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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
2
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
9
Location, Original
Location, afterVirtual Displacement
Deriving the EOM
10
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
11
Some Clarifications[regarding the “Rigid Body” concept]
12
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
13
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.
14
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
16
Virtual Work: Dealing with Internal Interaction Forces
17
Dealing with Active Forces
18
Dealing with Active Torques
19
Virtual Work: Dealing with Constraint Reaction Forces
20
Virtual Work: Dealing with Constraint Reaction Torques
21
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)
22
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)
23
On the meaning of [B¹sP ]T ¢F
P
i
j
O
Y
X
O
y
x
r
Ps
G -RF
L -RFPr
F
25