ME451 Kinematics and Dynamics of Machine Systems

1. ME451 Kinematics and Dynamics of Machine Systems Dynamics of Planar Systems May 07, 2009 EOM in non-Cartesian Reference Frames~ not in textbook~ Quote of the day: Never mistake motion for action. Ernest Hemingway © Dan Negrut, 2009ME451, UW-Madison

2. Before we get started… • Last Time • Inverse Dynamics Analysis • Equilibrium Analysis • Covered two examples • Today: • Discuss about formulating the EOMs using a set of generalized coordinates (GCs) different than the Cartesian GCs • The idea is that you use precisely the same ingredients that you use in the Cartesian case • You need only one additional thing – a “bridging” function that tells you how the Cartesian GCs are obtained in terms of the new GCs that you prefer to use • Looking ahead • Final Exam is one week from Friday • Bring your take-home component • Bring your final project if you chose to do one 2

3. Course Evaluation • Please comment on what you didn’t like about this class • Please comment on what you liked about this class • Please comment on the use of ADAMS in this class • Any departing thoughts you might have for me 3

4. Today lecture’s Question: • You have a mechanism and are interested in finding its time evolution • What if you want to express the equations of motion (EOM) in a set of generalized coordinates that is not Cartesian? • Example: Simple Pendulum • Why not use the angle  to express the time evolution of the pendulum? 4

5. Non-Cartesian GCs: Further comments • Benefits of working with non-Cartesian GCs: • You might be able to formulate the problem as an ODE problem as opposed to a DAE problem as it is almost always the case when you use Cartesian coordinates • For ODE problems you don’t have the complications deriving from the use of Newmark’s formulas, and using any classical numerical integration formula will do • The dimension of the problem in general is reduced • Typically, you can reduce it all the way to having a number of differential equations equal to the number of degrees of freedom associated with your mechanism • Example: • For the simple pendulum you’d have one second order ODE • For the double pendulum you’d have two second order ODEs 5

6. Non-Cartesian GCs: Further comments(Cntd.) • Drawbacks of working with non-Cartesian GCs: • You might lose the ability to compute the reaction forces since you eliminate their presence from the problem • This goes back to trading the DAE for an ODE problem (there are no Lagrange multipliers associated with an ODE problem…) • The expression that the ODE problem assumes is tremendously messy • Ex: Slider-Crank 6

7. EOM for Slider Crank: The coefficients d1, d2, and d3 above are defined as:

8. Nomenclature & Notation OLD NEW 8

9. [The Important Point] Nomenclature & Notation • Recall that we call the Cartesian GCs “q” and the new GCs “w” • There is a relationship between the two of them • Specifically, the Cartesian GCs are obtained based on the new GCs through a function : Cartesian GCs New GCs

10. [Example, Simple Pendulum]Computation of  • Scenario 1: • What is the expression of the function (w)? • Observation 1: There are no constraints on w • can assume any value • Observation 2: For any , if I compute q=(w), it turns out that q satisfies the constraint equations 10

11. [Example, Simple Pendulum]Computation of  • Scenario 2: (the x and y of the CM) • What is the expression of the function (w)? • Observation: Note that there are constraints on x and y: 11

12. Taxonomy • Based on how we select the generalized coordinates w, we can be in one of TWO cases: • 1) There are no constraints that the new GCs must satisfy • These GCs are called Lagrangian GCs • They are equal in number to the number of degrees of freedom • See Scenario 1 on previous slide, the angle  was not subject to any constraints • 2) There are some constraints that the new GCs must satisfy • See Scenario 2 on previous slide • w must satisfy in this case some nonlinear algebraic equations 12

13. The Lagrangian GCs Case • NOTE: For any selection of w, the set of Cartesian GCs q is consistent, that is, it satisfies the original set of constraint equations • Go back to variational form of the equations of motion: • Here w can change in an arbitrary fashion, no constraints that need to be satisfied • How is a virtual displacement in q related to a virtual displacement in w? 13

14. The case of Lagrangian Gen. Coordinates(Cntd.) • Note that the acceleration in q and w are related: • Simple manipulation lead to • From here, since w is arbitrary (no constraints acting on w), 14

15. Example [AO]Simple Pendulum • Simple Pendulum: • Mass 20 kg • Length L=2 m • Force acting at tip of pendulum • F = 30 sin(2 t) [N] • ICs: hanging down, starting from rest • FIND EQUATION OF MOTION IN TERMS OF  • NOTE: • This is a Lagrangian generalized coordinates scenario 15

16. The case of non-Lagrangian and non-Cartesian Gen. Coordinates(this is pretty thick…) 16

17. The case of non-Lagrangian and non-Cartesian Gen. Coordinates • Here wcannot change in an arbitrary fashion, it must satisfy constraints • How is a virtual displacement in q related to a virtual displacement in w? • It can be proved that if w represents a consistent virtual displacement, that is, … then by taking q = ww you got yourself a consistent virtual displacement q, that is, 17

18. The case of non-Lagrangian and non-Cartesian Gen. Coordinates • Proof of previous result is skipped but it relies on the following observation: • If w is a consistent configuration, that is … then q= (w) is also a consistent configuration, that is 18

19. The case of non-Lagrangian and non-Cartesian Gen. Coordinates • Note: Here w must satisfy at all times • Variational form of the equations of motion: • The above holds only when w is consistent, that is • Applying Lagrange’s theorem, one ends up with the following: 19

20. What GCs do people use? • ADAMS uses Cartesian GCs, as does the second most widely used commercial package out of US • Most widely used commercial packages out of Asia, RecurDyn, uses non-Cartesian GCs • For open tree topologies they use Lagrangian GCs • Most widely used commercial packages out of Europe, SimPack, uses non-Cartesian GCs • My take on this: • I prefer Cartesian GCs • EOMs are larger, but sparser • EOMs are easy to formulate, I let the computer deal with their solution • Where are non-Cartesian GCs widely used? • In robotics • When you have open tree topologies • In these cases you can actually easily select a set of Lagrangian GCs and end up with an ODE problem for your EOM.