 3 3 Gyroscopic effects 3 3 1 Fixed frame models 3 3 2 Rotating frame models 3 4 Frequency domain analysis full model 3 4 1 Campbell diagrams full model 3 4 2 Blade with centrifugal stiffening 1 Q oo N 10 11 12 12 3 4 3 Complex modes 3 4 4 Forced frequency response to unbalanced load 3 5 Solvers for models with cyclic symmetry l SEN ERE IX Gia hea Se Oe a E cepa Stare ates aes ek 3 0 6 Forced frequency response to unbalanced load Tic y AMET Lol ee ao Ee FK ROESSOS Ames 3 7 1 Simple examplel 207 q Ev eee MT 4 Validation 4 1 Rigid disk examplel
111. odes for rim tesselation If you choose the penalty approach here is a working example where the edges of sector edges are assumed coincident thus allowing an automated search of inter section with the FindIntersect option 4 load two disk example with space between disks cfedemo cyclic testrotor 7 10 3 NoRim RimH 1 blade cf 2 reset RimStepi find nodes at the matching interfaces ni fe cyclicb DisplayAllEdges FindIntersect epsl 2 cf RimStep2 build rims as springs fe cyclicb MeshRimStep2 epsl 1 kp 1e12 slavedisk 1 3 masterdisk 2 cf n1 cf sel groupall colordatagroup edgealpha 05 alpha 1 def fe cyclicb shaft eig 0 cf mdl sel disk1 groupall disk2 groupall disk3 groupall rim 3 1 ROTOR MESHING 25 cf def fe_cyclicb DisplaySel cf def sel fecom ColorDataEvalTanz 3 1 7 View meshes for cyclic symmetry Figure 3 4 Sample viewing mesh for post processing multi stage computations Meshing tools also include procedures to build viewing meshes from the finite element mesh fe_cyclicb MeshRimLine2Patch cf sel aims to build viewing meshes made of surface elements connecting selected nodes of the true mesh of the rotor sel can be e an array of lines connecting nodes of the inner disk parts in the 2D cut of the rotor returned by a previous fe_fmesh 3dLineInit Note that t
112. odic 2 that DOF come in repetitive groups except for mono where the concept of mono harmonic modeshapes assumes that structures are periodic e egy and egyfrac provides means for energy based computations Option egy displays the fraction of energy in each existing harmonic within each disk so that the total amount of energy within each disk is 1 Option egyfrac displays the fraction of energy in each existing harmonic within each disk so that that total amount of energy within the rotor is 1 The default displays these quantities disk by disk from top to bottom and for each disk all the possible harmonics are displayed from bottom to top as depicted in the figure below e sortbyd groups these quantities first by harmonics from bottom to top and then by disk from top to bottom with the appropriate number of zeros for non present harmonics typically when 6 gt N 2 for a given disk as displayed in the figure below cf def demo_cyclic buildstep1 4 sdtweb demo cyclic stepi Curve fe cyclicb fourier 1 50 mono egyfrac cf fe def subdef def def data 2 2 15 xxx Whfe cyclicb fourier mono egyfrac cf 3 Curve 4 sdtveb demo cyclic step4 multi harmonic analysis cf defledemo cyclic buildstep4 4 fe cyclicb fourier 7 25 red egyfrac cf 11 cf def fe cyclicb fourier 7 25 red egyfrac sortbyd cf 13 cf def o4fe_cyclicb See also Section obsolete Purpose
113. ommand generates a non trivial extrusion of a face mesh follwing the face normal at each node to generate a volumic layer model nl mesh Extrudelayer thick Val model model is an SDT model with shell elements a surface definition Command option thick specifies the extrusion thickness Command option setmat allows specifying a specific MatId to the output Command option setpro allows specifying a specific ProId to the output nl meshi4 StackClean This call cleans up a model stack when mesh modifications have been performed It cleans up stack entries definition that became incoherent with some mesh modifica tions model nl mesh StackClean model Command option rmuns removes stack entries that could not be sorted out Command option rmmod removse stack entries affected by the model modifications See also s l p spring fe gmsh spfmex utils Purpose OfactOptim This command can be used to set spfmex parameters in order to optimize compu tation speed for factorization and or solving spfmex_utils OfactOptim ki R ofact 1 1u ki is the matrix that is used for the optimization RO is a data structure defining options with following fields nCompt Number of computation for result averaging maxDomain Max size of blocks of the elimination tree fraction of matrix size maxZeros Max number of zeros in the blocks of the resolution tree fraction of matrix size refi
114. ommand option sel is specified and a selection of elements is provided the shapes are recovered to that selection only Selections are cell arrays with the typical entries e Sel diski sel1 where sel1 is either a string to select a subset of el ements in the true mesh or a list of elements to build a reduced viewing mesh e Sel selg where selg is a string that selects elements of the global mesh selg is often eltname SE so that only disks are displayed In the same fashion mono harmonic static responses are returned by ShaftSolveStat This is of particular interest to compute the static deformation under the centrifu gal loading known to have a Fourier harmonic coefficient of 0 and built with command LoadCentrifugal within fe cyclic Model Initialization cf demo cyclic testrotor 7 10 blade cf 2 reset Mono harmonic Solutions model stack_set cf mdl info EigOpt 5 5 1e3 11 1e 81 def fe cyclicb shaft Teig 2 model demo cyclic RefcheckDisk710 def non regression check cf Stack disk1 fe_case cf Stack disk1 park blade innode r gt 20 Curve fe_cyclicb fourier 1 13 mono egyfrac cf def check energies iiplot Curve colormap flipud hot cf def fe cyclicb Display cf def static responses sdtweb freqcyclicitcyclic static cf Stack info Omega struct data 1000 unit RPM cf Stack cu
115. or an unbalanced load and for an asynchronous load rotating at Q speed The unbalanced load excites the forward modes 3296 RPM whereas the asynchronous load excites the backward modes 2785 RPM and 4697 RPM Frequencies match those computed as critical frequen cies in the Campbell diagram 4 3 2 1D models in a rotating body fixed frame While this representation is not very classical it corresponds to the nominal choice when doing time integration of a rotor that is not axisymmetric 4 4 3D rotor The same rotor as described in lalanne see fig 4 3 is meshed using hexa8 elements Use modeled rotor TestVolShaftDiskMd1 58 CHAPTER 4 VALIDATION Figure 4 11 3d model of Lalanne rotor Matrices are defined in the local rotating frame We described the unbalanced load by a static load and we use in following example the same procedure as for local frame 1d rotor at w 0 model d_rotor testvolshaftdiskmdl 4 Assemble nominal matrices model DOF model fe_caseg assemble se matdes 2 1 7 8 model model DOF fe_case gettdof model Campbell diagram model stack_set model info eigopt 5 20 1e31 fe_rotor campbell critical model linspace 0 20000 30 4 Unbalanced mass or asynchronous load mb 1e 4 db 0 15 mass distance to axis s 0 f0 1 7 s 1 unbalanced load s lt gt 1 asynchronous load om sort 2789 2750 10 2820 11760 10 11840 linspace 0 200
116. pic matrix in the global fixed frame There is no centrifugal softening in this frame 3 3 1 Fixed frame models The massi and beami elements gyroscopic matrices are only available in the global fixed frame MatType 70 The rotation axis is assumed to be the axis of the beam for beami elements Moments of inertia must be equal axisymmetry For 3 4 FREQUENCY DOMAIN ANALYSIS FULL MODEL 29 massi elements the rotation axis is assumed to be the one whose rotation inertia is different from the 2 others that must be equal massi Elt format n mxx myy mzz ixx izz EltId One can build simple models of 1d rotor using massi elements to represent rigid disk and beamt to represent the shaft One can find an example of such a rotor in d rotor TestShaftDiskMdl See section for more details on how to mesh such a rotor For volume and shell elements the formulation of gyroscopic matrices in global fixed frame is unclear and thus not currently implemented 3 3 2 Rotating frame models For all volume elements one can compute gyroscopic MatType 7 and centrifu gal softening MatType 8 matrices in the local rotating frame Elements under development are e nass Only point masses with same mass along the 3 translation DOF and no rotation inertia are considered Other are ignored e beami Not supported e shell Not supported In that case rotation axis must be given as a vector in info Omega stack entry in the model For
117. projections of the surfaces on a plane Cover The Cover call is degined to mesh the interstice between two closed planar contours when one fully contains the other The call is performed as newModel opt largeContour nl mesh cover model eltsel large j eltsel 11 F model is a standard SDT model Variables eltsel_large and eltsel small are FindElt calls defining the element selection of the respectively large surface and small surface the small being contained in the large The output newModel is the mesh generated from the surface contours opt outputs additional information about the mesh generation it is a struct containing fields NodeAdd specifiying the potential nodes added in the interstice space meshed nodeEdgeSel1 specifying the Nodeld of the nodes located on the eltsel large contour nodeEdgeSe12 specifying the Nodeld of the nodes located on the eltsel small contour and tname the name of the temporary file containing the generated mesh largeContour provides the original contour in beam elements of the eltsel large selection Command option merge allows merging the interstice mesh with the inner mesh of the eltsel_smal1 selection e Command option quad allows generating proper quadratic meshes e Command option snartSize allows generating an interstice mesh with a char acteristic length in coherence with the contour mesh lengh nl_meshi37 Command option lcval allows setting the charact
118. r 3d The next section illustrates meshing capabilities supported computations are described next Frequency domain analyses section 3 4 Campbell diagram building Direct computation of critical speeds response to unbalanced mass asynchronous load and harmonic loads on bearings e Time domain simulation accounting for non linear bearings is under develop ment these are performed with the shaft in a rotating frame and the stator fixed 3 1 Rotor meshing 3 1 1 Meshing utilities 1D The SDT Rotor toolbox supports analysis of 1D models of symmetric rotors com posed of e shafts represented by beam1 element see sdtweb beam1 and sdtweb p_beam The rotation axis is taken to be that of the beam e disks represented by massi see sdtveb mass1 The rotation axis is the one whose moment of inertia is different from the 2 others that are equal e bearings supports by celas You can generate a beam model of your rotor by providing a skyline points not on the axis defining the radius at various locations Use NaN to define 2 segments See rotorid Skyline for more details 18 CHAPTER 3 TOOLBOX TUTORIAL xy 0 0 E35 111 1541 1 1 1 21522 2115 3 0 fipgure l plot xy 1 xXy 20 5 4 Mesh as beams molderotorid skylineToBeam xy Add bearings as spring elements molderotorid AddBearing D F 123 k 1e4 keep mold 10 1 0 01 molderotorid AddBearing DOF 123 k 1e4 keep
119. re upper blade corner Seli eltname SE disk1 selface amp withnode NodeId 154 Seli fesuper SeBuildSel initrot cf Sel1 Sel2 eltname SE disk2 selface amp withnode NodeId 154 Sel2 fesuper SeBuildSel initrot cf Se12 4 and do restore xF1 fesuper SeDef Sel1 cna 1 xF xF2 fesuper SeDef Se12 cna11 xF plot responses ci iiplot 3 Xr iicom ci curveZXF XF Disk1 struct w xF data xf xF1 def dof xF1 D F XF Disk2 struct w xF data xf xF2 def dof xF2 D F iicom subMagPha iicom ci IIxOnly Disk1 Disk2 ii plp def data fe cyclicbos Fourier ind phys mono red egy egyfrac sortbyd This command allows to perform a 3D Fourier analysis of given modeshapes The maximum norm of each harmonic is plotted against the harmonic coefficient The plot is different when dealing with a single modeshape or a set of modeshapes Accepted options are e ind is an optional selection of deformations See also the alternate oubDef phys mono and red are used to distinguish between an analysis of physical modeshapes full 3D mono harmonic modeshapes the DET step is omitted and generalized modeshapes reduced multi harmonic model In all cases the user has to check that the physical or reduced models are geometrically peri
120. rve StaticState fe eyclicb shaftsolvestatic 0 cf mdl 4 Pre stressed modes dp fe cyclicb shaft Teig 2 mode1 36 CHAPTER 3 TOOLBOX TUTORIAL def data dp data After full rotor assembly restitution is performed using _SeRestit 3 5 4 Campbell diagrams First example of the beam with single disk Model Initialization model2D rotor2d test simpledisk back cf feplot rotor2d buildFrom2D model2D SE cf Stack disk1 enforce boundary cond on sector and assemble SE fe_case SE FixDof Base z 1 01 cf Stack disk1 SE Compute matrix coefficients for a multi stage rotor range struct data 0 1 1000 2 pi 60 unit RPM fe_cyclicb polyassemble noT cf range Now run a mono harmonic computation returning reduced model cf Stack info Omega struct data range data 1 unit RPM cf Stack info EigOpt 5 20 01 define eigenvalue options MVR fe_cyclicb shafteig 1 ReAssemble 2 NoN buildMVR cf MVR Stack end end 1 0 4 1 0 2 skip problem with geometric softenii rc struct data linspace 0 8000 2 pi 50 unit RPM hist fe rotor Campbell MVR rc fe rotor plotCampbell hist struct fig 100 axProp ylim 0 250 Another example will be needed to treating the multi stage case This example needs further validat
121. s and initialize non linearities related outputs this is a callback generated by nl_spring Output Init FinalCleanUpFcn field is modified to perform cleanup on non linearities as well this is realized through the ExitFcn command option offe simulfe t imeCle see e timelTimetpt using ExitFcn nl spring fe timeCleanUp uonl spring TimeOutput ptions Fine tuning offe time output can be achieved by specifying an info OutputOptions case entry Accepted fields for the OutputOptions structure are FnlAlTT if defined and equal to 1 non linear loads are saved at all time steps e Fnlliplot if defined and equal to 1 non linear loads are displayed in an iiplot figure as curve FNL If the display timer associated with this figure does not stop automatically you can stop it with cingui TimerStop mkl utils Non linearities are treated by mk1 utils mex file In order to make proper distri bution of nk1 utils please send us result of ver cd fullfile matlabroot bin getenv MATLAB ARCH 115 lt mkl For more details see sdtweb mkl utils rheo2NL OBSOLETE Use now nl spring NI NL nl_spring rheo2NL model D F offset This command is used to convert rheological data into a structure of data under standable for NLforce command DOF is the list of the DOF coherent with u and v arguments of NLforce command Offset is optional It is a structure of data with fie
122. sing these formulas XXX In this example Centrifugal softening in global fixed frame Not 0 In GLOBAL frame gyroscopic matrix DG do not modify unbalanced response because only translation dof are affected In LOCAL frame gyroscopic D do not modify unbalanced response XOL inv md K2 w md K4 u 1 01 local Unbalanced mass local H global Amplitude m 1000 2000 3000 4000 5000 6000 7000 8000 9000 Rotation velocity RPM 200 T T T T 400 o Frequency Hz 3 o Frequency Hz N Q o o e 8 9 0 1 2 0 1 2 3 4 5 6 3 4 5 6 Rotation speed 1000 RPM Rotation speed 1000 RPM Figure 4 2 Top Unbalanced mass response amplitude computed in local and global frame Bottom left Campbell diagram in local rotating frame right in global frame 50 CHAPTER 4 VALIDATION 4 2 Simple 2DOF model of shaft with disk This section is about the simple 2 DOF rotor model described in chapter 2 of Lalanne and Ferraris 7 Z L 3 s 2L 3 Figure 4 3 Simple model of rotor The disk is rigid the shaft is described by 2 sin shape functions f y sin 77 where L is the length of the shaft Bearing at y 5 is represented by 2 additional stiffness and damping Kbearing kzz kez 25 crx CTZ kec kzz The 2 DOF are considered in the global fixed frame 2 22 E 0 a
123. so that SeDefinit is performed for every restitution fe eyclicb Display cf def defines a full disk selection within ep1ot def fe_cyclicb Display cf def is used to recover full motion from mono harmonic solutions For large models restitution on the full shaft model may be very costly remember that one vector for 1e6 DOF requires 7 6 MB and DisplaySel is typically preferred fe eyclicb DisplayAllEdges cfL sel1 displays 2D cuts of the disks specified in the cell array sel with a typical entry diskz to display only disk i default displays all disks This cut is basically the projection of the right interface or equivalently the left interface when meshes are compatible to the plane 0 0 Such a view is particularly well suited to the definition of the inter stage rim nodes in MeshAddRim as well as to the construction of viewing meshes in MeshRimLine2Patch Note finally that this view keeps the elements in their original groups cf demo cyclic buildstepO fe_cyclicb DisplayAllEdges cf fecom colordatag fe eyclicb DisplayFirst cfL sel1 provides a simple command to dis play the first sector of each stage cf can be replaced by a model resulting from fe cyclicb DiskFromSectorl A selection can also be specified to restrict the view to a subset of stages cf demo cyclic buildstepO fe_cyclicb DisplayFirst cf disk2 fe_cyclicb DisplaySkin cf
124. tdispay If command option sel is specified and a selection of elements is provided the shapes are recovered to that selection only Selections are cell arrays with the typical entry Sele disk1 sel11 where se11 is either a string to select a subset of elements in the true mesh or a list of elements to build a reduced viewing mesh Model Initialization cfedemo cyclic testrotor 7 blade cf 2 reset Mono harmonic Solutions model stack_set cf mdl info EigOpt 5 5 1e3 11 1e 81 def fe cyclicb shaft Teig 5 model partial display and computation of strain energy Sel disk1 withnode r lt 130 fe_cyclicb Displaysel cf def Sel enerkdens if 1 2 total display if needed cf sel groupall cf def fe_cyclicb Display cf def end cf Stack info StressCritFcn fe_cyclicb StressRR dfull STRESS fe_cyclicb Displaysel cf def Sel stress gstate z STRESS GroupInfo 1 5 figure 1 plot squeeze z Y 1 1 7 This will be extended to full disk computations in section 3 5 SOLVERS FOR MODELS WITH CYCLIC SYMMETRY 35 3 5 3 Multi stage harmonic mode computations Call shaftTeig in fe cyclicb allow to compute the specified mono harmonic solutions multi stage solutions for which disks share the same Fourier harmonic coefficient in a single job Mono harmonic eigensolutions are recovered using Dispay If c
125. ting a new non linearity nl fun m s st nl_fun ParamEdit e db returns default NLdata fields for a non linearity This allows integrated building of non linearities in a model This function can call ParamEdit to allow interactive setup This call must return a NLdata field and is of the type NLdata nl_fun db data 0 e Renumbering capability must return the non linearity written for the new renumbered nodes elements dof The call performed by feutilbfor example is of the type NL nl_fun renumber NL nind nind is the renumbering vector The designed n1 fun template is given in the non linear toolbox sdtweb nl_fun m 1 It is a functional non linear function computing a zero non linear force The def inition of a non linearity using nl fun in a standard SDT model is given in the following 4 A standard SDT model modelestruct Node f1 000 000 2 000 00 1 Elt Inf abs celas 12346 0 4 0 10 4 linear celas 133 Define a non linearity of type nl fun model nl spring SetPro ProId 100 model nl fun db data0 ZEquivalent to 4 model stack_set model pro nl_fun 4 struct Cd 100 fe mat p spring SI 1 type s spring i NLdata struct type nl_fun data 4 Define the case model fe_case model FixDof base 1 model fe_case model DofLoad in struct DOF 2 03 d
126. to obtain tangent damping and stiffness Internal states are integrated using an independent finite differences explicit scheme with the same step of time as the main scheme or a subsampling opt NLsteps times At the first residue computation the initial internal states are computed according to initial condition in terms of displacements and velocities through a time inte gration until variation of speed between the 2 last computed steps is lower than opt RelTol Force on the observation DOF F displacement Qc and velocity dQc of the internal DOF displacement and velocity observations are stored in the NL output The command n1 spring dbtype is a database of generalized maxwell rheological models type can be zener standard viscoelsatic model Parameter k0 k1 and c1 can be given as a string of the form db zener kO k0 ki k1 c1 ci in the command The example of the standard viscoelastic model is detailed here as an illustration The standard viscoelastic model also known as Zener model is composed by a spring Ko in parallel with another spring K and a serial dashpot Ci as displayed figure Non linearities listi17 Figure 5 2 Standard viscoelastic model In the Laplace domain the relation between the relative load and the relative displacement is given by m KoK14 Ko K1 Cis K 1 s z F s K s X s K t s S 5 3 where p and z are respectively the pole and the zero of the model p 5 4
127. to the 6 DOFs at node A by I Abn E 5 16 Node A node is free to rotate The linearized stiffness thus corresponds to an axial stiffness in the direction of the rod The computation of the stiffness is however based on the current position of the extremity nodes a difficulty in model manipulations is thus to translate these nodes data describing this link is stored in model stack as a p spring pro entry Stiff ness and damping are stored respectively as 3rd and 5th column of the data il field standard linear spring see sdtweb p_spring NL information is stored in the data NLdata field which has itself following fields e type string rod e sel aFindElt command to find associated celas of rod1 type proid100 e ulim build tabulated law from ulim to ulim Default is 1e3 e lab label nl modaldmp Implementation of modal damping Although modal damping is not a non linear feature in itself its implementation needs makes it behave as a non linearity The concept is to provide shapes defined on a part of a model with associated damping ratios nl modaldmp handles the kinematic projection on the model which can contain superlements In the case where superelements are used and concerned with modal damping the shapes provided must be written on the physical DOF of the superelements The set of shapes must be stacked in model with a valid ID field It is a common deformation SDT
128. trix to sector n the relation to to the Fourier DOFs is given by a row Eh the product ET En is not block diagonal as a result one has coupling between the Fourier harmonics 2 3 2 Fourier transform for shaft computations To simplify computations XXX 2 39 3 Solutions in periodic media This really does not apply to rotors but is implemented in and thus documented here One considers a model whose properties are spatially periodic For a physical response known at regularly spaced positions nAz one can compute the its fourier transform Foo Ul 5 u nAz e Jn 5A 2 16 n Cco 2 3 FOURIER ANALYSIS OF STRUCTURES 13 U r is a complex shape defined on the mesh of the repeated cell One actually uses two cells to represent the real and imaginary parts of U The wave number k varies in the 2 interval or any interval of the same length since U k is periodic in the wavelength domain Given the fourier transform U amp on can recover the physical motion by com puting the inverse Fourier transform Az fiz u nAx x 2 17 For a mono harmonic response fixed wave number k the spatial transform is given by u nAz Ue 222 using the continuity condition linked to the fact that ujgi nAz uright n 1 Az one thus has cz U er U k eI Hence in the real imaginary format one has the constraint equation c cos k Az c sin amp Ax c Re U sin Az
129. ub NL to be called Following example XXX need to geenralize psa08 ShaftRebuild and ShaftTime XXX define a zener link in the bearing of simple 2beam example cf feplot 2 t nlspring 2beam build psa08 ShaftRebuild cf define large rotation DOF psa08 ShaftTime 2000 1 1e 5 0 cf Omega rpm N rot dt theta0 moi il1 2 6 0 7 No gyroscopic because of shaft beami opt RotFollow 1 Follow velocity and not iterations ri stack get mo1 pro celas103 GetData 4 define zener NLdata nl maxvell db zener k0 125e3 ki 50e3 c1 0 4 ri NLdata RelF type nl maxvell r NLdata RelF NLdata NLdata molestack set mo1 pro celas103 r1 def fe time opt mo1 rodi XXXEB following needs to be checked The rod1 non linear connection is a simple penalized rigid link One considers two nodes A and B see figure 5 4 z Figure 5 4 Large rotation rod functional representation Currently one can introduce masses at points and B mass2 elements should be used to account for the actual position of the center of gravity The global non linear load associated with the rod is thus ke HzB za l Lo 42 24 5 15 xp cAHI which accounts for a load proportional to the length fluctuation around Lo penalized rod model 122Non linearities list When linearizing one considers a strain energy given by k gg q with the motion at node A being related
130. uch that the stiffness at rotation speed 0 is approximated by K Q Y K2 5 2 Qi The zCoef uses velocity Omega in rad s This example now treats computation at variable rotation speeds Model Initialization model demo_cyclic testblade cf feplot model 4 Compute matrix coefficients for a multi stage rotor range struct data 0 0 1 0 0 800 0 0 1600 unit RPM Assembling in the feplot figure allows memory offload fe_cyclicb polyassemble noT cf range 75 cyclic X struct data linspace 0 1600 10 unit RPM fe rotor Campbell cf1 model X Another example will be needed to treat the multi stage case Model Initialization model demo_cyclic testrotor 7 10 blade model fe_cyclicb shaftRimAsSe model cf feplot model Compute matrix coefficients for a multi stage rotor range struct data 0 0 1 0 0 800 0 0 1600 unit RPM 4 Assembling in the feplot figure allows memory offload fe cyclicb polyassemble noT cf range 4 Nov run a mono harmonic multi speed computation cf Stack info Omega struct data range data 1 unit RPM def fe cyclicb shafteig ReAssemble 2 NoN cf Sel diski groupall disk2 groupall fe_cyclicb DisplaySel cf def Sel 4 Reduce the full model fe_cyclicb ShaftPrep svdtruncate mseqi handle norestit cf def fesuper fassemb
131. ur provided By default gmsh is forced not to add nodes to the lines defining the contour to mesh Rivet This command generates rivet drills in a specified contour A model containing a beam contour can be provided or an E1tSel string generating a surface selection see section and the selface option on a bigger model A data structure providing the origins and rivet radiuses and washer or rivet head radiuses The mesh generated between both radiuses is structured The data structure must contain fields nl meshi3 e Orig providing the rivet centers in an x y z matrix e radHole providing the rivet hole radius either a scalar if all rivets have the same radius or a line vector providing each rivet radius separately e radWash providing the rivet washer or head radius either a scalar if all rivets have the same washer radius or a line vector providing each rivet washer radius separately and can optionnaly contain fields e plane To directly provide the contour plane normal to define the drillings in an nx ny nz matrix e Ns To define the number of mesh segments in the rivet to washer radius area default 10 either a scalar if all rivet heads have the same properties or a line vector defining the property for each rivet separately e Nr To define the number of mesh radial nodes in the rivet to washer radius area default 2 either a scalar if all rivet heads have the same properties or a line vector
132. ure 4 4 Campbell diagram Symmetric rotor without bearing Unbalanced mass Asynchronous load 0 50 A o Theoretical o 0 radial def amplitude m o radial def amplitude m 3 Theoretical SDT 1 0 10 L 4 i 4 N nt 2L J 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Rotation speed RPM Rotation speed RPM Figure 4 5 Response to an unbalanced load left and asynchronous load right Symmetric rotor without bearing Maximum response for unbalanced mass is obtained for the exact critical rota tion speed of forward mode computed in the Campbell 3089 RPM This speed is 52 CHAPTER 4 VALIDATION the same as in Lalanne theoretical expression Bearing stiffness along z is now taken in account so that rotor is not symmet ric kzz 5eb and kr kzz kzx 0 There is no damping in the bearing Chearing 0 beta 0 ri struct om linspace 0 9e3 101 s 1 0 kee Bab Exz 0 Ezx cxx 1e2 beta czz be2 beta cxz 0 czx 0 Ried rotor TestLalanne2DOF ri eo o pbbbbbbb 4444444 Frequency Hz Q o gt o I eo 1 2 3 4 5 6 E 8 9 Rotation speed 1000 RPM Figure 4 6 Campbell diagram Asymmetric rotor with z stiffness bearing Now the 2 modes are not at the same frequency for Q 0 Unbalanced m
133. w t sin w t Rp Rp zeros 2 zeros 2 Rpp w 2 R Global matrices MG R nd K 1 R DG 2 R md K 1 Rp R w md K 3 R KG R md K i Rpp R w md K 3 Rp R w 2 md K 4 R R md K 2 R 4 compare to theoretical values m pi R2 2 R1 2 3e 2 7800 48 CHAPTER 4 VALIDATION Ive0 5xmx R272 4R172 Iu 0 5 Iv m h 2 12 DGth zeros 4 DGth 4 3 Iv DGth 3 4 Iv DGth w DGth KGth diag O O 1 1 KGth w 2 Iv 2 KGth if norm KG R md K 2 R KGth norm KGth gt 5e 2 error Cent soft mismatch end if norm DG DGth norm DGth gt 5e 2 error gyro matrix mismatch end unbalanced mass XOL inv md K 2 w 2 md K 4 w 2 1 01 local XOL abs XOL 1 XOG inv w72 MG 1i w DG KG w 2 1i 1 01 global X0G abs X0G 1 Q 1 j1 XOL Q 2 j1 X0G end figure 4 semilogy wrange 60 2 pi Q 1 wrange 60 2 pi Q 2 legend local global title Unbalanced mass setlines xlabel Rotation velocity RPM ylabel Amplitude m if norm Q 1 Q 2 norm Q 1 gt 1e 8 error Global Local error end 4 check massi gyroscopic matrix 70 mdmass struct Node 1 Elt Inf abs massi 0 1mm m Iu Iv Iu mdmass fe_case mdmass fixdof Base 1 02 1 051 mdmass fe_caseg assemble se matdes 2 1 70 mdmass if norm w mdmass K 3 DGth norm DGth gt 1e 8 sdtw _err mass1 gyro 70 unmatch check
134. x 10 100 b E F 80 2 po Ba d F 60 r s pp 3 ag p 40 s E 134 4 4 8 344444 29 5 10 15 20 02 04 06 08 1 12 14 16 18 2 Rotation speed 1000 RPM Rotation speed RPM x 10 80 b E r 60 n Rz p 2 10 4444444444 44 p b a 40 14 y E 788 4 5 pr 74e E ET 4444 44 44 20 D 3 p R 5 10 15 20 02 04 06 08 1 12 14 16 18 2 Rotation speed 1000 RPM Rotation speed RPM x 10 Figure 4 8 Campbell left and Responses to unbalanced load right for different damping top 3 1 middle 8 15 and bottom 26 Asymmetric rotor with z bearing stiffness and damping 54 CHAPTER 4 VALIDATION Backward and forward mode can cross each other in the Campbell diagram Asymmetry leads to the excitation of the backward mode Damping leads to a more spread resonance response 4 3 1D models 4 3 1 1D example in a fixed frame This first example treats the simple case taken from 6 of a shaft with a rotating disk at one third the length In order to compare this model to the simple 2 DOF model of Lalanne see section 4 2 we project the matrices on the 2 sin shape function of the shaft Relative error for mass matrix is 0 01 4 59 for stiffness and 1 64 for gyroscopic matrix Campbell for the projected model and the Lalanne 2 DOF model are almost the same For the full 1d model not projected the increasing of the frequency of Irst forward whirl mode tends to an asymptot
135. xed axially and radially on a set of nodes and first point only in tangential direction Display Display commands group tools to build mesh views specific to disk assemblies e def ENER fe_cyclicb Displaysel cf def Sel enerkdens is used to recover mono harmonic solutions on a partial selection For details on mono harmonic solutions see section Examples can be found in section 3 5 2 section m Sel is a cell array specifying how each stage is displayed In the example from section 3 5 1 one uses Sel EltName SE Keep only SE for display no interstage rim Idisk1 1 2 groupall all elements from sectors 1 2 disk2 1 3 groupall Z2 all elements from sectors 1 3 See sdtweb fesuper SeBuildSel for details on partial superelement display and more examples on the way to define Se1 The last command can be any valid e stressl command Without output argument the result is displayed fe cyclicbss cf defledemo cyclic buildstepi def fe def subdef def def data 2 gt 0 remove fixed edge solu fe cyclicb CDisplay cf def fecom ColorDataEvalA fe eyclicb DisplaySel cf def cf Stack VievMesh fecom ColorDataEvalA During the command one defines SE cGLO corresponding to a rotation by one sector And the SE Alpha for the harmonic shift fesuper sedefinit rot cf is then used to define a restitution by disk The SeRestit then contains the def
136. y each stage is modeled using a superelement called diski see fe eyclic DiskFromSectorl Coupling between stages is done using elements The most consistent approach is to use a physical area that is properly meshed for the full 360 degrees but this may be difficult in particular when the mesh refinement is notably different between the two stages so that a node to surface penalized connection is also implemented and an example given at the end of this section Two steps are required 1 the first step is a manual declaration of the nodes that belong to the two regarding surfaces The declaration of the nodes in the 2D cut provided by fe cyclicb DisplayfAllEdges command is sufficient and therefore recom mended 2 the second step is the automatic reconstruction of the rims as volumes using automated 3D tessellation or penalty based node to surface bilateral contact in the command Note that fe shapeoptim can be used to local deform the disk in order to allow rim meshing 24 CHAPTER 3 TOOLBOX TUTORIAL 4 load two disk example with space between disks cfedemo cyclic TestForMeshRimVol RimStepi select nodes at the matching interfaces fe_cyclicb DisplayAllEdges cf start cursor to pick values fe fmesh 3dlineinit ni 12 18 24 1127 1133 1139 RimStep2 build rims and tessellate fe cyclicb MeshRimStep2 epsl 1 cf n1 cf sel EltName SE fecom showpatch Figure 3 3 Selected disk edge n
137. ype sparse opt Opt 4 1e 4 opt Opt 5 0 6e4 NSteps Initial impact model fe_case model DofLoad In struct def 1e3 D F 2 02 curve sprintf TestStepZ 15g opt Opt 4 5 Without gyroscopic effect def0 fe_time opt model 7 compute With NL gyroscopic effects model stack_set model pro DofKuva1005 7 gyroscopic effects struct 417 1006 Pe mat p spring BI D 0 O O 0 Ol type p spring NLdata struct type DofKuva lab gyroscopic effect Dot 11 Dofuya 10 1 0 MatTyp T7O factor 1 1000 60 2 pi exponent 1 uva 0 1 01 def fe time opt model compute 4 Display results as curves ci 11plot iicom sub 2 47 iicom ci IIxOnly disp 1 disp 2 D ilefa c ci Btack disp J DOF 24 02 03 ind iicom cl sprintf caxi chcAl cax2 chcfi 11 1 11 2 comgui ImWrite comgui cd o sdt cur rotor plots shaftdisk_withanc display deformation cf feplot model display model cf def def0 7 without gyrocopic effects cf def def 7 with gyroscopic effects 42 CHAPTER 3 TOOLBOX TUTORIAL 3 7 8 Other representations of bearings In the fixed basis the simplest bearing model is a linear spring ce1as element In rotating basis using a spring to model beaing is not possible using a celas element The n1 spring RotCenter non linearty is usefull to mo

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