Nondimensionalization

This technique can simplify and parameterize problems where measured units are involved.

In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit.

For example, if a system has an intrinsic resonance frequency, length, or time constant, nondimensionalization can recover these values.

One of the simplest characteristic units is the doubling time of a system experiencing exponential growth, or conversely the half-life of a system experiencing exponential decay; a more natural pair of characteristic units is mean age/mean lifetime, which correspond to base e rather than base 2.

Many illustrative examples of nondimensionalization originate from simplifying differential equations.

This is because a large body of physical problems can be formulated in terms of differential equations.

Measuring devices are practical examples of nondimensionalization occurring in everyday life.

Then, the absolute value of the measurement is recovered by scaling with respect to the standard.

Suppose a pendulum is swinging with a particular period T. For such a system, it is advantageous to perform calculations relating to the swinging relative to T. In some sense, this is normalizing the measurement with respect to the period.

However, they are generally chosen so that it is convenient and intuitive to use for the problem at hand.

For example, if x is a quantity, then xc is the characteristic unit used to scale it.

As an illustrative example, consider a first order differential equation with constant coefficients:

This new equation is not dimensionless, although all the variables with units are isolated in the coefficients.

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.

Choosing this substitution allows xc to be determined by normalizing the coefficient of the forcing function:

The general nth order linear differential equation with constant coefficients has the form:

This is because the roots of its characteristic polynomial are either real, or complex conjugate pairs.

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.

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.

Define Suppose the applied force is a sinusoid F = F0 cos(ωt), the differential equation that describes the motion of the block is

The first characteristic unit corresponds to the total charge in the circuit.

The first variable corresponds to the maximum charge stored in the circuit.

The Ω can be considered as a normalized forcing function frequency.

The Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator is

is in fact (coincidentally) the ground state energy of the harmonic oscillator.

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.