Question

For the given values of m, c, k, and f(t), assume the forced vibration is initially at equilibrium. For t 0, find the motion x(t), and identify the steady-periodic xs(t) and transient Xtrt) parts m 1, c4, k 5, f(t) 20 cos(3t) x(t) ain(3)cos (3t) xsp(t)= xtr(t)

Answer 1

m ck f(t) 4x5 20 cos(3t)

4 5=0 D2 4D 5 (D 2)2 -12 D =-2 t i

So and

r(t)c1e2 cos(t)ce2 sin(t)

A cos(3t B sin(3t) 4x 5 (-4B 12.A) sin(3t) + (t) (12B-4A) cost (3t)

Thus, we have
12B 4A 20, 4B12A 0 B=-3A
so that
3 -40.A 20 A B =
meaning

is the particular solution

(t)cos(3 - sin(3t) 2

The general solution is

3 -2t C cos(t)esn(t)cos(3t)sin(3t) -2t

Initial conditions imply so

r(0)'(0)0 0,-2c1+c2- l

C2 C1

Thus, $x(t)=-\frac{1}{2}e^{-2t}\cos(t)+\frac{7}{2}e^{-2t}\sin(t)+\frac{1}{2}\cos(3t)-\frac{3}{2}\sin(3t)$

Where transient solution is and steady state solution is

sin(t) -2t cos(t) --e Ttrans (t) e 2

$x_{\text{steady}}(t)=\frac{1}{2}\cos(3t)-\frac{3}{2}\sin(3t)$