# Buckingham π theorem

Buckingham π theorem This article needs additional citations for verification. Bitte helfen Sie mit, diesen Artikel zu verbessern, indem Sie zuverlässige Quellen zitieren. Nicht bezogenes Material kann angefochten und entfernt werden. Quellen finden: "Buckingham π theorem" – news · newspapers · books · scholar · JSTOR (April 2022) (Erfahren Sie, wie und wann Sie diese Vorlagennachricht entfernen können) Edgar Buckingham circa 1886 In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π1, p2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.) The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.

The Buckingham π theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (zum Beispiel, pressure and volume are linked by Boyle's law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and the theorem would not hold.

Inhalt 1 Geschichte 2 Aussage 3 Bedeutung 4 Nachweisen 4.1 Outline 4.2 Formal proof 5 Beispiele 5.1 Speed 5.2 The simple pendulum 5.3 Other examples 6 Siehe auch 7 Verweise 7.1 Anmerkungen 7.2 Zitate 7.3 Literaturverzeichnis 7.4 Original sources 8 External links History Although named for Edgar Buckingham, the π theorem was first proved by French mathematician Joseph Bertrand[1] in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem ("the method of dimensions") became widely known due to the works of Rayleigh. The first application of the π theorem in the general case[Hinweis 1] to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892,[2] a heuristic proof with the use of series expansions, zu 1894.[3] Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892,[4] then in 1911—apparently independently—by both A. Federman[5] und d. Riabouchinsky,[6] and again in 1914 by Buckingham.[7] It was Buckingham's article that introduced the use of the symbol " {Anzeigestil pi _{ich}} " for the dimensionless variables (or parameters), and this is the source of the theorem's name.

Statement More formally, die Nummer {Anzeigestil p} of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, und {Anzeigestil k} is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.

In mathematischer Hinsicht, if we have a physically meaningful equation such as {Anzeigestil f(q_{1},q_{2},Punkte ,q_{n})=0,} wo {Anzeigestil q_{1},Punkte ,q_{n}} are the {Anzeigestil n} independent physical variables, and they are expressed in terms of {Anzeigestil k} independent physical units, then the above equation can be restated as {Anzeigestil F(Pi _{1},Pi _{2},Punkte ,Pi _{p})=0,} wo {Anzeigestil pi _{1},Punkte ,Pi _{p}} are dimensionless parameters constructed from the {Anzeigestil q_{ich}} durch {displaystyle p=n-k} dimensionless equations — the so-called Pi groups — of the form {Anzeigestil pi _{ich}=q_{1}^{a_{1}},q_{2}^{a_{2}}cdots q_{n}^{a_{n}},} where the exponents {Anzeigestil a_{ich}} are rational numbers. (They can always be taken to be integers by redefining {Anzeigestil pi _{ich}} as being raised to a power that clears all denominators.) Significance The Buckingham π theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown. Jedoch, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful".

Two systems for which these parameters coincide are called similar (as with similar triangles, they differ only in scale); they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one. Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions.

Proof Outline It will be assumed that the space of fundamental and derived physical units forms a vector space over the rational numbers, with the fundamental units as basis vectors, and with multiplication of physical units as the "vector addition" Betrieb, and raising to powers as the "scalar multiplication" Betrieb: represent a dimensional variable as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). Zum Beispiel, the standard gravity {Anzeigestil g} has units of {displaystyle D/T^{2}=D^{1}T^{-2}} (distance over time squared), so it is represented as the vector {Anzeigestil (1,-2)} with respect to the basis of fundamental units (distance, time).

Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical-units vector space.

Formal proof Given a system of {Anzeigestil n} dimensional variables {Anzeigestil q_{1},Punkte ,q_{n}} (with physical dimensions) in {Anzeigestil k} fundamental (basis) Maße, the dimensional matrix is the {displaystyle ktimes n} Matrix {Anzeigestil M} whose {Anzeigestil k} rows are the fundamental dimensions and whose {Anzeigestil n} columns are the dimensions of the variables: das {Anzeigestil (ich,j)} th entry (wo {Anzeigestil 1leq ileq k} und {displaystyle 1leq jleq n} ) is the power of the {Anzeigestil i} th fundamental dimension in the {Anzeigestil j} th variable. The matrix can be interpreted as taking in a combination of the dimensions of the variable quantities and giving out the dimensions of this product in fundamental dimensions. So the {displaystyle ktimes 1} (Säule) vector that results from the multiplication {Anzeigestil M{Start{bMatrix}a_{1}\vdots a_{n}Ende{bMatrix}}} consists of the units of {Anzeigestil q_{1}^{a_{1}},q_{2}^{a_{2}}cdots q_{n}^{a_{n}}} in terms of the {Anzeigestil k} fundamental independent (basis) units.[Hinweis 2] A dimensionless variable is a quantity that has all of its fundamental dimensions raised to the zeroth power (the zero vector of the vector space over the fundamental dimensions). The dimensionless variables are exactly the vectors that belong to the kernel {displaystyle ker M} of this matrix.[Hinweis 2] By the rank–nullity theorem, a system of {Anzeigestil n} vectors (matrix columns) in {Anzeigestil k} linearly independent dimensions (the rank of the matrix is the number of fundamental dimensions) leaves a nullity {displaystyle p=dim(ker M)} befriedigend {displaystyle p=n-k,} where the nullity is the number of extraneous dimensions which may be chosen to be dimensionless.

The dimensionless variables can always be taken to be integer combinations of the dimensional variables (by clearing denominators). There is mathematically no natural choice of dimensionless variables; some choices of dimensionless variables are more physically meaningful, and these are what are ideally used.

The International System of Units defines {displaystyle k=7} base units, which are the ampere, kelvin, zweite, metre, kilogram, candela and mole. It is sometimes advantageous to introduce additional base units and techniques to refine the technique of dimensional analysis. (See orientational analysis and reference.[8]) Examples Speed This example is elementary but serves to demonstrate the procedure.

Suppose a car is driving at 100 km/h; how long does it take to go 200 km?

This question considers {Darstellungsstil n=3} dimensioned variables: distance {Anzeigestil d,} time {Anzeigestil t,} und Geschwindigkeit {Anzeigestil v,} and we are seeking some law of the form {displaystyle t=operatorname {Duration} (v,d).} These variables admit a basis of {displaystyle k=2} Maße: time dimension {Anzeigestil T} and distance dimension {displaystyle D.} Thus there is {displaystyle p=n-k=3-2=1} dimensionless quantity.

The dimensional matrix is {Anzeigestil M={Start{bMatrix}1&0&;;;1\0&1&-1end{bMatrix}}} in which the rows correspond to the basis dimensions {Anzeigestil D} und {Anzeigestil T,} and the columns to the considered dimensions {Anzeigestil D,T,{Text{ und }}v,} where the latter stands for the speed dimension. The elements of the matrix correspond to the powers to which the respective dimensions are to be raised. Zum Beispiel, the third column {Anzeigestil (1,-1),} besagt, dass {displaystyle V=D^{0}T^{0}V^{1},} represented by the column vector {Anzeigestil mathbf {v} =[0,0,1],} is expressible in terms of the basis dimensions as {displaystyle V=D^{1}T^{-1}=D/T,} seit {displaystyle Mmathbf {v} =[1,-1].} For a dimensionless constant {displaystyle pi =D^{a_{1}}T^{a_{2}}V^{a_{3}},} we are looking for vectors {Anzeigestil mathbf {a} =[a_{1},a_{2},a_{3}]} such that the matrix-vector product {displaystyle Mmathbf {a} } equals the zero vector {Anzeigestil [0,0].} In der linearen Algebra, the set of vectors with this property is known as the kernel (or nullspace) von (the linear map represented by) the dimensional matrix. In this particular case its kernel is one-dimensional. The dimensional matrix as written above is in reduced row echelon form, so one can read off a non-zero kernel vector to within a multiplicative constant: {Anzeigestil mathbf {a} ={Start{bMatrix}-1\;;;1\;;;1\Ende{bMatrix}}.} If the dimensional matrix were not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant, replacing the dimensions by the corresponding dimensioned variables, may be written: {displaystyle pi =d^{-1}t^{1}v^{1}=tv/d.} Since the kernel is only defined to within a multiplicative constant, the above dimensionless constant raised to any arbitrary power yields another (equivalent) dimensionless constant.

Dimensional analysis has thus provided a general equation relating the three physical variables: {Anzeigestil F(Pi )=0,} oder, vermieten {Anzeigestil C} denote a zero of function {Anzeigestil F,} {displaystyle pi =C,} which can be written in the desired form (which recall was {displaystyle t=operatorname {Duration} (v,d)} ) wie {displaystyle t=C{frac {d}{v}}.} The actual relationship between the three variables is simply {displaystyle d=vt.} Mit anderen Worten, in diesem Fall {Anzeigestil F} has one physically relevant root, and it is unity. The fact that only a single value of {Anzeigestil C} will do and that it is equal to 1 is not revealed by the technique of dimensional analysis.

The simple pendulum We wish to determine the period {Anzeigestil T} of small oscillations in a simple pendulum. It will be assumed that it is a function of the length {Anzeigestil L,} the mass {Anzeigestil M,} and the acceleration due to gravity on the surface of the Earth {Anzeigestil g,} which has dimensions of length divided by time squared. The model is of the form {Anzeigestil f(T,M,L,g)=0.} (Note that it is written as a relation, not as a function: {Anzeigestil T} is not written here as a function of {Anzeigestil M,L,{Text{ und }}g.} ) Es gibt {displaystyle k=3} fundamental physical dimensions in this equation: time {Anzeigestil t,} Masse {Anzeigestil m,} and length {Anzeigestil ell ,} und {displaystyle n=4} dimensional variables, {Anzeigestil T,M,L,{Text{ und }}g.} Thus we need only {displaystyle p=n-k=4-3=1} dimensionless parameter, bezeichnet durch {Anzeigestil pi ,} and the model can be re-expressed as {Anzeigestil F(Pi )=0,} wo {Anzeigestil pi } wird von gegeben {displaystyle pi =T^{a_{1}}M^{a_{2}}L^{a_{3}}g^{a_{4}}} for some values of {Anzeigestil a_{1},a_{2},a_{3},a_{4}.} The dimensions of the dimensional quantities are: {displaystyle T=t,M=m,L=ell ,g=ell /t^{2}.} The dimensional matrix is: {Anzeigestil mathbf {M} ={Start{bMatrix}1&0&0&-2\0&1&0&0\0&0&1&1end{bMatrix}}.} (The rows correspond to the dimensions {Anzeigestil t,m,} und {Anzeigestil ell ,} and the columns to the dimensional variables {Anzeigestil T,M,L,{Text{ und }}g.} Zum Beispiel, the 4th column, {Anzeigestil (-2,0,1),} states that the {Anzeigestil g} variable has dimensions of {Anzeigestil t^{-2}m^{0}ell ^{1}.} ) We are looking for a kernel vector {displaystyle a=left[a_{1},a_{2},a_{3},a_{4}Rechts]} such that the matrix product of {Anzeigestil mathbf {M} } an {Anzeigestil a} yields the zero vector {Anzeigestil [0,0,0].} The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant: {displaystyle a={Start{bMatrix}2\0\-1\1Ende{bMatrix}}.} Were it not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant may be written: {Anzeigestil {Start{ausgerichtet}pi &=T^{2}M^{0}L^{-1}g^{1}\&=gT^{2}/Lend{ausgerichtet}}.} In fundamental terms: {displaystyle pi =(t)^{2}(m)^{0}(Ell )^{-1}links(ell /t^{2}Rechts)^{1}=1,} which is dimensionless. Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant.

In diesem Beispiel, three of the four dimensional quantities are fundamental units, so the last (welches ist {Anzeigestil g} ) must be a combination of the previous. Beachten Sie, dass wenn {Anzeigestil a_{2}} (the coefficient of {Anzeigestil M} ) had been non-zero then there would be no way to cancel the {Anzeigestil M} value; deshalb {Anzeigestil a_{2}} muss Null sein. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass {displaystyle M.} (In the 3D space of powers of mass, time, and distance, we can say that the vector for mass is linearly independent from the vectors for the three other variables. Up to a scaling factor, {Anzeigestil {vec {g}}+2{vec {T}}-{vec {L}}} is the only nontrivial way to construct a vector of a dimensionless parameter.) The model can now be expressed as: {displaystyle Fleft(gT^{2}/Lright)=0.} Assuming the zeroes of {Anzeigestil F} are discrete and that they are labelled {Anzeigestil C_{1},C_{2},Punkte ,} then this implies that {displaystyle gT^{2}/L=C_{ich}} for some zero {Anzeigestil C_{ich}} of the function {displaystyle F.} If there is only one zero, nennen {Anzeigestil C,} dann {displaystyle gT^{2}/L=C.} It requires more physical insight or an experiment to show that there is indeed only one zero and that the constant is in fact given by {displaystyle C=4pi ^{2}.} For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle. The above analysis is a good approximation as the angle approaches zero.