energy = f(amplitude, freq)

[MarkUMSU]

What is the relationship to express power required to force an object to vibrate a varying amplitude and frequency for a known mass, spring constant, and damping value?

Thanks

 

[BobM3]

Just want to verify, it's power that you want (or energy)? It it's energy then energy in = energy lost by damping. If it's power then I imagine you want to get the amplitude from zero to its steady state value in a certain amount of time?

 

[electricpete]

If the motion is sinusoidal:

<p(t)> = Frms * Vrms * cos(theta) where
<p(t)> = average value of power
Frms = Fpeak/sqrt2
Vrms = Vpeak/sqrt2
theta = time angle between F and V

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[electricpete]

F = force
V = velocity

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[Denial]

If your system (mass = m, spring stiffness = k) is vibrating so its displacement is given by y = a*sin(wt)
(where w is not necessarily its natural frequency) then its velocity is given by v = aw*cos(wt).

From these formulae it follows that when the object is at its equilibrium position (t = 0, ?/w, 2?/w, ...) the system's total energy is m*(aw)²/2 (since it is entirely kinetic). Similarly when the object is at one of its two extreme positions (t = ?/(2w), 3?/(2w), 5?/(2w), ...) the system's total energy is k*a²/2 (since it is entirely potential).

When the characteristics of the vibration (a and/or w) are changing with time, then the changes in the system's total energy are the result of energy losses due to damping and/or work done on the system by some external force (or work done by the system on an external force). To take things further you need to know the damping model and the external force as a function of time. You can then set up the appropriate differential equation: this might have an analytical solution, or might require a numerical solution. Standard textbooks on dynamics will present the derivation and solution for the case where the damping is "viscous" and the external force varies sinusoidally.

[Note that if we have no damping and no external force, the system's total energy will be constant. We can then equate the two energy formulae above and solve for the system's natural frequency, hopefully getting w = ?(k/m).]

 

[Denial]

Problem driving the fancy features. It all looked fine in "preview mode", but Murphy seems to be in the ascendant today. All "?" symbols in the upper part of my previous post are meant to be the symbol pi ([pi;]). The final "?" is meant to be the square root symbol ([radic;]).

 

[Denial]

Yes, I know I left out the &. More Murphy.

 

[electricpete]

By conservation of energy, the average power in is the power dissipated in the damper.

If the system is
Mass
| |
| |
K C
| |
| |
Ground

where sinusoidal force is applied to the mass in steady state:

<p(t)>=v * F = v^2 * c = w^2*x^2*c

where w = 2*Pi*f
f = frequency
F = force (rms)
v = velocity (rms)
x = displacement (rms)


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[electricpete]

<p(t)> = average power

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[electricpete]

<p(t)> = average power input to the system by the external force

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[electricpete]

just to roll it into a tighter bundle:

average power input is
<p(t)>=(2*pi*f*x)^2 * c
where f is frequency and x is rms displacment.

I assumed sinusoidal steady state, linear time invariant system in configuration shown above.

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[MarkUMSU]

Hey Guys
Thanks for you help.