ME451 Kinematics and Dynamics of Machine Systems Dynamics of Planar Systems November 10, , starting 6.3 © Dan Negrut, 2011 ME451, UW-Madison TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A AA A A A A AA “I am the greatest, I said that even before I knew I was. “ Muhammad Ali

Before we get started… Last Time Derived EOM for one body Discussed how to determine the net effect of a concentrated (point) force on the equations of motion That is, we learned how to compute a generalized force Q induced by a point force F P Looked into inertia properties of 2D geometries Center of mass Parallel axis theorem Mass moment of inertia for composite geometries Today Example Look into two types of concentrated forces/torques: TSDA/RSDA Formulating the EOM for an entire mechanism (start discussion) Next Tuesday: TA will discuss how to improve the implementation of simEngine2D 2

Generalized Force Induced by a Point Force The fundamental idea : Whenever some new force shows up, figure out the virtual work that it brings into the picture Then account for this “injection” of 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 3 (Keep this in mind when you solve Problem 6.2.1)

[Review of material from last lecture] Concentrated (Point) Force Setup: At a particular point P, you have a point-force F P acting on the body 4 General Strategy: Step A: write the virtual work produced by this force as a results of a virtual displacement of point P Step B: express this additional virtual work in terms of body virtual displacements End with this: Start with this:

Point Force point P) [Review, cntd.] How is virtual work computed? How is the virtual displacement of point P computed? (we already know this…) 5 The step above: expressing the virtual displacement that the force “goes through” in terms of the body virtual displacements  r and  

Notation & Nomenclature Matrix-vector notation for the expression of d’Alembert Principle: “generalized forces”: Q above 6

Example Tractor model: derive equations of motion 7 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) Elaborate on the nature of the “concentrated forces” that we introduced. A closer look at the “concentrated” forces reveals that they could be Forces coming out of translational spring-damper-actuator elements Forces coming out of rotational spring-damper-actuator elements Reaction forces (due to the presence of a constraint, say between body and ground) 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 Where are we going with this? By the end of the 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. Points 1) and 2) above are important pieces of the puzzle. 8

Scenario 1: 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 9 Translational spring, stiffness k Zero stress length (given): l 0 Translational damper, coefficient c Actuator (hydraulic, electric, etc.) – symbol used “h”

Scenario 1: TSDA (Cntd.) General Strategy: Step A: write the virtual work produced by this force as a results of a virtual displacement of point P Step B: express additional virtual work in terms of body virtual displacements 10 Force developed by the TSDA element: Alternatively,

Scenario 2: RSDA 11 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 2: 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 this additional virtual work in terms of body virtual displacements 12 Torque developed by the TSDA element: Notation: – torque developed by actuator – relative angle between two bodies

End: Discussion regarding Generalized Forces Begin: Variational Equations of Motion for a Planar System of Bodies (6.3.1) 13

A Vector-Vector Multiplication Trick… [One Slide Detour] Given two vectors a and b, each made up of nb smaller vectors of dimension 3… …the dot product a times b can be expressed as 14

Matrix-Vector Approach to EOMs For body i the generalized coordinates are: Variational form of the Equations of Motion (EOM) for body i (matrix notation): 15 Generalized force, contains all external (aka applied) AND internal (aka reaction) forces… Arbitrary virtual displacement Generalized Mass Matrix:

EOMs for the Entire System Assume we have nb bodies, and write for each one the variational form of the EOMs 16 Sum them up to get… Use matrix-vector notation… Notation used:

A Word on the Expression of the Forces Total force acting on a body is sum of applied [external] and constraint [internal]: 17 IMPORTANT OBSERVATION: We want to get rid of the constraint forces Q C since we do not know them (at least not for now) For this, we need to compromise…

Constraint Forces… Constraint Forces Forces that show up in the constraints present in the system: revolute, translational, distance constraint, etc. They are the forces that ensure the satisfaction of the constraint (they are such that the motion stays compatible with the kinematic constraint) KEY OBSERVATION: The net virtual work produced by the constraint forces present in the system as a result of a set of consistent virtual displacements is zero Note that we have to account for the work of *all* reaction forces present in the system Here is what this buys us: 18 …provided  q is a consistent virtual displacement

Consistent Virtual Displacements What does it take for a virtual displacement to be “consistent” [with the set of constraints] at a fixed time t * ? Say you have a consistent configuration q, that is, a configuration that satisfies your set of constraints: Ok, so now you want to get a virtual displacement  q such that the configuration q+  q continues to be consistent: Apply now a Taylor series expansion (assume small variations): 19

Getting Rid of the Internal Forces: Summary 20 Our Goal: Get rid of the constraint forces Q C since we don’t know them For this, we had to compromise… We gave up our requirement that holds for any arbitrary virtual displacement, and instead requested that the condition holds for any virtual displacement that is consistent with the set of constraints that we have in the system, in which case we can simply get rid of Q C : provided… This is the condition that it takes for a virtual displacement  q to be consistent with the set of constraints NOMENCLATURE: Constrained Variational Equations of Motion