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

(a)

Let z + R = t

=> z' + R' = t'

=> z'' = t'' ( Assumption : R'' = 0 )

Mt'' + ct' + kt = W

Solving the homogenous equation by plugging t = erm

=> Mr2  + cr + k = 0

=> r1,2 = 0.5( ( -c + ( c2 - 4kM ) ) / M )

=> t = c1er1m + c2er2m

=> t' = c1r1er1m + c2r2er2m

From the given condition t(0) = 0 , t'(0) = 0

=> c1+ c2 = 0 => c1= -c2

and c1r1+ c2r2 = 0

=> r1 = r2

=> t = 0

=> z + R = 0

Car would remain in the situation of rest as both roadway and vertical distance would cancel each other

=> c1r1- c1r2 = 0

=> r1 = r2

=> c2 = 4kM

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.