ME451 Kinematics and Dynamics of Machine Systems

1. ME451 Kinematics and Dynamics of Machine Systems Newton-Euler EOM 6.1.2, 6.1.3October 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), Learn@UW (11:59pm) • Adams 3 – due Wednesday (Oct. 16), Learn@UW (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 • Our toolbox provides a relationship between forces and accelerations (Newton’s 2nd law) – but that applies for particles only • Idea: look at a body as a collection of infinitesimal particles • Consider a differential mass at each point on the body (located by ) • For each such particle, we can write • What forces should we include? • Distributed forces • Internal interaction forces, between any two points on the body • Concentrated (point) forces

4. A Model of a Rigid Body • We model a rigid body with distance constraints between any pair of differential elements (considered point masses) in the body. • Therefore the internal forces on due to the differential mass on due to the differential mass satisfy the following conditions: • They act along the line connectingpoints and • They are equal in magnitude, opposite in direction, and collinear

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 purelygeometric concept: • Does not depend on actual forces • Is a property of the particular constraint • The real (true) displacement coincideswitha virtual displacement only if theconstraint does not change with time Virtual displacements Actual trajectory

6. Variational EOM for a Rigid Body (1)

7. The Rigid Body Assumption:Consequences • The distance between any two points and on a rigid body is constant in time:and therefore • The internal force acts along the line between and and therefore is also orthogonal to :

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 ofVariations in Generalized Coordinates (1/2)

13. Virtual Displacements in terms ofVariations in Generalized Coordinates (2/2)

14. 6.1.2, 6.1.3 Variational EOM with Centroidal CoordinatesNewton-Euler Differential EOM

15. Centroidal Reference Frames • The variational EOM for a single rigid body can be significantly simplified if we pick a special LRF • A centroidalreference 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 • The variational EOM of a rigid body with a centroidal body-fixed reference frame were obtained as: • Assume all forces acting on the body have been accounted for. • Since and are arbitrary, using the orthogonality theorem, we get: • Important: The Newton-Euler equations are valid only if all force effects have been accounted for! This includes both appliedforces/torques and constraint forces/torques(from interactions with other bodies). Leonhard Euler (1707 – 1783) Isaac Newton (1642 – 1727)

20. Tractor Model[Example 6.1.1] • Derive EOM under the following assumptions: • Traction (driving) force generated at rear wheels • Small angle assumption (on the pitch angle) • Tire forces depend linearly on tire deflection: