Transcript Equilibrium Analysis

ME451
Kinematics and Dynamics
of Machine Systems
Equilibrium Analysis
November 8, 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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Equilibrium Analysis
Wrap-up ME451
Assignment due dates:
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Example: numerical solution of a dynamics problem
Inverse Dynamics
Project 1 – today
Matlab 8 / ADAMS 5 – Wednesday, November 13
Matlab 9 / ADAMS 6 – Wednesday, Novemebr 20
Project 2 – Wednesday, November 27
Final Project – Wednesday, December 11
Final Exam:
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Tuesday, December 17, 7:45-9:45am in ME2109
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Equilibrium Analysis
Equilibrium Analysis: The Idea
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A mechanical system is in equilibrium if the following conditions hold:
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Equivalently, the system is at rest with zero acceleration
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So what does it take to be in this state of equilibrium?
 The system must be in a certain configuration 𝐪
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The reaction forces (in other words, the Lagrange Multipliers 𝛌) should
assume certain values
What does “certain” mean?
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Equilibrium Analysis: The Math
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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:
Equilibrium Analysis: The Math
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Approach 1
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Simply solve the nonlinear system to find 𝐪 and 𝛌
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Problem: need a good initial guess
Approach 2
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Add damping in the system and perform dynamic analysis until it stops
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Brute force, but effective
Approach 3
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Cast it as an optimization problem (minimize potential energy)
Problem: works for conservative systems only
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Example: Equilibrium Analysis
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Consider a pendulum connected to ground through a
revolute joint and a rotational spring-damper.
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Length: 𝐿 = 1
Mass: 𝑚 = 10
Polar moment of inertia: 𝐽′ =? ? ?
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RSDA spring coefficient: 𝑘 = 25
RSDA free angle: 𝜃0 = 0
RSDA damping coefficient: 𝑐 =? ? ?
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Gravitational acceleration: 𝑔 = 9.81
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All units are SI
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RSD
End ME451
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Summary of ME451
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Pick a set of Cartesian generalized coordinates
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Formulate the following sets of equations
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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:
Solve the appropriate set of equations for Kinematics or Dynamics
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What was ME451 good for?
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Understand how to pose a Kinematics/Dynamics problem (modeling)
“A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant.”
Manfred Eigen
“…all models are approximations. Essentially, all models are wrong, but some are useful.”
George E.P. Box
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Understand how to solve a Kinematics/Dynamics problem (simulation)
“The purpose of computing is insight, not numbers.”
Richard Hamming
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Improve your MATLAB programming skills
“The process of preparing programs for a digital computer is especially attractive, not only because it can be economically
and scientifically rewarding, but also because it can be an aesthetic experience much like composing poetry or music.”
Donald E. Knuth