Question

F(N) 2. A 15 kg oscillator with a stiffness of k = 960 N/m and damping coefficient c = 60 Ns/m is driven by a square- wave excitation F(t) shown in the figure. Determine and plot the steady state response for 12 s using 100 terms in the Fourier series solution. 100 -100

Answer 1

Given

M=15 kg

K=960 N/m

C=60 Ns/m

X(12)=?

from the besic of the vibration we must say that behaviour of output or response is same as the behaviour of exitation or input

by seeing the force on time domane

we can say that the the responce will show same behaviour on time domane

formula for the

responce of vibration system are as follow

X(t)= (F/K)/((1-r^2)+(2*z*r)^2)^.5

z= ratio of damping coefficient to critical damping coefficient

natural frequeancy Wn=(K/M)^.5=8 rad/s

z=C/(2*M*Wn) = 60/(2*15*8) =0.25

Force(t)=F*sin(2*3.14*(X(t)-X(0))/T)..........(A)

X(t) = responcr at time t

X(0) = responce at t=0

T = time period

Equat eqn (A) = F*sin(W*t)

W=2*3.14*(2-0)/T^2

W=3.14

X(0)=(100/960)/((1-0.3925^2)+(2*.25*.3925)^2)^.5=0.1199 m

X(12) = magnitude (0.1199) m

plot

X-Axis = time

Y-Axis = Responce in meter