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ME451 Kinematics and Dynamics of Machine Systems Dynamics of Planar Systems March 24, 2009 Chapter 6 © Dan Negrut, 2009 ME451, UW-Madison Before we get started… Last Time Discussion about numerical methods Emphasis placed on numerical integration methods We will rely on numerical integration methods when numerically solving the IVP associated with the set of Newton-Lagrange equations of motion Pointed out the difference between ODE and IVP Discussed explicit integration methods: Euler, RK4, Predictor-Corrector Discussed implicit integration methods: Backward Euler Discussed about MATLAB offering when it comes to solution of IVPs The solution of this problem gives the time evolution of the mechanical system Today Start Chapter 6: Dynamics Work on Sections 6.1.1 and 6.1.2 HW (due March 31): 6.1.1, 6.1.2, 6.1.3, and 6.1.4 ADAMS Component available on the website 2 Purpose of Chapter 6 At the end of this chapter you should understand what “dynamics” means and how you should go about carrying out a dynamics analysis We’ll learn a couple of things: How to formulate the equations that govern the time evolution of a system of bodies in planar motion These equations are differential equations and they are called equations of motion As many bodies as you wish, connected by any joints we’ve learned about… How to compute the reaction forces in any joint connecting any bodies in the mechanism Understand how to properly handle the applied (aka, external) forces to correctly use them in formulating the equations of motion 3 The Idea, in a Nutshell… First part of the class: Kinematics You have as many constraints (kinematic and driving) as generalized coordinates No spare degrees of freedom left Position, velocity, acceleration found as the solution of algebraic problems (both nonlinear and linear) We do not care whatsoever about forces applied to the system, we are told what the motions are and that’s enough for the purpose of kinematics Second part of the class: Dynamics You only have a few constraints imposed on the system You have extra degrees of freedom The system evolves in time as a result of external forces applied on it We very much care about forces applied and inertia properties of the components of the mechanism (mass, mass moment of inertia) 4 Some clarifications Dynamics key question: how can I get the acceleration of each body of the mechanism? Why is acceleration so relevant? If you know the acceleration you can integrate it twice to get velocity and position information for each body How is the acceleration of a body “i ”measured in the first place? You attach a reference frame on body “i ” and measure the acceleration of the body reference frame with respect to the global reference frame: The answer to the key question: To get the acceleration of each body, you first need to formulate the equations of motion Remember F=ma? Actually, the proper way to state this is ma=F, which is the “equation of motion” and is the most important piece of the dynamics puzzle 5 Equations of motion of ONE planar RIGID body Framework: We are dealing with rigid bodies (flexible bodies covered in ME751) For this lecture, we’ll consider only one body We’ll extend to more bodies in two weeks… What are we after? Proving that for one body with a reference frame attached at its center of mass location the equations of motion are: r is the position of the body reference frame is the orientation of the body reference frame 6 More clarifications… Centroidal reference frame of a body A reference frame located right at the center of mass of that body How is this special? It’s special since a certain integral vanishes... What is J’ ? Mass moment of inertia NOTE: Textbook uses misleading notation 7 [Cntd.] More clarifications… Applied (external) force per unit mass acting at point P: Then F defined as: Internal force per unit mass that a mass at location R exerts on point P: Total internal force exerted on point P by all other points R in the body: 8 KEY CONCEPT: Virtual Displacement VERY important concept, but rather esoteric and hard to grasp It’s important because it comes up when one starts talking about taking a “variational approach” to deriving the equations of motion Virtual Displacement, definition: a tiny change applied to the position and orientation (generalized coordinates) associated with the reference frame of a body In other words, the location and orientation of the reference frame are slightly changed Since the reference frame is attached to the body, it means that the body is slightly nudged when is subjected to a virtual displacement 9 KEY CONCEPT: Virtual Displacement Changes in x, y, and of body fixed reference frame are arbitrary but infinitesimally small Changes are denoted by x , y , and , respectively … using generalized coordinates notation … Y NOTE: The virtual displacement is applied with the understanding that the time is held fixed; that is, the concept of time does not exist xi yi i Oi O X 10 KEY CONCEPT: Virtual Displacement Applying a virtual displacement leads to changes in the value of any function that depends on location/orientation of a body, since its very location/orientation has slightly changed The change in function as a result of applying a virtual displacement follows the rules of differential calculus However, you have to replace the “d” operator with the ““ operator to clearly indicate that the change is a consequence of a virtual displacement (time is held fixed) Example: What is the change in position of point P on body i when the body is the subject of a virtual displacement? 11 [Example] Virtual Displacement Related… Indicate how the change in the quantities below that are a consequence of applying a virtual displacement to the generalized coordinates q The dimensions of the vectors and matrix above such that all the operations listed can be carried out. 12 Deriving Newton’s Equations for one body in planar motion Assumptions: The body is rigid The body undergoes planar motion We start ab-initio, that is, we only know that for a point, Newton noticed that See derivation in the textbook 13