1. ME451 Kinematics and Dynamics of Machine Systems Introduction to Dynamics 6.1 October 09, 2013 Radu Serban University of Wisconsin-Madison

2. 2 Before we get started… Last Time: Concluded Kinematic Analysis Today: Towards the Newton-Euler equations for a single rigid body Assignments: Matlab 4 – due today, Learn@UW (11:59pm) Adams 2 – due today, Learn@UW (11:59pm) Submit a single PDF with all required information Make sure your name is printed in that file Midterm Exam Friday, October 11 at 12:00pm in ME1143 Review session: today, 6:30pm in ME1152 Midterm Feedback Form emailed to you later today Anonymous Complete it and return on Friday

3. 3 Kinematics vs. Dynamics Kinematics We include as many actuators as kinematic degrees of freedom – that is, we impose KDOF driver constraints We end up with NDOF = 0 – that is, we have as many constraints as generalized coordinates We find the (generalized) positions, velocities, and accelerations by solving algebraic problems (both nonlinear and linear) We do not care about forces, only that certain motions are imposed on the mechanism. We do not care about body shape nor inertia properties Dynamics While we may impose some prescribed motions on the system, we assume that there are extra degrees of freedom – that is, NDOF > 0 The time evolution of the system is dictated by the applied external forces The governing equations are differential or differential-algebraic equations We very much care about applied forces and inertia properties of the bodies in the mechanism

4. 4 Dynamics M&S Dynamics Modeling Formulate the system of equations that govern the time evolution of a system of interconnected bodies undergoing planar motion under the action of applied (external) forces These are differential-algebraic equations Called Equations of Motion (EOM) Understand how to handle various types of applied forces and properly include them in the EOM Understand how to compute reaction forces in any joint connecting any two bodies in the mechanism Dynamics Simulation Understand under what conditions a solution to the EOM exists Numerically solve the resulting (differential-algebraic) EOM

5. 5 Roadmap to Deriving the EOM Begin with deriving the variational EOM for a single rigid body Principle of virtual work and D’Alembert’s principle Consider the special case of centroidal reference frames Centroid, polar moment of inertia, (Steiner’s) parallel axis theorem Write the differential EOM for a single rigid body Newton-Euler equations Derive the variational EOM for constrained planar systems Virtual work and generalized forces Finally, write the mixed differential-algebraic EOM for constrained systems Lagrange multiplier theorem (This roadmap will take several lectures, with some side trips)

6. 6 What are EOM? In classical mechanics, the EOM are equations that relate (generalized) accelerations to (generalized) forces Why accelerations? If we know the (generalized) accelerations as functions of time, they can be integrated once to obtain the (generalized) velocities and once more to obtain the (generalized) positions Using absolute (Cartesian) coordinates, the acceleration of body i is the acceleration of the body’s LRF: How do we relate accelerations and forces? Newton’s laws of motion In particular, Newton’s second law written as

7. 7 Newton’s Laws of Motion 1 st Law Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. 2 nd Law A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. 3 rd Law To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction. Newton’s laws are applied to particles (idealized single point masses) only hold in inertial frames are valid only for non-relativistic speeds Isaac Newton (1642 – 1727)

8. Variational EOM for a Single Rigid Body 6.1.1

9. 9 Body as a Collection of Particles

10. 10 Forces Acting on a Differential Mass dm(P)

11. 11 Newton’s EOM for a Differential Mass dm(P)

12. 12 A Model of a Rigid Body

13. 13 [Side Trip] Virtual Displacements A small displacement (translation or rotation) that is possible (but does not have to actually occur) at a given time In other words, time is held fixed A virtual displacement is virtual as in “virtual reality” A virtual displacement is possible in that it satisfies any existing constraints on the system; in other words it is consistent with the constraints Virtual displacement is a purely geometric concept: Does not depend on actual forces Is a property of the particular constraint The real (true) displacement coincides with a virtual displacement only if the constraint does not change with time Actual trajectory Virtual displacements

14. 14 [Side Trip] Calculus of Variations (1/3)

15. 15 [Side Trip] Calculus of Variations (2/3)

16. 16 [Side Trip] Calculus of Variations (3/3)

17. 17 Virtual Displacement of a Point Attached to a Rigid Body

18. 18 The Rigid Body Assumption: Consequences

19. 19 [Side Trip] An Orthogonality Theorem

20. 20 Variational EOM for a Rigid Body (1)

21. 21 Variational EOM for a Rigid Body (2)

22. 22 [Side Trip] D’Alembert’s Principle Jean-Baptiste d’Alembert (1717– 1783)

23. 23 [Side Trip] Principle of Virtual Work Principle of Virtual Work If a system is in (static) equilibrium, then the net work done by external forces during any virtual displacement is zero The power of this method stems from the fact that it excludes from the analysis forces that do no work during a virtual displacement, in particular constraint forces D’Alembert’s Principle A system is in (dynamic) equilibrium when the virtual work of the sum of the applied (external) forces and the inertial forces is zero for any virtual displacement “D’Alembert had reduced dynamics to statics by means of his principle” (Lagrange) The underlying idea: we can say something about the direction of constraint forces, without worrying about their magnitude

24. 24 [Side Trip] PVW: Simple Statics Example