Question

The frequency fd of a damped harmonic oscillator is 100 Hz, and the ratio of the amplitude of two successive maxima is one half.

a. What is the undamped frequency f0 of this oscillator?
b. What is the resonant frequency fr?

Answer 1

Step-by-step explanation:

The ratio of amplitude of two successive maxima is
`e^{-YT_(d) }` .

`e^{-YT_(d) } = (1)/(2)`

by taking log we get
`Y = (1)/(T_(d) ) ln2 = f_(d) ln2` ∵
`f_(d) = (1)/(T_(d) )`

(a) we know that

`W_(o) = (W_(d)^(2) + Y^(2))^{(1)/(2) }`

or
`f_(o) = [f_(d)^(2) + ((Y)/(2\pi ))^(2) ]^{(1)/(2) }` ∴ W= 2πf

`f_(o) = f_(d)[ 1+ ((ln2)/(2\pi ))^(2)]^{(1)/(2) }` ∵
`Y = f_(d)ln2`

we got
`f_(0) = 100.6 Hz`

(b)

we know

`(W_(d)^(2) - Y^(2))^{(1)/(2) } = W_(r)`

or
`f_(r) ^(2) = [ f_(d) ^(2) -((Y)/(2\pi ) )^(2)]} ^{(1)/(2) }` ∵ W =2πf

`= f_(d)[ 1- ((ln2)/(2\pi ))^(2)]^{(1)/(2) }`

`f_(r) = 99.4 Hz`