A parametric oscillator is a
harmonic oscillator whose physical properties vary with time. The equation of such an oscillator is :\frac{d^{2}x}{dt^{2}} + \beta(t) \frac{dx}{dt} + \omega^{2}(t) x = 0 This equation is linear in x(t). By assumption, the parameters \omega^{2} and \beta depend only on time and do
not depend on the state of the oscillator. In general, \beta(t) and/or \omega^{2}(t) are assumed to vary periodically, with the same period T. If the parameters vary at roughly
twice the
natural frequency of the oscillator (defined below), the oscillator phase-locks to the parametric variation and absorbs energy at a rate proportional to the energy it already has. Without a compensating energy-loss mechanism provided by \beta, the oscillation amplitude grows exponentially. (This phenomenon is called
parametric excitation,
parametric resonance or
parametric pumping.) However, if the initial amplitude is zero, it will remain so; this distinguishes it from the non-parametric resonance of driven simple
harmonic oscillators, in which the amplitude grows linearly in time regardless of the initial state. A familiar experience of both parametric and driven oscillation is playing on a swing. Rocking back and forth pumps the swing as a
driven harmonic oscillator, but once moving, the swing can also be parametrically driven by alternately standing and squatting at key points in the swing arc. This changes moment of inertia of the swing and hence the resonance frequency, and children can quickly reach large amplitudes provided that they have some amplitude to start with (e.g., get a push). Standing and squatting at rest, however, leads nowhere.
Transformation of the equation We begin by making a change of variable :q(t) \ \stackrel{\mathrm{def}}{=}\ e^{D(t)} x(t) where D(t) is the time integral of the damping coefficient :D(t) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{2} \int_{0}^{t} \beta(\tau) \, d\tau . This change of variable eliminates the damping term in the differential equation, reducing it to :\frac{d^{2}q}{dt^{2}} + \Omega^{2}(t) q = 0 where the transformed frequency is defined as :\Omega^{2}(t) \ \stackrel{\mathrm{def}}{=}\ \omega^{2}(t) - \frac{1}{2} \frac{d\beta}{dt} - \frac{1}{4} \beta^{2}(t). In general, the variations in damping and frequency are relatively small perturbations :\beta(t) = \omega_{0} \big[b + g(t) \big] :\omega^{2}(t) = \omega_{0}^{2} \big[1 + h(t) \big] where \omega_{0} and b are constants, namely, the time-averaged oscillator frequency and damping, respectively. The transformed frequency can then be written in a similar way as :\Omega^{2}(t) = \omega_{n}^{2} \big[1 + f(t) \big], where \omega_{n} is the
natural frequency of the damped harmonic oscillator :\omega_{n}^{2} \ \stackrel{\mathrm{def}}{=}\ \omega_{0}^{2} \left( 1 - \frac{b^{2}}{4} \right) and :f(t) \ \stackrel{\mathrm{def}}{=}\ \frac{\omega_{0}^{2}}{\omega_{n}^{2}} \left[ h(t) - \frac{1}{2\omega_{0}} \frac{dg}{dt} - \frac{b}{2} g(t) - \frac{1}{4} g^{2}(t) \right]. Thus, our transformed equation can be written as :\frac{d^{2}q}{dt^{2}} + \omega_{n}^{2} \big[1 + f(t) \big] q = 0. The independent variations g(t) and h(t) in the oscillator damping and resonance frequency, respectively, can be combined into a single pumping function f(t). The converse conclusion is that any form of parametric excitation can be accomplished by varying either the resonance frequency or the damping, or both.
Solution of the transformed equation Let us assume that \ f(t)\ is sinusoidal with a frequency approximately twice the natural frequency of the oscillator: : f(t) = f_{0} \sin (2\omega_{p}t) where the pumping frequency \ \omega_{p} \approx \omega_{n}\ but need not equal \ \omega_{n}\ exactly. Using the method of
variation of parameters, the solution \ q(t)\ to our transformed equation may be written as :\ q(t)\ =\ A(t)\ \cos (\omega_{p} t)\ +\ B(t)\ \sin (\omega_{p} t)\ where the rapidly varying components, \ \cos (\omega_{p}t)\ and \ \sin (\omega_{p}t)\ , have been factored out to isolate the slowly varying amplitudes \ A(t)\ and \ B(t) ~. We proceed by substituting this solution into the differential equation and considering that both the coefficients in front of \ \cos (\omega_{p}t)\ and \ \sin (\omega_{p}t)\ must be zero to satisfy the differential equation identically. We also omit the second derivatives of \ A(t)\ and \ B(t)\ on the grounds that \ A(t)\ and \ B(t)\ are slowly varying, as well as omit sinusoidal terms not near the natural frequency, \ \omega_{n}\ , as they do not contribute significantly to resonance. The result is the following pair of coupled differential equations: : 2 \omega_{p} \frac{\ \operatorname d A\ }{\operatorname d t } = \frac{1}{2} f_0\ \omega_{n}^{2}\ A - \left( \omega_{p}^{2} - \omega_{n}^{2} \right)\ B\ , :2\omega_{p} \frac{\ \operatorname d B\ }{\operatorname d t } = \left( \omega_{p}^{2} - \omega_{n}^{2} \right)\ A - \frac{1}{2} f_0\ \omega_{n}^{2}\ B ~. This
system of linear differential equations with constant coefficients can be decoupled and solved by
eigenvalue/
eigenvector methods. This yields the solution :\begin{bmatrix} A(t) \\ B(t) \end{bmatrix} = c_1\ \vec{V_1}\ e^{\lambda_1 t} + c_2\ \vec{V_2}\ e^{\lambda_2 t} where \ \lambda_1\ and \ \lambda_2\ are the eigenvalues of the matrix : \frac{ 1 }{\ 2 \omega_p\ }\begin{bmatrix} \frac{1}{2} f_0\ \omega_{n}^{2} & - \left( \omega_{p}^{2} - \omega_{n}^{2} \right) \\ + \left( \omega_{p}^{2} - \omega_{n}^{2} \right) & -\frac{1}{2} f_0\ \omega_{n}^{2} \end{bmatrix}\ , \ \vec{V_1}\ and \ \vec{V_2}\ are corresponding eigenvectors, and \ c_1\ and \ c_2\ are arbitrary constants. The eigenvalues are given by : \lambda_{1,2} = \pm \frac{ 1 }{\ 2 \omega_p\ } \sqrt{\Bigl( \tfrac{ 1 }{ 2 } f_0\ \omega_n^2 \Bigr)^2 - \Bigl( \omega_{p}^{2} - \omega_{n}^{2} \Bigr)^2 \;} ~. If we write the difference between \ \omega_p\ and \ \omega_n\ as \ \Delta \omega = \omega_p - \omega_n\ , and replace \ \omega_p\ with \omega_n everywhere where the difference is not important, we get : \lambda_{1,2} = \pm \sqrt{\Bigl( \tfrac{ 1 }{ 4 }f_0\ \omega_n \Bigr)^2 - \Bigl( \Delta \omega \Bigr)^2 \;} . If \ \bigl| \Delta \omega \bigr| then the eigenvalues are real and exactly one is positive, which leads to
exponential growth for \ A(t)\ and \ B(t) ~. This is the condition for parametric resonance, with the growth rate for \ q(t)\ given by the positive eigenvalue \ \lambda_1 = + \sqrt{ \Bigl(\tfrac{1}{4} f_0\ \omega_n \Bigr)^2 - \Bigl( \Delta \omega \Bigr)^2 \;} ~. Note, however, that this growth rate corresponds to the amplitude of the transformed variable \ q(t)\ , whereas the amplitude of the original, untransformed variable \ x(t) = q(t)\ e^{-D(t)}\ can either grow or decay depending on whether \ \lambda_1 t - D(\ t\ )\ is an increasing or decreasing function of time, \ t ~.
Intuitive derivation of parametric excitation The above derivation may seem like a mathematical sleight-of-hand, so it may be helpful to give an intuitive derivation. The q equation may be written in the form :\frac{d^{2}q}{dt^{2}} + \omega_{n}^{2} q = -\omega_{n}^{2} f(t) q which represents a simple harmonic oscillator (or, alternatively, a
bandpass filter) being driven by a signal -\omega_{n}^{2} f(t) q that is proportional to its response q(t). Assume that q(t) = A \cos (\omega_{p} t) already has an oscillation at frequency \omega_{p} and that the pumping f(t) = f_{0} \sin (2\omega_{p}t) has double the frequency and a small amplitude f_{0} \ll 1. Applying a
trigonometric identity for products of sinusoids, their product q(t)f(t) produces two driving signals, one at frequency \omega_{p} and the other at frequency 3 \omega_{p}. :f(t)q(t) = \frac{f_{0}}{2} A \big[ \sin (\omega_{p} t) + \sin (3\omega_{p} t) \big] Being off-resonance, the 3\omega_{p} signal is attenuated and can be neglected initially. By contrast, the \omega_{p} signal is on resonance, serves to amplify q(t), and is proportional to the amplitude A. Hence, the amplitude of q(t) grows exponentially unless it is initially zero. Expressed in Fourier space, the multiplication f(t)q(t) is a convolution of their Fourier transforms \tilde{F}(\omega) and \tilde{Q}(\omega). The
positive feedback arises because the +2\omega_{p} component of f(t) converts the -\omega_{p} component of q(t) into a driving signal at +\omega_{p}, and vice versa (reverse the signs). This explains why the pumping frequency must be near 2\omega_{n}, twice the natural frequency of the oscillator. Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the -\omega_{p} and +\omega_{p} components of q(t). ==Parametric resonance==