Mechanical Engineer''''s Reference Book 2011 Part 17 pps

Series and transforms 17/13 wt Figure 17.13 Square wave ut Figure 17.14 Triangular wave Wl Figure 17.15 Sawtooth wave Figure 67.16 Pulse wave 17.3.11 Square wave 17.3.12 Triangular wave sin(2rt - 1)wt (2n - 1)’ 19.3.13 Sawtooth wave 17.3.14 Pulse wave 17.3.15 Fourier transforms Basic formulae: 1: U(f)exp(j2Tff)df = u(t) % U(f) = Change of sign and complex conjugates: u(t)exp(-j2~rjf)dt i‘ u(-t) % (U(-f), u*(r) ts U*(-f) Time and frequency shifts (T and 4 constant): u(t - T) ts U(f)exp(-j2~f~)exp(j2~@)u(t) s U(f - 4) Scaling (T constant): u(t/T) ts 7-qf.r) u(t)*v(t) % U(f)V(f)> u(t)v(t) % W*V(f) Products and convolutions: Differentiation: u’(t) % j27rfU(f), -j2vcu(t) 3 U’(f) du(t.a)/da s a( u - f,a)/da Integration (U(0) = 0, a and 6 real constants): l:U(T)dT 3 u(f)/JzTrf v(t,a)da ts lb V(f,a)da I’ Interchange of functions: u(t) % u(-f) Dirac delta functions: S(t) % 1 Rect(t) (unit length, unit amplitude pulse, centred on 1 = 0): rect(t) % sin .irfi~f Gaussian distribution: exp(- Trt’) % exp( -@) Repeated and impulse (delta function) sampled waveforms: exp(j2vfot) % S(f - fo) 17.3.16 Laplace transforms , Among other applications, these are used for converting from the time domain to the frequency domain. x, = jo x(t)exp(-st) 17/14 Engineering mathematics 1 . -L .,' ,,. __ Function Transform Remarks e-al sin wt cos wt sinh wt cosh wt t" H(t) H(t - T) X(t - T)ff(t - T) S(t - T) exp( -at)x(t) exp(-at) sin wt exp(-at) cos wt d2x(t) dt2 - x"(t) dnx(t) dx" - X(")(t) S s= + 02 1 - exp(-s.r) S exp( -s~)T(s) exp( -s~) sT(s) - x(0) s4(s - SX(0) - x'(0) Heaviside step function Shift in t Dirac delta function Shift in s s"i(s) - s"-'x(o) - s"-Zx'(O) AX = b in which A is the matrix of the coefficients at,, and x and b are the column matrices (or vectors) (xl . . . x,J and (b, . . . bn). In this case the matrix A is square (n x n). The equations can be solved unless two or more of them are not independent, in which case det A = IAi = 0 and there then exist non-zero solutions xi only if b = 0. If det A # 0, there exist non-zero solutions only if b # 0. When det A = 0. A is singular. 17.4.2 Matrix arithmetic If A and B are both matrices of m rows and n columns they are conformable. and A t B = C where Ci, = Aij t Bij 17.4.2.1 Product If A is an m X n matrix and B an n X I, the product AB is defined by In this case, if 1 # m. the product BA will not exist. 17.4.2.2 Transpose The transpose of A is written A' or At and is the matrix whose rows are the columns of A, i.e. (A'Iij = (Alii A square matrix may be equal to its transpose, and it is then said to be symmetrical. If the product AB exists, then (AB)' = B'A' 17.4.2.3 Adjoint The adjoint of a square matrix A is defined as B, where (B)ii = (Alii (0) - X("-')(o) -sx("-2) and Ai; is the cofactor of aii in det A. Convolution integral 6' x1(v)xz(t - u)dv+ fl(s)f&) 17.4 Matrices and determinants 17.4.1 Linear simultaneous equations The set of equations allxl + aI2x2 + . . . + alnx, = bl aZ1x1 + a22x2 + . . . + aznx,, = bl 17.4.2.4 Inverse If A is non-singular, the inverse A-' is given by A-' = adj Ndet A and A-'A = AA-' = 1 the unit matrix. (AB)-' = B-~A-] if both inverses exist. The original equations Ax = b have the solutions x = A-'b if the inverse exists. 17.4.2.5 Orthogonality A matrix A is orthogonal if AAt = 1. If A is the matrix of a coordinate transformation X = AY from variables yi to variables xi, then if A is orthogonal X'X = Y'Y, or n anlxl + ~"2x2 + . . . + annx, = b, x: = yj! ,=I i= 1 Matrices and determinants 1771 5 If two rows or two columns are interchanged, the numerical value of the determinant is unaltered, but the sign will be changed if the permutation of rows or columns is odd. If two rows or two columns are identicai, the determinant is zero. If each element of one row or one column is multiplied by k, so is the value of the determinant. If any row or column is zero, so is the determinant. If each element of thepth row or column of the determinant c, is equal to the sum of the elements of the same row or column in determinants ars and b,, then 17.4.3 Eigenvalues and eigenvectors The equation Ax = Ax where A is a square matrix, x a column vector and A a number (in general complex) has at most n solutions (x, A). The values of h are eigenvalues and those of x eigenvectors of the matrix A. The relation may be written (A - hl)~ = 0 so tha? if x # 0, the equation A - AI = 0 gives the eigen- vaiues. If A is symmetric and real, the eigenvalues are real. If A is symmetric, the eigenvectors are orthogonal. If A is not symmetric, the eigenvalues are complex and the eigenvectors are not orthogonal. 17.4.4 Coordinate transformation Suppose I and y are two vectors related by the equation y = Ax when their components are expressed in one orthogonal system. and that a second orthogonal system has unit vectors ul, u2, , . . , u, expressed in the first system. The components of x and y expressed in the new system will be x' and y', where XI = U'X, yf = U'y ' is the orthogonal matrix whose rows are the unit vectors u\. u;, etc. Then y' = U'y = U'Ax = IJ'Ax = U'AIJx' or = pl'x' where A' = U'AU Matrices 19 and A' are congruent. 17.4.5 Determinants The determinant is defined as fellows. The first suffix in a, refers to the row, the second to the column which contains ars. Denote by M, the determinant left by deleting the rth row and sth column from D, then k=l gives the value of D in terms of determinants of order n - 1. hence by repeated application, of the determinant in terms of the elements a,~s. 17.4.6 ]Properties of determinants If the rows of larsl are identical with the columns of ibsrl, arr = b,, and larsl = IbV that is, the transposed determinant is equal to the original. The addition of any multiple of one row (or column) to another row (or column) does cot alter the value of the determinant. 17.4.6.1 Minor If row p and column q are deleted from larsl. the remaining determinant M,, is called the minor of a,,. 17.4.6.2 Cofactor The cofactor of ap4 is the minor of aq4 prefixed by the sign which the product M,,a,, would have in the expansion of the determinant, and is denoted by Ap,: A,, = (- l)P+qMp, A determinant a,, in which a,, = a,, for al! i and j is called symmetric. whilst if a,, = -a,[ for all i and j, the determinant is skew-symmetric. It follows that a,, = 0 for all i in a skew- symmetric determinant. 17.4.7 Numerical solution of linear equations Evaluation of a determinant by direct expansion in terms of elements and cofactors is disastrously slow. and other methods are available, usually programmed on any existing computer system. 17.4.7.1 triangular or to diagonal form The system of equations may be written Reduction of determinant or matrix to upper . . fl22 . . all2 ' ' The variable x1 is eliminated from the last n - 1 equations by adding a multiple -ajl/all of the first row to the ith, obtaining all a12 . . . where primes indicate altered coefficients. This process may be continued by eliminating x2 from rows 3 to n, and so on. 17/16 Engineering mathematics Eventually the form will become A linear differential equation is one which is linear in the dependent variable and its derivatives, having the general form all a12 . ' ' a In 0 ai2 . . . ain 0 0 a;,, . . . x,, can now be found from the nth equation, substituted in the (n - 1)th to obtain x,-1 and so on. Alternatively the process may be applied to the system of equations in the form Ax = Ib where I is the unit matrix, and the same operations carried out upon 1 as upon A. If the process is continued after reaching the upper triangular form, the matrix A can eventually be reduced to diagonal form. Finally, each equation is divided by the corresponding diagonal element of A, thus reducing A to the unit matrix. The system is now in the form Ix = Bb and evidently 6 = A-'. The total number of operations required is O(n3). 17.5 Differential equations A differential equation is an equation involving a dependent variable and its derivatives with respect to one or more independent variables. An ordinary differential equation is one in which there, is only one independent variable - conventionally x or t. A partial differential equation is one in which there are several independent variables. 17.5.1 Notation and definitions An ordinary differential equation with y as dependent variable and x as independent variable has the general form f[x;y,%.$, . . . }=0 where f{ } represents some specified function of the argu- ments. Solving a differential equation involves obtaining an explicit expression for y as a known function of x. The order of a differential equation is the order of the highest derivative appearing in it. Thus d'y dy du2 dx - + 3 - + 6y = 6 is a second-order equation. A differential equation of order n has a general solution containing n arbitrary constants. Speci- fied values of the dependent variable and/or its derivatives which allow these arbitrary constants to be determined are called boundary conditions or (when the independent variable is t and the values are given at t = 0) initial conditions. Boundary conditions in which the dependent variable or its derivatives are assigned zero values are called homogeneous boundary conditions. A solution in which the arbitrary constants take definite values is called a particular solution. wherepo (x) . . . p,(x) and f(x) are specified functions of x. If f(x) # 0 the differential equation is said to be inhomogeneous. If f(x) = 0, so that the differential equation is said to be homogeneous. In a partial differential equation the independent variables are normally variables defining spatial position plus (possibly) time. A particular solution of a partial differential equation requires the definition of a solution region with a bounding curve or bounding surface, together with the specification of suitable boundary conditions on that curve or surface. A partial differential equation, like an ordinary differential equation, may be linear or non-linear, and a linear partial differential equation may be homogeneous or inhomogeneous. Boundary conditions, specifying values of the dependent variable and/or its derivatives, may also be homogeneous or inhomogeneous. 17.5.2 Ordinary differential equations: analytical solutions Simple analytical solutions exist for first-order linear differential equations and for linear equations of higher order with constant coefficients. 17.5.2.1 First-order linear equations A first-order linear differential equation has the general form pl(x)(dyldx) + po(x)y = f(x), which can be written as This equation has the general solution (17.3) (17.4) where C is an arbitrary constant. The function known as the integrating factor. is 17.5.2.2 Linear equations with constant coefficients Homogeneous equations A second-order homogeneous linear differential equation with constant coefficients has the general form d2Y dY dx* dx a- + 6- + cy = 0 The general solution is y = CleAIX + C2ehzx (17.5) (17.6) where A,, A? are the roots of the auxiliary equation aA2 + bh + c = 0 and C1, C2 are arbitrary constants. If the roots of the auxiliary equation are complex, with values AI = a + Ip, A? = a - ip, it is more convenient to write the general solution of the differential equation in the form y = e"(C1 cos px + C2 sin px) (17.7) Differential equations 17/17 tions. This procedure generates a linear differential equation (with order equal to the sum of the orders of the original equations) for one of the dependent variables: after solution of this equation the other dependent variables can be obtained by back-substitution. Inserting the initial or boundary conditions A linear differential equation of order n has a general solution If the roots are equal. i.e. hl = h2 = A, say, then the general solution is y = e"(C'l + @) (17.8) where again Ci, C2 are arbitrary constants. The solution of third- and higher-order homogeneous equations follows a similar pattern, the auxiliary equation being a polynomial equation in h of appropriate degree. Inlzomogeneous equations A second-order inhomogeneous linear differential equation with constant coefficients has the general form d'y dy dx- dx a 7 + b - + cy = f(x) (17.9) where f(x) is a specified function. The general solution of equation (17.9) is the general solution of the homogeneous equation (17.5) containing two arbitrary constants (this solution is called the coinpleinentary funcrion) plus a function (ca!led the particuhr integral) which, when substituted into equation (17.9), gives the correct function f(x) on the right- hand side. For many simple right-hand sides the particular integral can be found by replacing y in the differential equation by a 'trial solution' containing one or more unknown parameters, here written as CY, p, etc. Right-hand side: f(.r) constant (Y Trial solution: y(x) x"(n integral) e kr xe k.r (a.r I- p)ekx x"e kx sin kx cos kx J axn + pxn-1 + . . . Lye kr (ax" + ox"-' + . . . )e';. a sin kx + p cos kx. (If only even differential co- 1 .( efficients occur in the differential equation then c sin kx or p cos kx is sufficient.) Equating the coefficients of the functions on the two sides of the equation gives the values of the parameters. This technique can also be used to solve equations of third and higher orders. If f(x) has the same form as one of the terms in the complemixtary function then the substitution y = uf(x) should be made, where II is an unknown function of x. This substitution generates a simple differential equation for u(x). Simziltnnroits linear differential equations The analysis of a linear mechanical or electrical system with several degrees of freedom may require the solution of a set of simultaneous linear differencial equations. in which there is one independent variabie (normally time) and several dependent variables. Iil 'cases where the equations have constant coefficients, as in the example dll dl' -+3-+u-v=r' dt dr d I' dr - 2ti + 3v = 0 the equations can be solved by a procedure very similar to the elimination method fo; soiving sets of linear algebraic equa- where &(x) is the particular integral and C,&(x) + C?!f?(x) + . . . + C,!f,(x) is the complementary function. Once this general solution has been found, the values of the n constants C1, . . . , C, can be obtained by imposing n boundary or initial conditions, i.e. n values of y and/or its derivatives at particular values of x. If all the boundary conditions are specified at a single value of x the problem is referred to as a one-point boundary-va!ue problem or, if the independent variable is t and the conditions are specified at t = 0, as an initial-value problem. Initial value problems can also be solved by the use of Laplace transforms (see Section 17.3.16). The Laplace transform method determines a particulzr solution of a differential equation. with the initial conditions inserted, rather than the general solution (17.10). Impulse and frequency responses: the convolution integral The solution of the differential equation d"Y dY a, - + . . . + al - + aoy = f(r) dt" dt (17.11) for a general function of time f(t) with homogeneous initial conditions d"-'Y - d"-2Y - - 'Y - y = 0 at I = 0 dt dtn-1 dtn-2 ' ' . can be obtained from the impulse response g(t), which is the solution of the differential equation with the same initial conditions when f(t) = 6(t). (6(t) is the Dirac 6-function, defined by the equations 1: G(t)dt = 1; 6(t) = 0 if t f 0.) The impulse response can be obtained by solving the homogeneous equation d nY dY a,-+ + al-+aoy=O dtn dt (17.12) with initial conditions (d"-'y)/(dt"-') = 1/a,,, (d"-2)/(dt"-2) . . . = dyldt = y = 0 at 1 = 0. Alternatively, it can be found by the use of Laplace transforms. The solution of equation (17.11) for an arbitrary right-hand side f(~) is given in terms of the impulse response g(t) by the convolution integra! - - (17.13) This integral is symmetric in the functions g and f, and can therefore be written in the alternative form Y(t) = ~'f(T)S(f - 7)dT (17.14) If f(t) = elw' and equation (17.11) represents a stable system (i.e. the complementary function has no exponential terms with positive real part) then as r + 33 the solution tends to the 'steady state' form y(f) = G(o)e'"'. The complex function 17/18 Engineering mathematics G(w) is called the frequency response of the system. It may be obtained from the differential equation by substituting the trial solution y = ae'"' or from the impulse response by the use of equation (17.13). The latter derivation gives the result G(w) = g(T)e-iWTdT (17.15) This equation states that the frequency response G(w) is the Fourier transform of the impulse response g(t) (see Section 17.3.15). 17.5.2.3 Linear equations with variable coefficients Second- and higher-order linear equations with variable coefficients do not, in general, have solutions which are expressible in terms of elementary functions. However, there are a number of second-order equations which occur fre- quently in applied mathematics and for which tables of solutions exist. Sub-routines for generating these solutions are available on most scientific computers. Two of the most important of these equations are 1 d'y dY Bessel's equation: x' - + x - + (A2x2 - n')y = 0 du2 dx (17.16) d'y dy (1 - x') - - 2x - dx2 dx Legendre's equation: + n(n + l)y = 0 (17.17) In certain other cases an equation with variable coefficients can be converted into one with constant coefficients by means of a change of variable. In general. however, solutions of linear differential equations with variable coefficients can only be obtained by approximate methods. 17.5.3 Ordinary differential equations: approximate solutions Appi >ximate solutions of differential equations can be obtained I v graphical, numerical or analytical methods. 17.5.3.1 A graphical method for first-order equations A graphical solution of the general first-order equation dyldx = f(x,y) can be obtained as follows. A series of curves f(x,y) = clr c2, . . . , cir . . . (termed isoclines) are drawn in the x, y plane, where the c's are suitable constants. On each isocline line-segments are drawn with slope equal to the associated value of ci: these segments give the direction of the solutions as they cross the isocline. The general form of these solutions can be obtained by joining up the segments to form continuous curves. A simple example is shown in Figure 17.17, which illustrates the solution of the differential equation dyldx = - x/y. The isoclines -x/y = cl, c2, . . . , ci, . . . are straight lines through the origin, and the segments which form part of the solutions are always perpendicular to the isoclines. It is clear from the figure that the solutions are circles centred on the origin: this is easily verified analytically. 17.5.3.2 Approximate numerical methods Derivatives and differences If a continuous function y(x) is sampled at a series of equally spaced points xo, . . , , x,, . . . , x~ to give a set of values yo, . . . , yn, . . . , y~ then it follows from the definition of a differential coefficient that Y Figure 17.17 Isoclines for the differential equation dyidx = - x/y (17.18) or alternatively (17.19) where h is the sampling inverval, as shown in Figure 17.18. Taking the difference of the two equations (17.18) and dividing by h gives (17.20) and the process can be continued in a similar way to give approximations to (d3yldx3),+,/?, etc. The quantities (y1 - yo), . . . (y,l+l - y,), . . . (y~ - y,+1) are termed the first yt ; Y"+l I Yn I I I I I I I Yn-9 I I I Xn -1 Xn X"+1 X Figure 17.18 Approximate representations of dyldx Differential equations 671 9 the solution over the next interval. The truncation error in a single step is O(h2). If the step-length h is kept constant over a given range 0 6 i 6 T the number of steps is Tlh, so that the trauncation error over the range is O(h). (The round-off error increases with the number of steps, so that there is an optimum value of h which minimizes the total error.) The accuracy of the Euler procedure can be improved by using equation (17.24) as a 'predictor' to obtain an approximate value Y:+~, which is then inserted in a suitable 'corrector' formula to generate a more accurate value of y,+l. A simple predictor/qorrector pair is Predictor y:+1 = y, + hf(f.,yn) (17.25) Corrector Y,+I = Yn + h(f(tn,yn) + f(tn+i.Yn*+i)l/2 One of the most popular predictorxorrector procedures is the Runge-Kutta. A single step of the procedure involves four evaluations of f(t,y) in accordance with the formulae differences of the set of values y,, tlhe quantities . . . (yn+l - 2yn + yn-l), . . . the second differences, and so on. The role of differences in numerical analysis is similar to that of differential coefficients in calculus. Two-poi,at boundary-value problems An approximate solution of the second-order linear differential equation (17.21) with boundary conditions y = yo at x = 0. y = YN at x = a can be found by dividing the solution range 0 6 x S a into N equal intervals and replacing the continuous function y(x) by a set of N + 1 quantities yrr = y(x,). (n = 0, . . . , N), where x, = nh and h = a/N. Replacing the differential coefficients in equatnon (17.21) by the approximations (17.19) and (17.20) gives PZ(X,)(.Y,+I - 2yv + yn-I) + hPl(xn)cVn+i - ~n-iP + h2po(.rn)yn 1 Axir) (n = 1,. . . ,N- 1) (17.22) Setting up an equation of this form at each of the points xl, . . . , x, ~ produces a set of n - 1 simultaneous linear algebraii: equations which can be solved for the unknown function values yl, . . . , yN-l (the values of yo and y~ which appear in these equations are known from the boundary conditions). intermediate values of y(x) can be found subse- quently by interpolation. Initial-volue problems The general first-order differential equation (17.23) with initial condition y = yo at t = fo can be solved by a step-by-step procedure in which approximate function values yl, y2, , . are computed successively at t = tl,t2, . . . The simplest step-by-step procedure is due to Euler and involves the replacement of the differentiad equation (17.23) by the approximation Y*,+I = ~n + hf(En,Yn) (n = 0, 1, 2, . . .) (17.24) where h is equal to the interval - t,. As shown in Figure 17.19 this procedure takes the tangent at each solution point as Y Yn+r Yn I I I I I I I L f" rn+l t Figure 1'7.19 Euler's approximate integration procedure a3 = hf(t, + h/2,y, + a2/2), a4 = hf(i,, + h,y, + a3) the final value of Y,+~ being Y,+~ = y, + {al + 2a2 + 2a3 + a4}/6 (1 7.26) The error per step is S(h5), so that the error over a given range of t is O(h4). A computer sub-routine for the Runge- Kutta procedure normally requires a user-supplied sub-routine to evaluate f(t,y) for specified values of t and y. An initial-value problem involving a differentia! equation of second or higher order can be solved by reducing the differential equation to a set of first-order equations. For example, the third-order non-linear equation can be solved by introducing the additional variables u and v and writing the equation as This set of first-order equations for the three variables u, v and y can be solved by any of the methods described above, the step-by-step procedure being carried forward simultaneously for each of the variables. 17.5.3.3 Approximate analytical methods An approximate solution of a linear differential equation can also be obtained by choosing a set of M basis functions B,,(x) and expressing the unknown solution y(x) as M y(x) = ClB](X) + , . . + CMBM(X) = rn-1 There are a number of methods based on this approach. They may be classified according to the choice of basis functions B,(x) and the procedure used to find the constants c,. The most important sets of basis functions are the integra: powers of x (which generate power-series approximations) and the harmonic functions sin mx and cos mx (which generate Fou- rier approximations). In the following account the equation to be solved is written as 2y = w(x) (17.28) where 9 represents a specified linear differential operator and w(x) is a specified function of x. It is assumed that a solution is required in an interval p S x < q and that sufficient homoge- 17/20 Engineering mathematics neous boundary conditions are specified at x = p and x = q to make the solution unique. It is further assumed that each of the approximating functions B,(.r), . . , , B,(x) satisfies these boundary conditions. In general the approximation (17.27) will not be capable of satisfying the differential equation (17.28) exactly, whatever values are assigned to the constants ci: there will be an error function (17.29) where b,(x) = Y{B,(x)}. Two procedures for finding sets of constants which make the error E(X) 'small' are Collocation and Galerkin's method. In the Collocation method the constants c, are obtained by making c(x) zero at a selected set of points xk (k = 1, . . , , M) in the interval p S x S q. This generates a set of M simultaneous equations M b,(Xk)C, = W(Xk) (k = 1, . , 1 M) (17.30) which can be solved for the M constants. In Galerkin's method the constants c, are obtained by making ~(x) orthogonal to the M basis functions B(x), m=l (k=l, , M) These equations can be written in the form m=l JP JP (k=l , M) (17.31) Equation (17.31), like equation (17.30), represents a set of M linear algebraic equations for the unknown constants c,. If the differential operator 2 is self-adjoint (a condition satisfied in most practical applications of the method) the coefficients [q Bk(x)bm(x)dx form a symmetric matrix. If, in addition, the functions B,(x) are chosen to be the normalized eigenfunctions of the differential operator 3, so that 3{B,(x)} = b,(x) = A,B,,(x), then equation (17.31) takes the simpler form Ck = lq Bk(X)W(X)&IAk (k = 1, . . . , M) (17.32) with each constant ck depending only on the corresponding function B&). 17.5.4 Partial differential equations Linear partial differential equations can be classified as elliptic, hyperbolic or parabolic. An elliptic differential equation is one in which the boundary conditions imposed on each segment of the boundary affect the solution at all points in the solution region or, conversely, one in which the solution at any point depends on the boundary conditions over the whole (b) Figure 17.20 Partial differential equation types: (a) elliptic, (b) hyperbolic boundary, as shown in Figure 17.20(a). The commonest elliptic equation is Laplace's equation J2$ J2b d'4 -+-+-=o Jx' dy' Jz' (17.33) which is the equation governing gravitational fields in free space, steady heat and electrical conduction. seepage flow in soils, etc. The inhomogeneous form of Laplace's equation is Poisson's equation (17.34) where u is a known function of position. This equation governs gravitational fields in regions containing distributed matter, heat conduction in the presence of distributed heat sources, etc. Another elliptic differential equation of interest to mechanical engineers is the bi-harmonic equation governing the bending of an initially flat plate: -+2-+-= a44 a44 a44 -qlD Jx' ax2Jy' Jy4 (17.35) where 4 is the transverse displacement of the plate, q is the known distribution of transverse load and D is a constant representing the stiffness of the plate. Equations (17.33)-(17.35) can also be written in the more general form V24 = 0, V24 = -cr, 044 = -q/D, where V2 is the Laplacian operator of vector calculus. This operator takes various forms, depending on the coordinate system (Carte- sian, cylindrical polar, spherical polar, etc.) used to define the solution region. A hyperbolic differential equation is one in which the boundary conditions on a segment of the boundary only affect a part of the solution region or, conversely, one in which the solution at any point only depends on the boundary conditions over part of the boundary, as shown in Figure 17.20(b). The commonest hyperbolic differential equation is the wave equation J'4 1 J24 1 J24 or, more generally. V24 = - - ax2 a dt2 a2 a7t' (17.36) - - _- - Differential equations 17/21 which governs the propagation of sound and other waves in both fluids and solids. Another common partial differential equation is the diffusion equation " (17.37) $4 I 84 - or, more generally, V24 = - - a2 a' at a at - - which governs, for example, the unsteady flow of heat in solids. The diffusion equation is an example of a parabolic differential equation. Such equations can be thought of as lying on the borderline between elliptic and hyperbolic forms. 17.5.4.1 Simple analytical solutions exist for linear partial differential equation:; with constant coefficients. For example, Laplace's eqaation in two dimensions is satisfied by both the real and imaginary parts of any analytic function f(z), where z is the complex variable x + jy. This fact allows many two- dimensional field problems to be solved by a technique known as conformal mapping. Similarly, the one-dimensional wave equatioil Analytical solutions: separation of variables a% 1 $4 ax2 a' at2 has solutions of the form f(x k at), where f is an arbitrary differentiable function. These solutions represent waves of arbitrary shape travelling along the x axis. Analytical solutions of linear partial differential equations can be obtained by using the method of separation of variables. For a differential equation whose dependent variable is + and whose independent variables are x and y this method involves assuming a solution of the form 4 = X(x)Y(y), where X is an unknown function of x only and Y is an unknown function of y only. Substitution of this solution into the differential equation yields ordinary differential equations for the functions X and Y, which can be solved by methods described in Section 17.5.2.2. Typical examples of separable solutions are the function - - - which satisfies both the two-dimensional Laplace equation and the homogeneous plate bending equation and the function which satisfies the one-dimensional diffusion equation. Separalble solutions always contain an arbitrary parameter h called the separation constant. The imposition of boundary conditions on a solution may result in only certain values of A being permissible. In such cases more general solutions can often be built up by combining a number of basic solutions involving these values of A. For example. the solution of the one-dimensional diffusion equation given above implies the existence of a more general solution 1 4 = e-'";'(A. cos A& + B, sin A&) n = which can be made to fit a variety of boundary conditions by suitable 'choice of the constants A,, and B,. Figure 17.21 A finite-difference mesh 17.5.4.2 Numerical solutions: the finite-difference method The finite-difference method for solving partial differential equations is similar to the numerical technique for solving ordinary differential equations with two-point boundary conditions described in Section 17.5.3.3. The following example shows how the method can be used to find the steady-state distribution of temperature within the L-shaped region shown in Figure 17.21 when the temperature variation on the boundary of the region is given. In this problem the temperature + satisfies the two-dimensional Laplace equation a2+ a2+ ax2 a)12 +-=O - with appropriate values of 4 specified on the boundary. The region is first covered with a uniform grid of squares, as shown in the figure. The intersections of the grid lines within the solution region are called nodal points and the values of 4 at these points are called nodal values: it is these values which are determined by the method. At each nodal point the partial derivatives which make up the differential equation are replaced by differences, using an appropriately amended version of equation (17.20). This operation converts the partial differential equation into a linear algebraic equation involving the nodal values at the chosen nodal point and its four nearest neighbours. If these points are labelled as shown in Figure 17.22 then the linear equation associated with the point p is (4q + 4r + 4s + 4, - 44,w2 = 0 (17.38) :+ t Figure 17.22 Nodal points associated with the difference equation for node p [...]... given by (17. 53) (17. 58) A special case of this probability law is wher, events are mutually exclusive, i.e the occurrence of one event prevents the other from happening Then P(A or B) = P(A) 17. 6.6.2 + P(B) - P(A and B) + P(B) (17. 59) If A and B are two events then the probability that they may occur together is given by (17. 60) P(A and B) = P ( A ) X P(BIA) or P(A and B) = P(B) x P(AIB) (17. 61) P(BIA)is... 1) x ( n - 2) x x 3 x 2 x 1 Using this the number of combinations of n items from a group of n is given by n! "C, = ~ r!(n - r)! (17. 48) (17. 45) X 17. 6.4 Skewness The distribution shown in Figure 17. 26 is symmetrical since the mean, median and mode all coincide Figure 17. 27 shows a skewed distribution It has positive skewness although if it bulges the other way, the skewness is said to be negative... part and m is negative if the line slopes the other way to that shown in Figure 17. 28 If an event A occurs n times out of a total of m cases then the probability of occurrence is stated to be v=mx+c The best straight line to fit a set of points is found by the method of least squares as (17. 51) (17. 52) where n is the number of points The line passes th1ough the mean values of x and y , Le X and 8 17. 6.7... m/s, 10 m/s and 15 m/s, then the mean speed would be given by the arithmetic mean as ( 5 10 15)/3 = 10 m/s + + 17. 6.3 Dispersion 17. 6.2 Averages 17. 6.3.1 Range and quai-tiles 17. 6.2.1 Arithmetic mean The arithmetic mean of n numbers x,, x2 x3 x , is given by xi + x? 4- x j + x, x=n + OK (17. 40) The arithmetic mean is easy to calculate and it takes into account all the figures Its disadvantages... equation (17. 63) P(B~T) = 0.2 0.2 X x 0.8 0.8 + 0.8 X 0.3 = 0.4 The probability of an event occurring m successive times is given by the Poisson distribution as c-"P = (nP)"X (17. 68) The mean P ( M ) and standard deviation P(S) of the Poisson distribution are given by P ( M ) = np P(S) = (np) (17. 69) (17. 70) Poisson probability calculations can be done by the use of probability charts as shown in Figure 17. 29... distribution as p ( m ) = 'lC,,pmq"-m (17. 65) Expectat ion Figure 17. 29 Poisson probability paper Y The binomial distribution is used for discrete events and is applicable if the probability of occurrence p of an event is constant on each trial The mean of the distribution B ( M ) and the standard deviation B(S) are given by B(M) = np (17. 66) B(S) = (npq)l" (17. 67) 17. 6.8.2 Poisson distribution The Poisson... chance 17. 6.10.3 Significance of correlation The significance of the product moment correlation coefficient of equations (17. 53) or (17. 54) can be tested at any confidence level by means of the standard error of estimation given by equation (17. 55) An alternative method is to use the Student t test of significance This is given by t= 0.050 Number of accidents Time (24 hour clock) r(n - 2)”’ (1 - ,.)I/ (17. 84)... imposes conditions of continuity on the temperature 4.If two elements have nodes p and q in common, as shown in Figure 17. 25, then they share the same Sum of nodal flows = 0 U Figure 17. 23 A finite-element mesh Figure 17. 25 Adding up the nodal flows in a finite-element mesh Statistics 17/ 23 nodal values +p and + q Furthermore, since 4 is linear within each element, 4 3s also continuous on their common... the negative sign, Table 17. 1 gives the probability of lamps not failing as 0.977 so that the probability of failure is 1 - 0.977 or 0.023 Therefore 5000 x 0.023 or 115 lamps are expected to fail after 800 hours 17. 6.8.4 Exponential distribution The exponential probability distribution is a continuous distribution and is shown in Figure 17. 31 It has the equation 1 Y = - e-x/2 (17. 74) where t is the mean... 0.7 then from Table 17. 2 the area beyond 0.7 is 0.497 which is the probability that the equipment will still be working after 700 hours 0.08 0.0 0 L 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 How many lamps will fail in the first 800 hours? from equation (17. 72) 1.9 2.0 2.1 2.2 2.3 2.4 25 2.6 2.7 2.8 2.9 3.0 Y X Figure 17. 31 The exponential curve Table 17. 2 Area under the exponeritiai . and transforms 17/ 13 wt Figure 17. 13 Square wave ut Figure 17. 14 Triangular wave Wl Figure 17. 15 Sawtooth wave Figure 67.16 Pulse wave 17. 3.11 Square wave 17. 3.12 Triangular. that Y Figure 17. 17 Isoclines for the differential equation dyidx = - x/y (17. 18) or alternatively (17. 19) where h is the sampling inverval, as shown in Figure 17. 18. Taking the. ~ (n - r)! (17. 49) Statistics 17/ 25 or Z(x - f)(y - y) [qx - q’qy - y)’]’!? r= (17. 54) Figure 17. 28 A scatlev diagram .6 Regression and correlation 17. 6.6.1 Regression

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