Advanced Mathematics and Mechanics Applications Using MATLAB phần 1 pptx

... FiniteElementandFiniteDifferenceMethods 10 .6.1MathematicalFormulation 10 .6.2DiscussionoftheCode 10 .6.3NumericalResults 10 .7VibrationModesofanEllipticMembrane 10 .7.1AnalyticalFormulation 10 .7.2ComputerFormulation 11 BendingAnalysisofBeamsofGeneralCrossSection 11 .1Introduction 11 .1. 1AnalyticalFormulation 11 .1. 2ProgramtoAnalyzeBeamsofGeneralCrossSection 11 .1. 3ProgramOutputandCode 12 ApplicationsofAnalyticFunctions 12 .1PropertiesofAnalyticFunctions 12 .2DeÞnitionofAnalyticity 12 .3SeriesExpansions 12 .4IntegralProperties 12 .4.1CauchyIntegralFormula 12 .4.2ResidueTheorem 12 .5PhysicalProblemsLeadingtoAnalyticFunctions 12 .5.1Steady-StateHeatConduction 12 .5.2IncompressibleInviscidFluidFlow 12 .5.3TorsionandFlexureofElasticBeams 12 .5.4PlaneElastostatics 12 .5.5ElectricFieldIntensity 12 .6BranchPointsandMultivaluedBehavior 12 .7ConformalMappingandHarmonicFunctions 12 .8MappingontotheExteriorortheInteriorofanEllipse 12 .8.1ProgramOutputandCode 12 .9LinearFractionalTransformations 12 .9.1ProgramOutputandCode 12 .10 Schwarz-ChristoffelMappingontoaSquare 12 .10 .1ProgramOutputandCode 12 .11 DeterminingHarmonicFunctionsinaCircularDisk 12 .11 .1NumericalResults 12 .11 .2ProgramOutputandCode 12 .12 InviscidFluidFlowaroundanEllipticCylinder 12 .12 .1ProgramOutputandCode 12 .13 TorsionalStressesinaBeamMappedontoaUnitDisk 12 .13 .1ProgramOutputandCode 12 .14 StressAnalysisbytheKolosov-MuskhelishviliMethod 12 .14 .1ProgramOutputandCode© ... animation’), pause 11 6: % print -deps penangle 11 7: 11 8:nt=length(th); z=zeros(nt ,1) ; 11 9: x=[z,sin(th)]; y=[z,-cos(th)]; 12 0: hold off, close 12 1: if trac 12 2: axis([ -1, 1, -1, 1]), axis square, ... plot(x,y,’-k’,x (1: end -1, :)’,y (1: end -1, :)’, 11 2: ’-k’,xo,yo,’-k’) 11 3: 11 4:% Add a title and axis labels 11 5: title([’Mapping of a Square Using a ’, 11 6: num2str(m),’-term Polynomial’]) 11 7: xlabel(’x axis’);...

... X=x-xd (1) ; 10 8: 10 9:if id==0 % y(x) 11 0: v=yd (1) +s1*X+m1/2*X.*X+ 11 1: powermat(x,xd,3)*C/6; 11 2: elseif id= =1 % y’(x) 11 3: v=s1+m1*X+powermat(x,xd,2)*C/2; 11 4: elseif id==2 % y’’(x) 11 5: v=m1+powermat(x,xd ,1) *C; 11 6: ... N=length(kn) -1; m=round(abs(t (1) )); 11 5: if -t (1) ==m, t=linspace (1, nd ,1+ N*m)’; end 11 6: z=[]; zp=[]; zplot=[]; 11 7: for j =1: N 11 8: k1=kn(j); k2=kn(j +1) ; K=k1:k2; 11 9: k=find(k1<=t & t<k2); 12 0: ... splined 10 8: % 10 9: 11 0:nd=length(xd); zd=xd(:)+i*yd(:); td= (1: nd)’; 11 1: if isempty(kn), kn= [1; nd]; end 11 2: kn=sort(kn(:)); if kn (1) ~ =1, kn= [1; kn]; end 11 3: if kn(end)~=nd, kn=[kn;nd]; end 11 4:...

... spans43: Width =1; E =1; a=0.5^2; Npts =10 0;44: h1=0.5; k1 =1; x1=linspace(0 ,1, Npts);45: h2 =1. 5; k2 =1; x2=linspace (1, 2,Npts);46: h3=2.5; k3 =1; x3=linspace(2,3,Npts);47: y1=(x1-h1).^2/a+k1; y2=(x2-h2).^2/a+k2;48: ... least 11 1: % square points used for the eta and xi 11 2: % directions 11 3: % nfuns - vector [meta,mxi] giving the number of 11 4: % approximating functions used for the eta and 11 5: % xi directions 11 6: ... DiskFigure 12 .2: Mapping |z| < 1 onto an Elliptic Disk© 2003 by CRC Press LLC−2 1 0 1 2−2 1. 5 1 −0.500.5 1 1.52−3−2 1 0 1 23real axisDiscontinuous Surface for imag( sqrt( z2 − 1 )...

... function of the variable xdata. 11 1: % The function has the form: 11 2: % 11 3: % y(x) = sum (1= >ntop) ( a(j)*x^(j -1) ) / 11 4: % ( 1 + sum (1= >nbot) ( b(j)*x^(j)) ) 11 5: % 11 6: % xdata,ydata - input ... tlim=[0 ,10 0]; nt=400; 10 8: end 10 9: n=size(m ,1) ; t=linspace(tlim (1) ,tlim(2),nt); 11 0: if nargin< ;10 , y0=zeros(n ,1) ; v0=y0; end 11 1: 11 2:% Determine eigenvalues and eigenvectors for 11 3: % the homogeneous ... (real or 11 7: % complex) 11 8: % ntop,nbot - number of series terms used in 11 9: % the numerator and the 12 0: % denominator. 12 1: % 12 2: % User m functions called: none 12 3: % 12 4: 12 5:ydata=ydata(:);...

... idface= [1 2 365463; 17 : 13 970000; 18 : 17 820000; 19 : 28930000; 20: 7 9 12 10 11 12 9 8; 21: 410 1260000; 22: 4 511 100000; 23: 5 612 110 000];24: polhdplt(x,y,z,idface, [1, 1 ,1] );25: [v,rc,vrr,irr]=polhedrn(x,y,z,idface)26:27:%=============================================28:29:function ... 0.95 1 1.05 1. 1 1. 15 -1. 5 -1 -0.500.5 1 1.5x axisy axisFourier Series for Harmonics up to Order 250Figure 6 .10 : Fourier Series for Harmonics up to Order 2500.9 0.95 1 1.05 1. 1 1. 15 -1. 5 -1 -0.500.5 1 1.5x ... 9.74 10 .77 12 8: 13 .06 15 .07 21. 60 25.49; 27.38 12 9: 31. 56 34.94 36.66 38.03 40.67 13 0: 41. 87; 48.40 51. 04 53.80 0 13 1: 000]’; 13 2: yft=[ 13 3: 0 0.92 -0.25 1. 00 -0.29 13 4: 0.46 -0 .16 ; -0.97...

... the Cable Midpoint0 5 10 15 20 25 10 16 10 14 10 12 10 10 10 −8 10 −6 10 −4 10 −2 10 0timeasymmetry errorGrowing Loss of Symmetry in Vertical DeflectionFigure 8 .13 : Growing Loss of Symmetry ... omegax*jaxial/jtrans 10 6: % 10 7: % zdot - the time derivative of z 10 8: % 10 9: % User m functions called: none 11 0: % 11 1: 11 2:z=z(:); r=z (1: 3); len=norm(r); ur=r/len; 11 3: 11 4:% Make certain ... the input velocity is 11 5: % perpendicular to r 11 6: v=z(4:6); v=v-(ur’*v)*ur; 11 7: vdot=-c1*(uz-ur*ur(3))+c2*cross(ur,v)- 11 8: ((v’*v)/len)*ur; 11 9: zdot=[v;vdot]; 12 0: 12 1:%============================================= 12 2: 12 3:%...

... values 11 6: % rho - vector of mass per unit volume 11 7: % values 11 8: inode= [1 1 2 2 3 3 4 3 4 5 6 7 5 6 6 6 7 7 7 11 9: 8 9 10 11 10 11 10 11 13 ]; 12 0: jnode=[3 4 3 4 4 6 6 7 7 6 7 8 9 9 10 11 10 ... 10 12 1: 11 12 12 10 11 12 13 13 14 14 14 ]; 12 2: elast=3e7*ones (1, 28); 12 3: area=ones (1, 28); rho=ones (1, 28); 12 4: 12 5:% Any points constrained against displacement 12 6: % are defined by: 12 7: ... end. 12 7: v=-[35 /12 ,-26/3 ,19 /2, -14 /3 ,11 /12 ]; 12 8: a (1, 1:5)=v; a(n,n: -1: n-4)=v; 12 9: a=diag(eiv/h^2)*a; 13 0: % Write homogeneous equations to make the 13 1: % slope and deflection vanish at x=len. 13 2:...

... bet=-kapa*c(nft)/ (1+ kapa); 16 6: sig=(-ti (1) +ti(2)-2*i*ti(3))/2; 16 7: 16 8:% Generate a and b coefficients using the 16 9: % Fourier coefficients of N+i*T. 17 0: a=zeros(np +1, 1); b=zeros(np+3 ,1) ; j= (1: np)’; 17 1: ... a(j +1) =c(nft +1- j); a (1) =alp; 17 2: a(2)=bet+c(nft); a(3)=sig+c(nft -1) ; 17 3: j=(3:np+2)’; b(j +1) =(j -1) .*a(j -1) -conj(c(j -1) ); 17 4: b (1) =conj(sig); b(2)=conj(bet); 17 5: b(3)=alp+a (1) -conj(c (1) );© ... interpolation 11 1: % through values fb corresponding to the 11 2: % points zb. Numerical evaluation of the 11 3: % integral is performed using a composite 11 4: % Gauss formula of arbitrary order. 11 5: % 11 6:...

... pd=max(ovrsiz/2*(pmax-pmin)); 31: if length(pmin)==232: range=pm( [1, 1,2,2])+pd*[ -1, 1, -1, 1];33: else34: range=pm( [1 1 2 2 3 3])+pd*[ -1, 1, -1, 1, -1, 1];35: endFunction curve2d 1: function [z,zplot,zp]=curve2d(xd,yd,kn,t)2: ... c=[3*c(: ,1) ,2*c(:,2),c(:,3)]; 10 8: sright=ppval(mkpp(b,c),xd(end)); 10 9: [b,c]=unmkpp(spline(xd,[endc(2);yd;sright])); 11 0: 11 1:case 4 11 2: % Slope at right end known. Compute left end 11 3: % slope. 11 4: [b,c]=unmkpp(spline(xd,yd)); 11 5: ... [b,c]=unmkpp(spline(xd,yd)); 11 5: c=[3*c(: ,1) ,2*c(:,2),c(:,3)]; 11 6: sleft=ppval(mkpp(b,c),xd (1) ); 11 7: [b,c]=unmkpp(spline(xd,[sleft;yd;endc(2)])); 11 8: 11 9:endFunction spterp 1: function [v,c]=spterp(xd,yd,id,x,endv,c)2:...

... FiniteElementandFiniteDifferenceMethods 10 .6.1MathematicalFormulation 10 .6.2DiscussionoftheCode 10 .6.3NumericalResults 10 .7VibrationModesofanEllipticMembrane 10 .7.1AnalyticalFormulation 10 .7.2ComputerFormulation 11 BendingAnalysisofBeamsofGeneralCrossSection 11 .1Introduction 11 .1. 1AnalyticalFormulation 11 .1. 2ProgramtoAnalyzeBeamsofGeneralCrossSection 11 .1. 3ProgramOutputandCode 12 ApplicationsofAnalyticFunctions 12 .1PropertiesofAnalyticFunctions 12 .2DeÞnitionofAnalyticity 12 .3SeriesExpansions 12 .4IntegralProperties 12 .4.1CauchyIntegralFormula 12 .4.2ResidueTheorem 12 .5PhysicalProblemsLeadingtoAnalyticFunctions 12 .5.1Steady-StateHeatConduction 12 .5.2IncompressibleInviscidFluidFlow 12 .5.3TorsionandFlexureofElasticBeams 12 .5.4PlaneElastostatics 12 .5.5ElectricFieldIntensity 12 .6BranchPointsandMultivaluedBehavior 12 .7ConformalMappingandHarmonicFunctions 12 .8MappingontotheExteriorortheInteriorofanEllipse 12 .8.1ProgramOutputandCode 12 .9LinearFractionalTransformations 12 .9.1ProgramOutputandCode 12 .10 Schwarz-ChristoffelMappingontoaSquare 12 .10 .1ProgramOutputandCode 12 .11 DeterminingHarmonicFunctionsinaCircularDisk 12 .11 .1NumericalResults 12 .11 .2ProgramOutputandCode 12 .12 InviscidFluidFlowaroundanEllipticCylinder 12 .12 .1ProgramOutputandCode 12 .13 TorsionalStressesinaBeamMappedontoaUnitDisk 12 .13 .1ProgramOutputandCode 12 .14 StressAnalysisbytheKolosov-MuskhelishviliMethod 12 .14 .1ProgramOutputandCode© ... else, p= [1; 0;0]; end 11 1: j=cross(k,p); nj=norm(j); 11 2: if nj~=0 11 3: j=j/nj; mat=[cross(j,k),j,k]; 11 4: else 11 5: mat=[[0 ;1; 0],cross(k,[0 ;1; 0]),k]; 11 6: end© 2003 by CRC Press LLC and obtain ... FiniteElementandFiniteDifferenceMethods 10 .6.1MathematicalFormulation 10 .6.2DiscussionoftheCode 10 .6.3NumericalResults 10 .7VibrationModesofanEllipticMembrane 10 .7.1AnalyticalFormulation 10 .7.2ComputerFormulation 11 BendingAnalysisofBeamsofGeneralCrossSection 11 .1Introduction 11 .1. 1AnalyticalFormulation 11 .1. 2ProgramtoAnalyzeBeamsofGeneralCrossSection 11 .1. 3ProgramOutputandCode 12 ApplicationsofAnalyticFunctions 12 .1PropertiesofAnalyticFunctions 12 .2DeÞnitionofAnalyticity 12 .3SeriesExpansions 12 .4IntegralProperties 12 .4.1CauchyIntegralFormula 12 .4.2ResidueTheorem 12 .5PhysicalProblemsLeadingtoAnalyticFunctions 12 .5.1Steady-StateHeatConduction 12 .5.2IncompressibleInviscidFluidFlow 12 .5.3TorsionandFlexureofElasticBeams 12 .5.4PlaneElastostatics 12 .5.5ElectricFieldIntensity 12 .6BranchPointsandMultivaluedBehavior 12 .7ConformalMappingandHarmonicFunctions 12 .8MappingontotheExteriorortheInteriorofanEllipse 12 .8.1ProgramOutputandCode 12 .9LinearFractionalTransformations 12 .9.1ProgramOutputandCode 12 .10 Schwarz-ChristoffelMappingontoaSquare 12 .10 .1ProgramOutputandCode 12 .11 DeterminingHarmonicFunctionsinaCircularDisk 12 .11 .1NumericalResults 12 .11 .2ProgramOutputandCode 12 .12 InviscidFluidFlowaroundanEllipticCylinder 12 .12 .1ProgramOutputandCode 12 .13 TorsionalStressesinaBeamMappedontoaUnitDisk 12 .13 .1ProgramOutputandCode 12 .14 StressAnalysisbytheKolosov-MuskhelishviliMethod 12 .14 .1ProgramOutputandCode©...