Nondimensionalization

Nondimensionalization steps

To nondimensionalize a system of equations, one must do the following:

1. Identify all the independent and dependent variables;
2. Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined;
3. Divide through by the coefficient of the highest order polynomial or derivative term;
4. Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1;
5. Rewrite the system of equations in terms of their new dimensionless quantities.

The last three steps are usually specific to the problem where nondimensionalization is applied. However, almost all systems require the first two steps to be performed.

Conventions

There are no restrictions on the variable names used to replace "x" and "t". However, they are generally chosen so that it is convenient and intuitive to use for the problem at hand. For example, if "x" represented mass, the letter "m" might be an appropriate symbol to represent the dimensionless mass quantity.

In this article, the following conventions have been used:

• t – represents the independent variable – usually a time quantity. Its nondimensionalized counterpart is ${\displaystyle \tau }$.
• x – represents the dependent variable – can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is ${\displaystyle \chi }$.

A subscript 'c' added to a quantity's variable name is used to denote the characteristic unit used to scale that quantity. For example, if x is a quantity, then x_c is the characteristic unit used to scale it.

As an illustrative example, consider a first order differential equation with constant coefficients: $${\displaystyle a\frac{dx}{dt}+bx=Af(t).}$$

1. In this equation the independent variable here is t, and the dependent variable is x.
2. Set ${\displaystyle x=\chi x_{\text{c}},t=\tau t_{\text{c}}}$. This results in the equation $${\displaystyle a\frac{x_{\text{c}}}{t_{\text{c}}}\frac{d\chi }{d\tau }+b{x}_{\text{c}}\chi =Af(\tau t_{\text{c}})\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}AF(\tau ).}$$
3. The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives $${\displaystyle \frac{d\chi }{d\tau }+\frac{b{t}_{\text{c}}}{a}\chi =\frac{A{t}_{\text{c}}}{a{x}_{\text{c}}}F(\tau ).}$$
4. The coefficient in front of ${\displaystyle \chi }$ only contains one characteristic variable t_c, hence it is easiest to choose to set this to unity first:

Subsequently,

1. The final dimensionless equation in this case becomes completely independent of any parameters with units: $${\displaystyle \frac{d\chi }{d\tau }+\chi =F(\tau ).}$$

Substitutions

Suppose for simplicity that a certain system is characterized by two variables – a dependent variable x and an independent variable t, where x is a function of t. Both x and t represent quantities with units. To scale these two variables, assume there are two intrinsic units of measurement x_c and t_c with the same units as x and t respectively, such that these conditions hold: $${\displaystyle \tau =\frac{t}{t_{\text{c}}}\Rightarrow t=\tau t_{\text{c}}}$$ $${\displaystyle \chi =\frac{x}{x_{\text{c}}}\Rightarrow x=\chi x_{\text{c}}.}$$

These equations are used to replace x and t when nondimensionalizing. If differential operators are needed to describe the original system, their scaled counterparts become dimensionless differential operators.

Differential operators

Consider the relationship $${\displaystyle t=\tau t_{\text{c}}\Rightarrow dt=t_{\text{c}}d\tau \Rightarrow \frac{d\tau }{dt}=\frac{1}{t_{\text{c}}}.}$$

The dimensionless differential operators with respect to the independent variable becomes $${\displaystyle \frac{d}{dt}=\frac{d\tau }{dt}\frac{d}{d\tau }=\frac{1}{t_{\text{c}}}\frac{d}{d\tau }\Rightarrow \frac{d^n}{d{t}^n}={\left(\frac{d}{dt}\right)}^n={\left(\frac{1}{t_{\text{c}}}\frac{d}{d\tau }\right)}^n=\frac{1}{{t_{\text{c}}}^n}\frac{d^n}{d{\tau }^n}.}$$

Forcing function

If a system has a forcing function ${\displaystyle f(t)}$ then $${\displaystyle f(t)=f(\tau t_{\text{c}})=f(t(\tau ))=F(\tau ).}$$

Hence, the new forcing function ${\displaystyle F}$ is made to be dependent on the dimensionless quantity ${\displaystyle \tau }$.

Linear differential equations with constant coefficients

First order system

Consider the differential equation for a first order system: $${\displaystyle a\frac{dx}{dt}+bx=Af(t).}$$

The derivation of the characteristic units to Eq. 1 and Eq. 2 for this system gave $${\displaystyle t_{\text{c}}=\frac{a}{b},x_{\text{c}}=\frac{A}{b}.}$$

Second order system

A second order system has the form $${\displaystyle a\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+cx=Af(t).}$$

Substitution step

Replace the variables x and t with their scaled quantities. The equation becomes

$${\displaystyle a\frac{x_{\text{c}}}{{t_{\text{c}}}^2}\frac{d^2\chi }{d{\tau }^2}+b\frac{x_{\text{c}}}{t_{\text{c}}}\frac{d\chi }{d\tau }+c{x}_{\text{c}}\chi =Af(\tau t_{\text{c}})=AF(\tau ).}$$

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

$${\displaystyle \frac{d^2\chi }{d{\tau }^2}+t_{\text{c}}\frac{b}{a}\frac{d\chi }{d\tau }+{t_{\text{c}}}^2\frac{c}{a}\chi =\frac{A{t_{\text{c}}}^2}{a{x}_{\text{c}}}F(\tau ).}$$

Now it is necessary to determine the quantities of x_c and t_c 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 t_c:

1. If ${\displaystyle t_{\text{c}}=\frac{a}{b}}$ the first order term is normalized.
2. If ${\displaystyle t_{\text{c}}=\sqrt{\frac{a}{c}}}$ 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 x_c to be determined by normalizing the coefficient of the forcing function: $${\displaystyle 1=\frac{A{t}_{\text{c}}^2}{a{x}_{\text{c}}}=\frac{A}{c{x}_{\text{c}}}\Rightarrow x_{\text{c}}=\frac{A}{c}.}$$

The differential equation becomes $${\displaystyle \frac{d^2\chi }{d{\tau }^2}+\frac{b}{\sqrt{ac}}\frac{d\chi }{d\tau }+\chi =F(\tau ).}$$

The coefficient of the first order term is unitless. Define $${\displaystyle 2\zeta \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{b}{\sqrt{ac}}.}$$

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. 2ζ is also known as the linewidth of the system. The result of the definition is the universal oscillator equation. $${\displaystyle \frac{d^2\chi }{d{\tau }^2}+2\zeta \frac{d\chi }{d\tau }+\chi =F(\tau ).}$$

Higher order systems

The general nth order linear differential equation with constant coefficients has the form: $${\displaystyle a_n\frac{d^n}{d{t}^n}x(t)+a_{n-1}\frac{d^{n-1}}{d{t}^{n-1}}x(t)+\dots +a_1\frac{d}{dt}x(t)+a_{0}x(t)=\sum _{k=0}^n{a}_k(\frac{d}{dt}{)}^{k}x(t)=Af(t).}$$

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

• ${\displaystyle x}$ = displacement from equilibrium [m]
• ${\displaystyle t}$ = time [s]
• ${\displaystyle f}$ = external force or "disturbance" applied to system [kg⋅m⋅s⁻²]
• ${\displaystyle m}$ = mass of the block [kg]
• ${\displaystyle B}$ = damping constant of dashpot [kg⋅s⁻¹]
• ${\displaystyle k}$ = force constant of spring [kg⋅s⁻²]

Suppose the applied force is a sinusoid F = F₀ cos(ωt), the differential equation that describes the motion of the block is $${\displaystyle m\frac{d^{2}x}{d{t}^2}+B\frac{dx}{dt}+kx=F_0\cos (\omega t)}$$

Nondimensionalizing this equation the same way as described under § Second order system yields several characteristics of the system:

• The intrinsic unit x_c corresponds to the distance the block moves per unit force

$${\displaystyle x_{\text{c}}=\frac{F_0}{k}.}$$

• The characteristic variable t_c is equal to the period of the oscillations

$${\displaystyle t_{\text{c}}=\sqrt{\frac{m}{k}}}$$

• The dimensionless variable 2ζ corresponds to the linewidth of the system.

$${\displaystyle 2\zeta =\frac{B}{\sqrt{mk}}}$$

• ζ itself is the damping ratio

Electrical oscillations

First-order series RC circuit

For a series RC attached to a voltage source $${\displaystyle R\frac{dQ}{dt}+\frac{Q}{C}=V(t)\Rightarrow \frac{d\chi }{d\tau }+\chi =F(\tau )}$$ with substitutions $${\displaystyle Q=\chi x_{\text{c}},t=\tau t_{\text{c}},x_{\text{c}}=C{V}_0,t_{\text{c}}=RC,F=V.}$$

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 $${\displaystyle L\frac{d^{2}Q}{d{t}^2}+R\frac{dQ}{dt}+\frac{Q}{C}=V_0\cos (\omega t)\Rightarrow \frac{d^2\chi }{d{\tau }^2}+2\zeta \frac{d\chi }{d\tau }+\chi =\cos (\mathrm{\Omega }\tau )}$$ with the substitutions $${\displaystyle Q=\chi x_{\text{c}},t=\tau t_{\text{c}},x_{\text{c}}=C{V}_0,t_{\text{c}}=\sqrt{LC},2\zeta =R\sqrt{\frac{C}{L}},\mathrm{\Omega }=t_{\text{c}}\omega .}$$

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 $${\displaystyle \left(-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+\frac{1}{2}m{\omega }^2{x}^2\right)\psi (x)=E\psi (x).}$$

The modulus square of the wavefunction |ψ(x)|² represents probability density that, when integrated over x, gives a dimensionless probability. Therefore, |ψ(x)|² has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute $${\displaystyle \tilde{x}\equiv \frac{x}{x_{\text{c}}},}$$ where x_c is some characteristic length of this system. This gives us a dimensionless wave function ${\displaystyle \tilde{\psi }}$ defined via $${\displaystyle \psi (x)=\psi (\tilde{x}{x}_{\text{c}})=\psi (x(x_{\text{c}}))=\tilde{\psi }(\tilde{x}).}$$

The differential equation then becomes $${\displaystyle \left(-\frac{{\hslash }^2}{2m}\frac{1}{x_{\text{c}}^2}\frac{d^2}{d{\tilde{x}}^2}+\frac{1}{2}m{\omega }^2{x}_{\text{c}}^2{\tilde{x}}^2\right)\tilde{\psi }(\tilde{x})=E\tilde{\psi }(\tilde{x})\Rightarrow \left(-\frac{d^2}{d{\tilde{x}}^2}+\frac{m^2{\omega }^2{x}_{\text{c}}^4}{{\hslash }^2}{\tilde{x}}^2\right)\tilde{\psi }(\tilde{x})=\frac{2m{x}_{\text{c}}^{2}E}{{\hslash }^2}\tilde{\psi }(\tilde{x}).}$$

To make the term in front of ${\displaystyle {\tilde{x}}^2}$ dimensionless, set $${\displaystyle \frac{m^2{\omega }^2{x}_{\text{c}}^4}{{\hslash }^2}=1\Rightarrow x_{\text{c}}=\sqrt{\frac{\hslash }{m\omega }}.}$$

The fully nondimensionalized equation is $${\displaystyle \left(-\frac{d^2}{d{\tilde{x}}^2}+{\tilde{x}}^2\right)\tilde{\psi }(\tilde{x})=\tilde{E}\tilde{\psi }(\tilde{x}),}$$ where we have defined $${\displaystyle E\equiv \frac{\hslash \omega }{2}\tilde{E}.}$$ The factor in front of ${\displaystyle \tilde{E}}$ 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 $${\displaystyle \frac{\hslash \omega }{2}\left(-\frac{d^2}{d{\tilde{x}}^2}+{\tilde{x}}^2\right)\tilde{\psi }(\tilde{x})=E\tilde{\psi }(\tilde{x}).}$$

Statistical analogs

In statistics, the analogous process is usually dividing a difference (a distance) by a scale factor (a measure of statistical dispersion), which yields a dimensionless number, which is called normalization. Most often, this is dividing errors or residuals by the standard deviation or sample standard deviation, respectively, yielding standard scores and studentized residuals.