Nondimensionalization

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 $${\displaystyle {\frac {d}{dt}}={\frac {d\tau }{dt}}{\frac {d}{d\tau }}={\frac {1}{t_{\text{c}}}}{\frac {d}{d\tau }}\Rightarrow {\frac {d^{n}}{dt^{n}}}=\left({\frac {d}{dt}}\right)^{n}=\left({\frac {1}{t_{\text{c}}}}{\frac {d}{d\tau }}\right)^{n}={\frac {1}{{t_{\text{c}}}^{n}}}{\frac {d^{n}}{d\tau ^{n}}}.}$$

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 ${\displaystyle \tau }$.

Substitution step

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

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

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

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

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: $${\displaystyle 1={\frac {At_{\text{c}}^{2}}{ax_{\text{c}}}}={\frac {A}{cx_{\text{c}}}}\Rightarrow x_{\text{c}}={\frac {A}{c}}.}$$

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 $${\displaystyle 2\zeta \ {\stackrel {\mathrm {def} }{=}}\ {\frac {b}{\sqrt {ac}}}.}$$

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

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 $${\displaystyle m{\frac {d^{2}x}{dt^{2}}}+B{\frac {dx}{dt}}+kx=F_{0}\cos(\omega t)}$$

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

Quantum harmonic oscillator

The Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator is $${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\right)\psi (x)=E\psi (x).}$$

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 $${\displaystyle \psi (x)=\psi ({\tilde {x}}x_{\text{c}})=\psi (x({\tilde {x}}))={\tilde {\psi }}({\tilde {x}}).}$$

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

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

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 $${\displaystyle E\equiv {\frac {\hbar \omega }{2}}{\tilde {E}}.}$$ 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 $${\displaystyle {\frac {\hbar \omega }{2}}\left(-{\frac {d^{2}}{d{\tilde {x}}^{2}}}+{\tilde {x}}^{2}\right){\tilde {\psi }}({\tilde {x}})=E{\tilde {\psi }}({\tilde {x}}).}$$