The steady-state amplitude of a weakly damped forced oscillator is given by
2 2 2 2 2 2/1
0
0
[( ) 4 ]
( )
ω −ω + ω
ω =
b
f
a
Depict it as a function of frequency and obtain expression for resonance frequency.
Also obtain expression for peak value of steady-state amplitude.
To depict the steady-state amplitude of a weakly damped forced oscillator as a function of frequency, we need to plot the equation given.
The equation for the steady-state amplitude of the weakly damped forced oscillator is:
A = (F0/m) / √((ω0^2 - ω^2)^2 + (2βω)^2)
Where:
A is the steady-state amplitude
F0 is the amplitude of the driving force
m is the mass of the oscillator
ω0 is the natural frequency of the oscillator
ω is the frequency of the driving force
β is the damping coefficient
To depict it as a function of frequency, you need to choose values for F0, m, ω0, and β. Then, plot the steady-state amplitude (A) on the y-axis against the frequency (ω) on the x-axis. You can vary the frequency over a range and observe how the amplitude changes.
Now, let's obtain the expression for the resonance frequency. The resonance frequency occurs when the steady-state amplitude is maximum. At resonance, the denominator of the equation becomes minimum. This happens when the term inside the square root is equal to zero:
(ω0^2 - ω^2)^2 + (2βω)^2 = 0
Simplifying this equation will give you the expression for the resonance frequency (ωr). Solve for ω:
(ω0^2 - ωr^2)^2 + (2βωr)^2 = 0
Now, to obtain the expression for the peak value of the steady-state amplitude, substitute the resonance frequency (ωr) into the equation for the steady-state amplitude (A). This will give you the maximum amplitude value.
A = (F0/m) / √((ω0^2 - ωr^2)^2 + (2βωr)^2)
By substituting the resonance frequency (ωr) into this equation, you can find the peak value of the steady-state amplitude.