Three massless rods are free to move on a common axis at the end of cach rod. Th

Question

Three massless rods are free to move on a common axis at the end of cach rod. The other end of each rod is attached to a mass m (all 3 masses are identical). Rods 1 and 2, 2 and 3 and 3 and 1 are connected by torsional springs with torque r (0 0a). where G0) is the instant angle between two adjacent rods, and Q 2 3 is the equilibrium angle for the spring, and K is a constant. Write the Lagrangian for this system, find the equation of motion for each mass and determine the normal mode frequencies and eigenvectors. What motion does each eigenvector describe?

Explanation / Answer

Since the motion of these three adjecent rods connected at the center is circular, if we take axis of rotation passing through the connected point of these three rods, therefore, the generalized coordinates will be r and the change in the position angle Q( in this solution, theta will be denoted as Q), therefore, the kinetic energy T of the mass which is connected at the other end of the rod will be T = 1/2 m [r`2 + r2 Q`2] and the potential energy in a circular motion is V = mgr , since the distance of the connnected from the center of the rod is r, Therefore

L = T - V = 1/2 m[r`2 + r2Q`2] - mgh , where Q` or r` are the derivatives of Q and r with respect to time t.

dL/dr` = mr` , dL/dr = mrQ`2 , where d/dr ot d/dQ represents the partial derivatives with respect to Q and r.

dL/dQ` = mr2 , dL/dQ = 0 , therefore, putting them into the lagrangian equation we get,

D/Dt(dL/dr`) - dL/dr = mr`` - mrQ`2 , where D/Dt denotes total derivative

D/Dt(dL/dQ`) - dL/dQ = 2mr r` , therefore, these are the two required equation of motions for each mass, since all the three masses are identical. Therefore,

mr`` - mrQ`2 = 0

mr``= mrQ`2

r`` = rQ`2 , therefore, normal mode frequencies will be w = +- (r)1/2.

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