ME451
Kinematics and Dynamics
of Machine Systems
Newton-Euler EOM
6.1.2, 6.1.3
October 14, 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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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
Principle of Virtual Work and D’Alembert’s Principle
Introduce centroidal reference frames
Derive the Newton-Euler EOM
Assignments:
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Matlab 5 – due Wednesday (Oct. 16), Learn@UW (11:59pm)
Adams 3 – due Wednesday (Oct. 16), Learn@UW (11:59pm)
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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
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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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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
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Variational EOM for a Rigid Body (1)
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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 ):
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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)
[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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Virtual Displacements in terms of
Variations in Generalized Coordinates (1/2)
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Virtual Displacements in terms of
Variations in Generalized Coordinates (2/2)
6.1.2, 6.1.3
Variational EOM with Centroidal Coordinates
Newton-Euler Differential EOM
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Centroidal Reference Frames
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The variational EOM for a single rigid body can be significantly simplified
if we pick a special LRF
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A centroidal reference frame is an LRF located at the center of mass
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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
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The variational EOM of a rigid body with a centroidal body-fixed reference frame
were obtained as:
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Assume all forces acting on the body have been accounted for.
Since 𝛿𝐫 and 𝛿𝜙 are arbitrary, using the orthogonality theorem, we get:
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Important: The Newton-Euler equations are
valid only if all force effects have been
accounted for! This includes both applied
forces/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]
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Derive EOM under the following assumptions:
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Traction (driving) force 𝑇𝑟 generated at rear wheels
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Small angle assumption (on the pitch angle)
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Tire forces depend linearly on tire deflection: