MEAM 535 Advanced Dynamics

Fall 1999

Homework #7

1. Two CDs are held together rigidly by an aluminum rod and the disks are let to roll in xy-plane as shown below. Assume that the aluminum rod can be approximated as two particles of mass m/2 at either end of the rod and that the length of the rod is L. The disks are assumed to be mass-less at this point. The initial configuration of the system is as follows. The midpoint of the rod is at the origin and the length of the rod is along the y-axis. The initial velocity of the midpoint of the rod is v0 in the x-direction. Write the equations of motion of this system using Lagrange's method:
   • by adjoining constraint(s)
   • by embedding constraint(s)
   • Also, solve the equations of motion and obtain motion of the midpoint of the rod.
   • For extra credit (both grads and undergrads), apply Hamilton's equations to get equations of motion.
   

2. The figure below shows a snakeboard. It has a board with two steerable pair of wheels. The wheel system is similar to the one in Problem 1 in this homework. A person can stand on this board and propel himself/herself by twistingt the torso. This action is modeled here by two masses connected by a mass-less rod that is hinged to the mid-point of the snakeboard. There is also a torsional spring at the hinge.
   • Write Lagrange's equations of motion.
   • Are there any ignorable coordinates? If so, apply the Routhian method and obtain the equations of motion.
   
   M = mass of the board
   m = mass of each spherical particle
   Ib = moment of inertia of the board about a vertical (z) axis through its center
   Iw = moment of inertia of each of the wheel subsytsems about the vertical axis through the steering pin-joint at the center of the rod of the wheel subsystem.
   Other things you need are in the figure.