1. ME451 Kinematics and Dynamics of Machine Systems
Variational EOM for Planar Systems
October 18, 2013
Radu Serban
University of Wisconsin-Madison

2. Before we get started… Last Time: Today: Assignments:
Properties of the polar moment of inertia
Virtual Work and Generalized Forces
Today:
Wrap-up generalized forces (TSDA & RSDA)
Constrained variational EOM (variational EOM for a planar mechanism)
Assignments:
Homework 8 – – due Wednesday, October 23 (12:00pm)
Matlab 6 and Adams 4 – due October 23, (11:59pm)
Monday (October 21) lecture – simEngine2D discussion – in EH 2261

3. Roadmap: Check Progress
What have we done so far?
Derived the variational and differential EOM for a single rigid body
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?
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

4. Virtual Work and Generalized Force
6.2
Virtual Work and Generalized Force

5. Including Concentrated Forces or Torques
Nomenclature:
𝐪 generalized accelerations
𝛿𝐪 generalized virtual displacements
𝐌 generalized mass matrix
𝐐 generalized forces
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

6. Including a Concentrated Force or Torque

7. (TSDA) Translational Spring-Damper-Actuator (1/2)
Setup
Compliant connection between points 𝑃 𝑖 on body 𝑖 and 𝑃 𝑗 on body 𝑗
In its most general form it can consist of:
A spring with spring coefficient 𝑘 and free length 𝑙 0
A damper with damping coefficient 𝑐
An actuator (hydraulic, electric, etc.) which applies a force ℎ(𝑙, 𝑙 ,𝑡)
The distance vector between points 𝑃 𝑖 and 𝑃 𝑗 is defined as and has a length of

8. (TSDA) Translational Spring-Damper-Actuator (2/2)
General Strategy
Write the virtual work produced by the force element in terms of an appropriate virtual displacement where
Express the virtual work in terms of the generalized virtual displacements 𝛿 𝐪 𝑖 and 𝛿 𝐪 𝑗
Identify the generalized forces (coefficients of 𝛿 𝐪 𝑖 and 𝛿 𝐪 𝑗 )
Note: positive 𝛿𝑙 separates the bodies
Hence the negative sign in the virtual work
Note: tension defined as positive

9. (RSDA) Rotational Spring-Damper-Actuator (1/2)
Setup
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:
A torsional spring with spring coefficient 𝑘 and free angle 𝜃 0
A torsional damper with damping coefficient 𝑐
An actuator (hydraulic, electric, etc.) which applies a torque ℎ( 𝜃 𝑖𝑗 , 𝜃 𝑖𝑗 ,𝑡)
The angle 𝜃 𝑖𝑗 from 𝑥′ 𝑖 to 𝑥′ 𝑗 (positive counterclockwise) is

10. (RSDA) Rotational Spring-Damper-Actuator (2/2)
General Strategy
Write the virtual work produced by the force element in terms of an appropriate virtual displacement where
Express the virtual work in terms of the generalized virtual displacements 𝛿 𝐪 𝑖 and 𝛿 𝐪 𝑗
Identify the generalized forces (coefficients of 𝛿 𝐪 𝑖 and 𝛿 𝐪 𝑗 )
Note: positive 𝛿 𝜃 𝑖𝑗 separates the axes
Hence the negative sign in the virtual work
Note: tension defined as positive

11. Generalized Forces: Summary
Question: How do we specify the terms 𝐅 and 𝑛 in the EOM?
Answer: Recall where these terms come from…
Integral manipulations (use rigid-body assumptions)
Redefine in terms of generalized forces and virtual displacements
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.
As such, to include a new force (or torque), we are interested in the contribution of this force on the virtual work balance.
Explicitly identify virtual work of generalized forces
Virtual work of
generalized
external forces
Virtual work of
generalized
inertial forces

12. 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
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

13. Variational Equations of Motion for Planar Systems
6.3.1
Variational Equations of Motion for Planar Systems

14. Variational and Differential EOM for a Single Rigid Body
The variational EOM of a rigid body with a centroidal body-fixed reference frame were obtained as:
Since 𝛿𝐫 and 𝛿𝜙 are arbitrary, using the orthogonality theorem, we get:
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).

15. Matrix Form of the EOM for a Single Body
Generalized Virtual Displacement
(arbitrary)
Generalized Force; includes all forces acting on body 𝑖:
This includes all applied forces and all reaction forces
Generalized Mass Matrix
Generalized Accelerations

16. [Side Trip] A Vector-Vector Multiplication Trick
Given two vectors 𝐚 and 𝐛, each made up of 𝑛𝑏 vectors, each of dimension 3
The dot product of 𝐚 and 𝐛 can be expressed as

17. Variational EOM for the Entire System (1/2)
Consider a system made up of 𝑛𝑏 bodies
Write the EOM (in matrix form) for each individual body
Sum them up
Express this dot product as

18. Variational EOM for the Entire System (2/2)
Matrix form of the variational EOM for a system made up of 𝑛𝑏 bodies
Generalized Virtual Displacement
Generalized Force
Generalized Mass Matrix
Generalized Accelerations

19. A Closer Look at Generalized Forces
Total force acting on a body is sum of applied (external) and constraint (internal to the system) forces:
Goal: get rid of the constraint forces 𝐐 𝑖 𝐶 which (at least for now) are unknown
To do this, we need to compromise and give up something…

20. Constraint Forces 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)
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
Therefore
provided q is a consistent virtual displacement

21. 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 ∗ ?
Start with a consistent configuration 𝐪; i.e., a configuration that satisfies the constraint equations:
A consistent virtual displacement 𝛿𝐪 is a virtual displacement which ensures that the configuration 𝐪+𝛿𝐪 is also consistent:
Apply a Taylor series expansion and assume small variations:

22. Constrained Variational EOM
Arbitrary
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

23. Lagrange Multipliers Mixed Differential-Algebraic Equations of Motion
6.3.2, 6.3.3
Lagrange Multipliers Mixed Differential-Algebraic Equations of Motion

24. [Side Trip] Lagrange Multiplier Theorem
Joseph-Louis Lagrange
(1736– 1813)

25. Lagrange Multiplier Theorem: Example (1/2)

26. Lagrange Multiplier Theorem: Example (2/2)

27. Mixed Differential-Algebraic EOM (1/)
Constrained Variational Equations of Motion
Condition for consistent virtual displacements
Lagrange Multiplier Form
of the EOM

28. Lagrange Multiplier Form of the EOM
Equations of Motion
Position Constraint Equations
Velocity Constraint Equations
Acceleration Constraint Equations
Most Important Slide in ME451

29. Mixed Differential-Algebraic EOM
Combine the EOM and the Acceleration Equation to obtain a mixed system of differential-algebraic equations
The constraint equations and velocity equation must also be satisfied
Question: Under what conditions can we uniquely calculate the generalized accelerations and Lagrange multipliers?

30. Constrained Dynamic Existence Theorem

31. Slider-Crank Example (1/5)

32. Slider-Crank Example (2/5)

33. Slider-Crank Example (3/5)
Constrained Variational Equations of Motion
Condition for consistent virtual displacements

34. Slider-Crank Example (4/5)
Lagrange Multiplier Form
of the EOM
Constraint Equations
Acceleration Equation

35. Slider-Crank Example (5/5)
Mixed Differential-Algebraic Equations of Motion
Constraint Equations
Velocity Equation