Question

For an object in damped harmonic motion with initial amplitude a, period 2π/ω, and damping constant c, find an equation that models the displacement y at time t for the following.
(a)
y = 0 at time t = 0
y= ?
(b)
y = a at time t = 0
y= ?

Answer 1

Answer:

  see below

Step-by-step explanation:

We presume the damping constant is the opposite of the multiplier of time in the exponential term. Then the equations are ...

(a)  y = a·e^(-ct)·sin(ωt)

(b)  y = a·e^(-ct)·cos(ωt)

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These are the standard equations for simple harmonic motion assuming there is no driving function.

  a = initial amplitude*

  c = damping constant**

  ω = frequency of oscillation in radians per second

  t = time in seconds

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* Of course, when y(0) = 0, the motion never actually reaches this amplitude because it is subject to decay before it can.

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** In electrical engineering, damping is often specified in terms of a time constant, the time it takes for amplitude to decay to 1/e (≈36.8%) of the original amplitude. If that time is represented by τ, then the exponential factor is e^(-t/τ).