3.S: Linear Oscillators (Summary)
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Linear systems have the feature that the solutions obey the Principle of Superposition, that is, the amplitudes add linearly for the superposition of different oscillatory modes. Applicability of the Principle of Superposition to a system provides a tremendous advantage for handling and solving the equations of motion of oscillatory systems.
Geometric representations of the motion of dynamical systems provide sensitive probes of periodic motion. Configuration space (q,q,t), state space (q,˙q,t) and phase space (q,p,t), are powerful geometric representations that are used extensively for recognizing periodic motion where q, ˙q, and p are vectors in n-dimensional space.
Linearly-damped free linear oscillator
The free linearly-damped linear oscillator is characterized by the equation
¨x+Γ˙x+ω20x=0
The solutions of the linearly-damped free linear oscillator are of the form
z=e−(Γ2)t[z1eiω1t+z2e−iω1t]ω1≡√ω2o−(Γ2)2
The solutions fall into three categories
x(t)=Ae−(Γ2)tcos(ω1t−β) | underdamped | ω1=√ω2o−(Γ2)2>0 |
x(t)=[A1e−ω+t+A2e−ω−t] | overdamped | ω±=−[−Γ2±√(Γ2)2−ω2o] |
x(t)=(A+Bt)e−(Γ2)t | critically damped | ω1=√ω2o−(Γ2)2=0 |
The energy dissipation for the linearly-damped free linear oscillator time averaged over one period is given by
⟨E⟩=E0e−Γt
The quality factor Q characterizing the damping of the free oscillator is define to be
Q=EΔE=ω1Γ
where ΔE is the energy dissipated per radian.
Sinusoidally-driven, linearly-damped, linear oscillator
The linearly-damped linear oscillator, driven by a harmonic driving force, is of considerable importance to all branches of physics, and engineering. The equation of motion can be written as
¨x+Γ˙x+ω20x=F(t)m
where F(t) is the driving force. The complete solution of this second-order differential equation comprises two components, the complementary solution (transient response), and the particular solution (steady-state response). That is,
x(t)Total=x(t)T+x(t)S
For the underdamped case, the transient solution is the complementary solution
x(t)T=F0me−Γ2tcos(ω1t−δ)
and the steady-state solution is given by the particular solution
x(t)S=F0m√(ω20−ω2)2+(Γω)2cos(ωt−δ)
Resonance
A detailed discussion of resonance and energy absorption for the driven linearly-damped linear oscillator was given. For resonance the maximum amplitudes occur at frequencies
Resonant system | Resonant frequency |
undamped free linear oscillator | ω0=√km |
linearly-damped free linear oscillator | ω1=√ω20−(Γ2)2 |
driven linearly-damped linear oscillator | ωR=√ω20−2(Γ2)2 |
The energy absorption for the steady-state solution for resonance is given by
x(t)S=Aelcosωt+Aabssinωt
where the elastic amplitude
Ael=F0m(ω20−ω2)2+(Γω)2(ω20−ω2)
while the absorptive amplitude
Aabs=F0m(ω20−ω2)2+(Γω)2Γω
The time average power input is given by only the absorptive term
⟨P⟩=12F0ωAabs=F202mΓω2(ω20−ω2)2+(Γω)2
This power curve has the classic Lorentzian shape.
Wave propagation
The wave equation was introduced and both travelling and standing wave solutions of the wave equation were discussed. Harmonic wave-form analysis, and the complementary time-sampled wave form analysis techniques, were introduced in this chapter and in appendix 19.9. The relative merits of Fourier analysis and the digital Green’s function waveform analysis were illustrated for signal processing.
The concepts of phase velocity, group velocity, and signal velocity were introduced. The phase velocity is given by
vphase=ωk
and group velocity
vgroup=(dωdk)k0=vphase+k∂vphase∂k
If the group velocity is frequency dependent then the information content of a wave packet travels at the signal velocity which can differ from the group velocity.
The Wave-packet Uncertainty Principle implies that making a precise measurement of the frequency of a sinusoidal wave requires that the wave packet be infinitely long. The standard deviation σ(t)=√⟨t2⟩−⟨t⟩2 characterizing the width of the amplitude of the wavepacket spectral distribution in the angular frequency domain, σA(ω), and the corresponding width in time σA(t), are related by :
σA(t)⋅σA(ω)⩾1
The standard deviations for the spectral distribution and width of the intensity of the wave packet are related by:
σI(t)⋅σI(ω)⩾12σI(x)⋅σI(kx)⩾12σI(y)⋅σI(ky)⩾12σI(z)⋅σI(kz)⩾12
This applies to all forms of wave motion, including sound waves, water waves, electromagnetic waves, or matter waves.