Ordinary Differential Equations question

Question

As the engineer in charge of the calibration software, you are to create the correct algorithm to analyze

the desk on which the printer is. Let us focus on some part of this.

Assume the machine is the object

producing an external vibration on the desk, a pure sine function, ranging from 1 to 10 Hz, with an

amplitude kept constant at 1 mm (for sake of simplicity). Consider the following information for the

desk, no damping considered, and assume the initial condition are x(0) = 0 and x’(0) = 0.

M = 70 kg ; k = 7 kN /m

a) Write a differential equation that would work for all frequencies in the range from 1 to 10 Hz.

b) What is the natural frequency of the desk?

c) Solve the differential equation found in a) for the case when the applied frequency is different

than the one found in b).

d) Do a graph of the amplitude of the response as a function of the applied frequency of 1.55, 1.58,

1.585, 1.6, 1.61, and 1.65 Hz. What do you understand from this graph? (You don’t need to be

exact here, we are looking for the general features/trend of the graph; you can neglect some

components of the exact amplitude)

Clearly state all your hypotheses!

**Reposted since nobody answered still**, will give 5 star review if you get it right with full solutions and explanations.

Answer 1

Let

f be the frequency of vibration produced by the machine,

A = 1 mm = 0.001 m be the amplitude of that vibration,

x(t) be the displacement of the desk at a time t,

M = 70 kg be the mass of the desk,

k = 7kN/m = 7000 N/m be the "spring" constant of the desk.

The challenge with this problem is interpreting the given statement.

The solution depends on the (unknown) force F(t)

with which the machine drives the vibration of the table.

We are told that it is a "pure sine function" with frequency f.

I interpret that as meaning

F(t) = F₀sin(2πft)

for some amplitude F₀,

and acting in the direction of displacement, i.e.,

in the direction in which the constant k was measured.

Instead of providing us with F₀, we are given the quantity A.

I do not interpret it as the amplitude of the resulting vibration of the desk,

as that would defeat the purpose of this exercise.

Instead I interpret it as the displacement the desk would experience

if a constant force of magnitude F₀ were applied.

Under this interpretation

F₀ = kA

(But my interpretation may not be what your teacher had in mind.)

Now we can start solving the problem.

The total force acting on the desk at a time t is composed of two forces:

the driving force F(t), and

the reaction of the desk to the displacement, which has magnitude -kx(t) (by Hooke's Law).

By Newton's second law this net force equals mass times acceleration.

Mx"(t) = F(t) - kx(t)

This can be rewritten as

Mx"(t) + kx(t) = kAsin(2πft)

or

x"(t) + (k/M)x(t) = (k/M)Asin(2πft) (1)

As is customary, I will write

x instead of x(t),

ω = 2πf

ω₀ = √(k/M) = √(7000/70) = 10 Hz

B = (k/M)A = (7000/70)×0.001 = 0.1 m/s²

Equation (1) becomes

x" + ω₀²x = Bsin(ωt) (2)

The solution to this equation depends on the relationship between ω and ω0.

If ω = ω₀ then the solution to the initial value problem is

x = (B/2ω)(sin(ωt)/ω - tcos(ωt)) (3)

If ω ≠ ω₀ then the solution to the initial value problem is

x = (B/(ω²-ω₀²))(sin(ω₀t)ω/ω₀ - sin(ωt)) (4)

The point is the following.

If you keep changing ω so as to approach ω₀, the amplitude of the oscillations will keep increasing.

When ω becomes equal ω0 (called resonance), solution (3) applies;

with that solution the amplitude of the oscillations increases with time.

In reality, during resonance the table does fly to infinity for two reasons.

First there is always some damping, and secondly

Hooke's Law applies only for small displacements.

But at resonance you do maximize the amplitude of the oscillations.