Write the equation of motion for this system Write the Lapla

Write the equation of motion for this system. Write the Laplace transformation of the equation of motion for this system. Assume zero initial conditions. Find the transfer function G(s) for this system. Assume zero initial conditions. Evaluate the transfer function at an argument of j omega. Evaluate the real component of G(j omega). Evaluate the imaginary component of G(j omega). Evaluate the magnitude of G(j omega). Evaluate the angle of G(j omega). Find the expression for the steady-state response of this system.

Solution

This is a forced Vibration

Forces on the system

Impressed Oscillating force = p(t) ….Downward

Inertia force = m ………upward

Damping Force = b…….upward

Spring force = kx …….upward

          m + b + kx – p(t) = 0

where m is the mass attached common to spring and damper.

L(m + b + kx – p(t) = 0 )

m(s^2 x(s) –sx(0) – (0) ) +b sx(s) - x(0) +k x(s) – p(s) = 0

At zero initial condition

x(0)=0, (0)=0

hence equation becomes

ms^2 x(s) + bsx(s)+kx(s)=p(s)

c)

Transfer Function of the system = output/input

G(s) = Displacement/Force applied = x(t)/p(t)

From equation in b

(ms^2 + bs+k) x(s) = p(s)

Then Transfer function

G(s) = x(s) /p(s)

G(s) = 1/[ ms^2 + bs+k]

d) Transfer function at an argument of jw

s=jw

G(s=jw) = 1/[ m(jw)^2 + b(jw)+k]

Complex form is

G(jw) = 1/[(k-mw^2 +jbw]

e) G(jw) = 1/k[1-m/k*w^2 +j b/k*w]

G(jw) = (1-(w/w_n)^2)/[sqrt((1-(w/w_n)^2)^2 +(2(w/w_n ))^2)] - j(2(w/w_n ))/[sqrt((1-(w/w_n)^2)^2 +(2(w/w_n ))^2)]

Real Component is = (1-(w/w_n)^2)/[sqrt((1-(w/w_n)^2)^2 +(2(w/w_n ))^2)]

f) Imaginary Component is =(2(w/w_n ))/[sqrt((1-(w/w_n)^2)^2 +(2(w/w_n ))^2)]