Harmonic oscillator

In the case ζ < 1 and a unit step input with x(0) = 0:

${\displaystyle {\frac {F(t)}{m}}={\begin{cases}\omega _{0}^{2}&t\geq 0\\0&t<0\end{cases}}}$

the solution is

${\displaystyle x(t)=1-\mathrm {e} ^{-\zeta \omega _{0}t}{\frac {\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right)}{\sin(\varphi )}},}$

with phase φ given by

${\displaystyle \cos \varphi =\zeta .}$

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

In electrical engineering, a multiple of τ is called the settling time, i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term overshoot refers to the extent the response maximum exceeds final value, and undershoot refers to the extent the response falls below final value for times following the response maximum.

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

In the case of a sinusoidal driving force:

${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {1}{m}}F_{0}\sin(\omega t),}$

where ${\displaystyle F_{0}}$ is the driving amplitude, and ${\displaystyle \omega }$ is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC-driven RLC circuits (resistor–inductor–capacitor) and driven spring systems having internal mechanical resistance or external air resistance.

The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude ${\displaystyle F_{0}}$, driving frequency ${\displaystyle \omega }$, undamped angular frequency ${\displaystyle \omega _{0}}$, and the damping ratio ${\displaystyle \zeta }$.

The steady-state solution is proportional to the driving force with an induced phase change ${\displaystyle \varphi }$:

${\displaystyle x(t)={\frac {F_{0}}{mZ_{m}\omega }}\sin(\omega t+\varphi ),}$

where

is the absolute value of the impedance or linear response function, and

${\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi }$

is the phase 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 the resonance, or resonant frequency ${\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}}$, the amplitude (for a given ${\displaystyle F_{0}}$) is maximal. This resonance effect only occurs when ${\displaystyle \zeta <1/{\sqrt {2}}}$, 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 (${\displaystyle F_{0}=0}$) damped harmonic oscillator and represent the systems response to other events that occurred previously. The transient solutions typically die out rapidly enough that they can be ignored.

The solution based on solving the ordinary differential equation is for arbitrary constants c₁ and c₂

${\displaystyle q_{t}(\tau )={\begin{cases}\mathrm {e} ^{-\zeta \tau }\left(c_{1}\mathrm {e} ^{\tau {\sqrt {\zeta ^{2}-1}}}+c_{2}\mathrm {e} ^{-\tau {\sqrt {\zeta ^{2}-1}}}\right)&\zeta >1{\text{ (overdamping)}}\\\mathrm {e} ^{-\zeta \tau }(c_{1}+c_{2}\tau )=\mathrm {e} ^{-\tau }(c_{1}+c_{2}\tau )&\zeta =1{\text{ (critical damping)}}\\\mathrm {e} ^{-\zeta \tau }\left[c_{1}\cos \left({\sqrt {1-\zeta ^{2}}}\tau \right)+c_{2}\sin \left({\sqrt {1-\zeta ^{2}}}\tau \right)\right]&\zeta <1{\text{ (underdamping)}}\end{cases}}}$

The transient solution is independent of the forcing function.

Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution:

${\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=\cos(\omega \tau )+\mathrm {i} \sin(\omega \tau )=\mathrm {e} ^{\mathrm {i} \omega \tau }.}$

Supposing the solution is of the form

${\displaystyle q_{s}(\tau )=A\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )}.}$

Its derivatives from zeroth to second order are

${\displaystyle q_{s}=A\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )},\quad {\frac {\mathrm {d} q_{s}}{\mathrm {d} \tau }}=\mathrm {i} \omega A\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )},\quad {\frac {\mathrm {d} ^{2}q_{s}}{\mathrm {d} \tau ^{2}}}=-\omega ^{2}A\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )}.}$

Substituting these quantities into the differential equation gives

${\displaystyle -\omega ^{2}A\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )}+2\zeta \mathrm {i} \omega A\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )}+A\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )}=(-\omega ^{2}A+2\zeta \mathrm {i} \omega A+A)\mathrm {e} ^{\mathrm {i} (\omega \tau +\varphi )}=\mathrm {e} ^{\mathrm {i} \omega \tau }.}$

Dividing by the exponential term on the left results in

${\displaystyle -\omega ^{2}A+2\zeta \mathrm {i} \omega A+A=\mathrm {e} ^{-\mathrm {i} \varphi }=\cos \varphi -\mathrm {i} \sin \varphi .}$

Equating the real and imaginary parts results in two independent equations

${\displaystyle A(1-\omega ^{2})=\cos \varphi ,\quad 2\zeta \omega A=-\sin \varphi .}$

Bode plot of the frequency response of an ideal harmonic oscillator

Squaring both equations and adding them together gives

${\displaystyle \left.{\begin{aligned}A^{2}(1-\omega ^{2})^{2}&=\cos ^{2}\varphi \\(2\zeta \omega A)^{2}&=\sin ^{2}\varphi \end{aligned}}\right\}\Rightarrow A^{2}[(1-\omega ^{2})^{2}+(2\zeta \omega )^{2}]=1.}$

Therefore,

${\displaystyle A=A(\zeta ,\omega )=\operatorname {sign} \left({\frac {-\sin \varphi }{2\zeta \omega }}\right){\frac {1}{\sqrt {(1-\omega ^{2})^{2}+(2\zeta \omega )^{2}}}}.}$

Compare this result with the theory section 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.

To solve for φ, divide both equations to get

${\displaystyle \tan \varphi =-{\frac {2\zeta \omega }{1-\omega ^{2}}}={\frac {2\zeta \omega }{\omega ^{2}-1}}\Rightarrow \varphi \equiv \varphi (\zeta ,\omega )=\arctan \left({\frac {2\zeta \omega }{\omega ^{2}-1}}\right)+n\pi .}$

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

Combining the amplitude and phase portions results in the steady-state solution

${\displaystyle q_{s}(\tau )=A(\zeta ,\omega )\cos(\omega \tau +\varphi (\zeta ,\omega ))=A\cos(\omega \tau +\varphi ).}$

The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions:

${\displaystyle q(\tau )=q_{t}(\tau )+q_{s}(\tau ).}$

For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients.

A simple 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 length ${\displaystyle l}$, where ${\displaystyle g}$ is the local acceleration of gravity, is

If the maximal displacement of the pendulum is small, we can use the approximation ${\displaystyle \sin \theta \approx \theta }$ and instead consider the equation

${\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{l}}\theta =0.}$

The general solution to this differential equation is

${\displaystyle \theta (t)=A\cos \left({\sqrt {\frac {g}{l}}}t+\varphi \right),}$

where ${\displaystyle A}$ and ${\displaystyle \varphi }$ are constants that depends on the initial conditions. Using as initial conditions ${\displaystyle \theta (0)=\theta _{0}}$ and ${\displaystyle {\dot {\theta }}(0)=0}$, the solution is given by

${\displaystyle \theta (t)=\theta _{0}\cos \left({\sqrt {\frac {g}{l}}}t\right),}$

where ${\displaystyle \theta _{0}}$ is the largest angle attained by the pendulum (that is, ${\displaystyle \theta _{0}}$ is the amplitude of the pendulum). The period, the time for one complete oscillation, is given by the expression

which is a good approximation of the actual period when ${\displaystyle \theta _{0}}$ is small. Notice that in this approximation the period ${\displaystyle \tau }$ is independent of the amplitude ${\displaystyle \theta _{0}}$. In the above equation, ${\displaystyle \omega }$ represents the angular frequency.

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:

${\displaystyle F(t)=-kx(t),}$

where F is the force, k is the spring constant, and x 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:

${\displaystyle F(t)=-kx(t)=m{\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}x(t)=ma,}$

the latter being Newton's second law of motion.

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by

${\displaystyle x(t)=A\cos \left({\sqrt {\frac {k}{m}}}t\right).}$

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

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 is then transferred into kinetic energy. The potential energy within a spring is determined by the equation ${\displaystyle U=kx^{2}/2.}$

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 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.