ME451 Kinematics and Dynamics of Machine Systems Newton-Euler EOM 6.1.2, October 14, 2013 Radu Serban University of Wisconsin-Madison

2 Before we get started… Last Time: Started the derivation of the variational EOM for a single rigid body Started from Newton’s Laws of Motion Introduced a model of a rigid body and used it to eliminate internal interaction forces Today: Principle of Virtual Work and D’Alembert’s Principle Introduce centroidal reference frames Derive the Newton-Euler EOM Assignments: Matlab 5 – due Wednesday (Oct. 16), (11:59pm) Adams 3 – due Wednesday (Oct. 16), (11:59pm) Submit a single PDF with all required information Make sure your name is printed in that file

3 Body as a Collection of Particles

4 A Model of a Rigid Body

5 [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

6 Variational EOM for a Rigid Body (1)

7 The Rigid Body Assumption: Consequences

8 Variational EOM for a Rigid Body (2)

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

10 [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

11 [Side Trip] PVW: Simple Statics Example

12 Virtual Displacements in terms of Variations in Generalized Coordinates (1/2)

13 Virtual Displacements in terms of Variations in Generalized Coordinates (2/2)

Variational EOM with Centroidal Coordinates Newton-Euler Differential EOM 6.1.2, 6.1.3

15 Centroidal Reference Frames The variational EOM for a single rigid body can be significantly simplified if we pick a special LRF A centroidal reference frame is an LRF located at the center of mass How is such an LRF special? By definition of the center of mass (more on this later) is the point where the following integral vanishes:

16 Variational EOM with Centroidal LRF (1/3)

17 Variational EOM with Centroidal LRF (2/3)

18 Variational EOM with Centroidal LRF (3/3)

19 Differential EOM for a Single Rigid Body: Newton-Euler Equations Isaac Newton (1642 – 1727) Leonhard Euler (1707 – 1783)

20 Tractor Model [Example 6.1.1]