Calculating Force Transmitted to Base of a Cantilever Beam

[Dustinsfl]

Homework Statement

How does one determine the amplitude of the force transmitted to the base of a beam?

Homework Equations

The Attempt at a Solution

The ODE modeling displacement is
$$
-0.000891(9.60875\sin(183.26t) - 323.778\sin(5.4386t))
$$

 

[haruspex]

How does one determine the amplitude of the force transmitted to the base of a beam?

In what set-up?

 

[Dustinsfl]

In what set-up?

What do you mean?

 

[haruspex]

What do you mean?

I mean, describe what's going on. What beam, what forces, ...?

 

[Dustinsfl]

I mean, describe what's going on. What beam, what forces, ...?

It is a cantilever beam with end load.

 

[haruspex]

It is a cantilever beam with end load.

OK, and what facts are you given - lengths, masses, moduli..? In the OP you mention ODE and quote an expression involving time, so I guess this is not a statics question. Is there some perturbation applied?

 

[Dustinsfl]

OK, and what facts are you given - lengths, masses, moduli..? In the OP you mention ODE and quote an expression involving time, so I guess this is not a statics question. Is there some perturbation applied?

A steel cantilever beam is $120$ in long by $1\times 1$ in$^2$ which has a motor that weighs $100$ lb$_f$ mounted at the end. The motor runs at 183.26 rad/sec. The motor has an unbalance of $0.1$ lb$_m$ located at a radius of $0.1$ in from the axis of the shaft. Assume that for the steel $E = 30\times 10^6$ psi, the density is $0.28$ lb$_m$/in$^3$, and that the damping ratio is $0.01$.

What I did was then:
First, let's convert Young modulus from psi to Pascals, 1 psi is $6894.76$ Pascals. Then $E = 2.07\times 10^{11}$ pascals. The moment of inertia is $I = \frac{bh^3}{12} = 3.47\times 10^{-8}$ m since 1 in is $0.0254$ m. The equivalent spring constant is
$$
k_{eq} = \frac{3EI}{\ell^3} = 760.985\text{ N/m}.
$$
The relation of mass with density is $\rho = \frac{m}{V}$. The volume of the cantilever beam is $V = 1^2(120) = 120$ in$^3$. Then $m = V\rho = \frac{120(0.28)}{2.2} = 15.273$ kg. The equivalent mass
$$
m_{eq} = 15.273 + \frac{100}{2.2}0.23 = 25.7275\text{ kg}.
$$
Then the natural frequency of the beam and the motor system is
$$
\omega_n = \sqrt{\frac{k_{eq}}{m_{eq}}} = 5.4386\text{ rad/sec}.
$$
Then I used a Laplace transform to determine $y(t)$

 

[haruspex]

OK, well there's a lot there I could not have guessed at.
Plugging all these numbers in straight away makes it unnecessarily hard to follow.

$I = \frac{bh^3}{12}$

what is b?

$m_{eq} = 15.273 + \frac{100}{2.2}0.23 = 25.7275\text{ kg}.$

Where does the 0.23 come from?

Then I used a Laplace transform to determine y(t)

And that is the time-dependent expression in the OP, right?
I assume you want the max stress at the beam support. In terms of the oscillation cycle, when will that occur?

 

[Dustinsfl]

OK, well there's a lot there I could not have guessed at.
Plugging all these numbers in straight away makes it unnecessarily hard to follow.

what is b?

Where does the 0.23 come from?And that is the time-dependent expression in the OP, right?
I assume you want the max stress at the beam support. In terms of the oscillation cycle, when will that occur?

b = 1in or 0.0254 m 0.23 is by definition of equivalent mass of a cantilever beam.

I don't know. I am trying to determine the force transmitter to the base.