# How to find cross sectional dimension of beam

1. Oct 16, 2014

### Dustinsfl

1. The problem statement, all variables and given/known data
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

2. Relevant 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}
3. 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.

Last edited: Oct 16, 2014
2. Oct 16, 2014

### SteamKing

Staff Emeritus
You need to understand what the variables in your formulas represent.

$\omega_n$ is the natural frequency of the beam, and it is a function of E and the moment of inertia of the beam. The problem statement doesn't specify a beam material or a cross section shape, so E and I have to be determined using some subjective criteria. (The length of the beam is already specified, so it is not considered here). You should be aware that I = bh3/12 is correct only for beams having a rectangular cross section.

ω is the frequency of excitation, and the problem statement has already stated that the electric motor operates at 1200 RPM. Can you calculate what ω is at this speed?

k is a little trickier. You have the motor placed at mid-span of the beam, which implies that there is a certain static deflection produced in the beam. The problem also wants you to consider the mass of the beam in solving this problem. The k you have specified is that for a beam with a uniform distributed load, but what about the effect of the motor on the deflection of the beam?

3. Oct 16, 2014

### Dustinsfl

omega was added to the OP 11minutes ago and the motor is not a mid span it is at the free end which is stated in the question. The k I specified comes from the book for a cantilever beam with end load which is the scenario we have.