Nondimensionalization

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.…

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. a\frac{dx}{dt} + bx = Af(t). In this equation the independent variable here is t, and the dependent variable is x. Set x = \chi x_\text{c}, \ t = \tau t_\text{c} . This results in the equation a \frac{x_\text{c}}{t_\text{c}} \frac{d \chi}{d \tau} + b x_\text{c} \chi = A f(\tau t_\text{c}) \ \stackrel{\mathrm{def}}{=}\ A F(\tau). The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives \frac{d \chi}{d \tau} + \frac{b t_\text{c}}{a} \chi = \frac{A t_\text{c}}{a x_\text{c}} F(\tau). The coefficient in front of \chi only contains one characteristic variable t c , hence it is easiest to choose to set this to unity first: {{NumBlk|:::| \frac{b t_\text{c}}{a} = 1 \Rightarrow t_\text{c} = \frac{a}{b}. | }} Subsequently, {{NumBlk|:::| \frac{A t_\text{c}}{a x_\text{c}} = \frac{A}{b x_\text{c}} = 1 \Rightarrow x_\text{c} = \frac{A}{b}. | }} The final dimensionless equation in this case becomes completely independent of any parameters with units: \frac{d \chi}{d \tau} + \chi = F(\tau). \tau = \frac{t}{t_\text{c}} \Rightarrow t = \tau t_\text{c} \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. 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 \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}. f(t) = f(\tau t_\text{c}) = f(t(\tau)) = F(\tau). Hence, the new forcing function F is made to be dependent on the dimensionless quantity \tau . a\frac{dx}{dt} + bx = Af(t). The derivation of the characteristic units to and for this system gave t_\text{c} = \frac{a}{b}, \ x_\text{c} = \frac{A}{b}. a \frac{d^2 x}{dt^2} + b\frac{dx}{dt} + cx = A f(t). 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 = A f(\tau t_\text{c}) = A F(\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 \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. If t_\text{c} = \frac{a}{b} the first order term is normalized. If 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: 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 \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 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. \frac{d^2 \chi}{d \tau^2} + 2 \zeta \frac{d \chi}{d\tau} + \chi = F(\tau) . a_n \frac{d^n}{dt^n} x(t) + a_{n-1} \frac{d^{n-1}}{dt^{n-1}} x(t) + \ldots + a_1 \frac{d}{dt} x(t) + a_0 x(t) = \sum_{k = 0}^n a_k \big( \frac{d}{dt} \big) ^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. 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 , the differential equation that describes the motion of the block is m \frac{d^2 x}{d t^2} + B \frac{d x}{d t} + kx = F_0 \cos(\omega t) Nondimensionalizing this equation the same way as described under yields several characteristics of the system: The intrinsic unit x c corresponds to the distance the block moves per unit force x_\text{c} = \frac{F_0}{k}. The characteristic variable t c is equal to the period of the oscillations t_\text{c} = \sqrt{\frac{m}{k}} The dimensionless variable 2ζ corresponds to the linewidth of the system. 2 \zeta = \frac{B}{\sqrt{mk}} ζ itself is the damping ratio R \frac{dQ}{dt} + \frac{Q}{C} = V(t) \Rightarrow \frac{d \chi}{d \tau} + \chi = F(\tau) with substitutions 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. L \frac{d^2 Q}{dt^2} + R \frac{d Q}{d t} + \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(\Omega \tau) with the substitutions 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}}, \ \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. \left(-\frac{\hbar^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 represents probability density that, when integrated over , gives a dimensionless probability. Therefore, has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute \tilde x \equiv \frac{x}{x_{\text{c}}}, where is some characteristic length of this system. This gives us a dimensionless wave function \tilde \psi defined via \psi(x) = \psi(\tilde x x_{\text{c}}) = \psi(x(\tilde x)) = \tilde \psi(\tilde x). The differential equation then becomes \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{2 m x_{\text{c}}^2 E}{\hbar^2} \tilde \psi(\tilde x). To make the term in front of \tilde x^2 dimensionless, set \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 \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 E \equiv \frac{\hbar \omega}{2} \tilde E. The factor in front of \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 \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). == References== == External links== Analysis of differential equation models in biology: a case study for clover meristem populations (Application of nondimensionalization to a problem in biology). Course notes for Mathematical Modelling and Industrial Mathematics Jonathan Evans, Department of Mathematical Sciences, University of Bath. (see Chapter 3). Scaling of Differential Equations Hans Petter Langtangen, Geir K. Pedersen, Center for Biomedical Computing, Simula Research Laboratory and Department of Informatics, University of Oslo.

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