Power absorbed by eccentric mass vibrator

[RiBeneke]

I am trying to find some theoretical basis to calculate the maximum power that could be required to drive a vibrator of the eccentric mass type. The applications I have in mind are mineral aggregate screening and feeding equipment.

There are recommendations from many sources on the power required for a particular piece of equipment with particular loads. It seems that these power requirements have been derived from experience and I have not been able to find the theory.

The relation between speed of rotation of the eccentric mass, its mass, its eccentricity and the resulting forces is well established in traditional formulae. From these one can calculate the resulting accelerations on the equipment to which it is attached.

My question relates to the power absorbed in this process. As I see it, there are 2 limiting cases, and the desired value will be somewhere between.

Case 1) The rotating shaft with eccentric mass is mounted on an infinitely heavy structure. The centre of the shaft remains in one place and the eccentric mass centre orbits around it. Work done = 0.

Case 2) ) The rotating shaft with eccentric mass is mounted on an weightless structure. The centre of the shaft performs a circular orbit and the eccentric mass centre remains in one position. Work done = 0.

Does anyone have a formula to calculate the intermediate real-life case of power absorbed with known solid structure mass ?

More importantly, does anyone have a formula to calculate the maximum theoretical power that could be absorbed in driving the eccentric shaft when the the structure on which it is mounted contains energy absorbing (damping, loose) material at the optimim mass to absorb the greatest power ?

Thanks
Richard Beneke

 

[electricpete]

If there is no damping in the structure, then the steady state power absorbed will be zero, regardless of masses. An ideal undamped oscillating mass spring system consumes no energy, only converts between kinetic and potential. I think even that ideal undamped system still consumes energy as you increase speed into resonance.

If you have a SDOF or higher order damped mass spring model in mind then you can very easily compute the energy dissipated in the damping element under sinusoidal steady state conditions.
That should be the easy part (I can give the SDOF equations if you want). Coming up with a realistic model would be the tough part.
Just my two cents.

 

[electricpete]

The basis for a solution of power consumed by damping would be based upon:

Force: Fdamp(t) = c * x(t)'
Power: P(t) = Fdamp(t) * v(t)
= c * x(t)' *x(t)'
= c * (x(t)') ^2
= c * v(t)^@
Assume sinusoidal steady state v = V*sin(w*t+theta)

Average Power <P(t)> = c*<[V*sin(w*t+theta)]^2>
= 0.5 c * V^2
= c * Vrms^2
where Vrms = V/sqrt(2)

 

[RiBeneke]

Hi electricpete

Thanks for the reply. I am a civil/structural engineer and I am not too familiar with damping formulae and the terms you use.
c ... is that a constant
x ... linear displacement .. could be calculated from basics in the absence of damping
V ... ? please explain further.

Tha actual vibratory path of a typical feeder etc is typically a distorted ellipse .. would that have a significant effect on the formula you suggets ?

Thanks

 

[Strong]

Try this link to a commercial product for ship vibrations:

http://www.gertsen-olufsen.dk/English/index.html

 

[electricpete]

c is the damping constant. (F=-c * dx/dt)
x is linear displacement
v(t) is velocity = dx(t)/dt
V is peak amplitude of sinusoidal v(t).
Equation v(t) = V*sin(w*t+theta)

&quot;The actual vibratory path of a typical feeder etc is typically a distorted ellipse .. would that have a significant effect on the formula you suggets ?&quot;

I think that a two-dimensional simple oval-shaped orbit problem could be decomposed into two single-dimensional problems (x and y directions).

 

[RiBeneke]

Thanks, Strong.
The equipment you refer to is interesting, but it does not give me the background theory I am looking for. I doubt that the manufacturers would divulge all the details.

Thanks, electricpete.
I will work through the maths, and ask again if I still have questions.
Normally if I design a feeder, I would determine the required amplitude and hence the g (acceleration) force required. Then select a suitable vibrator motor from the likes of Uras, Deltech etc. However I have never found a tool to determine whether the motor power that results from the selection is in fact the optimum. Recently I did some preliminary work on a project that would require 4 x 11 kW motors, and I cannot just rely on the vibrator motor catalogue for this one.

Thanks again.