MEAM 535 Advanced Dynamics

Fall 1999

Homework #5

1. The figure below shows a schematic of a car garage door in a home. The door consits of four segments (A, B, C, and D) that are constrained to move in an inverted L-shaped fixed slot (E) using sliding-pin joints. Further, the segments A and B, B and C, and C and D are connected to each other using pin joints.
   • Using Grubler's formula, find the degrees-of-freedom of this system.
   • If point P is moving at a constant speed v0, find the velocity and acceleration of points Q, R, S, and T.
     Find also the angular velocity vectors of bodies A, B, C, and D in fixed frame E.
   

2. Look around and identify a simple real dynamic system (spring-mass systems, pendulum, a generic robot and such do not qualify; problem 1 in this homework does qualify). If you are an engineer (or proud to be one), you will take apart something that appeals to you. Now, model that system to understand how many rigid bodies are there in that system, how many constraints (and their type) and degrees-of-freedom exist in that system, and what assumptions (idealizations) you need to make to analyze that system.

   There will be a penalty if your system closely resembles a system found in any text-book. The intent here is to encourage you to think creatively about real systems and learn modeling techniques.

   For the system you have chosen,

   • provide a schematic that describes it adequately.
   • identify rigid bodies (if there are flexible (elastic) bodies, identify them too -- but avoid them if possible as we haven't discussed them yet.)
   • identify constraints and their type ( extra credit for those who come with nonholonomic and rehonomic-holonomic constraints.)
   • compute the degrees of freedom using Grubler's formula or otherwise
   • idetify a set of generalized coordinates
   • identify external forces acting on the system

3. Use d'Alembert's principle to derive the equations of motion of the system in problem 3 of homework 1. Neglect friction.

4. The following problem is for graduate students only. Undergraduate, if they solve it, get extra credit.
   
   For a scleronomic-holonomic system, derive the conservation of energy principle (V+T=constant) using d'Alembert's principle.