I need help solving the following second order differential equation, used for m

Question

I need help solving the following second order differential equation, used for mechanical vibration analysis of a simple car model and in properly assigning m, c, k and W.

mz'' + cz' + kz = W - cR' - kR, where m, c, k and w are appropriately assigned.

m= mass, c= damping coefficient, k = stiffness coefficient, z = vertical distance (positive downwards), R = roadway (positive upwards).

The initial conditions are:

(1) Car at rest --> Z''(0) = Z'(0) = R'(0) = R(0) = 0

(2) Car traveling on a smooth roadway at constant velocity --> Z''(t') = Z'(t') = R'(t') = R(t') = 0 for all t' < 0

Any help would be greatly appreciated.

Explanation / Answer

Car at rest

At initial condition given Z''(0) = Z1 (0) = R1(0) = R(0) =0

Then we wii get the equation as KZ = W

K,Z matrices depends upon the degree of the freedom

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