First order system Consider the differential equation for a first order system: a\frac{dx}{dt} + bx = Af(t). The derivation of the characteristic units to and for this system gave t_\text{c} = \frac{a}{b}, \ x_\text{c} = \frac{A}{b}.
Second order system A second order system has the form a \frac{d^2 x}{dt^2} + b\frac{dx}{dt} + cx = A f(t).
Substitution step Replace the variables
x and
t with their scaled quantities. The equation becomes a \frac{x_\text{c}}{{t_\text{c}}^2} \frac{ d^2 \chi}{d \tau^2} + b \frac{x_\text{c}}{t_\text{c}} \frac{d \chi}{d \tau} + c x_\text{c} \chi = A f(\tau t_\text{c}) = A F(\tau) . This new equation is not dimensionless, although all the variables with units are isolated in the coefficients. Dividing by the coefficient of the highest ordered term, the equation becomes \frac{d^2 \chi}{d \tau^2} + t_\text{c} \frac{b}{a} \frac{d \chi}{d \tau} + {t_\text{c}}^2 \frac{c}{a} \chi = \frac{A {t_\text{c}}^2}{a x_\text{c}} F(\tau). Now it is necessary to determine the quantities of
xc and
tc so that the coefficients become normalized. Since there are two free parameters, at most only two coefficients can be made to equal unity.
Determination of characteristic units Consider the variable
tc: • If t_\text{c} = \frac{a}{b} the first order term is normalized. • If t_\text{c} = \sqrt{\frac{a}{c}} the zeroth order term is normalized. Both substitutions are valid. However, for pedagogical reasons, the latter substitution is used for second order systems. Choosing this substitution allows
xc to be determined by normalizing the coefficient of the forcing function: 1 = \frac{A t_\text{c}^2}{a x_\text{c}} = \frac{A}{c x_\text{c}} \Rightarrow x_\text{c} = \frac{A}{c}. The differential equation becomes \frac{d^2 \chi}{d \tau^2} + \frac{b}{\sqrt{ac}} \frac{d \chi}{d\tau} + \chi = F(\tau). The coefficient of the first order term is unitless. Define 2 \zeta \ \stackrel{\mathrm{def}}{=}\ \frac{b}{\sqrt{ac}}. The factor 2 is present so that the solutions can be parameterized in terms of
ζ. In the context of mechanical or electrical systems,
ζ is known as the
damping ratio, and is an important parameter required in the analysis of
control systems. 2
ζ is also known as the
linewidth of the system. The result of the definition is the
universal oscillator equation. \frac{d^2 \chi}{d \tau^2} + 2 \zeta \frac{d \chi}{d\tau} + \chi = F(\tau) .
Higher order systems The general
nth order linear differential equation with constant coefficients has the form: a_n \frac{d^n}{dt^n} x(t) + a_{n-1} \frac{d^{n-1}}{dt^{n-1}} x(t) + \ldots + a_1 \frac{d}{dt} x(t) + a_0 x(t) = \sum_{k = 0}^n a_k \big( \frac{d}{dt} \big) ^k x(t) = Af(t). The function
f(
t) is known as the
forcing function. If the differential equation only contains real (not complex) coefficients, then the properties of such a system behaves as a mixture of first and second order systems only. This is because the
roots of its
characteristic polynomial are either
real, or
complex conjugate pairs. Therefore, understanding how nondimensionalization applies to first and second ordered systems allows the properties of higher order systems to be determined through
superposition. The number of free parameters in a nondimensionalized form of a system increases with its order. For this reason, nondimensionalization is rarely used for higher order differential equations. The need for this procedure has also been reduced with the advent of
symbolic computation.
Examples of recovering characteristic units A variety of systems can be approximated as either first or second order systems. These include mechanical, electrical, fluidic, caloric, and torsional systems. This is because the fundamental physical quantities involved within each of these examples are related through first and second order derivatives.
Mechanical oscillations Suppose we have a mass attached to a spring and a damper, which in turn are attached to a wall, and a force acting on the mass along the same line. Define • x = displacement from equilibrium [m] • t = time [s] • f = external force or "disturbance" applied to system [kg⋅m⋅s−2] • m = mass of the block [kg] • B = damping constant of
dashpot [kg⋅s−1] • k = force constant of spring [kg⋅s−2] Suppose the applied force is a sinusoid , the differential equation that describes the motion of the block is m \frac{d^2 x}{d t^2} + B \frac{d x}{d t} + kx = F_0 \cos(\omega t) Nondimensionalizing this equation the same way as described under yields several characteristics of the system: • The intrinsic unit
xc corresponds to the distance the block moves per unit force x_\text{c} = \frac{F_0}{k}. • The characteristic variable
tc is equal to the period of the oscillations t_\text{c} = \sqrt{\frac{m}{k}} • The dimensionless variable 2
ζ corresponds to the linewidth of the system. 2 \zeta = \frac{B}{\sqrt{mk}} •
ζ itself is the
damping ratio Electrical oscillations First-order series RC circuit For a series
RC attached to a
voltage source R \frac{dQ}{dt} + \frac{Q}{C} = V(t) \Rightarrow \frac{d \chi}{d \tau} + \chi = F(\tau) with substitutions Q = \chi x_\text{c}, \ t = \tau t_\text{c}, \ x_\text{c} = C V_0, \ t_\text{c} = RC, \ F = V. The first characteristic unit corresponds to the total
charge in the circuit. The second characteristic unit corresponds to the
time constant for the system.
Second-order series RLC circuit For a series configuration of
R,
C,
L components where
Q is the charge in the system L \frac{d^2 Q}{dt^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \cos(\omega t) \Rightarrow \frac{d^2 \chi}{d \tau^2} + 2 \zeta \frac{d \chi}{d\tau} + \chi = \cos(\Omega \tau) with the substitutions Q = \chi x_\text{c}, \ t = \tau t_\text{c}, \ \ x_\text{c} = C V_0, \ t_\text{c} = \sqrt{LC}, \ 2 \zeta = R \sqrt{\frac{C}{L}}, \ \Omega = t_\text{c} \omega. The first variable corresponds to the maximum charge stored in the circuit. The resonance frequency is given by the reciprocal of the characteristic time. The last expression is the linewidth of the system. The Ω can be considered as a normalized forcing function frequency.
Quantum mechanics Quantum harmonic oscillator The
Schrödinger equation for the one-dimensional time independent
quantum harmonic oscillator is \left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x). The modulus square of the
wavefunction represents probability density that, when integrated over , gives a dimensionless probability. Therefore, has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute \tilde x \equiv \frac{x}{x_{\text{c}}}, where is some
characteristic length of this system. This gives us a dimensionless wave function \tilde \psi defined via \psi(x) = \psi(\tilde x x_{\text{c}}) = \psi(x(\tilde x)) = \tilde \psi(\tilde x). The differential equation then becomes \left(-\frac{\hbar^2}{2m} \frac{1}{x_{\text{c}}^2} \frac{d^2}{d \tilde x^2} + \frac{1}{2} m \omega^2 x_{\text{c}}^2 \tilde x^2 \right) \tilde \psi(\tilde x) = E \, \tilde \psi(\tilde x) \Rightarrow \left(-\frac{d^2}{d \tilde x^2} + \frac{m^2 \omega^2 x_{\text{c}}^4}{\hbar^2} \tilde x^2 \right) \tilde \psi(\tilde x) = \frac{2 m x_{\text{c}}^2 E}{\hbar^2} \tilde \psi(\tilde x). To make the term in front of \tilde x^2 dimensionless, set \frac{m^2 \omega^2 x_{\text{c}}^4}{\hbar^2} = 1 \Rightarrow x_{\text{c}} = \sqrt{\frac{\hbar}{m \omega}} . The fully nondimensionalized equation is \left(-\frac{d^2}{d \tilde x^2} + \tilde x^2 \right) \tilde \psi(\tilde x) = \tilde E \tilde \psi(\tilde x), where we have defined E \equiv \frac{\hbar \omega}{2} \tilde E. The factor in front of \tilde E is in fact (coincidentally) the
ground state energy of the harmonic oscillator. Usually, the energy term is not made dimensionless as we are interested in determining the energies of the
quantum states. Rearranging the first equation, the familiar equation for the harmonic oscillator becomes \frac{\hbar \omega}{2} \left( -\frac{d^2}{d \tilde x^2} + \tilde x^2 \right) \tilde \psi(\tilde x) = E \tilde \psi(\tilde x). == Statistical analogs ==