Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. 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. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units. Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or time constant, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by differential equations. One important use is in the analysis of control systems. 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. Consider the following:
List of dynamical systems and differential equations topics List of partial differential equation topics Differential equations of mathematical physics Although nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is normalization in statistics. Measuring devices are practical examples of nondimensionalization occurring in everyday life. Measuring devices are calibrated relative to some known unit. Subsequent measurements are made relative to this standard. Then, the absolute value of the measurement is recovered by scaling with respect to the standard.
Rationale 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. Measurements made relative to an intrinsic property of a system will apply to other systems which also have the same intrinsic property. It also allows one to compare a common property of different implementations of the same system. Nondimensionalization determines in a systematic manner the characteristic units of a system to use, without relying heavily on prior knowledge of the system's intrinsic properties (one should not confuse characteristic units of a system with natural units of nature). In fact, nondimensionalization can suggest the parameters which should be used for analyzing a system. However, it is necessary to start with an equation that describes the system appropriately.
Nondimensionalization steps To nondimensionalize a system of equations, one must do the following:
Identify all the independent and dependent variables; Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined; Divide through by the coefficient of the highest order polynomial or derivative term; Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1; 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 τ {\displaystyle \tau } . x – represents the dependent variable – can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is χ {\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 xc is the characteristic unit used to scale it. As an illustrative example, consider a first order differential equation with constant coefficients:
a d x d t + b x = A f ( t ) . {\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}
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