The amplitude of the driving force is 0.554 N and the amplitude of the oscillato

Question

The amplitude of the driving force is 0.554 N and the amplitude of the oscillator's steady-state motion in response to this driving force is 0.812 m. What is the oscillator's damping constant?

Engineers can determine properties of a structure that is modeled as damped spring oscillator-such as a bridge-by applying a driving force to it. A weakly damped spring oscillator of mass 0.250 kg is driven by a sinusoidal force at the oscillator's resonance frequency of 33.4 Hz. Find the value of the spring constant. Number 11010.14 N/ m The amplitude of the driving force is 0.554 N and the amplitude of the oscillator's steady-state motion in response to this driving force is 0.812 m. What is the oscillator's damping constant? Number kg/s

Explanation / Answer

If you have a damped oscillator of mass m with natural frequency Wo and damping constant z and you are forcing the oscillator with an external driving force

F(t) = Fo sin(W t)

the response of the system is

X(t) = Fo sin(W t + phi) / ( m H W)


where

H = sqrt ( ( 2 Wo z)^2 + ( (Wo^2 - W^2) / W )^2 )

and

phi = atan ( (Wo^2 - W^2) / ( 2 W Wo z) )

If W = Wo

H = 2 Wo z

and so

X(t) = Fo sin(Wo t + phi) / ( m 2 Wo^2 z)

The peak amplitude is

Xpealk = Fo / ( m 2 Wo^2 z)

z = Fo / ( m 2 Wo^2 Xpeak)

Wo = 2 pi 33.4 Hz = 209.752 Hz

z = 0.554 N / ( 0.250 kg (209.752 Hz)^2 0.812 m)

z = 6.20 10^-5 KG/SEC

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