As shown in the figure below, three beads of mass m, 2m, and m are threaded onto
ID: 2075507 • Letter: A
Question
As shown in the figure below, three beads of mass m, 2m, and m are threaded onto the three frictionless rods that are in parallel and separated each other by a distance d. The beads are connected by springs with spring constant k (assuming that the unstretched lengths of the springs are zero (l_0 = 0)). The beads can slide on the rods freely. The system is placed horizontally on a table. a) Work out kinetic energy matrix and potential energy matrix b) Find normal mode frequencies c) Work out the modal matrix A and normal coordinates and describe normal modes. c) Work out the modal matrix A and normal coordinates and describe normal modes.Explanation / Answer
a) The kinetic energy matrix:
First calculate the KE of the whole system
KE = (1/2) M1*V1 ^2 +(1/2)* M2*V2^2 +(1/2)*M2*V3^2
M1 = M3 = m and M2 = 2m
KE = 1/2 [m*V1^2 +2m*V2^2+ m*V3^2]
Matrix :
m 0 0
(1/2) [ 0 2m 0 ]
0 0 m
Potential enegy matrix
PE = (1/2) ( kx1^2+k(x2-x1)^2+k(x3-x2)^2+kx3^2)
PE matrix = 2k -k 0
(0.5) ( -k 2k -k)
0 -k k
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