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ME451 Kinematics and Dynamics of Machine Systems Dynamics of Planar Systems: Chapter 6 November 1, 2011 © Dan Negrut, 2011 ME451, UW-Madison TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A Computer science education cannot make anybody an expert programmer any more than studying brushes and pigment can make somebody an expert painter. Eric Raymond

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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

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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 DAlemberts 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, well have to regard each rigid body as a collection of particles that are rigidly connected to each other and that together make up the body 3

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Virtual Work and Virtual Displacement 4

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5 [Small Detour, 2 slides]

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[Example] Calculus of Variations 7 The 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

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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: 8

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Calculus of Variations in ME451 Virtual Displacement of a Point P Attached to a Body 9 Location, Original Location, after Virtual Displacement

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Deriving the EOM 10

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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, well work our way up to determining the EOM for one body Then, well learn how to deal with a collection of bodies that are interacting through kinematic joints and/or friction & contact 11

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Some Clarifications [regarding the Rigid Body concept] 12

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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 13

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Types of Forces & Torques Acting 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

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Virtual Work: Dealing with Inertia Forces 15

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Virtual Work: Dealing with Mass-Distributed Forces 16

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Virtual Work: Dealing with Internal Interaction Forces 17

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Dealing with Active Forces 18

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Dealing with Active Torques 19

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Virtual Work: Dealing with Constraint Reaction Forces 20

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Virtual Work: Dealing with Constraint Reaction Torques 21

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Deriving Newtons Equations for a body with planar motion NOTE: You should be able to derive Newtons 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

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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 dAlemberts Principle Inertia forces Internal forces Other distributed forces (gravity) 23

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