Advanced Mathematics and Mechanics Applications Using MATLAB phần 5 ppsx

... 294 ADVANCED MATH AND MECHANICS APPLICATIONS USING MATLAB 0 5 10 15 20 25 0.20. 25 0.30. 35 0.40. 45 0 .5 x coordinatetimeHorizontal Position of the ... versus ( θ’(τ) / ω ) for Case OneFigure 8 .5: θ versus (θ(τ)/ω) for Case One© 2003 by CRC Press LLC0 5 10 15 20 25 30 35 40 45 5000. 05 0.10. 15 0.20. 25 Solution Error For Implicit 4th Order ... on fast 51 : % computers 52 : tic; 53 : for j=1:jmr; 54 : [tmr,xmr]=udfrevib(m,k,dsp,vel,t0,tfin,ntim); 55 : end 56 : tcpmr=toc/jmr; 57 : 58 :% Second order implicit results 59 : i2=10; h2=h/i2; tic;60:...

... 42.4439 61.7893 54 .1 856 74.1827279: 29.2909 24.2702 43.8439 63. 152 4 55 .7297 75. 6493280: 30.4733 25. 7482 45. 23 15 64 .50 84 57 . 257 7 77.1067281: 31. 650 1 27.1990 46.6081 65. 856 4 58 .7709 78 .55 55 282: 32.8218 ... 62.04 85 271: 19 .55 46 38.6843 32.1897 52 .0077 71.4639 63.6114272: 20.8070 39. 952 6 33.7166 53 .4 352 72. 850 6 65. 159 3273: 22.0470 41.21 35 35. 2187 54 . 851 7 74.2302 66.6933274: 23.2 758 42.4678 36.6990 56 . 257 6 ... 36.6990 56 . 257 6 75. 6032 68.21422 75: 24.4949 43.7 155 38. 159 8 57 . 653 8 76.9699 69.7230276: 25. 7 051 44. 957 7 39.6032 59 .0409 78.33 05 71.22 05 277: 26.9074 21.2117 41.0308 60.4194 52 .6241 72.70 65 278:...

... positions shown’) 53 : trac=input( 54 : ’in the animation, otherwise input 0 > ? ’); 55 : 56 :% Specify the initial conditions and solve the 57 : % differential equation using ode 45 58: theta0=0; ... LLC3 .5. 3TheStructuralDynamicsEquation3.6ComputingNaturalFrequenciesforaRectangularMembrane3.7ColumnSpace,NullSpace,OrthonormalBases,andSVD3.8ComputationTimetoRunaMATLABProgram4MethodsforInterpolationandNumericalDifferentiation4.1ConceptsofInterpolation4.2Interpolation,Differentiation,andIntegrationbyCubicSplines4.2.1ComputingtheLengthandAreaBoundedbyaCurve4.2.2Example:LengthandEnclosedAreaforaSplineCurve4.2.3GeneralizingtheIntrinsicSplineFunctioninMATLAB4.2.4Example:ASplineCurvewithSeveralPartsandCorners4.3NumericalDifferentiationUsingFiniteDifferences4.3.1Example:ProgramtoDeriveDifferenceFormulas5GaussIntegrationwithGeometricPropertyApplications 5. 1FundamentalConceptsandIntrinsicIntegrationToolsinMATLAB 5. 2ConceptsofGaussIntegration 5. 3ComparingResultsfromGaussIntegrationandFunctionQUADL 5. 4GeometricalPropertiesofAreasandVolumes 5. 4.1AreaPropertyProgram 5. 4.2ProgramAnalyzingVolumesofRevolution 5. 5 ... LLC3 .5. 3TheStructuralDynamicsEquation3.6ComputingNaturalFrequenciesforaRectangularMembrane3.7ColumnSpace,NullSpace,OrthonormalBases,andSVD3.8ComputationTimetoRunaMATLABProgram4MethodsforInterpolationandNumericalDifferentiation4.1ConceptsofInterpolation4.2Interpolation,Differentiation,andIntegrationbyCubicSplines4.2.1ComputingtheLengthandAreaBoundedbyaCurve4.2.2Example:LengthandEnclosedAreaforaSplineCurve4.2.3GeneralizingtheIntrinsicSplineFunctioninMATLAB4.2.4Example:ASplineCurvewithSeveralPartsandCorners4.3NumericalDifferentiationUsingFiniteDifferences4.3.1Example:ProgramtoDeriveDifferenceFormulas5GaussIntegrationwithGeometricPropertyApplications 5. 1FundamentalConceptsandIntrinsicIntegrationToolsinMATLAB 5. 2ConceptsofGaussIntegration 5. 3ComparingResultsfromGaussIntegrationandFunctionQUADL 5. 4GeometricalPropertiesofAreasandVolumes 5. 4.1AreaPropertyProgram 5. 4.2ProgramAnalyzingVolumesofRevolution 5. 5 Computing Solid Properties Using Triangular Surface Elements and UsingSymbolicMath 5. 6NumericalandSymbolicResultsfortheExample 5. 7GeometricalPropertiesofaPolyhedron 5. 8EvaluatingIntegralsHavingSquareRootTypeSingularities 5. 8.1ProgramListing 5. 9GaussIntegrationofaMultipleIntegral 5. 9.1Example:EvaluatingaMultipleIntegral6FourierSeriesandtheFastFourierTransform6.1DeÞnitionsandComputationofFourierCoefÞcients6.1.1TrigonometricInterpolationandtheFastFourierTransform6.2SomeApplications6.2.1UsingtheFFTtoComputeIntegerOrderBesselFunctions6.2.2DynamicResponseofaMassonanOscillatingFoundation6.2.3GeneralProgramtoPlotFourierExpansions7DynamicResponseofLinearSecondOrderSystems7.1SolvingtheStructuralDynamicsEquationsforPeriodicForces7.1.1ApplicationtoOscillationsofaVerticallySuspendedCable7.2DirectIntegrationMethods7.2.1ExampleonCableResponsebyDirectIntegration©...

... 0 5 10 15 20 25 30 35 40 45 50−10−8−6−4−202468x 10− 15 timey(2)RESPONSE VARIABLE NUMBER 2Figure 3.6: Motion of Mass 2 in Threemass Model© 2003 by CRC Press LLC 5 0 5 5 0 5 −8−6−4−202468x ... boundary and zero gradient elsewhere. 50 : % 51 : % func - function specifying the function 52 : % value between zero and alp 53 : % radians 54 : % alp - angle between zero and pi which 55 : % specifies ... its axis along 52 : % the z axis. 53 : % 54 : % rb,rt,h - the base radius,top radius and 55 : % height 56 : % n - vector of two integers defining the 57 : % axial and circumferential grid 58 : % increments...

... list47:48:% 49: 50 :if isempty(nquad), nquad=10; end 51 : if isempty(mparts), mparts=1; end 52 : 53 :% Compute base points and weight factors 54 : % for the single interval [-1,1]. (Ref: 55 : % ’Methods ... zcentr.2.9638 4.2 350 0.10 45 axx axz azz 53 .8408 1.7264 1.3101Results Using Function SRFVVolume = 59 .1 056 Rg = [0.91016 0.91016 0.13749]Inertia Tensor =1.0e+003 *0 .57 68 0.1167 -0.00900.1167 0 .57 68 -0.0090-0.0090 ... -0.00900.1167 0 .57 68 -0.0090-0.0090 -0.0090 1.0999>> 5 0 5 5 0 5 −0 .5 00 .5 11 .5 x axisVOLUME OF REVOLUTIONy axisz axisFigure 5. 4: Partial Volume of RevolutionProgram for Properties...

... Press LLC184 ADVANCED MATH AND MECHANICS APPLICATIONS USING MATLAB −2−10120123400 .5 11 .5 22 .5 33 .5 4x axisSurface Plot of a General Polyhedrony axisz axisFigure 5. 6: Surface ... Order 250 Figure 6.10: Fourier Series for Harmonics up to Order 250 0.9 0. 95 1 1. 05 1.1 1. 15 -1 .5 -1-0 .5 00 .5 11 .5 x axisy axisSmoothed Fourier Series for Harmonics up to Order 250 Figure ... long13: m=1; k=36;14: 15: % Use damping equal to 5 percent of critical16: c=. 05* (2*sqrt(m*k));17:© 2003 by CRC Press LLC0.9 0. 95 1 1. 05 1.1 1. 15 -1 .5 -1-0 .5 00 .5 11 .5 x axisy axisFourier...

... conditions, the magnitudes,284: % and whether values are for the© 2003 by CRC Press LLC 55 : disp(’ ’) 56 : 57 :nlsq=[300,300]; nfuns=[ 25, 25] ; 58 : 59 :v=input([’Input the major and minor ’, 60: ’semi-diameters ... 212/ 255 ]) 250 : colormap([1 1 0]) 251 : drawnow, shg 252 : end 253 : pause(1); 254 : end 255 : end 256 : 257 :%================================================== 258 : 259 :function [u,x,y]=modeshap( 260: ... y2 y3]’; 50 : EIvalue=E*Width/12*h.^3; 51 : mn=min(EIvalue); EIvalue=EIvalue./mn; 52 : else 53 : % Beam width and Young’s modulus 54 : BeamWidth=[]; BeamE=[]; Depth=[]; DepthX=[]; 55 : end 56 : elseif...

... T=[T,0;T,360]; end 151 : 152 :% Expand the boundary stresses in a Fourier 153 : % series 154 : f=pi/180; nft =51 2; np=min(np,nft/2-1); 155 : thta=linspace(0,2*pi*(nft-1)/nft,nft); 156 : 157 :% Interpolate ... closnes/bigz<.01 152 : disp(’ ’) 153 : disp([’WARNING! SOME DATA VALUES ARE ’, 154 : ’EITHER NEAR OR ON’]); 155 : disp([’THE BOUNDARY. COMPUTED RESULTS ’, 156 : ’MAY BE INACCURATE’]); disp(’ ’) 157 : end 158 : ... k=1:np+1)’); 52 : fprintf(’\nPsi=sum(b(k)*z^(-k+1), k=1:np+3)’); 53 : fprintf(’\n’); 54 : fprintf(’\nCoefficients defining stress ’); 55 : fprintf(’function Phi are:\n’); 56 : disp(a(:)); 57 : fprintf(’Coefficients...

... 54 3: 0.70 45 3.2903 0.8263 54 4: 3.2932 0.7074 0.82 95 5 45: 0.7783 3.4977 0.3767 54 6: 3.4994 0.7800 0.3 755 54 7: 0.0026 3.0000 2.9934 54 8: 0.0028 1.0000 3.0001 54 9: 0.7034 0.7107 -2.0000 55 0: 1 .51 39 ... -2.0000 55 0: 1 .51 39 1 .53 20 -0.7382]; 55 1: 55 2:%=================================================© 2003 by CRC Press LLC612 ADVANCED MATH AND MECHANICS APPLICATIONS USING MATLAB Routine Chapter ... d=sqrt(d); 53 4: r=[x(i);y(i);z(i)]; R=[X(j);Y(j);Z(j)]; 53 5: 53 6:%================================================= 53 7: 53 8:function R=rads 53 9: % R=rads 54 0: % Radii for the problem solutions 54 1: 54 2:R=[...

... dx=w(2)-w(1); 151 : ytop=ymin+.8*dy; Dy=.0 65* dy; 152 : xlft=xmin+0.04*dx; 153 : text(xlft,ytop,label) 154 : grid off, shg 155 : 156 :%============================================= 157 : 158 :function ... LLC3 .5. 3TheStructuralDynamicsEquation3.6ComputingNaturalFrequenciesforaRectangularMembrane3.7ColumnSpace,NullSpace,OrthonormalBases,andSVD3.8ComputationTimetoRunaMATLABProgram4MethodsforInterpolationandNumericalDifferentiation4.1ConceptsofInterpolation4.2Interpolation,Differentiation,andIntegrationbyCubicSplines4.2.1ComputingtheLengthandAreaBoundedbyaCurve4.2.2Example:LengthandEnclosedAreaforaSplineCurve4.2.3GeneralizingtheIntrinsicSplineFunctioninMATLAB4.2.4Example:ASplineCurvewithSeveralPartsandCorners4.3NumericalDifferentiationUsingFiniteDifferences4.3.1Example:ProgramtoDeriveDifferenceFormulas5GaussIntegrationwithGeometricPropertyApplications 5. 1FundamentalConceptsandIntrinsicIntegrationToolsinMATLAB 5. 2ConceptsofGaussIntegration 5. 3ComparingResultsfromGaussIntegrationandFunctionQUADL 5. 4GeometricalPropertiesofAreasandVolumes 5. 4.1AreaPropertyProgram 5. 4.2ProgramAnalyzingVolumesofRevolution 5. 5 Computing Solid Properties Using Triangular Surface Elements and UsingSymbolicMath 5. 6NumericalandSymbolicResultsfortheExample 5. 7GeometricalPropertiesofaPolyhedron 5. 8EvaluatingIntegralsHavingSquareRootTypeSingularities 5. 8.1ProgramListing 5. 9GaussIntegrationofaMultipleIntegral 5. 9.1Example:EvaluatingaMultipleIntegral6FourierSeriesandtheFastFourierTransform6.1DeÞnitionsandComputationofFourierCoefÞcients6.1.1TrigonometricInterpolationandtheFastFourierTransform6.2SomeApplications6.2.1UsingtheFFTtoComputeIntegerOrderBesselFunctions6.2.2DynamicResponseofaMassonanOscillatingFoundation6.2.3GeneralProgramtoPlotFourierExpansions7DynamicResponseofLinearSecondOrderSystems7.1SolvingtheStructuralDynamicsEquationsforPeriodicForces7.1.1ApplicationtoOscillationsofaVerticallySuspendedCable7.2DirectIntegrationMethods7.2.1ExampleonCableResponsebyDirectIntegration© ... values 150 : % in rows 1, 2 and 3. 151 : % n - the number of points at which 152 : % properties are to be evaluated along 153 : % the curve 154 : % 155 : % R - a 3 by n matrix with columns 156 : % containing...