Consider the vibration model of a machine part shown in Figu

Consider the vibration model of a machine part shown in Figure 1. The disks both roll without slipping along the horizontal surface. Using any valid method, derive the equation(s) of motion of the system. Your Anal answer should be in matrix form and expressed symbolically in terms of the provided variables. Using the equation(a) of motion derived in Question (1) with k_1 = 10 N/m, k_2 = 15 N/m, a = 0.5 m, b = 0.75 m, and first natural frequency of 0.3202 rad/s along with mode shape vectors {U}_1 = {1 -0.8633}, and {U}_2 = {1 0.4633} Determine the value of the moments of inertia J_1 and J_2 for each disk. Determine the value of the second natural frequency

Solution

The problem is similar to the coupled spirng mass system, here one has rolling instead of sliding.

Take the displacement of the first disk as x1, that of the second as x2

If moment of Inertias are I1 and I2, these are balanced by the torques provided by the springs.

write the eqns as

I1*(theta1)\" = -k1a*(theta1) + k2 *( b*theta2 - b*theta1)

I2*(theta2)\" = -k2*(b*theta2-btheta1)

Matrix form: [ I1( theta1)\"] [ -k1a+k2b k2b][theta1]

[ i2(THETA2)\"] = [k2b -k2b][theta2]

Step 2: this is an eigenvale problem. The moments of Inerta must be calculated ( for a disc= Mr^2 )

Step 3:: The second natural freq canbe found from the given value of 1st natural frequency knowing the ratio of the diagonal entries of the mass matrix

ie : [ 1st diag entry- given 1st freq]/[ 2nd diag - unknwn second freq] = .4633/.8633