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ADAMS/Solver Primer

1. x of the root is provided a new configuration x hopefully closer to the root x is computed as 0 X gd Ors f OY where f is the derivative of the function with respect to the variables on which it 2 depends This strategy of updating the value of x is obtained by linearizing the function f at the point x Thus FD FEO Faa 3 Rather than solving f x 0 the root of the right hand side expression in Eq 3 is sought This leads to a linear equation and its root is denoted by x FE POS which is the configuration update defined in Eq 2 The algorithm continues by setting x lt x and performing another iteration as indicated in Eq 2 and illustrated in Figure 1 The method just described is called Newton Raphson or sometimes simply f x F x 0 gt x x Newton In general this approach leads to a quadratic rate of convergence to the root x provided the iterative process is started close enough to the root The drawback of the method is that as indicated in Eq 2 the new update x requires the computation of the derivative both the function f x and the derivative f x At the cost of one function and one derivative evaluation per iteration this method is considered to be rather expensive An alternative is to only update the derivative once in a while and to recycle the derivative computed at an older configuration for several iterations Th
2. Figure 3 31 7 2 Initial Condition Analysis Recall that for an Initial Condition analysis we are assembling the parts and constraints such that all of the constraints are satisfied for both positions and velocities as well as determining the initial accelerations and reaction forces in the model In this example we will look at the constrained optimization problem for solving the position initial conditions First we need to develop the constraint equations so that we can determine a consistent set of generalized coordinates For our two dimensional pendulum we currently have only a revolute joint to provide the constraint equations The revolute joint attaches the body of the pendulum to ground and ensures that ADAMS View 2005 0 0 0 x File Edit View Build Simulate Review Seltigas Tools Help gravity R pendulum y b DE Figure 4 R C R C or equivalently 32 C R C R 0 Where e The constant R is a vector which locates the pin revolute joint in the ground reference frame The variable C is a vector which locates the center of mass of the pendulum in the ground reference frame The variable R C is the vector from the center of mass of the pendulum to the pin revolute joint attaching it to ground The condition imposed by the mechanism can be rewritten in vector form as the constraint equation B x y 9 x R C R i y R C R j 0 R C lcos R C
3. Note that because we are building a two dimensional problem in ADAMS View the THETA above is not the same as the one in our two dimensional model Substituting the initial configuration into our optimization problem produces 10 0 1 OTfd 0 01 0 0 1l 0 0 0 10 0 1 d 0 10 0 0 ofa 6 1 4 1 01 1 0 O 4 2 0 0 2 By inspection you can see that 6222 1 d 4 i A _ aT A x x d 6 l 5 y y d 2 2 0 6 0 d 0 2 0 35 This puts our pendulum center of mass position x y at 5 0 at 0 degrees orientation horizontal And since the half length of the pendulum J is one this is the correct location given our revolute joint is at 4 0 gt ADAMS View 2005 0 0 i lol xj File Edit View Build Simulate Review Settings Tools Help Last_Run Time 0 0000 Frame 01 7 3 Dynamic Analysis For the dynamic analysis of a multi body system the inertial forces the constraining forces the potential forces and any externally applied forces must be kept in equilibrium The Euler Lagrange equations are used by ADAMS Solver to generate the equations of motion and as listed previously are the following d aL L oy 16 HE Sara q is the column matrix of n generalized coordinates of the rigid bodies which describe the configuration of the system at any given instant in time L defines the Lagrangian which is the difference between the kinetic energy T of the mechanical system
4. in the local Cartesian reference frame Q q q t e R the generalized force acting on the body Obtained by projecting the applied force F upon the generalized coordinates Typically Q 47 where with v being the velocity of the point of application P of the external force F the projection operators are computed like av I 48 a 48 0 mk 2 49 at 49 22 5 2 Formulation of Equation Of Motion in ADAMS The Lagrange formulation of the equations of motion leads to the following second order differential equations d 0K oK T A Ha vere s Considering the choice of generalized coordinates in ADAMS i e the definition of q as in Eq 8 Eq 50 is rewritten for a rigid body as i aK aK d ou B op amp a E It is worth pointing out that when dealing with a full system of rigid bodies connected through joints the system Equation Of Motions EOM are obtained by simply stacking together the EOM for the bodies in the system 51 da m f PA m a Since d K M 52 a x oe OK eon a 53 F aa with the angular momenta defined as _ OK B JB 54 JBC 54 the EOM of Eq 51 are reformulated in ADAMS as R T Mu 1 II f 55 peeks l a dE The first order differential equations above are called in what follows kinetic differential equations and they indicate how external forces determine the time variation of t
5. lsin where and j are unit vectors along the x and y coordinate axes in the global ground reference frame We can alternatively express this vector constraint in matrix notation ey P x y 0 ee ley a The Jacobian matrix for is the 2 by 3 array 1 O0 sing 0 1 0 1 lcos Let s now formulate the constrained optimization problem for our constraint equations Www p 1 0 Ta 0 Ow oO 0 1 fla 0 0 0 w Isin6 Icos d 0 1 0 lsin 0 0 A x Icos R 0 1 Icos 0 0 J4 Ly Jsin R For example let s say we have a broken joint and the pendulum is horizontal length 2l 2 and we have decided to fix the orientation such that 33 gt ADAMS iew 2005 0 0 File Edit View Build Simulate Review Settings Tools Help pendulum gravity 8 0 horizontal w 10 fixed R 4 5 broken x 6 R 0 broken y 2 w w 1 free l 1 In other words our pendulum is horizontal and the pendulum center of mass position x y is at 6 2 while the revolute joint location is at 4 0 This puts the end of the pendulum of length 2 at 5 2 which is not coincident with the revolute location as required to make the constraint consistent Notice that we have put a large weight for the orientation of the pendulum which is achieved by fixing orientation like so 34 A Modify Body Pasition Initial Conditions
6. Substituting the velocity equations into our system of DAE s for the pendulum yields m A mU A mg IU Alsin A l cos U i x 0 U y U x lcos R y lsin R Thus the second order system has been reduced to a first order system with the unknowns S g we S wo Be x This system is well posed at this point but ADAMS Solver adds set of equations to make the solution more computationally simple and to increase the scarcity of the equations The angular momentum of the system is calculated explicitly and added to the system of equations while the translational momentum is not calculated explicitly since it is a relatively easy quantity to compute Thus for our pendulum the angular momentum p will be introduced to the system of equations as OL eri oq gt p 10 p IU 0 Now our final set of equations is 42 mU A mU A g Pp t Alsin Al cos p Iu U i 0 Dg U 0 x lcos R y lsin R with these unknowns Note the scarcity in the relationships between the variables in the system of DAE s It is of considerable importance in the algorithms to invert the matrices for the solution process as well as in forming the analytical derivatives in the C Solver as opposed to the FORTRAN solver which numerically perturbs the system to get the derivatives To express this system of DAE s in a more compact form we can rewrite the e
7. UX 0 UX U X x lcosO R y Ilsin R In turn the Jacobian is used to effectively steer Newton Raphson to a static configuration by adjusting the generalized coordinates i e positions orientation and reaction forces 48 The Jacobian Matrix K XX AlcosO AJsin sind Sh KR x XK lsin 0 1 l cos 0 l cos 0 i Keep in mind that there as static uses Newton Raphson to find the equilibrium solution it makes no consideration to find a minimum energy configuration Thus it is possible to find an equilibrium position where the pendulum is straight down stable or straight up unstable Use of ADAMS Linear to look at the eigenvalues of the system would reveal positive eigenvalues when the pendulum is straight up and negative eigenvalues when down One option is to use the EQUILIBRIUM DYNAMIC which allows an exit criteria for equilibrium based on kinetic energy of the system 49 8 Ways and Means for Improving your MSC ADAMS Simulation 8 1 Settings in the INTEGRATOR Statement This section discusses some of the INTEGRATOR statement attributes that have proven to be the most effective in changing the behavior of the integrator in ADAMS Solver a b c d e By far the most used integrator in ADAMS is GSTIFF It comes in at least two flavors the I3 and the SI2 the ADAMS SolverFortran has I1 as well The I3 GSTIFF formulation has been around for more than two decades As such it ha
8. and the potential energy V The first expression represents the inertial forces the second expression represents the potential forces the third expression represents the constraint i e joints and motions forces and Q represents the externally applied forces The general form of the this equation appears a bit difficult to understand as a whole but examining the components of the equations piece by piece will help in understanding how it generates the equations of motion The application of the mechanical system consisting of a single moving part will illustrate the general case 36 For the two dimensional pendulum the generalized coordinates q are Q II D x More generally q contains all of the n coordinates of the bodies that make up a mechanical system For a two dimensional system n 3N where N number of rigid bodies The Euler Lagrange equations above require the Lagrangian to generate the inertial and potential forces L T V For our two dimensional pendulum the kinetic and potential energy are respecitively T A mx my 16 V mgy Let s take the Euler Lagrange equations one term at a time starting with the left most term for the inertial forces oL First notice that Ja is the momentum and is an n x1 array ran Oi ipsa aL aL my aq oF aL Lae The inertia forces on the body in the respective directions q are then 37 aL
9. directly with a modeling error and is only a symptom Note that the output shown before is specific to GSTIFF and other algorithms solver modes produce different but qualitatively similar output Although not appearing in the above eprint output another piece of information that the Solver supplies to the user is the occurrence of a singular Jacobian matrix This is in conjunction for instance with the correction phase in GSTIFF or when trying to find an equilibrium configuration and the Newton Raphson method runs into a singular Jacobian 63 VMNPO SING_MTX The system matrix has become numerically singular with this pivot order Equation for the singular row MOTION 270_1032 MO0T10320301 MEERE row MOTION 10320301 CONSTR Variable for the singular column PART 270_1032 PAR10323 TERA col PART 10323 Theta PIVON Magnitudeen seca aoa ob etecacd ete Secs 4 05476E 17 SOLVE REGEN Symbolic refactorization attempted at time 1 7274 When solving for the Newton Raphson corrections see section 1 1 the Jacobian needs to be factored to determine its LU factorization The pivots in the LU factorization the diagonal entries in U are chosen at the beginning of the simulation and they are re used time and again upon a new call for the solution of this system Especially for long simulations after a while the original pivot sequence results in a singular matrix This requires a refactorization to com
10. generalized coordinates The choice of generalized coordinates is not unique for instance one could choose Cartesian coordinates or spherical coordinates Likewise one could choose either global or relative coordinates in which the configuration of a part position orientation speeds is defined relative to the origin or another part respectively The key observation is that once a set of generalized coordinates is selected it is expected that it will uniquely define the configuration of each part of the system at a given instance of time In this context in ADAMS Solver the position of a rigid body is defined by three Cartesian coordinates x y and z p y 6 K The orientation of a rigid body is defined by a set of three Euler angles that correspond to the 3 1 3 sequence rotation y 0 and respectively These three angles are stored in an array y E 9 7 0 The set of generalized coordinates associated with rigid body i in ADAMS is denoted in what follows by P q 8 Based on this choice of generalized coordinates the body longitudinal and angular velocity are obtained as u p 9 Be BC 10 where sinsin 0 cos B cos sin 0 sin 11 cos 1 0 and is the body angular velocity expressed in the body fixed coordinate system Equation 11 is important as it defines the relationship between the angular velocity of the body an intrinsic characteristic of the body and the choice of gene
11. if they are not redundant It is highly advised for the ADAMS modeler to eliminate redundant constraints While sometimes the redundant constraint s can be benign they can often have detrimental effects on the simulation as the reaction forces may not be calculated as intended causing such things as poor simulation performance unexpected eigenvectors from ADAMS Linear etc More information on this can be found in Section 8 3 2 Velocity Initial Condition Analysis The velocity IC analysis is a direct and simple application of the algorithm employed for the position IC analysis It is direct because it is applied exactly as presented before and it is simple because the constraint equations that need to be satisfied are already in a linear form Therefore there is no need to linearize them as was the case with the 17 position constraint equations see Eq 28 and the solution is guaranteed to be found in one iteration Thus the convex constrained optimization problem solved to retrieve the initial velocities g minimizes the cost function Fl F 4 40 W 4 4 68 subject to the linear velocity kinematic constraint equations of Eq 20 d t 4 8 4 4 0 39 As in the displacement IC analysis the weight diagonal matrix W has several very large positive entries that ensure that the user prescribed initial velocities are not changed significantly by the optimization algorithm From here on the same procedure previously
12. integration error is proportional to A and this quantity multiplied by a scaling factor should be less than the user prescribed ERROR Since the value of the step size h is typically less than 1 0 it s easy to see that the higher the order the integrator works at the larger the step size h can be and still not violate the user selected ERROR 53 In the end if you can run a simulation at large step sizes and high order it is guaranteed that your simulation will finish quickly and the quality of the results is going to be very good Finally please note that even when a function looks smooth what is important is how many derivatives of the function continue to be smooth For instance if a cubic spline is used to represent a motion then the function will have two continuous derivatives The third derivative will be already discontinuous and therefore GSTIFF will be prevented to increase the integration order to high values An easy fix to this problem is to actually specify the motion as a velocity rather than a position level motion This simple change will potentially increase the GSTIFF integration order by one a very positive change 2 Choose the modeling units wisely A poor choice of modeling units leads in different equations to entries that are vastly different in magnitude The end result is that during the solution sequence the Solver will encounter matrices that have very large condition numbers If the situation is not ext
13. is obtained from a Taylor expansion based linearization of the non linear constraint equations q4 P qo t dot 4 Ao 43 Since the number of constraints is equal to the number of generalized coordinates the matrix q t is square As the constraint equations were assumed to be independent this matrix is also invertible Based on an explicit integrator e g Forward Euler an initial starting configuration q is determined and the iterative algorithm proceeds at each iteration j2 0 by finding the correction AV a4 A 8 a n 44 Then q q AV and the iterative process is repeated until the correction A and or the residual at t become small enough As with position IC analysis ADAMS can fail to find q if the linearization at the initial guess turns out to be a poor approximation of the non linear manifold In these situations the remedy lies in decreasing the simulation step size causing q to lie closer on the manifold to the last consistent configuration qo 4 2 Velocity Level Kinematic Analysis Velocity kinematic analysis is straightforward as the velocity kinematic constraint equations are linear in velocity With q already available from the position kinematic 20 analysis the non singular matrix q is evaluated and the linear system of Eq 20 is solved for the new velocity q 4 3 Acceleration Level Kinematic Analysis Acceleration kinematic a
14. it closer to the actual solution This iterative process can and should be monitored for each model that is problematic Below is shown an actual portion of the output that you get by having DEBUG EPRINT and running a dynamics analysis with the GSTIFF integrator Integration step No 7 Order 2 Cumulative iterations of From 5 000000000E 04 Step 8 33333E 05 the corrector 315 Iteration Residual or equation error Change in the Variable Jacobian count Maximum Element ID Equation Maximum Element ID Variable new 1 4 7E 05 Part 10 Phi Torque 6 2E 05 Revdnt 13 LAM yes 2 2 5E 00 Part 11 Psi Torque 5 0E 02 RtnMot 2 LAM no 3 2 0E 03 Part 11 Psi Torque 2 6E 01 Revdnt 15 LAM no 4 1 3E 04 Part 10 Theta Torque 1 4E 04 RevJnt 13 LAM no Integration step No 8 Order 2 Cumulative iterations of From 5 833333333E 04 Step 8 33333E 05 the corrector 319 Iteration Residual or equation error Change in the Variable Jacobian count Maximum Element ID Equation Maximum Element ID Variable new 1 4 1E 05 Part 10 Theta Torque 6 7E 05 RtnMot 2 LAM yes 2 9 0E 01 Part 11 Psi Torque 5 5E 01 RtnMot 2 LAM no 3 9 2E 05 Part 1l Psi Torque 2 1E 02 Revdnt 15 LAM no 4 1 6E 05 Part 11l Psi Torque 2 7E 05 RtnMot 2 LAM no Integration step No 9 Order 2 Cumulative iterations of From 6 666666667E 04 Step 8 33333E 05 the corrector 323 Iteration Residual or equation error Change in the Variab
15. the MOTION statement in ADAMS Solver User Manual see also Knowledge Base Article 9752 Avoid defining a MOTION as a function of variables i e states A cubic spline CUBSPL may in general work better on motions than the Akima spline The derivatives of the Akima are not as nice as those of the cubic spline they are more useful in forces rather than motions see Knowledge Base Article 7534 Consider filtering data to smooth out spline data for motions FORCES Remember advice number 1 above keep the expression of the force as smooth as possible and avoid discontinuities If using data approximate forces with smooth continuous spline functions Make sure velocities are correct in force expressions For example in this damping function c VX 12 25 25 the fourth marker is missing This marker defines the reference frame in which the time derivatives are taken and this may be important CONTACTS Contacts should penetrate before statics Models with impacts should have slight penetration in model position when doing statics 55 b c d e g a b c d All tires should penetrate the road Models with tires should have slight penetration in model position when doing statics For example if only rear tires penetrate the static position could be a handstand Contact properties are model dependent See the CONTACT statement documentation and Knowledge Base Article 10170 for a starting po
16. the Newton Raphson algorithm diverges see section 1 1 a Allow MAXIT to assume large values 200 for example This is not going to affect the speed of convergence but it will give the Solver quite a few chances to find an equilibrium configuration b Unless you start close to the equilibrium configuration use a STABILITY value c that is large 0 1 to 2 0 rather than the 1 E 5 which is the default A large value of the STABILITY parameter is an indication that you are not in a hurry to reach the equilibrium but you are content with the Solver taking small steps towards the equilibrium configuration Large values of the STABILITY attribute indicate that you allow the Solver to be very aggressive in making corrections in the model configuration Itis a good idea to experiment with the value of this settings as in our experience we found it to have quite an effect in the success failure of the static analysis Use the attribute ALIMIT to limit the value of the orientation update that the Solver is allowed to take The default is 30D but for problematic models more conservative values such as ALIMIT 10D can actually help with the overall convergence For more challenging models you might want to also change the default value of the TLIMIT attribute although in our experience this had a smaller impact FRICTION a b ADAMS has its own friction model implemented However if you are not satisfied with the existing friction mo
17. the dynamic analysis case due to the predictor employed by the integrator It is one of the roles of the predictor to produce a good initial starting point for the Newton Raphson algorithm The convenience of a predictor is not available in the STATIC approach and therefore the user has to either a provide a good starting point by setting up the model to be in a configuration close to equilibrium or b change the EQUILIBRIUM parameters to control the convergence of the Newton Raphson method see sections 8 2 and 8 3 for a discussion in this sense 6 2 The DYNAMIC Approach The DYNAMIC approach is only available in ADAMS SolverF77 and it is used less frequently The idea is that rather than solving a non linear system to find the equilibrium configuration as in the STATIC approach the Solver falls back on the integrator to find the equilibrium configuration Virtual damping is artificially added to the system and the assumption is that in a finite amount of time the system will settle in an equilibrium configuration due to energy dissipation resulting from the virtual and numerical damping associated with the approach The reader is referred to section 5 and the ADAMS Solver Manual for details about how a dynamic analysis is carried out The DYNAMIC approach in ADAMS SolverF77 is not the first choice for finding an equilibrium configuration However the user is encouraged to fall back on it when the STATIC which is the preferre
18. to values that are less than the width of the impulse e Use a conservative value on HMAX combined with a lax ERROR setting if you want to cause the integrator to acts like a fixed step integrator There is a consensus that preventing frequent step size changes improves the quality of the solution f Spikes in results output may arise from changes in step size Reduce HMAX or try setting HINIT HMAX Run with SI2 instead or use the INTERPOLATE ON feature g ADAMS Solver uses a body 313 rotation sequence psi theta phi When the second rotation angle theta becomes zero the model is in a singular configuration there is an infinite number of configurations for the psi and phi angles which lead to the same overall orientation likewise the time derivative of the orientation angles run into problems in a singular configuration When the system finds itself into a configuration very close to a singular one internally the solver changes the orientation of the body principal axes This operation always requires an integrator restart As a rule of thumb if the z axis of the part principal inertia axis is parallel to z axis of ground there will be an Euler singularity If you have prior knowledge about how the orientation of the bodies in the model is going to change time maybe you can define the axes of the IM marker such that Euler singularities are avoided Euler singularities lead to simulation slow down and spikes in results since the int
19. turn out to be redundant The benign case is when a redundant constraint is consistent An example of such a case is when ADAMS is presented with one constraint equation that looks like xX y 0 35 as well as a different constraint equation that reads like 2x 2y 0 36 The constraint in Eq 36 does not add anything to the picture when the first equation is satisfied the second one is automatically satisfied too Thus the second equation is redundant but consistent and throughout the simulation ADAMS will monitor this equation to make sure that the redundant constraint continues to be consistent The solver will fail if the previous constraint equation is replaced by 2x 2y 1 37 Equations 35 and 37 cannot be simultaneously satisfied for any x y e R When incompatible redundant constraint equations are found ADAMS Solver will carry out an LU factorization with full pivoting and inform the user about encountering this situation ADAMS will stop the simulation upon finding incompatible redundant constraints because from a modeling perspective there is something qualitatively wrong with the system being simulated Redundant constraints are usually encountered when too many joints are used to model the mechanical system and the number of constraint equations generated by these joints exceeds the number of generalized coordinates of the model In what follows two or more constraint equations will be called independent
20. used for position IC analysis is employed Note that because of the linearity of the velocity kinematic constraint equations the algorithm is guaranteed to converge in one iteration An exception to the linearity of this problem is if the ADAMS user includes a GCON generalized constraint statement which defines a non linear velocity constraint In this case obviously a non linear solver is used to generate the consistent set of IC s 3 3 Force and Acceleration Initial Condition Analysis In the absence of friction forces the acceleration IC analysis requires the solution of the linear system assembled from the equations of motion EOM and the acceleration kinematic constraint equations of Eq 17 The resulting system has the form wey bree i q where M is the generalized mass matrix Note that this is a linear system and the iterative process involved typically converges in one iteration The user cannot directly prescribe any initial acceleration for the force acceleration IC analysis The reaction force and initial accelerations are evaluated based on the computed initial position initial velocity and the applied force acting on the system at the initial time For a more detailed explanation of how the EOM are obtained the first row in the Eq 40 see the Section on Dynamic Analysis in ADAMS Besides q the solution of the linear system above also provides the Lagrange multipliers The constraint force and torque induced
21. 0 0 eesceesseeceeneeceseeeeeeeeesaeeeeseeeeeeees 59 8 5 Debugging Support in ADAMS Sol Vet ee eeeeeeceeeeseeeeeeeeeesaeeeeneeeeeneeeeeaeees 61 SOA WEBUGIEPRIN T ireen a neee aseara rikin 61 9 3 2 DEBUG RHSDUMP aTe niches rE TEE EURAN eet Aes 64 8 3 3 DEBUG JMDUMP nri nonnina aa a E Ti A aa e 65 Appendix A Integration Jacobian sesssseseseseseeessersssrerseeeseressersseressesseresseessereseeesseess 66 1 Introduction The purpose of this document is to introduce the ADAMS user to the theory that underlies the ADAMS Solver A basic understanding of the theory and the ways in which the solver works can help the user make good modeling decisions or choose a set of simulation parameters that will lead to faster and more reliable simulations This document provides in section 1 1 a very brief introduction to the Newton Raphson method for the solution of non linear systems Virtually all analysis modes in ADAMS use this algorithm and the user is strongly encouraged to understand the basics This is essential for understanding the informational and debugging messages that ADAMS Solver provides why an equilibrium analysis is challenging how changing the Jacobian evaluation pattern is going to impact the convergence properties of the solver and list goes on In the first part of the document sections 2 through 6 the focus is on equation formulation and solution The reader is introduced to the concept of generalized coordin
22. 67 f F u C p e f n x n T u p f n x t d u p f n X t Note that unlike Eqs 56 through 60 this set of non linear equations was modified to include the ADAMS DIFF elements which for the purpose of this discussion were assumed to have the form x d u C p f n x t 0 Here x is the state associated with the model DIFFs and for simplicity the DIFFs were assumed in explicit form Setting x 0 in the definition of the DIFFs leads to the last equation in Eq 67 It is important to underline that the DIFFs should have been present 28 in the dynamic analysis discussion in section 5 2 as well and they were omitted there only to keep the presentation simple In the STATIC approach an equilibrium configuration y u lr C pe Af nx is found as the solution of the algebraic non linear system of Eq 67 and to this end the Newton Raphson algorithm of section 1 1 is used Recall that for this algorithm to work the initial starting point of the iteration sequence should be close enough to the solution This is precisely what makes an equilibrium analysis challenging Unless the user ensures that the initial mechanical system configuration is close to equilibrium the algorithm might fail NOTE The numerical solution of the dynamic analysis discussed in detail in section 5 3 also requires the solution of a similar non linear system during the integration corrector stage The situation is more tractable in
23. ADAMS Solver Primer Ann Arbor August 2004 Dan Negrut Dan Negrut mscsoftware com Andrew Dyer Andrew Dyer mscsoftware com Tess EPO GUC OM fee cc ont ss eres Seon ests valet T ices Banaras A EE tee OE RETE 3 1 1 Solving Nonlinear Equations The Newton Method cee eeeeeseeeseeeeneeeees 3 2 Definitions Notations ConventionS sseessssseeesssssssssseeeresssssssetrrersssssseerressssssseee 9 2 1 Generalized Coordinates used in ADAMS ssssesseessessesesssseressrssrssressrssrssressessresse 9 2 27 Jomts m ADAMS rieni ipia s a AER EAEE ned oe 11 2 3 lt WIENS in ADAMS one kes ic ao socal aae sand R a A TE REA is 12 3 Initial Condition Analysis seseeeeseneseeeseeeeseeessressreesseesseresseesseesseressetsseresseessresseeesee 13 3 1 Position Initial Condition Analysis ccescceeescecesseeceeeeeeeeeeeesaeeeesaeeeeneeeesaes 13 3 2 Velocity Initial Condition Analysis sseeseeseseeessersseeesssesseeesseesseresseesseeeseeesse 17 3 3 Force and Acceleration Initial Condition AnalySis eesceesseeeseeeeeneeeeeees 18 4 Kinematic Analysiss cieciem a E a R 20 4 1 Position Level Kinematic Analysis cesseccesscccesneeeeseceeesneeeeseeeeeneeeeeeeeeees 20 42 Melocity Level Kinematic Analysis 2 2 cctceciecciseeninnsactiieieneaends 20 4 3 Acceleration Level Kinematic Analysis ccssccssecccesececesceesseceesseeeeseeeens 21 5 gt Dy amic Analysis cid czcctan ebay l
24. MS computes the value of d and given q computes the solution of the convex optimization problem as q q d 34 The configuration induced by the new set of generalized coordinates obtained as in Eq 34 corresponds to the linearized problem of Eqs 29 and 30 Therefore while the solution satisfies the conditions of Eq 30 it might be that it does not satisfy the original non linear system constraint equations induced by the system s joints as in Eq 26 If this is the case then the configuration q just obtained is set to be the new q and another iteration starting with the linearization of Eq 28 is carried out Typically after a couple of iterations the solution of the linear convex optimization problem will satisfy the non linear constraint equations of the mechanical system The approach fails when the linearization in Eq 28 is a poor approximation of the non linear manifold solution of the original constraint equations However even when the manifold is highly non linear the solution sequence will converge if the starting point q is close enough to the final solution For this reason it is essential for the algorithm to have a good starting point 15 To gain an understanding of how the weights w help to keep the user defined initial conditions at their prescribed values consider a case with only one constraint Likewise assume that the system has two generalized coordinates x and y and that the constraint e
25. Pon q Q 1 eee which defines the numerical relationship between a generalized coordinate and its first derivative h is the time step for the first order backward differentiation formula BDF For the higher order approximation the equation above becomes 44 aes 4 4 q hB where the coefficient B depends upon the order of the formula From the expression above Rais mle for each unknown q in the state vector The Newton Raphson algorithm for the solution of a system of non linear algebraic equations requires the formulation of the Jacobian matrix for which the entry in the iy row and j column is the partial derivative of the i equation with respect to the r unknown Thus after converting our DAE s with our BDF method the Jacobian matrix is calculated to be o i hp 1 hp 1 Alcos 4A lsin Isin l cos 0 hp I 1 1 1 hp 1 1 hp 1 1 poor hp 1 Isin i 1 cos 0 where the entries that are not filled in are identically zero here B 1 for Backward Euler Following the process outlined in Section 5 the Jacobian is evaluated and inverted repeatedly as specified by PATTERN to solve the equations of motion Notice that several terms have the step size h in the denominator When the step size is small these terms become very large and this produces an ill conditioned Jacobian Many options exist for dealing with this scenario and are described in more detail i
26. T dx mx mx d oL d L d x Gil de aoe E N 10 16 aL Lae The second term from the left is se which is an n x 1 array and indicates the sensitivity q of the Lagrangian for the mechanical system to one of the coordinates These are the potential forces note that ADAMS Solver includes only the potential force due to gravity in this term all other potential forces are actually included in Q DA Ox 0 L oL mg dq oy 0 L Lae So what we have thus far is m 0 O x 0 0 m O mg A Q 0 0 zi 0 This is a system of second order differential equations Note that this system will be converted to a set of first order differential equations later by introducing velocity variables The forces in the translational direction are determined by the first two equations In the third equation the inertial load due to the angular acceleration is balanced by the net torque about the center of mass of the pendulum Since there are no applied forces Q in this model the last part of the Euler Lagrange equations to determine is the constraint forces 38 The Constraint Equations The systems of equations that are generated thus far include five unknowns x y 0 as well as the Lagrange multipliers constraint forces A and A However we currently only have three equations and thus the solution to the five unknowns is not completely determined The additional equations required are the constra
27. are since this approach can sometimes lead to results that are different than the actual solution 8 2 Settings in the EQUILIBRIUM Statement Static analyses in ADAMS are mostly carried out using the STATIC rather than the DYNAMIC approach and the focus of the discussion below is reflecting this bias It should be pointed out that finding an equilibrium configuration is a challenging task particularly when the initial configuration of the system if far from the equilibrium configuration When the STATIC approach is used the equilibrium configuration is obtained as a solution of a non linear system and therefore if the initial guess is far from the solution the Newton Raphson algorithm diverges see section 1 1 As a general piece of advice always consider turning on the debug information during the static analysis having DEBUG EPRINT in your command file before the static simulation turns on the debug information you can turn it off with DEBUG NOEPRINT afterwards see section 8 5 1 for more information on DEBUG EPRINT 51 a b c d e ALIMT is one way to tame large corrections in the Newton Raphson iterations you can verify how large the Newton Raphson corrections are by inspecting the DEBUG EPRING output Every single time the largest angle orientation correction that is suggested by the Newton Raphson algorithm exceeds the value ALIMIT all the corrections are scaled by a value that renders the largest orient
28. ates as well as the equations that govern the most representative types of analyses that can be performed by the ADAMS Solver i e Initial Condition Analysis Kinematic Analysis Dynamics Analysis and Static Quasi static Analysis These equation sets are rather abstract and therefore this discussion is followed in section 7 by a simple test case that reinforces in a simplified two dimensional framework the process of deriving the equations for each analysis type Section 8 discusses ways in which an ADAMS Solver simulation can be made to run faster and more robustly Subsections 8 1 and 8 2 are focused on dynamics and statics simulations as they are typically more challenging The discussion will cover relevant parameters associated with the INTEGRATOR and EQUILIBRIUM statements Subsection 8 3 provides a set of recommendations and good practices for ADAMS modeling Subsection 8 4 discusses the issue of validating your results in ADAMS while subsection 8 5 explains how the user can get useful debug information from ADAMS Solver 1 1 Solving Nonlinear Equations The Newton Method The Newton Method is widely used by ADAMS Solver and is an important numerical algorithm to understand in order to produce models that lead to robust and fast simulations In one dimension the Newton Raphson algorithm finds the root x of a non linear equation f x 0 1 where the function f R R is assumed to be differentiable If an initial approximation
29. ation correction equal to ALIMIT As an example consider that ALIMIT 30D which is actually the default value and the first correction in the Newton Raphson algorithm suggests an angle correction of 45D If this is the largest angular correction all the corrections will be scaled by a factor 1 5 45D 30D Therefore indirectly all the corrections including translational corrections are scaled down As a rule of thumb if the static analysis fails try setting ALIMIT 10D or even to a smaller value However you should also increase MAXIT since as you restrict the solver to small corrections you will have to allow for more iterations to reach the final equilibrium configuration ERROR regards the correction convergence threshold in other words the static analysis will not be considered successful if the most recent Newton Raphson iteration produces a state correction that is larger in magnitude than ERROR IMBALANCE regards the residual in satisfying the non linear equations that need to be satisfied for equilibrium see section 6 If your model has a hard time finding an equilibrium try the funnel approach first run a static analysis with lax ERROR and IMBALANCE values a large value for STABILITY say 1 0 see below a conservative value for ALIMIT say 10D and allow for a large number of iterations say MAXIT 250 If this attempt succeeds decrease ERROR IMBALANCE and STABILITY by a factor of 10 and run a second static analy
30. by joint j on body i are computed as Pe D r j AY 41 vV L 18 o l ON ne r j aO 42 In Eqs 41 and 42 the superscript C indicates that the quantities are expressed in a Cartesian coordinate system v is the Cartesian velocity of body 7 is the global angular velocity p represents the set of constraint equations associated with joint j An explanation of why the reaction forces expressed in the global reference frame are computed as indicated in Eqs 41 and 42 is beyond the scope of this discussion but it suffices to say that they are obtained by projecting the Lagrange multipliers A along the Cartesian translational and rotational velocities v and of body i respectively 19 4 Kinematic Analysis Typically for a kinematic analysis to be carried out a number of independent constraint equations equal to the number of generalized coordinates in the model must be prescribed For the mechanism to actually change its configuration in time some of these constraints must be motions i e they depend on time 4 1 Position Level Kinematic Analysis Given the position of the system at time t the problem here is to determine the position at time gt t Because of the non linear nature of the constraint equations in Eq 19 a Newton Raphson iterative method is used in ADAMS to compute q at time t To understand how this method works and what its limitations are first note that it
31. d approach fails to find an equilibrium configuration 6 3 The STATIC_HOLD Attribute STATIC_HOLD is exclusively an attribute of the DIFF statement command and it is irrelevant in any analysis mode except statics Consider for example the definition of the following ADAMS DIFF DIFF 10 IC 2 0 STATIC_HOLD FUNCTION DIFF 10 DX 23 11 29 In this example DX represents the distance along the X axis between markers 23 and 11 assumed to be defined somewhere else in the model If the STATIC_HOLD is not present in the above definition during an equilibrium analysis the Solver will adjust the value of DIFF 10 that is the state associated with the DIFF such that the expression DIFF 10 DX 23 11 will become zero This is because at equilibrium the time derivative of any quantity should be zero and therefore DIFF1 10 which according to the DIFF definition is equal to DIFF 10 DX 23 11 should be zero Thus the value IC 2 0 will be overridden by the solver and the actual value of the state associated with the DIFF after the statics analysis i e the true IC value will be DX 23 11 Sometimes this is what the user wants but sometimes keeping the value of IC to what was originally set is preferable This is precisely what STATIC_HOLD does Under these circumstances for all purposes the equilibrium analysis proceeds as though the DIFF element is not present in the model Furthermore whenever the quantity DIFF 10 is referenced during the
32. d very small numbers Be wary when your model contains numbers like 1e 23 or le 20 This is most likely a symptom of a bad units choice see the UNITS section above and it typically results in matrices with large condition numbers Extend the range of spline data beyond the range of need Don t write function expressions that can potentially divide by zero The quick and dirty way to do this is to use the MAX function as in FUNCTION 8 MAX 0 01 your_function However there are more intelligent ways of doing this since the function MAX is discontinuous which leads to problems of a different nature remember the advice number 1 above Add moderate damping in your model so frequencies can die off This will speed up your simulation Avoid very high damping rates High damping is causing a rapid decay in response which is difficult for an integrator to follow Avoid toggles dual solutions or bifurcations since they lead to discontinuities Don t use 1 0 for the exponent in IMPACT or BISTOP functions This creates a corner i e a non smooth function Try 2 2 for the exponent instead note that this parameter depends on units The ADAMS SolverC has support for 2D parts A 2D part adds fewer equations to the overall problem and especially for such elements as chains and belts this can results in more robust and faster simulations 8 4 Validating your Simulation Results Before checking on the quality of results and ensuring that the
33. del new friction models such as LuGre elasto plastic etc can be implemented using ADAMS function and user subroutine support Note that unlike the ADAMS Solver C the ADAMS SolverF77 does not allow friction in joints connecting flexible bodies DEBUGGING a b While trying to debug or validate your simulations always turn on DEBUG EPRINT You should only turn off the DEBUG EPRINT command when you are comfortable with the results and satisfied with the CPU time required to complete a simulation Understand the output sent out to the user and see where the Solver has a hard time advancing the simulation Monitor and correct any warnings sent out by the solver 58 c d e g h i dD Try to understand the system that you are simulating from a physical standpoint Use building blocks of concepts that worked in the past Add enhancements to the model using a crawl walk run approach Test with small models to isolate problems Have graphics for visualizing motion Look at damping terms as a source of errors Incorrect sign missing terms are typical mistakes Models should have no warnings during simulation e g redundant constraints spline functions etc Understand numerical methods and particularly understand your integrator Look for results which become very large in magnitude this could indicate a discontinuity MISCELLANEOUS a b c d e g h Avoid very large numbers an
34. egrator is restarted at a small step size and at order one h In the ADAMS SolverC if your model already runs fine and you are after gaining some extra speed ups try to set the Jacobian evaluation pattern to F like PATTERN F This will effectively change the strategy of evaluating the Jacobian Recall that if you don t specify anything the Jacobian evaluation PATTERN is T F F F T F F F T F However if you set PATTERN F the integrator will take over the task of deciding how often the Jacobian needs to be evaluated adaptive Jacobian mode It was noticed that for some models this strategy leads to speed ups of up to 15 Note that i if you want to actually never have the Jacobian re evaluated unless a convergence failure occurs helpful for linear models you can specify PATTERN F F ii the adaptive Jacobian mode works only for GSTIFF both I3 and SI2 in the ADAMS SolverC i Inthe ADAMS SolverC 4 if your model has high frequencies or discontinuities you might want to use the new HHT integrator In our experiences quite often the HHT integrator produced sizeable speed ups in comparison with GSTIFF I3 and SI2 57 SIMULATION STATICS Remember that the static analysis if a very challenging analysis particularly when the initial configuration of the system if far from equilibrium The equilibrium configuration is typically obtained as a solution of a non linear system and therefore if the initial guess is far from the solution
35. element might be easily noticed in a large value in a very large initial residual in satisfying the force balance equation Note that each equation that is formulated in ADAMS is given a name and this name along with the violation residual associated with that equation is sent out to the user 64 b Inspect all the values of the state to find any state that has an abnormally large or small value ADAMS Solver has names associated with every variable such that the user can monitor name values state information c Inspect the values of the corrections computed at the end of each Newton iteration This information is of limited use but if an extremely large value is noticed or the opposite when a zero value is obtained where a non zero correction is expected is an indication that either 7 some residual is not correctly computed see a above or ii some Jacobian entries are not computed correctly Note that the amount of information that is sent out to the user in DEBUG RHSDUMP mode is large Keep this in mind when running long simulations with large models Only turn this feature on at the beginning of the simulation for a short while or right before the model runs into a problem that you try to diagnose 8 5 3 DEBUG JMDUMP The information obtained with the DEBUG JMDUMP command statement is primarily used for debugging of the code by ADAMS developers It is basically a dump of the Jacobian matrix assembled by the Solver for the
36. ep A stable algorithm that qualitatively captures all the relevant details characteristic to higher order methods Backward Euler integration formula replaces the derivative y at time with 1 1 y N 7 Vo 61 Based on Eq 61 an Initial Value Problem IVP g y t y t y is solved by finding y t at time t gt f as the solution y of the discretization algebraic non linear system 1 1 yi Fo Ble i 0 62 The system of equations in Eq 62 is called a discretization system since the derivative in the original IVP problem was discretized using the integration formula of Eq 61 Since almost always the function g is non linear a non linear algebraic system needs to be solved to retrieve y This is done in ADAMS by using a Newton Raphson type iterative algorithm Based on the implicit Euler discretization formula introduced above all the first order time derivatives that appear in the equations of motion in Eqs 56 through 60 are discretized to produce a set of algebraic non linear equations In the index 3 approach the position kinematic constraint equations are appended to these equations along with the force function definition F and T This appending of the force functions is done for the sole purpose of increasing the number of unknowns and thus inducing a larger but yet sparser Jacobian matrix Thus after the implicit Euler based discretization Eqs 19 and 56 through 60 along with the f
37. es almost unchanged while adjusting the free to change generalized coordinates Notice that the solution of the problem means minimizing the cost function while satisfying the constraints In matrix notation the constrained optimization problem reads For qe R minimize f a 5 a 4 W a a 25 subject to q 1 0 26 In Eq 25 W is a diagonal matrix of weights W diag w Wy W 27 while the constraints of Eq 26 that must be satisfied in the optimization problem are exactly the position kinematic constraints of Eq 19 ADAMS approximates the non convex optimization problem of Eqs 25 and 26 by a succession of convex problems that are guaranteed to have a solution which can potentially be found in one iteration Thus the set of non linear constraint equations of Eq 26 is linearized in the vicinity of q q t q 4 q a 4 28 With Equation 28 replacing Eq 26 and the notation d q q the now convex optimization problem reads Minimize f d Zawa 29 14 Subject to q 1 q d 0 30 To solve the convex constrained optimization problem of Eqs 29 and 30 define the optimization Lagrangian F d f d 4 q 4 a a 1 The optimality conditions for this problem are OF 0 ee 32 an on which lead to the following linear system of equations w a H 0 33 l o JAJ e a Based on Eq 33 ADA
38. es dee cantuieatosaae deh a e sdnas ts nade E EE ety eae SETS 22 5 1 Nomenclature Conventions Definitions cccccccccccccseeseseccccccseesseseeeecesseeees 22 5 2 Formulation of Equation Of Motion in ADAMS 0 eeeeeeeeseeeeteeeteeeees 23 5 3 Numerical Solution for Dynamic Analysis Jacobian Computation 24 Ge Statics Analysis nanona ne a caches vaste xd E E N 28 Gale The STATIC approaches reeet ao e aer E E E ESE Sea 28 0 27 The DYNAMIC Approach om n are EE ean E N TEE 29 6 3 The STATIC HOLD Ati ute acc i hiia ri aaen aA EEEn EE ERE Tias 29 7 A Simple Pendulum Example n soseseeeesesesesesseeseessseeseressrrssoressresseesseressereseeeseee 31 7 1 The Simple Pendulum Model s iccccsnteccceatieins wihiestieanin cineca 31 dae _ Imtal Copdit n Analysis s e anr rea e sea ae gos I R das 32 Taa Dynami Analysis min ee E EE EE ee aa a EEE 36 Conversion to First order System o ssc cies eee scsssendseon tees esntgvenccev snes cabs teoesavarssavesveavesaenes 41 T Kinematic Analysis crenn n ee Lae eG 46 Dis Stac Analysis enn a a a a aoe a 48 8 Ways and Means for Improving your MSC ADAMS Simulation eee 50 8 1 Settings in the INTEGRATOR Statement 0 0 cece eesecceeseceeeneeeeeeeeeeseeeeeneees 50 8 2 Settings in the EQUILIBRIUM Statement 00 eee eeeeeeseecsneeeseeeseeeeaees 51 8 3 Rules of Thumb for Creating Robust Models in ADAMS eee eeeeeeeeteees 53 8 4 Validating your Simulation Results 0
39. etween two thin skin geometries ERROR is the parameter that allows you to indicate how accurate you want the results to be Always remember that ERROR means different things for different integrators Itis only by using an integrator for a while in conjunction with your problem that you will get an intuition about what ERROR setting strikes the right compromise between accuracy and efficiency Too loose of a value might produce results that towards the end of the simulation become very inaccurate or sometimes it can lead to outright simulation failure Too strict of a tolerance can result in large simulation CPU times and or simulation failure if the integrator tries to reduce the step size to very small values to meet the accuracy demand The INTERPOLATE attribute when set to ON can improve the quality of the results by preventing some of the spikes Typically switching to 50 INTEROPLATE ON in GSTIFF slightly slows down the simulation The HHT integrator in ADAMS SolverC only marginally slows down when in interpolate mode g MAXIT is an attribute not recommended to change unless you have a very good reason to do so One thing that has been reported to be useful was to set MAXIT 7 and PATTERN F F By doing this the Solver only re evaluates the integration Jacobian when the corrector fails to converge which happens after 7 iterations in this case This has resulted in fast simulations for models that are close to linear or which do not cha
40. f freedom between the two bodies connected by this joint The collection of all constraints induced by the joints present in the model is denoted by T T q 8 a a amp 4 8 a a amp a 05 where nj is the number of joints in the model and m is the sum of the number of constraints induced by all joints Note that qe R while Be R Typically m lt n i e the number of generalized coordinates is larger than the number of constraints they must satisfy By taking one time derivative of the position kinematic constraint equations of Eq 15 the velocity kinematic constraint equations are obtained as q 0 16 By taking yet another time derivative of Eq 16 the acceleration kinematic constraint equations are obtained as 4 4 T 17 Equations 15 through 17 can be seen as conditions that the generalized coordinates array q along with its first and second time derivatives must satisfy This is to ensure 11 that the evolution of the mechanical system makes sense i e the mechanism is assembled and the parts move such that the constraints imposed by joints are obeyed at every time 2 3 Motions in ADAMS From a mathematical perspective motions indicate that a generalized coordinate of the system or an expression depending on generalized coordinates explicitly depends on time As an example consider a simple pendulum connected to ground through a revolute joint A moti
41. g ADAMS SolverC you can provide the partial derivatives from within your subroutine in order to improve the overall quality of the Jacobian The function call SYSPAR is the means through which the user can feed the partials back into the ADAMS SolverC As indicated earlier better partials result in better Newton Raphson convergence If you receive errors in your model for the purpose of debugging eliminate user subroutines so as they are not the source of error Make sure that your compiler is compatible with the current version of MSC ADAMS SIMULATION INTEGRATORS a When applicable perform an initial static analysis first Note that a static solution may be more difficult to find than a dynamic solution If you care only about the dynamic solution and cannot find static equilibrium then either increase your equilibrium error tolerance or forget about the static simulation 56 b If GSTIFF won t start it is most likely a problem with initial conditions Try to set HINIT to values that are conservative and thus the integrator will get a chance to slowly ease into the simulation c Read the documentation about the new corrector implemented in ADAMS and use it when having problems with discontinuities Understand the benefits and drawbacks associated with CORRECTOR NEW seeting d Don t let the integrator step over important events Short duration events such as an impulse can be captured by setting the maximum step size HMAX
42. he translational and angular momenta Finally the time variation of the generalized coordinates is related to the translational and angular momenta by means of the kinematic differential equations By assembling the kinetic and kinematic differential equations ADAMS generates a set of 15 equations for 23 each rigid body that provide the information necessary to find a numerical solution for the dynamic analysis of a mechanical system These equations are as follows Ma II f 0 56 T B JBC 0 57 OR te R T C o 0 n 0 58 dE p u 0 59 C 0 60 5 3 Numerical Solution for Dynamic Analysis Jacobian Computation Equations 56 through 60 indicate how the generalized coordinates reaction forces and applied forces variables change in time What is missing in this picture is the fact that the solution of this system of differential equations must also satisfy the kinematic constraint equations of Eqs 19 through 21 From a numerical standpoint this is what makes the dynamic analysis of a mechanical system challenging There is a multitude of methods for solving the assembly of differential constraint equations This document is not concerned with these methods and it only provides a glimpse at what ADAMS does to address this problem In this context it is worth mentioning that this assembly of differential and constraint equations forms what is called a set of Differential Algebraic Equations DAE A DAE has an
43. index associated with it index defined as the number of times the DAE s must be differentiated to get the system into ODE s and rule goes that the higher the index the more challenging the numerical solution of the DAE becomes In particular the DAE induced by the dynamic analysis problem in mechanical system simulation has index 3 which is considered high In the ADAMS solver there are two more reliable methods for solution The most common one is a direct index 3 DAE solver in which associated to the differential equations induced by 56 through 60 are the position kinematic constraint equations of Eq 19 This is how the solver GSTIFF I3 works in ADAMS 24 A second more refined algorithm reduces the original index 3 problem to an analytically equivalent yet numerically different index 2 DAE problem Thus instead of considering the position the velocity level kinematic constraint equations of Eq 20 are solved for along with the kinematic differential equations In the ADAMS solver this algorithm is called SI2 and while typically slower than the index 3 approach it turns out to be more accurate and robust In what follows the index 3 approach is presented in a reasonable amount of detail In an attempt to keep the presentation simple the index 3 DAE will be integrated via an order 1 implicit integration formula which converts the DAEB s into a set of algebraic equations This formula is the backward Euler formula a one st
44. int Adjustment of the properties to match experimental results is expected For contact models set the INTEGRATOR attribute HMAX 0 01 a good initial guess If problems are encountered some times a smaller 0 001 values is better Note that many times problems come from the use of thin shells If the input geometry is very thin there is a possibility that one geometry may completely pass through another resulting in invalid volume of intersection calculations This can result in missed contacts pass through or generation of unusually high contact force generation It s not a bad idea to set INTEGRATOR HINIT also if there is a possibility of sudden movement at the beginning of the simulation Set HINIT to a small value e g 1 E 7 to give the integrator a chance to ease into the simulation Don t turn contact friction on if possible until everything else is working The contact penetration depth parameter CONTACT DMAX can force the integrator to take extremely small step sizes if it is too low it should not usualy be less than 0 1 mm or 0 004 in SUBROUTINES Always use an ADAMS function over a subroutine if possible The quality of the partial derivatives is incomparably better when a function for a force for instance is defined via a statement rather than a user subroutine Consequently the quality of the Jacobian in a Newton Raphson algorithm is going to be better which typically results in better convergence properties If usin
45. int equations which are provided by joints and motions in mechanical systems In ADAMS users can also create general constraints by using the GCON statement with the C Solver For our two dimensional pendulum we currently have only a revolute joint to provide the constraint equations The revolute joint attaches the body of the pendulum to ground and ensures that ADAMS View 2005 0 0 lol x File Edit View Build Simulate Review Seltigas Tools Help pendulum T gravity R wi jee Bile Figure 5 R C R C or equivalently 39 C R C R 0 Where e The constant R is a vector which locates the pin revolute joint in the ground reference frame The variable C is a vector which locates the center of mass of the pendulum in the ground reference frame The variable R C is the vector from the center of mass of the pendulum to the pin revolute joint attaching it to ground The condition imposed by the mechanism can be rewritten in vector form as the constraint equation B x y 0 x R C R i t y R C R j 0 R C lcos R C lsin where and j are unit vectors along the x and y coordinate axes in the global ground reference frame We can alternatively express this vector constraint in matrix notation x R C R P x y 0 y R C R The Jacobian matrix for is the 2 by 3 array 1 0 sin p 0 1 0 1 Icos Associated with each constraint is a Lagrange
46. is method is called Newton like and the convergence rate is linear Referring to the ADAMS Solver documentation consider the INTEGRATOR statement One of its attributes is the PATTERN setting which enables the user to dictate how often the derivative is to be updated Computing the derivative also called the Jacobian is in general an expensive operation and should be done as seldom as possible On the other hand if it s done too seldom the speed of convergence will suffer as there will be degradation from quadratic for Newton Raphson to linear rate of convergence for Newton like methods Figure 1 shows a geometric interpretation of the Newton Raphson method starting with the configuration x the nonlinear function is locally approximated by a straight line The new configuration x is taken to be the place where the straight line intersects the x axis the solution of the linear equation of Eq 3 Moving on the new tangent is determined i e a new derivative f x is 2 computed and the new configuration x is obtained The Newton Raphson iteration sequence is thus carried out as 0 X KEN Waaa Fa FO O Ee Le FO until either the change in the configuration or the value of the of the function Fo f x is below a specified threshold either exit criteria could be used in general Figure 1 Newton Raphson Method Note in Figure that the computed configurations get closer and c
47. ivalent to 52 having a very large step size in the integrator Especially for challenging models that fail to find equilibrium setting a larger value for STABILITY effectively makes the static algorithm be less aggressive Recall the funnel approach discussed above see c above In the beginning take large STABILITY values and when you get closer to the solution be more aggressive and take smaller values A large STABILITY value is in the range 0 1 5 A small value is 1 E 5 which is the default Note when experimenting with the STABILITY setting always turn on DEBUG EPRINT to be able to judge how this attribute is influencing the convergence of the static analysis for your model g TLIMIT qualitatively plays the same role that ALIMIT does except that it monitors the corrections in the positions rather than orientations It has been noted that in general changing the default setting of this attribute does not have a great impact on the robustness of a static analysis or at least not as much as changing the ALIMIT attribute 8 3 Rules of Thumb for Creating Robust Models in ADAMS Below are presented a series of rules of thumb or best practices for modeling that will lead to more robust and faster ADAMS simulations 1 Avoid discontinuities This is the most important advice that can be provided It cannot be stressed enough that by far discontinuities are the root cause of most simulation problems in ADAMS Avoid them Exa
48. le Jacobian count Maximum Element ID Equation Maximum Element ID Variable new dL 2 6E 05 Part 10 Theta Torque 3 3E 05 Revdnt 13 LAM yes 2 1 4E 01 Part 11 Psi Torque 5 9E 00 Revdnt 13 LAM no 3 7 7E 06 Part 1l1 Psi Torque 5 5E 02 TrnJdnt 2 LAM no 62 8 6E 06 Part 11l Psi Torque 2 0E 03 Trndnt 2 LAM no The information reported in this example is as follows a b c d e g h The current integration step 7 8 9 etc Note that this number increases provided the time step was successful Although it is not the case in this example suppose step 8 would fail Then the integration step number will not be incremented to 9 unless the integrator can advance the simulation from time T 5 833333333E 04 where the integrator finds itself at the beginning of step 8 The integration order stays 2 over these three integration steps as indicated in the output Recall that for GSTIFF the integration order is anywhere from 1 to 6 a higher integration order indirectly indicates a smooth simulation The current simulation time is indicated after the word From it says that for instance the integration step 9 finds the simulation at time 6 666666667E 04 ready to take an integration step of Step 8 33333E 05 To advance the simulation to time 6 666666667E 04 see integration step 9 the integrator reports that it has taken a total number of 323 corrector iterations Recall that in GSTIFF a non linear sys
49. lem is larger than one This situation that is discussed in greater detail below In general ADAMS Solver deals with system of nonlinear equations in multiple unknowns Thus the root x from the simple example above is replaced by the n dimensional vector q IR and the system that needs to be solved assumes the form fa f ra 0 4 f Q LAA Qualitatively the approach is identical to the one dimensional case the non linear function f R R is locally linearized at a point q and the linear approximation is equated to zero The resulting linear system is solved for a new value q that is hopefully closer to the root q The linearization of the function produces f q fq F q q q where F q the Jacobian at q is defined as q q q q q The new configuration q is computed by solving the linear problem fq F 4 q 0 o of F q F which leads to 1 q q Fq fq 5 for the multi dimensional case This is the analog of Eq 2 2 Definitions Notations Conventions 2 1 Generalized Coordinates used in ADAMS ADAMS Solver is used to simulate the time evolution of a mechanical system At each moment of time ADAMS Solver is capable of indicating the position orientation and speeds associated with each part in the model The position and orientation of a part is monitored by means of what is called
50. loser to the point where the non linear function f x intersects the horizontal axis that is it gets closer and closer to the solution of f x 0 In Figure 2 the Newton like method is illustrated by adding dotted lines to the solid Newton Raphson lines from Figure 1 The dotted line represents the slope of the curve as computed at the point x which is used to find not only x as is the case 2 4 with the Newton Raphson method but also x x and x 4 Note that the slope is re and the iterative process continues with finding x evaluated at x Figure 2 Newton like Method The interesting thing to notice is that the slope of the line stays the same for four consecutive iterations in ADAMS syntax this would be equivalent to specifying a PATTERN T F F F this is by the way the default setting for this attribute in the ADAMS Solver INTEGRATOR statement f x a 1 0 _ ERR O 3 4 8 f x Oe La 4 5 _ 4 fx ie ae Fe As suggested by Figure 2 the Newton Raphson algorithm represented with solid line takes two iterations to achieve nearly the same value obtained by the Newton like approach after four iterations This is a nice way to see the superior convergence rate associated with the Newton Raphson method However as pointed our earlier the efficacy of this convergence rate must be balanced by the expensive of evaluating the Jacobian especially when the dimension of the prob
51. lution the iterative process might fail to converge This is more likely to happen with dynamic analysis than with other types of analysis as it is clear that the system that needs to be solved at each integration step is highly non linear If the iterative process fails the integration step size is decreased and another step is attempted ADAMS users are familiar in this context with messages informing them that the step size was decreased too much and yet the convergence was not attained Getting such a message is a bad omen as typically the user will have to revisit the model make modifications in the simulation defining parameters or to try a different integrator like SI2 for example More details on this can be found in Section 8 Finally although the discretization formula used to convey the dynamic analysis solution message was backward Euler it conceptually captures the essence of the ADAMS solution sequence ADAMS typically uses higher order integrators whenever the signals sent over by the problem being solved suggest that this would improve performance The expression of the integration Jacobian is qualitatively the same with very minor and insignificant changes for example the denominator of the fraction 1 h would become 1 hp where is an integration formula specific coefficient What is important to remember here is that the use of a more sophisticated integration formula serves in the end the same purpose namely to
52. m the new set of equations mU A mU A mg Pp t Alsin Al cos p Iu Ux Us U 98 x lcosO R y Ilsin R x R Csinat 46 The addition of another constraint equation to the DAE s has brought a fundamental change to the nature of the problem Again notice that the number of generalized coordinated equals the number of independent constraint equations and thus the dynamics problem is now a kinematic problem characterized by zero degrees of freedom Since the motion of the system is completely described throughout all time adding forces and torques to the system will not affect the motion of this system Adding forces and torques will change the required forces for the constraint equations to impose this motion however The Jacobian Matrix The motion of a kinematics problem is completely defined by the constraint equations in the system Although it is still possible to form the Jacobian matrix and integrate the full system of equations it is no longer necessary We only need to solve the three by three system of non linear algebraic constraint equations that is the final three equations in the system above Beginning with an initial estimate of Xx y 0 the Newton Raphson algorithm will be used to solve for the position of the mechanism at any given time t Note that the initial estimate for the first time step is the design position and the second time step is the solution from the first ti
53. me step However for every time step afterward a linear prediction based on the last two solutions is used to come up with the first estimate for Newton Raphson The Jacobian matrix for our kinematic system is 1 0O ZJsin 0 1 Ilcos 1 0 0 The kinematic solution generates values only for the displacement variables To compute values for the remaining elements of the state vector Y the constraint equations are differentiated once and evaluated with the computed values of the variables to obtain the velocities Next the equations are differentiated and evaluated a second time to obtain the accelerations Finally the force balance equations are used to compute values for the constraint forces 47 7 5 Static Analysis We can generate the set of static equations for the simple pendulum by starting with the dynamics equations we formed previously In order to find a static solution our pendulum must have at least one degree of freedom and thus our kinematic problem in the last section cannot find a static solution What needs to be done to form our static problem is to set all time varying terms to zero since our objective is to find a set of states where the system does not change with time So starting from our dynamics equations m A mU A mg Pp t Alsin Al cos p lU U j 0 U y U 9 x Icos R y lsin R Remove all derivatives leaves mK A myx A mg K 4 1 sin A cos 0 p Iu
54. mples of discontinuous functions are MIN MAX DIM MOD IF etc Discontinuous displacement velocity or acceleration motions cause corrector and integrator failures and produce large integration errors Likewise discontinuous forces cause corrector failures To understand the reason why discontinuities are causing problems recall that an integrator like GSTIFF is a variable order integrator The order of the integration formula used by GSTIFF can be anywhere between one and six and always the higher the order the better the results will be The user does not have control over how the integrator chooses its order since this is something determined inside the integrator based on some optimality conditions However the user can help the integrator in succeeding to choose high orders five or six by making sure that there are no discontinuities in the model The rule is that for the integrator to work at order p where p for GSTIFF is between one and six all the functions appearing in the model motions forces variables etc should have at least p continuous derivatives Thus for instance if you hope to have a simulation during which GSTIFF works at order six then you need to have all the functions in the model have at least seven continuous derivatives If you invest the time and make sure that the system is modeled right you will be rewarded by a large integration order The reason why you want a large integration order is that the local
55. multiplier that provides the constraint reaction forces Using the constraint equations and the Lagrange multipliers final term in the Euler Lagrange equations can be formed 1 0 A B A 0 1 7 A A Isin Icos Alsin A lcos 40 The first two entries are constraint forces and the third is the torque In summary the power of the formulation process that uses the Euler Lagrange equation is illustrated by the conversion of the constraint equations into the coupling terms The dynamics of the mechanism are governed by e Three force balance equations e Two constraint equations Combining the force balance and constraint equations gives us our mechanical pendulum m O O x 0 A O m O mg A 0 0 0 T 4 0 Alsin Alcos Or equivalently by carrying out the matrix algebra mx A my A mg 16 Alsin AJcos 0 x lcosO R y Ilsin R This set of DAEB s for our initial value problem is well defined with five equations and five unknowns Conversion to First order System The system above contains second order differential equations The formulation used by ADAMS Solver GSTIFF formulation makes use of the method of introducing an intermediate variable for each higher order derivative to reduce the order of the system to first order Let Here the components of U are eve ll D S x 41 These are kinematic differential equations also known as the velocity equations
56. n Section 8 45 7 4 Kinematic Analysis The motion of the simple pendulum described previously is approximately sinusoidal only for very low amplitude oscillatory motion We can for the sake of this discussion impose a restriction that the motion be exactly sinusoidal By controlling the horizontal or vertical motion of the center of mass we are imposing another constraint upon the system With an additional constraint the remaining single degree of freedom is removed and the motion of the constrained pendulum becomes purely kinematic Another way of saying this is that since we have three degrees of freedom and three independent constraint equations the motion of the pendulum is completely prescribed by the constraint equations Sinusoidal motion could be achieved mechanically by attaching a slotted rod constrained to slide back and forth to a pin at the center of mass The connection between the rod and the body of the pendulum would permit motion in the vertical direction while enforcing sinusoidal motion in the horizontal direction We will achieve the effect mathematically by adding a constraint equation between ground and the center of mass of the pendulum The constraint equation for the motion generator is x R Csinat where the line of action is along the x axis and R is the location of the revolute joint for the pendulum in the ground reference frame With the addition of this constraint equation motion to our syste
57. nalysis is immediate as at time it is found as the solution of the linear system of Eq 21 Notice that the same matrix that is factored for velocity kinematic analysis is used for a forward backward substitution sequence to solve for the generalized accelerations q Once q is available the Lagrange multipliers associated with the set of constraints acting on the system are computed as the solution of the linear system XN F Mq 45 This equation is identical to the first row of the linear system of Eq 40 and in fact represents precisely the equations of motion More information on how Eq 45 is obtained is provided in the next Section 21 5 Dynamic Analysis 5 1 Nomenclature Conventions Definitions In addition to the definitions and notations introduced at the beginning of this document the following quantities will be used in formulating the rigid body equations of motion M generalized mass matrix J generalized inertia matrix expressed about the principal local reference frame K kinetic energy defined as l r l ts Kone Ma J 46 Xe R array of Lagrange multipliers The number m of Lagrange multipliers is given by the number of constraint equations induced by joints connecting a body to other bodies in the system f F q q t H R the vector of applied forces f is the vector of applied forces expressed in the global reference frame while n represent the applied torque expressed
58. nge much their configuration in time h If you set PATTERN T then the jacobian is evaluated at each iteration very expensive but the rate of convergence is theoretically quadratic since the resulting method is Newton Raphson see section 1 1 On the other hand if PATTERN F F then the jacobian is only evaluated when the corrector fails to converge Note one important difference between the FORTRAN and C solvers If PATTERN F in the C solver that means that the integrator is given control how often to evaluate the Jacobian Remember that doing it too often is in general expensive doing it too seldom might lead to convergence failure The C Solver tries to reach a compromise with an adaptive Jacobian pattern and as indicated this strategy kicks in by setting PATTERN F To conclude in C Solver you can obtain the FORTRAN PATTERNG F behavior by setting PATTERN F F i The attribute KMAX is not something that is usually modified It has been noticed that for models that have very high mechanical stiffness such as models with flexible bodies setting KMAX 2 sometimes results in shorter simulation times j The CORRECTOR attribute might come handy for models where the corrector has a hard time converging the Newton Raphson process By choosing the MODIFIED corrector the integrator will accept more readily a system configuration as being converged This is useful for models with contacts and discontinuities Use this feature with c
59. on might impose that the angle associated with its unique degree of freedom will change in time like sin 107 t Generally a motion is represented as a time dependent constraint equation q t 0 18 Revisiting the definition of the position velocity and acceleration kinematic constraint equations both joints and motion constraints may in general be written as q t 0 19 4 1 4 4 1 20 4 1 4 2 4 4 1 21 Equations 20 and 21 are obtained by taking one and respectively two time derivatives of the position kinematic constraint equation of Eq 19 A set of generalized coordinates is said to be consistent if it satisfies the position kinematic constraint equations Likewise a set of generalized velocities is considered consistent if for a consistent position configuration q satisfies the velocity kinematic constraint equations of Eq 20 12 3 Initial Condition Analysis Initial Condition IC Analysis is concerned with determining a consistent configuration for the mechanical system model at the beginning of the simulation During IC analysis the mechanism must be assembled and the velocities of the parts in the mechanism must be consistent To be assembled the generalized coordinates q must satisfy all constraint equations at time f i e amp q 7 0 22 while for the generalized velocities to be consistent they must satisfy the velocity kinematic constrain
60. orce torque definition equations assume the following form 25 Mu Mu 1I f 0 h h r B JB 0 1 _ 1l OK T r rT 01a 0 n 0 ge Be ey 63 nP 7 Po 1 1 e e C 0 ogee a p e 7 0 f F u C p e f n 1 0 n T u p e f n 0 The unknowns in this non linear system are u I G p A f n The subscript 1 indicating the time step was dropped for convenience Introducing the array u T p y 64 E f 7 the non linear system of Eq 63 is rewritten as W y 0 65 A Newton Raphson type algorithm finds the solution of this system First a prediction y of the solution is computed typically by using a predictor constructed around an explicit integrator Once an initial guess of the solution is provided iterations Yy y A ays y E de 66 yt y AY are carried out until the correction A are small enough note that the residual Y y i could potentially be used as an exit criteria but is not 26 The Newton Raphson method requires the computation of the Jacobian P Ya which is obtained from Eq 63 With the notation introduced in Eq 64 the expression of the Jacobian y is provided in the Appendix The same remarks made in conjunction with the iterative Newton Raphson algorithm used for IC analysis and for Kinematic analysis are applicable here Thus if the initial guess i e the predicted value of y is too far away from the so
61. ot an issue d Dummy parts should be massless 0 0 or unassigned and not le 20 4 Keep an eye on the JOINT elements in the model In most simulations the use of JOINTs in the model is not leading to robustness issues However avoid redundant constraints ADAMS Solver tries to eliminate them by 54 looking at pivots in a constraint jacobian which are in no particular order As a result the physical meaning may be disregarded In general model that has redundant constraints is an example of poor modeling practices and it s an indication that the nature of the system being modeled has not been understood Redundant constraints can cause eigenvectors from ADAMS Linear to produce unwanted results as the DOF kept may not be the ones you are interested in for the linear analysis 5 MOTIONS Ideally motions should be defined through functions that are as smooth as possible i e that have a large number of continuous derivatives A simulation in which all the inputs are smooth will run with high integration orders a b c d e a b c a Remember the advice number 1 above keep the expression of the MOTION as smooth as possible and avoid discontinuities Avoid using SPLINE functions in a MOTION definition they only have two continuous derivatives If you have to use a spline function in a motion though you are better off defining the spline at the velocity rather than position level For more information see
62. pare gdiff message files between the two simulations Ensure that Solver is doing more work taking more steps because of the reduced error tolerance value If you are not seeing more integration steps being taken in the simulation with the reduced error tolerance value then most likely you need to decouple the effects of step size and integration error control In this situation either your output step size DTOUT or HMAX setting is smaller than the integration step size that would otherwise be required by your ERROR setting and is thus controlling the accuracy of your results To decouple DTOUT or HMAX and ERROR make DTOUT and or HMAX much bigger fewer steps bigger steps and use ERROR to do the study as before until you return to case a above and find that the reduced error tolerance value did cause solver to do more work At this point reset your error tolerance to the larger of the ERROR values from the two simulations that proved that you had a converged solution the simulations that gave the same results At this point you can be comfortable that you have achieved a converged solution for displacements NOTE Dealing with corrector integrator problems will most likely happen is handled in the ADAMS Solver documentation under the INTEGRATOR statement command section More information is available in the MSC ADAMS Knowledge Base at http support adams com kb or in section 8 1 of this document Follow steps thro
63. purpose of carrying out Newton Raphson iterations The information sent out indicates each partial derivative that is computed in the process of assembling the Jacobian that is a value is provided along with information regarding what equation this partial derivative is of and with respect to what variable the partial is taken The information output with DEBUG JMDUMP is typically useful for small models and even then analyzing the output obtained with DEBUG EPRINT and DEBUG RHSDUMP is more useful and should be considered first Note that for large models the amount of information generated can become very quickly overwhelming 65 Appendix A Integration Jacobian This appendix contains the integration Jacobian as generated during dynamic analysis of a mechanical system The integrator used was backward Euler the DAE solution is based on an index 3 approach I is the identity matrix of appropriate dimension M 0 0 aya nt ya m t m 0 0 1 B I 0 K 0 0 0 e 0 I x ana a an 0 n K eo o m a 0 0 1 0 0 0 0 0 0 A 0 i 0 0 0 h 0 0 0 0 0 0 F 0 F F F F I kR F T 0 lt r aT AT S S ER 66
64. pute a new pivot sequence in the LU factorization and possibly a decrease of the integration step size The user is flagged when the solver runs into such a scenario as indicated in the ADAMS SolverF77 message above DEBUG EPRINT is an extremely useful tool in diagnosing simulation problems By inspecting eprint output for problematic models you can find the root cause of your problem and thus be able to fix the problem yourself see the discussion in sections 8 1 through 8 3 or call the ADAMS hotline and provide the support engineer with additional information that can help him her diagnose the issue To conclude always start the process of tinkering with a model simulation by having the DEBUG EPRINT on You are encouraged to have eprint on and experiment with different settings of the INTEGRATOR EQUILIBRIUM etc statements commands This is the way to gain understanding of what parameters influence what aspect of the simulation and eventually become a power ADAMS Solver user 8 5 2 DEBUG RHSDUMP The DEBUG RHSDUMP statement command is intended for the power user who has a basic understanding about the concept of ADAMS state residual Newton Raphson correction This is the type of information that DEBUG RHSDUMP will output and it will help the user to quickly a Inspect all the entries in the array of force balance residuals constraint violation etc to check that the values are as expected For instance a modeling error in an obscure
65. quation they must satisfy is B x y x y 0 Obviously this constraint equation is satisfied by an infinite number of pairs like 2 4 2 5 6 25 etc but in this simple case the user would like to specify that the value of x is x 1 while the value y is free to change As the value for y is not prescribed assume our initial guess takes y 6 Since the user prescribed a value for x the weight associated with this generalized coordinate is large i e w 10 Since there is no condition imposed on the second generalized coordinate w 1 Notice that prescribed in the discussion above refers to coordinates specified to be exact in the ADAMS modeling language as in the ADAMS adm file With this the matrix of Eq 33 assumes the form 10 0 2a O 0 1 lij d 0 2 1 02 5 Then d 10 d 5 A 5 and Xj 14 10 1 Ye 6 5 1 As can be easily verified xe s Vic 2 10 10 The non linear constraint equation is very well satisfied and the correction applied in the user prescribed initial condition x is negligible order 10 In this manner the weights w bias the solution toward changes in one subset of generalized coordinates rather than another It is worth noting that during position IC analysis ADAMS checks the compatibility of the constraint equations induced by the joints and motions present in the model During 16 this constraint analysis some of the constraint equations might
66. quations as G Y Y t 0 where G represents the column matrix of function of the unknowns and where 43 1S the column matrix of the unknowns Again the differential equations in the system are first order and the system as a whole is nonlinear Notice that the system is implicit as opposed to an explicit form where Y f t and a set of ODE s could be solved For the implicit integration algorithm the DAE s are transformed into a system of non linear algebraic equations by approximating each derivative in the column matrix Y with a backward differentiation formula BDF with GSTIFF For numerically stiff systems characterized by over damped high frequency eigenvalues and under damped low frequency eigenvalues the BDF methods provide the largest stability region among the multi step integration methods The Jacobian Matrix One of the most expensive and difficult steps of solving the dynamics problem is in generating and inverting the Jacobian matrix In this section we will form the Jacobian matrix to give you a sense of both the structure and individual entries In later sections this can help you understand how the model is solved and how problems can occur in the simulation We need to convert our DAE s into an algebraic set of equations so that we can use Newton Raphson to solve the problem To accomplish this the backward differentiation formula s can be used The first order BDF is
67. ralized coordinates which as mentioned earlier can be chosen in a variety of ways Finally note the relationship between the time derivative of the body orientation matrix A and angular velocity A 12 where the orientation matrix A is defined in terms of the 3 1 3 Euler rotation sequence y 0 and as cos cos sinywcos sing cosysing sinycos cosg sinysind A sinycos cosycos sing sinwsing coswcos cos cosy sin sin sing sin 8 cos cos which is used to compute the derivatives for rotating bodies in three dimensional space Note that represents the skew symmetric operator applied to a vector quantity In general for a vector a a a a 0 ay a a a 0 a a aq 0 Also a vector quantity marked with an over bar indicates that the vector is expressed in a local body reference frame For an entire mechanical system models containing nb bodies the vector 10 asla ay lt a Slap ai ae a 13 with n 6 nb will describe at a given time the position and orientation of each body in the system 2 2 Joints in ADAMS Joints in ADAMS are regarded as constraints that act among some of the coordinates q through q of Eq 13 From a mathematical perspective such a constraint assumes the expression q 0 14 which is simply an algebraic constraint For example a revolute joint acting between two bodies would induce a set of five constraints to allow one degree o
68. remely bad a matrix with large condition number will lead to very poor quality corrections in the Newton Raphson algorithm for example see section 1 1 and therefore large number of iterations for convergence If the condition number is very large the corrector stage of the solution can fail altogether or the matrix is identified singular by the linear sparse solver To prevent situations like these a Choose units so that model states displacements and velocities assume reasonable values For example choosing mm for displacements of a rocket which travels thousands of kilometers is a poor choice b Choose time units appropriate to the phenomena being studied If duration of dynamic event time is short consider using MILLISECONDS units 3 If it s possible avoid dummy parts any part with zero or very small mass Keep in mind these recommendations a Sometimes dummy parts are useful but if possible avoid using them b Avoid connecting dummy parts with compliant connections BEAM BUSHING etc If the mass is very small qualitatively acceleration Force mass very large number Thus under certain circumstances small masses moments of inertia introduce high frequencies and or jolts into the system which has detrimental effects on the solver performance c If you must use dummy parts then constrain all its degrees of freedom since with no degrees of freedom for the dummy part qualitatively acceleration Force mass is n
69. replace a first order time derivative with a linear combination of future and past values of the unknown which is what backward Euler formula does in a very basic way through Eq 62 27 6 Statics Analysis The ADAMS SolverF77 supports two methods for finding the equilibrium configuration of a mechanical system the STATIC approach and the DYNAMIC approach see the ADAMS Solver manual the EQUILIBRIUM statement command discussion for more details The ADAMS SolverC currently only supports the STATIC approach 6 1 The STATIC approach From an algorithmic perspective the STATIC approach is based on transforming the equilibrium problem into an equivalent problem that requires the solution of a non linear system of algebraic equations As such this method employs a Newton Raphson algorithm to find the solution of the non linear system of equations The key observation in the STATIC approach is that at equilibrium while all the typical equations equations of motion constraint equations DIFF equations force definition equations etc must be satisfied the time derivative of any quantity that appears in these equations should be zero at equilibrium there is no change in time in the value of a variable and therefore its derivative should be zero Based on this observation considering Eqs 56 through 60 and setting all the derivatives to zero the following set of algebraic equations are obtained a A 0 T B JBC 0
70. results are converged the assumption is that you have applied the recommended debugging techniques to eliminate all modeling errors see section 8 5 for debugging support in ADAMS Likewise it is assumed that you have followed to the extent possible the rules 59 of thumb for creating robust ADAMS models see section 8 3 With these tasks out of the way set DTOUT END STEPS in the output statement command to suit your time sampling simulation requirements For a correlation study i e correlation with test data this value is typically driven by the sampling rate of the test data For a given DTOUT or END STEPS combo take the following annotated steps 1 2 Start with an initial error tolerance value for the INTEGRATOR statement The starting point could be the Solver default value of 1E 3 Build Run simulation with DEBUG EPRINT turned on with the initial estimate of error value Reduce error by a factor of your choosing a factor of 10 is commonly used Build Run simulation with DEBUG EPRINT turned on with the reduced error value Compare the results between the simulations with original and reduced ERROR values Check results particularly displacements first if you are using I3 integrator s You will be looking at one of the following two scenarios a If results don t match reduce error tolerance further from the current value by a factor of your choosing and start again from Step 2 b If results match closely com
71. s experienced many improvements and it is now considered to be the most robust integrator It doesn t mean however that there aren t models that will not run with GSTIFF but run with WSTIFF In fact in case you are not successful to run your simulation GSTIFF you should try to use WSTIFF or in ADAMS SolverC the new HHT integrator Speaking of the SI2 versus the I3 GSTIFF integrators it is worth noting that the latter is typically less picky when it comes to discontinuities handling The SI2 integrator gives nicer and smoother results but they usually come at a higher CPU time unless you relax the ERROR setting when compared to the value that is used for the same model simulated with GSTIFF A good starting point for ERROR when using SI2 is 10 times the ERROR for I3 The HINIT setting comes handy if you know that at the beginning of the simulation there are high transients Don t by shy about setting HINIT to small values if there are no high transients the integrator will quickly increase the value of the integration step size However if there are high transients a small value of HINIT e g 1 E 7 might help the integrator prevent rejected or failed time steps in the beginning of the simulation The HMAX setting is useful if you want to d1 get the integrator to basically run at constant step size set a small value for HMAX and a large value for ERROR d2 prevent the integrator from skipping over very short events like contact b
72. sis If this is successful tighten even further the simulation settings The idea is that you run a succession of static analyses with more and more conservative settings thus rather than finding the equilibrium configuration in one shot you use a funnel approach to finding it MAXIT indicates the maximum number of Newton Raphson iterations allowed to find an equilibrium configuration Allow large values for MAXIT particularly so if you have changed the default values for ALIMIT by decreasing its value or STABILITY by increasing it Note that a very large value for MAXIT does not hurt Have DEBUG EPRINT on and watch on the screen how your residuals and corrections hopefully decrease with each iteration Having a large MAXIT will increase your chances of success PATTERN is by default set to true therefore the Jacobian is evaluated at each iteration This makes the algorithm a true Newton Raphson method see section 1 1 This setting has not proved to be very helpful if changed and the CPU penalty of evaluating the Jacobian at each iteration is not prohibitive since in general statics analysis do not take long simulation times STABILITY is a factor that can greatly impact the success or failure of your static analysis An analogy can be made between the roles that the STABILITY and the integration step size play in the static and dynamic analysis respectively Having a very small value for the STABILITY attribute would be equ
73. static analysis by another modeling element for instance a force definition the value returned would be that indicated by the IC 2 0 in our example while the quantity DIFF1 10 will be zero The key observation when STATIC_HOLD is present is that even if the Solver indicates that the static analysis converged the residual in satisfying the DIFF equation is not zero In our case the residual would be 2 0 DX 23 11 which would be zero only by chance Finally if the STATIC_HOLD attribute is not present in the definition of a DIFF when the static analysis will have converged the value of the state associated with the DIFF would not be 2 0 but precisely the value DX 23 11 The function DIFF1 10 would evaluate to 0 0 and there would be a small residual in satisfying the DIFF but it would be at the most as large as the one specified in the IMBALANCE setting associated with the EQUILIBRIUM statement command 30 7 A Simple Pendulum Example 7 1 The Simple Pendulum Model A simple two dimensional pendulum model is chosen as an example since the ideas for the two dimensional model are representative of the three dimensional space while keeping the equations as simple as possible The model is contains one rigid part the body of the pendulum and ground A pin or revolute joint connects the pendulum body to ground gt ADAMS iew 2005 0 0 o x File Edit View Build Simulate Review Settings Tools Help pendulum gravity
74. t equation q 1 q q 23 3 1 Position Initial Condition Analysis During IC analysis the user might want to fix the value of some of the generalized coordinates q q etc from the generalized coordinate array Eq 13 In other words if the user prescribes the position of a body in the system to be q 0 9 q 1 0 q 0 the solver should assemble the mechanism and at the same time do its best to satisfy the prescribed conditions This IC analysis is solved in ADAMS via an optimization approach The constrained optimization problem solved minimizes the cost function 0 1 1 Pde 5M N 4 PE a 24 subject to the constraint equations q 0 In Eq 24 the values w are weight factors while q can be regarded as the initial generalized coordinates inducing an initial T configuration of the system Notice that this configuration q a qg ql j need not be consistent i e it may not satisfy the constraints 13 The user may prescribe that some of the entries in the q array i e q q etc are to be regarded exact As it will be justified shortly the corresponding weights w w etc will be given large values i e values like 10 The remaining weights namely the ones corresponding to generalized coordinates q that the user did not specify are given values of 1 With this approach the constrained optimization problem solution will keep the user imposed IC valu
75. t informative messages and warnings during both stages a 61 and b Acting on the warnings sent out by the solver is going to make for better models and more robust fast simulations As an example of a warning message you might get START WARNING An Out of range X value has been used in the spline calculation Extrapolation is required for curve 1 of SPLINE 123 The X value 100 000000 should be between 50 000000 and 55 000000 END WARNING In this case the user defined the spline curve anticipating that the free parameter will always assume values anywhere between 50 and 55 Nevertheless the model configuration is such that the spline is being evaluated for a value of 100 This is an indication that either the spline range initial assumption was wrong and the bounds should be altered or that somewhere else there could possibly be problem in the model that led to unreasonable values for the spline parameter There are some differences in the amount and type of information that is put out by the ADAMS SolverC and ADAMS SolverF77 Either one though sends out information summarizing the progress towards finding a solution The algorithms employed are specific to each analysis type position IC statics kinematics dynamics etc but the common denominator is that the solution is always computed as a set of successive iterations that each corrects the previous system configuration to bring
76. tem is solved to retrieve the configuration of the system during implicit integration As such an iterative Newton like method see section 1 1 is employed to find out the solution These iterations are indicated by the Iteration count which is reset at the end of each integration step In this example notice that each of the three integration steps took the same number that is 4 of iterations in the corrector stage in order to converge to a solution During each iteration the solver reports the largest violation in satisfying the equations that it s solving for the given analysis mode dynamics in this case As it can be seen initially the largest violation is quite large 2 6E 05 for integration step 9 The Solver also reports which equation registered this large imbalance In our example for integration step 9 this imbalance occurs in the torque balance equation for part 10 The last piece of information provided is the value of the largest correction that the Newton like iterative solver computed Recall from section 1 1 that these corrections gradually bring the system configuration towards the correct one towards the solution In our example notice that the corrections become gradually smaller This is the expected behavior and convergence failure occurs otherwise Look for trends in f and g as a starting point for investigating your model Keep in mind that the equations or correction reported most may have nothing to do
77. ugh 5 if you want to converge the acceleration results when using an 3 integrator the accelerations are known to sometimes display spikes which typically are more difficult to eliminate In our experience the acceleration spikes can be eliminated or reduced by slowly decreasing the value 60 of HMAX and or ERROR while switching to INTERPOLATE ON in the INTEGRATOR statement command In case this approach does not work you might also want to read the Knowledge Base Article 12400 which presents a slightly different alternative for converging the acceleration results See also section 8 3 of this document for other suggestions 8 If both the displacement and accelerations have been converged it is recommended to run the simulation with SI2 as a last step in this process the idea being that all integrators should converge to the same solution albeit with different settings If S12 will run with the settings determined by steps 1 through 7 for I3 then it will most likely be converged to the same solution This is because SI2 will converge displacements with a larger setting of ERROR than needed with I3 Typically velocities will also be converged during the ERROR part of the process steps 1 through 5 when using SI2 The advantages of SI2 are documented in the ADAMS Solver documentation but the main points to be noted are i SI2 is more accurate for velocities and accelerations and ii the Jacobian matrix is very stable at small step si
78. zes something to keep in mind when you are reducing HMAX with I3 One caveat with SI2 is that motions must be smooth and twice differentiable Non smooth motions which cause infinite accelerations are a common cause of failures with SI2 As pointed out in section 8 3 you should always strive to provide smooth inputs to your model be it of force of motion type On a final note keep in mind that sometimes in spite of all your efforts you won t be able to converge the results Almost always either a modeling error or a non smooth input function is responsible for this 8 5 Debugging Support in ADAMS Solver The user can get a wealth of information about the states and internal doings of the Solver as it advances in time the configuration of the model There are three main statements commands that expose this information to the user a DEBUG EPRINT b DEBUG RHSDUMP c DEBUG JMDUMP 8 5 1 DEBUG EPRINT This statement command is used to obtain a synopsis of the processes that take place during a solution sequence A solution sequence should be regarded as any action or algorithm invoked by the Solver to advance the simulation or to execute a user issued command For example issuing a simulate statics command will cause the Solver to a set up the set of equations that needs to be solved to find an equilibrium configuration and b employ and algorithm to find the solution of the assembled equation set DEBUG EPRINT will send ou

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