# Forced vibration system response

1. Jan 8, 2010

### gowrath

I designed a spring and mass system (damping unknown) that is vibrated by applying a sinusoidal force to the mass using an electrodynamic shaker.

The mass is estimated to be 100 Kg.
The spring is actually 4 springs, each with a k value of 63.55 N/mm, supporting a platform (one spring at each corner) with the mass.

Attached is the system's frequency response curve.

If the shaker is set to vibrate at frequencies above the system's natural frequency, where the amplification ratio is very low, is it possible to control the acceleration that the mass is exposed to or will the inability of the system to respond fast enough, and thus have very small displacements, limit the acceleration?

Is it reasonable to expect the electrodynamic shaker system, provided the feedback correction is snappy enough, to increase the driving force and be able to execute the desired displacements (resulting in desired accelerations) or will the frequency response characteristics of the system prevent this?

Thanks.

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2. Jan 8, 2010

### Bob S

I think that if the shaker acceleration, and not the amplitude, is fixed, then you will get beat frequencies between the shaker frequency and the natural resonant frequency ω0=sqrt(k/m). See the thread
Post #5, and the two attachments
https://www.physicsforums.com/attachment.php?attachmentid=22300&d=1260059684
and
https://www.physicsforums.com/attachment.php?attachmentid=22303&d=1260064087
In this thread, the particular solution was for a constant sinusoidal driving force. The beat (difference) frequency is a dominant feature of the amplitude in this solution.
Bob S

3. Jan 10, 2010

### Bob S

Comment on my previous post. In my second of three attachments, I derive the solution for a harmonic oscillator driven by an off-frequency force (acceleration). My solution is for the undamped case. The particular solution (Eq(3)) is the steady-state solution (w/o damping). Eq (8) is the complete solution, where the first term matches the initial conditions to the steady state. They are:
1) a(t) = (F/m) sin ωt (so the forced acceleration is zero at t=0 (when it is turned on);
2) y(t)=0 at t=0; and
3) y'(t)=0 at t=0 for same reasons.

There is no remaining beat frequency after the first term in Eq(8), for matching the initial conditions, is damped out.

Bob S