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ME451 Kinematics and Dynamics of Machine Systems

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. Before we get started…. Last Time Aborted ADAMS tutorial

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ME451 Kinematics and Dynamics of Machine Systems

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  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, 2009ME451, UW-Madison

  2. Before we get started… • 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

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