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ME451
Kinematics and Dynamics
of Machine Systems
Variational EOM for Planar Systems
October 18, 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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Wrap-up generalized forces (TSDA & RSDA)
Constrained variational EOM (variational EOM for a planar mechanism)
Assignments:
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Properties of the polar moment of inertia
Virtual Work and Generalized Forces
Homework 8 – 6.2.1 – due Wednesday, October 23 (12:00pm)
Matlab 6 and Adams 4 – due October 23, Learn@UW (11:59pm)
Monday (October 21) lecture – simEngine2D discussion – in EH 2261
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Roadmap: Check Progress
What have we done so far?
 Derived the variational and differential EOM for a single rigid body
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These equations are general but they must include all forces applied on the
body
These equations assume their simplest form in a centroidal RF
Properties of the polar moment of inertia
What is left?
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Define a general methodology for including external forces, concentrated at a
given point 𝑃 on the body
 Virtual work and generalized forces
Elaborate on the nature of these concentrated forces. These can be:
 Models of common force elements (TSDA and RSDA)
 Reaction (constraint) forces, modeling the interaction with other bodies
Derive the variational and differential EOM for systems of constrained bodies
6.2
Virtual Work and Generalized Force
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Including Concentrated Forces or Torques
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Nomenclature:
 𝐪
generalized accelerations
 𝛿𝐪
generalized virtual displacements
 𝐌
generalized mass matrix
 𝐐
generalized forces
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Recipe for including a concentrated force in the EOM:
 Write the virtual work of the given force effect (force or torque)
 Express this virtual work in terms of the generalized virtual
displacements
 Identify the generalized force
 Include the generalized force in the matrix form of the variational EOM
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Including a Concentrated Force or Torque
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(TSDA)
Translational Spring-Damper-Actuator (1/2)
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Setup
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Compliant connection between points 𝑃𝑖 on body 𝑖 and 𝑃𝑗 on body 𝑗
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In its most general form it can consist of:
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A spring with spring coefficient 𝑘 and free length 𝑙0
A damper with damping coefficient 𝑐
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An actuator (hydraulic, electric, etc.) which applies a force ℎ(𝑙, 𝑙, 𝑡)
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The distance vector between points
𝑃𝑖 and 𝑃𝑗 is defined as
and has a length of
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(TSDA)
Translational Spring-Damper-Actuator (2/2)
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General Strategy
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Write the virtual work produced by the force element in terms of an appropriate virtual
displacement
Note: positive 𝛿𝑙
separates the bodies
where
Hence the negative sign
in the virtual work
Note: tension defined as
positive
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Express the virtual work in terms of the generalized virtual displacements 𝛿𝐪𝑖 and 𝛿𝐪𝑗
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Identify the generalized forces (coefficients of 𝛿𝐪𝑖 and 𝛿𝐪𝑗 )
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(RSDA)
Rotational Spring-Damper-Actuator (1/2)
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Setup
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Bodies 𝑖 and 𝑗 connected by a revolute joint at 𝑃
Torsional compliant connection at the common point 𝑃
In its most general form it can consist of:
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A torsional spring with spring coefficient 𝑘 and
free angle 𝜃0
A torsional damper with damping coefficient 𝑐
An actuator (hydraulic, electric, etc.) which
applies a torque ℎ(𝜃𝑖𝑗 , 𝜃𝑖𝑗 , 𝑡)
The angle 𝜃𝑖𝑗 from 𝑥′𝑖 to 𝑥′𝑗
(positive counterclockwise) is
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(RSDA)
Rotational Spring-Damper-Actuator (2/2)
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General Strategy
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Write the virtual work produced by the force element in terms of an appropriate virtual
displacement
Note: positive 𝛿𝜃
𝑖𝑗
separates the axes
where
Hence the negative sign
in the virtual work
Note: tension defined as
positive
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Express the virtual work in terms of the generalized virtual displacements 𝛿𝐪𝑖 and 𝛿𝐪𝑗
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Identify the generalized forces (coefficients of 𝛿𝐪𝑖 and 𝛿𝐪𝑗 )
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Generalized Forces: Summary
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Question: How do we specify the terms 𝐅 and 𝑛 in the EOM?
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Answer: Recall where these terms come from…
Integral manipulations (use rigid-body assumptions)
Redefine in terms of generalized forces and virtual displacements
Explicitly identify virtual work of generalized forces
D’Alembert’s Principle effectively says that,
upon including a new external force, the body’s
generalized accelerations must change to
preserve the balance of virtual work.
Virtual work of
generalized
external forces
As such, to include a new force (or torque), we
are interested in the contribution of this force on
the virtual work balance.
Virtual work of
generalized
inertial forces
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Roadmap: Check Progress
What have we done so far?
 Derived the variational and differential EOM for a single rigid body
 Defined how to calculate inertial properties
 Defined a general strategy for including external forces
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Concentrated (point) forces
Forces from compliant elements (TSDA and RSDA)
What is left?
 Treatment of constraint forces
 Derive the variational and differential EOM for systems of constrained
bodies
6.3.1
Variational Equations of Motion for Planar Systems
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Variational and Differential EOM
for a Single Rigid Body
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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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Since 𝛿𝐫 and 𝛿𝜙 are arbitrary, using the orthogonality theorem, we get:
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Important: The above 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).
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Matrix Form of the EOM for a Single Body
Generalized Force;
includes all forces
acting on body 𝑖:
This includes all
applied forces and all
reaction forces
Generalized Virtual
Displacement
(arbitrary)
Generalized
Mass Matrix
Generalized
Accelerations
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[Side Trip]
A Vector-Vector Multiplication Trick
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Given two vectors 𝐚 and 𝐛, each made up of 𝑛𝑏 vectors, each
of dimension 3
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The dot product of 𝐚 and 𝐛 can be expressed as
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Variational EOM for the Entire System (1/2)
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Consider a system made up of 𝑛𝑏 bodies
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Write the EOM (in matrix form) for each individual body
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Sum them up
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Express this dot product as
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Variational EOM for the Entire System (2/2)
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Matrix form of the variational EOM for a system made up of 𝑛𝑏 bodies
Generalized Virtual
Displacement
Generalized
Force
Generalized
Mass Matrix
Generalized
Accelerations
A Closer Look at Generalized Forces
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Total force acting on a body is sum of applied (external) and
constraint (internal to the system) forces:
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Goal: get rid of the constraint forces 𝐐𝐶𝑖 which (at least for now) are
unknown
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To do this, we need to compromise and give up something…
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Constraint Forces
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Constraint Forces
 Forces that develop in the physical joints present in the system:
(revolute, translational, distance constraint, etc.)
 They are the forces that ensure the satisfaction of the constraints (they are
such that the motion stays compatible with the kinematic constraints)
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KEY OBSERVATION: The net virtual work produced by the constraint forces
present in the system as a result of a set of consistent virtual displacements is
zero
 Note that we have to account for the work of all reaction forces present in the
system
 This is the same observation we used to eliminate the internal interaction
forces when deriving the EOM for a single rigid body
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Therefore
provided q is a consistent virtual displacement
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Consistent Virtual Displacements
What does it take for a virtual displacement to be consistent (with the set of
constraints) at a given, fixed time 𝑡 ∗ ?
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Start with a consistent configuration 𝐪; i.e., a configuration that satisfies the
constraint equations:
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A consistent virtual displacement 𝛿𝐪 is a virtual displacement which ensures that
the configuration 𝐪 + 𝛿𝐪 is also consistent:
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Apply a Taylor series expansion and assume small variations:
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Constrained Variational EOM
Arbitrary
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Arbitrary
Consistent
We can eliminate the (unknown) constraint forces if we compromise to only
consider virtual displacements that are consistent with the constraint equations
Constrained Variational
Equations of Motion
Condition for consistent
virtual displacements
6.3.2, 6.3.3
Lagrange Multipliers
Mixed Differential-Algebraic Equations of Motion
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[Side Trip]
Lagrange Multiplier Theorem
Joseph-Louis
Lagrange
(1736– 1813)
Lagrange Multiplier Theorem: Example (1/2)
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Lagrange Multiplier Theorem: Example (2/2)
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Mixed Differential-Algebraic EOM (1/)
Constrained Variational
Equations of Motion
Condition for consistent
virtual displacements
Lagrange Multiplier Form
of the EOM
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Lagrange Multiplier Form of the EOM
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Equations of Motion
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Position Constraint Equations
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Velocity Constraint Equations
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Acceleration Constraint Equations
Most Important Slide in ME451
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Mixed Differential-Algebraic EOM
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Combine the EOM and the Acceleration Equation
to obtain a mixed system of differential-algebraic equations
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The constraint equations and velocity equation must also be satisfied
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Question: Under what conditions can we uniquely calculate the
generalized accelerations and Lagrange multipliers?
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Constrained Dynamic Existence Theorem
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Slider-Crank Example (1/5)
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Slider-Crank Example (2/5)
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Slider-Crank Example (3/5)
Constrained Variational
Equations of Motion
Condition for consistent
virtual displacements
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Slider-Crank Example (4/5)
Lagrange Multiplier Form
of the EOM
Constraint Equations
Acceleration Equation
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Slider-Crank Example (5/5)
Mixed Differential-Algebraic
Equations of Motion
Constraint Equations
Velocity Equation