ME451 Kinematics and Dynamics

of Machine Systems

Dynamics: ReviewNovember 4, 2013

Radu SerbanUniversity of Wisconsin, Madison

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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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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 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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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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Variational EOM for a Rigid Body

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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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Virtual Displacements in terms ofVariations in Generalized Coordinates (1/2)

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Virtual Displacements in terms ofVariations in Generalized Coordinates (2/2)

Variational EOM with Centroidal CoordinatesNewton-Euler Differential EOM

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Variational EOM with Centroidal LRF

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Variational EOM with Centroidal LRF

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Differential EOM for a Single Rigid Body:Newton-Euler Equations

The variational EOM of a rigid body with a centroidal body-fixed reference frame were obtained as:

Assume all forces acting on the body have been accounted for. Since and are arbitrary, using the orthogonality theorem, we get:

Important: The Newton-Euler equations are valid only if all force effects have been accounted for! This includes both appliedforces/torques and constraint forces/torques(from interactions with other bodies).

Isaac Newton(1642 – 1727)

Leonhard Euler(1707 – 1783)

Virtual Work and Generalized Force

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Calculating Generalized Forces

Nomenclature: generalized accelerations generalized virtual displacements generalized mass matrix generalized forces

Recipe for including a 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 Point Force

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Including a Torque

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Setup Compliant connection between points on body and on body In its most general form it can consist of:

A spring with spring coefficient and free length A damper with damping coefficient An actuator (hydraulic, electric, etc.) which applies a force

The distance vector between points and is defined as

and has a length of

(TSDA)Translational Spring-Damper-Actuator (1/2)

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(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 and

Identify the generalized forces (coefficients of and )

Note: tension defined as positive

Note: positive separates the bodies

Hence the negative sign in the virtual work

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Setup Bodies and connected by a revolute joint at Torsional compliant connection at the common point In its most general form it can consist of:

A torsional spring with spring coefficient and free angle

A torsional damper with damping coefficient An actuator (hydraulic, electric, etc.) which

applies a torque

The angle from to (positive counterclockwise) is

(RSDA)Rotational Spring-Damper-Actuator (1/2)

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(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 and

Identify the generalized forces (coefficients of and )

Note: tension defined as positive

Note: positive separates the axes

Hence the negative sign in the virtual work

Variational Equations of Motion for Planar Systems

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Matrix Form of the EOM for a Single Body

Generalized Force; includes all forces acting on body :This includes all

applied forces and all reaction forces

Generalized Virtual Displacement

(arbitrary)

Generalized Mass Matrix

Generalized Accelerations

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Variational EOM for the Entire System

Matrix form of the variational EOM for a system made up of bodies

Generalized Force

Generalized Virtual Displacement

Generalized Mass Matrix

Generalized Accelerations

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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

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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 ?

Start with a consistent configuration ; i.e., a configuration that satisfies the constraint equations:

A consistent virtual displacement is a virtual displacement which ensures that the configuration is also consistent:

Apply a Taylor series expansion and assume small variations:

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Constrained Variational EOM

We can eliminate the (unknown) constraint forces if we compromise to only consider virtual displacements that are consistent with the constraint equations

Arbitrary Arbitrary Consistent

Constrained Variational Equations of Motion

Condition for consistent virtual displacements

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Lagrange Multiplier Theorem

Joseph-Louis Lagrange

(1736– 1813)

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Mixed Differential-Algebraic EOM

Constrained Variational Equations of Motion

Condition for consistent virtual displacements

Lagrange Multiplier Formof the EOM

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Lagrange Multiplier Form of the EOM

Equations of Motion

Position Constraint Equations

Velocity Constraint Equations

Acceleration Constraint Equations

Most Important Slide in ME451

Initial Conditions Reaction Forces

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Initial Conditions

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Reaction Forces

Numerical Integration

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Basic Concept IVP

In general, all we can hope for is approximating the solution at a sequence of discrete points in time Uniform grid (constant step integration) Adaptive grid (variable step integration)

Basic idea: somehow turn the differential problem into an algebraic problem (approximate the derivatives)

IVP in dynamics: What we calculate are the accelerations Oversimplifying, we get something like

This is a second-order DE which needs to be integrated to obtain velocities and positions

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Simplest method: Forward Euler

Starting from the IVP

Use the simplest approximation to the derivative

Rewrite the above as

and use ODE to obtain

Forward Euler Methodwith constant step-size

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FE: Geometrical Interpretation

IVP

Forward Euler integration formula

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Stiff Differential Equations

Problems for which explicit integration methods (such as Forward Euler) do not work well Other explicit formulas: Runge-Kutta (RK4), DOPRI5, Adams-

Bashforth, etc.

Stiff problems require a different class of integration methods: implicit formulas The simplest implicit integration formula: Backward Euler (BE)

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BE: Geometrical Interpretation

IVP

Forward Euler integration formula

Backward Euler integration formula

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Forward Euler vs. Backward Euler

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Stability of a Numerical Integrator

The problem: How big can the integration step-size be without the numerical solution

blowing up?

Tough question, answered in a Numerical Analysis class

Different integration formulas, have different stability regions

We would like to use an integration formula with large stability region: Example: Backward Euler, BDF methods, Newmark, etc.

Why not always use these methods with large stability region? There is no free lunch: these methods are implicit methods that require the

solution of an algebraic problem at each step

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Accuracy of a Numerical Integrator

The problem: How accurate is the formula that we are using? If we decrease , how will the accuracy of the numerical solution improve? Tough question, answered in a Numerical Analysis class

Examples: Forward and Backward Euler: accuracy RK45: accuracy

Why not always use methods with high accuracy order? There is no free lunch: these methods usually have very small stability regions Therefore, they are limited to using very small values of

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Implicit Integration: Conclusions

Stiff problems require the use of implicit integration methods

Because they have very good stability, implicit integration methods allow for step-sizes that could be orders of magnitude larger than those needed if using explicit integration

However, for most real-life IVPs, discretization using an implicit integration formula leads to another nasty problem:

To find the solution at the new time, we must solve a nonlinear algebraic problem

This brings back into the picture the Newton-Raphson method (and its variants) We have to deal with providing a good starting point (initial guess), computing the

matrix of partial derivatives, etc.

Putting it all together: Mechanism Analysis

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Dynamics Modeling & Simulation

We are given a mechanism… Describe how we would model this mechanism. How many bodies? How many

GCs? What kinematic constraints would we use? What is KDOF? Write down the

equations that model those constraints. What external forces act on the mechanism? Write down the corresponding

generalized forces. Assemble the resulting EOM. What is the initial configuration? Write down the additional conditions used to

specify initial conditions. Calculate the initial positions and velocities. Calculate the accelerations and Lagrange multipliers at the initial time. How would you solve these equations? Assuming we have the solution to the Dynamics problem, in particular the

Lagrange multipliers, calculate constraint reaction forces. If the mechanism is kinematically driven, what is the interpretation of the constraint

forces/torques corresponding to a driver constraint?

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Mechanism Analysis

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Mechanism AnalysisModel

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Mechanism Analysis: Kinematics

Constraint Equations

Acceleration EquationVelocity Equation

Jacobian

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Mechanism Analysis: Generalized Forces

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Mechanism Analysis: Equations of Motion

Lagrange Multiplier Formof the EOM

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Mechanism Analysis: DAEs

Acceleration Equation

Velocity Equation

Constraint Equations

EOM

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Find a set of consistent initial conditions (ICs) so that the mechanism starts in an “all stretched out” configuration, with body 1 having an angular velocity .

Kinematic constraint equations

Additional conditions (for position ICs)

Solve for the initial positions

Mechanism Analysis: Position ICs

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Find a set of consistent initial conditions (ICs) so that the mechanism starts in an “all stretched out” configuration, with body 1 having an angular velocity .

Jacobian and velocity equation

Additional conditions (for position ICs)

Solve for the initial positions

Mechanism Analysis: Velocity ICs

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The accelerations and Lagrange multipliers at the initial time can be directly obtained using the EOM and acceleration equation (once consistent initial conditions for positions and velocities are available):

Mechanism Analysis: Accelerations and Lagrange Multipliers at Initial Time

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Mechanism Analysis: Inverse Dynamics

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Mechanism Analysis: Inverse Dynamics

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The “reaction force” associated with the driver constraint provides the force/torque required to impose the prescribed motion

Driver constraint and Jacobian blocks

Mechanism Analysis: Inverse Dynamics

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