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Encyclopedia > Nondimensionalization

Nondimensionalization refers to the partial or full removal of units from a mathematical equation 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. The word unit means any of several things: One, the first natural number. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ... Measure can mean: To perform a measurement. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... For the Macintosh operating system, which was called System up to version 7. ... The International System of Units (symbol: SI) (for the French phrase Système International dUnités) is the most widely used system of units. ... An extensive property is a property of matter that depends on amount. ... An intensive property is a property of matter that does not depend on amount. ...


Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonant 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. This article is about resonance in physics. ... In general English usage, length (symbol: l) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth) is... The RC time constant, usually denoted by the Greek letter τ (tau), is a parameter that characterizes the frequency response of a resistance-capacitance (RC) circuit. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... A control system is a device or set of devices that manage the behavior of other devices. ...


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:

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. This is a list of dynamical system and differential equation topics, by Wikipedia page. ... This is a list of partial differential equation topics, by Wikipedia page. ... Differential equations are a basic tool for understanding the physical world. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...


Measuring devices are practical examples of nondimensionalization occurring in everyday life. Most measuring devices are first 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. Captain Nemo and Professor Aronnax contemplating measuring instruments in Twenty Thousand Leagues Under the Sea In physics and engineering, measurement is the activity of comparing physical quantities of real-world objects and events. ...

Contents

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. A gravity pendulum is a weight on the end of a rigid rod, which, when given some initial lift from the vertical position, will swing back and forth under the influence of gravity over its central (lowest) point. ... A period is an arbitrary interval of time. ... Broadly, normalization (also spelled normalisation) is any process that makes something more normal, which typically means conforming to some regularity or rule, or returning from some state of abnormality. ...


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 natural units of a system to use, without relying heavily on prior knowledge of the system's intrinsic properties. 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:

  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 definition of the characteristic unit for each variable so that the coefficients of as many terms becomes unity;
  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.


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

  1. In this equation the independent variable here is t, and the dependent variable is x.
  2. Set . This results in the equation
  3. The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives
  4. The coefficient in front of χ only contains one characteristic variable tc, hence it is easiest to choose to set this to unity first:
    Subsequently,
  5. The final dimensionless equation in this case becomes completely independent of any parameters with units:

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 xc and tc with the same units as x and t respectively, such that these conditions hold: In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

The latter two 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.


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.
  • x - represents the dependent variable - can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is χ.

A subscripted 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.


Differential operators

Consider the relationship

The dimensionless differential operators with respect to the independent variable becomes

Forcing function

If a system has a forcing function f(t), then

Hence, the new forcing function F is made to be dependent on the dimensionless quantity τ.


Linear differential equations with constant coefficients

First order system

Let us consider the differential equation for a first order system:

The derivation of the characteristic units for this system gives

Second order system

A second order system has the form

Substitution step

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

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

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:

  1. If the first order term is normalized.
  2. If 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:

The differential equation becomes

The coefficient of the first order term is unitless. Define

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 A control system is a device or set of devices that manage the behavior of other devices. ... The Q factor or quality factor is a measure of the quality of a resonant system. ... A harmonic oscillator is a mechanical system in which there exists a returning force F directly proportionate to the displacement x, i. ...

Higher order systems

The general n-th order linear differential equation with constant coefficients has the form:

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. Roots is: The plural of Root Roots (album) Roots (TV miniseries), a mini-series based on a novel by Alex Haley Roots: The Saga of an American Family, a novel by Alex Haley Roots Canada Ltd. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... The word real has many different meanings: Real is something that exists, in the physical sense. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...


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 like this

 Displacement <-ve x --- x=0 --- +ve x > || / Wall /====| Spring /------ /====|/////////| | /====| Dashpot | Mass |<-----Force /====|------[===========| | /====| ------/ 

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 F = F0 cos(ωt), the differential equation that describes the motion of the block is

Nondimensionalizing this equation the same way as described under second order system yields several characteristics of the system.


The intrinsic unit xc corresponds to the distance the block moves per unit force

The characteristic variable tc is equal to the period of the oscillations

and the dimensionless variable 2ζ corresponds to the linewidth of the system. ζ itself is the damping ratio.

Electrical oscillations

First order series RC cirucit

For a series RC attached to a voltage source An RC circuit or RC network consists of a resistor R and a capacitor C, either in series (a series RC circuit) or in parallel (a parallel RC circuit). ... A power supply unit (sometimes abbreviated power supply or PSU) is a device that supplies electrical power to a device or group of devices. ...

with substitutions

The first characteristic unit corresponds to the total charge in the circuit. The second characterstic unit corresponds to the time constant for the system. Charge is a word with many different meanings. ... The RC time constant, usually denoted by the Greek letter τ (tau), is a parameter that characterizes the frequency response of a resistance-capacitance (RC) circuit. ...


Second order series RLC circuit

For a series configuration of R,C,L components where Q is the charge in the system

with the substitutions

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


Nonlinear differential equation example

Since there are no general methods of solving nonlinear differential equations, each case has to be considered on an individual basis when nondimensionalizing.


Quantum harmonic oscillator

The Schrödinger equation for the one dimensional time independent quantum harmonic oscillator is In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

The wavefunction ψ itself represents probability, which is in a sense already dimensionless and normalized. Therefore, there is no need to nondimensionalize the wavefunction. However, it should be rewritten as a function of a dimensionless variable. Furthermore, the variable x has units of length. Hence substitute In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared...

The differential equation becomes

To make the term in front of χ² unitless, set

Hence, the fully nondimensionalized equation is

The nondimensionalization factor for the energy is the same as the ground state of the harmonic oscillator. Usually, the energy term is not made dimensionless because a primary emphasis of quantum mechanics is determining the energies of the states of a system. Rearranging the first equation, the familiar equation for the harmonic oscillator is In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... Quite literally, quantum state describes the state of a quantum system. ...

See also

The Buckingham π theorem is a key theorem in dimensional analysis. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... In the physical sciences, a dimensionless number (or more precisely, a number with the dimensions of 1) is a quantity which describes a certain physical system and which is a pure number without any physical units; it does not change if one alters ones system of units of measurement... In physics, Planck units are physical units of measurement originally proposed by Max Planck. ... This is a list of dynamical system and differential equation topics, by Wikipedia page. ... This is a list of partial differential equation topics, by Wikipedia page. ... Differential equations are a basic tool for understanding the physical world. ... An RLC circuit is a kind of electrical circuit composed of a resistor (R), an inductor (L), and a capacitor (C). ... An RC circuit or RC network consists of a resistor R and a capacitor C, either in series (a series RC circuit) or in parallel (a parallel RC circuit). ...

External links

Application of nondimensionalization to a problem in biology http://www.rsnz.org/publish/nzjar/1998/59.php


Chapter 3 of these notes has a discussion on the merits of nondimensionalization http://www.maths.bath.ac.uk/~masjde/MSc/CourseNotes/MA50176.pdf


References


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