1. ME451 Kinematics and Dynamics of Machine Systems
Basic Concepts in Planar Kinematics - 3.1
Absolute Kinematic Constraints – 3.2
Relative Kinematic Constraints – 3.3
February 10, 2009
© Dan Negrut, 2009 ME451, UW-Madison

2. Before we get started… Today Last Time
Aborted ADAMS tutorial
Prior to that: started discussion on the topic of Kinematics Analysis
Today
Continue discussion on Kinematics Analysis
Start talking about geometric constraints
Real life counterpart: joints between bodies
HW Assigned: 3.3.2, 3.3.4, 3.3.5, ADAMS problem
Due on Tu, Feb. 17
ADAMS component available for download from class website 2

3. Motion: Causes How can one set a mechanical system in motion?
For a system with ndof degrees of freedom, specify NDOF additional driving constraints (one per degree of freedom) that uniquely determine q(t) as the solution of an algebraic problem (Kinematic Analysis)
Specify/Apply a set of forces acting upon the mechanism, in which case q(t) is found as the solution of a differential problem (Dynamic Analysis)
Ignore this for now… 3

4. Example 3.1.1 A pin (revolute) joint present at point O
Specify the set of constraints associated with this model
A motion 1=4t2 is applied to the pendulum
Use Cartesian coordinates 4

5. Kinematic Analysis Stages
Position Analysis Stage
Challenging
Velocity Analysis Stage
Simple
Acceleration Analysis Stage OK
To take care of all these stages, ONE step is critical:
Write down the constraint equations associated with the joints present in your mechanism
Once you have the constraints, the rest is boilerplate 5

6. Once you have the constraints… (Going beyond the critical step)
The three stages of Kinematics Analysis: position analysis, velocity analysis, and acceleration analysis they each follow *very* similar recipes for finding for each body of the mechanism its position, velocity, and acceleration, respectively
ALL STAGES RELY ON THE CONCEPT OF JACOBIAN MATRIX:
q – the partial derivative of the constraints wrt the generalized coordinates
ALL STAGES REQUIRE THE SOLUTION OF A SYSTEM OF EQUATIONS
WHAT IS DIFFERENT BETWEEN THE THREE STAGES IS THE EXPRESSION OF THE RIGHT-SIDE OF THE LINEAR EQUATION, “b” 6

7. The details… As we pointed out, it all boils down to this:
Step 1: Before anything, write down the constraint equations associated with your model
Step 2: For each stage, construct q and the specific b , then solve for x
So how do you get the position configuration of the mechanism?
Kinematic Analysis key observation: The number of constraints (kinematic and driving) should be equal to the number of generalized coordinates
This is a prerequisite for Kinematic Analysis
IMPORTANT: This is a nonlinear systems with nc equations and nc unknowns that you must solve to find q 7

8. Getting the Velocity and Acceleration of the Mechanism
Previous slide taught us how to find the positions q
At each time step tk, generalized coordinates qk are the solution of a nonlinear system
Take one time derivative of constraints (q,t) to obtain the velocity equation:
Take yet one more time derivative to obtain the acceleration equation:
NOTE: Getting right-hand side of acceleration equation is tedious 8

9. Example 3.1.1 A motion 1=4t2 is applied to the pendulum
Formulate the velocity analysis problem
Formulate the acceleration analysis problem 9