ME451 Kinematics and Dynamics

of Machine Systems

Introduction to Dynamics6.1

October 09, 2013

Radu SerbanUniversity of Wisconsin-Madison

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Before we get started… Last Time:

Concluded Kinematic Analysis

Today: Towards the Newton-Euler equations for a single rigid body

Assignments: Matlab 4 – due today, Learn@UW (11:59pm) Adams 2 – due today, Learn@UW (11:59pm)

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

Midterm Exam Friday, October 11 at 12:00pm in ME1143 Review session: today, 6:30pm in ME1152

Midterm Feedback Form emailed to you later today Anonymous Complete it and return on Friday

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Kinematics vs. Dynamics

Kinematics We include as many actuators as kinematic degrees of freedom – that is, we

impose KDOF driver constraints We end up with NDOF = 0 – that is, we have as many constraints as

generalized coordinates We find the (generalized) positions, velocities, and accelerations by solving

algebraic problems (both nonlinear and linear) We do not care about forces, only that certain motions are imposed on the

mechanism. We do not care about body shape nor inertia properties

Dynamics While we may impose some prescribed motions on the system, we assume

that there are extra degrees of freedom – that is, NDOF > 0 The time evolution of the system is dictated by the applied external forces The governing equations are differential or differential-algebraic equations We very much care about applied forces and inertia properties of the bodies

in the mechanism

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Dynamics M&S

Dynamics Modeling

Formulate the system of equations that govern the time evolution of a system of interconnected bodies undergoing planar motion under the action of applied (external) forces These are differential-algebraic equations Called Equations of Motion (EOM)

Understand how to handle various types of applied forces and properly include them in the EOM

Understand how to compute reaction forces in any joint connecting any two bodies in the mechanism

Dynamics Simulation

Understand under what conditions a solution to the EOM exists Numerically solve the resulting (differential-algebraic) EOM

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Roadmap to Deriving the EOM

Begin with deriving the variational EOM for a single rigid body Principle of virtual work and D’Alembert’s principle

Consider the special case of centroidal reference frames Centroid, polar moment of inertia, (Steiner’s) parallel axis theorem

Write the differential EOM for a single rigid body Newton-Euler equations

Derive the variational EOM for constrained planar systems Virtual work and generalized forces

Finally, write the mixed differential-algebraic EOM for constrained systems Lagrange multiplier theorem

(This roadmap will take several lectures, with some side trips)

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What are EOM?

In classical mechanics, the EOM are equations that relate (generalized) accelerations to (generalized) forces

Why accelerations? If we know the (generalized) accelerations as functions of time, they

can be integrated once to obtain the (generalized) velocities and once more to obtain the (generalized) positions

Using absolute (Cartesian) coordinates, the acceleration of body i is the acceleration of the body’s LRF:

How do we relate accelerations and forces? Newton’s laws of motion In particular, Newton’s second law written as

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Newton’s Laws of Motion

1st LawEvery body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

2nd LawA change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

3rd LawTo any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction.

Newton’s laws are applied to particles (idealized single point masses) only hold in inertial frames are valid only for non-relativistic speeds

Isaac Newton(1642 – 1727)

Variational EOM for a Single Rigid Body6.1.1

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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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Forces Acting on a Differential Mass dm(P)

External distributed forces Described using a force per unit mass:

This type of force is not common in classical multibody dynamics; exception: gravitational forces for which

Applied (external) forces Concentrated at point For now, we ignore them (or assume they are folded into )

Internal interaction forces Act between point and any other point on the body, described using

a force per units of mass at points and Including the contribution at point of all points on the body

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Newton’s EOM for a Differential Mass dm(P)

Apply Newton’s 2nd law to the differential mass located at point P, to get

This is a valid way of describing the motion of a body: describe the motion of every single particle that makes up that body

However It involves explicitly the internal forces acting within the body (these are

difficult to completely describe) Their number is enormous

Idea: simplify these equations taking advantage of the rigid body assumption

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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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[Side Trip]

Calculus of Variations (1/3)

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[Side Trip]

Calculus of Variations (2/3)

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[Side Trip]

Calculus of Variations (3/3)

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Virtual Displacement of a Point Attached to a Rigid Body

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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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[Side Trip]

An Orthogonality Theorem

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Variational EOM for a Rigid Body (1)

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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