MEAM 535 Advanced Dynamics

Fall 1999

Homework #6

1. The following problem is from Baruh's book (problem 4.26) and is reproduced here for your convenience.
   The Figure below depicts a simpified illustration of a spacecraft to which a robot arm is attached at the center of mass. The robot arm moves by a moment T exerted to it at the pin joint by a motor on the spacecraft. Considering only plane motion for both the spacecraft and the robot arm, derive the equations of motion by
   • (a) separating the two masses, writing force and moment balances, and eliminating constraints.
   • (b) using Lagrange's equations.
   

2. Two particles are constrained to move in the xy-plane such that the line joining them always passes through the origin.
   • Identify the constraint(s) and their type, and generalized coordinates.
   • Write the equations of motion using the Lagrange's equations.
   • Also, obtain as many integrals of motion as possible analytically.

3. 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.
   An inextensible string is attached to segment A using a moving pulley, a fixed pulley, and a spring as shown. The spring has spring constant of k. The spring has free length when all four segments are horizontal. The mass of each segment is m and the length is L.
   
   • (a) Write the equation of motion using Lagrange's equations.
   • (b) Comment on the static equilibrium of this system. For the configuration shown, what value of theta gives static equlibrium condition.
   
   Anything else you can say about this system will get you extra credit.