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## Homework Statement

If the electric motor of problem 3.66 is to be mounted at the free end of a steel cantilever beam of length ##5## m and the amplitude of vibration is to be limited to ##0.5## cm, find the necessary cross-sectional dimension of the beam. Include the weight of the beam in the computations.

The electric motor has mass of ##75## kg and speed ##1200## rpm at its mid spin. A rotating force of magnitude ##F_0 = 5000## N

## Homework Equations

\begin{align}

I &= \frac{1}{12}bh^3\\

E &= \frac{m\omega_n^2\ell^3}{3I}\\

k_{eq} &= \frac{3EI}{\ell^3}\\

\omega_n &= \sqrt{\frac{k}{m}}

\end{align}

## The Attempt at a Solution

Since the amplitude of vibration is limited to 0.5cm,

$$

\frac{1}{200} = \sqrt{1 + (2\zeta r)^2}\lvert H(i\omega)\rvert

$$

where

$$

\lvert H(i\omega)\rvert = \frac{1}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}

$$

where ##r = \frac{\omega}{\omega_n}##, ##2\zeta\omega_n = \frac{c}{m}##

I solved for zeta in terms of r but I dont think that is leading any where

$$

\zeta = \pm\frac{\sqrt{\omega^4 - 2\omega^2\omega_n^2 - 39999\omega_n^4}}{2\omega\omega_n\sqrt{39999}}

$$

We can convert rpm to radians per sec by

$$

1200rpm\frac{2\pi}{1rev}\frac{1m}{60s} = 40\pi

$$

That is ##\omega = 40\pi## and our general equation of motion is

$$

m\ddot{x} + c\dot{x} + kx = 5000e^{40\pi it}

$$

The mass in the EOM is mass of motor plus mass of beam. How would I find the mass of the beam and the damping coefficient? If I can get these two pieces of information, I should be good to go.

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