ME451 Kinematics and Dynamics

of Machine Systems

Newton-Euler EOM6.1.2, 6.1.3

October 14, 2013

Radu SerbanUniversity of Wisconsin-Madison

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Before we get started…

Last Time: Started the derivation of the variational EOM for a single rigid body Started from Newton’s Laws of Motion Introduced a model of a rigid body and used it to eliminate internal interaction

forces

Today: Principle of Virtual Work and D’Alembert’s Principle Introduce centroidal reference frames Derive the Newton-Euler EOM

Assignments: Matlab 5 – due Wednesday (Oct. 16), Learn@UW (11:59pm) Adams 3 – due Wednesday (Oct. 16), Learn@UW (11:59pm)

Submit a single PDF with all required information Make sure your name is printed in that file

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Body as a Collection of Particles

Our toolbox provides a relationship between forces and accelerations (Newton’s 2nd law) – but that applies for particles only

Idea: look at a body as a collection of infinitesimal particles

Consider a differential mass at each point on the body (located by )

For each such particle, we can write

What forces should we include? Distributed forces Internal interaction forces, between any two points on the body Concentrated (point) forces

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A Model of a Rigid Body We model a rigid body with distance constraints between any pair of

differential elements (considered point masses) in the body.

Therefore the internal forces

on due to the differential mass on due to the differential mass

satisfy the following conditions: They act along the line connecting

points and They are equal in magnitude,

opposite in direction, and collinear

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[Side Trip]Virtual Displacements

A small displacement (translation or rotation) that is possible (but does not have to actually occur) at a given time In other words, time is held fixed A virtual displacement is virtual as in “virtual reality” A virtual displacement is possible in that it satisfies any existing

constraints on the system; in other words it is consistent with the constraints

Virtual displacement is a purelygeometric concept: Does not depend on actual forces Is a property of the particular constraint

The real (true) displacement coincideswith a virtual displacement only if theconstraint does not change with time

Actualtrajectory

Virtualdisplacements

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Variational EOM for a Rigid Body (1)

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The Rigid Body Assumption:Consequences

The distance between any two points and on a rigid body is constant in time:

and therefore

The internal force acts along the line between and and therefore is also orthogonal to :

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Variational EOM for a Rigid Body (2)

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[Side Trip]D’Alembert’s Principle

Jean-Baptiste d’Alembert(1717– 1783)

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[Side Trip]Principle of Virtual Work Principle of Virtual Work

If a system is in (static) equilibrium, then the net work done by external forces during any virtual displacement is zero

The power of this method stems from the fact that it excludes from the analysis forces that do no work during a virtual displacement, in particular constraint forces

D’Alembert’s Principle A system is in (dynamic) equilibrium when the virtual work of the sum

of the applied (external) forces and the inertial forces is zero for any virtual displacement

“D’Alembert had reduced dynamics to statics by means of his principle” (Lagrange)

The underlying idea: we can say something about the direction of constraint forces, without worrying about their magnitude

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[Side Trip]PVW: Simple Statics Example

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Virtual Displacements in terms ofVariations in Generalized Coordinates (1/2)

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Virtual Displacements in terms ofVariations in Generalized Coordinates (2/2)

Variational EOM with Centroidal CoordinatesNewton-Euler Differential EOM

6.1.2, 6.1.3

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Centroidal Reference Frames

The variational EOM for a single rigid body can be significantly simplified if we pick a special LRF

A centroidal reference frame is an LRF located at the center of mass

How is such an LRF special?

By definition of the center of mass (more on this later) is the point where the following integral vanishes:

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Variational EOM with Centroidal LRF (1/3)

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Variational EOM with Centroidal LRF (2/3)

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Variational EOM with Centroidal LRF (3/3)

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Differential EOM for a Single Rigid Body:Newton-Euler Equations

The variational EOM of a rigid body with a centroidal body-fixed reference frame were obtained as:

Assume all forces acting on the body have been accounted for. Since and are arbitrary, using the orthogonality theorem, we get:

Important: The Newton-Euler equations are valid only if all force effects have been accounted for! This includes both appliedforces/torques and constraint forces/torques(from interactions with other bodies).

Isaac Newton(1642 – 1727)

Leonhard Euler(1707 – 1783)

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Tractor Model[Example 6.1.1] Derive EOM under the following assumptions:

Traction (driving) force generated at rear wheels Small angle assumption (on the pitch angle)

Tire forces depend linearly on tire deflection: