ME451 Kinematics and Dynamics

of Machine Systems

Dynamics of Planar Systems: Chapter 6November 1, 2011

© Dan Negrut, 2011ME451, UW-Madison

“Computer science education cannot make anybody an expert programmer any more than studying brushes and pigment can make somebody an expert painter.”

Eric Raymond

Before we get started…

Last Time Discuss “Singular Configurations of Mechanisms” (Section 3.7) Start the “Dynamics Analysis” part of the course (Chapter 6)

Today: Virtual displacements Look at all types of forces we might deal with in ME451 and determine the virtual work they lead to Start derivation of EOM of one body (pp. 200 of textbook)

HW (due on November 3 at 11:59 PM): ADAMS MATLAB

Quick Remarks: Exam coming up on Nov. 3 during regular class hour Exam Review on Nov. 2, starting at 6PM in room 1153ME

Note that the review room is the one next door

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Two Principles

Principle of Virtual Work Applies to a collection of particles States that a configuration is an equilibrium configuration if and only if

the virtual work of the forces acting on the collection of particle is zero

D’Alembert’s Principle For a collection of particles experiencing accelerated motion you can still

fall back on the Principle of Virtual Work when you also include in the set of forces acting on each particle its inertia force

NOTE: we are talking here about a collection of particles Consequently, we’ll have to regard each rigid body as a collection of particles

that are rigidly connected to each other and that together make up the body3

Virtual Work and Virtual Displacement

4

P

i

j

O

Y

X

O

y

x

r

Ps

G -RF

L -RFPr

PF

5

[Small Detour, 2 slides]

[Example]

Calculus of Variations

7The dimensions of the vectors and matrix above such that all the operations listed can be carried out.

Indicate the change in the quantities below that are a consequence of applying a virtual displacement q to the generalized coordinates q

Calculus of Variations in ME451

In our case we are interested in variations of kinematic quantities (locations of a point P, of A matrix, etc.) due to a variations in the location and orientation of a body.

Variation in location of the L-RF:

Variation in orientation of the L-RF:

As far as the change of orientation matrix A(Á) is concerned, using the result stated two slides ago, we have that a variation in the orientation leads to the following variation in A:

r ¡ ! r + ±r

Á ¡ ! Á+ ±Á

±A =dAdÁ

¢±Á = B ¢±Á8

Calculus of Variations in ME451Virtual Displacement of a Point P Attached to a Body

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Location, Original

Location, afterVirtual Displacement

Deriving the EOM

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Some Clarifications

Assumptions: All bodies that we work with are rigid*

The bodies undergo planar motion

We will use a full set of Cartesian coordinates to position and orient a body in the 2D space

Start from scratch, that is, from the dynamics of a material point First, we’ll work our way up to determining the EOM for one body Then, we’ll learn how to deal with a collection of bodies that are interacting

through kinematic joints and/or friction & contact

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Some Clarifications[regarding the “Rigid Body” concept]

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Road Map [2 weeks]

Introduce the forces present in a mechanical system Distributed Concentrated

Express the virtual work produced by each of these forces

Apply principle of virtual work and obtain the EOM

Eliminate the reaction forces from the expression of the virtual work

Obtain the constrained EOM (Newton-Euler form)

Express the reaction (constraint) forces from the Lagrange multipliers

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Types of Forces & TorquesActing on a Body

Type 1: Distributed over the volume of a body Examples:

Inertia forces Internal interaction forces Etc.

Type 2: Concentrated at a point Examples:

Action (or applied, or external) forces and torques Reaction (or constraint) forces and torques Etc.

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P

i

j

O

Y

X

O

y

x

r

Ps

G -RF

L -RFPr

PF

Virtual Work: Dealing with Inertia Forces

15

Virtual Work: Dealing with Mass-Distributed Forces

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Virtual Work: Dealing with Internal Interaction Forces

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Dealing with Active Forces

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Dealing with Active Torques

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Virtual Work: Dealing with Constraint Reaction Forces

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Virtual Work: Dealing with Constraint Reaction Torques

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Deriving Newton’s Equations for a body with planar motion

NOTE: You should be able to derive

Newton’s equations for a planar rigid body on your own (closed books)

Overall, the book does a very good job in explaining the derivation the equations of motion (EOM) for a rigid body

The material is straight out of the book (page 200)

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EOM: First Pass

For now, assume that there are no concentrated forces

Do this for *one* body for now

We are going to deal with the distributed forces and use them in the context of d’Alembert’s Principle Inertia forces Internal forces Other distributed forces (gravity)

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On the meaning of [B¹sP ]T ¢F

P

i

j

O

Y

X

O

y

x

r

Ps

G -RF

L -RFPr

F

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