a) The forces on the particle are shown below. The forces from the dampers (orange) are proportional to the velocity, hence always point in the same direction and can effectively be combined. The spring force (blue) is a restoring force proportional to the displacement. Only gravity is consistently pointing down. The others can be in either direction.

b)

Spring / length / natural length / extension / stiffness / / Hi
PA / / l0 / / k / i /
Damper / length / rate of change of length / damping constant / / Ri
PA / / / r1 / i /
PB / / / r2 / i /

c) Newton’s second law gives

Letting and , we have

.

In equilibrium, , so , so

Now letting gives

Here y is the position of the box and q is the displacement of the particle from its equilibrium position within the box. So the first term represents the acceleration of the box with respect to the stationary floor, and the remaining three terms represent the motion of the particle with respect to its equilibrium position in the box.

d) When the machine is static, we have

,

This means the string is stretched downward 0.98 mm by the mass (below it’s natural position).

e) The damping ratio is , so the system is strongly damped.

f) This is an example of phasor addition. To prove the general case, we first rewrite

For our specific case, we have so

g) Our differential equation is

where , , and .

If we try a sinusoidal solution,

which gives

From our phasor-addition result, this becomes

For this equation to hold, the frequency, phase and amplitude of the cosine term must all be identical:

, , and .

Plotting vs for the given parameters,

/ parameter / value
k / 2000
m / 0.2
r1 / 20
r2 / 50
r / 70
w0 / 100
z / 1.75

h) To find the maximum value of as a function of , we differentiate and set the result equal to zero:

For , the resonant frequency of the system is

,

with an amplitude gain of .

i) The frequency range for which is within 20% of the maximum is found from numerically examining the function: