Download Kinematics vs. Dynamics - Welcome to the Simulation Based

ME451
Kinematics and Dynamics
of Machine Systems
Introduction to Dynamics
6.1
October 09, 2013
Radu Serban
University of Wisconsin-Madison
Before we get started…
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Last Time:
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Today:
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Concluded Kinematic Analysis
Towards the Newton-Euler equations for a single rigid body
Assignments:
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Matlab 4 – due today, Learn@UW (11:59pm)
Adams 2 – due today, Learn@UW (11:59pm)
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Midterm Exam
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Submit a single PDF with all required information
Make sure your name is printed in that file
Friday, October 11 at 12:00pm in ME1143
Review session: today, 6:30pm in ME1152
Midterm Feedback
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Form emailed to you later today
Anonymous
Complete it and return on Friday
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Kinematics vs. Dynamics
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Kinematics
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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
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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
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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
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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
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Begin with deriving the variational EOM for a single rigid body
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Consider the special case of centroidal reference frames
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Newton-Euler equations
Derive the variational EOM for constrained planar systems
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Centroid, polar moment of inertia, (Steiner’s) parallel axis theorem
Write the differential EOM for a single rigid body
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Principle of virtual work and D’Alembert’s principle
Virtual work and generalized forces
Finally, write the mixed differential-algebraic EOM for
constrained systems
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Lagrange multiplier theorem
(This roadmap will take several lectures, with some side trips)
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What are EOM?
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In classical mechanics, the EOM are equations that relate
(generalized) accelerations to (generalized) forces
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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
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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
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1st Law
Every 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.
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2nd Law
A change in motion is proportional to the motive force impressed and takes place
along the straight line in which that force is impressed.
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3rd Law
To 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.
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Newton’s laws
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are applied to particles (idealized single point masses)
only hold in inertial frames
are valid only for non-relativistic speeds
Isaac Newton
(1642 – 1727)
6.1.1
Variational EOM for a Single Rigid Body
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Body as a Collection of Particles
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Our toolbox provides a relationship between forces and accelerations
(Newton’s 2nd law) – but that applies for particles only
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Idea: look at a body as a collection of infinitesimal particles
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Consider a differential mass 𝑑𝑚(𝑃) at each point 𝑃 on the body (located
by 𝐬 𝑃 )
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For each such particle, we can write
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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)
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External distributed forces
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Described using a force per unit mass:
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This type of force is not common in classical multibody dynamics;
exception: gravitational forces for which 𝐟d 𝑃 = 𝐠
Applied (external) forces
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Concentrated at point 𝑃
For now, we ignore them (or assume they are folded into 𝐟d 𝑃 )
Internal interaction forces
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Act between point 𝑃 and any other point 𝑅 on the body, described
using a force per units of mass at points 𝑃 and 𝑅
Including the contribution at point 𝑃 of all points 𝑅 on the body
Newton’s EOM for a Differential Mass dm(P)
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Apply Newton’s 2nd law to the differential mass 𝑑𝑚(𝑃) located at point P,
to get
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This is a valid way of describing the motion of a body: describe the
motion of every single particle that makes up that body
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However
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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
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We model a rigid body with distance constraints between any pair of
differential elements (considered point masses) in the body.
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Therefore the internal forces
𝐟𝑖 𝑃, 𝑅 𝑑𝑚 𝑅 𝑑𝑚(𝑃) on 𝑑𝑚 𝑃 due to the differential mass 𝑑𝑚 𝑅
𝐟𝑖 𝑅, 𝑃 𝑑𝑚 𝑃 𝑑𝑚(𝑅) on 𝑑𝑚 𝑅 due to the differential mass 𝑑𝑚 𝑃
satisfy the following conditions:
 They act along the line connecting
points 𝑃 and 𝑅
 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
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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 purely
geometric concept:
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Does not depend on actual forces
Is a property of the particular constraint
The real (true) displacement coincides
with a virtual displacement only if the
constraint does not change with time
Virtual
displacements
Actual
trajectory
[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
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The distance between any two points 𝑃 and 𝑅 on a rigid body is constant
in time:
and therefore
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The internal force 𝐟𝑖 𝑃, 𝑅 𝑑𝑚 𝑃 𝑑𝑚(𝑅) acts along the line between 𝑃 and
𝑅 and therefore is also orthogonal to 𝛿(𝐫 𝑃 − 𝐫 𝑅 ):
[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)
[Side Trip]
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D’Alembert’s Principle
Jean-Baptiste d’Alembert
(1717– 1783)
[Side Trip]
Principle of Virtual Work
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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
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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)
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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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