Advanced Mathematics and Mechanics Applications Using MATLAB phần 8 pptx

10.7.2 Computer Formulation The program elipfreq was written to compute frequencies and mode shapes for an elliptic membrane.. least squares points used, and the number of terms used in

f (η) =

N



k=1 cos(η(k − 1)) a k , g(ξ) =

The analogous approximations for the modes anti-symmetric about the x-axis are:

f (η) =

N



k=1 sin(ηk) a k , g(ξ) =

Let us choose a set of collocation points η i , i = 1, , n , and ξ j , j = 1, , m

Then substituting the series approximation for f(η) into the differential equation

gives the following over-determined system of equations:

where A is a column matrix consisting of the coefÞcients a k A similar equation

results when the series for g(ξ) is substituted into the differential equation for the

radial direction It reduces to

E B − α B + λ G B = 0.

The parameter α can be eliminated from the last two equations to yield a single

eigenvalue equation

W E  + C W = λ ( −W G  + D W ) where W = A B , and the tic mark indicates matrix transposition By addressing

the two-dimensional array W in terms of a single index, the eigenvalues λ and the

modal multipliers deÞned by W can be computed using the function eig Then the

values of the other eigenvalue parameter α can also be obtained using the known

λ, W combinations The mathematical developments just given are implementedbelow in a program which animates the various natural frequency vibration modesfor an elliptic membrane

10.7.2 Computer Formulation

The program elipfreq was written to compute frequencies and mode shapes for

an elliptic membrane The primary data input includes the ellipse semi-diameters, aßag indicating whether even modes, odd modes, or both are desired, the number of

least squares points used, and the number of terms used in the approximation series.Natural frequencies and data needed to produce modal surfaces are returned Theprogram also animates the various mode shapes arranged in the order of increasingfrequency The modules employed are described in the following table.

elipfreq reads data, calls other computational

mod-ules, and outputs modal plots

frqsimpl forms the matrix approximations of the

Math-ieu equations and calls eigenrec to generate

frequencies and mode shapes

eigenrec solves the rectangular eigenvalue problem

plotmode generates animated plots of the modal

func-tions

modeshap computes modal function shapes using the

approximating function series

funcxi approximating series functions in the xi

2) results obtained from the commercial PDE toolbox from MathWorks employing

triangular Þnite element analysis The elliptic coordinate formulation is singular for

a circular shape, but a nearly circular shape with a = 1 and b = 0.9999 causes no

numerical difÞculty Figure 10.14shows how well frequencies from elipfreq with

nlsq=[200,200] and nfuns=[30,30] compare with the roots of J n (r) The Þrst Þfty

frequencies were accurate to within 0.8 percent and the Þrst one hundred frequencies

were accurate to within 5 percent The function pdetool from the PDE toolbox was

also used to compute circular membrane frequencies with a quarter circular shapeand 2233 node points The Þrst two hundred even mode frequencies from this modelwere accurate to within 1 percent for the Þrst one hundred frequencies and to within 7

percent for the Þrst 200 frequencies Since the function pdetool would probably give comparable accuracy for an elliptic membrane, results from elipfreq were compared

with those from pdetool using an ellipse with a = 1 and b = 0.5 The percent

difference between the frequencies from the two methods appears inFigure 10.15

This comparison suggests that the Þrst Þfty frequencies produced by elipfreq for the

elliptic membrane are probably accurate to within about 2 percent

The various modal surfaces of an elliptic membrane have interesting shapes The

program elipfreq allows a sequence of modes to be exhibited by selecting vectors of

frequency numbers such as 1:10 or 10:2:20 Two typical shapes are shown inFigures10.16and 10.17 The particular modes shown have no special signiÞcance besidestheir esthetic appeal A listing of some interactive computer output and the source

code for elipfreq follows.

y axis

Figure 10.16: Surface for Anti-Symmetric Mode Number 98

y axis

Figure 10.17: Surface for Symmetric Mode Number 99

Interactive Input-Output for Program elipfreq

>> elipfreq;

VIBRATION MODE SHAPES AND FREQUENCIES

OF AN ELLIPTIC MEMBRANE

Input the major and minor semi-diameters > ? 1,.5

Select the modal form option

1<=>even, 2<=>odd, 3<=>both > ? 1

The computation takes awhile Please wait

Computation time = 44.1 seconds

Number of modes = 312

Highest frequency = 116.979

Press return to see modal plots

Give a vector of mode indices (try 10:2:20)

10: % a separation of variables formulation in elliptical11: % coordinates.

12: %

15: % nlsq - two-component vector giving the number

18: % nfuns - two-component vector giving the number of

21: % type - use 1 for even modes symmetric about the

25: %

26: % frqs - a vector of natural frequencies

28: % modes - a three dimensional array in which

32: % indx - a vector telling whether each

37: % alpha - a vector of eigenvalue parameters in

42: % cptim - the cpu time in seconds used to

45: % noplot - enter any value to skip mode plots

46: %

47: % User m functions called:

50:

51: if nargin==0

52: disp(’ ’)

53: disp(’VIBRATION MODE SHAPES AND FREQUENCIES’)

77: indx=ones(length(frqs),1);

78: [frqso,modeso,x,y,alphao,cpto]=frqsimpl( 79: a,b,2,nlsq,nfuns);

107: % a,b - ellipse major and minor semi-diameters

108: % type - numerical values of one or two for modes

110: % nlsq - vector [neta,nxi] giving the number of least

113: % nfuns - vector [meta,mxi] giving the number of

116: % frqs - natural frequencies arranged in increasing

118: % Modes - modal surface shapes in the ellipse

119: % x,y - coordinate points in the ellipse

120: % alpha - vector of values for the eigenvalues in the

157: % Compute values of the second eigenvalue

158: % parameter in Mathieu’s equation

159: alpha=zeros(1,nmax); tic;

160: s=size(modes); s=s(1:2); Vxi=Vxi’;

161:

162: % Obtain the modal surface shapes

163: Neta=91; Nxi=25; Modes=zeros(Neta,Nxi,nmax);

170: frqs=sqrt(2*frqs)/h; cptim(3)=toc;

171:

172: %==============================================173:

174: function [eigs,vecs,Amat,Bmat]=eigenrec(A,B,C,D)175: % [eigs,vecs,Amat,Bmat]=eigenrec(A,B,C,D)

176: % Solve a rectangular eigenvalue problem of the177: % form: X*A+B*X=lambda*(X*C+D*X)

178: %

179: % A,B,C,D - square matrices defining the problem.

182: % eigs - vector of eigenvalues

183: % vecs - array of eigenvectors where vecs(:,:,j)

186: % Amat,

187: % Bmat - matrices that express the eigenvalue

189: %

190: n=size(B,1); m=size(A,2); s=[n,m]; N=n*m;191: Amat=zeros(N,N); Bmat=Amat; kn=1:n; km=1:m;192: for i=1:n

209: function plotmode(a,b,x,y,eigs,modes,indx)210: %

211: % plotdmode(a,b,x,y,eigs,modes,indx)

212: % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

213: % This function makes animated plots of the214: % mode shapes of an elliptic membrane for215: % various frequencies

216: % a,b - major and minor semi-diameters217: % x,y - arrays of points defining the

219: % eigs - vector of sorted frequencies

220: % modes - array of modal surfaces for

222: % indx - vector of indices designating

230: while isempty(jlim), disp(’ ’)

231: disp([’Give a vector of mode ’,

233: jlim=input(’(input 0 to stop > ? ’);

259: function [u,x,y]=modeshap(

261: %

262: % [u,x,y]=modeshap(a,b,type,modemat,nxi,neta,H)263: % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~264: % This function uses the eigenvectors produced by265: % the rectangular eigenvalue solver to form modal266: % surface shapes in cartesian coordinates.

267: % a,b - major and minor semi-diameters

268: % type - 1 for even, 2 for odd

269: % modemat - eigenvector matrix output by eigenrec270: % nxi,neta - number of radial and circumferential

274: % u,x,y - modal surface array and corresponding

278:

279: if nargin<7, H=1; end

280: if nargin<6, neta=81; end; if nargin<5, nxi=22; end281: h=sqrt(a^2-b^2); r=atanh(b/a); x=[]; y=[];

282: xi=linspace(0,r,nxi); eta=linspace(-pi,pi,neta);283: if nargout>1

284: [Xi,Eta]=meshgrid(xi,eta); z=h*cosh(Xi+i*Eta);285: x=real(z); y=imag(z);

292:

293: %==================================================294:

303: % type - 1 for even valued, 2 for odd valued

304: % xi - vector of radial coordinate values

305: % f,f2 - matrix of function and second derivative

307:

308: xi=xi(:); nxi=length(xi); R=atanh(b/a);

309: if type==1, N=pi/R*(1/2:n); f=cos(xi*N);

310: else, N=pi/R*(1:n); f=sin(xi*N); end

311: f2=-repmat(N.^2,nxi,1).*f;

312:

313: %==================================================314:

321: % n - number of series terms used

322: % type - 1 for even valued, 2 for odd valued

323: % xi - vector of circumferential coordinate values324: % f,f2 - matrix of function and second derivative

325: % values

326:

327: eta=eta(:); neta=length(eta);328: if type==1, N=0:n-1; f=cos(eta*N);329: else, N=1:n; f=sin(eta*N); end330: f2=-repmat(N.^2,neta,1).*f;

con-of these entities is facilitated by use con-of singularity functions [9] The singularity

function of order n is denoted by < x − x0> nand is deÞned as

< x − x0> n dx = < x − x0> n+1

n + 1 .

The special case where n = −1 is appropriate for describing a concentrated load.

The term < x − x0> −1 means the limit as  → 0 of the following function

< x − x0> −1 dx = < x − x0>0.Analyzing the loads and deformations in the beam requires computation of the shear,

moment, slope, and deßection designated as v(x), m(x), y  (x), and y(x) The beam

lies in the range 0 ≤ x ≤ L A total of four end conditions are imposed at x = 0

and x = L Normally, two conditions will be speciÞed at each end; so, two known conditions applicable at x = 0 need to be found during the solution process Along with the end conditions, interior supports may exist at x = r , 1≤  ≤ N s

un-Displacements y  will occur at supports, and the reactions R , as well as four end

conditions, needed to cause the deßections will have to be determined during theanalysis Within the beam span, the applied loading will consist of known external

loads described as w e (x)and the support reactions Fundamentals of Euler beamtheory developed in standard textbooks [9, 102] imply the following differential andintegral relations:

< x − r  >1k(x) dx;

V) Deßection

y(x) = y0+ y 

0x + m0

 x0

 x0

k(x) dx dx +

v0

 x0

 x0

x k(x) dx dx +

 x0

 x0

 x0

< x − r  >1k(x) dx dx

where E(x)I(x) is the product of the Young’s modulus and the cross section ment of inertia, y0, y 

mo-0, v0, m0, are the left-end values of the deßection, slope, shear

and moment respectively The property k(x) will be spatially variable unless EI is

constant, which yields the following simple formulas

 x0

The external loading conditions employed here can handle most practical situations

It is assumed that several concentrated loads F  act at positions f , 1 ≤  ≤ N f.Distributed loads are described by linearly varying ramp loads A typical ramp load

starts at position p  with intensity P  and varies linearly to magnitude Q at position

q  The ramp load is zero unless p  ≤ x ≤ q  A total of N r ramp loads may be

present Instances where P  = Q can also occur, implying a uniformly distributedload The general external loading chosen can be represented as

and each summation extends over the complete range of pertinent values Similarly,integration using the properties of singularity functions yields

The single and double integrals given earlier involving m e (x) and k(x) can easily

be evaluated exactly when EI is constant, but these are not needed here Since

k(x) will generally be spatially variable in the target problem set, the integrations

to compute y  (x) and y(x) are best performed numerically Leaving the number of

integration increments as an independent parameter allows high accuracy evaluation

of all integrals whenever this is desirable Typically, problems using several hundredintegration points only require a few seconds to solve using a personal computer.Completing the problem solution requires formulations and solution of a system

of simultaneous equations involving v0, m0, y 

0, y0, R1, , R N s The desiredequations are created by specifying the displacement constraints at the supports, aswell as four of eight possible end conditions To present the equations more conciselythe following notation is adopted:

 x

0

k(x) dx = K1(x) ,

 x0

 x0

 x0

x k(x) dx dx = L2(x),

 x

0 m e (x) k(x) dx = I1(x) ,

 x0

and it is evident from their deÞnitions that both J1(x, r )and J2(x, r )both equal

com-at x = 0 and pin supported com-at x = L would require y(0) = 0, y (0) = 0, m(L) = 0,

and y(L) = 0 In general, conditions imposed at x = 0 have an obvious form since only v0, m0, y0, or y 

0are explicitly involved To illustrate a typical right endcondition, let us choose slope, for example This yields

11.1.2 Program to Analyze Beams of General Cross Section

A program to solve general beam problems was written which tabulates and plots

the shear, moment, slope, and deßection The driver program vdb deÞnes the data,

calls the analysis functions, and outputs the results Six functions that implement themethods given in this section were written Understanding the program details canbest be achieved by studying the code closely The program was checked extensivelyusing examples from several texts and reference books The three span beam havingparabolically tapered haunches shown inFigure 11.2 was analyzed previously by

Arbabi and Li [5] The program vdb was used to analyze the same problem and

produces the results inFigure 11.3, which agree well with the paper

We believe that the computer program is general enough to handle a wide variety

of practical problems Some readers may want to extend the program by addinginteractive input or input from a data Þle Such a modiÞcation is straightforward

11.1.3 Program Output and Code

Output from Arbabi and Li Example

Analysis of a Variable Depth Elastic Beam

-Title: Problem from Arbabi and Li

Beam Length: 3

Number of integration segments: 301

Print frequency for results: 10

| left deflection 0.0000e+000

| right deflection 0.0000e+000

EI values are specified

Solution time was 0.55 secs.

Reactions at Internal Supports:

| X-location Shear Moment Theta Delta

| -

Variable Depth Beam Program

1: function vdb

2: % Example: vdb

3: % ~~~~~~~~~~~~

4: %

5: % This program calculates the shear, moment,

6: % slope, and deflection of a variable depth

7: % indeterminate beam subjected to complex

8: % loading and general end conditions The

9: % input data are defined in the program

10: % statements below.

11: %

12: % User m functions required:

13: % bmvardep, extload, lintrp, oneovrei,

14: % sngf, trapsum

16: clear all; Problem=1;

17: if Problem == 1

18: Title=[’Problem from Arbabi and Li’];

19: Printout=10; % Output frequency

20: BeamLength=3; % Beam length

21: NoSegs=301; % # of beam divisions for

37: % EI or beam depth specification

38: EIorDepth=1; % 1=EI values specified

53: % Beam width and Young’s modulus

54: BeamWidth=[]; BeamE=[]; Depth=[]; DepthX=[];

78: I=BeamWidth*Depth.^3/12; Imin=min(I); L1=36;79: k1=BeamE*Imin/L1; k2=k1/2; k3=k1;

80: t0=10.46/k1; t1=15.33/k1; t2=22.24/k1;

82: fprintf(’\n\nValues from reference’);

84: fprintf(’\n Theta (x= 36): %12.4e’,t1);85: fprintf(’\n Theta (x=108): %12.4e’,t2);86: fprintf(’\n Theta (x=144): %12.4e\n’,t3);

133: fprintf(’\n |%4.0f %12.4e %12.4e’,

135: end

136: end

137: fprintf(’\n\nConcentrated Forces: (%g)’, 138: NoExtForce);

139: if NoExtForce > 0

141: fprintf(’\n | - - -’);142: for i=1:NoExtForce

143: fprintf(’\n |%4.0f %12.4e %12.4e’,

150: fprintf(’ X-end Load’);

151: fprintf(’\n | - - -’);152: fprintf(’ - -’);

170: fprintf(’\nEI values are specified’);

172: fprintf(’\n | - - -’);173: for i=1:NoEIorDepths

174: fprintf(’\n |%4.0f %12.4e %12.4e’,

188: fprintf(’\n |%4.0f %12.4e %12.4e’,

205: fprintf(’\nReactions at Internal Supports:’);

207: fprintf(’\n | - -’);208: for i=1:NoIntSup

238: plot(x,V,’k-’); grid; xlabel(’x axis’);

239: ylabel(’Shear’); title(’Shear Diagram’);

240: subplot(2,2,2);

241: plot(x,M,’k-’); grid; xlabel(’x axis’);

242: ylabel(’Moment’); title(’Moment Diagram’)243: subplot(2,2,3);

244: plot(x,Theta,’k-’); grid; xlabel(’x axis’);245: ylabel(’Slope’); title(’Slope Curve’);

257: function [V,M,Theta,Delta,Reactions]=

258: bmvardep(NoSegs,BeamLength,Force,ExtRamp, 259: EndCond,IntSup,EIdata,BeamProp)

260: % [V,M,Theta,Delta,Reactions]=bmvardep 261: % (NoSegs,BeamLength,Force,ExtRamp,EndCond, 262: % IntSup,EIdata,BeamProp)

263: % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~264: %

265: % This function computes the shear, moment,266: % slope, and deflection in a variable depth267: % elastic beam having specified end conditions,268: % intermediate supports with given

269: % displacements, and general applied loading,270: % allowing concentrated loads and linearly271: % varying ramp loads.

272: %

273: % NoSegs - number of beam divisions for

275: % BeamLength - beam length

276: % Force - matrix containing the magnitudes

279: % ExtRamp - matrix containing the end

282: % EndCond - matrix containing the type of

285: % left or right ends

286: % IntSup - matrix containing the location

288: % EIdata - either EI or depth values

289: % BeamProp - either null or beam widths

290: %

294: % Delta - vector of deflection values295: % Reactions - reactions at interior supports296: %

297: % User m functions required:

298: % oneovrei, extload, sngf, trapsum

299: 300:

% -301: if nargin < 8, BeamProp=[]; end

302: % Evaluate function value coordinates and 1/EI303: x=linspace(0,BeamLength,NoSegs)’;

304: kk=oneovrei(x,EIdata,BeamProp);

305:

306: % External load contributions to shear and307: % moment interior to span and at right end308: [ve,me]=extload(x,Force,ExtRamp);

309: [vv,mm]=extload(BeamLength,Force,ExtRamp);310:

311: % Deflections and position of interior supports312: ns=size(IntSup,1);

330: % slope/deflection at the right end

331: ss=smat(NoSegs,ns+3); yy=ymat(NoSegs,ns+3);332:

333: % Equations to solve for left end conditions334: % and internal reactions

335: ns4=ns+4; j=1:4; a=zeros(ns4,ns4);

336: b=zeros(ns4,1); js=1:ns; js4=js+4;

337:

338: % Account for four independent boundary

339: % conditions Usually two conditions will be340: % imposed at each end.

375: % are affected by end conditions, external

376: % loads, and support reactions.

406: end

407:

408: %=============================================409:

420: % of shear, moment, slope and deflection all421: % equal zero when x=0.

422: %

424: % Force - concentrated force matrix

425: % ExtRamp - distributed load matrix

465: end

466: end

467:

468: %=============================================469:

477: %

479: % EIdata - EI or depth values

480: % BeamProp - null or width values

% -487: if size(EIdata,1) < 2 % uniform depth case488: v=EIdata(1,1);

510: % User m functions required: none

511: 512:

% -513: if nargin < 3, n=0; end

514: x=x(:); nx=length(x); x0=x0(:)’; n0=length(x0);515: x=x(:,ones(1,n0)); x0=x0(ones(nx,1),:); d=x-x0;516: s=(d>=zeros(size(d))); v=d.*s;

538: %

539: % a,b - limits of integration

540: % y - integrand that can be a vector-valued

548: % n - the number of function values used to

552: %

553: % v - integral value

554: %

555: % User m functions called: none

556: 557:

% -558: if isstr(y)

559: % y is an externally defined function

560: x=linspace(a,b,n)’; h=x(2)-x(1);

561: Y=feval(y,x); % Function values must vary in

565: m=size(Y,2);

566: else

567: % y is column vector or a matrix

568: Y=y; [n,m]=size(Y); h=(b-a)/(n-1);

575: % function y=lintrp(xd,yd,x)

576: % See Appendix B

Chapter 12

Applications of Analytic Functions

12.1 Properties of Analytic Functions

Complex valued functions of a single complex variable are useful in various ciplines such as physics and numerical approximation theory The current chap-ter summarizes a number of attractive properties of analytic functions and presentssome applications in which MATLAB is helpful Excellent textbooks presenting thetheory of analytic functions [18, 75, 119] are available which fully develop varioustheoretical concepts employed in this chapter Therefore, only the properties whichmay be helpful in subsequent discussions are included

dis-12.2 DeÞnition of Analyticity

We consider a complex valued function

F (z) = u(x, y) + iv(x, y) , z = x + iy

which depends on the complex variable z The function F (z) is analytic at point z if

it is differentiable in the neighborhood of z Differentiability requires that the limit

exists independent of how|∆z| approaches zero Necessary and sufÞcient conditions

for analyticity are continuity of the Þrst partial derivatives of u and v and satisfaction

of the Cauchy-Riemann conditions (CRC)