Question

Hello, can someone assist with Forced Vibrations using differential equations. Photo of problem attached,

Answer 1

For my governing equation, I have:
\[u'' + \frac{ 1 }{ 4 }u = 14 \cos(wt)\]

Answer 2

where u(0)=0 and u'(0)=0

Answer 3

\[\omega _{0} = \sqrt{k/m} =\sqrt{\frac{ 1 }{ 8 } * 2} = \sqrt{\frac{ 1 }{ 4 }} = \frac{ 1 }{ 2 }\]

Answer 4

I think that at equilibrum, we can write:
\[k\Delta z = mg \Rightarrow \frac{k}{m} = \frac{g}{{\Delta z}} = \frac{{32}}{2} = 16\]
so, I got this equation of motion:
\[\ddot z + 16z = 14\cos \left( {\omega t} \right)\]
we have the phenomenon of resonance, when the subsequent condition holds:
\[\sqrt {\frac{k}{m}} = \omega \]
being: \(\Omega = \sqrt {\frac{k}{m}} \) the natural frequency