Question

Help setting up an ODE please?

A small object of mass 4kg is attached to an elastic spring with spring-constant 64 N/m, and is acted upon by an external force \[F(t)=Acos ^{3}(\omega t)\]
Find all values of omega at which resonance occurs.

Now, since there is no damping give, I assume we are in a "forced free vibration" situation. So I think the form of the differential equation will be:

\[y''+\omega^2_{0}y=\frac{ F _{0} }{ m }\cos(\omega _{0})t\]
correct?

Answer 1

Now, correct me if I am wrong, but I believe resonance occurs every time \[\omega = \omega_0\]

Furthermore, the book says \[\omega_0 = \sqrt{\frac{ k }{ m }}\] which in this case means \[\omega_0 = \sqrt{\frac{ 64 }{ 4 }} = 4\]

Thus resonance occurs when omega = 4.

Answer 2

I believe the next step is setting up and solving the differential equation, and this is where I could use some help. I think the correct case to describe this problem is "Forced Free Vibration" so it should look like: \[y'' + \omega_0^2 y = \frac{ A }{ m }\cos^3(\omega_0 t)\] does this seem good so far?

Answer 3

correct ! go ahead

Answer 4

substitute wo =4

Answer 5

so I have \[y'' + 16y = \frac{ A }{ 4 }\cos^3(4t)\]

Answer 6

yes

Answer 7

find the complimentary function/particular integral

Answer 8

Im not sure how to do that?