A mass-spring system has b/m = ω0/2, where b is the damping constant and ω0 the natural frequency. How does its amplitude A when driven at a frequency 12% above ω0 compare with its amplitude at resonance Ares at ω0?
The amplitude at resonance (ω₁ = ω₀) ia Ar =F₀/b•ω₀,
A/Ar=A/( F₀/b•ω₀)= { b•ω₀/m}/sqrt{( ω₁²- ω₀²)²+(b²•ω₁²/m²)}=
=1/sqrt{(m•ω₀/b)²•[(ω₁²/ω₀²)-1]²+(ω₁²/ω₀²)}.
If m•ω₀/b =2, and ω₁/ω₀ =1.12 (12% above resonance),
and ω₁²/ω₀²= 1.2544, then
A/Ar =1/sqrt{4•(1.2544 -1)²+1.2544} =
=0.81.
To compare the amplitude A when driven at a frequency 12% above ω0 with the amplitude at resonance Ares at ω0, we can use the concept of impedance in the mass-spring system.
Impedance is a measure of the opposition that a circuit presents to the flow of alternating current. In the case of a mass-spring system driven by an external force, the impedance relates the applied force to the resulting displacement or amplitude.
The impedance of a mass-spring system can be given by the formula Z = sqrt(R^2 + (ωm - ω)^2), where R is the resistance, ω is the angular frequency of the driving force, and ωm is the resonant angular frequency.
At resonance, when ω = ωm, the impedance simplifies to Zres = R.
Now, let's consider the case where the system is driven at a frequency 12% above ω0. We can denote this frequency as ωdriven = 1.12ω0.
The impedance at this driven frequency can be expressed as Zdriven = sqrt[R^2 + (ωm - ωdriven)^2].
Comparing Zres and Zdriven, we can determine the relationship between the amplitudes Ares and A:
Ares / A = Zres / Zdriven = R / sqrt[R^2 + (ωm - ωdriven)^2].
Since the given expression b/m = ω0/2 relates the damping constant b to the natural frequency ω0, we can express R in terms of ω0:
R = b/(2m) = (ω0 / 2)(1 / ω0) = 1/2.
Plugging this into the equation for the ratio of amplitudes:
Ares / A = (1/2) / sqrt[(1/2)^2 + (ωm - ωdriven)^2].
Now, we need to find ωm in terms of ω0. The resonant angular frequency ωm can be written as ωm = sqrt(ω0^2 - (b/2m)^2).
Using the given relation b/m = ω0/2, we can substitute ω0/2 for b/m:
ωm = sqrt(ω0^2 - (ω0/2)^2).
Simplifying further:
ωm = sqrt(4ω0^2/4 - ω0^2/4) = sqrt(3ω0^2/4) = ω0 * sqrt(3)/2.
Now, we can plug in the values for R and ωm into the expression for the ratio of amplitudes:
Ares / A = (1/2) / sqrt[(1/2)^2 + (ω0 * sqrt(3)/2 - 1.12ω0)^2].
Calculating this expression will give you the comparison between the amplitude A when driven at a frequency 12% above ω0 and the amplitude at resonance Ares at ω0.