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3.2 Damping
In real life there would be some damping of the oscillations that is not accounted for in the idealized equations of the previous posting. An earlier posting on shock excitation discusses the effects of damping (also see this and this).
Fig. 6. This animation shows the motion of a damped mass-spring resonator. The mass is submerged in a container of water to add resistance to its motion. A graph is provided to show the damped harmonic motion, whose amplitude decrease with time. The motion is described by the first equation shown in (23) below. The envelope of this decay is shown as a blue curve. The envelope is the e−t/τ factor in (23). Mouse over the figure to see the motion. Click on the figure to restart it. |
As we discussed in the earlier postings, damping to a mechanical resonator is often the result of friction usually adding an additional force of the form ffriction = − Rv where R is a friction coefficient (or resistance to motion) and v is the velocity (of the bob in this case). Note that sometimes damping is due also to coupling of the resonator to some other device allowing energy to escape via that route.
With friction, the total force on the bob (ignoring gravity, which just causes an offset of sag as discussed in the previous posting), we have:
. (20)
Adding our new friction force to (8) and remembering that velocity is the first time derivate of position (i.e. v = dx/dt) yields:
. (21)
This can be rearranged in the standard form of A d2x/dt2 + B dx/dt + C = 0 as:
. (22)
We have previously solved (22) in the posting on shock excitation. The results are shown in Table 6 of that posting. We next adapt those results to the mass/spring resonator.
We note that A of Table 6 is the coefficient of the second derivative which equals m in (22). Similarly B, the coefficient of the first derivative becomes R and C becomes k . Making these substitutions in the results of Table 6 yields solutions in the form of sines, cosines and complex exponentials:
, (23)
where the decay angular frequency ωdecay , the undamped angular frequency ω0 , and the decay time τ are given by:
. (24)
If a person wishes to derive these results again, they could substitute any one of the forms in (23) into (22) and solve for the constants ωdecay , ω0 , and τ . The above mentioned earlier postings discuss the derivation in considerable detail.
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