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Viscoelastic vibration toolbox, User Manual

1. The command can also be used with reduced models so set a viscoelastic parameter variation _visc entry in zCoef for a stiffness parameter that is already existing 91 fevisco 92 Fluid merge matrix makereduced These commands are meant to allow the generation of a coupled fluid structure model based on a reduced model used for viscoelastic structure predictions result of a fexf direct command for example A complete example is treated in section The steps are e add the fluid as a superelement to the solid model mCP fevisco fluidmerge modelS modelF e assembly of the fluid structure coupling and fluid matrices with the FluidMatrix command fevisco FluidMatrix cf FluidInterfaceSel SolidInterfaceSelec The fluid interface selection applies to the elements of the fluid superelement renumbered consistently for the combined model The result is stored in the feplot model stack as two assembled superelements which you can access with cf Stack fluid and cf Stack FSC When built with fevisco one also has coup Node coup E1t with the fluid interface as fsc elements and the solid interface as quad4 elements to allow verification of the matching Note that the coup matrix can also be imported from an external reference code that often exists in the automotive industry fevisco FluidMatrix cf coup with coup K coup sdof coup fdof describing the coupling matrix solid and fluid
2. i where nglob is defined in NASTRAN using PARAM G and the loss factors for the viscoelastic parts using the GE values of the respective components For viscoelastic analysis NASTRAN requests a complex curve T w given as two tables NASTRAN applies the PARAM G to all elements the matrix called K in NASTRAN lingo is equal to K Kvi as a result the variable viscoelastic stiffness which are here proportional to G w Ginom 1 ingtoh T w ninom hence given the complex modulus the table is given by 73 3 Toolbox tutorial 74 Ninom Gi nom inom al Age 1 ta ERED iy 3 6 Generation of variable coefficients to be used in the toolbox from NASTRAN input is obtained with MVR zCoefFcn fevisco nastranzCoefv1 model 3 5 4 Parametric model from NASTRAN element matrices An alternative to the step12 procedure Viscoelastic computations are performed on anfupcom model typically imported from NASTRAN after a SOL103 real eigenvalue run with PARAM POST 4 to export the element matrices Thus for a NASTRAN run UpModel dat which generated UpModel op2 and where the upcom superelement is saved in UpModel mat a typical script begins with Up nasread UpModel dat BuildOrLoad In some cases one may want to use reduced basis vectors that are generated by an external code typically NASTRAN The problem is then to generate the reduced model MVR without asking to generate the appropriate basis You can sim
3. 50 10 10 Reduced frequency Figure 4 4 Modulus vs frequency and temperature WriteNastran This utility is meant to help users generate complex modulus tables for use in NAS TRAN The tables are generated as two TABLED1 bulk entries if only one ID is 104 Loss factor 10 10 Reduced frequency specified in the command the real and imaginary parts are written with sequential numbers The command specifies the table identifier default 1 and filename Note that this only supports a single temperature since NASTRAN does not handle multi dimensional tables For more details on writing TABLED1 entries see WriteCurve fname horzcat nas2up tempname b1k m_visco horzcat WriteNastran 101 201 fname 10 10 100 10 isd112 1993 type fname HandNomo FileName Displays the bitmap image of a nomogram and guides the user through the process needed to obtain a tabular version suitable for inclusion inm visco The main steps of the procedure are as follows e Rect define the location of the axis in the bitmap by selecting two of its corners e Elim click on the text of ytick labels on the left of the axis to define the modulus range and the maximum axis value for the loss factor range e wlim click on the text of ytick labels on the right of the axis to define the minimum frequency displayed on the true frequency axis left y axis and the min and max values for th
4. 1 fo o X the X t dt ymaxy o X t e X t n X T 1 3 This definition in terms of dissipated energy extends the complex modulus definition to cases with small non linearities In such cases the dissipation may be a structural rather than a material effect and may depend on amplitude or history so that its use as a constitive parameters may not be relevant Figure 1 1 Elliptical stress strain cycle The raw measure of viscoelastic characteristics gives a complex modulus which typ ically has the characteristics shown in figure 1 Modeling viscoelastic materials Oe Storage Modulus Pa 3 Loss factor aX E 28 oO j No y k a5 10 10 10 10 5 0 5 5 0 5 Frequency Log Hz Frequency Log Hz Figure 1 2 Sample constitutive law for a viscoelastic material The solution of frequency domain problems frequency response functions com plex mode extraction requires the interpolation of measurements between frequency points shown as a solid line and the extrapolation in unmeasured low and high frequency areas shown in dotted lines in the figure For interpolation a moving average weighted for 2 or 3 experimental points is easily implemented and fast to compute For extrapolation at low frequencies one will assume a real asymptote E 0 i e 7 0 0 from basic principles since the relaxation function is real its Fourier transform is even and thus real at 0 For
5. br amp lab cf Ri fe2xf frfzr cf compare mean and frf ri squeeze mean abs R1 2 Y 72 2 r2 squeeze R1 1 Y if norm ri r2 1 gt 1e 10 error Mismatch in mean computation end e impe computes the impedance for an input u w u w Hw u w 4 8 Following example computes a set of responses cf fevisco TestPlateLoadMVR feplot MV cf md1 redefine sensors il feutil findnode x 0 amp z 0 epslie 3 MV MV fe_case MV SensDof Sensors i1 03 MV fe_case MV DofLoad Inputs i1 1 2 03 MV fe_case MV setcurve Inputs input input reset reduced sensor representation fe sens r lab cf fe_sens br amp lab cf MV stack_set MV info Freq 30 5 32 Target frequencies define the time dependent load data X 1 linspace 0 200 1e 4 201 data Xlab 1 fe_curve datatypecell freq data Y ones 1 length data X 1 MV fe_curve MV set input data define model selection for energy computation moi feutil rmfield MV GetData file wd Opt Stack moi Flt feutil selelt matid103 mo1 define responses to be computed MV stack_set MV info MifDes frf syd struct 7 ind 2 svdu 1 usvd fet sacs struct out 1 2 fei vel struct out 2 feta 3 gt enerk moi enerm moi mea
6. 109 nasread 110 PARAM POST 4 or PARAM POST 5 see BuildUp command below BuildUp BuildOrLoad A standard use of FEMLink is to import a model including element matrices to be used later with upcom You must first run NASTRAN SOL103 with PARAM POST 4 to generate the appropriate op2 file note that you must include the geometry in the file that is not use PARAM OGEOM NO Assuming that you have saved the bulk file and the op2 result in the same directory with the same name different extension then Up nasread FileName blk buildup reads the bulk and op2 file to generate a superelement saved in FileName mat It is necessary to read the bulk because linear constraints are not saved in the op2 file during the NASTRAN run If you have no such constraints you can read the op2 only with Up upcom load FileName Up nasread Up FileName op2 The BuildOrLoad command is used to generate the fupcon file on the first run and simply load it if it already exists nasread FileName blk BuildOrLoad result in global variable Up OUTPUT4 binary out nasread FileName output4 reads output4 binary output format for ma trices The result out is a cell array containing matrix names and values stored as MATLAB sparse matrices All double precision matrix types are now supported If you encounter any problem ask for a patch which will be provided promptly OUTPUT4 text out
7. 5 amp y gt 5 MAP cf feplot cf model model fecom colordatamat view 1 3 3 2 Meshing foam fillings When meshing foam filling of geometrically complex parts generating a mesh of the part and foam can be difficult The problem is solved through the following steps 1 mesh the part usually done in a CAD environment then imported into SDT and mesh approximate foam volume 2 defines a foam expansion map giving an expansion direction for external nodes of the approximate foam volume In this phase one typically defines the connected surface of the foam model defines normals on this surface and possibly corrects these normals The result is a vector map structure with field ID node numbers of connected nodes and normal giving the associated directions see fe_case map 3 expand the foam to touch the initial part mesh and generate MPC connections to connect the foam and the underlying mesh This is done automatically with the ConnectionStickThenSolid command Example femesh reset model femesh testubeam model feutil objecthexa 101 101 model Ra 5 0e26 0 0S0 2 080 0 275 8 813 cf feplot 2 cf model model fecom colordatamat alpha 1 Build normal MAP for connected foam surface MAP feutil getnormal node MAP model Node feutil selelt matid 101 amp selfacek amp withnode y gt 49 amp z gt 0 amp z lt 2 5 model Define the facing elements Facing
8. A 0 2 51 and uses standard orthonormality conditions Vja E Px jn et Vja A xd Ajdjx 2 52 In the complex plane one should distinguish complex poles associated with vibration modes and real poles which can be related with a damping ratio above 1 super critical damping that is not common or to material relaxation be linked to poles of the complex modulus To understand this distinction one considers the vis coelastic oscillator 2 9 The viscoelastic behavior leads to one real pole 8 whereas supercritical damping corresponds to gt 1 and leads to two real poles A Cwtwy 2 1 2 53 In a viscoelastic computation where the constitutive law contains real poles the root locus of the solution is similar to that shown in figure 2 3 One must thus distinguish the classical spectrum of vibration modes and the real poles associated with material relaxation T T T T T or T og 4 2000 4000F o 4 Relaxation P 6000 J 8000 Vibration 1 0000 L L L L L L L 2 1 5 1 0 5 0 0 5 1 1 5 2 Im a x10 Figure 2 3 Poles of a viscoelastic beam in traction It is important to note that the number of real poles is directly associated with the number of DOFs in the internal states q The number of these poles will the increase with mesh refinement In practice one thus cannot expect to compute the real poles and associated modes in the area of relaxation poles This selection of conver
9. model Description Viscoelastic materials are described by a structure with fields pl Reference elastic properties used for initial model assembly This should include the loss factor name Reference name for the material type Always set to m_visco since this is the material handling function unit Unit system see fe_mat convert TO Reference temperature G Description of the modulus as a function of frequency This can be a matrix with three columns giving frequencies real and imaginary parts of the modulus or a string that will be evaluated to determine the modulus To sort frequencies use mat G sortrows mat G at Description of the frequency shift as a function of temperature This can be a matrix with two columns giving temperature and frequency shift r or a string that will be evaluated to determine the shift factor nomo is a cell array containing information need to create the nomogram See details under the nomo command Accepted commands are default info database dbval These commands are used to select materials Info lists available materials Default gives the first material m_visco searches its own database and for mvisco_ m func tions in the MATLAB search path The mvisco_3m function is given to illustrate the contents of a user database file To start a selection process use cf feplot m_visco database cf This will give you a list of materials in the database In the Mat tab select material
10. of each parameter in opt points When used for selected parameters by giving indp the unused parameters are set to their nominal value fevisco Rangegrid opt edge elevel Up indp generates a uniform grid by di viding the range of each parameter in opt points One then only retains points that are on an edge level elevel defined by the fact that elevel parameters are equal to their minimal or maximal value Range fevisco RangeBuildMinMax Up indp resp RangeBuildMinNom resp RangeBuildNomMax generates a grid whose first design point is defined with all parameters at their maximum resp nominal resp maximum value and following design points one for each parameter in indp or one for each of all parameters otherwise with one parameter at the minimum resp minimum resp nominal value and others at maximum value resp nominal resp maximum fevisco Rangecubeedge opt Up indp retains the one dimensional edges of the hypercube defined by parameters selected in indp fevisco RangeRandopt Up indp creates a random experiment on indp with opt design points Samples calls are as follows hcube fevisco RangeVect 4 0 10 par 0 0 2 1 2 typ cur 1e0 min 1e 2 max 1e1 log O 1 2 12 typ cur 1le 1 min 1e 2 max 1e1 1log 01 120 typ cur 1 min 1 max 2 lin Uniform 3 by 4 grid on parameters 2 and 3 hcube fevisco RangeGrid 3 4 par 2 3 hcube val Face centers on all parame
11. pz 2 Fo Fo E1 and the frequency where this maximum is reached between the pole and the zero um VP By ee 1 13 The maximum loss factor is higher when the pole and zero are well separated and the low and high frequency moduli differ This is consistent with the fact that materials that dissipate well also have storage moduli that are strongly frequency dependent 1 12 Figure 1 6 illustrates problems encountered with three parameter models The pa rameters of the standard viscoelastic model where chosen to match the low and high frequency asymptote and the frequency of the maximum in the loss factor One clearly sees that the slope is not accurately represented and more importantly that the loss factor maximum and evolution in frequency are incorrect These limitations have motivated the introduction of more complex models detailed in the following sections 11 1 Modeling viscoelastic materials 12 200 i x J Loss factor Storage modulus dB Frequency Figure 1 6 amplitude and loss factor of ISD112 overlaid with a 3 parameter model 1 2 3 High order rational models The first generalization is the introduction of higher order models in the form of rational fractions All classical representations of rational fractions have been con sidered in the literature Thus fractions of high order polynomials decompositions in products of first order polynomials giving pol
12. rational fractions of relatively high order This representation is practical only for reduced models Mp TT MT where the strategy for the selection of reduction basis T will be detailed in chapter 2 2 3 Second order models with internal states If the state space form is more compact a priori the operators available in a given FEM code may make its manipulation more difficult A classical solution is thus to build a second order model of the form usual in mechanics and thus easily manip ulated with a mechanically oriented code The implementation of internal states however requires the definition of multiple fields at the same node which is not easily implemented in all software packages The Anelastic Displacement field method considers a modulus representation of the form 2 33 which leads to a model of the form 0 0 0 M0 J a 0 0 s a 0 0 Em En 2 35 Ke Eos Kri Kyi tae Kon q y Kyl E vl cee 0 qv1 _ F Eo w a 0 0 0 oe Er Kon dun The absence of a mass associated with internal states q can lead to problems with certain solvers An alternative is the GHM method which represents the modulus as E s r ET ee 2 36 and defines internal states by quj eR ETL One will note that not all rational jj j fractions can be represented in the form 2 36 The introduction of q fields in the previous section corresponds to the classical thermodynamics theory of materials with augmented poten
13. that has no well established solution Practice is thus to measure a compression E w or a shear modulus G w and to assume a constant Poisson s ratio although this is known to be approximate Section analyzes the main representations of complex moduli Section lists representations used to account for the effect of environmental factors temperature pre stress Domains of applicability for different models are given at the end of the chapter For more details Ref a detailed account of viscoelasticity theory Refs present most practical representations of viscoelastic behavior 1 2 Representing complex modulus Even though viscoelastic constitutive laws are fully characterized by the measure ment of a complex modulus As will be detailed in the next chapter numerical solvers are often associated with specific analytical representations of moduli This section illustrates the main classes of modulus representations 1 2 1 Non parametric tabular representations At each frequency the complex modulus describes an elliptical stress strain relation For coswt one has o Re E 1 ine E coswt nsinwt as shown in figure 1 1 One calls storage modulus the real part of the complex modulus Es Re and loss factor the ratio of the imaginary and real parts n Im E Re The loss factor can also be computed as the ratio of the power dissipated over a cycle divided by 27 by the maximum strain energy
14. within the Case tab feplot 2 mdl File Edit Desktop Window Help 44e D A Model Mat ElProp Stack Cases simul i Pa visco2 Elt selection Matld 1002 urrent Min 0 5 2 3 5 2 Sample parametric study in SDT Typically one starts predictions with the a first order approximation to generate a reduced model MVR which is then used to approximate the frequency response functions Rred Up fevisco testplate upreset Up fevisco testplate up 70 Up stack_rm Up mat Up fevisco addmat 101 Up First area ISD112 1993 Up fevisco addmat 103 Up Second area ISD112 1993 MV fevisco makemodel matid 101 103 Up cf feplot MV cf Stack info Range 10 30 cf Stack info Freq 30 5 500 Target frequencies Hz cf Stack info EigOpt 6 40 2 pi 30 2 11 fe2xf directfirst zCoef0 cf ci iiplot cf Stack curve RESP iicom challDesign point This is a call to the new strategy for pole tracking as function of ten RO struct ind 7 20 Temp 20 10 100 Freq logspace 2 4 20 Po fevisco poletemp cf First area ROQ fe2xf plotpolesearch Po This is a sample direct computation of the viscoelastic response at 3 frequencies fevisco testplate matrix cf feplot MV cf md1 cf Stack info Freq 5 5 40 Rf
15. 36 ea ee e ee ee rn ro 37 2 3 3 Model reduction methods 39 error oer 40 ook oes Bee ae ee 42 EOSS 43 2 4 1 Mesh convergence and non conformity 47 2 5_ Thermal considerations s s sos sosse 48 2 0 1 Thermal model oaoa aa a 48 2 0 2 Heat source due to viscoelastic behavior 49 pu ah Gave eae ER 50 2 5 4 Thermo elastic damping 51 2 Viscoelastic FEM models 24 The design of viscoelastic treatments is typically composed of a series of steps which are outlined in this section e definition of dynamic objectives e potential area localization by combination of technological constraints on place ment and sensitivity analysis on treatment potential This chapter analyzes properties of models used to represent damped structures In the case of linear viscoelasticity these models have the general form Z s T a s bly xna U 8 naxi 2 1 y s nsxa elnsxw 19 8 bx where physical loads b u s are decomposed into vectors b characterizing spatial localization and u s characterizing time frequency responses and vector y rep resents outputs physical quantities to be predicted assumed to depend linearly in DOFs q Section 2 1 details the case of models with viscous and structural damping Although one saw in the last chapter that these models did not allow a correct representation of material behavior over a wide frequency band they are more easily acce
16. A s u s c a s c 2 s b u s 2 41 which involve the inverse of the dynamic stiffness Z s In the very general case where complex moduli are supposed to be analytic functions in the complex plane locally regular the dynamic stiffness Z s is also an analytic function Observation and input matrices c and b being constant the poles i e singularities of H s correspond to non zero solutions of ZA viv 0 and tje IZO 0 2 42 which defines the generalized non linear eigenvalue problem associated with a visce lastic model Near a given pole analytic functions have a unique Laurent s development 5 7 55 f Aos Aa 2 43 TN 2o aF avec ak 57 q s s s where y is a abitrary closed direct contour of the singularity As a result near an isolated pole A one has T giay WD Yje Z REWI aj s j where the normalization coefficient a depends on the choice of a norm when solv ing 2 42 and is determined by O 1 2 44 O Z s aj vyja a i Vip 2 45 To simplify writing it is desirable to use a 1 which si the usual scaling for constant matrix eigenvalue problems described in the next section Having determined the set of poles in a given frequency band having normalized the associated modes so that a 1 one obtains a first order development in s s ewjp Wi b H s e203 gt a H z 2 46 where the c Z 0 b terms correspond t
17. DOFs Solid DOFs correspond to rows of coup K fluid DOFs to columns The fevisco fluidCheckCon cf can be used to check the connectivity of the coupling matrix The distance between each coupled fluid node and structural nodes involved in the coupling is computed and the associated links are shown for verification compute the fluid modes and generate a reduced coupled model mRCoup df1u fevisco fluidMakeReduced mCP dsol1 which can then be used for predictions with fex as illustrated in section Feplot fevisco feplotEnerK matid1001 matidi 2 3 displays the strain energy of the viscoelastic part of the structure first selection here MatId 1001 as one object and the rest of the structure element selection in third argument as a wire frame feplotStress provides stress cuts within volume Accepted options are e MatId desired material e DefLen desired material e Rule desired material e v element direction 1 for x 2 y 3 z 4 z revolution normal MakeModel Matid i A parametric model is described by a type 1 superelement as detailed in section The MakeModel command generates this model based on a type 3 superelement where parameters are set see commands one then generates a model for parametric studies or where viscoelastic materials have been properly declared using addmat commands note that you can start with parameters and use the par2visc command to declare those parameters to co
18. MVR reduced model e ThermoiModes computes associated complex modes First a finite element model md1 must be created with element associated to struc ture properties m_ elastic and p_solid Thermal properties m_heat and p_heat must be present in model material and element property stack Matrices must be assembled as following RO comp12 Thermo1Assemble md1 RO RO is the coupling data structure with fields e MatId Matid_struct MatId_therm first is the MatId of the structure ma terial property and second the thermal one e ProId ProId_struct ProId_therm TO Matid_struct MatId_therm TO is the thermal coupling C Inf if no coupling Then reduced MVR can be build using MVR comp12 ThermoiBuild reduce md1 RO If command option reduce is given model is reduced according to real modeshapes Then eig options must be given in RO EigOpt Complex coupled thermoelastic modes can then be computed using def comp12 Thermo1Modes MVR zc compute modes 2 5 4 Thermo elastic damping To perform coupled thermoelastic modes computation 3 commands are available in comp12 ThermoiAssemble assembles structure matrices thermal matrices and coupled thermoelastic ma trices ThermoiBuild builds MVR reduced model ThermoiModes computes associated complex modes First a finite element model md1 must be created with element associated to struc ture properties m elastic and p solid Thermal properti
19. Pit _ bh REE ECP e Eo e from which one determines an expression of the response of mode 1 given by biu i p 1 e1 22 aad 133 2 19 with 2 127215 e1 2 20 52 118 w 82 y228 w2 and Tasot 2 21 s y118 w s yo28 w3 The T matrix being positive definite one has 2721 711722 lt 1 Thus the e1 term can be close to 1 and coupling be significant if and only if both factors in the denominator are small simultaneously that is min 1w4 awa w4 wol lt 1 2 22 Similarly the eg ter is only significant in special cases of closely spaced modes or such that b1 lt be In practice the validity of the modal damping assumption is thus linked to the frequency separation criterion 2 22 which is more easily verified for small damping levels In cases where modal damping is not a good approximation that is criterion 2 22 is not verified a generalization of the modal damping model is the use of a viscous damping matrix that is non diagonal in principal coordinates One the generally talks of non proportional damping since condition is not verified The use of non proportionally damped models will be discussed in section 2 3 4 For groups of modes that do no verify condition 2 22 one can still use an as sumption of modal damping by block where the off diagonal terms of C are considered for j and k in the same group of modes Between modes of differ en
20. S e P R A S e R e TR A RK E which gives a simple way to estimate the loss factor associated with the local stress strain state 2 69 Despite the local nature of dissipation one may want to verify that the integral over the volume ot the dissipated energy is equal to the input power Noting H w the transfer collocated with the input the equivalent power input in the structure is given by pw for RE a t 2 70 49 2 Viscoelastic FEM models 50 In general there is no direct way to measure the maximum strain energy in a structure and thus no experimental definition of a global or system loss factor For models the energy can be computed but there is not particular reason for the local energy to reach a maximum value at all points simultaneously One can thus search for the maximum strain energy in the system using e qw mar 210 8 K a 0 pa 8 q w R K B qlw R q w R A R q w In the case of normal modes the strain energy is equal to the square of the mode pulsation so that the MSE methods can be meaningful see section 2 1 3 An illustration of global loss factor use for uniform and non uniform material loss can be found in in t_visco ThermoPower 2 5 3 Cantilever plate example To perform coupled thermoelastic modes computation 3 commands are available in comp12 e ThermoiAssemble assembles structure matrices thermal matrices and coupled thermoelastic matrices e ThermoiBuild builds
21. a MatId and an ProID while in NASTRAN they only have a ProlId with the element property information pointing to material entries While this command is typically used indirectly while writing a full model you may want to access it directly For example model demosdt demogartfe nas2up Writeplil 1 model1 properties are implemented somewhat differently in NASTRAN and SDT thus for a il row giving ProID type Coordm In Stress Isop Fctn In NASTRAN In is either a string or an integer If it is an integer this property is the same in il If it is a string equal to resp TWO or THREE this property is equal to resp 2 or 3 in il In NASTRAN Stress is either a string or an integer If it is an integer this property is the same in il If it is a string equal GAUSS this property is equal to 1 in il In NASTRAN Isop is either a string or an integer If it is an integer this property is the same in il If it is a string equal FULL this property is equal to 1 in il If Fctn is equal to FLUID in the NASTRAN Bulk file it is equal to 1 in il and elements are read as flui elements e MAT9 and m_elastic 3 differ by the order of shear stresses yz zx Gay in SDT and xy yz zz in NASTRAN The order of constitutive values is thus different which is properly corrected in SDT 6 5 masread lufread lufwrite 117
22. a state vector combining displacements and velocities leading to a model of the form Gt 0 I q 0 7 MK MC l4 Mtb tu ai y t ed 0 q This model is rarely used as such because M is often singular or for a consistent mass leads to matrices M K and M C that are full When using generalized coordinates one can however impose a unit mass matrix and this easily use the model form Some authors also prefer this form to define eigenvalue problems but never build the matrices explicitly so that the full matrices are not built The other standard representation is the generalized state space model Peo tiye lo Sligo a w e o 3 which preserves symmetry and allow the definition of simple orthogonality conditions on complex modes One will now introduced generalized state space models for the case of viscoelastic constitutive laws A rational fraction that does not go to infinity at high frequencies and having distinct poles can be represented by a sum of first order rational fractions E s E J 2 Be Fe ea j 1 By introducing the intermediate relaxation field quj Grant one can rewrite 2 29 as a state space model of higher size M O O 0 M 0 s 0 0 M 0 M 0 0 0 q P 2 34 Ke Boo Kvi 0 Ky RE Kon sq F EM 0 wM sao 0 qvi 0 E M 0 0 M i n s LB qon One saw in the previous chapter that real constitutive laws could be represented with 33 2 Viscoelastic FEM models 34
23. be computed e irange specifies indices of design points to be used in the experiment specified by the info Range entry 86 e frac adds energy fraction output for each mode in the history cf fevisco TestBeam MVR MVR cf Stack MVR ind 1 4 indices of target modes range struct val logspace 3 0 30 1 1i lab Visco MVR stack_set MVR info Range range hist fe2xf PoleRange MVR ind Display and reformat figure fe2xf plotpolesearch kHz cf1 NomInd 10 20 hist cingui objset 1 Position NaN NaN 560 250 line linestyle none To see the modeshapes should only be used with a small number of point hist def fe2xf PoleRange MVR ind Generally recompute one point here 10 as shown in search and display cf def fe2xf PoleRange irange 10 20 def MVR ind fecom ch2 6 PlotPoleSearch Is the general interface to display pole histories Accepted options are e cf specifies the figure to use e khz uses kHz units e nomind i specified design point numbers to be highlighted by a marker e ShowDef displays in feplot the mode shapes associated with design points in NomInd in a call of the form fe2xf plotpolesearch cf1 nomind 15 leg E showdef hist MVR cf e leg s specifies a legend type Currently e is accepted for a modulus range legend If a hist faces is defined these a
24. deformation locking Re E Frequency Frequency Figure 1 4 Complex modulus associated to standard models a Maxwell s model b vicous damping Kelvin Voigt s model c structural damping d standard viscoelastic solid The structural damping model approximates the modulus by a complex constant Although it does not represent accurately any viscoleatic material such as the one shown in figure it leads to a good approximation when the global behavior of the structure is not very sensitive to the evolutions of the complex modulus with frequency This model is generally adapted for materials with load damping metals concrete The standard viscoelastic model has the main characteristics of real materials low and high frequency asymptotes maximum dissipation at the point of highest slope of the real part of the modulus 1600 oP ff 4 a 1400 fi n tlt l i 4 if it 1200 fi i i if 1000 bbe l ae 10 10 OO 10 10 10 mo 10 frequency frequency Figure 1 5 Complex modulus associated to standard viscoelastic solid model This model is characterized a nominal level HO and two pairs of related quantities a pole where the modulus flattens out and a zero z EoE1 Eo 1 C1 1 11 Wm tim V1 12 giving the inflexion point where the modulus starts to augment or the maximum loss factor p z _1 Ey Me 2
25. display your full model in a figure Re duced models can be generated with fe2xf direct commands or with a NASTRAN DMAP step12 reduced matrices are stored in the model stack as an SE MVR entry XXX 65 3 Toolbox tutorial 66 The problems handled by and computations at multiple frequencies and de sign points Frequencies are either stored in the model using a call of the form model stack set model info Freq w hertz_colum or given explicitly as an argument the unit is then rad s Design points temperatures optimization points are stored as rows of the info Range entry see for generation When computing a response fe2xf zCoef starts by putting frequencies in a local variable w which by convention is always in rd s and the current design point row of info Range entry or row of its val field if it exists in a local variable par zCoef2 end 4 is then evaluated to generate weighting coefficients zCoef giving the weighting needed to assemble the dynamic stiffness matrix 3 4 For example in a parametric analysis where the coefficient par 1 stored in the first column of Range One defines the ratio of current stiffness to nominal Kvucurrent par 1 x Kv nominal as follows external to fexf zCoef Klab mCoef zCoef0 zCoefFcn M 0 a e a Ke O 1 1 ixfe_def DefEta Ky O al perc y model struct K cel1 1 3 mode
26. generation of this interface cannot be performed using a single feutil command then the element matrix should be provided instead of a selection command aselection for the solid interface In the example one retains all elements but in practical applications one will often eliminate parts of the model weld points stiffeners You can again provide the selection as an element matrix The result is stored as a fsc superelement It is obtained by estimating trans lations at the fluid interface nodes trough MPCs see and strandard integration of the fluid structure coupling elements Its DOFs combine DOF s of the solid model and pressure DOF s on the fluid superelement the MPCs are eliminated The final step is to generate a reduced coupled model This requires defining applied loads required sensors and a modal model structure in vacuum de fined trough its modes def as done here or a parametric reduced model MVR You can start by adding additional pressure sensors then use the call cf Stack fluid stack_set cf Stack fluid Info EigOpt 6 10 cf mdl fe_case cf mdl SensDof Sensors 65 01 1 19 DofLoad Point load 65 01 dsol fe_eig modelS 6 20 1e3 cf Stack info FluidEta 1 Fluid loss set to 10 cf Stack info DefaultZeta 01 Structure loss set to 2 fevisco fluidMakeReduced cf md1 dsol Acoustic loads are not yet co
27. handles parametric models where various areas of the model are associated with a scalar coefficient weighting the model matrices stiffness mass damping The first step is to define a set of parameters which is used to decompose the full model matrix in a linear combination The elements are grouped in non overlapping sets indexed m and using the fact that element stiffness depend linearly on the considered moduli one can represent the dynamic stiffness matrix of the parameterized structure as a linear combination of constant matrices Z Gm 8 8 M Pm Km 3 3 Parameters are case stack entries defined by using commands which are identical to commands for an superelement A parameter entry defines alelement selection and a type of varying matrix Thus model demosdt demoubeam model fe_case model par k 1 1 10 Top withnode z gt 1 fecom proviewon fecom curtabCase Top highlight the area zcoef The weighting coefficients in 3 4 are defined formally using the cf Stack info zCoef cell array viewed in the figure and detailed below feplot 2 mdl File Edit Desktop Window Help 4 9 QQB Model Mat ElProp Stack lcases Simul nastrancother ca a zCoef help info NastranJobE T p info NasJobOpt l mCoef zCoef zCoefF cn V 0 w 2 info Range 3 SEMVR Ke 1 0 005 i way Hil mn par 2 par 3 par 4 v amp Remove Th
28. is partially supported You are expected to have ssh and scp installed on your com puter On windows it is assumed that you have access to these commands using CYGWIN You first need to define your preferences setpref FEMLink CopyFcn scp setpref FEMLink RunNastran nastran setpref FEMLink RemoteShell ssh setpref FEMLink RemoteDir tmp2 nastran setpref FEMLink RemoteUserHost user myhost com setpref FEMLink DmapDir fullfile fileparts which nasread dmap You can define a job handler customized to your needs and still use the nas2up calls for portability by defining setpref FEMLink NASTRANJobHandler FunctionName You can then run a job using nas2up joball BulkFileName dat Additional arguments can be passed to the RunNastran command by simply adding them to the joball command For example nas2up joball BulkFileName dat struct RunOptions memory 1GB It is possible provide specific options to your job handler by storing them as a info NasJobOptentry in your model Stack nas2up JobOptReset resets the default The calling format in various functions that use the job handling facility is then model stack_set info NasJob0pt nas2up jobopt nas2up joball step12 dat model RunOpt RunOpt
29. models typically used for viscoelastic studies are described in sec tion 3 1 where sample applications are treated AddMat Par2Visc model fevisco addmat MatId model NameForMatId mat is used to prop erly append a viscoelastic material with the given MatId to the model During this definition one sets the reference elastic material in model p1 the viscoelastic mate rial definition in model Stack mat NameForMatId and a parameter definition in model CStack par NameForMatId If mat p1 does not contain a reference material one is created using a call tofm_visco RefMat The tag NameForMatId will be used for parametric model generation and zCoef calls if you change material you should just modify the model Stack mat NameForMatId data but not the parameter or the reference material fevisco Par2Visc cf uses the NameForMatId tag to check the existence of viscoelastic materials associ ated with matrices and modifies the entries accordingly cf fevisco testplateLoadMV feplot cf Stack info Freq 500 10 4000 cf Stack info Range 10 20 define material with a unit conversion mat m_visco convert INSI m_visco database Soundcoat DYAD609 h MatId for original name of parameter cf mdl fevisco addmat 101 cf mdl Constrained 101 mat fevisco par2visc cf cf Stack zCoef falxt zeoet ct
30. performed As an alternate format fevisco WriteInclude NewModel 01dModel FileName writes elements nodes and properties of NewModel that are not in 01dModel Case entries of the NewModel are also written For example model fevisco testplate model fe_case model remove Drilling tname nas2up tempname include dat fevisco writeinclude MatId 101 102 model tname ceig direct The ceig command supports advanced complex eigenvalue solutions for constant viscous and or hysteretic damping It is normally called through fe_ceig where the solvers are documented The direct command supports various direct frequency response solvers It is normally called through fexf where the solvers are documented NASTRAN Utilities for interfacing with NASTRAN MVR zCoefFcn fevisco nastranzCoefv1 model MVR builds the proper zCoefFcn to reproduce jobs that would otherwise run as SOL108 with one viscoelastic material fevisco nastranEtaGlob model return the global loss factor known PARAM G for NASTRAN with proper default handling NastranEig implements a call to NASTRAN as fe_eig method 50 To use this capability first use the call fevisco NastranEig that will set preferences See also section 99 4 3 m_visco mvisco_3m Purpose Material function for support of viscoelastic materials This function is part of the viscoelastic tools Syntax out m_visco command
31. pl dbval 2 Aluminum therm define element properties md1l il p_solid dbval 1 d3 3 structure md1 il p_heat mdl il dbval 2 d3 3 therm mdl feutil lin2quad md1 RO struct build options RO MatId 1 2 structure therm RO ProId 1 2 TO 1 2 structure therm Temperature if coupling Inf Assemble elementary matrices according to coupling defined in RO RO comp12 Thermo1Assemble md1 RQ Build MVR with coupling RO EigOpt 5 40 1 3 MVR comp12 Thermo1Build reduce md1 R0O Compute associated complex modes def comp12 ThermoiModes MVR compute modes cf feplot mdl cf def def 53 2 Viscoelastic FEM models 54 Toolbox tutorial Seine bobs ies be eee ie he eae A 57 e A 57 3 2 1 Introducing your own nomograms 57 CP 58 Sate ha enh 59 3 3 1 Generation of sandwich models 59 Bon aei ee Cae had 60 SEUEN 61 a 61 3 4 1 Parametric models zCoef 61 EE E eS 64 3 4 3 Input definitions ooa a 67 3 4 4 Sensor definitions oao a a 67 3 4 9 Format reference ooo a a 68 3 4 6 Response post processing options 69 3 5 Sample setup for parametric studies 69 3 5 1 Model parameterization o ooo a aa 69 3 5 4 Parametric model from NASTRAN element matrices 74 3 6 Sample setup for parametric studies 74 3 6 1 Performance in modulus loss pla
32. the data structure combining the observation equations of all your sensors MV cf fevisco TestPlateLoadMVR cf Stack MVR redefine sensors il feutil findnode x 0 amp z 0 epslie 3 MV MV fe_case MV SensDof Sensors i1 03 reset reduced sensor representation fe_sens cr amp lab cf cf Stack MVR 3 4 5 Format reference A viscoelastic model see section for the typical generation procedure stores the information needed to compute 3 4 in a data structure generic type 1 su perelement handled by fe super with the following fields Opt Options characterizing the model In particular the second row de scribes the type of each matrix in model K 1 for stiffness 2 for mass K a cell array of matrices giving the constant matrices in 3 4 One normally uses M Ke Kym These matrices should be real and only the combination coefficients should be complex Klab a cell array giving a labels for matrices in the K field kd optional static preconditioner nominally equal to ofact k0 use to compute static response based on a known residue DOF DOF definition vector associated with matrices in K Mt a cell array of data structures describing viscoelastic materials This cell array is generated automatically by MakeModel but can also be built with direct calls to For example model Mt m_visco database ISD112 Stack standard model stack where one defi
33. to have its stiffness dominated by either shear or compression Based on this assumption one considers the frequency dependence of either the shear or Young s modulus The error associated to neglecting the true variation of other moduli is assumed to be negligible methodologies to treat cases where this is not true are not addressed here 3 4 3 Input definitions Inputs at DOFs are declared as fe_case entries If you have stored full basis vectors when building a reduced model MVR TR field you may often be interested in redefining your inputs To do so you can use fe_case commands to define your loads and MVR fe_sens br amp lab MVR model to rede fine the reduced inputs accordingly 3 4 4 Sensor definitions MV fevisco testplate MV fe_case MV SensDof BasicAtDof 1 246 03 Surface velocity RRA 4 strain sensors along a line pos linspace 5 1 4 pos 2 75 pos 3 006 data struct Node pos dir ones size pos 1 1 1 0 0 MV fe_case MV SensStrain ShearInLayer data Sens fe_case MV sens 67 3 Toolbox tutorial If you have stored full basis vectors when building a reduced model MVR TR field you may often be interested in redefining your sensors To do so you can use the fe_case commands to set your sensors and fe_sens cr amp lab cf to redefine the reduced sensors accordingly this calls Sens fe_case sens model to generate
34. u w one load are supported at this time e usvd computes the singular values of y H u w which corresponds to the quadratic norm of the response on Hw vsxna uw va 4 7 e ener k m computes the strain energy enerk or the kinetic energy enerm in a selection of elements after recovering the physical displacement to as sociated DOFs The recovery basis must fit in memory copt must give a model 83 fe2xf 84 e mean computes the mean quadratic response of one or multiple sensor sets A sensor set is defined by a vector of indices in the cell array copt group which each row has the form SetName Indices Following example computes the mean quadratic response along the node nor mals of previous model using the fe2xf function and correlates it to FRF as an illustration cf fevisco TestPlateLoadMVR feplot MV cf md1 MAP feutil getnormalnode map cf md1 ind find ismember MAP ID feutil getnode MatId1 cf mdl MAP ID MAP ID ind MAP normal MAP normal ind tdof MAP ID MAP ID MAP normal cf mdl fe_case cf mdl remove Sensors SensDof Sensors tdof cf Stack info MifDes mean vel struct group main 1 size tdof 1 mean velocit frf vel all response il feutil findnode x 0 amp z 0 epslie 3 MV MV fe_case MV DofLoad Inputs i1 1 2 03 fe_sens crklab cf fe_sens
35. 1 2 4 Fractional derivative models 1 3 Environmental factors 1 3 1 Influence of temperature 1 3 2 Other environment factors 1 4 Determining the complex modulus 1 5 Conclusion Viscoelastic FEM models 2 1 Viscous and structural damping 2 1 1 Properties of the damped 1 DOF oscillator 2 1 2 Real modes and modal damping 2 1 3 Selection of modal damping coefficients 2 2 Viscoelastic models 2 2 1 Frequency domain representation with variable coefficients 2 2 2 State space representations 2 2 3 Second order models with internal states 2 2 4 Fractional derivatives 2 3 Spectral decomposition and reduced models 2 3 1 Complex modes of analytical models 2 3 2 Complex mode eigenvalue problems with constant matrices 2 3 3 Model reduction methods 2 3 4 Equivalent viscous damping 2 3 5 Case of viscoelastic models NID OW 13 14 14 19 19 20 23 24 25 27 30 31 32 32 34 35 35 36 37 39 40 42 CONTENTS Ae ah oe eh a Se eal dee Fe yeatee 43 2 4 1 Mesh convergence and non conformity 47 2 5 Thermal considerations ooo a a ee 48 2 5 1 Thermal modell aoaaa aa a 48 2 5 2 Heat source due to viscoelastic behavior 49 aagana e ee ee eee 50 2 5 4 Thermo elastic damping 208 51 3 Toolbox tutorial 55 s maai an Ba e al
36. ASTRAN Scalar points are treated as standard SDT nodes with the scalar DOF being set to DOF 01 this has been tested for nodes DMIG and MPC OUTPUT2 binary model nasread FileName output2 reads output2 binary output format for tables matrices and labels You can omit the output2 command if the file names end with 2 The output model is a model data structure described in section If deformations are present in the binary file the are saved OUG i entries in the stack see section With no output argument the result is shown in feplot Warning do not use the FORM FORMATTED in the eventual ASSIGN OUTPUT2 state ment The optional out argument is a cell array with fields the following fields hame dname data trl Header data block name table matrix or label label Data block name table matrix or NASTRAN header label cell array with logical records tables matrix matrix empty label Trailer 7 integers followed by record 3 data if any for table and ma trix date for label Translation is provided for the following tables GEOM1 GEOM2 GEOM4 GPDT KDICT MPT OUG OEE OES nodes with support for local coordinates and output of nodes in global coordinates elements with translation to SDT model description matrix see bulk command translates constraints MPC OMIT SPC and rigid links RBAR RBE1 RBE2 RBE3 RROD to SDT model description matrix with us
37. D601 G 2 23e 006 Pa e 0 75 3 00e 001 Hz 20 C EAR C 1002 E 9 99e 006 Pa e 0 93 3 00e 001 Hz 20 C ISD112 1993 5 53e 005 Pa e 0 99 3 00e 001 Hz 20 C ISD130 1999 4 16e 006 Pa e 1 02 3 00e 001 Hz 20 C EAR C 1105 E 2 60e 007 Pa e 1 04 3 00e 001 Hz 20 C ISD112 1999 1 19e 006 Pa e 1 04 3 00e 001 Hz 20 C antiphon 13 G 1 51e 007 Pa e 1 13 3 00e 001 Hz 20 C EAR C 2003 G 1 15e 008 Pa e 1 29 3 00e 001 Hz 20 C EAR C 1002 G 9 34e 006 Pa e 1 82 3 00e 001 Hz 20 C You can also select your material by opening an feplot figure with all materials and selecting graphically The InfoTarg command can then be used to select materials in a plots For example giving a target frequency and a range of temperatures is achieved with m_visco database open FEPLOT figure with database RO struct wtarg 20 ttarg 10 5 20 40 60 80 figure 1 m_visco info targ 23 20 2 0 2 cf RQ 101 m_visco mvisco_3m 102 mat m_visco database Name tries to find an optimal match for Name in the database and returns the associated material mat m_visco database unit TM Name converts to unit system TM see fe_mat Convert Command option matid can be used to specify a MatId for the material Without output argument material is added to the stack of the feplot figure one can give the cf pointer as a second input argument If Name is not given all materials are returned mat m_visco dbval is used to
38. Surface matid 1 amp selface amp facing gt 0 0 0 1 mo2 fevisco ConnectionStickThenSolid model Foam MAP FacingSurface cf model mo2 fecom colordatamat alpha 3 3 3 3 Exporting submeshes to NASTRAN Once the sandwich model generated it can be exported to NASTRAN There two typical strategies rewriting the whole model using naswrite FileName model or generating an include file The second strategy is more adapted when testing multiple viscoelastic configura tions since it is more robust at preserving all options of the original file The Zevisco WriteInclude command is meant for that purpose It lets you select newly meshed viscoelastic parts using any selection their MatId for example and generates the NASTRAN bulk containing the associated nodes elements material and element property cards RBE2 entries connected to the selected elements 3 4 Parametric models structure reference 3 4 1 Parametric models zCoef Different major applications use families of structural models Update problems where a comparison with experimental results is used to update the mass and stiff ness parameters of some elements or element groups that were not correctly modeled initially Structural design problems where component properties or shapes are op timized to achieve better performance Non linear problems where the properties of elements change as a function of operating conditions and or frequency visco
39. Viscoelastic Vibration Toolbox For Use with MATLAB User s Guide Etienne Balm s Version 1 0 Jean Michel Lecl re How to Contact SDTools 33 1 44 24 63 71 SDTools 44 rue Vergniaud 75013 Paris France http www sdtools com comp soft sys matlab http www openfem net support sdtools com suggest sdtools com info sdtools com Phone Mail Web Newsgroup An Open Source Finite Element Toolbox Technical support Product enhancement suggestions Sales pricing and general information Viscoelastic Vibration Toolbox on July 2 2013 Copyright 1991 2013 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 Modeling viscoelastic materials 2 1 1 Introduction 1 2 Representing complex modulus 1 2 1 Non parametric tabular representations 1 2 2 Simple rheological models 1 2 3 High order rational models
40. age modulus The associated temperature is called transition temperature Note that this can be confusing since the transition temperature depends on the frequency used to generate figure 1 8 In the rubberlike region storage modulus and loss factor are both characterized by relatively small values and low temperature dependence The fourth region cor responds to a fluid state This region is rarely considered because of its inherent instability For damping applications one typically uses viscoelastic materials in the transition region This choice is motivated by the fact that the loss factors presents a maximum in this area thus allowing an efficient use of the material damping properties 15 1 Modeling viscoelastic materials 16 Real E D Temperature C Frequency Hz Figure 1 9 Variations of ISD 112 modulus in the frequency temperature domain In the frequency temperature domain figure illustrates the existing inverse relation of the effects of temperature and frequency Experimentally one often finds that by shifting isotherm curves along the frequency axis by a given factor r one often has good superposition This property motivates the introduction of the reduced frequency a T w and a description of the complex modulus under the form E w T E a T w 1 16 The validity of this representation is called the frequency temperature superposition principle and the curve is called a master curve V
41. and constrained layer models The demo basic_sandwich generates curves for the validation of shell volume shell model used to represent constrained layer treatments as first discussed in section 2 4 The demo cut_optim illustrates the classical result that there is an optimal length for a simple constrained layer treatment 2 134 5 Hz 1800 T T T T T T T a ro gt Flexion 1 1600 N a N Flexion 2 f Ss i Torsion 1 f P m Flexion 3 woop Y a Seer a a7 o Torsion 2 4 pa a a E g 2 Th a 7 1200F g Sg p a N x y j amp bsg A oe 3 Dos a 8 1000 4 2 e OS ax ay Pas j A 5 Ee as 2 cook ag Ten aee aa Da RR ae a 400 eax woo 4 200 ee weer 4 aaa ee oe oe ee o 1 i L 1 L f L L L 0 01 0 2 03 04 05 Cut position L Figure 3 2 Free layer model validity 79 3 Toolbox tutorial 80 Toolbox Reference eRe eae eee 4 1 fe 2xf ee ee 82 ee ea ae eee 91 cg cites 100 ee ee eee ee ae 107 eee ee ee eee 112 4 1 fe2xf Purpose Syntax Description 82 Direct computation of frequency response functions This function is part of the viscoelastic tools cases fe2xf command model case frfzr file FileName This command supports direct frequency response computations and post processing for reduced models For a given input the response is fully characterized by the response at DOFs q state vector in control theory which dependen
42. arious authors have given thermodynamic justifications to the frequency temper ature superposition principle These are limited to unique polymers For polymer blends which have significant advantages it is not justified 10 10 mo Fr quence Hz 10 10 Module de stockage E Pa facteur de perte Mh 2 10 10 j 4 Fr quence r duite wof Figure 1 10 Reduced frequency nomogram The superposition principle is used to build a standard representation called nomo gram which simplifies the analysis of properties as a function of temperature T and frequency w The product way corresponds to an addition on a logarithmic scale One thus defines true frequencies on the right vertical axis and isotherm lines allow ing to read the reduced frequency graphically on the horizontal axis For a frequency wj and a temperature Tk one reads the nomogram in three step shown in figure 1 10 e 1 one seeks the intersection P of the horizontal w line and the of sloped Th line e 2 the abscissa of point P gives the reduced frequency wj r Tk e 3 the intersection of the vertical line at that reduced frequency with the storage modulus F and loss factor 7 master curves gives their respective values at Wj Ty Various parametric expression have been proposed to model the temperature shift factor ar The empirical equation of Williams Landel Ferry called WLF equa 17 1 Modeling viscoelastic materia
43. ate space model of the form a s s A a s B u s 2 40 where the state vector will combine fractional derivatives of the displacement s gq where k 1 2p 1 and of the internal state qx Gena see for example In practice the number of blocs in the state vector being proportional to p constant matrix representations are thus limited to small values of p This limits practical uses of fractional derivative models to frequency domain response and non linear eigenvalue computations see section 2 3 Spectral decomposition and reduced models For an input b u s characterized by the frequency domain characteristics of u s and spatial content of b component mode synthesis and substructuring methods provide approximations of the solution of problems 2 2 2 34 or 2 35 35 2 Viscoelastic FEM models 36 For damped problems one should distinguish e exact spectral decompositions using complex modes as treated in ections 2 3 1 and 23 3 e model reduction methods which only seek to approximate the transfer spec trum by projecting the model on bases built using solutions of problems that are simpler to solve than the complex eigenvalue problem These approaches are detailed in sections 2 3 3 and 2 3 1 Complex modes of analytical models All the problems that where introduced earlier can in the frequency domain be represented as frequency response computations of the form y s
44. ce on the load is defined by an evolution equation equation of dynamics in mechanics ZW vx 9 tax llvxwa U bv 4 1 Unit inputs b describe the spatial content of loads see more details in section 3 4 3 The frequency content is described at each frequency by specifying an input u or the associated covariance matrix X u Cases with enforced displacements require a few additional manipulations discussed in but leads to similar equation forms Outputs are characterized by a vector y that is supposed to be linearly related to DOFs through an observation equation see more details in section 3 4 4 YWw Ins lelvsxn ta bn 4 2 where the y components can be translations rotations normal velocities strains The first step of an analysis is to define the input shape matrix b and possibly the inputs u The second step is to define the outputs fe2xf provides a fairly large set of response processing options for full and reduced models through specification of a info MifDes entry in the model stack This entry should contain a cell array each row describing a response processing with post_name copt post_name is the string identifier of the post frf svd and copt contains the options related to the post see below For example MVR fevisco TestPlateLoadMVR MVR stack_set MVR info Freq 30 1 500 MVR stack_set MVR info MifDes frf RESP f
45. der the assumption that the strain energy is sufficiently uniform to be represented as a spring Q71 the inverse of the quality factor corresponds to a loss factor For a damper following the 3 parameter law of a standard viscoelastic solid 1 8 the system has a pair of complex poles A A and one real pole 8 1 1 T H s 2 8 mea nse m s 2Cws w 1 4 and the model characteristics depend on those poles as follows p p w p xu k xp kK m Xu 2 9 For low damping of the conjugate pair of poles that is 1 and 6 and w in the same frequency range p and z are close which leads to a small maximum loss factor and a response that is very similar to that of the oscillator with viscous damping 2 3 Figure 2 2 shows that for viscous structural and viscoelastic dynamic stiffness mod els for the oscillator c the dynamic flexibilities a b are almost exactly overlaid This is linked to the fact that the real parts of the dynamic stiffness coincide natu rally since they are they not viscous or structural or little viscoelastic influenced by the damping model and the imaginary parts d are equal at resonance This equivalence principle is the basis for the Modal Strain Energy MSE method that will be detailed in section 2 1 3 4 i E a ge J 5 B at z 3 s Z Foe Z 4 rf a a a a B Q x 10 10 10 10 a ee l A Frequency Rea
46. e columns of the cell array which can be modified with the feplot interface give e the matrix labels Klab which must coincide with the defined parameters the values of coefficients in 3 4 for the nominal mass typically mCoef 1 0 O iga 1 the real valued coefficients zCoef0 in 3 4 for the nominal stiffness Ko the values or strings zCoefFcn to be evaluated to obtain the coefficients for the dynamic stiffness 3 4 Given a model with defined parameters matrices model fe_ def zcoef default mod defines default parameters zcoef fe_def zcoef model returns weigthing coefficients for a range of values using the frequencies see Freq and design point stack entries 63 3 Toolbox tutorial Frequencies are stored in the model using a call of the form model stack_set model info F1 Design points temperatures optimization points are stored as rows of the info Range entry see for generation When computing a response fe_def zCoef starts by putting frequencies in a local variable w which by convention is always in rd s and the current design point row of info Range entry or row of its val field if it exists in a local variable par zCoef2 end 4 is then evaluated to generate weighting coefficients zCoef giving the weighting needed to assemble the dynamic stiffness matrix 3 4 For example in a parametric analysis where the coefficient par 1 stored in the first column of Ra
47. e diagram building rules in any course in controls The modulus slopes found in true materials are generally such that it is necessary to introduce many poles to accurately approximate experimental mea surements This motivated the use of fractional derivatives described in the next section 1 2 4 Fractional derivative models The use of non integer fractions of s allows a frequency domain representation with an arbitrary slope A four parameter model is thus proposed in Emin Emax E s E 1 15 s max 1 s w 13 1 Modeling viscoelastic materials 14 where the high Eng and low Emin frequency moduli are readily determined and the w and a coefficients are used to match the frequency of the maximum loss factor and its value Properties of this model are detailed in One can of course use higher order fractional derivative models To formulate constant matrix models one is however bound to use rational exponents see sec tion 2 2 4 The main interest of fractional derivative models is to allow fairly good approxi mations of realistic material behavior with a low number of parameters The main difficulty is that their time representation involves a convolution product One will find in Ref a recent analysis of the thermomechanical properties of fractional derivative models as well as a fairly complete bibliography 1 3 Environmental factors For usual damping materials the complex modulus depends not o
48. e law coefficients one can represent the dynamic stiffness of a viscoelastic model as a linear combination of constant matrices for independent complex moduli in the same material their may be more than one matrix associated to a given material Kyi Eo Z E 8 M8s Ke iKei sC Y E s T 00 Eo a 2 29 This representation is the basis for the development of solvers adapted for structures with viscoelastic materials For frequency domain computations it is rather inefficient to reassemble Z at each operating point E s Two solutions can be implemented easily On can store the various M Ke Kei C Ky matrices and evaluate the weighted sum at each operating point or store element matrices and reassemble with a weighing coefficient associated with the material property of each element 2 2 2 State space representations One discusses here state space representations associated with analytical represen tations of the complex modulus discussed in section State space models are first order differential equations assumed here with constant coefficients with the standard form amp t A x B u 2 30 y t C x P u t l The matrices are called A transfer B input C observation and D direct feed trough The first equation is the evolution equation while the second is called the observation equation The usual technique for time integration of mechanical models is to define
49. e of GPL and CSTM to obtain nodes in global coordinates reading of element mass MDICT MELM and stiffness KDICT KELM matrix dictionaries and transformation of a type 3 su perelement handled by This is typically obtained from NASTRAN with PARAM POST 4 To choose the file name use Up file FileName Up nasread Up Output2 op2 material property tables transformation of shapes modes time response static response as curve entries in the stack possibly multiple if various outputs are requested Note by default deformations are in the SDT global coordinate system basic in NASTRAN terminology You may switch to output in the local global in NASTRAN terminology using PARAM OUGCORD GLOBAL To avoid Out of Memory errors when reading deformations you can set use a smaller buffer sdtdef OutOfCoreBufferSize 10 in MB When too large def def is left in the file and read as a v_handle object that lets you access deformations with standard indexing commands Use def def def def to load all To get the deformation in the stack use calls of the form def stack_get model curve OUG 1 getdata tables of element energy tables of element stresses or strains This translation allows direct reading translation of output generated with NAS TRAN PARAM POST commands simply using out nasread FileName op2 For model and modeshapes use PARAM POST 1 For model and element matrices use
50. e one the dual subspace T is zero this congruent transformation corresponds to a Ritz Galerkin analysis Transfer functions are the approximated by 7 1 H s d Z s b eT T Z s T 779 2 55 One can note that for a non singular transformation T when q T qr is bijective the input u output y relation is preserved One says that the transfer functions are objective quantities they are physical quantities that are uniquely defined while DOFs q are generally not objective Classical bases used for model reduction combine modes and static responses to characteristic loads On distinguishes e bases containing free modes and static responses to applied loads b z T ie J R par 61 ona era Men a 2 56 ll a j For component mode synthesis component model reduction to prior to a cou pled system prediction free modes have been used by MacNeal and 39 2 Viscoelastic FEM models 40 many others e bases containing static displacements associated with displacement enforced on an interface and fixed interface modes T oe i ey For CMS the use of static terms only is called Guyan condensation Adding fixed interface modes leads to the Craig Bampton method e damped modes can be considered as elastic models with an external damping load Static correction for the effects of damping loads can then be incorpo rated The first order correction pro
51. e reduced frequency x axis e at click on the isotherm that is always generated to enter the isotherm edition mode You can then use the following keys to edit the at values n to add new isotherm values Left and right arrows to move the offset of the current isotherm Up and down arrow to select a different isotherm If the isotherm slope is not correct check the Elim and wlim values and pos sibly adjust the axis rectangle using uU to move the rectangle along x or x vV to move the rectangle along y or y xXyY to resize the rectangle e eta click on the green line with circle markers Adjust values with your mouse Use left and right arrows to select the points and q to exit e G click on the blue line with circle markers Adjust values with your mouse Use left and right arrows to select the points and q to exit 105 m_visco mvisco_3m At any point during the procedure you can press the i key to generate a screen printout of the material This can then be included in or a mvisco_ m database file With a material saved in a database can superpose the original figure and the nomogram using m_visco HandNomo MatName while having the image file in the current directory See also section 106 4 4 nasread Purpose Syntax Description Read results from outputs of the MSC NASTRAN finite element code This function is part of FEMLink out nasread FileName Command reads bulk data d
52. e sdtdef ter e def precomputed def For use with fe_reduc use an info Fe2xfBuild entry Following example builds a reduced model using RO list model fevisco TestCantilever cf feplot model mat m_visco convert INSI m_visco database Soundcoat DYAD609 cf mdl fevisco addmat 101 cf mdl visco mat RO struct list eig 5 10 0 0 1 1 gie 5 10 0 0 1 i s first oo ah fe2xf build cf RO Another example with a call using a parameter Range 89 fe2xf See also 90 model fevisco TestCantilever cf feplot model mat m_visco convert INSI m_visco database Soundcoat DYAD609 cf mdl fevisco addmat 101 cf mdl visco mat Range struct val 0 1 1 0 1 1 lab m k visco RO struct EigOpt 5 10 0 Range Range f 2xf build cf RO When starting from a model without pre defined matrices cf md1 K does not exist one assembles mass stiffness and parameter matrices assemble matdes 2 1 1 se NoT and removes the nominal parameter matrices from the base stiffness The nominal model is thus associated with coefficients 1 for all parameters section B23 4 2 fevisco Purpose Syntax Description User interface function for support of viscoelastic materials This function is part of the viscoelastic tools out fevisco command model The solid and fluid
53. e2xf horzcat frfzr file nas2up tempname RESP mat MVR R1 RESP 1 R1 xf abs R1 xf fe_curve plot figi1 ylog xtight R1 These low level commands are used given a reduced model to compute FRFs for a frequency parameter range The reduced model uses the standard fields used to describe parametric models see section 3 4 5 It must at least contain the following fields Range cr br zCoefFcn K Given a FileName results are saved every 30 seconds to allow post processing during the evaluation Without output argument one can specify the identifier of an iiplot or feplot figure whose stack will store results of computation using a cf i command option Accepted post processing options given in an info MifDes entry e frf computes the transfer function for unit inputs 1 H w lysxna lelysxn Zlvxn lvxwa 4 3 Possibly select inputs and outputs by their indices with copt in and copt out e frfu computes the response to specified loads ns H nsxwatu tna 4 4 Possibly select outputs by their indices with copt out e svd computes the singular values of H the first one is sometimes called the spectral radius or LH w gx na 4 5 Possibly select of a subset of singular values with copt ind e svdu computes the singular values of H weighted by the input level given in u w onl w lwsxwva ue ya ara 4 6 Possibly select of a subset of singular values with copt ind Only vectors
54. e_step12 dat file and run the job using cf feplot the model should be displayed in feplot cf mdl nas2up JobOpt cf md1 Init NasJobOpt entry to its default fevisco writeStep12 run RootName cf fecom cf Save FileName save your model for reload where the write command edits the nominal job files found in getpref FEMLink DmapDi If you need to edit the bulk file for job specific aspects of the Case Control Section definition of MPC and SPC for example omit the run do your manual edits then run the job for example nas2up joball memory 8GB RootName_st You can also pre specify a series of EditBulk entries so that your job can run automatically For example edits insert pOL 103 gt gt GEOMCHECK NONE replace SPC S107 7 SPG ii model stack_set model info EditBulk edits Once the NASTRAN job done you should have locally the files RootName_mkekvr op4 reduced matrices RootName_USETT op2 degree of freedom set and info needed to build Case T RootName_TR op4 basis vectors defined on DOFs that are needed sensor and input DOF You are now ready to build the reduced parameterized model using cf feplot fecom cf Load FileName MVR fevisco BuildStep12 RootName model If the files are not automatically copied from the NASTRAN server machine the BuildStep12 cpfrom makes sure the result file are co
55. eck NASTRAN input direct reading of model and result information in OUTPUT2 and OUTPUTG4 files generated using NASTRAN PARAM POST lt cards This is the most efficient and accurate method to import NASTRAN results for post processing visualization with a te model fapcom handling with fnor2ss or parameterized model handling with upcom Results in the 06 text file no longer supported Available commands are Bulk file model nasread FileName bulk reads NASTRAN bulk files for nodes grid points element description matrix material and element properties and coordinate transformations MPC SPC DMIG SETS Use BulkNo for a file with no BEGIN BULK card Unsupported cards are displayed to let you know what was not read You can omit the bulk command when the file name has the dat or bdf extension Each row of the bas bas output argument contains the description of a coordinate system The following table gives a partial conversion list For an up to date table use nas2up convlist 107 nasread 108 NASTRAN SDT CELAS1 CELAS2 RBAR celas RBE2 igi rigid RBE3 i rbe3 in Case CONROD bar1 CBAR CBEAM CROD beami CBUSH cbush CSHEAR quad4 CONM1 CONM2 mass2 CHEXA hexa8 hexa20 CPENTA penta6 pentald CTETRA tetra4 tetrai0 CTRIA3 CTRIAR trias CTRIAG6 i tria6 CQUAD4 CQUADR quad4 CQUAD8 quadb Details on properties are given under naswrite WritePLIL N
56. ed femesh plotelt fecom colordatamat Offsets are handled using rigid links between the shell neutral fiber and upper lower surfaces By default element normals at the center are used to define thickness you can also use normals at node by inserting a node option in the command Notes on sandCom format The first layer uses 0 material property it means that the original material property of the first layer is used For the volume layer 101 is used and only the thickness needs to be specified 002 The last layer is a shell property 102 with a thickness of 001 with an offset of 005 The command allows more accurate control of normals used for the sandwich gen eration Nominally the normals are generated using the commands MAPE feutil get normalmap node elt MAPN feutil get normalmap node node elt where elt is the element selection for the sandwich generation You can provide your own maps using with a call of the form fevisco sandCom FEnode FEel0 treated MAP It is then expected that the provided MAP has an MAP opt field where opt 1 1 leads to sandwich generation with offset at element center element normal map and opt 1 2 uses a normal map at nodes MatSplit model fevisco MatSplit MatId model Split material MatId into 6 material corresponding to the 6 main directions Element are also repeated 6 times Thermo Computation of the power dissipated within a visc
57. efficients on the stiffness matrix of elements within a given selection NASTRAN set nas_sdtserv This is used to overload basic SDT functionality through NASTRAN calls You must first set local preferences setpref SDT ExternalEig Callback used by fe_eig method 50 gt mode fevisco nastran eig m k model DOF opt model Case 3 9 Advanced connection models 3 9 1 Screw models Proper representation of screws is a classical difficulty that can be much alleviated using automated procedures The models used here consider the screw as a circular beam with diameter equal to that of the hole Rigid screw models use rings of rigid connections One only takes a few nodes on the screw to be masters and big lumps of model nodes are forced to follow the associated rigid body motion Flexible screw models assume that the responses of the nodes on the beam are dependent on the response of a number of other nodes in the model This depen dency is represented as a weighted sum which in terms of loads corresponds to a distribution of loads on a number of nodes 3 9 2 Physical point with rotations This type of connection finds the element a given node is connected to see feutild match and uses shape functions of underlying element to estimate motion at the node Rotations are determined using local derivatives of the shape functions at the physical point 3 10 Validation examples 3 10 1 Validity of free
58. elastic behavior geometrical non linearity etc A family of models is defined see for more details as a group of models of the general second order form where the matrices composing the dynamic stiffness depend on a number of design parameters p Z p s M p C p s K p 3 1 Moduli beam section properties plate thickness frequency dependent damping node locations or component orientation for articulated systems are typical p pa rameters The dependence on p parameters is often very non linear It is thus often desirable to use a model description in terms of other parameters a which depend non linearly on the p to describe the evolution from the initial model as a linear 61 3 Toolbox tutorial combination a 2 p s a5 p Zja s 3 2 j l with each Zj s having constant mass damping and stiffness properties Plates give a good example of p and a parameters If p represents the plate thickness one defines three a parameters t for the membrane properties t for the bending properties and t for coupling effects p parameters linked to elastic properties plate thickness beam section properties frequency dependent damping parameters etc usually lead to low numbers of a parameters so that the a should be used In other cases p parameters representing node positions configuration dependent properties etc the approach is impractical and p should be used directly par 62 SDT
59. erated by T of the representative elastic problem TF KoT w T7 MT djr 0 2 60 one can verify this uncoupling if wir is within the band of interest the decoupling is not verified One can always choose bases Tm and T so as to diagonalize the reference prob lem Furthermore by writing AZ s Z s K s M the transfer function is approximated by s wim TEAZ s Tmn T Z s T TT Z s Tm s win TT AZ 8 T ral 2 61 The Modal Strain Energy method MSE with damping ratio given by 2 28 cor H s cT 41 2 Viscoelastic FEM models 42 responds to the following approximation TRAZ s Tm 8 E 8 Mwn 2 62 One can easily generalize this approximation by building an equivalent viscous damp ing matrix by enforcing Tm AZ iw Tm iw jp 2 63 for a characteristic frequency wjr for diagonal terms j k wjr wkr 2 otherwise For a viscous damping model it is a simple projection computation of TECTm For a structural damping model there is a degree of approximation The validity of this approximation is discussed in where it is shown that building the equivalence in generalized coordinates and using term by term characteristic frequencies is efficient For coupling terms TT Z s T damping only has low influence see the discussion on non proportional damping in section 2 1 2 and can thus be neglected For residual terms TT AZ s T damping can be neglected for frequenc
60. erred here is to simply interpolate between points of an ar table 1 3 2 Other environment factors Between the other environmental factors influencing the behavior one essentially distinguishes non linear effects static and dynamic and history effects exposition to oil high temperatures vacuum Non linear dynamic effects are very hard to characterize since high amplitude varia tions of the induced strain are typically correlated with significant energy dissipation and thus temperature changes The effects of level and temperature changes are then coupled for materials of interest which are typically in the transition region Exper imentally such non linear studies are thus limited to the rubber like region The effects are similar to those of temperature although of smaller magnitude Non linear static effects that is effects linked to a static prestress assumed to be constant are significant and easier to characterize It is known to be essential when considering machinery suspensions or constrained viscoelatic sandwiches where press forming induced significant pre stress History effects are generally associated with extreme solicitations that one seeks to avoid but whose probability of occurrence is non zero As for temperature one generally represent the effect of other environmental fac tors as shift factors although the superposition hypothesis may not be as well verified 1 4 Determining the complex mod
61. es m_heat and p heat must be present in model material and element property stack Matrices must be assembled as following RO comp12 Thermo1Assemble md1 RO RO is the coupling data structure with fields MatId Matid_struct MatId_therm first is the MatId of the structure material property and second the thermal one ProId ProId_struct ProId_therm TO Matid_struct MatId_therm TO is the ther mal coupling C Inf if no coupling Then reduced MVR can be build using MVR comp12 ThermoiBuild reduce md1 R0O If command option reduce is given model is reduced according to real modeshapes Then eig options must be given in RO EigOpt 51 2 Viscoelastic FEM models 52 Complex coupled thermoelastic modes can then be computed using def comp12 ThermoiModes MVR zc compute modes Following full example can be found in comp12 numeric It illustrates the com putation of thermo elastic damping in a simple rectangular plate example TO 20 coupling temperature Build model Nz 5 md1l femesh sprintf testhexa8 divide 10 10 i Nz h 2e 3 md1 Node 5 7 mdl Node 5 7 diag 304 192 h md1 Elt feutil set groupall matid 1 md1 md1 Elt feutil set groupall proid 1 md1 define material properties md1l pl m_elastic dbval 1 Aluminum structure mdl feutil setmat 1 alpha 22e 6 TO 20 md1 XXX alpha at what temperature md1 pl m_heat mdl
62. es and zeros sums of low order fractions 1 S hilsY G zi ae E s Eg Ex Sity a ee 1 14 A ea aaa O i 1 i 1 are just a few of various equivalent representations Figure 1 7 Examples of generalized damping models a Kelvin s chain b gener alized Maxwell model Figure shows two classical representations Maxwells generalized model intro duces a pole Aj E Cj associated to settling time 7 C E and high fre quency stiffness for each spring dashpot branch The physical meaning of each representation is identical However each representa tion may have advantages when posing equations for implementation in a numerical solver For example section will discuss models using particular represen tations of the modulus of order 2 so called GHM models for Golla Hughes Mc Tavish and order 1 so called ADF models for Anelastic Displacement Field This section will also show how these models correspond to the formalism of materials with internal states that are commonly used in non linear mechanics Rational fractions have a number of well known properties In particular the modu lus slope at a given frequency s iw is directly characterized by the position of poles and zeros in the complex plane see Bod
63. for dry fric tion dissipation associated with drag in a viscous fluid a plastic spring or small impacts nj w 2 25 While the modal strain energy method is typically associated with the modal damp ing assumption it can be easily extended to account for frequency dependent non di agonal one says non proportional damping matrices Thus using q T qr 1 6nm qr leads to a model of the form s 2 F Um Z s 9 w 0 5 I LF 9 Fak 2 26 This is typically referred to as a modal solution In NASTRAN for example you will find modal complex eigenvalue SOL110 frequency response SOL111 transient SOL112 For more general viscoelastic models it is always possible to define pseudo normal modes solutions of M Re K w 45 0 2 27 to normalize the using a condition similar to 2 45 and to define an equivalent damping ratio at resonance by _ 1 8 Bm w 1 95 a 5 2 51 RecK a 4 2 28 Experimental and design damping ratio The other classical approach is to use modal damping ratio determined experimen tally Identification techniques of experimental modal analysis give methods to determine these ratios For correlated modes when a one to one match between test and analysis is estab lished one thus uses a damping ratio Cjrest while typically preserving the analysis frequency For uncorrelated mode
64. gence area is the aspect that needs to be accounted for in the development of partial eigenvalue solvers for damped problems A second consequence of the increase in the number of real poles is the potential impossibility to compute modes with supercritical damping The author s experience is that this limitation is mostly theoretical since in practice modes with supercritical damping are rare The complex modulus being the Fourier transform of a real valued relaxation func tion should be symmetric in frequency E w E w For a modulus represen tation that does not verify this hypothesis as is the case of structural damping only poles with positive imaginary parts have a meaning For response synthesis one will thus take the conjugates j pj of modes computed with positive imaginary parts The computation of complex modes can be used to approach transfer functions using the developments 2 44 and 2 46 Orthogonality conditions given above correspond to the a 1 normalization 2 3 3 Model reduction methods To simulate the dynamic response it is not useful and rarely possible from a nu merical cost standpoint to use the full model for direct time simulations see section Model reduction methods modal analysis substructuring component mode synthesis seek an approximate solution within a restricted subspace One thus assumes atnxi T nxnrR IR NRx1 2 54 and seek solution of 2 1 whose projection on th
65. generate equivalent elastic materials This is not currently functional The 3ParWLF material is a standard viscoelastic model with WLF type temperature dependence Its constitutive law is the characterized by 1 iwg z ae iwr p G G and wp war wl0 P T c2 T To 4 9 The parameters retained to characterize this model are GO etamax wmax cl 2 TO DispMat m_visco dispmat mat is used to generate a text display of a viscoelastic material describe by the mat data structure RefMat m_visco refmat mat model uses data on temperature range and frequency in the model to create a nomo The nomo command generates a standard nomogram plot The following entries in the cell array mat nomo are used If using a mat G table that contains raw experimental data that is non smooth mat m_visco nomo smooth mat will generate a smooth table at Eeta gives the log10 of Emin Emax etamax Note that for units in Pa a shift to MPa is performed in the plot W gives the log10 of Wmin WRmin WRmax Increasing Wmin shifts the show isotherms right file gives the name of the bitmap file used to digitize the nomogram When this file is in the current directory the nomogram and the bitmap are overlaid This allows editing of tabular values as described under fevisco handnomo T is used to specify isotherm positions in the nomogram For example m_visco nomo isdii2 1999 4 mat m_visco
66. he architecture is fully compatible there is no simplified mechanism to parameterize the mass The first step of a study is thus to define parameters For all models this can be done using fe_case par commands orfupcom Par commands for an upcom superelement model demosdt demoubeam model fe_case model par Top withnode z gt 1 fecom proviewon fecom curtabCase Top highlight the area If the parameters correspond to viscoelastic materials one needs to declare which of the initially elastic materials are really viscoelastic This is done using fevisco AddMat calls which associate particular MatId values with viscoelastic materials se lected in the database Up fevisco testplate upreset cf feplot Up Up cf mdl1 Up stack_rm Up mat Up fevisco addmat 101 Up First area ISD112 1993 Up fevisco addmat 103 Up Second area ISD112 1993 cf Stack info Range 20 cf Stack info Freq logspace 1 3 30 reset default zCoef Fcn and display fe2xf zCoef default cf fe2xf zCoef cf Viscoelastic materials are then considered as parameters by Eevisco The full constant matrices M Ke Kym can be assembled using fevisco MakeModel or with a NASTRAN DMAP Stepi2 for implementations with other software such as ABAQUS ANSYS or SAMCEF please contact us The normal mode of operation is to
67. hicknesses along the element normal h or along the normal at nodes hi This distinction is important for relatively coarse meshes of press formed parts as the floor panel of figure Advanced options meshing options let you preserve thickness either at element center or nodes and possibly control the normal map used as a meshing support For stiff layers shells are preferred over volumes because volume element formula tions are sensitive to shear locking when considering high aspect ratio dimensions of the element large compared to thickness 43 2 Viscoelastic FEM models 44 For soft layers the use of a volume element both necessary because shell elements will typically not correctly represent high shear through the thickness and accept able because almost all their energy is associated with shear so that they will not lock in bending Note that shear corrections used in some FEM codes to allow bending representation with volumes may have to be turned off to obtain appro priate results Finally there are doubts on how to properly model the through the layer compression stiffness of a very thin viscoelastic layer this can have significant effects on curved layers The demo basic_sandwich generates curves for the validation of shell volume shell model used to represent constrained layer treatments as first discussed in The idea is to vary the properties of a central volume layer between a very soft modulus and the
68. high frequencies one uses a complex asymptote E 4 oo There are mathematical limitations on possible values for this asymptote but in practice there are also other physical dissipation mechanisms that are not represented by the viscoleastic law so that the real objective is to avoid numerical pathologies when pushing the model outside its range of validity The main advantage of non parametric representations of constitutive laws is to allow fully general accounting of behavior that are strongly dependent on frequency temperature pre stress Moreover the direct use of experimental data avoids standard steps of selecting a representation and identifying the associated parame ters Since good software to perform those steps is not widespread avoiding them is really useful Disadvantages are very few and really mostly linked to the extraction of complex modes and as a consequence time response simulations using modal bases And even these difficulties can be circumvented 1 2 2 Simple rheological models The classical approach in rheology is to represent the stress strain relation as a series of springs and viscous dashpots Figure 1 3 shows the simplest models E E E AN E 1 in0 MN W W SES Figure 1 3 Material damping models with 2 or 3 parameters a Maxwell s model b vicous damping Kelvin Voigt s model c structural damping d standard vi
69. idity of the modal damping assumption which leads to models where the response is decomposed in a sum of independent oscillators e techniques used to estimate equivalent viscous damping models 2 1 1 Properties of the damped 1 DOF oscillator This section illustrates the properties of the single degree of freedom oscillator with a viscoelastic stiffness shown in figure 2 1 K s Figure 2 1 Oscillator with a viscoelastic stiffness For a viscous damping K s k cs the load to displacement transfer is given by B 1 B 1 m ms estk s x s Hyisels 2 3 25 2 Viscoelastic FEM models 26 whose poles root of the transfer denominator are A CWpntiwg wd wn4 1 wn 4 k m 2 4 For structural damping K s k 1 in the load to displacement transfer is given by H _ 1 H ikn cs nls ms k 1 in ms cs k ms k 1 in its pole with a positive imaginary part is identical to that of the viscous model for n 2 2 6 which leads to a difference Hy Huise that is zero at resonance wn The pole with a negative imaginary part is unstable positive real part which is a classical limitation of the structural damping model 2 5 Damping is also defined by a quality factor which can be measured in a shaking table as the ratio between the acceleration of the mass at resonance and the acceleration of its base The value is approximately i Q 5 2 7 Un
70. ilar equations can of course be developed for applications where the fluid is represented using boundary elements Acoustic stiffness on a loudspeaker The procedure is divided in the following steps e declare the fluid as a superelement in the structure model A typical call would take the form modelS modelF fevisco fluidtest this generates demo models cf feplot fevisco fluidmerge mode1S modelF 75 3 Toolbox tutorial 76 fecom curtabStack SE fluid cf sel 1 innode y gt 0 amp eltname SE colordatamat Note that the solid model mode1F will often be read from NASTRAN as mode shapes PARAM POST 2 or superelement PARAM POST 4 If node numbers in the fluid and solid are coincident the call automatically shifts fluid node numbers assembly of the fluid structure coupling and fluid matrices with the FluidMatrix command fevisco fluidmatrix cf SelElt selface Fluid interface in fluid SE SelElt EltName quad4 Solid interface in structure cf Stack fsc see the coupling superelement Arguments of the command are a pointer to the figure containing the solid model and fluid as a superelement a series of femesh commands that allows the selection of the fluid interface for which a fluid structure property is defined In the example one selects the fluid eltname flui and keeps its exterior boundary SelFace If the
71. ions stores text options to be added to the nastran command RunOpt BackWd can be used to specify an additional relative directory for the JobCpFrom command RunOpt RemoteRelDir can be used to specify the associated input for the JobCpTo command nas2up JobCpTo LocalFileName RemoteRelDir puts copies files to the remote directory or to fullfile RemoteDir RemoteRelativeDir if specified nas2up JobCpFrom RemoteFileName fetches files The full remote file name is given by fullfile RemoteDir RemoteFileName Any relative directory is ig nored for the local directory Here is a simple script that generates a model runs NASTRAN and reads the result wd sdtdef tempdir 114 model demosdt demoubeam 2mat cf feplot model fe_case model dofload Input struct DOP 849 015 360 015 241 017365 03 det 15 15 151 D 100 model nas2up JobOpt mode1 model stack_set model info Freq 20 2 150 write bulk but do not include eigenvalue options naswrite new fullfile wd ubeam bdf stack_rm model info EigOpt generate a job by editing the reference mode dat file fname ubeam dat edits Set PARAM POST 2 replace include model bdf include ubeam bdf replace EIGRL nas2up writecard 1 1 0 0 30 ijji EIGRL
72. is the only one applicable to test the effect of pre stress in sandwich structures Building a test rig with no perturbing modes being quite difficult modulus char acterization is always performed on fairly narrow frequency bands The frequency temperature superposition hypothesis is thus made to create master curves over a wide frequency band The next possible step is the determination of the coefficients of an analytical representation Identification tools developed in control theory ARMA models are suited for rational fraction models It is however difficult to enforce a good reproduction of quantities that are typically judged as important low and high frequency moduli maximum loss factor In other case a non linear optimization is readily implemented using optimization tools available in MATLAB 1 5 Conclusion For a given complex modulus it is difficult to validate the fact that general ther modynamic principles are verified Ref gives an analysis based on the fact that each component of a generalized maxwell model must verify the second principle of thermodynamics and thus be associated with positive stiffness and damping coeff cients In practice the criteria that are used to judge the quality of a constitutive model are the accurate reproduction of experimental complex modulus measurements within the tested frequency temperature range and the likelihood of extrapolations Fig ure illustrates the fact that wi
73. isd112 1993 mat sdsetprop mat nomo w 1 0 12 Eeta 4 8 2 type G Pani ge T 20 Os 120 3 m_visco nomo mat ees e w Temperature S E g3 ge e o Ta 10 10 10 10 Loss factor T Figure 4 3 Nomogram of the ISD112 The at command is used to generate shift factor using calls of the form at m_visco at T mat where mat can be a data structure or a string matched against the m_visco database 103 m_visco mvisco_3m interp unit SI showrange The interp command is used to generate the modulus at arbitrary frequency temperature design points using calls of the form G m mvisco interp w T mat where mat can be a data structure or a string matched against the m_visco database An optional unit conversion code see fe mat convert can be given when needed The InterpShowRange command generates a standard display giving modulus vari ations for a given frequency temperature range This allows the user to validate the usefulness of a particular material for the operating conditions given by this range For example w logspace 1 3 100 default T 0 10 50 default mat m_visco isd1i12 1993 cf Stack info Freq in Hz cf Stack info Range G m_visco interp unit MM w T mat m_visco interp ShowRange w T mat m_visco interp cf MatName is also acceptable
74. k_get cf mdl info Freq getdata T stack_get cf mdl info Range getdata Mat stack_get cf mdl mat m_visco nomo cf 3 3 Viscoelastic device meshing tools 3 3 1 Generation of sandwich models Starting from an undamped structure without treatment you often want to gen erate models for viscoelastic patches applied to the structure This is done using MakeSandwich commands Modeling issues associated with this meshing are discussed in section Sample meshes are listed with fevisco Test For example the generation of a three layer sandwich with the original layer 0 01 thick leading to a 0 005 offset a volume of thickness 0 002 and a second 0 01 thick shell looks like model femesh testquad4 divide 10 12 model Elt feutil orient 1 n 0 O 1 model sandCom makesandwich shell 0 0O 005 volume 101 002 shell 102 005 005 treated withnode x gt 5 amp y gt 5 model fevisco sandCom model treated cf feplot cf model model fecom colordatamat You can also specify normals using a map model femesh testquad4 divide 3 4 model Elt feutil orient 1 n 0 O 1 model 59 3 Toolbox tutorial 60 sandCom makesandwich shell 0 O 005 volume 101 2 use a normal map that specify the direction of extrusion MAP feutil getnormal map node model MAP normal 1 2 model fevisco sandCom model withnode x gt
75. l H x10 c d K s S x oO E y aa x cose o o o o o Frequency Figure 2 2 Low sensitivity of the dynamic flexibility to the damping model x viscous o structural standard viscoelastic 2 1 2 Real modes and modal damping For an elastic model normal modes are solution of the eigenvalue problem see ref for more details M oj WF Kl yy Oitnxr 0 nx 2 10 associated with elastic properties sometimes called the underlying conservative problem They verify two orthogonality conditions in mass 65 IM on e6jx 2 11 and stiffness 63 M bx Puyo 2 12 27 2 Viscoelastic FEM models 28 There are different standard scaling for normal modes and one will assumed that they are scale so as to obtain uj 1 which greatly simplifies equation writing The other standard scaling often used in experimental modal analysis sets one DOF node direction of to unity u is then called the generalized mass at this DOF The basis of normal modes is classically used to build reduced model by congruent transformation 2 55 with q T qr 1 6nm qr In the resulting coor dinates called principal coordinates the mass and stiffness matrices are diagonal 2 11 2 12 conditions But this is not the case for the viscous and hysteretic damping matrices TCT and TT DT The modal damping assumption also called Basile s hypothesis in French ter minology co
76. l stack_set model info zCoef zCoef model stack_set model info Range struct val 152 3 ab par l PDY Within fe2xf w 1 10 2 pi frequencies in rad s Range stack_get model info Range getdata for jPar 1 size Range val 1 Range jPar jPar zCoef fe2xf zcoef model w Range disp zCoef some work gets done here end To use a viscoelastic material you can simply declare it using an AddMat command and use a _visc entry in the zCoefFcn column the stack name for the material and the zCoef matrix name must match cf fevisco testplateLoadMV feplot define material with a unit conversion mat m_visco convert INSI m_visco database Soundcoat DYAD609 h MatId for original name of parameter cf mdl fevisco addmat 101 cf mdl Constrained 101 mat cf Stack zCoef 4 4 _visc Third coefficient will use material with name cf Stack zCoef 4 1 fe2xf zcoef cf 500 10 4000 struct val 5 lab T At time of computation the matrix coefficient in 3 4 is found as Saca where the reference modulus is found in the cf Stack Constraint 101 pl entry The choice of E or G is based on the existence of a type E or type G entry in the mat nomo field For such materials one assumes Poisson s ratio to be real and the considered viscoelastic material
77. l with sets of point loads model demosdt Demo ubeam dofload noplot Define the desired frequencies for output model stack_set model info Freq struct ID 101 data linspace 0 10 12 fid 1 fid fopen FileName edits nas2up writefreqload fid model fprintf s n edits end Main bulk to be modified with EditBulk hf close fid Write Curve Set SetC Uset Write commands are used to WriteCurve lets you easily generate NASTRAN curve tables WriteSet lets you easily generate NASTRAN node and elements sets associated with and selection commands WriteSetcC formats the sets for use in the case control section rather than the bulk WriteUset generates DOFs sets model demosdt demogartfe fid 1 display on screen otherwise use FOPEN to open file nas2up WriteSet fid 3000 model findnode x gt 8 selections zone_1 group 1 zone_2 group 2 3 nas2up WriteSet fid 2000 model selections st nas2up WriteSet 1 2000 model selections curves curve Sine fe_curve testsin id1 linspace 0 pi 10 gt curve Exp fe_curve testexp id100 linspace 0 1 30 nas2up WriteCurve fid curves DOF feutil getdof model nas2up WriteUset U4 fid DOF 1 20 See also WritePLIL The WritePLIL is used to resolve identifier issues in MatId and ProId elements in SDT have both
78. ls 18 tion C1 T To C2 T To is often used Various papers state that C1 17 4 et Co 51 6 are realistic values for many materials with changing To but this claims seems mostly unfounded Log ar 1 17 On should also cite the wp model based on Arrhenius equation used in thermo dynamics to quantify the relation between the rate of a chemical reaction and it s temperature Ea 1 1 Loglar R 1 18 where T is the temperature in degree Kelvin R 8 314 x 10 k Jmol K is constant of perfect gas and Ea corresponds to the activation energy of the reaction This relation is less used than the WLF equation or other models uniquely based on AT but the reason is probably only linked to easiness in the determination of parameters 50 Log o 17 4AT 51 6 AT 0 14AT 34 2 Log a 25AT 89 7 AT o1 144T 45 5 A ej T ee o T Facteur de d calage en temp rature Log a N e Oo T T I 0 L L L L L L L 40 30 20 10 0 10 Ecart de temp rature AT T T C Figure 1 11 Frequency shift factor ar and reduced frequency Figure shows typical ay curves and their expression as a function of AT differ ence between T and a given reference temperature To One clearly sees that these curves mostly differ in their low temperature behavior In practical applications one can usually adjust parameters of any law to be appropriate The solution pref
79. ls 46 4 5 4b x p P ee L got a 35 z F p 2 3F 1 25 Pa yt J i a t 2 Fs J if 1 57 a 1 i L L L L L L 1 0 20 40 60 80 100 120 140 160 180 200 Frequency Hz 30 T ost Shell volume 4 Laminated plate T o 20 Z oO 15F H ear pe SES 4 10 L L L L 1 10 10 10 10 10 10 10 Figure 2 8 Free layer model validity The element degree does not seem critical to obtain accurate predictions of the response The use of multiple elements through the viscoelastic layers has also been considered by some authors but the motivation for doing so is not understood For press formed sandwiches there are further unknowns in how the forming process affects the core thickness and material properties In particular most materials used for their high damping properties are also very sensitive to static pre stress For a simple folded plate figure illustrates how the modal frequencies and energy distribution in the viscoelastic layer are modified if the shear modulus is multiplied by 10 in the fold Such behavior was found in tests and motivated the study in Ref where the effect of static pre stress is measured experimentally Overall predicting the effects press forming or folding sandwiches is still a very open issue 7 78 13 Hz 7 74 57 Hz 25 0 5 Figure 2 9 Energy density in the viscoelastic layer of a simple folded sandwich plate T
80. me or creates it if it does not exist Use option newFileName to force deletion of an existing file You can also provide a handle fid to a file that you opened with fopen fid 1 can be used to have a screen output EditBulk Supports bulk file editing Calls take the form nas2up EditBulk InFile edits Outfile where InFile and OutFile are file names and edits is a cell array with four columns giving command BeginTag EndTag and data Accepted commands are Before inserts data before the BeginTag Insert inserts data after the EndTag Remove removes a given card Warning this does not yet handle multiple line cards Set used to set parameter and assign values For example edits Set PARAM POST 2 rootname my_job f 0 OUTPUT4 sprintf s_mkekvr op4 rootname NEW 40 DELETE OUTPUT4 sprintf s_TR op4 rootname NEW 41 DELETE edits end 1 1 4 set ASSIGN f0 When writing automated solutions the edits should be stored in a stack entry info EditBulk model 112 naswrite FileName model the nominal call it writes everything possible nodes elements material properties case information boundary conditions loads etc For example naswrite 1 femesh testquad4 The following information present in model stack is supported e curves as TABLED1 cards if some curves are declared in the model Stack see f
81. model sandwich structures building higher order shell models or connecting multiple elements The main problem with the higher order element approach is that developing good shell elements is very difficult so that most developments for sandwiches will not perform as well as state of the art shell elements The multiple element strategy is also the only available for immediate implementation into industrial FEM software To properly account for shear effects in the viscoelastic layer the offsets between the neutral fiber and the shell surface are most of the time essential Rather than defin ing offsets for shell elements rigid links between the shell nodes and the volume element are used here as shown in figure Although this generates additional nodes 4 node layers for a single constrained layer model this strategy accommo dates all possible layer configurations During resolution the model is smaller since all viscoelastic volume nodes are constrained Bee S a a re Fat Shell Rigid links Sandwich shell FEM Model Figure 2 6 Problems with thickness def Figure 2 5 Shell volume shell model for initions in shells with significant curva sandwiches ture Automated layer mesh generation from a selected area of a nominal shell model is a basic need supported by the fevisco MakeSandwich commands Figure 2 6 illustrates the fact that for curved shells the use of flat elements generates a distinc tion between layer t
82. n vel struct group all 1 10 impet OH compute responses fe2xf frfzr cf The commands fe_sens br amp lab and fe_sens crklab are used to compute respectively the reduced load matrix br and the reduced observation matrix cr and their associated labels If a load is time dependent the associated curve must be linked to it before calling fe_sens br amp lab However there is no need to call again fe_sens br amp lab if the curve is changed provided that the name of the curve stays the same direct Full First zCoef0 Iter Reduced direct commands are used for direct frequency response computations It is as sumed that loads and sensors are defined using entries in the model case XF def fe2xf DirectFull model calls a direct full order complex sparse solver With no output argument the FRFs are displayed in iiplot model is a structure array that describes the impedance of the model whose response is to be computed It is typically generated using the fevisco MakeModel command which describes the fields used by fe2xf direct commands see section 3 5 DirectFirst builds a reduction basis containing nominal normal modes and a first order correction for damping effects DirectFirst zCoef0O generates zCoef0 based on the actual contents of zCoef rather than the values stored in the info zCoef stack entry 85 fe2xf DirectReduced assumes the model already con
83. nas2up editbulk mode dat edits fullfile wd fname cd wd type fname nas2up joball fname model1 cg feplot 4 mo1 nasread ubeam op2 Wop4 Matrix writing to OUTPUT4 format You provide a cell array with one matrix per row names in first column and matrix in second column The optional byte swapping argument can be used to write matrices for use on a computer with another binary format kv speye 20 ByteSwap 0 No Byte Swapping needed nas2up wop4 File op4 kv kv ByteSwap For ByteSwap you can also specify ieee le for little endian Intel PC or ieee be depending on the architecture NASTRAN will be running on You can omit speci fying ByteSwap at every run by setting setpref FEMLink OutputBinaryType ieee le WriteFreqLoad 115 naswrite 116 edits naswrite Target bdf WriteFreqLoad model or the equivalent nas2up call when the file is already open as show below writes loads defined in model and generated with Load fe_load model as a series of cards FREQ1 for load frequen cies TABLED1 for the associated curve RLOAD1 to define the loaded DOFs and DAREA for the spatial information about the load The return edits argument is the entry that can be used to insert the associated subcase information in a nominal bulk The identifiers for the loads are supposed to be defined as Case Stack j1 end ID fields Generate a mode
84. nasread FileName output4 Output4 text files are also supported with less performance and no support for non sequential access to data with the SDT v_handle object Supported options full assumes that the matrix to be read should be stored as full default sparse transpose transpose data while reading hdf save data in a hdf file Reading is performed using buffer setpref SDT OutOf for a 100MB buffer It is useful to overcome the 2GB limit on 32 bit Matlab see sdthdffor details about how to build v_handle on hdf file f06 output obsolete ASCII reading in f06 files is slow and often generates round off errors You should thus consider reading binary OUTPUT2 and OUTPUTG4 files which is now the only supported format You may try reading matrices with nasread FileName matprt tables with nasread F tabpt and real modes with vector mdof nasread filename vectortype yp Supported vectors are displacement displacement applied load vector oload and grid point stress gpstress See also FEMLink 111 4 5 naswrite Purpose Formatted ASCII output to MSC NASTRAN bulk data deck This function is part of FEMLink naswrite FileName node elt pl il Syntax i naswrite FileName command naswrite newFileName command naswrite fid command Description naswrite appends its output to the file FileNa
85. ne 74 3 7 Fluid structure coupling 75 3 7 1 Summary of theory 75 3 8 NASTRAN Generation of the parametric model 77 3 9 Advanced connection models 78 3 9 1 Screw models 00004 3 9 2 Physical point with rotations 3 10 Validation examples 2 6 3 10 1 Validity of free and constrained layer models 3 1 Viscoelastic modeling tools The viscoelastic modeling tools provide packaged solutions to address the following problems Generation of sandwich structures Handling of tabulated and analytical representations of viscoelastic constitu tive laws Parameterization of FEM models to allow multiple viscoelastic materials Approximate solutions for the viscoelastic response Vibroacoustic response generation These tools are not distributed with SDT Please contact SDTools for licensing information 3 2 Representing viscoelastic materials 3 2 1 Introducing your own nomograms You can introduce your own nomograms in the m_visco database By simply defining an mvisco_ m file mvisco_3m m serves as a prototype The data structure defines a reference elastic material in mat pl complex modulus and shift factor tables an finally additional properties stored in mat nomo which will be better documented later mat pl 1 fe_mat m_elastic SI 1 1e6 2 1 49 49 1500 1e6 mat name ISD112 1993 mat type m_visc
86. nes viscoelastic materials see AddMat frequencies zCoef reduced model range see Range as well as the case that contains descriptions for loads sensors parameters Additional fields used than may be used by reduced models are 68 br input shape matrix for reduced model per input shape matrix for reduced model 3 4 6 Response post processing options For a given reduced or full model you may want to post process the computed frequency responses before saving them This is in particular important to analyze responses on large sensor sets panel velocities stresses which would require a lot of storage space if saved at all frequencies The list XXX 3 5 Sample setup for parametric studies 3 5 1 Model parameterization fevisco handles parametric models with variable stiffness expressed as linear com binations of constant matrices 3 4 The initial step of all parametric studies is to define the parameter sets For example a selection based on ProId model fe_case model par FrontStruts k ProId 168 par Backstruts k Prold 3047 The PREDIT_mat model XXX 69 3 Toolbox tutorial Figure 3 1 The PREDIT mat model For example in the PREDIT_mat model one can define each of the 4 squares of viscoelastic materials on the pane as a parameter using ProId cf feplot model MV cf mdl XXX One can then visualize each parameter using feplot model properties windows
87. nge One defines the ratio of current stiffness to nominal Kucurrent par 1 Kv nominal as follows external to fexf zCoef Klab mCoef zCoef0 zCoefFcn M 0 A es Ke O 1 1 ixfe_def DefEta Ky 0 1 par 1 model struct K cel1 1 3 model stack_set model info zCoef zCoef model stack_set model info Range stract val 15253 iab 44 per Within fe2xf w 1 10 2 pi frequencies in rad s Range stack_get model info Range getdata for jPar 1 size Range val 1 Range jPar jPar zCoef fe2xf zcoef model w Range disp zCoef some work gets done here end 3 4 2 Parametric models zCoef The viscoelastic tools handle parametric models where various areas of the model are associated with a scalar coefficient weighting the model matrices stiffness mass damping The first step is to define a set of parameters which is used to decompose the full model matrix in a linear combination The elements are grouped in non overlapping sets indexed m and using the fact that element stiffness depend 64 linearly on the considered moduli one can represent the dynamic stiffness matrix of the parameterized structure as a linear combination of constant matrices Z Gim 5 8 M Ke X Pm Kum 3 4 By convention Ke represents the stiffness of all elements not in any other set While t
88. nly of frequency but also of other environmental factors such as temperature prestress etc The following sections discuss these factors and the representation in constitutive laws 1 3 1 Influence of temperature Temperature is the environmental factor that has the most influence on viscoelastic material characteristics At various temperatures these materials typically have four different regions shown in figure glassy transition rubberlike and fluid Depending on the considered material the operating temperature can be in any of the four regions For polymer blends each polymer can be in a different region a b c Module de stockage E et facteur de perte n T l l l l l l NI l l I l l l l l l l 1 Temp rature Figure 1 8 Evolution of complex modulus with temperature at a fixed frequency Regions a vitrous b transition c rubberlike d fluid In the first region associated with low temperatures the material in its glassy state is characterized by a storage modulus that reaches its maximum value and has low variations with temperature The loss factor is very small and diminishes with temperature Material deformations are then small The transition region is characterized by a modulus decreasing with temperature and a loss factor peaking in the middle of the region Typically the maximum corresponds to the point of maximum slope for the stor
89. nsidered in this procedure e Reduced basis computations are then performed using low level calls cont inuedoc ci XF iiplot cf Stack info Freq linspace 10 250 1024 ci Stack curve coupled fe2xf frfzr cf cf Stack zCoef 4 6 4 w 2 1e 3 1e 3 1e 3 ci Stack curve no fluid fe2xf frfzr cf iicom iixonly coupled no fluid 3 8 NASTRAN Generation of the parametric model The viscoelastic vibration toolbox is distributed with a set of NASTRAN DMAP and sample files that are used to implement some solutions steps within the NASTRAN environment The following parameters are used in those DMAP ViNset number of sets defining the element selections Vidut type of desired output O returns the full matrices of the linear combination 1 returns a first order reduced model ViSet1 ID for the first set Sets are assumed to be written in sequential order Vil0SET ID for set containing the list of nodes that need to be saved for input output calculations ViShift Frequency shift used to handle cases with rigid body modes By default the shift is the complex double precision number 10 fo_by_set The objective of this DMAP fo_by_set dmp associated with sample job file vfirst_step and the associated WriteStep12 and BuildStepi2 commands is to gen TT 3 Toolbox tutorial 78 erate a first order reduction of a model where parameters are co
90. nsists in an approximation of the response where the off diagonal terms of the damping matrices in principal coordinates are neglected In practice one even further restricts the model to viscous damping in principal coordinates since the result can then be integrated in the time domain On thus has s A s 2 jw Ao fe s I LF 9 Fa 213 where the approximation is linked to the fact of neglecting coupling terms described by Fa s 2 w 47 Cs DI lel 7 s 2 14 Damping is exactly modal Fy 0 if the matrices C and or D are linear combina tions of products of M and K C X ox M K 2 15 Rayleigh damping typically called proportional damping where C a M K 2 16 is thus often used This is a behavior model that can be easily adjust to predict correct damping levels for two modes since 65 aay to 2 17 Over a wide frequency band it is however very unrealistic since T gives high damping ratio for low frequencies w and si for high frequencies Rayleigh damping is thus really inappropriate to accurate damping studies When one starts from a local description of dissipation in the materials or interfaces modal damping is an approximation whose validity needs to be understood To do so one should analyze whether the coupling force 2 14 can be neglected To validate the domain of validity of this hypothesis one considers a two DOF sytem 1 0 Yu 12 wt 0
91. o mat unit SI mat TO 0 mat G Freq Re G Im G 1 1 72688e 004 3 51806e 003 10 2 33865e 004 5 35067e 003 57 3 Toolbox tutorial 58 100 3 49390e 004 8 25596e 003 1000 5 76323e 004 1 67974e 004 10000 1 03151e 005 5 72383e 004 1e 005 2 10295e 005 1 79910e 005 1e 006 6 59947e 005 6 57567e 005 1e 007 2 06023e 006 1 95406e 006 1e 008 5 83327e 006 3 57017e 006 1e 009 1 48629e 007 5 60247e 006 1e 010 3 25633e 007 7 33290e 006 te O011 6 16925e 007 5 40189e 006 te 012 1 01069e 008 2 48077e 006 I at at l 4 T at 10 1 32885e 007 O 9 16273e 005 10 1 14678e 005 20 2 45660e 004 30 9 00720e 003 40 2 99114e 003 50 1 27940e 003 60 7 10070e 002 70 2 88513e 002 80 1 96644e 002 90 1 37261e 002 100 1 03674e 002 110 6 84906e 001 120 4 66815e 001 is mat nomo w 1 0 12 Eeta 4 9 2 unit SI www www 3m com file ISD_112_93 png Rect 145 35 538 419 type 1G 3 2 2 Selecting a material for your application cf feplot m_visco database cf select all materials m_visco info cf m_visco deffreq cf set frequencies vector for all the material m_visco defT cf set temperature vector for all the material cf Stack info Freq logspace 3 10g10 15e3 300 range of interest cf Stack info Range 20 Temperature of interest m_visco nomo cf list with all nomograms Freq stac
92. o the exact static response to loads associated with the input shape matrix b This static correction term is well known in component mode synthesis applications and is analyzed in section It is often useful to consider a representation using residual flexibility 2NM few p wrab 2NM eal pyTab H s d Z9 b y a ae be i ZO e So tipte j 1 J j l J 2 47 2 3 2 Complex mode eigenvalue problems with constant matrices The solution of the non linear eigenvalue problem 2 42 is difficult see section Solution algorithms are thus greatly simplified by restating the problem as a classical first order eigenvalue problem with constant matrices For models with viscous and structural damping 2 2 or viscoelastic models of form 2 35 on thus generally solves the eigenvalue problem associated with 2 32 K that is C M 0 M 0 0 M 1 Oj 0 2 48 Because of the block form of this problem one can show that 0 2 49 and one thus gives the name complex mode both to 6 and wj Aj The existence of 2N eigenvectors that diagonalize the matrices of 2 48 is equivalent to the verification of two orthonormality conditions al 7 A 6 p AT MY TM 0 2 50 al aa 9 r Kp AdTMYA Pal For a model represented in is state space form such as 2 34 one solves the left 37 2 Viscoelastic FEM models 38 and right eigenvalue problems E A 9p 0 et 0j a E A
93. oelastic layer for forced response at a selected number of frequencies StrainMap This is an experimental command to compute membrane strains with an offset to a given shell surface This result can be used for placement 97 fevisco 98 smap fevisco StrainMap OffSet model def SurfaceSelection fevisco StrainPlot smap def Test A few test cases are provided with fevisco to allow example documentation TestPlate generates the model of a square plate with partial coverage using con strained layer damping on one side and free layer damping on the other side TestCantilever generates the model of a cantilevered constrained layer damping treatment WriteInclude fevisco WriteInclude Selection model FileName is used to generate a NAS TRAN bulk with name FileName containing e the associated nodes e the elements selected by Selection e the rigid links and other case information connected to the nodes used by these elements e the material and element property information used by the selected elements This command is used to write the bulk for sandwich structures generated by MakeSandwich commands Usually specific MatId are used to identify materials for the sandwich so that the selection is simply a list of material identifiers for example WriteInclude MatId 1001 1002 When called with a selection an at tempt to select the case information associated with the nodes of the selection is
94. of strain energy maps for two levels of refinement When considering free placement of damping devices see section one is rapidly faced with the problem of incompatible meshes For discrete connections where loads are transmitted at isolated points with at most one point on a given element of the supporting structure the problem is very much related to that of the repre sentation of weld spots and strategies that use the underlying shape functions are most effective see SDT feutilb MpcFromMatch command 2 5 Thermal considerations In this section one addresses models needed to evaluate the steady state temperature field for a forced harmonic response 2 5 1 Thermal model xxx detail xxx Exchange on free structure interfaces Exchange at the internal viscoelastic metal interfaces 2 5 2 Heat source due to viscoelastic behavior Assuming a forced harmonic response one has X t R e X w e 0 X t R 0 X w e 2 65 where stress is related to strain through the complex modulus o X w A X w e X w 2 66 or in the time domain o X t R A w e w e 2 67 Integrating the power dissipated over a period of the forced response one obtains the spatial distribution of dissipated power p X w fo o X t X t dt for R A e etot T R iwe X w e t 2 68 r S 6 TS A S 6 R E S A R C Note that the maximum strain energy during the cycle is given by e X w marg e X t R A e X t
95. ons 1 2 2 Simple rheological models 1 2 3 High order rational models 0 2 1 2 4 Fractional derivative models tam oir Sh soi 4s Bah sata Be Ie dh 1 3 1 Influence of temperature 1 3 2 Other environment factors 1 4 Determining the complex modulus 1 5 Conclusion 20 22 20 ee ee eee OND DH 1 Modeling viscoelastic materials 1 1 Introduction Linear viscoelasticity assumes that stress is a function of strain history This translates into the existence of a relaxation function h t given by co o t elt r h r dr 1 1 0 Using Laplace transform one sees that this hypothesis is equivalent to the existence of a complex modulus A s transform of h t such that a s A s e s A s iA s e s 1 2 From a practical point of view one can solve viscoelasticity problems as elasticity problems with a complex modulus that depends on frequency This property is known as the elastic viscoelatic equivalence principle For a strain tensor the number of independent coefficients in A is identical to that in Hookes law for an elastic material for the same reasons of material invariance For homogeneous and isotropic materials one thus considers a Young s modulus and a Poisson coefficient that are complex and frequency dependent The separate measurement of E w et v w is however a significant experimental challenge
96. op High stiffness viscoelastic in the fold Bottom equal stiffness in the fold and elsewhere A final difficulty is to deal properly with boundary conditions of the skin layers Since differential motion of the skins plays a major role in the effectiveness of the core the boundary conditions of each layer has to be considered separately This is easily illustrated by the generation of cuts in constraining layers and cut_optim demo 2 4 1 Mesh convergence and non conformity As illustrated in figure the dissipation if often localized on a fairly small sub part of the structure It is thus quite important to validate the accuracy of predictions obtained with various mesh refinements Figure 2 10 illustrates a convergence study where a constrained layer damping treatment is refined and one compares the strain energy density maps for two levels of refinement The strain energy maps clearly indicate edge effects which are typical of constrained layer treatments In such studies the author s have usually found that the distribution of constraints is well predicted and energy fractions strain energy in the viscoelastic compared to total strain energy in the model predicted with the fine and coarse meshes do not show AT 2 Viscoelastic FEM models 48 significant differences 42d 1466 Hz 4 2d 1466 Hz Figure 2 10 Zoom on the refined mesh of a constrained layer damping treatment placed on a volume model Comparisons
97. or the format e Fixed DOFs as SPC1 cards if the model case contains FixDof and or KeepDof entries FixDof AutoSPC is ignored if it exists e Multiple point constraints as MPC cards if the model case contains MPC entries e coordinate systems as CORDi cards if model bas is defined see for the format The obsolete call naswrite FileName node elt pl il is still supported node elt You can also write nodes and elements using the low level calls but this will not allow fixes in material element property numbers or writing of case information femesh reset femesh testquad4 fid 1 fid fopen FileName naswrite fid node FEnode naswrite fid node FEnode hf close fid Note that node 4 which is a group identifier in SDT is written as the SEID in NASTRAN This may cause problems when writing models from translated from other FEM codes Just use model Node 4 0 in such cases dmig DMIG writing is supported through calls of the form naswrite fid dmigwrite NAME mat mdof For example naswrite 1 dmigwrite KAAT rand 3 1 3 01 A nastran dmig entry in model Stack where the data is a cell array where each row gives name DOF and matrix will also be written You can then add these matrices to your model by adding cards of the form K2GG KAAT to you NASTRAN case 113 naswrite job NASTRAN job handling on a remote server from the MATLAB command line
98. pied back Here is a complete example of this procedure cd sdtdef tempdir if isunix return end Don t test on windows wd getpref FEMLink DmapDir fullfile fileparts which nasread dme copyfile fullfile wd fo dmp copyfile fullfile wd sdt dmp Irm ubeam_step12 MASTER ubeam_step12 DBALL ubeam_ 0 9 model demosdt demoubeam cf feplot model fe_case model dofload Input struct DOP 349 01 360 01 241 01 365 03 det 1 151 lt 1 71D 100 model fe_case model par Top withnode zai P model fe_case model sensdof Sensors 360 01 cf Stack info EigOpt 5 20 1e3 11 model nas2up JobOpt model Init NasJobOpt entry to its default cf Stack info Freq 20 2 150 fevisco writeStep12 write run ubeam mode1 fecom save ubeam_param mat save before MVR is built you may sometimes need to quit Matlab here if NASTRAN is long cf feplot Load ubeam_param mat reload if you quit matlab fevisco BuildStep12 ubeam cf fecom save ubeam_param mat save after MVR is built When dealing with a model that would have been treated through a SOL108 full order direct frequency response the parametric model returns a B matrix This is actually related to the Ke and Ky matrices by B NglobKe T 5 Ninom Kyi 3 5
99. ply call the DirectReduced command Up fevisco testplate upreset Up stack_rm Up mat Up fevisco addmat 101 Up First area ISD112 1993 Up fevisco addmat 103 Up Second area ISD112 1993 def fe_eig Up 6 10 1e3 MVR fevisco makemodel matid 101 103 Up def MVR stack_set MVR info Freq 20 5 90 RESP fe2xf directreduced MVR Result in XF 1 3 6 Sample setup for parametric studies 3 6 1 Performance in modulus loss plane See d_visco ScriptEhPerf 3 7 Fluid structure coupling This section illustrates the use of Fluid commands 3 7 1 Summary of theory One considers fluid structure interaction problems of the form lala E FIT with q the displacements of the structure p the pressure variations in the fluid and F the external load applied to the structure where Jog Tij UW eij Ou da gt dq Kq Jos psu dudx gt dq Mq on Jap VpVSpda dp Fp 3 8 oe Jor popdx gt dp Kpp Jy pou ndx gt q Cp Full order equations for the coupled problem are impractical One thus considers model reduction for both the solid and fluid parts of the model Given a reduction basis T for the structure one builds a reduction basis containing fluid modes within the bandwidth of interest and static corrections for the effects of vectors in Ts Thus the resulting basis for the fluid model is TA paneer FY CT r 3 9 Sim
100. posed in r r0 A 1p Ko brna 2 58 2 3 4 Equivalent viscous damping Section showed that the modal damping assumption was justified under the hypothesis of modal separation 2 22 When dealing with a real basis it is of ten possible to compute an equivalent viscous damping model approximating the damped response with good accuracy To build this equivalence one distinguishes in reduction bases 2 56 2 57 or other blocs Tm associated with modes whose resonance is within the frequency band of interest and T associated with residual flexibility terms As shown in figure one is interested in the modal contribution of the first while for the others only the asymptotic contribution counts exact nor stat exact residual Stat retained modes F_max Figure 2 4 Transfer function decomposition into modal contributions and residual terms For a real basis reduction one is thus interested in an approximation of the form TpZ s Tm Tr Z s Tr ir H s eT 2 59 TE Z s Tm TT Zs i The objective in distinguishing Tm and T is to guarantee that T Z s T does not present singularities within the band of interest it represents residual flexibility not resonances To validate this hypotheses one defines a reference elastic problem characterized by a stiffness Ko one will use Ko Re Z wo Mwg avec wo with typically better results for a higher frequency value By computing modes in a subspace gen
101. re used to connect points of the parametric study For an example see fe2xf PoleRange 87 fe2xf 88 zCoef This command is used to generate the weighting coefficients in 3 4 more details are given in section 3 4 2 model fe2xf zcoef default model is used to analyze the model and define a default info zCoef entry fe2xf zcoef model displays the values If a parameter range is defined you can specify the parameter to use with the jpar i option and all parameters with jpar 1 DefList When reading shapes for the purpose of generating reduced models def fe2xf DefList roo reads all files matching root mat and combines the shapes in a def def field Selection of diameters and frequency range during the read process is peformed using RO struct Fmax 8000 di fe_cyclicb DefList root mat RO Optional argument RO can have following fields e Fmax defines maximal frequency of retained vectors e Fmin defines minimal frequency of retained vectors e First6RB defines upper frequency tolerance of rigid body modes Then only the first 6 occurences of rigid bodies are kept and next removed Build fe2xf build cf RO This command can be used to build reduced model MVR cf is a handle to the feplot figure that contains the initial model Reduced model built is stored in cf Stack SE MVR RO is a data structure with fields defining the option
102. rrespond to viscoelas tic materials see section 3 4 2 You can also parameterize elastic materials using gt MakeModel MatId 7 where the additional elastic materials are given by ti The example given in section 3 5 4 uses commands fevisco testplate up cf feplot Up cf mdl This is a simple test case Up stack_rm Up mat Up fevisco addmat 101 Up First area ISD112 1993 Up fevisco addmat 103 Up Second area ISD112 1993 MV fevisco makemodel matid 101 103 Up If you have a reduced basis in def MVR fevisco makemodel matid 101 103 Up de is used to generate a reduced model See section 3 5 4 The resulting data structure MV contains all the stack entries needed for viscoelastic response computations see section 3 4 5 parameters defined in the case stack and viscoelastic materials defined as mat entries and as elastic materials in MV p1 this is needed to define the reference modulus value the selection of a reference E or G is based on the content of mat nomo 93 fevisco Write Build Step12 This command is used to generate parameterized reduced models in one NASTRAN run as detailed in section 3 8 From the command line you can use fevisco WriteStep12 run cf md1 to start the job and fevisco BuildStep12 cf md1 to build the parametric model With the command option write writestep12 command writes the bulk model before
103. s e list cell array that defines reduction operations see below One can also rebuild MVR from existing def providing RO list as a cell array of file fname strings or cell array of def data structures e Range is an alternative way to define the list of reduction operation As many mode computations according to options defined in the RO EigOpt field as rows in RO Range val are performed See for how to define a range data structure field val with as many rows as design points and as many columns as matrices field lab is a cell array with string labels for each column e EigOpt defines mode computation options see fe_eig e Save contains 1 if mode computation results are to be saved during the re duction process If 2 the MVR is not built and modes are stored for delayed build call if 0 or absent modes are not saved e Root is a string that begins the filenames where modes are saved if RO Save 1 or 2 Basis building calls RO list contains a cell array with entries e cig EigOpt entries in first column and the coefficients to be used to generate the current stiffness in the second column e first entries in first column to generate a first order correction based on the last eigenvalue problem solved and stiffness defined by coefficients given in the second column e file fname reads subspace from def def in fname This can be used in con junction with a first run where data is saved using RO Root fullfil
104. s one uses values determined as design criteria C 107 fora pure metallic component 107 for an assembled metallic structure a few percent in the medium frequency range or a civil engineering structure Each industry typically has rules for how to set these values for example for nuclear plants 2 2 Viscoelastic models This section details models of structures used to account more precisely for the constitutive behavior of various viscoelastic materials For frequency response computations section shows how the complex dynamic stiffness is built as a weighted sum of constant matrices associated with the various materials For eigenvalue computations or time responses on the full model the introduction of state space models section 2 2 2 or second order models with internal states section 2 2 3 allow constant matrix computations Such formulations could be used for frequency domain solutions but they are higher order and the increase in DOF count limits their usefulness For the case of fractional derivative models section 2 2 4 only modeshape computations are accessible 31 2 Viscoelastic FEM models 32 2 2 1 Frequency domain representation with variable coefficients For a structure composed of elastic and viscoelastic materials frequency domain computations only require the knowledge of the complex modulus F s T o9 for each material By using the fact that stiffness matrices depend linearly on the constitutiv
105. s you want to see use the delete key or the Remove material context menu to remove entries You can also add materials to the stack manually with m_visco database unit TM Densil cf 100 m_visco database unit TM Smactane 50_G cf By default the maximum loss factor and associated modulus and frequency are displayed for the material reference temperature You can order the list by giving a target InfoTarg freq temp sort frequency temperature sort by modulus 1 or loss factor 2 Thus gt gt m_visco info targ30 20 2 CorningGlass0010 G 2 87e 009 Pa e 0 02 3 00e 001 Hz 20 C ChicagoVitreous CV325 E 1 03e 008 Pa e 0 06 3 00e 001 Hz 20 C Soundcoat DYAD609 G 6 21e 008 Pa e 0 08 3 00e 001 Hz 20 C BarryControl H 326 G 5 90e 006 Pa e 0 08 3 00e 001 Hz 20 C PolyIsoButene E 1 15e 006 Pa e 0 11 3 00e 001 Hz 20 C LordCorp BTR G 6 78e 006 Pa e 0 15 3 00e 001 Hz 20 C Densil G 2 67e 007 Pa e 0 22 3 00e 001 Hz 20 C Soundcoat DYAD606 G 5 16e 008 Pa e 0 24 3 00e 001 Hz 20 C Smactane 50_2 G 8 21e 006 Pa e 0 41 3 00e 001 Hz 20 C Smactane 70 7 4 46e 007 Pa e 0 44 3 00e 001 Hz 20 C BI2F15_Jul03 4 91e 006 Pa e 0 44 3 00e 001 Hz 20 C ISD110 G 3 21e 007 Pa e 0 51 3 00e 001 Hz 20 C Smactane 50_1 E 1 52e 006 Pa e 0 52 3 00e 001 Hz 20 C T9D113 G 5 18e 004 Pa e 0 60 3 00e 001 Hz 20 C 3M 468 G 3 95e 005 Pa e 0 65 3 00e 001 Hz 20 C LordCorp LD 400 G 4 35e 008 Pa e 0 70 3 00e 001 Hz 20 C Soundcoat DYA
106. scoelastic solid Maxwell s model is composed of a spring and a dashpot in series Each element carries the same load while strains are added The stress strain relation is in the frequency domain written as TEE 7 where T 1 5 is a settling time that is characteristic of the model and is associated to a pole at w E C 1 r The viscous model called Kelvin Voigt Model is composed of a spring and a dashpot in parallel Each element undergoes the same elongation but strains add Damp ing is then represented by the addition of strains proportional to deformation and deformation velocity In the frequency domain the modulus has the form E s Eo 1 Bs 1 6 Structural damping also called hysteretic damping is introduced by considering a complex stiffness that is frequency independent The complex modulus is thus 1 Modeling viscoelastic materials 10 characterized by constant storage moduli and loss factors E s Eo 1 jn 1 7 The standard viscoelastic solid has a constitutive law given by 1 1 t Fok Eo E Cis 1 s z o 0 m hta Je E ities as leading to a time domain representation E o t Ci a t Foe E Ce 1 9 Figure 1 4 shows the frequency domain properties of these models Maxwell s model is only valid in the high frequency range since its static stiffness is zero The viscous damping model is unrealistic because the loss factor goes to infinity at high frequency there
107. skin modulus For a very soft value figure shows convergence to the asymptotic value of a single skin plate For a modulus of the viscoelastic core equal to that of the skins one should converge to the frequency of a plate model with thickness equal to the sum of skins viscoelastic core Figure 2 7 shows that the high modulus asymptote is slightly higher for the shell volume shell model This is due to shear locking in the very thin volume layer well known and documented problem that low order volumes cannot represent bending properly This difficulty can be limited by using volume element with shear locking protection The figure also shows that damping is optimal somewhere between the low and high modulus values 2 5 A HHH Ft ant Pry Re 4 f F L 2p E T ae ue 1 _ o b amp pi 5 t F A 1 5F a al 1 L 1 L i 0 50 100 150 200 250 300 Frequency Hz 50 45 gt J 40H 4 N amp g 357 J 2 U 30 F J 25 J 20 Figure 2 7 Constrained layer model validity Figure illustrates the validity of a shell volume model as compared to a single shell based on composite shell theory Figure illustrates that the results are nearly identical provided that volume elements with proper shear locking protection are used For a standard isoparametric volume a shell volume model tends to be be to stiff shear locking associated with bending 45 2 Viscoelastic FEM mode
108. ssible and can be used to illustrate the difference in needs when modeling vibration damping in a structure and dissipation in a material Section 2 2 details strategies that can be used to create models with realistic repre sentation of viscoelastic materials Section 2 3 deals with issues of spectral decompositions and model reduction which are central to the analysis of the vibratory behavior of damped structures The objective of this chapter is to introduce equations used to solve viscoelastic problems The next chapter will focus on numerical techniques used for the resolu tion 2 1 Viscous and structural damping In this section one analyses the properties of classical models with viscous and structural damping of the form Ms Cs K iD yyy 4 8 ba Z 8 9 5 blyx na 4 bax y s Nsx1 lelusxn 19 8 x1 2 2 where matrices are assumed to be constant This type of models does not allow a correct representation of the local behavior of damping treatments constitutive laws detailed in the preceding chapter At the level of a complete structure it is however often possible to represent the effect of various damping mechanisms by a viscous or structural model One then uses a global behavior model It does not necessarily have a local mechanical meaning but this does not lower its usefulness The main results introduced in this section are e properties of the damped oscillator e the conditions of val
109. starting job For in MATLAB operation you can use the fexf DirectFirst command Range Vect Face Grid CubeEdge rand Range commands generate standard experiments series of design points that can then be used to evaluate model properties at these points These are defined based on the upcom parameter matrix obtained with upcom parcoefpar where each row describes the acceptable range of a parameter type cur min max vtype type is not used here and vtype 2 indicates a logarithmic variation The output is an hcube data structure with fields val matrix where each rows gives values of all parameters at a particular design point param indices of the parameters that actually change during the experiment edge connectivity matrix used to define lines connecting different design points of the experiment Figure 4 1 Positions of exact mode evaluations in parameter space a Hypercube face center b Classical 2 factorial plan 94 fevisco Rangeface 1 2 par indp generates a design points at the orthogonal projection of the nominal point given by par 2 on the lower and or upper faces defined by the parameter range defined by par 3 4 ParFace 1 only generates points of faces with minimum parameter values Face 0 is the nominal point The optional third argument indp is used to enforce variations on a subset of parameters fevisco Rangegrid opt Up indp generates a uniform grid by dividing the range
110. t groups the reasoning developed above remains valid and the error induced by neglecting damping coupling terms is small 29 2 Viscoelastic FEM models 30 2 1 3 Selection of modal damping coefficients Modal Strain Energy method The Modal Strain Energy method MSE is a classical approximation base on the choice of an equivalent viscous damping coefficient chosen by evaluating the loss factor for a cycle of forced response along a particular real mode shape As we saw for the particular case of section 2 1 1 this is an appropriate choice since it leads to a near perfect superposition of the transfer function for an isolated mode A general approach that can also be considered for a weakly non linear system is to compute the ratio called loss factor of energy dissipated over an enforced motion cycle q t cos wt divided by 27 times the maximum elastic energy associated to that deformation a T W 7 i T 9i K 4 and to impose at each resonance frequency wj of the elastic problem the equality of this loss factor with that of the model with modal damping 2 26 my wy g W 2 24 For a model where a viscous and or structural damping model is associated with each component element m the loss factor of mode j is thus obtained as a weighted sum of loss factors in each component j Ze Lo DI oj w 0 CI 0 J mMm Eim 07 KI 4 For non linearities there exists classical results of equivalent damping
111. tains a reduced model as illustrated in section 3 5 4 These commands are illustrated in section 3 5 2 FrfPoleSearch xf po fe2xf frfpolesearch rmodel w ind This low level command is used given a reduced model to track poles in the same range The reduced model uses the standard fields used to describe parametric models see section 3 4 5 It must at least contain the following fields Range cr br zCoefFcn K The pole tracking algorithm assumes that the modeshape is well approximated by the reduction vector with the same index PoleTemp Interpolation based search for dependence of poles on temperature the FrfPoleSearch is a slower alternative xxx example needed xxx RO struct ind 1 10 selected modes for output Temp 50 110 temperatures Freq logspace 10g10 1000 10g10 20e3 30 target range of frequencie Po fe2xf poletemp MV2 Visco RO hist fe2xf PoleRange MVR ind fe2xf plotpolesearch hist PoleRange This command supports a parametric study on the poles of frequency invariant damped model Variable coefficients must be defined by a zCoes stack entry The mass column in particular should be filled correctly and matrix types should be found in model Opt 2 so that the stiffness can be found by setting to zero coefficients of mass matrices in zCoef Accepted options are e Real specifies that real and not complex modes should
112. ters hcube fevisco RangeFace 1 2 par hcube val MakeSandwich layers Tools for the generation of multi layer sandwich models fevisco makesandwich layer generation FEnode FEel0 treated MAPN 95 fevisco For each layer the makesandwich command specifies e element nature shell or volume e desired material property for the meshed layer for the starting layer use 0 e thickness of volumes and distances of faces to neutral fiber for shells For shells one has two distances from the bottom layer to the neutral axis and from the neutral axis to the top layer The supporting model can be specified by its nodes and elements as shown above or using a model data structure Treated is an optional FindNode command that allows generation of a sandwich for a part of the original model only original surface i SHELL SHELL 0 102 element normal i Figure 4 2 Example of a MakeSandwich command For example the generation of a three layer sandwich with the original layer 0 01 thick leading to a 0 005 offset a volume of thickness 0 002 and a second 0 01 thick shell looks like femesh reset femesh testquad4 FEel0 feutil orient 1 n 0 0 1 FEnode FEel10 96 femesh divide linspace 0 1 10 linspace 0 1 12 sandCom makesandwich shell O O 005 volume 101 002 shell 102 005 0C treated withnode x gt 5 amp y gt 5 FEnode FFelt fevisco sandCom FEnode FEel0 treat
113. th rational fractions one can fairly easily satisfy the first objective reproduce the modulus in a narrow band in bold but extrapo lations can be difficult thin lines are here of poor quality 10 Module de Stockage Pa Facteur de perte Figure 1 12 Complex modulus of the TA viscoelastic and estimation with a 3 pole model whose frequencies are indicated as vertical lines On the other hand one should note that the determination of the complex modulus is often difficult and that the frequency temperature superposition principle is an hypothesis It just happens to be reasonably valid in many cases In conclusion for the material stand point accurate representations are either tab ulated our point of view is that this is actually the most practical representation rational fractions with a sufficient degree or fractional derivative models The fol lowing chapters will address difficulties in using these representations for modeling damped structures 21 1 Modeling viscoelastic materials 22 Viscoelastic FEM models O a aii 2 1 Viscous and structural damping 24 2 1 1 Properties of the damped 1 DOF oscillator 25 poner 27 vier 30 2 2 Viscoelastic models 4 2e0e 31 eficients es ek a ew ae cts By i Roa we ae 32 2 2 3 Second order models with internal states 34 2 2 4 Fractional derivatives 35 L 35 Soe
114. tial including internal states The firs row in equation indeed corresponds to the representation of stress in viscoleastic materials in the form o Ene q Evo _ qvi 2 37 i In practice the qy are only non zero for viscoelastic elements The direct use of matrices assembled following must thus be done with solvers capable of elim inating unused DOFs When using reduced models the problem does not normally occur since reduced basis vectors are typically non zero over the whole structure and thus lead to none zero internal states In the time domain this formalism is more easily dealt with because the time evo lution of internal states can easily be computed by time integration of the relation between the qui and q For a model of form 2 33 the evolution of the internal state is thus given by dit w Gi E a 2 38 which can be easily integrated this approach is used in ABAQUS for example This formalism corresponds to the separate treatment of bloc rows in or 2 35 which is simple in the time domain but leads to a non linear problem in the frequency domain unless operators are defined implicitly as in Ref 2 2 4 Fractional derivatives The internal state formalism can also be used to represent fractional derivative constitutive laws if one uses non integer but rational derivatives For a common denominator p one will use a modulus of the form p Ek E s Emaz 2 39 s eae 2 39 and build a st
115. ull def fe2xf directfull MV cf def def NEED REVISE iiplot Rfull Given the reduced model MVR you can then track poles through your parametric range using MVR Range 0 10 50 Hist fe2xf frfpolesearch MVR fe2xf plotpolesearch Hist Generate standard plot of result 3 5 3 Parametric model generated within NASTRAN fo_by_set DMAP The following gives a simple example of a beam separated in two parts The step12 calls run NASTRAN with the fo_by_set DMAP where an eigenvalue computation is used to generate a modal basis that is then enriched so called step 1 before generating a parametric reduced model step 2 The steps of the procedure are the following 71 3 Toolbox tutorial 72 load the initial model into MATLAB using a nasread command Generate throughnaswrite commands or manual editing a RootName_bulk bdf file that contains bulk data information EIGRL and and possibly PARAM cards NOTE elements that will be parameterized should have the loss factor of their material set to zero Define element groups associated with parameters see section 3 5 1 These can be easily checked using feplot open the Edit Model properties menu and go to the Case tab cf sel selection 1 2 commands Once the job is written elements sets will be written in a parameter_sets bdf file using nas2up WriteSetC commands Sensors using SensDof case entries see section 3 4 4 Generate the RootNam
116. ulus A number of methods and a few standards exist for the experimental determi nation of the complex modulus Only the main test categories will be listed here e traction compression tests under sinusoidal excitation are used to measure the properties of materials that are sufficiently stiff to allow testing without com bination of the material sample with metallic components With significant experimental precautions this technique has also been applied to films De pending on the experimental setup one will determine the complex modulus 19 1 Modeling viscoelastic materials 20 directly on isofrequencies or isotherms e Oberst and sandwich beams are used to determine the properties of a viscoelastic layer glued onto a metallic beam Stiff materials work in traction compression in a free layer while soft ones work in shear in a metal visco metal sandwich Vibration analysis of the beam gives resonance frequencies and as sociated damping ratio By inverse analysis of analytical or numerical solu tions one determines the complex modulus at the resonances under the modal damping assumption see section 2 1 3 Using changes in the beam plate di mensions and temperatures one can obtain a large number of points on the nomogram e shear tests allow the direct determination of the complex modulus of films between two rigid surfaces This test is more representative of material solici tation for sandwich structures It
117. y analyses For transient analyses the presence of high frequency undamped modes linked to the wjr induces non physical oscillations since these modes are introduced to approximate low frequency contributions and not high frequency resonances It is thus good practice to introduce a significant modal damping for residual modes For example one uses 1 2 which leads to assume TF AZ iw Tr 8 VZwn 2 64 2 3 5 Case of viscoelastic models The reduction is also applicable for viscoelastic models detailed in section In deed all matrices used in the formulation of the dynamic stiffness can be projected State space or second order representations can be generated by replac ing each matrix by its reduced version M by TT MT etc This reduction form will be used for eigenvalue solvers discussed in section The equivalent viscous damping model building strategy detailed in the previous sec tion cannot be generalized since the real part of T AZ s T undergoes significant variations as the storage modulus changes with frequency In other terms the wm associated with Ko differ possibly significantly for frequencies of the non linear eigenvalue problem Re K w ws M 0 For modal synthesis or tran sient response computations one will thus prefer representations associated with the spectral decomposition as detailed in section section 2 4 Meshing of sandwich models Two main strategies have been considered to
118. y bed he BO ae 57 ine GP oe ee ak ee R 57 3 2 1 Introducing your own nomograms 57 aorageaoan 58 Se ae eee ee Boe E 59 3 3 1 Generation of sandwich models 0 59 Bohne aea ac est Stas crags eee Oe Se tha 60 race 61 a e aa a REL EE e Be 61 3 4 1 Parametric models zCoef o oaoa aaa 61 E ey Ges eae ea at 64 3 4 3 Input definitions 2 2 2 0 00 0 ee ee 67 3 4 4 Sensor definitions 2 ee 67 3 4 5 Format reference o oo 68 hae Gaede oe Ac 69 Pace Boe od Gee ee ee 69 3 0 1 Model parameterization 0 00002 eee 69 Lee Gu aREGS Bae es 70 3 5 3 Parametric model generated within NASTRAN fo_by_set DMAP 71 3 0 4 Parametric model from NASTRAN element matrices 74 ee ee eee 74 3 6 1 Performance in modulus loss plane 74 3 7 Fluid structure coupling ooo el 75 3 7 1 Summary of theory oaa 00502 ee 75 3 8 NASTRAN Generation of the parametric model aoaaa 77 3 9 Advanced connection models 2 0002 2a 78 3 9 1 Screw models 0 0 0 ee ee 78 Pep Hott hak he ee 78 eee See BAR ewe He Seats So es ee eS 78 PODR 79 4 Toolbox Reference 81 ee ee ee ee ee 82 ER E nee ee A 91 POSERTE 100 E E oh te amp Oe he 107 dees ee eG Se 112 117 CONTENTS Modeling viscoelastic materials 1 1 Introduction 2 2 228 1 2 Representing complex modulus 1 2 1 Non parametric tabular representati

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