How to determine Vibration Amplitude

[Su Solberg]

Hello Every one,

When I am looking at the design of a vibration screen, I found that the claims from different source have different equation.

Please have a look on the attachment.

You can see "Claim 2 " is a more general approach but need to know spring constant k.
while "Claim 1" vaild when driving frequency>>system natural frequency, from ASME shale shaker comittee.

However, I failed to prove Claim 1 by using the principle of Claim 2.

Please tell me where is the problem.

Thanks for your kind help ^_^

 

[minger]

What exactly are you trying to find? The amplitude of displacement, force magnitude? For an unbalance load, sure, the unbalance load is simply:
$$F = mr\omega^2$$
However, the displacement will be dependent on the entire system.

 

[Su Solberg]

What exactly are you trying to find? The amplitude of displacement, force magnitude? For an unbalance load, sure, the unbalance load is simply:
$$F = mr\omega^2$$
However, the displacement will be dependent on the entire system.

Thanks for your concern.
I am only concentrate on "The maximum amplitude of displacement" in ideal case for my question.
I have the force magnitude, unbalance load, unbalance load's angular velocity, system mass, system's spring constant already.

I am wondering whether Claim 1 is valid for driving frequency>> system natural frequency, because I am pretty sure Claim 2 is correct.

 

[minger]

I mean, I'm not real sure what claim 1 is from, it seems to be an empirical expression based on experience...maybe?

Claim 2 is more analytic, but involves damping. It is a chart that says how your displacement will look near your natural frequencies in the presence of damping.

If you are away from natural frequencies, then a force at a given frequency applied to a mass on a spring should be fairly trivial.

 

[Su Solberg]

I mean, I'm not real sure what claim 1 is from, it seems to be an empirical expression based on experience...maybe?

Claim 2 is more analytic, but involves damping. It is a chart that says how your displacement will look near your natural frequencies in the presence of damping.

If you are away from natural frequencies, then a force at a given frequency applied to a mass on a spring should be fairly trivial.

Thanks.
I have similar mind with you too.

For " (eccentric mass * eccentric radius)/ system mass ", do you think it can be derrived
or just an empirical expression?

Thanks for your help.

p.s. if any other have interest, please join the discussion.

 

[minger]

Yes, your forcing function as I mentioned is analytic. The force generated from a rotating unbalance load is:
$$F = mr\omega^2$$
That can be derived from dynamics equations.

OK, I'll bite. If we assume that your rotating unbalance is causing force only in the direction of the resisting spring, that is the unbalance only causes force in one direction, then the equation of motion is:
$$\frac{W}g \ddot{x} = W - (W+kx) + P\sin \omega t$$
Introduce the following notation:
$$p^2 = \frag{kg}{W}$$
and
$$q = \frac{Pg}{W}$$
The equation of motion becomes:
$$\ddot{x} + p^2 x = q\sin \omega t$$
The particular solution is obtained by assuming that x is proportional to sin wt, by taking:
$$x = C_3 \sin \omega t$$
Chossing the magnitude of the constant such that id satisfies the equation of mtion, we get:
$$C_3 = \frac{q}{p^2 - \omega^2}$$
So, the particular solution is:
$$x = \frac{q \sin\omega t}{p^2 - \omega^2}$$
Adding this particular solution to the general solution of the homogeneous equation, we get:
$$x = C_1\cos pt + C_2\sin pt + \frac{q \sin\omega t}{p^2 - \omega^2}$$
The first two terms represent free vibrations, and the third term represents the forced vibration of the system. Using the notation from above and ignoring the free vibrations, we obtain a steady state forced vibration equation:
$$x = \left(\frac{P}{k}\sin\omega t\right)\left(\frac{1}{1- \omega^2/p^2}\right)$$
The absolute value of the second term is often called the magnification factor:
$$\beta = | \frac{1}{1-\omega^2/p^2} |$$
You'll find that if you plot beta against w/p, you'll get your plot from your Claim 2.

p.s. The homogenous equation defined earlier in the book is:
$$x = C_1\cospt + C_2 \sin pt$$

p.p.s. This post paraphrased from Timoshenko's "Vibration Problems in Engineering".

Hopefully this helps, good luck,