Harmonic oscillator
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A Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force <math>F</math> proportional to the displacement <math>x</math>:
- <math> F = -k x \, </math>
where <math>k</math> is a positive constant.
If <math>F</math> is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as damped. In such situation, the frequency of the oscillations is smaller than in the non-damped case, and the amplitude of the oscillations decreases with time.
If an external time-dependent force is present, the harmonic oscillator is described as driven.
Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators (see RLC circuit).
This article discusses the harmonic oscillator in terms of classical mechanics. See the article quantum harmonic oscillator for a discussion of the harmonic oscillator in quantum mechanics.
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Simple harmonic oscillator
The simple harmonic oscillator has no driving force, and no friction (damping), so the net force is just
- <math> F = -k x \, </math>
Using Newton's Second Law
- <math> F = m a = -k x \, </math>
The acceleration, <math>a</math> is equal to the second derivative of <math>x</math>.
- <math> m \frac{d^2x}{dt^2} = -k x </math>
If we define <math>{\omega_0}^2 = k/m</math>, then the equation can be written as follows,
- <math> \frac{d^2x}{dt^2} + {\omega_0}^2 x = 0</math>
and has the general solution
- <math> x = A \cos {(\omega_0 t + \phi)} \, </math>
where the amplitude <math>A \,</math> and the phase <math>\phi \,</math> are determined by the initial conditions.
Alternatively, the general solution can be written as
- <math> x = A \sin {(\omega_0 t + \phi)} \, </math>
where the value of <math>\phi \,</math> is shifted by <math>\pi/2 \,</math> relative to the previous form;
or as
- <math> x = A \sin{\omega_0 t} + B \cos{\omega_0 t} \, </math>
where <math>A \,</math> and <math>B \,</math> are the constants which are determined by the initial conditions, instead of <math>A \,</math> and <math>\phi \,</math> in the previous forms.
The frequency of the oscillations is given by
- <math> f = \frac{\omega_0}{2\pi} </math>
The kinetic energy is
- <math>T = \frac{1}{2} m \left(\frac{dx}{dt}\right)^2 = \frac{1}{2} k A^2 \sin^2(\omega_0 t + \phi)</math>.
and the potential energy is
- <math>U = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \cos^2(\omega_0 t + \phi)</math>
so the total energy of the system has the constant value
- <math>E = \frac{1}{2} k A^2</math>
Driven harmonic oscillator
This satisfies the nonhomogeneous second order linear differential equation
- <math>\frac{d^2x}{dt^2} + {\omega_0}^2x = A_0 \cos(\omega t).</math>
Example: AC LC (inductor-capacitor) circuit.
Note that this is true if the driving force applied is itself sinusoidal. The term on the right side of the equal sign corresponds to this applied driving force.
Damped harmonic oscillator
This satisfies the equation
- <math>\frac{d^2x}{dt^2} + b/m \frac{dx}{dt} + {\omega_0}^2x = 0.</math>
Example: weighted spring underwater (where the damping force exerted by the water is proportional to b).
Damped, driven harmonic oscillator
This satisfies the equation
- <math>m\frac{d^2x}{dt^2} + r \frac{dx}{dt} + kx= F_0 \cos(\omega t).</math>
The general solution is a sum of a transient (the solution for damped undriven harmonic oscillator, homogeneous ODE) that depends on initial conditions, and a steady state (particular solution of the unhomogenous ODE) that is independent of initial conditions and depends only on driving frequency, driving force, restoring force, damping force, and inertial moment of the oscillator (see also kernel and image).
The steady-state solution is
- <math> x(t) = \frac{F_0}{Z_m \omega} \sin(\omega t - \phi)</math>
where
- <math> Z_m = \sqrt{r^2 + \left(\omega m - \frac{k}{\omega}\right)^2}</math>
is the absolute value of the impedance
- <math> Z = r + i\left(\omega m - \frac{k}{\omega}\right) </math>
and
- <math> \phi = \arctan\left(\frac{\omega m - \frac{k}{\omega}}{r}\right)</math>
is the phase of the oscillation relative to the driving force.
One might see that for a certain driving frequency, <math> \omega </math>, the amplitude (relative to a given <math>F_0</math>) is maximal. This occurs for the frequency
- <math> {\omega}_r = \sqrt{\frac{k}{m} - \frac{r^2}{4 m^2}} </math>
and is called resonance of displacement.
In summary: at a steady state the frequency of the oscillation is the same as that of the driving force, but the oscillation is phase-offset and scaled by amounts that depend on the frequency of the driving force in relation to the preferred (resonant) frequency of the oscillating system.
Example: RLC circuit.
Full mathematical definition
Most harmonic oscillators, at least approximately, solve the differential equation:
- <math>\frac{d^2x}{dt^2} + b/m \frac{dx}{dt} + {\omega_0}^2x = A_0 \cos(\omega t) </math>
where t is time, b is the damping constant, ωo is the characteristic angular frequency, and Aocos(ωt) represents something driving the system with amplitude Ao and angular frequency ω. x is the measurement that is oscillating; it can be position, current, or nearly anything else. The angular frequency is related to the frequency, f, by
- <math> f = \frac{\omega}{2 \pi}.</math>
Important terms
- Amplitude: maximal displacement from the equilibrium.
- Period: the time it takes the system to complete an oscillation cycle. Opposite of frequency.
- Frequency: the number of cycles the system performs per unit time (usually measured in hertz = 1/s).
- Angular frequency: <math> \omega = 2 \pi f </math>
- Phase: how much of a cycle the system completed (system that begins is in phase zero, system which completed half a cycle is in phase <math> \pi </math>).
- Initial conditions: the state of the system at t = 0, the beginning of oscillations.
Simple harmonic oscillator
A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is:
- <math>\frac{d^2x}{dt^2} + {\omega_0}^2x = 0.</math>
Physically, the above never actually exists, since there will always be friction or some other resistance, but two approximate examples are a mass on a spring and an LC circuit.
In the case of a mass attached to a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that:
- <math> -k x = ma \, </math>
- where k is the spring constant
- m is the mass
- x is the position of the mass
- a is its acceleration.
Because acceleration a is the second derivative of position x, we can rewrite the equation as follows:
- <math> -k x = m \frac{d^2 x}{d t^2}. </math>
The most simple solution to the above differential equation is
- <math>x = A \cos(\omega t + \delta) \, </math>
and the second derivative of that is
- <math>\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t + \delta)</math>
- where A is the amplitude, δ is the phase shift, and ω is the angular frequency.
Plugging these back into the original differential equation, we have:
- <math> -A k \cos(\omega t +\delta) = -A m \omega^2 \cos(\omega t + \delta). \, </math>
Then, after dividing both sides by <math>-A \cos(\omega t + \delta) \, </math> we get:
- <math>k = m \omega^2 \,</math>
or, as it is more commonly written:
- <math>\omega = \sqrt{\frac{k}{m}}. </math>
The above formula reveals that the angular frequency ω of the solution is only dependent upon the physical characteristics of the system, and not the initial conditions (those are represented by A and δ). We will label this ω as ωo from now on. This will become important later.
Universal oscillator equation
The equation
- <math>\frac{d^2q}{d \tau^2} + 2 \zeta \frac{dq}{d\tau} + q = 0</math>
is known as the universal oscillator equation since all second order linear oscillatory systems can be reduced to this form. This is done through nondimensionalization.
If the forcing function is f(t) = cos(ωt) = cos(ωtcτ) = cos(ωτ), where ω = ωtc, the equation becomes
- <math>\frac{d^2q}{d \tau^2} + 2 \zeta \frac{dq}{d\tau} + q = \cos(\omega \tau).</math>
The solution to this differential equation contains two parts, the "transient" and the "steady state".
Transient solution
The solution based on solving the ordinary differential equation is for arbitrary constants c1 and c2 is
<math>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 \ \mbox{(overdamping)} \\ e^{-\zeta\tau} (c_1+c_2 \tau) = e^{-\tau}(c_1+c_2 \tau) & \zeta = 1 \ \mbox{(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 \ \mbox{(underdamping)} \end{cases}</math>
The transient solution is independent of the forcing function. If the system is critically damped, the response is independent of the damping.
Steady-state solution
Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution:
- <math>\frac{d^2 q}{d\tau^2} + 2 \zeta \frac{dq}{d\tau} + q = \cos(\omega \tau) + i\sin(\omega \tau) = e^{ i \omega \tau} .</math>
Supposing the solution is of the form
- <math>\,\! q_s(\tau) = A e^{i ( \omega \tau + \phi ) } . </math>
Its derivatives from zero to 2nd order are
- <math>q_s = A e^{i ( \omega \tau + \phi ) }, \ \frac{dq_s}{d \tau} = i \omega A e^{i ( \omega \tau + \phi ) }, \ \frac{d^2 q_s}{d \tau^2} = - \omega^2 A e^{i ( \omega \tau + \phi ) } .</math>
Substituting these quantities into the differential equation gives
- <math>\,\! -\omega^2 A e^{i (\omega \tau + \phi)} + 2 \zeta i \omega A e^{i(\omega \tau + \phi)} + A e^{i(\omega \tau + \phi)} = (-\omega^2 A \, + \, 2 \zeta i \omega A \, + \, A) e^{i (\omega \tau + \phi)} = e^{i \omega \tau} .</math>
Dividing by the exponential term on the left results in
- <math>\,\! -\omega^2 A + 2 \zeta i \omega A + A = e^{-i \phi} = \cos\phi - i \sin\phi . </math>
Equating the real and imaginary parts results in two independent equations
- <math>A (1-\omega^2)=\cos\phi \qquad 2 \zeta \omega A = - \sin\phi.</math>
Amplitude part
Squaring both equations and adding them together gives
- <math>\left . \begin{matrix}A^2 (1-\omega^2)^2 = \cos^2\phi \\ (2 \zeta \omega A)^2 = \sin^2\phi \end{matrix} \right \} \Rightarrow A^2[(1-\omega^2)^2 + (2 \zeta \omega)^2] = 1. </math>
By convention the positive root is taken since amplitude is usually considered a positive quantity. Therefore,
- <math>A = A( \zeta, \omega) = \frac{1}{\sqrt{(1-\omega^2)^2 + (2 \zeta \omega)^2}}.</math>
Compare this result with the theory section on resonance, as well as the "magnitude part" of the RLC circuit. This amplitude function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Note that the variables in these equations ought to be identified before showing the equation.
Phase part
To solve for φ, divide both equations to get
- <math>\tan\phi = - \frac{2 \zeta \omega}{ 1 - \omega^2} = \frac{2 \zeta \omega}{\omega^2 - 1} \Rightarrow \phi \equiv \phi(\zeta, \omega) = \arctan \left( \frac{2 \zeta \omega}{\omega^2 - 1} \right ). </math>
This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Full solution
Combining the amplitude and phase portions results in the steady-state solution
- <math>\,\! q_s (\tau) = A(\zeta,\omega) \cos(\omega \tau + \phi(\zeta,\omega)) = A\cos(\omega \tau + \phi).</math>
The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions
- <math>\,\! q(\tau) = q_t (\tau) + q_s (\tau).</math>
For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients.
Relationship to RLC circuit
Comparing a mechanical harmonic oscillator with an RLC circuit, the following correspond:
- F (force) <math>\Leftrightarrow</math> V (electric potential)
- x (position) <math>\Leftrightarrow</math> Q (charge)
- k (spring constant) <math>\Leftrightarrow \frac{1}{C}</math> (electrical elastance (reciprocal of capacitance)
- v (velocity) <math>\Leftrightarrow</math> I (electric current)
- b (damping factor) <math>\Leftrightarrow</math> R (electrical resistance)
- a (acceleration) <math>\Leftrightarrow \frac{dI}{dt} \,</math> (rate of change of current)
- m (mass) <math>\Leftrightarrow</math> L (inductance)
Applications
The problem of the simple harmonic oscillator occurs frequently in physics because of the form of its potential energy function:
- <math>V(x) = \frac{1}{2} k x^2. </math>
Given an arbitrary potential energy function <math>V(x)</math>, one can do a Taylor expansion in terms of <math>x</math> around an energy minimum (<math>x = x_0</math>) to model the behavior of small perturbations from equilibrium.
- <math>V(x) = V(x_0) + (x-x_0) V'(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3</math>
Because <math>V(x_0)</math> is a minimum, the first derivative evaluated at <math>x_0</math> must be zero, so the linear term drops out:
- <math>V(x) = V(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3</math>
The constant term is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
- <math>V(x) \approx \frac{1}{2} x^2 V^{(2)}(0) = \frac{1}{2} k x^2</math>
Thus, given an arbitrary potential energy function <math>V(x)</math> with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.
Examples
Simple Pendulum
Image:Simple pendulum height.jpg
Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is given by
- <math>{d^2\theta\over dt^2}+{g\over \ell}\theta=0</math>
Solution to this equation is given by:
- <math>\theta(t) = \theta_0\cos\left(\sqrt{g\over \ell}t\right) \quad\quad\quad\quad |\theta_0| \ll 1</math>
where <math>\theta_0</math> is the largest angle attained by the pendulum. Period, the time for one complete oscillation (time for the bob to return to its starting position), is given by <math>2\pi</math> divided by whatever is multiplying the time in the argument of the cosine
- <math>T_0 = 2\pi\sqrt{\ell\over g}\quad\quad\quad\quad |\theta_0| \ll 1</math>
Pendulum swinging over turntable
Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion. Consider a long pendulum swinging over the turntable of a record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line. It is possible to change the frequency of rotation of the turntable in order to have a perfect synchronization with the motion of the pendulum.
The angular speed of the turntable is the pulsation of the pendulum.
In general, the pulsation-also known as angular frequency, of a straight-line simple harmonic motion is the angular speed of the corresponding circular motion.
Therefore, a motion with period T and frequency f=1/T has pulsation
<math>\omega=2\pi\cdot f = \frac{2\pi}{T}</math>
In general, pulsation and angular speed are not synonymous. For instance the pulsation of a pendulum is not the angular speed of the pendulum itself, but it is the angular speed of the corresponding circular motion.
Spring-mass system
When a spring is stretched or compressed by a mass, the spring develops a restoring force. The Hooke's Law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length.
<math>Fs \left( t \right) =kx \left( t \right) </math>
where Fs is the force, k is the spring constant, and the x is the displacement of the mass with respect to the equilibrium position.
This relationship shows that the distance of the spring is always opposite to the force of the spring.
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:
<math> m \frac {d^{2}}{d{t}^{2}} x \left( t \right) +kx(t)=0 </math>
If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by: <math> x \left( t \right) =A\cos \left( (\sqrt {k/m}) t\right) </math>
- Energy variation in the spring-damper system
In terms of energy, all systems have two types of energy, potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which then is transferred into kinetic energy. The potential energy within a spring is determined by the equation <math> U = 1/2\,k{x}^{2} </math>
When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximum potential energy, the kinetic energy of the mass is zero. When the spring is released, the spring will try to reach back to equilibrium, and all its potential energy is converted into kinetic energy of the mass.
References
See also
de:Harmonischer Oszillator fi:Harmoninen värähtelijä he:אוסצילטור הרמוני it:moto armonico ja:調和振動子 pl:Oscylator harmoniczny sl:Nihanje