An automobile is modeled as a simple four-degree of freedom system as shown in the accompanying figure. The parameters of the model are as follows:

c = equivalent damping coefficient for two shock absorbers (both front and back).

k1 = equivalent stiffness of two springs (both front and back).

k2 = equivalent stiffness of two tires (both front and back).

m1 = combined mass of automobile body and motor.

I2 = mass moment of inertia of combined mass of body and motor about its mass center G.

m2 = equivalent mass of two wheels (both front and back).

Assuming small amplitudes of oscillation for the system, verify that the stiffness matrix K and damping matrix C are, respectively,

K = [2*k1, k1*(I2 - I1), -k1, -k1;

k1*(I2 - I1), k1*(I1^2 + I2^2), k1*I1, -k1*I2;

-k1, k1*I1, (k1+k2), 0;

-k1, -k1*I2, 0, (k1+k2);]

C = [2c, c*(I2 - I1), -c, -c;

c*(I2 - I1), c*(I1^2 + I2^2), c*I1, -c*I2;

-c, c*I1, c, 0;

-c, -c*I2, 0, c;]

Assume that the m1 = 40 kg, m2 = 1800 kg, I = 5000 kg*m^2, I1 = 3.2 m, I2 = 2.1 m, k1 = k2 = 90,000 N/m, and c = 800 Nm/s;

1. Determine natural frequencies and the mode shapes.
2. Considering the base displacement x1(t) = 0.1*sin(66t) applied to the right side only.