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Kinematics, Dynamics, and Design of Machinery, 2 Ed.

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1. 3 1 is where the user specifies the cam follower type and motion specification conditions Fig 3 1 Window for specifying the follower type and motion specification This first screen allows the program s user to design the cam follower system and to specify the follower s motion program The direction of the cam s rotation cam base circle radius follower type and follower parameters may all be input here From a pull down menu shown in Fig 3 2 AQ the follower type can be chosen from the four available The follower motion can be specified segment by segment in the box in the center of the screen shown in Fig 3 1 A follower motion type can be specified for each segment from the pull down menu shown in Fig 3 3 In the specification window in Fig 3 1 beta gives the duration of the cam rotation corresponding to the type of displacement chosen A continuous rotation cam is assumed Therefore theta is assumed to start at 0 and to end at 360 The input angles are in degrees Note that any of the values beta Start and End may be input where Start is the initial angle for the cam rotation and End is the end value for the range being considered Once a value is input the values that can be computed are computed Similarly the follower deflection is assumed to start and stop at zero For any given segment either the ending value can be given or the total deflection Again after eac
2. A 4 Relationship Among IC Centrodes IC Tangent and IC Velocity The relative motion between two rigid bodies is equivalent to two curves called centrodes rolling on each other as discussed in Chapter 2 One centrode is fixed to one body and the second is fixed to the other body This is represented in Fig A 13 for the coupler of a four bar linkage The point of contact is the instant center and the centrodes are the paths of the instant centers on the two bodies The instant center IC tangent is the common tangent to the two centrodes The IC velocity is the instantaneous velocity with which the IC shifts it is along the IC tangent Note that the point that has the IC velocity will belong to neither of the rigid bodies being considered Relative to the two bodies the IC is at a different location for each relative position of the two bodies This situation is shown in Fig A 14 In that figure the instant center I 3 is in a different location relative to links 1 and 3 for each position of the linkage The path of the instant center is defined by the path of point Is relative to the frame where link 5 is the ball captured between the two yokes in Fig A 14 This path will be the fixed centrode For any instantaneous position the location of point Z5 coincides with the instant center 3 and the velocity of Z5 is the IC velocity discussed above Moving centrode IC Tangent fixed to Link 3 Fixed Centrode fixed to Link 1 Fi
3. An example is shown in Fig 2 26 The variables that must be input are shown in a figure if the Definition button is selected The resulting figure is shown in Fig 2 27 All of the push buttons are the same as in the four bar routine The capability of moving the coupler point continuously using mouse dragging is also included in this program 2 6 2 The Analysis Window for Slider Crank Program The analysis window is the same as the previous cases An example is shown in Fig 2 28 Either 1 2 3 or 4 figures can be plotted The animation must be stopped before the figures can be changed To resume the animation select the Start button To change the linkage design select the return button and return to the design window Any of the linkage parameters can be changed before returning the analysis window As in the case of the four bar linkage program the coupler curve for the slider crank program is shown dashed This feature allows the user to visualize the relative speed of the coupler point as it moves along the coupler curve 99 Fig 2 25 Design window for the slider crank program slider driving 22 9424 Fig 2 26 Design window with second assembly mode me Fig 2 27 Window showing variable definitions for slider crank analysis program Fad iW Fig 2 28 Analysis window for the slider crank analysis program 2 7 Program for Analyzing a Stevenson s Six Bar Linkage SixBarAnalysis This routine analyz
4. For example as was shown in Section 4 2 5 the points in a lamina that lie on a straight line in three specified positions of that lamina lie on a circle The corresponding result when the positions become infinitesimally separated is that at any instant in the motion of a lamina the points whose paths have inflections that is the points whose paths are locally straight lie on a circle called the inflection circle The inflection circle passes through the instantaneous center and is tangent to the same line as the fixed and moving centrodes which are the loci of the successive positions of the instantaneous center relative to the fixed and moving reference frames The pole triangle collapses into the instantaneous center A 2 Two Infinitesimally Separated Positions Specifying two design positions infinitesimally separated from one another is equivalent to specifying a position of a lamina and the velocity state of the lamina as it moves through that position The velocity state can be specified by specifying the velocity vo of the point O in the moving lamina that is instantaneously coincident with the origin together with the angular velocity of the lamina The velocity of any other point A is then given by VA Vo VaAi0 Vo WX FAO A 1 where rx is the directed line OA Let us choose any point C in the moving lamina as a circle point We seek a crank with circle point at C such that the path of C produced by that crank is tangent to
5. S 120 100 rocker angle deg 400 500 600 700 input angle degree o T A 2 3 20 1 g 10 o 3 E 19 8 3 9 20 400 500 600 700 400 500 600 700 input angle degree input angle degree No of 4 Fig mechanism Fg2 _rockeranag 1 start _5top Speed g 4 Fg3 1ockervel Fid4 _ rocker ace f Return Fig 2 36 Analysis window for four bar linkage coupler point atlas program 2 9 Program for Generating Atlas of Coupler Curves for Slider Crank Mechanism HRSliderCrankDesign This routine is to generate the coupler curves for a slider crank mechanism The program is called HRSliderCrankDesign after Hrones and Nelson who developed an atlas of coupler curves A uniform grid of coupler points is assumed for the coupler and the user can choose one point for analysis by following a sequence of selecting grid dimension grid density and row and column numbers The program uses three windows The first is a design window where the linkage and coupler point grid is defined The next is an animation window that displays the coupler curves for the points identified in the analysis window One of the coupler points can be selected for further analysis The third window is the analysis window for the mechanism with the single coupler point that is selected 2 9 1 The Design Window for Slider Crank Coupler Point Analysis Program The design window is shown in Fig 2 37 In the design window
6. Selecting the zoom region Fig 4 7 Figure after zoom option 4 3 Rigid Body Guidance Using a Crank Slider Linkage RBGCrankSliderDesign This routine is used for the design of slider crank linkages with the crank as the driver The center point circle point and slider point are the inputs The user can specify three coupler positions and angles The pole locations between each two coupler positions are calculated and shown as a marker in the graph The circle of sliders corresponding to the three image poles for position 1 is shown in black After all of the input data are provided the linkage can be animated to determine if it moves through all of the positions identified 53 The program is structured in two windows The first window is the design window where all of the input data are identified The second window is the animation window where the linkage can be verified The need to distinguish between a crank slider and a slider crank is due to the rectification process The forbidden regions for the slider point and the circle point for the crank are different for the two cases and the two cases are treated in separate subprograms The design window for the case when the crank is the driver is shown in Fig 4 8 In the design window frames are utilized to group four types of geometry the center points circle points slider point and coupler positions Editable boxes for the user input of three coupler positions are
7. Eq A 7 Equation A 5 gives y2 p n aa 74 Substituting from Eqs A 1 and A 3 for v4 and a with origin at the instantaneous center A a A I aL YA w Y I Fig A 6 The inflection circle for given J aj r w and y4 v 0 k xran Va rjr Now or r isk a inn VA OOTA T TA I and from Eq A 3 a y art a k x ras WTA SO r n a Le a 04 02 1 FAI Referring to Fig A 6 FA 1 Al Ya ray COSY A so n aa ar COSY A T and 2 OTi p W2Pra 7 a COSyVa Now if D is the diameter of the inflection circle Eq A 9 gives and so Z Tir pe Tar Deosy 4 A0 75 Equation A 10 is one form of the Euler Savary equation The Euler Savary Equation is very useful because given the instantaneous center and inflection circle it can be used to locate the center point corresponding to any given circle point or vice versa The inflection circle is readily constructed for a given four bar linkage and it is therefore more convenient to work with the inflection circle than with the variables w and aj The geometric meaning of the Euler Savary Equation is discernible by referring to Fig A 7 Let A be the point whose path curvature is sought If we use directed line segments 14 7 points from I to A and r4 4 points from A to A Also rjg D cosy where J is the location where a ray from I to A crosses the inflection circle Hence if A is the center o
8. Guidance eeeeee 49 4 2 1 1 Visual Aid To Identify Limits for Center Points 00 eee eeeeeseceeeeeeeeeeeeeeeeees 49 42 N2 Rectification venti sec ee et es ei tes Ate e lee Sse a dake E ede atid 49 4 2 2 Analysis Window for Four Bar Linkage for Rigid Body Guidance eee 50 4 2 3 Zooming Feature in Analysis Window cccessecesececeeneeceeececsceecseeeesaeeeenaeeeenaeeees 51 4 3 Rigid Body Guidance Using a CrankSliderLinkage RBGCrankSliderDesign 53 4 3 1 Rectification When the Crank Is the Driver ceeeesceeseceeeeeeeeceeecseenseeesneeeneeees 54 4 3 2 Analysis Window when the Crank Is the Driver ee eeeeeeceeseeeseeceeeeeeeeeneeeneeens 55 4 4 Rigid Body Guidance Using a SliderCrank Linkage RBGSliderCrankDesign 56 4 4 1 Rectification When the Slider Is the Driver cee eee eeseeeseceeeeeeeeeeeeceseenseeeeeeeeneeens 56 4 4 2 Analysis Window When the Slider is the Driver eee ceeeseeessceeeeeeeeeteeeesteeeenaeees 57 4 5 Rigid Body Guidance Using a Elliptic Trammel Linkage RBGEITrammelDesign 59 4 5 1 Design Window for Elliptic Trammel Linkage eee eceeeeseeeeeeeeceeeeeesteeeeneeeenaeees 59 4 5 2 Analysis Window for Elliptic Trammel Mechanism cccceeseceeseeeeeeteeeeeteeeenaeees 59 4 6 Situations When Rectification Procedure Fails ee eeeceeseeeseecsseceseeeseeeeseecsaeenseesseeennees 60 4 1 ROTETCNCOS ceed giccc bce sada ha te ceccdis ca iea te enap t
9. Jg However calculations are required to locate these two points The Bobillier constructions allow the inflection circle to be determined without calculations The Bobillier constructions are graphical solutions of the Euler Savary equation for a four bar linkage That is they permit the location of the center point corresponding to a given circle point for three infinitesimally separated positions A 6 1 Bobillier s Theorem Bobillier s theorem states that the angle between the centrode tangent at the instantaneous center of the coupler relative to the base of a four bar linkage and one of the cranks is equal to the angle between the other crank and the collineation axis The collineation axis is the line joining the instantaneous center of the coupler relative to the base to the instantaneous center of one crank relative to the other as shown in Fig A 15 This theorem permits easy location of the centrode 83 tangent A line normal to the centrode tangent at the instant center gives a locus for the center of the inflection circle Collineation Axis Centrode Tangent Fig A 15 Statement of Bobillier s Theorem The theorem states that the angles marked 6 are equal Proof For a four bar linkage we can find the IC tangent for 3 by a simple relative velocity analysis Referring to Fig A 16 let the instant center location for 3 be designated simply as Z Also let Z5 be a point on rigid body 5 which traces the path of
10. N2 The addendum constant a2 a3 for each gear is 1 and the dedendum constant b2 b3 is 1 2 Find the tooth form which is conjugate to gear 2 so that there will be a constant velocity ratio between the two gears Solution To find the conjugate tooth form we must first find an expression for the coordinates of points on the gear tooth and for the components of the normal vectors Figure B 3 shows an enlarged view o gear 2 The equations for the gear are similar to those for the hob in Example 8 5 93 Before developing the equations it is useful to compute several parameters These are A2 Addendum of gear 2 B2 Dedendum of gear 2 2 12 0 24 ft P y tooth angle 25 l gear 2 tooth thickness at tip 4 4 2A tany 4 2 0 2 tan25 0 128 ft 2D 2 5 nm 20 ft i O2 B2 Fig B 3 One tooth from pin gear le gear 2 tooth thickness at tip l 2Btany 2 0 24 tan25 0 538 ft 2D 2 5 From that figure b Ca l f z ee Usps Gre B 8 We need consider only one side of the driving tooth because only one side will contact the corresponding tooth on gear 3 for a given direction of rotation We can reflect the tooth about its centerline to find the other half 94 Fig B 3 One tooth from pin gear The number of teeth on gear 2 is N2 d Dy 2065 100 The pitch radius of gear 3 is given by and the center distance is given by Eq B 4 as Ca n
11. cross The coupler point is mouse moveable and the center of the path is dynamically updated as the coupler point is moved The inflection circle changes with the position of the linkage Moving the green crank with the mouse will change the position of the linkage Simply click the mouse near the joint between the crank and the coupler and drag the link A second position is shown in Fig 2 45 In the frame corresponding to the point coordinates A designates the coupler point and Astar denotes the center of path The x and y numerical values for A may be input A graphical description of the input variables is displayed when the Definitions button is selected This is shown in Fig 2 46 35 Fig 2 44 Design window for the inflection circle routine Fig 2 45 Inflection circle when crank and coupler positions changes 36 File Edit View Insert Tools Window Help a Inflection4bar Analysis Pid ee IC Tangent Close Fig 2 46 Window showing variable definitions for inflection circle program 2 12 Program for Analyzing the Shaking Force in a Slider Crank Program ShakeAnalysis This routine analyzes the slider crank mechanism for position velocity and acceleration for one degree increments of the crank In addition the shaking force is computed at each angle increment for both the given value of the counter balance weight and for zero counter balance weight The optimum value of the counte
12. descriptions are presented in the order in which they appear in the program main menu The programs require version 6 0 or higher of MATLAB Either the full version or student version of MATLAB may be used Some of the routines available in MATLAB tend to change with the version number and the previous versions of MATLAB do not have some of the routines that are employed by the new version of the kinematic programs In general the descriptions consist of a brief overview of the purpose of the program followed by a description of the input and output windows The programs are menu driven so the inputs can be changed interactively 1 1 Types of Programs Available A brief description of each type of program in alphabetical order is given in the following Cam Design Program for cam design with axial cylindrical faced follower axial flat faced follower oscillating cylindrical faced follower and oscillating flat faced follower Centrode Plot Program for computing the centrode for a four bar linkage Cognate Drawing Programs for computing the cognate linkages for a four bar linkage Coupler Curve Gen Programs for computing the coupler curves of a four bar linkages and slider cranks Crank Rocker Design Program for crank rocker design zf Four Bar Analysis Gear Drawing Inflection Circle Prog Rigid Body Guidance SC Shaking Force Six Bar Mechanism Slider Crank Analysis Program for the analysis of a four bar linkage with ei
13. driven slider point is chosen the driver circle point can be identified However even if the driven slider point is chosen outside of the yellow shaded region it is possible to choose the location of the driver circle point such that branching will still occur In the design window colored linear regions are shown radiating from the driven slider point The driver circle point must be chosen to lie outside of the colored regions Sometimes the acceptable region is very small or even nonexistent If there is no linear region that is free of color then there is no solution that will be free of the branch problem In the example shown in Fig 4 8 the linkage chosen has the driver circle point outside the forbidden region Therefore that linkage will not have a branch problem Crankslider Design Window for Rigid Body Guidance crank length 1 9054 coupler length 1 542 M offset 0 35252 slider angle 14 8742 center point ab p E Pe EE x y ae lt lt Astar 1 739 1 732 l circle point y N A 0 15507 1 5246 A N slider point x y oe x i 3 i entered coord 13 13 d slider coord 1 3654 1 268 d we coupler point E H 5 4 En a x y angle gt Z lt E post 0 o 45 Ba D o lt i aL pos2 135 pos3 3 i i i i i i i ix 2 x a E Fig 4 8 The
14. generate only external gears To see the entire gears select the analysis button 5 2 2 Analysis Window for Arb2thDesign Program The analysis window is shown in Fig 5 4 In the analysis window 1 2 3 or 4 plots can be displayed Options for the plots are an animated view of the gear teeth the tooth form for the generating tooth form and the entire gear for both the generating gear and generated gear In the individual windows the gears are plotted as large as possible so they are not normally plotted to the same scale The buttons for starting stopping and changing the speed of the animation is the same as in the previous programs By changing the display to a single figure as is shown in Fig 5 5 it is possible to observer the details of tooth meshing as the teeth come into and leave contact File Edit View Insert Tools Window Help Parameter No of 4 Fig meshinacear Fg2 jeneratinatootr _ Start Stop Speed j _ Fig3 generating gear Fia4 zoniugated gear Return Fig 5 4 Analysis window for the Arb2thDesign program 5 3 GeardrAnalysis Program This program will generate an involute tooth form given the geometry of the generating rack and the equations given in Section 10 12 are programmed The program uses two windows a design window where the input data are identified and an analysis window The analysis window is described first 63 Fig 5 5 Closeup of gear mesh when one figure
15. instantaneous center I between the moving body and the frame Then vg becomes zero and dg a That is ag becomes the acceleration of the point in the moving body which is at the instantaneous center Equation A 6 then becomes k X Fast x ar ak X LA WP 4 1 0 or k x raj x aj k x raj xrar O or k xrar xar w rajp k 0 Let the angle between a and r4 z be y4 see Fig A 5 where y4 is measured from a to r4 Then k X asp X ay FA 1 sin YA m 2 k A 7 since lk x ra ra and the angle between k x rar and a is y4 2 2 Hence a TA 1 k ay ray cos ya k 0 or A YA ay Tal k I k x Tay Fig A 5 The geometry of the vectors in Eq A 7 Za TAII mp2 COSY A A 8 This is the equation of a circle passing through I with diameter a D a A 9 The center of the circle through I is located on a line from I and in the a direction This circle is called the inflection circle and is represented in Fig A 6 The inflection circle can be viewed as the limit of the image pole circle for three finitely separated positions as those positions become infinitesimally close Just as the image pole circle circle of sliders is the locus of circle points whose three positions lie on a straight line the inflection circle is the locus of points whose paths are locally straight We now seek an expression for the radius of curvature of the path of any point A in terms of the variables used in
16. not applied In the analysis window 1 or 2 plots can be shown and animated The plotting options 50 are the animations for the two assembly modes By animating both assembly modes the effect of branching can be illustrated For example in assembly mode 1 in Fig 4 3 the coupler moves through position 2 and in assembly mode 1 the coupler moves through positions 1 and 3 Figures 4 4 and 4 5 are the design and analysis windows for a linkage chosen to avoid the branching problem When this linkage is animated assembly mode 1 goes through all of the positions Fig 4 3 The analysis window for the four bar linkage program for rigid body guidance 4 2 3 Zooming Feature in Analysis Window To zoom in or out in the analysis window the mouse is used To zoom in draw a marquee around the area that is to fill the window and click the mouse This is shown in Fig 4 6 The local window will redraw in a zoomed view as shown in Fig 4 7 To zoom out and return to the original view click the right mouse button An alternative way to activate the zoom feature is to place the mouse cursor on the figure and click the mouse The program will zoom relative to the point where the cursor is located A left mouse click will zoom in and a right mouse click will zoom out 51 Fig 4 4 Selection of driver circle point in acceptable region Fig 4 5 Analysis window showing that the linkage goes through all positions 52 Fig 4 6
17. point C in the coupler and Applying Eq A 3 AtD ra i 8 J JEFE and F2 32i 107 a Using this data C and D are located as shown in Fig A 4 Note that a minus sign on either p Or Pp would indicate that the center of curvature is located in the n direction Figure A 4 The solution to Example A 3 velocity and acceleration fields A 3 3 Inflection Circle We found that for three finitely separated positions there are an infinite number of points whose three positions all lie on the same straight line and that they are distributed on a circle which passes through all three image poles Let us seek the equivalent result for 3 infinitesimally separated positions namely the locus of points that for a given velocity and acceleration state have paths with locally infinite radius of curvature Another way of stating this is the locus of points whose paths have points of inflexion at the instant of passing through the design position Looking at Eq A 5 we see that p approaching infinity implies n a4 0 In the general case a will be nonzero Then since n is normal to v4 this implies that vy and a have the same or opposite directions Hence VA x44 0 Applying Eqs A 1 and A 3 vo wk xrajo x ao ak xrajo ra o 0 A 6 273 For the analysis we may select the origin of coordinates to be at any location that we like It will simplify the results if we move the origin to the
18. provided The user can either input the positions numerically or move the locations and angles of the three coupler positions by mouse dragging The GUI implementation also allows users to drag any circle center or slider point continuously with its coordinates updated dynamically To be able to recognize corresponding points on the plot and data in the editable boxes three different colors red blue green are used for the coupler positions The slider point can be input either through the mouse by dragging the mouse cursor around the circle of sliders of though the input boxes The slider points must lie exactly on the circle of sliders Therefore the program will correct any user numerical input to force the point to the nearest point on the circle To do this the program identifies a straight line from the point to the center of the circle and finds the nearest intersection of that straight line with the circle The value input by the user is designated by entered coord and the value of the corresponding slider point is designated by slider coord As in the case of a four bar linkage it is common to find that a slider crank linkage designed using the basic procedure outlined in Section 6 3 6 of the textbook does not function as assumed It is common to find that they do not guide the rigid body through all three positions unless the assembly mode is changed In such cases when the linkage is animated the rigid body will pass th
19. routines along with a detailed user s manual are included elsewhere on this CD Theoretical background information on some of the routines is included in that user s manual 2 2 Crank Rocker Design Program CRDesign The objective in the design of a crank rocker mechanism is to determine the lengths of the crank rocker and coupler for a given rocker angle and time ratio The routine has two major windows a design window and an analysis window The design window accepts the user input data including the rocker angle theta time ratio and one link length The second window is the analysis window that displays the final mechanism and shows the animated motion 2 2 1 The Design Window for Crank Rocker Program The design window for the crank rocker program is shown in Fig 2 1 In the window associated parameters are grouped together by a frame to visually indicate their relationship Also either or options are given by providing a radio button set which are also grouped together by a frame The design window uses the definitions given in Section 6 5 of the textbook The program also gives the definitions of some of he variables if the Definition button is selected clicked on The definition window shown in Fig 2 2 then appears A brief description of the program is given if the Jnfo button is pressed The information file is displayed in Fig 2 3 The actual analysis uses the procedure explained in Section 6 5 4 of the textbook The
20. since the direction of the tangent to the centrodes at the contact point instant center location is a purely geometric quantity A 6 2 First Bobillier Construction Given the centrode tangent and inflexion circle construct the center of curvature of the path of any nominated point The steps are given below and the construction is shown in Fig A 18 85 Steps 1 Select the circle point C 2 Locate point J on the opposite end of the diameter of the circle from 7 3 Draw line CI and construct the normal to CI at 1 4 Locate point G at the intersection of the normal to CZ at J and line CJ 5 Construct the normal to the centrode tangent through point G 6 Locate center point C at the intersection of the normal to the centrode tangent through G and line CI Inflection Circle Centrode Tangent a Fig A 18 First Bobillier construction Proof rca 1C p rcc D Wand yc Z JIC Triangles JJC and C GC are similar so C C_C G IC D and ZGC I ZJIC yc Therefore C G C I _ C C IC_ P tcn Also giving pD cos prcn or T ver toy D cosy c 86 which is the Euler Savary equation Eq A 10 A 6 3 Second Bobillier Construction Given the inflexion circle and the instantaneous center find the center points corresponding to two nominated circle points The steps are given below and the construction is given in Fig A 19 Steps 1 Select circle points C and D and draw line C
21. the inputs and the user can specify three coupler positions and angles The pole locations between each two coupler positions are calculated and shown as a marker in the graph The circle of sliders corresponding to the three image poles for position 1 is shown in black After all of the input data are provided the linkage can be animated to determine if it moves through all of the positions identified Figure 4 10 contains the same input information as Fig 4 8 but now the slider is the driver and the crank is the driven link Note that some of the rectification regions have changed The only difference between the design procedures for a slider crank and crank slider is in the rectification process 4 4 1 Rectification When the Slider Is the Driver If the slider is the driver the crank is the driven link so that the crank circle point is considered first The three image pole circles define acceptable locations for the circle point Note that the locations of these image pole triangles are the same regardless of which link is chosen as the driver The unacceptable positions for the driven circle point are shown shaded in yellow If the circle point is chosen in the yellow shaded area the linkage will have a branch problem and be unacceptable regardless of where the slider point is chosen After the driven circle point is chosen the driver slider point can be identified In the design window colored linear regions are shown radiating
22. the path of C required by the velocity state That is the circular path of C produced by the crank should have the velocity vector vc tangent to it Clearly any point on the normal to vc through C can serve as the center point C Example A 1 Synthesis of Linkage for Specified Velocity of Point in Coupler Problem Synthesize a four bar linkage to give the coupler point at the origin a velocity of one unit per secon in the X direction when the angular velocity is 4 rad sec counter clockwise Solution 67 Let the four bar linkage be defined in the usual manner with link 2 as the driver and link 3 as the coupler From the problem statement point O3 is the coupler point at the origin coordinates relative to the frame are 0 0 In the following the subscript 3 will be dropped because it is understood that all points being considered are in the coupler vo li w 4k rad sec For the four bar linkage we need to select two circle points and for this we will choose points C 1 1 and D 0 Then Vc Vo VciI0 Vo WX co li 4k x i J 314 4j where i j k are orthogonal unit vectors in the x y and normal directions respectively Also vyp li 4kx 2i i 8j Points C and D and velocities vc and vp are plotted on Fig A 1 The normals to vc and vp at those points were drawn and C and D were selected on those normals The resulting linkage is C CD D Compare this procedure to that used for two finitely separated po
23. velocity and normal acceleration in the following sie Len 1ClOc a C3 The location of Oc is along the normal vector in the direction of a This is shown in Fig A 3 A 3 2 Synthesis Using the Center of Curvature at a Point and Along a Path To synthesize a linkage to move a lamina through three infinitesimally separated positions we can take any point in that lamina find the direction of its path and the radius of curvature of that path and hence the center of curvature of the path By locating the center point C at the center of curvature we get a crank which gives the required path direction and path curvature in the design position Repeating this procedure for a second crank we generate a four bar linkage which gives the required velocity and acceleration states while passing through the design position Example A 3 Synthesis of a four bar linkage for three infinitesimally separated positions of a point in the coupler Problem The velocity state of a lamina is to be as in Example A 2 That is v 1 in s in the x direction 4 rad s counter clockwise In addition ag is to be 20 in s in the y direction and is to be 10 rad s clockwise Solution Choose C at position 1 1 and D at 2 0 as before then v 3i 4j in s and vp i 8 in s From the problem statement a 10k rad s and ay 10j Therefore applying Eq A 3 gives ac 10j 10 k x i j 16 i j 6i 16j At
24. Cam Design fT PP a THE QHIQ STATE UNIVERUTY ir y Guidance Design RBG4bar Design RBGCrankslider Design FBGSlider CrankDesign RBGElTrammelDesign Gear Design il ol TIA AT Click to expand this topic helen ll ww ei Py Fig 1 5 Subtopics under Rigid Body Guidance Design File Edit View Insert Tools Window Help KINEMATIC PROGRAM MENU Linkage Design Cam Design Rigid Body Guidance Design The Gear category has two routines One draws involute gears and the other will generate the gear tooth form for an arbitrary conjugate gear The conjugate gears are animated Geardr Analysis Click to expand this topic Close far Fig 1 6 Subtopics under Gear Design 1 4 MATLAB Graphics Window As the programs are run three windows will be of interest The first is the MATLAB command window This is the window that has the MATLAB command prompt and is where any errors are identified by the program The second window is the graphics window identifying the program options This is the window shown in Figs 1 1 1 6 When you launch a program by selecting the program and clicking on Run a second graphic will appear This is the window where the input data are changed and where some of the design analysis results are presentation If the program includes an animation feature the animation feature will appear in a third window Note that if several windows are open while running the programs some of th
25. D 2 Join points C and D to the instantaneous center J to locate points J and Jp at the intersections of the junction lines with the inflexion circle 3 Join points J and Jp to locate point E at the intersection of lines CD and J Jp Fig A 19 The second Bobillier construction 4 Join Ito E 5 Draw a line parallel to J Jp through Z Its intersection with line CD gives point Z The collineation axis is line ZZ 6 Draw a line through Z parallel to JE Its intersections with lines CZ and DI give the center points C and D respectively Proof Triangle JCE is similar to triangle C CZ 87 SO 4C JE Cec C2 Also triangle JJcE is similar to triangle C IZ SO hence Now sO IE__ Ue CZ CI IC _ Ik C C 7 CI IC cy C C p Uc Deosyc C I p r tca _ Deosyc 7 tn P P Icn fea Dcosy c A similar proof holds for point D Bobillier s second construction is of greater importance when used in reverse It then becomes a means of constructing the inflexion circle of a given four bar linkage The steps are given below and the construction is shown in Fig A 20 Fig A 20 The second Bobillier construction used in reverse to find the inflection circle of a given four bar linkage Steps 1 Locate the instantaneous centers J and Z and draw the collineation axis JZ 2 Draw a line through Z parallel to line A B Its intersection with line AB is E 3 Draw a line through E pa
26. Fig 2 35 displays the coupler curves for the coupler points identified in the design window One of these coupler points can be selected via the mouse for further analysis The color of the coupler point is changed when it is selected In the animation window the Analysis button is not available until the user selects the exact coupler point using the mouse 2 8 3 The Analysis Window for Four Bar Coupler Point Analysis Program The analysis window Fig 2 36 is the same as the other analysis windows Up to four plots can be displayed The plot options shown in Fig 2 36 are the mechanism the rocker angle the rocker velocity and the rocker acceleration The coupler curve shown in the analysis window is dished where each dash corresponds to 5 degrees of crank rotation This gives a means of coordinating the travel of the coupler point to the position of the crank Also the length of the dashes gives a visual comparison of the relative speed of the coupler point as it moves along the coupler curve ry ry Py ry ey K Fig 2 33 Design window for four bar linkage coupler point atlas program 28 Fig 2 34 Window showing variable definitions for four bar linkage coupler point atlas program 2 ae a YI Fig 2 35 Animation window for four bar linkage coupler point atlas program 29 File Edit View Insert Tools Window Help Parameter E
27. KINEMATIC PROGRAMS BASED ON MATLAB S GUI To Supplement the Textbook Kinematics Dynamics and Design of Machinery 2 Ed By K J Waldron and G L Kinzel 1996 03 by K Waldron and G Kinzel Department of Mechanical Engineering OHIO UNIVERSITY Table of Contents Section Page Wale sol COTE Is ee icc asiiz beste cea ed nena ech eee cons adenine ie eared cena ee i MOSAIC COND iessen eea scute EE a E e til bap a aE a a Ea yi deadl Ea 1 Ll Types of Programs Available irssi iieri iy ioni K E 1 1 2 Prost ann Installations nenne e ea E E T E E EE E EA SENE e 2 1 3 Running the Programs essnee r a n e evant alates 2 1 3 1 Programs Under LinkageDesign sista vendelasstacaet shun acta sede denesnstenssaoamdeadaencyscasien dens sacheoeenele 3 1 3 2 Programs Under Cam Design lt sssscccscsasuscisivupcessarivanscavavescsanayeesnaaaseascatisaacaucimassmaaessaneis 4 1 3 3 Programs Under Rigid Body Guidance Design cccccesscesessceeessceteseceeseceesneeeesneeeenaeees 4 1 3 4 Programs Under Gear Designs saciccsssccvicecnsy seaselsdiavaiacthonepesteacansdgionsenabenigarsuveasacdsoouetate 4 LAMA TEA B Graphics WINdOW ss cte snnesycasldsjes iioii sanoe ir eE ei E E E E 6 1 5 Help in Using MATLAB sy cscssssssssiysnscisivesusssansvscavasaustaany ces taalvasnseann coveeeapansiaransadeacavauaseannessavens 6 2 0 Programs under Linkage Design anino eves asvvesiasbasends ae anstavtatidesina ewes vesinliadnds 7 2 NIUE OG CU OM Gad ood ah an
28. The variables that must be input are shown in a figure if the Definition button is selected The resulting figure is shown in Fig 2 23 All of the push buttons are the same as in the cognates GUI routine which actually becomes the standard push button sets for all the subsequent design windows The capability of moving the coupler point continuously using mouse dragging is also included in this program Fig 2 20 Design window for the four bar design coupler represented by line 19 Fig 2 21 Coupler represented by triangle Fig 2 22 Design window with coupler curve for both assembly modes 20 Fig 2 23 Window showing variable definitions for four bar analysis program Fig 2 24 Analysis window for the four bar analysis program 2 5 2 The Analysis Window for Four Bar Program The analysis window is the same as those for the previous two analysis windows An example is shown in Fig 2 24 Either 1 2 3 or 4 figures can be plotted The animation must be stopped before the figures can be changed To resume the animation select the Start button To change 21 the linkage design select the Return button and return to the design window Any of the linkage parameters can be changed before returning the analysis window Note that the coupler curve in Fig 2 24 is dashed One dash corresponds to 5 degrees of crank rotation By observing the lengths of the dashes it is possible to estimate the relative speed of th
29. a uniform grid is created When a point is selected its grid marker changes to a hollow circle adding visual assistance The user can specify both the length and height of the coupler rectangular grid and the number of rows and columns of points In addition the user can select the specific grid points that will be analyze further These are identified in the animation range The push button set of Definitions Zoom Out and Zoom In buttons is moved to the space below the plot because all the grid creation options are arranged to be close together The definitions page gives a description of most of the input variables This page is shown in Fig 2 38 The GUI program checks the valid range for the grid to avoid interrupting execution If invalid data are inputted the previous data are retrieved and an error message is shown in the status bar below the plot 30 E w i OOt se eee t OOt te we p ry ry ry ry ry D ry p nl o n cE Fig 2 37 Design window for slider crank mechanism coupler point atlas program Fig 2 38 Window showing variable definitions for slider crank mechanism coupler point atlas program 31 2 9 2 The Animation Window for Slider Crank Coupler Point Analysis Program The animation window Fig 2 39 displays the coupler curves for the coupler points identified in the design window One of these coupler points can be selected for furth
30. aa EE Eak Sraa SEE EA RETEA ve acauei eee EE aa anes uae 60 5 0 Program for Displaying Gears sissies sivesseesaycsscataysanasany e pe E a e Ea E SE 61 S KO ELi 0 aa 0 TEE A EE O E Ur eater eae Aeon oem rere 61 5 2 ALD 2thDesionm Prostam ninoi veancsalsvesesaanibagenas Uonmaaviad cade avn A aa shaadi es 61 5 2 1 Design Window for Arb2thDesign Program eeceeeecceesseceesseceesneeeeseceeneeeeneeeees 61 5 2 2 Analysis Window for Arb2thDesign Program ccceescceseseceesseceesteceeneceenseeeneeeens 63 5 3 Greardt Amal ysis Prota e i sed vin eisint riasa nied EAE AaS EE deste tongs EE RENE esiet 63 5 3 1 Design Window for GeardrAnalysis Program eeeseseesseessesssesseeesseesseesseesseeesseee 64 5 3 2 Analysis Window for GeardrAnalysis Program cccescceesseeceeeeeceteeeesteeeeneeeenaes 66 Appendix A Procedure for Euler Savary Equation ccescceecceescecsecesneceseceseeesseecaecnseesseeeeneees 67 A Ino dC ON ced cece ect et le eeee ed ieee ede cee 67 A 2 Two Infinitesimally Separated Positions ccssesseeseccesseeseseesoneescettencenseesotsesceteees 67 A 3 Three Infinitesimally Separated Positions scccsenseccesseeesscetoneeecenececesteesoneeenerseeee 70 A 3 1 Center of Curvature of Path of Moving Point Relative to Frame cceeeeceeeeneeees 70 A 3 2 Synthesis Using the Center of Curvature at a Point and Along a Path 12 A 3 3 Inflection Circle ose cae sae hot de het
31. and B is the center of curvature of the path of B3 relative to the frame Therefore in this problem B3 is at infinity in the direction indicated in Fig A 12 Locate the instant center by finding the intersection of the rays through BB and AA To fin the intersection the angle is required From geometry this is given by p cos 22 81 869 Then rar rax pxtan 45 1tan 36 869 0 750 Inflection JA Circle Fig A 12 Inflection circle for Example 4 11 2 Next find J4 using Eq A 13 That is ra za AL AI A Therefore A is between A and J4 as shown in Fig A 12 2 Qto 3 062 in the direction of r4 To find the location of Jg we cannot use Eq A 13 because B is at infinity Instead we can use the form of the equation given by Eq A 12 That is gas ld FIBI VB TB 1 or OE 1 _ __ Y Tga COS 45 or rypit 13 4 cos 45 1 cos 36 869 1 250 or 1 250 in the opposite direction of rg Therefore J is between B and Jg as shown in Fig A 12 To locate the center C we must first find Jc by drawing a ray from C through Z as shown in Fig 81 A 12 We can measure rj jc directly to be 0 711 Also rc 1 197 Then from Eq A 13 2 2 a 0 752 in the same direction as rjjjc U eg OT SOLD Therefore C is between C and I The location is shown in Fig A 12 The approximate path of C is also drawn in Fig A 12
32. are the mechanism the rocker angle for the basic four bar linkage the dwell angle and the angular velocity for the rocker of the output dyad An example is shown in Fig 2 32 The animation must be stopped before the figures can be changed To resume the animation select the Start button To change the 26 linkage design select the return button and return to the design window Any of the linkage parameters can be changed before returning the analysis window File Edit View Insert Tools Window Help Parameter EN wi 4 bar rocker angle deg 0 100 200 300 crank angle degree s g gt 25 0 2 2 so amp 0 15 5 10 Je 2 z i E 5 s 0 4 o pi 0 100 200 300 0 100 200 300 crank angle degree crank angle degree No of 4 Fig mechanism Fig g parrocker ane _ start Stop Speed 4 y Fid3 dwellrockerana Fig4 dwell rocker vel Return f J Fig 2 32 Analysis window for the six bar analysis program 2 8 Program for Generating Atlas of Coupler Curves for Four Bar Linkage HRCrankRockerAnalysis This routine is to generate the coupler curves for a four bar mechanism The program is called HRCrankRockerAnalysis after Hrones and Nelson who developed an atlas of coupler curves A uniform grid of coupler points is assumed for the coupler and the user can choose one point for analysis by following a sequence of selecting grid dimension grid density and row and column numbers The pro
33. change Peta Aelter Canon Also users om input the values of link lengths and angles from the keyboard In the malysis step users can see the mimation from the initial position to the final position of the doable lever The Close button closes the Double Lever Disign window while the Animate button starts the Linkage mimation Fig 2 17 The information window for the double rocker design routine Al GS Fig 2 18 The definition window for the double rocker design routine 17 If the Analysis button is chosen the linkage is animated The animation will show that the design requirements are not met if a branch problem exists 2 4 2 The Analysis Window for the Double Rocker Design Program The analysis window of the double rocker design routine has the same layout as that of the crank rocker routine except that only two plots are available The results for one and two plots are shown in Figs 2 16 and 2 19 To change from one to two plots or vice versa it is necessary to press stop first if the animation is running Then change the number of figures using the button indicated The figures plotted are selected from the two titles for each of Fig 1 and Fig 2 To change the speed of the animation click on either the or button Press multiple times to make a large change in the speed Four Bar Analysis Window Fig 2 19 The window for the double rocker design rout
34. cote cd Bnd E ote ae EEEE 13 A 3 4 Different Forms for the Euler Savary Equation ccccessccecssceeseeeeesneeeeseeeenaeeees 76 A 4 Relationship Among IC Centrodes IC Tangent and IC Velocity eee eeeeeseeeeeeeeeees 82 A 5 Analytical Form for Euler Savary Equation cccsscccsssccecseeceeseeceeseeceesaeeeesaeeeenaeeeeaeeees 83 AO The Bobillier On ste tons ise tees teeens ct Feastaiole aus caecara ie ecule dante vnnselsbuaaengedetiaaatesuads 83 A 6 1 Bobillier s The Oren surisi snari ir un iiaiai neuen 83 A 6 2 First Bobillier Construction seesssessesesssesseesessrssresseserssresseseresresseserssressessresresseesee 85 A 6 3 Second Bobillier Construction eesseseesseeeeeseeesseesseserssressessrssresseserssressesstesresseeseee 87 Appendix B Procedure for Drawing Conjugate Tooth Form seseseeseeseserssseereesessrsresseesseseresreses 89 B 1 General Conjugate Tooth Forms ssssssessssssessssessseesseesseessesessseesseessersseessseessseesseesseessees 89 B 1 1 Required Geometric Parameters sssessesseeeseeeeeseesseesseesseressessseesseessesseeesseeesseesss 89 B 1 2 Determination of the Point of Contact esseseesseeeeesesesseresseseresressessrssresseseresresseesee 89 B 1 3 Coordinate Transformations seeeseseeseeseeeseeeeseesseserssresseseresresseserssressesseesresseeeeee 92 1i MATLAB PROGRAMS BASED ON GUI to Supplement the Textbook Kinematics Dynamics and Design of Machine
35. design window for the crank slider design program crank driving 4 3 2 Analysis Window When the Crank Is the Driver The three coupler positions in the analysis window Fig 4 9 use the same colors as in the design window Itis therefore possible to identify which position is missed if the rectification procedure is not applied In the analysis window 1 or 2 plots can be shown and animated The plotting options are the animations for the two assembly modes By animating both assembly modes the effect of branching can be illustrated In this example the design was based on the rectification procedure and the coupler passes through all of the positions The zooming feature is also available in the RBGCrankSliderDesign program To zoom in locate a rectangular marquee around the figure and click the left mouse button To zoom out click the right mouse button Alternatively simply locate the cursor about the new center of the figure and click the left mouse button to zoom out and the right mouse button to zoom in 55 RBG Crankslider Analysis Window No of figs node Start Stop Speed Return Fig 4 9 The analysis window for the example in Fig 4 8 crank driving 4 4 Rigid Body Guidance Using a Slider Crank Linkage RBGSliderCrankDesign This routine is used for the design of slider crank linkages with the slider as the driver Again the center point circle point and slider point are
36. e been completed The center of curvature of the path is in the direction of the normal component of acceleration In Eq A 5 the normal component of acceleration can be plus or minus If it is plus it is in the m direction and if it is minus it is in the n direction Example A 2 Center of Curvature of the Path that a Point on the Coupler of a Slider Crank Mechanism Traces on the Frame Problem Identify a procedure whereby we can locate the center of curvature of the path traced on the frame by points on the coupler of a slider crank mechanism Solution Consider the slider crank mechanism in Fig A 3 and assume that the path of C3 is of interest The center of curvature of the path is a purely geometric quantity and therefore the actual values used for the velocity and acceleration analysis are arbitrary Also the choice of the driver is arbitrary Fig A 3 Center of curvature of path of C3 on the frame Because the velocity of C3 is tangent to the path that C3 traces on link 1 the velocity vector for C3 indicates the direction of the tangent to the path The center of curvature for the path will be on a line through C and normal to the velocity of C3 From the acceleration analysis we can determine lac and resolve the acceleration into two components which are in the direction of yo tangent and normal to vc Then ag a3 aC3 The radius of curvature of the path is calculated by using the magnitudes of the
37. e coupler point as the linkage moves 2 6 Program to Analyze a Slider Crank linkage SliderCrankAnalysis This routine analyzes a slider crank mechanism in which the link driver slider coupler or crank can be specified Associated analysis plots for the position of the output link and coupler and the velocity of the output link are shown in the animation As in the previous cases the slider crank program is structured with a design window and an analysis window 2 6 1 The Design Window for Slider Crank Program The design window is shown in Fig 2 25 Again the design window has several radio button sets to set different features of the program A frame groups each set Otherwise the design window is similar to that of the four bar program In the slider crank program the coupler point must be identified by a triangle The radio buttons are associated with the following options a Crank coupler or slider driven The crank coupler or slider can drive the linkage b Assembly mode Either the 1 or 1 linkage assembly mode can be analyzed for positions and velocities The assembly mode will have different meanings depending on which link is the driver c One mode or two The linkage can be analyzed and the coupler curve displayed for either one mode or two If only one mode is chosen the coupler curve for that assembly mode only will be shown If both modes are chosen the coupler curve for both assembly modes will be displayed
38. e editable angles theta0 theta phi0 phi and link lengths frame crank coupler rocker are grouped together on the right hand side of the window If a linkage that will change branch is chosen the message Branch Problem Please input other values is displayed at the bottom of analysis window Different values can be input either by typing in new values in the input boxes or by dragging the end points of the two rockers In the drawing the green link is taken as the driver corresponding to theta and theta0 16 Two buttons Zoom in and Zoom out scale the plots because the parts of the mechanism might go outside of the plot window when the user drags the rocker points If the nfo button is chosen general information about a double rocker is presented as shown in Fig 2 17 If The Definitions button is chosen a generic double rocker linkage is displayed as shown in Fig 2 18 is displayed Double Lever Design Double Lever Design Double Lever Design is a program to design dable lever mechmiane ad to male the result based The program contains two steps design md analysis The nomenclature used by the proga is that given in the textbook Kirematics Dynamics and Design of Machmiisms by Kenneth Waldron and Gary Kirzel In the design step the voricbles are the four link lengths imam rk ene my md rodar od far onglar froe angle md it chmas rocker angle and its change of the program features i t users can drag the muse to
39. e i Bo a E E E E serra de tis EE E E Ain pera EE ta 7 2 2 Crank Rocker Design Program CRDesign e sssessssssesssesesssessseesseresseesseresseessesseesseeesseee 7 2 2 1 The Design Window for Crank Rocker Program eeseseseeeseerseseerrssressesessrssressesee 7 2 2 2 The Analysis Window for Crank Rocker Program eessseeeeseseseessreeseessesrrssresseseee 9 2 3 Program for Generating Cognate Linkages CognateAnalySis eeeeeseeseceseeeeeeeeneees 12 2 3 1 The Design Window for Cognate Program cecceesececsseceeeseeeeseeeesseeeeeeeeseeeenaes 12 2 3 2 The Analysis Window for Cognate Program ccceesceecseceesseceeeeeecseeeeesseeeesaeeeeaaes 12 2 4 Program for Designing a Double Rocker Four Bar Linkage DoubleRockerDesign 15 2 4 1 The Design Window for Double Rocker Design Program cceeseceesseceenteeeeeee 16 2 4 2 The Analysis Window for Double Rocker Design Program csscceesseeeesteeeenes 18 2 5 Program to Analyze a Four bar Linkage FourbarAnalySis 0 ceseessceeseeeseeeeseeeeeeeeeeees 18 2 5 1 The Design Window for Four bar Program eeeescesecesseeeneeceeceseeesseesneeenseesnees 18 2 5 2 The Analysis Window for Four Bar Program cescesscesseeessecseceeeeeeneesseenseeesnees 21 2 6 Program to Analyze a Slider Crank linkage SliderCrankAnalysis e cessseeeseeeseeeeeeees 22 2 6 1 The Design Window for Slider Crank Program eeceseeseeeseseseses
40. e input data are identified the generated tooth form and gear is shown in the analysis window 5 3 2 Analysis Window for GeardrAnalysis Program The analysis window is shown in Fig 5 8 In the analysis window 1 or 2 plots can be displayed Options for the plots are the generated gear tooth form and the entire gear By displaying one figure with the gear only it is possible to show visually how undercutting appears on the gear Fig 5 8 Analysis window for GeardrAnalysis program 66 Appendix A Procedure for Euler Savary Equation A 1 Introduction The information in this Appendix was originally contained in the main textbook however it was removed because of limited space The entire development is given here although the MATLAB programs apply to only part of what is presented Another way of generating a point path with desired properties is to use curvature theory This provides a way of precisely controlling the trajectory in one position of a lamina For example the direction and curvature of the path of a given point can be controlled in a given position The expectation is that the path will retain a similar curvature at all positions near to the designated point Curvature theory is actually closely related to the theory of motion generation through a series of finitely separated positions It can be thought of as the limiting case in which the design positions become infinitesimally separated There are many similarities
41. e known tooth form These parameters include the pitch radii tooth numbers and gear type 1 e internal or external and a mathematical function for the gear tooth form on the known gear If the function is not known directly it is possible to fit a spline to a set of points describing the tooth form Ultimately it is necessary to be able to compute the normal vector to the known gear tooth at each location B 1 2 Determination of the Point of Contact Assume that the known gear is gear 2 and the unknown gear is gear 3 Each gear will have a coordinate system attached as shown in Fig B 1 and the local gear geometry will be defined relative to the coordinate system fixed to each gear To satisfy the fundamental law of gearing the line normal to the tooth surfaces must pass through the pitch point as shown by line AP in Fig B 1 The line segment AP is a straight line which has the following equation y mxt b Here m is the slope of the line which is the direction of the normal to the known gear at point A and b is the y intercept An expression for b can be found by recognizing that the line passes through point A Therefore 89 Line of Centers Fig B 2 Line through pitch point and normal to tooth profile ya mxa b or b ya mxa If the slope of the normal is represented by 90 NA m 2 gt Na then an expression for the line segment is given by y x xXa Jna ya B 1 Two special cases e
42. e shown in Figs 2 5 2 8 The animation is continuous until the Stop button is selected To change options select the Stop button and make a change by changing either the number of plots or the items to be plotted Then press Start To change the item that is plotted press on the title button and select from the list presented The animation can be speeded up or slowed down by pressing the plus and minus buttons respectively To return to the design window select the Return button Users can easily switch between the design and analysis windows at any time Fig 2 5 Various output options for crank rocker analysis 4 plots 10 Fig 2 6 Various output options for crank rocker analysis 3 plots Fig 2 7 Various output options for crank rocker analysis 2 plots 11 File Edit View Insert Tools Window Help Parameter transmission angle deg ao o 50 100 150 200 250 300 350 400 crank angle degree No of _ 1 Fig Tansmitionang F92 ays Start Stop Speed ay ay FIGS popin oe Fig4 5 Return Fig 2 8 Various output options for crank rocker analysis 1 plot 2 3 Program for Generating Cognate Linkages CognateAnalysis This routine takes the basic four bar linkage geometry and the location of the coupler point as input It then determines Robert s linkage as well as the three individual cognate linkages The equations are developed from Section 6 6 3 of the textbook T
43. e textbook as a change of branch Considerable research has been devoted to identifying at the beginning of the design process linkages that do not have the change in branch Waldron and his student have developed a relatively simple procedure that has been implemented in the four bar linkage and slider crank mechanism routines 1 4 The procedure identifies acceptable regions for locating the two circle points under most circumstances and that procedure has been implemented in the programs in this section Avoiding the branch problem is a two step process and the regions in the two steps are different In the first step the circle point for the driven crank is chosen In the design window the driving crank is shown in green and the driven crank is shown in black in order to distinguish between the two The three image pole circles define acceptable locations for the circle point for the driven crank The distances between successive image poles define the diameters of these three circles There are three image poles P12 P13 and P 3 and these are the same points used to draw the circle of sliders in position 1 The unacceptable positions for the driven circle point are shown shaded in yellow If the circle point for the driven crank is chosen in the yellow shaded area the linkage will have a branch problem and be unacceptable The user can References are given in Section 4 6 at the end of the chapter 49 experiment with
44. e windows may become hidden under other windows 1 5 Help in Using MATLAB A brief overview of the use of the kinematic programs is given in the following sections When describing how to use the programs it is assumed that the user is familiar with the basics of MATLAB For details consult the MATLAB Users Manual supplied by Mathworks Alternatively MATLAB has an excellent help facility To obtain help on any topic in the library simply type help and MATLAB will present a series of topics on which help may be obtained By typing help and then the name of the topic a description of that topic is displayed Also a list of subtopics on which help can be obtained is displayed If the name of the subtopic is known it is possible to type help followed by the subtopic name anytime that the MATLAB command prompt gt gt appears in the MATLAB window 2 0 Programs under Linkage Design 2 1 Introduction The descriptions given in the following will be limited to explaining how to run the programs available It is assumed that the user will not be routinely modifying the code and therefore except in a few cases little theoretical information on the programs will be given However the programs based on the MATLAB GUI use essentially the same analytical routines used in the original set of MATLAB programs written for the first edition of the textbook Kinematics Dynamics and Design of Machinery by K J Waldron and G L Kinzel The original set of
45. ed by the circle as in Fig 3 6 Cam Angle Acceleration 25 3 04483 Next Pe ak Next Segment Minimize Follower Acceleration 0 40 80 120 160 200 240 280 320 360 Fig 3 6 Peak Acceleration Selection If the result of the optimization is undesirable the Reset button can be used to reset the follower motion to that specified in the first screen When the follower motion is acceptable the Animation button advances the program to the animation window 3 4 Cam Follower Animation Window This third and final screen is displayed in Fig 3 7 On this third screen the cam follower system can be animated using the animate button and the speed of the cam can be adjusted using the speed controls These speed controls include a button for increasing the speed and a button for decreasing the speed One click of either button changes the speed by 10 of its current value The Reset Speed button resets the speed to its initial value From the animation screen the Motion Plots button will return the program to the previous screen shown in Fig 3 4 Finally the Output Cam Profile button outputs the cam profile to a text file titled cam _ profile and located in the same directory as the program The coordinates of the points are given by ordered triples of numbers x y z where z is always 0 0 3 5 Radial Roller Follower Example In this example the follower s motion is defined by a dwell from 0 to 90 Then the follower r
46. effect of branching can be illustrated In this example the design was based on the rectification procedure and the coupler passes through all of the positions The zooming feature is also available in the RBGSliderCrankDesign program To zoom in locate a rectangular marquee around the figure and click the left mouse button To zoom out click the left mouse button aS oe 1 4702 1 1778 ee es ee a Fig 4 11 Selection of the slider and circle points that will give no branch problem Fig 4 12 The analysis window when the when the slider is the driver 58 4 5 Rigid Body Guidance Using an Elliptic Trammel Linkage RBGEITrammelDesign This routine is used for the design of a double slider mechanism or elliptic trammel for rigid body guidance For this design two slider points must be chosen on the circle of sliders The program is structured in two windows The first window is the design window where all of the input data are identified The second window is the animation window where the linkage can be verified The design process does not depend on which slider is chosen as the input Once the slider points are chosen only one assembly mode is possible Therefore rectification is not an issue 4 5 1 Design Window for Elliptic Trammel Linkage The design window for the elliptic trammel is shown in Fig 4 13 In the design window frames are utilized to group three types of geometry the slider point
47. eight The optimum counterbalance is the one that produces the minimum value for the maximum shaking force magnitude a eee res oaa E Fig 2 47 Design window for the shaking force routine Fig 2 48 Window showing variable definitions for shaking force program 38 Fig 2 49 Analysis window for the shaking force routine 39 3 0 Program for Cam Design 3 1 Introduction The cam design program is called Cam2 This program allows the user to select the cam follower to be either translating or oscillating and also allows flat faced or roller faced followers The follower motion types included in the program are uniform harmonic cycloidal and polynomial The program also includes two procedures for optimizing the follower motion Finally the program generates the cam profile using the procedures described in the textbook This manual will include only a description of how to use the program It is assumed that the reader is familiar with the material in Chapter 8 of the textbook The program itself was written by Michael Stevens as part of the research associated with his MS thesis His thesis is entitled Interactive Design of Plate Cams with Optimal Acceleration Characteristics and was completed in 2002 It is available through The Ohio State University 3 2 Cam Follower and Motion Specification Window The program employs a graphical user interface and has three separate windows The first of these windows Fig
48. elated by the ratio of the pitch radii As p increases decreases for external gears The resulting relationship is 0 ip Oo B 6 The coordinates must be transformed through four sets of coordinate systems local xy to global XY to global TS and finally to local ts Referring to Fig B 8 the xy and XY coordinate systems pertain to gear 2 whereas the TS and ts systems refer to gear 3 The x coordinate axis is along the center line of the tooth in gear 2 while the f axis is along the centerline of the contacted gear on gear 3 The three successive transformations are given in the following rh ind ewe OF dso ejir td Lue coelis The overall transformation is t cos sin x C4 cos B7 n hee GONG at p Casin0 CY These equations define the conjugate tooth form relative to gear 3 Example B 1 Conjugate Tooth Form for Straight Toothed Gearing Problem One tooth form which has been used on very large gears such as the ring gear on draglines is straight toothed form This is the same form as is used on a simple rack except that the pitch curve is a circle instead of a straight line Therefore the conjugate tooth form is not an involute For the problem assume that gear 2 has a pitch diameter d2 of 20 feet and the diametral pitch D is 5 teeth per foot of pitch diameter The tooth surface is inclined with the centerline at an angle of 25 The mating gear gear 3 is an external gear with 30 teeth
49. en raja 3 333 in the direction 2 Next compute Jg From Eq A 13 Je i _ 2 291 0 That is Jg is located at B We BIB could have determined this by inspection because point B traces a straight path on the frame Therefore B must be on the inflection circle by definition Given J J and Jg the inflection circle can be drawn as shown in Fig A 11 Inflection Circle Fig A 11 Inflection circle for slider crank mechanism in Example A 4 Example A 5 Inflection Circle and Radius of Curvature Problem Determine the radius of curvature of the path that point C3 in Fig A 12 traces on the frame Link 3 rolls on link 4 without slipping The dimensions for the linkage are as follows AA 1 cm B A 1 AC 2 cm and the radius of the roller is 0 2 cm Solution To solve the problem we first need to find the inflection circle As in Examples A 2 and A 3 we need to find three points lying on the inflection circle to define it Three points which can be foun from the information given are J J4 and Jg Point B3 is not indicated directly on the drawing however we can locate B3 by visualizing the path that B traces on link 3 That path is a straight line therefore the center of curvature of the path is at infinity Points B and B switch roles when 80 we invert the motion and make link 3 the reference and allow the frame to move Thus B3 is the center of curvature of the path of B relative to link 3
50. er analysis The color of the coupler point is changed when it is selected In the animation window the Analysis button is not available until the user selects the exact coupler point using the mouse 2 9 3 The Analysis Window for Slider Crank Coupler Point Analysis Program The analysis window Fig 2 40 is the same as the other analysis windows Up to four plots can be displayed The plot options shown in Fig 2 40 are the mechanism the slider distance the magnitude of the coupler velocity and the magnitude of the coupler acceleration The coupler curve shown in the analysis window is dished where each dash corresponds to 5 degrees of crank rotation This gives a means of coordinating the travel of the coupler point to the position of the crank Also the length of the dashes gives a visual comparison of the relative speed of the coupler point as it moves along the coupler curve 2 10 Program for Analyzing Four Bar Linkage Centrodes This routine generates the fixed and moving centrodes for the coupler of a four bar linkage given the linkage geometry The program consists of a design window where the linkage geometry is defined and an animation window where the motion is animated 2 10 1 Design Window for Centrode Program The GUI layout for the design window Fig 2 41 is quite simple compared to other examples Different centrodes can be generated by changing the link lengths for the four bar linkage A simple definition of terms is dis
51. er where n is specified by the user and is larger than 6 Values allowed are 7 8 9 and 10 The optimization procedure uses six of the polynomial coefficients to match the position slope and curvature conditions at both ends of the segment The remaining coefficients are then selected to minimize the maximum acceleration in the range of the segment The second optimization procedure involves beginning with an initial profile and then fitting a set of splines to the acceleration curve The control points of the splines are used as the design variables in optimizing the acceleration curve for minimum acceleration The control points are selected such that continuity is maintained in position velocity and acceleration at both ends of the segment Both procedures work well and it normally does not matter which procedure is chosen for the optimization Typically the improvement in the acceleration curve is modest but it is worth the effort because it is fast and all functional requirements are satisfied 42 The segment of the curve to optimize can be selected by clicking the Next Segment button until the desired segment is selected As segments are selected they change from blue to red By clicking the Minimize button the selected segment will be minimized using the specified optimization procedure Also the numerical value of the acceleration peaks can be displayed by clicking the Next Peak button until the desired peak acceleration is indicat
52. ering The Ohio State University 206 West 18th Avenue Columbus Ohio 3210 Ph 614 292 6864 Fax 614 292 3163 Copyright 2003 The Ohio State University All Rights Reserved Fig 1 1 Main screen after typing mainmenu File Edit View Insert Tools Window Help This menu show the index for a series of Cam Design kinematic programs Those programs adopt a Rigid Body Guidance Design graphical user interface to provide a friendly Gear Design environment for users Through the programs users can see the designed mechanism and its animation immediately These menu includes some basic mechanisms in Click to expand this topic Fig 1 2 Menu of program types The plus sign before each of the topics indicates that there are subtopics Note that the Run button on the bottom right hand side of the window cannot be actuated until one of the subtopics is selected To select a program first click on the topic you want and then click on the desired program To run the program click on the Run button Note that you cannot run the programs by simply double clicking on them When you are done with the programs click on the Close button to terminate the program 1 3 1 Programs Under LinkageDesign Under Linkage Design there are 10 subtopics as shown in Fig 1 3 These subtopics corresponding to individual programs are 1 Crank Rocker Design CRDesign 2 Cognate determinations of a four bar linkage CognateAnalys
53. es a Stevenson type six bar The analysis is conducted by treating the six bar as an assembly of a four bar a rigid body and a dyad As in the previous cases the program is structured into a design window where the linkage information is input and an analysis window 24 where the output is displayed graphically This program is intended as an analysis program for the linkages designed using the procedure given in Section 6 6 of the textbook 2 7 1 The Design Window for Six Bar Program The design window is shown in Fig 2 29 Again the GUI layout is similar to the others except for the additional required input data The location of the third bushing is an input from the user and this program makes it mouse movable The coupler point is also mouse moveable The assembly mode refers to the output dyad of the six bar linkage The opposite assembly mode corresponding to Fig 2 29 is shown in Fig 2 30 The variables that must be input are shown in a figure if the Definition button is selected The resulting figure for the six bar linkage is shown in Fig 2 31 The remaining push buttons are the same as in the previous programs Fig 2 29 The Design window for the six bar design 25 Fig 2 30 The second assembly mode for six bar output dyad Fig 2 31 Window showing variable definitions for six bar program 2 7 2 The Analysis Window for Six Bar Program The analysis window can display 1 2 3 or 4 plots The plot options
54. es given in Chapter 3 of the textbook To begin make the following substitutions 1 tan2 a 1 tan2 2tan a 1 cant T tan A YA Xana B n C nna Then the equation to be solved is A Bsing Ccosp 0 A H a Jde 1 T2 1 T2 and the solution is _ B JVB2 A24 C2 i A C and go 2tan Tt Note that all points on the known gear may not be possible choices for a contact point If the candidate point chosen is an impossible contact point B A C will be negative To locate the angle for all possible points x4 and ya it is only necessary to begin at one end of the known contour and increment x until the other end is reached The increments of x need not be uniform B 1 3 Coordinate Transformations Once the point of contact is located it becomes necessary to transform the coordinates from gear 2 to gear 3 The transformation will involve the following parameters C4 center distance for two gears 0o initial angle for axis t on gear 3 when the angle is zero N gt number of teeth on gear 2 N3 number of teeth on gear 3 92 The center distance is given by Ca nh in B 4 66s where 1 is equal to 1 for an external gear and 1 for an internal gear The initial angle O for the t axis on gear 3 is x minus the angle that subtends an arc which is one half of the tooth thickness measured at the pitch circle This angle is equal to b 1 N a B 5 The angles O and are r
55. f curvature of the path of point A then P TA A and 2 TAIT raa FAI VIA 1 Now raf Tet TAQ SO TA A r Al A AlI A 11 TAII FA JA that can be viewed as the geometric form of the Euler Savary Eq A 10 A 3 4 Different Forms for the Euler Savary Equation The Euler Savary Equation can be expressed in several different ways and the different forms are useful depending on the known quantities when a problem is formulated For example another form can be derived from Eq A 11 as follows Tir Va FA A FA I 11 A aJa Var HJA or rur rua iur rusa TR CADA Oraa a rAr Simplifying rain 117A Cua ranr ma ra O Now division by 14 1 1 7 11 4 gives S ete era ae Tyas uga YANI or Pe aera ee A 12 Fjar Vain F 76 ay Inflection Circle Fig A 7 The geometric relationship of the inflection circle with a center and circle point pair A A Some of the different forms for the Euler Savary Equation are summarized in Table A 1 The terms used in Table A 1 are shown schematically in Fig A 8 Most of the forms can be derived directly from Eq A 11 as was done in the case of Eq A 12 however several of the forms are based on Oy and Of the centers of curvature of the moving and fixed centrodes corresponding to the instant center These forms are derived by Hall Each form of the equation is based on a x Fixed Centrode A Op Movin
56. feature that is not covered in the textbook This will be discussed briefly when the topic is covered to describe the use of the programs 4 2 Rigid Body Guidance Using a Four Bar Linkage RBG4barDesign This routine is used for the design of four bar linkages with either center points or circle points as input The user can specify three coupler positions and angles The pole locations between each two coupler positions are calculated and shown as a marker in the graph The circle of sliders corresponding to the three image poles for position is shown in black After all of the input data are identified the linkage can be animated to determine if the linkage moves through all of the positions identified The program is structured in two windows The first window is the design window where all of the input data are identified The second window is the animation window where the linkage can be verified 48 4 2 1 Design Window for Four Bar Linkage for Rigid Body Guidance The design window is shown in Fig 4 2 In the design window frames are utilized to group three types of geometry the center points circle points and coupler positions Editable boxes for user input of the three coupler positions are provided The user can either input the positions numerically or move the locations and angles of the three coupler positions by mouse dragging The GUI implementation also allows users to drag any circle or center point continuousl
57. from the driven circle point The slider point must be chosen to lie on the parts of the circle of sliders that are outside of the colored regions Again the acceptable regions may be very small or even nonexistent If there is no linear region that is free of color then there is no solution that will be free of the branch problem In the example shown in Fig 4 10 the linkage chosen has the driver slider point inside the forbidden 56 colored region Therefore that linkage will have a branch problem This illustrates the importance of identifying the actual driver since if the crank is the driver there will be no branch problem as illustrated in the example from Section 4 3 As indicated in Fig 4 10 only a small part of the circle of sliders is in the acceptable region We have chosen a different set of circle and slider points in Fig 4 11 The resulting linkage does not have a branch problem 1 739 0 15507 S Fig 4 10 The design window for the slider crank design program slider driving 4 4 2 Analysis Window When the Slider is the Driver The three coupler positions in the analysis window Fig 4 12 use the same colors as in the design window Itis therefore possible to identify which position is missed if the rectification procedure is not applied In the analysis window 1 or 2 plots can be shown and animated The plotting options are the animations for the two assembly modes By animating both assembly modes the
58. g A 13 Location of instant center Iy3 and centrodes for a four bar linkage 82 Fig A 14 The path of 213 can be traced by 5 as shown Here link 5 is the ball captured by the two yokes on links 2 and 4 A 5 Analytical Form for Euler Savary Equation The approach used in Examples A 3 A 5 uses one of the forms of the Euler Savary Equation given in Table A 1 These equations lend themselves to the graphical solution of the Euler Savary Equation To use the equations we must establish a positive direction and identify that direction in the calculations When programming the equations it is convenient to work initially with points or absolute vectors rather than relative vectors From the absolute vectors the vectors in Table A 1 can be established For example FAI A Faa Ira Ta FAJI Fal Irs ri rysa Fasal Fa ral ru Ful W nl etc With these substitutions the equations in Table A 1 can be programmed easily to compute the unknowns MATLAB routines for the most common calculations are on the disk included with this book Combinations of these routines can be used to write programs for finding the inflection circle and determining the center of curvature of selected points on different links A routine for making these calculations are given for a four bar linkage A 6 The Bobillier Constructions As indicated in Example A 3 if we have a four bar linkage we can determine the inflection circle by locating J4 and
59. g Centrode Instant Center lt IC Tangent IC Normal Inflection Circle A 2Hall A S Kinematics and Linkage Design Balt Publishers West Lafayette IN 1961 TI Fig A 8 Summary of terms for Euler Savary equation Table A 1 Summary of forms of Euler Savary equation Using the ray I A different forms of the Euler Savary Equation are 2 _ IMI 1 1 1 TA A eck FA JA Tomir Yorn I 2 umanmi a ee ee FAI TIAIT Fjar Vain YA TA A a TAII ee Yair Ty 1 COSY A TJA J1 FA 1 2 1 1 1 1 TAIL COSY A VIAIT TAIT TAI A ToM I JOFII Yair A I VA A single ray through the instant center Z Therefore relative to the ray each vector can be treated as a signed number One direction from I can be taken arbitrarily as positive distances in the other direction are automatically taken as negative Examples of different locations of circle points and center points are shown in Fig A 9 Fig A 9 Locations for different points according to the Euler Savary equation Example A 4 Locating the Inflection Circle for Four Bar Linkage Problem Locate the inflection circle for the four bar linkage shown in Fig A 10 Solution To find the inflection circle we need to find three points lying on it Three points that can be found from the information given are J J and Jg First locate the instant center J From Chapter 3 the location is where a
60. g lamina as that lamina passes through the design position This is done by resolving the acceleration of that point into components tangent to and normal to its path Let n be a unit vector normal to the path that A traces on the frame and let t be a unit vector tangent to that path We know that the velocity of the point will be tangent to the path that the point traces on the frame Therefore t is in the v direction and n is directed so the k x t n Then the acceleration of point A can be written as as d4 a at a n When the acceleration is expressed in terms of the normal and tangential components it is the normal component which is a function of velocity and geometry An expression for this component was derived in Section 3 3 2 when coincident points were considered In particular the acceleration of A can be rewritten as E R aa aht a A 4 where p is the radius of curvature of the path that the point A traces on the frame Equation A 4 is derived in most undergraduate engineering mechanics texts and a detailed derivation is given by Hall If we take the dot product of n with each side of Eq A 4 we get y2 n aa a p or 2 2 VA _ Y are A 5 Hall A S Kinematics and Linkage Design Balt Publishers West Lafayette IN 1961 70 Equation A 5 allows us to locate the center of curvature of the path of any point in a linkage once the basic velocity and acceleration analyses hav
61. gn Under Cam Design there is one program Cam2 as shown in Fig 1 4 This program designs the follower displacement schedule and generates the cam profile for one of four types of followers translating roller follower translating flat faced follower oscillating roller follower and oscillating flat faced follower 1 3 3 Programs Under Rigid Body Guidance Design Under Rigid Body Guidance Design there are three subtopics as shown in Fig 1 5 These subtopics correspond to individual programs written for designing linkages for three position rigid body guidance The three programs are 1 Rigid body guidance or motion generation using a four bar linkage RBG4barDesign 2 Rigid body guidance or motion generation using a crank slider mechanism RBGCrankSliderDesign 3 Rigid body guidance or motion generation using a slider crank mechanism RBGSliderCrankDesign 4 Rigid body guidance or motion generation using a double slider or elliptic trammel mechanism RBGEITrammelDesign 1 3 4 Programs Under Gear Design Under Gear Design there are two programs as shown in Fig 1 6 The first program Arb2ThDesign will compute and draw the tooth profile conjugate to an arbitrarily specified tooth 4 form and the second routine GeardrDesign will draw an involute profile given the parameters of the hob used to generate the gear form THF OHIO STATE UNIVERAITY Nn to expand this topic ol hs Bh lana EB Fig 1 4 Subtopic under
62. gram uses three windows The first is a design window where the linkage and coupler point grid is defined The next is an animation window that displays the coupler curves for the points identified in the analysis window One of the coupler points can be selected for further analysis The third window is the analysis window for the mechanism with the single coupler point that is selected 2 8 1 The Design Window for Four Bar Coupler Point Analysis Program The design window is shown in Fig 2 33 In the design window a uniform grid is created When a point is selected its grid marker changes to a hollow circle The user can specify both the length and height of the coupler rectangular grid and the number of rows and columns of points In addition the user can select the specific grid points that will be analyze further These are identified in the Animation range The push button set of Definitions Zoom Out and Zoom In buttons is moved to the space below the plot because all the grid creation options are arranged to be close together The Definitions page gives a description of most of the input variables This page is shown in Fig 2 34 27 The GUI program checks the valid range for the grid to avoid interrupting execution If invalid data are inputted the previous data are retrieved and an error message is shown in the status bar below the plot 2 8 2 The Animation Window for Four Bar Coupler Point Analysis Program The animation window
63. h value is input the program will compute any value that it can based on the inputted value Follower Radial Flat z Radial Flat Oscillating Flat Radial Roller Oscillating Roller Fig 3 2 Follower Type Pull Down Menu Polynomial Fig 3 3 Follower Motion Type Pull Down Menu After the cam and follower parameters are specified and the follower motion is defined the cam and follower can be displayed By clicking the Refresh Drawing button shown in Fig 3 1 the user can update the drawing to show the cam follower system defined by the current inputs When the cam follower system and motion program is satisfactory select the Motion Plots button to advance the program to its next window 3 3 Motion Plots Window The motion plots window is shown in Fig 3 4 This second screen shows the follower s displacement velocity acceleration and jerk plots In addition this screen allows the option to apply an optimization procedure to the follower s motion The optimization method can be chosen from the pull down menu displayed in Fig 3 5 4 Fig 3 4 Cam motion plots window Position Polynomial Position Polynomial Acceleration Spline Fig 3 5 Optimization Method Pull Down Menu The optimization procedure minimizes the maximum acceleration in any given segment of the curve Two methods can be used The first method Position Polynomial approximates the segment selected by a polynomial of n ord
64. hat from Eq H 2 whatever solution is used Pa Iniol is constant That is regardless of the angular velocity the ratio of the velocity of the point at the origin to that angular velocity in the design position is constant Put another way dpo d 69 is constant where po is the position vector from the origin of the coordinate system to the coupler point O which has coordinates momentarily of 0 0 It is convenient to say that we are specifying the velocity state of the moving body but it is more precise to say that we are specifying the derivative of the position of a point on the coupler with respect to the coupler angle A 3 Three Infinitesimally Separated Positions A 3 1 Center of Curvature of Path of Moving Point Relative to Frame Specifying three infinitesimally separated design positions is equivalent to specifying a position of the moving lamina and its velocity and acceleration states in that position In addition to the velocity of the point in the moving lamina coincident with the origin and the angular velocity we must specify the acceleration ao of the point at the origin and the angular acceleration of the moving lamina The acceleration of any point A in the moving lamina can then be found a4 do Aajo do A X Taio 0 X W X Taro do Q X Tajo Faso A 3 Given the velocity and acceleration states of the moving lamina we can find the radius of curvature of the path of any point in the movin
65. he program is also structured in two windows a design window Fig 2 9 and an analysis window Fig 2 10 In the analysis window up to four plots can be displayed Typically these show each cognate separately along with Robert s linkage 2 3 1 The Design Window for Cognate Program The GUI displays the four bar linkage and coupler curve on the left hand side of the design window The editable link lengths frame crank coupler rocker coupler point radius are grouped together on the right hand side of the window The angle between the coupler point radius and coupler and the frame angle are shown at the bottom of the window The non editable link lengths are grouped in another frame on the right hand side of the design window The radio button applies to the assembly mode desired The first assembly mode is shown in Fig 2 9 and the second in Fig 2 10 The user also can change the coupler point and curve by dragging the coupler curve around the screen Two buttons Zoom in and Zoom out scale the plots because the parts of the mechanism and or coupler curve might go outside of the plot window when the user drags the coupler point If the nfo button is chosen the Robert s linkage information shown in Fig 2 11 is displayed 2 3 2 The Analysis Window for Cognate Program The analysis window of the cognate design routine has the same layout as that of the crank rocker routine The only difference is the plot contents For this routine the a
66. in 104 4 3 13 ft The initial angle 09 is given by Eq B 5 as rli L 41 L 60 x a1 3 037 rad To find the conjugate gear form it is only necessary to increment 6 from 0 to Az B and compute the x y coordinates of the points and normals using Eq B 8 The angle corresponding to the selected point can then be found by solving Eq B 3 using the procedure given above The angles 6 fora given value of is given by Eq B 6 0 i 4 amp 26 0 209 h 3 Knowing 0 and the coordinates of the conjugate point on gear 3 are given by Eq 6 39 or thea Siete ea Once the values of t s on gear 3 are known for each value of x y on gear 2 the tooth form on ge 3 can be computed Clearly this procedure is best done using a computer program to determine the tooth profile of gear 3 95
67. ine when two plots are chosen 2 5 Program to Analyze a Four bar Linkage FourbarAnalysis This routine analyzes a four bar linkage for which either a crank or the coupler can be specified as the driver Associated analysis plots for the angular position of the rocker and coupler and the velocity of the rocker are shown in the animation As in the previous two cases the four bar program is structured with a design window and an analysis window 2 5 1 The Design Window for Four bar Program The design window is shown in Fig 2 20 The design window has several radio button sets to set different features of the program A frame groups each set Otherwise the design window is similar to that of the cognates GUI routine The radio buttons are associated with the following options 18 a Line or triangle The coupler can be drawn using either a line or a triangle The coupler is represented by a line in Fig 2 20 and by a triangle in Fig 2 21 b Crank or coupler driven Either the crank or the coupler can drive the linkage c One mode or two The linkage can be analyzed and the coupler curve displayed for either one mode or two If only one mode is chosen the coupler curve for that assembly mode only will be shown If both modes are chosen the coupler curve for both assembly modes will be displayed This is shown in Fig 2 22 d Assembly mode Either the 1 or 1 linkage assembly mode can be analyzed for positions and velocities
68. is 3 Design of double rocker linkage DoubleRockerDesign 4 Simple four bar linkage analysis FourbarAnalysis 5 Simple slider crank linkage analysis SliderCrankAnalysis 6 Six bar analysis program SixbarAnalysis 7 Simulation of Hrones amp Nelson coupler curve atlas for four bar linkages HR CrankRockerAnalysis 8 Simulation of Hrones amp Nelson coupler curve atlas for slider crank linkages HRSliderCrankAnalysis 9 Display of four bar linkage centrode curves CentrodeDesign 10 Display of four bar linkage inflection circle and calculation of center of curvature Inflection4barAnalysis 11 Analysis of shaking forces in slider crank mechanism ShakeAnalysis Mechanical i KINEMATIC PROGRAM MENU Linkage Design The likage design and analysis category has a total CRDesign of ten kinematic routines It is mainly used for CognateAnalysis the synthesis and analysis of linkages DoubleRockerDesign brief description of the routines is FourbarAnalysis provided here SliderCrankAnalysis SixbarAnalysis 1 CRDesign is for crank rocker design HrCrankRockerAnalysis 2 CognateAnalysis is for the cognate analysis of a HrSliderCrankAnalysis four bar linkage CentrodeAnalysis p Inflection4barAnalysis d ShakeAnalysis Click to expand this topic Cam Design Rigid Body Guidance Design Gear Design Fig 1 3 Subtopics under Linkage Design 1 3 2 Programs Under Cam Desi
69. ises with cycloidal motion during the rotation of the cam from 90 to 180 The follower then dwells for 60 of cam rotation and then returns with simple harmonic motion for the cam rotation from 270 to 360 The amplitude of the follower translation is 2 cm and the follower radius is cm The base circle radius of the cam is 4 cm and the offset is 0 5 cm Finally the cam s direction of rotation is clockwise Once the follower motion is entered into the design program the Segment Data looks like Fig 3 8 The cam and follower data appears as it does in Fig 3 9 Finally the cam follower system is shown in Fig 3 10 The motion plots for the follower motion specified in Fig 3 8 look like those in Fig 3 11 43 Fig 3 7 Cam follower animation window Fig 3 8 Segment Data for Radial Roller Example Fig 3 9 Cam and Follower Data for Radial Roller Example Fig 3 10 Cam Follower System for Radial Roller Example The motion program can be optimized In this example the acceleration splines method of optimization is applied to the rise segment The optimization parameters are shown in Fig 3 12 After the optimization the peak acceleration is reduced from 5 09 to 4 70 The optimized motion plots are displayed in Fig 3 13 45 Fig 3 11 Motion Plots for Radial Roller Example Acceleration Spine 12 Fig 3 12 Optimization Inputs for Radial Roller Example 46 Fig 3 13 Optimi
70. n extension of AA intersects the line defined by BB Next find Ja This can be found by rewriting Eq A 11 as 78 TAIJA at A 13 From the geometry given in Fig A 10 raz ABsin 30 2 Substituting numbers into Eq p A 13 rasza Au a in the direction of r a This locates J between A and A Next Al A compute Jg using From the geometry given in Fig A 10 m 1 ABcos 30 2 3 Substituting numbers into Eq 2 Ba _ 2W3P A 13 again gives B I gt V3 in the direction of rg g This locates Jg between B F IB B 48 and B also Given J J4 and Jz the inflection circle can be drawn as shown in Fig A 10 Example A 5 Inflection Circle for Slider Crank Mechanism Problem Locate the inflection circle for the slider crank mechanism shown in Fig A 11 The li dimensions are AA 2 m and AB 4 m Solution Again to find the inflection circle we need to find three points lying on it Three points which can be found from the information given are J J4 and Jg First locate the instant center I using the procedure given in Chapter 3 The distance AB is given by AB 2c0s30 12 2 3 968 and r4 is given by 79 rar 3 968 cos30 4 582 Also rar 4 582 2 2 582 and B I ra z TA A sin 30 2 291 Tat _ 2 5822 FAJ A of rasa Therefore A is between A and J as shown in Fig A 11 Next find J4 using Eq A 13 For the values giv
71. nalysis window has all of the mechanism plots More than one instance of a graphic object can be generated if one cognate 12 linkage is chosen for more than one plot Several different plot options are shown in Figs 2 12 2 14 Fig 2 9 The GUI design window for the cognates routine Fig 2 10 Linkage for second mode 13 Fig 2 11 Information page for cognates program MQ Cognate Fig 2 12 Animation of single linkage 14 MO Cognate Fig 2 14 The GUI analysis window for the cognates routine 2 4 Program for Designing a Double Rocker Four Bar Linkage DoubleRockerDesign This routine facilitates the design of a four bar linkage as a double rocker The input information are the initial positions of the input and output links rockers and the input and output rocker angles The equations are developed from Section 6 2 2 of the textbook The program is also structured in two windows a design window Fig 2 15 and an analysis window Fig 2 16 In the analysis window one or two plots can be displayed These show the animated linkage and a plot of the output angle as a function of the input angle 15 Fig 2 15 The GUI design window for the double rocker design routine Fig 2 16 The GUI analysis window for the double rocker design routine 2 4 1 The Design Window for the Double Rocker Design Program The GUI displays the four bar linkage on the left hand side of the design window Th
72. only is displayed 5 3 1 Design Window for GeardrAnalysis Program The design window is shown in Fig 5 6 Half of the generating hob or rack tooth form is displayed in the graphics window and the generating tooth information is shown in the frame to the right of the figure o z9 s Fig 5 6 Design window for GeardrAnalysis program Geometric information on the generated tooth form is shown if the Definition button is selected 64 This is shown in Fig 5 7 For the generating hob or rack the user may input the following 1 Addendum constant for rack 2 Dedendum constant for rack 3 Radius of tip of rack tooth 4 Radius of fillet of rack tooth 5 Pressure angle in degrees 6 Diametral pitch for the generated gear the user may input the following 1 Number of teeth 2 Addendum constant Fig 5 7 The definitions window for the GeardrAnalysis program The gear tooth coordinates are generated numerically Therefore it is necessary to identify the number of points in the different regions of the hob Five regions are identified and the user may input the number of points in each region The accuracy generally increases with the number of points 1 Number of points in region of rack tip land 2 Number of points in region of rack tip radius 3 Number of points in region of rack flank 4 Number of points in region of rack base radius 5 Number of points in region of rack bottom land 65 After th
73. output oscillation angle theta must be input along with either alpha or Q The radio button identifies the specific input variable The user enters the value by moving the cursor over the value given and retyping a new value To actually enter the value the return key must be pressed on the computer keyboard The locus for one extreme location for the output pivot B2 is the blue arc To select new designs either input the angle beta directly or click on and drag the green arrow to change the beta values Here beta is the counterclockwise angle between the positive X axis and the green vector see Fig 2 2 The design is automatically updated as beta is changed In addition the transmission angle range is shown and updated dynamically in the status bar at the bottom of the design window The program also has an optimization feature If the Optimization button is selected the program will determine the value for beta that optimizes the transmission angle For the input values shown in Fig 2 1 the optimized output values are shown in Fig 2 4 Fig 2 1 The design window for the crank rocker design routine Rocker Tengih Fig 2 2 The definitions window for the crank rocker design routine Crank Rocker Design Search Demos B Begin Here Crank Rocker Design Default Topics gt Release Notes for Release 13 k paa ie Crank Rocker Design MATLAB gt amp Control System Toolbox CRDesign is a program to de
74. ow for the elliptic trammel routine 4 6 Situations When Rectification Procedure Fails As indicated earlier in most circumstances the rectification procedure implemented in the programs identifies the regions that will give unacceptable linkages However after using the programs the observant student will notice that the four bar and slider crank programs will sometimes identify linkages that will not move through all of the positions The problem that is not addressed is the circuit defect 5 This occurs when the two assembly modes of linkage are separated for all positions of the driver link When this happens it will be obvious from the animation that the linkage cannot move through the range of motion identified without disassembly To resolve the problem choose a different set of circle center of slider points 4 7 References l Waldron K J Range of Joint Rotation in Planar Four Bar Synthesis for Finitely Separated Positions Part I The Multiple Branch Problem ASME Paper No 74 DET 108 Mechanisms Conference New York 1974 Waldron K J Solution Rectification in Three Position Motion Generation Synthesis Department of Mechanical Engineering The Ohio State University pp 301 306 Chuang J C Strong R T and Waldron K J Implementation of Solution Rectification Techniques in an Interactive Linkage Synthesis Program Journal of Mechanical Design Trans ASME Vol 103 1981
75. played if the Definitions button is selected The window is shown in Fig 2 42 2 10 2 Analysis Window for Centrode Program The analysis window Fig 2 43 animates the motion of the linkage This shows that apparent rolling of the moving centrode on the fixed centrode as the linkage moves Eey Fig 2 39 Animation window for slider crank mechanism coupler point atlas program Fig 2 40 Analysis window for slider crank mechanism coupler point atlas program 33 Fig 2 41 Design window for the centrodes routine Fig 2 42 Window showing variable definitions for centrodes program 34 File Edit View Insert Tools Window Help Fig 2 43 Analysis window for the centrodes routine 2 11 Program for Analyzing Path Curvature Inflection4bar Analysis This routine graphically displays the solution of the Euler Savary equation for a four bar linkage The Euler Savary equation gives a relationship between points in the coupler of a four bar linkage and their centers of path curvature Because the theory for path curvature is not covered in the textbook a brief description of the procedure is given in Appendix A The inflection circle routine is developed in a single window of the GUI program as shown in Fig 2 44 In the graphics window the four bar linkage and the inflection circle is displayed The coupler point is designated by a green circle and the center of the coupler point s path by a red
76. pp 657 664 Waldron K J Graphical Solution of the Branch and Order Problems of Linkage Synthesis for Multiply Separated Positions Journal of Engineering for Industry Trans ASME Series B Vol 99 1977 pp 591 597 Mirth J A and Chase T Circuits and Branches of Single Degree of Freedom Planar Linkages ASME Journal of Mechanical Design vol 115 no 2 pp 223 230 1993 60 5 0 Program for Displaying Gears 5 1 Introduction The gear group includes two programs as shown in Fig 5 1 The first program Arb2thDesign determines the tooth form that is conjugate to a straight sided tooth The second program GeardrAnalysis draws a gear tooth given the geometry of the generating rack Each of the programs will be discussed separately File Edit View Insert Tools Window Help Mechanical Sie KINEMATIC PROGRAM MENU Linkage Design Cam Design Rigid Body Guidance Design RBG4har Design RBGCrankslider Design RBGElTrammelDesign The Gear category has two routines One draws involute gears and the other will generate the gear tooth form for an arbitrary conjugate gear The conjugate gears are animated Arb2thDesign GeardrAnalysis Click to expand this topic Fig 5 1 Programs available under gear design 5 2 Arb2thDesign Program This program displays two windows a design window where the input data are identified and an analysis window The analysis windo
77. r balance weight is also determined As in the cases of the majority of the programs the shaking force program is divided into a design window and an analysis window 2 12 1 The Design Window for Slider Crank Shaking Force Program The design window is shown in Fig 2 47 The basic mechanism is a slider crank and the input motion is similar to that for the slider crank program in Section 2 6 This routine is focused on the calculation of the shaking force the counter balance weight and its optimization A large space below the plot is utilized to output the numerical results associated with the shaking force The output data are in blue for emphasis In addition to the link lengths the acceleration of gravity and the weights of the crank coupler piston and counter balance weight must be input It is assumed that the weights and the acceleration of gravity are in consistent units As in the case of the other programs a graphics window that is displayed when the Definitions button is selected defines most of the variables This window is shown in Fig 2 48 2 12 2 The Analysis Window for Slider Crank Shaking Force Program The analysis window is shown in Fig 2 49 Again up to four plots can be displayed at one time The display options are the mechanism the polar shaking force diagram for no counterbalance weight the shaking force diagram for the given counterbalance weight and the shaking force for 37 the optimum counterbalance w
78. rallel to JZ Its intersections with lines A A and B B are points Ja Jp respectively 4 Draw a circle through points J J4 Jg This is the inflexion circle 88 Appendix B Procedure for Drawing Conjugate Tooth Form B 1 General Conjugate Tooth Forms The fundamental law of gearing requires that when two gears are in contact the angular velocity ratio is inversely proportional to the lengths of the two line segments created by the intersection of the common normal to the two contacting surfaces and the line of centers This ratio is constant if the common normal intersects the centerline at a fixed point the pitch point The tooth forms satisfying this condition are said to be conjugate The flat sided rack and involute tooth form are one example of conjugate tooth forms however there are an infinite number of other tooth forms which can be conjugate In this section we will generalize the procedure given in Section 10 12 of the textbook to develop a procedure for finding the tooth form which is conjugate to a general tooth form The information in Appendix B was originally contained in the textbook however it was removed because of page constraints Therefore the entire development is given here The program that will draw a conjugate gear is described in Chapter 5 of this manual B 1 1 Required Geometric Parameters Several parameters must be known about both gears to determine the unknown tooth form which is conjugate to th
79. reeseseresresesesesereses 22 2 6 2 The Analysis Window for Slider Crank Program seeeeseeseereeseesessresrersersresresseees 22 2 7 Program for Analyzing a Stevenson s Six Bar Linkage SixBarAnalysis ceeeeeeee 24 2 7 1 The Design Window for Six Bar Program esceecessecsseceseeesneeeseecaeeeseessaeenaeenees 25 2 7 2 The Analysis Window for Six Bar Program esseseseseeseesessresressessresressersersresreesesee 26 2 8 Program for Generating Atlas of Coupler Curves for Four Bar Linkage HRCrankRockerAnalysis enrr roseo ia koerse DE E EE EEA REEE A ENSE 21 2 8 1 The Design Window for Four Bar Coupler Point Analysis Program eeeeeeeeeee 27 2 8 2 The Animation Window for Four Bar Coupler Point Analysis Program 28 2 8 3 The Analysis Window for Four Bar Coupler Point Analysis Program 06 28 2 9 Program for Generating Atlas of Coupler Curves for Slider Crank Mechanism CHR SliderCrank Design is 4s3sss5 snegscavanacsanadvenasadiyaaseavenceseeasavenscansywaeseany caueeei O E 30 2 9 1 The Design Window for Slider Crank Coupler Point Analysis Program 30 2 9 2 The Animation Window for Slider Crank Coupler Point Analysis Program 32 2 9 3 The Analysis Window for Slider Crank Coupler Point Analysis Program 32 2 10 Program for Analyzing Four Bar Linkage Centrodes ee eeseeecceeseeeeseeseeeeseeeneeenseensees 32 2 10 1 Design Window for Centrode P
80. rogram sessessessesseeresseseresssresseserssressesseesressesee 32 2 10 2 Analysis Window for Centrode Program essesseseseseesessessresressersresressessesseesressesee 32 2 11 Program for Analyzing Path Curvature Inflection4barAnalysis cesceesseceseeeeeeeeeees 35 2 12 Program for Analyzing the Shaking Force in a Slider Crank Program ShakeAnalysis 37 2 12 1 The Design Window for Slider Crank Shaking Force Program ccsscceesseeees 37 2 12 2 The Analysis Window for Slider Crank Shaking Force Program cccesseeees 37 3 0 Prosram for Cam Desig scrissero aee e aaa K A Oo e Ao a E 40 S SEVERE CUNO PAATE NE E 40 3 2 Cam Follower and Motion Specification WindoW ssesessseesssessessseessesessseesseesseesseeessees 40 Section Page 3 3 Motion Plots Window s j aci nesac huts neientiendee adie init le 41 3 4 Cam Follower Animation Wid OW eesseseesssesesseseresressessresressessesstessesseeseesseeesesesseesreseese 43 3 5 Radial Roller Follower Example iajssecssyssnesadsvestiacypasdebasjaveaasscdecsesveazeasstexteadsvasestnubseeenasdeved 43 4 0 Program for Rigid Body Guidance isysvsssasisscesassnssccacnpassanaavcancavsyasesensdesseacipashesyavscassaaaveaseavayaeens 48 41 htrod thon sser aare seus te eset EE ea ier aaee ae aE Ee E eE aa ind he S ERS 48 4 2 Rigid Body Guidance Using a Four Bar Linkage RBG4barDesign ce eeeeeeeeeeeee 48 4 2 1 Design Window for Four Bar Linkage for Rigid Body
81. rough 1 or 2 positions in one assembly mode and 2 or 1 positions in the other assembly mode When this happens the linkage design is unacceptable This problem was referred to in the textbook as a change of branch The slider crank program uses a procedure developed by Waldron to identify linkages that are unacceptable in the initial stages of the design The procedure is similar to that used for four bar linkages 4 3 1 Rectification When the Crank Is the Driver Avoiding the branch problem is a two step process and the regions in the two steps are different The slider point is considered first because it is the driven link Since the slider point is really just a circle point with the corresponding center point at infinity the slider point can be chosen in the same way that the driven circle point was chosen in the case of the four bar linkage The main restriction is that only points on the circle of sliders can be chosen The three image pole circles identify acceptable locations for the slider point The distances between successive image poles define the diameters of these three circles There are three image poles Pj P13 and P 3 and these are the same points used to draw the circle of sliders in position 1 The unacceptable positions for the driven slider point are shown shaded in yellow in the program If the slider point is chosen in the yellow shaded area the linkage will have a branch problem and be unacceptable 54 After the
82. ry 2 Ed 1 0 Introduction In the first edition of the textbook entitled Kinematics Dynamics and Design of Machinery by K J Waldron and G L Kinzel a set of MATLAB programs were written to supplement the textbook These programs were written so that the input was command driven This means that the user must input information in response to prompts These original programs are included in a separate folder on this CD The programs are written using a fairly simple programming structure and either the students or the instructor can modify them easily The original programs will work with version 5 0 or higher or MATLAB While the original programs generally work well they are more difficult to use then mouse driven programs Therefore most of the programs were rewritten to incorporate a graphical user interface GUI that is mouse driven The new programs are much easier to use than the original ones however the programming structure is much more complex than the original programs and considerable MATLAB programming expertise is required to make modifications in them Therefore in this user s manual we have not attempted to define the internal structure of the programs We will only explain how to use them However the source code for the programs is provided on the disk for those who are experienced in programming using the MATLAB GUI This manual gives a description of the MATLAB programs written to support the textbook The program
83. s and lines and coupler positions Editable boxes for user input of three coupler positions are provided The user can either input the positions numerically or move the locations and angles of the three coupler positions by mouse dragging The GUI implementation also allows users to drag any slider point continuously with its coordinates updated dynamically To be able to recognize corresponding points on the plot and data in the editable boxes three different colors red blue green are used for the coupler positions Elliptic Trammel Design for Rigid Body Guidance r3 slider 1 angle slider 2 angle 4 3442 14 4374 8 9434 slider 1 point K y entered coord 3 1 09 slider coord 2 8855 2 3182 slider 2 point x y onbsciesth 0 97 2 69 slider coord 1 7764 1 8219 coupler point post 0 0 45 pos2 3 0 135 0 1 2 3 4 5 poss 2 2 Fig 4 13 The design window for the elliptic trammel routine 4 5 2 Analysis Window for Elliptic Trammel Mechanism The three coupler positions in the analysis window Fig 4 14 use the same colors as in the design window In the analysis window only one plot is shown and animated because only one assembly position is possible 59 RBG Elliptic Tramme l Analysis Window Fig 4 14 The analysis wind
84. sign a crank rocker mechanism and gt Fuzzy Logic Toolbox to analyze the result The program contains two windows a design gt B image Processing Toolbox window and an analysis window The nomenclature used by the program gt amp Mu Analysis and Synthesis Toolbox is that given in the textbook Kinematics Dynamics and Design of gt Neural Network Toolbox Machinery by Kenneth Waldron and Gary Kinzel gt amp Y Optimization Toolbox f F f In the design window the variables are the angles theta and gt G Signal Processing Toolbox beta the time ratio and one of the link lengths The time ratio P amp statistics Toolbox may be input either through the alpha angle or through the Q value gt B system Identification Toolbox One of the program features is that users can drag the mouse along gt BWavelet Toolbox the arc defining the limit positions for the output link ina gt BSimutink continuous fashion In addition the transmission angle range is gt Statefiow shown and updated dynamically De RE In the analysis window users can control the number of plots up to four and the contents of each plot ten options are provided Furthermore the animation speed can be easily adjusted by clicking on the speed buttons Two buttons connect the two windows The Return button in the analysis window closes the analysis window and brings back the design window while the Analysis button in the design window open the analysis window and s
85. sitions Note that this linkage will give a different velocity state for each value of angular velocity for the coupler Therefore an infinite number of velocity states are possible The instant center for the coupler is shown in Fig A 2 Notice that C CI and D D I are collinear This corresponds to the result that a crank subtends angle 012 2 at the pole P 2 As 0 2 approaches zero the pole becomes co linear with the circle and center points and becomes the instantaneous center of rotation I O Yo 1 Pp D Fig A 1 The solution of Example 4 6 68 N O Yo 1 Fig A 2 The location of the instantaneous center for the velocity field of Example 4 6 In the case of two finitely separated positions we found that it was also possible to move the moving lamina through the two design positions using only a single pivot between the fixed and moving planes as shown in Fig 4 14 This point was the pole for those two positions Correspondingly the required velocity state can be generated by means of a single pivot at the instantaneous center of the motion The location ry of the instantaneous center relative to the origin is obtained from Eq A 1 by letting vc 0 to get 0 vo w x ryo or 0 w x Vot w x x Fo w x Vo F10 or wxvo kxvo tio z A 2 w2 w where ok requiring that w be positive counter clockwise In Example A 1 above _kxdi _ jj This is shown in Fig A 2 Notice t
86. tart the linkage animation Every time the Analysis button is pressed all of the design parameters are updated to start a new animation There is a status bar at the bottom of the design window If the chosen values for the variables cannot be used to create a crank rocker mechanism an error message will be shown in the status box aM Fig 2 3 The help window of the crank rocker design routine pae Ile I 2 i 7 Fig 2 4 Optimized linkage for input values in Fig 2 1 2 2 2 The Analysis Window for Crank Rocker Program After the design is finalized i e beta is selected the Analysis button can be selected In the analysis window shown in Fig 2 5 users have control of the number of plots up to four shown and the contents of each plot nine options The nine options that are plotted as a function of the crank angle are 1 Rocker angular position 2 Rocker angular velocity for a constant crank angular velocity of 1 rad sec 3 Rocker angular acceleration for a constant crank angular velocity of 1 rad sec 4 Copular angular position 5 Coupler angular velocity for a constant angular velocity of 1 rad sec for the crank 6 Coupler angular acceleration for a constant angular velocity of 1 rad sec for the crank 7 Input torque output torque This gives the mechanical advantage for the linkage 8 Transmission angle degrees 9 Mechanism plot The nine options ar
87. the instant center as shown in Figs A 14 and A 16 Then the following relationships apply lyg lvn v n 144 V5 4 yia n 4 12 V14 anypt in system2 4 anypt in system4 withzero velocity relative to System 2 vy h42 vy 4 4 Fig A 16 Velocity polygon for determining the velocity of the instant center I5 Now lyy is perpendicular to line AJ yy is perpendicular to line BI lvi is parallel to line AJ y15 14 is parallel to line BI 84 and lv 4 s perpendicular to the line from J4 to 713 Line ZD Because of the right angles indicated the ends of vectors vp and vy lie on a circle with yj as the g g 2 v 5 diameter A detailed representation of the angles involved is shown in Fig A 17 Because quadrilateral Iacd is inscribed in a circle two observations can be made from plane geometry a Opposite angles of the quadrilateral are supplementary b All angles inscribed by the same chord segment or equal segments are equal Therefore y Pp 2 p 2 yenga Then y L 5 B p y E 0 and B p nta Fig A 17 Details of the velocity polygon in Fig A 16 Also triangles dic and dac contain a common chord line as a side Therefore 9 7 which requires that a 6 Comparing Figs A 15 and A 16 it is clear thata P 0 which proves the theorem A consequence of the Bobillier theorem is that the direction of the velocity of I5 is a purely geometric quantity as it should be
88. ther a crank or the coupler as driver Programs for drawing gears given the geometry of the cutter Program for computing the inflection circle of a four bar linkage Program for design of linkage for three positions for rigid body guidance Program for computing the shaking force for a slider crank mechanism Program for analyzing a six bar linkage Program for the analysis of a slider crank linkage with the slider coupler or crank as driver 1 2 Program Installation To install the programs simply copy the folder entitled GUI Based Kinematic Programs from the CD to a folder on your hard disk The programs can be run directly from the CD but they will be slower than if they were copied to the hard disk After the programs are copied to the hard disk open MATLAB and set the MATLAB path to the folder where the programs reside 1 3 Running the Programs To run the programs open MATLAB and at the command prompt type mainmenu The screen shown in Fig 1 1 will appear Click on the Continue button and the screen in Fig 1 2 will appear As indicated in Fig 1 2 the programs are arranged under four general headings Linkage Design Cam Design Rigid Body Guidance Design and Gear Design File Edit View Insert Tools Window Help THE OHIO STATE UNIVERSITY Mechanical Engineering Kinematic Design and Analysis Programs written by Yueh Shao Chen Sung Lyul Park Michael Stevens and Gary L Kinzel Department of Mechanical Engine
89. this by choosing linkages with the driven circle point in the yellow region and then animating the result This can be done easily by dragging the driven circle point with the mouse Regardless of where the driver circle point is chosen the linkage will have a branch problem After the driven circle point is chosen the driver circle point can be identified However even if the driven circle point is chosen outside of the yellow shaded region it is possible to choose the location of the driver circle point such that branching will occur In the design window colored linear regions are shown radiating from the driven circle point The driver circle point must be chosen to lie outside of the colored regions Sometimes the acceptable region is very small or even nonexistent If there is no linear region that is free of color then there is no solution that will be free of the branch problem In the example shown in Fig 4 2 the linkage chosen has the driver circle point in the forbidden region Therefore that linkage will have a branch problem as will be apparent when the linkage is animated Fig 4 2 The design window for the four bar linkage design for the rigid body guidance 4 2 2 Analysis Window for Four Bar Linkage for Rigid Body Guidance Those three coupler positions in the analysis window Fig 4 3 use the same colors as in the design window It is therefore possible to identify which position is missed if the rectification procedure is
90. w is described first 5 2 1 Design Window for Arb2thDesign Program The procedure used to generate the tooth profile is given in Appendix B of this manual The design window is shown in Fig 5 2 The generated tooth form is displayed in the graphics window and the generating tooth information is shown in the frame to the right of the figure Geometric information on the generating tooth form is shown if the Definition button is selected This is shown in Fig 5 3 For the generating gear the user may input the following 1 Number of teeth 2 Number of points describing the generating tooth The more points the higher the accuracy of the generated tooth 3 Addendum constant a 4 Flank angle for generating tooth see Fig 5 3 61 Fig 5 2 Design window for Arb2thDesign program Fig 5 3 The definitions window for the Arb2thDesign program The generated gear information is summarized in the frame below that for the generating gear For the generated gear the user may input the following 1 Number of teeth 2 Addendum constant a 3 Dedendum constant b 62 The final piece of information that the user may input is the diametral pitch Once the input data are established the following information is displayed 1 Center distance C 2 Tooth thickness at the pitch circle t 3 Addendum length on generating gear A Buttons are provided for an internal or external gear however currently the program can
91. xist for the line The first occurs when the line is horizontal Then y ya regardless of x When the line is vertical then x x a for all values of y Note that in Eq B 1 we assume that the components of the normal vector are known If only an equation for the tooth profile is known we can obtain the normal to the curve at any point by differentiation For example if the tooth profile is given by y F x then the slope of the normal vector is given by 1 dy dx where the derivative is evaluated at the point of interest If x and y are given as parametric expressions for example y rsin0 x rcos0 then the slope of the normal can be computed from Referring to Fig B 2 the line AP through P can be written relative to the coordinate system attached to gear 2 as XP h cosh yp pn sing B 2 where rp is the pitch circle radius of gear 2 The negative sign on yp is present because p is negative Substituting Eqs B 2 into Eq B 1 gives n sing ncos xana ya or nsing nna coso ya xana 0 B 3 In a typical problem both x4 and ya will be specified This will correspond to the contact point location for both gears although the x4 and ya specified will be relative to gear 2 We must find the coordinates relative to gear 3 to find the point on gear 3 which is conjugate to the point on the gear However to do this we must first find the angle 91 The angle can be found using the procedur
92. y with its coordinates updated dynamically To be able to recognize corresponding points on the plot and data in the editable boxes three different colors red blue green are used for the coupler positions As the circle and center points are moved using the mouse the linkage will change shape The current link lengths along with the Grashof type are continuously updated When a Grashof type 2 linkage is indicated the number 2 is printed in red 4 2 1 1 Visual Aid To Identify Limits for Center Points In Fig 4 2 a red polygon is shown made up of dashed lines This is provided as a visual aid to the user if there are locations where center points are or are not permitted The coordinates of the corners of this polygon are provided in the editable boxes below the picture The user may change any of the points This polygon is for visual purposes only It has no direct affect on the equations used in the design procedure 4 2 1 2 Rectification As indicated in Section 6 3 6 when linkages are designed using the basic procedure outlined in Section 6 3 of the textbook it is common to find that they do not guide the rigid body through all three positions unless the assembly mode is changed In such cases when the linkage is animated the rigid body will pass through 1 or 2 positions in one assembly mode and 2 or 1 positions in the other assembly mode When this happens the linkage design is unacceptable This problem was referred to in th
93. zed Motion Plots for Radial Roller Example 47 4 0 Program for Rigid Body Guidance 4 1 Introduction In rigid body guidance or motion generation the coupler of a linkage is guided through a series of positions The programs in this set address three positions and three different programs are available as shown in Fig 4 1 Mechanical ie KINEMATIC PROGRAM MENU Linkage Design Cam Design The rigid body guidance category includes three routines one for a four bar linkage one for a slider crank linkage and one for a double slider linkage A flexible user input from either the keyboard or mouse dragging is provided to change the locations of circle points center points and slider points RBGSlider Crank Design RBGEITrammelDesign Gear Design click to expand this topic aa Close far Fig 4 1 Programs available for rigid body guidance The first program RBG4barDesign is for the design of four bar linkages The second RBGCrankSliderDesign is for crank rockers and rocker crank linkages and the third RBGEITrammelDesign is for double slider or elliptic trammel linkages Each of these will be discussed separately in the following The basic programs follow the procedures and use the nomenclature in Section 6 3 of the textbook Therefore the basic design procedure will not be discussed here However the programs RBG4barDesign and RBGCrankSliderDesign both include a rectification

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