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Harmonic oscillator

From Wikipedia, the free encyclopedia
Physical system that responds to a restoring force inversely proportional to displacement
This article is about the harmonic oscillator in classical mechanics. For its uses inquantum mechanics, seequantum harmonic oscillator.

Part of a series on
Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}

Inclassical mechanics, aharmonic oscillator is a system that, when displaced from itsequilibrium position, experiences arestoring forceFproportional to thedisplacementx:F→=−kx→,{\displaystyle {\vec {F}}=-k{\vec {x}},}wherek is apositiveconstant.

The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such asclocks and radio circuits.

IfF is the only force acting on the system, the system is called asimple harmonic oscillator, and it undergoessimple harmonic motion:sinusoidaloscillations about theequilibrium point, with a constantamplitude and a constantfrequency (which does not depend on the amplitude).

If a frictional force (damping) proportional to thevelocity is also present, the harmonic oscillator is described as adamped oscillator. Depending on the friction coefficient, the system can:

  • Oscillate with a frequency lower than in theundamped case, and anamplitude decreasing with time (underdamped oscillator).
  • Decay to the equilibrium position, without oscillations (overdamped oscillator).

Theboundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is calledcritically damped.

If an external time-dependent force is present, the harmonic oscillator is described as adriven oscillator.

Mechanical examples includependulums (withsmall angles of displacement), masses connected tosprings, andacoustical systems. Otheranalogous systems include electrical harmonic oscillators such asRLC circuits. They are the source of virtually all sinusoidal vibrations and waves.

Simple harmonic oscillator

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Main article:Simple harmonic motion
Mass-spring harmonic oscillator
Simple harmonic motion

A simple harmonic oscillator is an oscillator that is neither driven nordamped. It consists of a massm, which experiences a single forceF, which pulls the mass in the direction of the pointx = 0 and depends only on the positionx of the mass and a constantk. Balance of forces (Newton's second law) for the system isF=ma=md2xdt2=mx¨=−kx.{\displaystyle F=ma=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=m{\ddot {x}}=-kx.}

Solving thisdifferential equation, we find that the motion is described by the functionx(t)=Asin⁡(ωt+φ),{\displaystyle x(t)=A\sin(\omega t+\varphi ),}whereω=km.{\displaystyle \omega ={\sqrt {\frac {k}{m}}}.}

The motion isperiodic, repeating itself in asinusoidal fashion with constant amplitudeA. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by itsperiodT=2π/ω{\displaystyle T=2\pi /\omega }, the time for a single oscillation or its frequencyf=1/T{\displaystyle f=1/T}, the number of cycles per unit time. The position at a given timet also depends on thephaseφ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the massm and the force constantk, while the amplitude and phase are determined by the starting position andvelocity.

The velocity andacceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.

The potential energy stored in a simple harmonic oscillator at positionx isU=12kx2.{\displaystyle U={\tfrac {1}{2}}kx^{2}.}

Damped harmonic oscillator

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Main articles:Mass-spring-damper model anddamping
Dependence of the system behavior on the value of the damping ratioζ
Phase portrait of damped oscillator, with increasing damping strength.
Video clip demonstrating a damped harmonic oscillator consisting of a dynamics cart between two springs. Anaccelerometer on top of the cart shows the magnitude and direction of the acceleration.

In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion. In many vibrating systems the frictional forceFf can be modeled as being proportional to the velocityv of the object:Ff = −cv, wherec is called theviscous damping coefficient.

The balance of forces (Newton's second law) for damped harmonic oscillators is then[1][2][3]F=−kx−cdxdt=md2xdt2,{\displaystyle F=-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}}=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}},}which can be rewritten into the formd2xdt2+2ζω0dxdt+ω02x=0,{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x=0,}where

Step response of a damped harmonic oscillator; curves are plotted for three values ofμ =ω1 =ω0√1 − ζ2. Time is in units of the decay timeτ = 1/(ζω0).

The value of the damping ratioζ critically determines the behavior of the system. A damped harmonic oscillator can be:

  • Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratioζ return to equilibrium more slowly.
  • Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating (although overshoot can occur if the initial velocity is nonzero). This is often desired for the damping of systems such as doors.
  • Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. Theangular frequency of the underdamped harmonic oscillator is given byω1=ω01−ζ2,{\textstyle \omega _{1}=\omega _{0}{\sqrt {1-\zeta ^{2}}},} theexponential decay of the underdamped harmonic oscillator is given byλ=ω0ζ.{\displaystyle \lambda =\omega _{0}\zeta .}

TheQ factor of a damped oscillator is defined asQ=2π×energy storedenergy lost per cycle.{\displaystyle Q=2\pi \times {\frac {\text{energy stored}}{\text{energy lost per cycle}}}.}

Q is related to the damping ratio byQ=12ζ.{\textstyle Q={\frac {1}{2\zeta }}.}

Driven harmonic oscillators

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Driven harmonic oscillators are damped oscillators further affected by an externally applied forceF(t).

Newton's second law takes the formF(t)−kx−cdxdt=md2xdt2.{\displaystyle F(t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}}=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}.}

It is usually rewritten into the formd2xdt2+2ζω0dxdt+ω02x=F(t)m.{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F(t)}{m}}.}

This equation can be solved exactly for any driving force, using the solutionsz(t) that satisfy the unforced equationd2zdt2+2ζω0dzdt+ω02z=0,{\displaystyle {\frac {\mathrm {d} ^{2}z}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} z}{\mathrm {d} t}}+\omega _{0}^{2}z=0,}

and which can be expressed as damped sinusoidal oscillations:z(t)=Ae−ζω0tsin⁡(1−ζ2ω0t+φ),{\displaystyle z(t)=Ae^{-\zeta \omega _{0}t}\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right),}in the case whereζ ≤ 1. The amplitudeA and phaseφ determine the behavior needed to match the initial conditions.

Step input

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See also:Step response

In the caseζ < 1 and a unit step input with x(0) = 0:F(t)m={ω02t≥00t<0{\displaystyle {\frac {F(t)}{m}}={\begin{cases}\omega _{0}^{2}&t\geq 0\\0&t<0\end{cases}}}the solution isx(t)=1−e−ζω0tsin⁡(1−ζ2ω0t+φ)sin⁡(φ),{\displaystyle x(t)=1-e^{-\zeta \omega _{0}t}{\frac {\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right)}{\sin(\varphi )}},}

with phaseφ given by

cos⁡φ=ζ.{\displaystyle \cos \varphi =\zeta .}

The time an oscillator needs to adapt to changed external conditions is of the orderτ = 1/(ζω0). In physics, the adaptation is calledrelaxation, andτ is called the relaxation time.

In electrical engineering, a multiple ofτ is called thesettling time, i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The termovershoot refers to the extent the response maximum exceeds final value, andundershoot refers to the extent the response falls below final value for times following the response maximum.

Sinusoidal driving force

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Steady-state variation of amplitude with relative frequencyω/ω0{\displaystyle \omega /\omega _{0}} and dampingζ{\displaystyle \zeta } of a driven harmonic oscillator. This plot is also called the harmonic oscillator spectrum or motional spectrum.

In the case of a sinusoidal driving force:d2xdt2+2ζω0dxdt+ω02x=1mF0sin⁡(ωt),{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {1}{m}}F_{0}\sin(\omega t),}whereF0{\displaystyle F_{0}} is the driving amplitude, andω{\displaystyle \omega } is the drivingfrequency for a sinusoidal driving mechanism. This type of system appears inAC-drivenRLC circuits (resistor–inductor–capacitor) and driven spring systems having internal mechanical resistance or externalair resistance.

The general solution is a sum of atransient solution that depends on initial conditions, and asteady state that is independent of initial conditions and depends only on the driving amplitudeF0{\displaystyle F_{0}}, driving frequencyω{\displaystyle \omega }, undamped angular frequencyω0{\displaystyle \omega _{0}}, and the damping ratioζ{\displaystyle \zeta }.

The steady-state solution is proportional to the driving force with an induced phase changeφ{\displaystyle \varphi }:x(t)=F0mZmωsin⁡(ωt+φ),{\displaystyle x(t)={\frac {F_{0}}{mZ_{m}\omega }}\sin(\omega t+\varphi ),}whereZm=(2ω0ζ)2+1ω2(ω02−ω2)2{\displaystyle Z_{m}={\sqrt {\left(2\omega _{0}\zeta \right)^{2}+{\frac {1}{\omega ^{2}}}(\omega _{0}^{2}-\omega ^{2})^{2}}}}is theabsolute value of theimpedance orlinear response function, andφ=arctan⁡(2ωω0ζω2−ω02)+nπ{\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi }

is thephase of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument).

For a particular driving frequency called theresonance, or resonant frequencyωr=ω01−2ζ2{\textstyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}}, the amplitude (for a givenF0{\displaystyle F_{0}}) is maximal. This resonance effect only occurs whenζ<1/2{\displaystyle \zeta <1/{\sqrt {2}}}, i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency.

The transient solutions are the same as the unforced (F0=0{\displaystyle F_{0}=0}) damped harmonic oscillator and represent the system's response to other events that occurred previously.

Parametric oscillators

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Main article:Parametric oscillator

Aparametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force.A familiar example of parametric oscillation is "pumping" on a playgroundswing.[4][5][6]A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequencyω{\displaystyle \omega } and dampingβ{\displaystyle \beta }.

Parametric oscillators are used in many applications. The classicalvaractor parametric oscillator oscillates when thediode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics,waveguide/YAG based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.

Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise is minimal, since a reactance (not a resistance) is varied. Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. For example, theOptical parametric oscillator converts an inputlaser wave into two output waves of lower frequency (ωs,ωi{\displaystyle \omega _{s},\omega _{i}}).

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits theinstability phenomenon.

Universal oscillator equation

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The equationd2qdτ2+2ζdqdτ+q=0{\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=0}is known as theuniversal oscillator equation, since all second-order linear oscillatory systems can be reduced to this form.[citation needed] This is done throughnondimensionalization.

If the forcing function isf(t) = cos(ωt) = cos(ωtcτ) = cos(ωτ), whereω =ωtc, the equation becomesd2qdτ2+2ζdqdτ+q=cos⁡(ωτ).{\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=\cos(\omega \tau ).}

The solution to this differential equation contains two parts: the "transient" and the "steady-state".

Transient solution

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The solution based on solving theordinary differential equation is for arbitrary constantsc1 andc2

qt(τ)={e−ζτ(c1eτζ2−1+c2e−τζ2−1)ζ>1 (overdamping)e−ζτ(c1+c2τ)=e−τ(c1+c2τ)ζ=1 (critical damping)e−ζτ[c1cos⁡(1−ζ2τ)+c2sin⁡(1−ζ2τ)]ζ<1 (underdamping){\displaystyle q_{t}(\tau )={\begin{cases}e^{-\zeta \tau }\left(c_{1}e^{\tau {\sqrt {\zeta ^{2}-1}}}+c_{2}e^{-\tau {\sqrt {\zeta ^{2}-1}}}\right)&\zeta >1{\text{ (overdamping)}}\\e^{-\zeta \tau }(c_{1}+c_{2}\tau )=e^{-\tau }(c_{1}+c_{2}\tau )&\zeta =1{\text{ (critical damping)}}\\e^{-\zeta \tau }\left[c_{1}\cos \left({\sqrt {1-\zeta ^{2}}}\tau \right)+c_{2}\sin \left({\sqrt {1-\zeta ^{2}}}\tau \right)\right]&\zeta <1{\text{ (underdamping)}}\end{cases}}}

The transient solution is independent of the forcing function.

Steady-state solution

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Apply the "complex variables method" by solving theauxiliary equation below and then finding the real part of its solution:d2qdτ2+2ζdqdτ+q=cos⁡(ωτ)+isin⁡(ωτ)=eiωτ.{\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=\cos(\omega \tau )+i\sin(\omega \tau )=e^{i\omega \tau }.}

Supposing the solution is of the formqs(τ)=Aei(ωτ+φ).{\displaystyle q_{s}(\tau )=Ae^{i(\omega \tau +\varphi )}.}

Its derivatives from zeroth to second order areqs=Aei(ωτ+φ),dqsdτ=iωAei(ωτ+φ),d2qsdτ2=−ω2Aei(ωτ+φ).{\displaystyle q_{s}=Ae^{i(\omega \tau +\varphi )},\quad {\frac {\mathrm {d} q_{s}}{\mathrm {d} \tau }}=i\omega Ae^{i(\omega \tau +\varphi )},\quad {\frac {\mathrm {d} ^{2}q_{s}}{\mathrm {d} \tau ^{2}}}=-\omega ^{2}Ae^{i(\omega \tau +\varphi )}.}

Substituting these quantities into the differential equation gives−ω2Aei(ωτ+φ)+2ζiωAei(ωτ+φ)+Aei(ωτ+φ)=(−ω2A+2ζiωA+A)ei(ωτ+φ)=eiωτ.{\displaystyle -\omega ^{2}Ae^{i(\omega \tau +\varphi )}+2\zeta i\omega Ae^{i(\omega \tau +\varphi )}+Ae^{i(\omega \tau +\varphi )}=(-\omega ^{2}A+2\zeta i\omega A+A)e^{i(\omega \tau +\varphi )}=e^{i\omega \tau }.}

Dividing by the exponential term on the left results in−ω2A+2ζiωA+A=e−iφ=cos⁡φ−isin⁡φ.{\displaystyle -\omega ^{2}A+2\zeta i\omega A+A=e^{-i\varphi }=\cos \varphi -i\sin \varphi .}

Equating the real and imaginary parts results in two independent equationsA(1−ω2)=cos⁡φ,2ζωA=−sin⁡φ.{\displaystyle A(1-\omega ^{2})=\cos \varphi ,\quad 2\zeta \omega A=-\sin \varphi .}

Amplitude part

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Bode plot of the frequency response of an ideal harmonic oscillator

Squaring both equations and adding them together givesA2(1−ω2)2=cos2⁡φ(2ζωA)2=sin2⁡φ}⇒A2[(1−ω2)2+(2ζω)2]=1.{\displaystyle \left.{\begin{aligned}A^{2}(1-\omega ^{2})^{2}&=\cos ^{2}\varphi \\(2\zeta \omega A)^{2}&=\sin ^{2}\varphi \end{aligned}}\right\}\Rightarrow A^{2}[(1-\omega ^{2})^{2}+(2\zeta \omega )^{2}]=1.}

Therefore,A=A(ζ,ω)=sgn⁡(−sin⁡φ2ζω)1(1−ω2)2+(2ζω)2.{\displaystyle A=A(\zeta ,\omega )=\operatorname {sgn} \left({\frac {-\sin \varphi }{2\zeta \omega }}\right){\frac {1}{\sqrt {(1-\omega ^{2})^{2}+(2\zeta \omega )^{2}}}}.}

Compare this result with the theory section onresonance, as well as the "magnitude part" of theRLC circuit. This amplitude function is particularly important in the analysis and understanding of thefrequency response of second-order systems.

Phase part

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To solve forφ, divide both equations to gettan⁡φ=−2ζω1−ω2=2ζωω2−1  ⟹  φ≡φ(ζ,ω)=arctan⁡(2ζωω2−1)+nπ.{\displaystyle \tan \varphi =-{\frac {2\zeta \omega }{1-\omega ^{2}}}={\frac {2\zeta \omega }{\omega ^{2}-1}}~~\implies ~~\varphi \equiv \varphi (\zeta ,\omega )=\arctan \left({\frac {2\zeta \omega }{\omega ^{2}-1}}\right)+n\pi .}

This phase function is particularly important in the analysis and understanding of thefrequency response of second-order systems.

Full solution

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Combining the amplitude and phase portions results in the steady-state solutionqs(τ)=A(ζ,ω)cos⁡(ωτ+φ(ζ,ω))=Acos⁡(ωτ+φ).{\displaystyle q_{s}(\tau )=A(\zeta ,\omega )\cos(\omega \tau +\varphi (\zeta ,\omega ))=A\cos(\omega \tau +\varphi ).}

The solution of original universal oscillator equation is asuperposition (sum) of the transient and steady-state solutions:q(τ)=qt(τ)+qs(τ).{\displaystyle q(\tau )=q_{t}(\tau )+q_{s}(\tau ).}

Equivalent systems

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Main article:System equivalence

Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (seeuniversal oscillator equation above). Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators – their output waveform, resonant frequency, damping factor, etc. – are the same.

Translational mechanicalRotational mechanicalSeries RLC circuitParallel RLC circuit
Positionx{\displaystyle x}Angleθ{\displaystyle \theta }Chargeq{\displaystyle q}Flux linkageφ{\displaystyle \varphi }
Velocitydxdt{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}}Angular velocitydθdt{\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t}}}Currentdqdt{\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}}Voltagedφdt{\displaystyle {\frac {\mathrm {d} \varphi }{\mathrm {d} t}}}
Massm{\displaystyle m}Moment of inertiaI{\displaystyle I}InductanceL{\displaystyle L}CapacitanceC{\displaystyle C}
Momentump{\displaystyle p}Angular momentumL{\displaystyle L}Flux linkageφ{\displaystyle \varphi }Chargeq{\displaystyle q}
Spring constantk{\displaystyle k}Torsion constantμ{\displaystyle \mu }Elastance1/C{\displaystyle 1/C}Magnetic reluctance1/L{\displaystyle 1/L}
Dampingc{\displaystyle c}Rotational frictionΓ{\displaystyle \Gamma }ResistanceR{\displaystyle R}ConductanceG=1/R{\displaystyle G=1/R}
DriveforceF(t){\displaystyle F(t)}Drivetorqueτ(t){\displaystyle \tau (t)}Voltagev{\displaystyle v}Currenti{\displaystyle i}
Undampedresonant frequencyfn{\displaystyle f_{n}}:
12πkm{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}12πμI{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {\mu }{I}}}}12π1LC{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {1}{LC}}}}12π1LC{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {1}{LC}}}}
Damping ratioζ{\displaystyle \zeta }:
c21km{\displaystyle {\frac {c}{2}}{\sqrt {\frac {1}{km}}}}Γ21Iμ{\displaystyle {\frac {\Gamma }{2}}{\sqrt {\frac {1}{I\mu }}}}R2CL{\displaystyle {\frac {R}{2}}{\sqrt {\frac {C}{L}}}}G2LC{\displaystyle {\frac {G}{2}}{\sqrt {\frac {L}{C}}}}
Differential equation:
mx¨+cx˙+kx=F{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=F}Iθ¨+Γθ˙+μθ=τ{\displaystyle I{\ddot {\theta }}+\Gamma {\dot {\theta }}+\mu \theta =\tau }Lq¨+Rq˙+q/C=v{\displaystyle L{\ddot {q}}+R{\dot {q}}+q/C=v}Cφ¨+Gφ˙+φ/L=i{\displaystyle C{\ddot {\varphi }}+G{\dot {\varphi }}+\varphi /L=i}

Application to a conservative force

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The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of anyconservative force, in the limit of small motions, behaves as a simple harmonic oscillator.

A conservative force is one that is associated with apotential energy. The potential-energy function of a harmonic oscillator isV(x)=12kx2.{\displaystyle V(x)={\tfrac {1}{2}}kx^{2}.}

Given an arbitrary potential-energy functionV(x){\displaystyle V(x)}, one can do aTaylor expansion in terms ofx{\displaystyle x} around an energy minimum (x=x0{\displaystyle x=x_{0}}) to model the behavior of small perturbations from equilibrium.

V(x)=V(x0)+V′(x0)⋅(x−x0)+12V″(x0)⋅(x−x0)2+O(x−x0)3.{\displaystyle V(x)=V(x_{0})+V'(x_{0})\cdot (x-x_{0})+{\tfrac {1}{2}}V''(x_{0})\cdot (x-x_{0})^{2}+O(x-x_{0})^{3}.}

BecauseV(x0){\displaystyle V(x_{0})} is a minimum, the first derivative evaluated atx0{\displaystyle x_{0}} must be zero, so the linear term drops out:V(x)=V(x0)+12V″(x0)⋅(x−x0)2+O(x−x0)3.{\displaystyle V(x)=V(x_{0})+{\tfrac {1}{2}}V''(x_{0})\cdot (x-x_{0})^{2}+O(x-x_{0})^{3}.}

Theconstant termV(x0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:V(x)≈12V″(0)⋅x2=12kx2.{\displaystyle V(x)\approx {\tfrac {1}{2}}V''(0)\cdot x^{2}={\tfrac {1}{2}}kx^{2}.}

Thus, given an arbitrary potential-energy functionV(x){\displaystyle V(x)} with a non-vanishingsecond derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.

Examples

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Simple pendulum

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Asimple pendulum exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude.

Assuming no damping, the differential equation governing a simple pendulum of lengthl{\displaystyle l}, whereg{\displaystyle g} is the localacceleration of gravity, isd2θdt2+glsin⁡θ=0.{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{l}}\sin \theta =0.}

If the maximal displacement of the pendulum is small, we can use the approximationsin⁡θ≈θ{\displaystyle \sin \theta \approx \theta } and instead consider the equationd2θdt2+glθ=0.{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{l}}\theta =0.}

The general solution to this differential equation isθ(t)=Acos⁡(glt+φ),{\displaystyle \theta (t)=A\cos \left({\sqrt {\frac {g}{l}}}t+\varphi \right),}whereA{\displaystyle A} andφ{\displaystyle \varphi } are constants that depend on the initial conditions.Using as initial conditionsθ(0)=θ0{\displaystyle \theta (0)=\theta _{0}} andθ˙(0)=0{\displaystyle {\dot {\theta }}(0)=0}, the solution is given byθ(t)=θ0cos⁡(glt),{\displaystyle \theta (t)=\theta _{0}\cos \left({\sqrt {\frac {g}{l}}}t\right),}whereθ0{\displaystyle \theta _{0}} is the largest angle attained by the pendulum (that is,θ0{\displaystyle \theta _{0}} is the amplitude of the pendulum). Theperiod, the time for one complete oscillation, is given by the expressionτ=2πlg=2πω,{\displaystyle \tau =2\pi {\sqrt {\frac {l}{g}}}={\frac {2\pi }{\omega }},}which is a good approximation of the actual period whenθ0{\displaystyle \theta _{0}} is small. Notice that in this approximation the periodτ{\displaystyle \tau } is independent of the amplitudeθ0{\displaystyle \theta _{0}}. In the above equation,ω{\displaystyle \omega } represents the angular frequency.

Spring/mass system

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Spring–mass system in equilibrium (A), compressed (B) and stretched (C) states

When a spring is stretched or compressed by a mass, the spring develops a restoring force.Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:F(t)=−kx(t),{\displaystyle F(t)=-kx(t),}whereF is the force,k is the spring constant, andx is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from flying off to infinity.

By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:F(t)=−kx(t)=md2dt2x(t)=ma,{\displaystyle F(t)=-kx(t)=m{\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}x(t)=ma,}the latter beingNewton's second law of motion.

If the initial displacement isA, and there is no initial velocity, the solution of this equation is given byx(t)=Acos⁡(kmt).{\displaystyle x(t)=A\cos \left({\sqrt {\frac {k}{m}}}t\right).}

Given an ideal massless spring,m{\displaystyle m} is the mass on the end of the spring. If the spring itself has mass, itseffective mass must be included inm{\displaystyle m}.

Energy variation in the spring–damping system

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In terms of energy, all systems have two types of energy:potential energy andkinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which is then transferred into kinetic energy. The potential energy within a spring is determined by the equationU=12kx2.{\textstyle U={\frac {1}{2}}kx^{2}.}

When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. Byconservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.

Definition of terms

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SymbolDefinitionDimensionsSI units
a{\displaystyle a}Acceleration of massLT−2{\displaystyle {\mathsf {LT^{-2}}}}m/s2
A{\displaystyle A}Peak amplitude of oscillationL{\displaystyle {\mathsf {L}}}m
c{\displaystyle c}Viscous damping coefficientMT−1{\displaystyle {\mathsf {MT^{-1}}}}N·s/m
f{\displaystyle f}FrequencyT−1{\displaystyle {\mathsf {T^{-1}}}}Hz
F{\displaystyle F}Drive forceMLT−2{\displaystyle {\mathsf {MLT^{-2}}}}N
g{\displaystyle g}Acceleration of gravity at the Earth's surfaceLT−2{\displaystyle {\mathsf {LT^{-2}}}}m/s2
i{\displaystyle i}Imaginary unit,i2=−1{\displaystyle i^{2}=-1}——
k{\displaystyle k}Spring constantMT−2{\displaystyle {\mathsf {MT^{-2}}}}N/m
μ{\displaystyle \mu }Torsion Spring constantML2T−2{\displaystyle {\mathsf {ML^{2}T^{-2}}}}Nm/rad
m,M{\displaystyle m,M}MassM{\displaystyle {\mathsf {M}}}kg
Q{\displaystyle Q}Quality factor——
T{\displaystyle T}Period of oscillationT{\displaystyle {\mathsf {T}}}s
t{\displaystyle t}TimeT{\displaystyle {\mathsf {T}}}s
U{\displaystyle U}Potential energy stored in oscillatorML2T−2{\displaystyle {\mathsf {ML^{2}T^{-2}}}}J
x{\displaystyle x}Position of massL{\displaystyle {\mathsf {L}}}m
ζ{\displaystyle \zeta }Damping ratio——
φ{\displaystyle \varphi }Phase shift—rad
ω{\displaystyle \omega }Angular frequencyT−1{\displaystyle {\mathsf {T^{-1}}}}rad/s
ω0{\displaystyle \omega _{0}}Natural resonant angular frequencyT−1{\displaystyle {\mathsf {T^{-1}}}}rad/s

See also

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Notes

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  1. ^Fowles & Cassiday (1986, p. 86)
  2. ^Kreyszig (1972, p. 65)
  3. ^Tipler (1998, pp. 369, 389)
  4. ^Case, William."Two ways of driving a child's swing". Archived fromthe original on 9 December 2011. Retrieved27 November 2011.
  5. ^Case, W. B. (1996). "The pumping of a swing from the standing position".American Journal of Physics.64 (3):215–220.Bibcode:1996AmJPh..64..215C.doi:10.1119/1.18209.
  6. ^Roura, P.; Gonzalez, J.A. (2010). "Towards a more realistic description of swing pumping due to the exchange of angular momentum".European Journal of Physics.31 (5):1195–1207.Bibcode:2010EJPh...31.1195R.doi:10.1088/0143-0807/31/5/020.S2CID 122086250.

References

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External links

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