ME451 Kinematics and Dynamics of Machine Systems Dynamics of Planar Systems March 31, , starting 6.3 © Dan Negrut, 2009 ME451, UW-Madison Quote of the Lecture: History will be kind to me, for I intend to write it. - Winston Churchill

Before we get started… [1] Coming up on Thursday Guest Lecturer Dr. Kevin Chang of British Aerospace Engineering (BAE), Land Division Last Time Wrapped up the derivation of the EOM for planar rigid bodies Recall that Looked into inertia properties of 2D geometries Center of mass Parallel axis theorem Mass moment of inertia for composite geometries Today Example, application of ma=F for rigid bodies Discuss the concept of generalized force HW (due on April 7): 6.1.5,

Before we get started… [2] HW Hint: Problem Use to find the location of the centroidal RF Also use the fact that the composite body has an axis of symmetry to reduce your effort Use and to find mass moment of inertia See also Example if you run into difficulties 3

Example Tractor model: derive equations of motion 4 Traction force T r Small angle assumption (pitch angle  ) Force in tires depends on tire deflection: P Q

What’s Left 30,000 Feet Perspective Two important issues remain to be addressed: 1) How do I take into account forces that act on the body when writing the equation of motion for that body? Concentrated forces (“Point forces”) Forces coming out of translational spring-damper-actuator elements Forces coming out of rotational spring-damper-actuator elements 2) We only derived the variational form of the equation of motion for the trivial case of *one* rigid body. How do I derive the variational form of the equations of motion for a mechanism with many components (bodies) connected through joints? Just like before, we’ll rely on the principle of virtual work Why do I care to do this? In one week we’ll be able to formulate the equations that govern the time evolution of an arbitrary set of rigid bodies interconnected by an arbitrary set of kinematic constraints. These two issues are important pieces of the puzzle. 5

Notation & Nomenclature We’ll formulate the problem in a concise fashion using matrix-vector notation: So what do I actually mean when I talk about “generalized forces”? ~ I mean the Q above ~ 6

Generalized Forces: Problem Context Q : How are F and n in the equations of motion obtained (specified)? After all, where are the original F and n coming from? 7

Generalized Forces: Problem Context F was the sum of all distributed forces F (P) acting per unit mass: n was the torque produced by the forces F (P) QUESTION: What happens when we don’t have distributed forces, such as F(P), but rather a force acting at a point Q of the body? (Eq ) (Eq ) 8

Generalized Forces The fundamental idea : Whenever some new force shows up, figure out the virtual work that it brings into the picture Then account for this additional virtual work in the virtual work balance equation: Caveat: Notice that for rigid bodies, the virtual displacements are  r and . Some massaging of the additional virtual work might be needed to bring it into the standard form, that is 9

Scenario 1: Point Force Setup: At a particular point Q, you have a point-force f acting on the body 10 General Strategy: Step A: write the virtual work produced by this force as a results of a virtual displacement of the body Step B: express force f produced additional virtual work in terms of body virtual displacements Recall:

Scenario 1: Point Force (Cntd.) How is virtual work computed? How is the virtual displacement of point Q computed? (we already know this…) 11 The step above is the challenging one: expressing the displacement that the force goes through in terms of the body virtual displacements  r and  

Scenario 2: TSDA (Translational-Spring-Damper-Actuator) – pp.216 Setup: You have a translational spring-damper-actuator acting between point P i on body i, and P j on body j 12 Translational spring, stiffness k Zero stress length (given): l 0 Translational damper, coefficient c Actuator (hydraulic, electric, etc.) – symbol used “h”

Scenario 2: TSDA (Cntd.) General Strategy: Step A: write the virtual work produced by this force as a results of a virtual displacement of the body Step B: express additional virtual work in terms of body virtual displacements 13 Force developed by the TSDA element: Notation: – force developed by actuator – distance between P i and P j

Scenario 3: RSDA 14 Rotational spring, stiffness k Rotational damper, coefficient c Actuator (hydraulic, electric, etc.) – symbol used “h” Setup: You have a rotational spring-damper-actuator acting between two lines, each line rigidly attached to one of the bodies (dashed lines in figure)

Example 3: RSDA (Cntd.) General Strategy: Step A: write the virtual work produced by this force as a results of a virtual displacement of the body Step B: express additional virtual work in terms of body virtual displacements 15 Torque developed by the TSDA element: Notation: – torque developed by actuator – relative angle between two bodies