Advanced Mathematics and Mechanics Applications Using MATLAB phần 8 pptx

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 ELLIPTICAL COORDINATE SYSTEM x axis y axis Figure 10.13: Elliptic Coordinate Grid and g  (ξ) − [α −λ cosh(2ξ)]g(ξ)=0, 0 ≤ ξ ≤ R where the eigenvalue parameters α and λ are determined to make f(η) have period 2π and make g(ξ) vanish at ξ = R. The modal functions can be written in terms of Mathieu functions as products of the form ce(η, q)Ce(ξ,q) for modes symmetric about the x-axis and se(η, q)Se(ξ, q) for modes anti-symmetric about the x-axis. The functions ce and se are periodic Mathieu functions pertaining to the circumferential direction, while Ce and Se are modiÞed Mathieu functions pertaining to the radial direction. The structure of these functions motivates using the following series approximation for the functions for even modes: f(η)= N  k=1 cos(η(k − 1)) a k ,g(ξ)= M  l=1 cos( πξ R (l − 1/2)) b l . © 2003 by CRC Press LLC The analogous approximations for the modes anti-symmetric about the x-axis are: f(η)= N  k=1 sin(ηk) a k ,g(ξ)= M  l=1 sin( πξ R l) b l . Thus the expressions for both cases take the form: f(η)= N  k=1 f k (η) a k and g(ξ)= M  l=1 g l (ξ) b l . 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: N  k=1 f  k (η i )a k + α N  k=1 f k (n i )a k − λ cos(2η i ) N  k=1 f k (η i )a k =0,i=1, ,n. Denote F as the matrix having f k (η i ) as the element in row i and column k. Then multiplying the last equation on the left by the generalized inverse of F gives a matrix equation of the form CA+ αA− λDA =0, 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 EB− αB+ λGB =0. The parameter α can be eliminated from the last two equations to yield a single eigenvalue equation WE  + CW = λ (−WG  + DW) where W = AB  , 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 implemented below in a program which animates the various natural frequency vibration modes for 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, a ßag indicating whether even modes, odd modes, or both are desired, the number of © 2003 by CRC Press LLC 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. The program also animates the various mode shapes arranged in the order of increasing frequency. 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 vari- able funceta approximating series functions in the eta vari- able The accuracy of the formulation developed above was assessed by 1) comparison with circular membrane frequencies known in terms of Bessel function roots and 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 =1and b =0.9999 causes no numerical difÞculty. Figure 10.14 shows 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 shape and 2233 node points. The Þrst two hundred even mode frequencies from this model were 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 =1and b =0.5. The percent difference between the frequencies from the two methods appears in Figure 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 in Figures 10.16 and 10.17. The particular modes shown have no special signiÞcance besides their esthetic appeal. A listing of some interactive computer output and the source code for elipfreq follows. © 2003 by CRC Press LLC 0 10 20 30 40 50 60 70 80 90 100 10 −3 10 −2 10 −1 10 0 10 1 COMPARING MEMBRANE FREQUENCIES FOR (a,b) = (1, 0.9999) WITH A CIRCLE Frequency Number Percent Difference Figure 10.14: Comparing Elipfreq Results with Bessel Function Roots 0 20 40 60 80 100 120 140 160 180 200 10 −2 10 −1 10 0 10 1 10 2 COMPARISON OF RESULTS FROM ELIPFREQ AND PDETOOL frequency number percent difference Figure 10.15: Comparing Elipfreq Results with PDE Toolbox © 2003 by CRC Press LLC −1 −0.5 0 0.5 1 −0.5 0 0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x axis ODD MODE 98, OMEGA = 43.85, B/A = 0.5 y axis u(x,y) Figure 10.16: Surface for Anti-Symmetric Mode Number 98 © 2003 by CRC Press LLC −1 −0.5 0 0.5 1 −0.5 0 0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x axis EVEN MODE 99, OMEGA = 41.37, B/A = 0.5 y axis u(x,y) Figure 10.17: Surface for Symmetric Mode Number 99 © 2003 by CRC Press LLC 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) enter 0 to stop > ? 1 Give a vector of mode indices (try 10:2:20) enter 0 to stop > ? 2:6 Give a vector of mode indices (try 10:2:20) enter 0 to stop > ? [20 25 30] Give a vector of mode indices (try 10:2:20) enter 0 to stop > ? 0 >> Elliptic Membrane Program 1: function [frqs,modes,indx,x,y,alpha,cptim]=elipfreq( 2: a,b,type,nlsq,nfuns,noplot) 3: % [frqs,modes,indx,x,y,alpha,cptim]=elipfreq( 4: % a,b,type,nlsq,nfuns,noplot) 5: % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 6: % This function computes natural frequencies and mode 7: % shapes for an elliptical membrane. Modes that are 8: % symmetrical or anti-symmetrical about the x axis are 9: % included. An approximate solution is obtained using © 2003 by CRC Press LLC 10: % a separation of variables formulation in elliptical 11: % coordinates. 12: % 13: % a,b - the ellipse major and minor semi- 14: % diameters along the x and y axes 15: % nlsq - two-component vector giving the number 16: % of least square points in the eta and 17: % xi directions 18: % nfuns - two-component vector giving the number of 19: % functions used to solve the differential 20: % equations for the eta and xi directions. 21: % type - use 1 for even modes symmetric about the 22: % x-axis. Use 2 for odd modes anti- 23: % symmetric about the x-axis. Use 3 to 24: % combine both even and odd modes. 25: % 26: % frqs - a vector of natural frequencies 27: % arranged in increasing order. 28: % modes - a three dimensional array in which 29: % modes(:,:,j) defines the modal 30: % deflection surface for frequency 31: % frqs(j). 32: % indx - a vector telling whether each 33: % mode is even (1) or odd (2) 34: % x,y - curvilinear coordinate arrays of 35: % points in the membrane where modal 36: % function values are computed. 37: % alpha - a vector of eigenvalue parameters in 38: % the Mathieu equation: u’’(eta)+ 39: % (alpha-lambda*cos(2*eta))*u(eta)=0 40: % where lambda=(h*freq)^2/2 and 41: % h=atanh(b/a) 42: % cptim - the cpu time in seconds used to 43: % form the equations and solve for 44: % eigenvalues and eigenvectors 45: % noplot - enter any value to skip mode plots 46: % 47: % User m functions called: 48: % frqsimpl eigenrec plotmode 49: % modeshap funcxi funceta 50: 51: if nargin==0 52: disp(’ ’) 53: disp(’VIBRATION MODE SHAPES AND FREQUENCIES’) 54: disp(’ OF AN ELLIPTIC MEMBRANE ’) © 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 > ? ’],’s’); 61: v=eval([’[’,v,’]’]); a=v(1); b=v(2); disp(’ ’) 62: disp(’Select the modal form option’) 63: type=input( 64: ’1<=>even, 2<=>odd, 3<=>both > ? ’); 65: disp(’ ’) 66: disp([’The computation takes awhile.’, 67: ’ PLEASE WAIT.’]) 68: end 69: 70: if type ==1 | type==2 % Even or odd modes 71: [frqs,modes,x,y,alpha,cptim]=frqsimpl( 72: a,b,type,nlsq,nfuns); 73: indx=ones(length(frqs),1)*type; 74: else % Both modes 75: [frqs,modes,x,y,alpha,cptim]=frqsimpl( 76: a,b,1,nlsq,nfuns); 77: indx=ones(length(frqs),1); 78: [frqso,modeso,x,y,alphao,cpto]=frqsimpl( 79: a,b,2,nlsq,nfuns); 80: frqs=[frqs;frqso]; alpha=[alpha;alphao]; 81: modes=cat(3,modes,modeso); 82: indx=[indx;2*ones(length(frqso),1)]; 83: [frqs,k]=sort(frqs); modes=modes(:,:,k); 84: indx=indx(k); cptim=cptim+cpto; 85: end 86: 87: if nargin==6, return, end 88: 89: % Plot a sequence of modal functions 90: neig=length(frqs); 91: disp(’ ’), disp([’Computation time = ’, 92: num2str(sum(cptim)),’ seconds.’]) 93: disp([’Number of modes = ’,num2str(neig)]); 94: disp([’Highest frequency = ’, 95: num2str(frqs(end))]), disp(’ ’) 96: disp(’Press return to see modal plots.’) 97: pause, plotmode(a,b,x,y,frqs,modes,indx) 98: 99: %============================================== © 2003 by CRC Press LLC 100: 101: function [frqs,Modes,x,y,alpha,cptim]=frqsimpl( 102: a,b,type,nlsq,nfuns) 103: % [frqs,Modes,x,y,alpha,cptim]=frqsimpl( 104: % a,b,type,nlsq,nfuns) 105: % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 106: 107: % a,b - ellipse major and minor semi-diameters 108: % type - numerical values of one or two for modes 109: % symmetric or anti-symmetric about the x axis 110: % nlsq - vector [neta,nxi] giving the number of least 111: % square points used for the eta and xi 112: % directions 113: % nfuns - vector [meta,mxi] giving the number of 114: % approximating functions used for the eta and 115: % xi directions 116: % frqs - natural frequencies arranged in increasing 117: % order 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 121: % Mathieu differential equation: 122: % u’’(eta)+(alpha-lambda*cos(2*eta))*u(eta)=0 123: % cptim - vector of computation times 124: % 125: % User m functions called: funceta funcxi 126: % eigenrec modeshap 127: if nargin==0 128: a=cosh(2); b=sinh(2); type=1; 129: nlsq=[200,200]; nfuns=[30,30]; 130: end 131: h=sqrt(a^2-b^2); R=atanh(b/a); neta=nlsq(1); alpha=[]; 132: nxi=nlsq(2); meta=nfuns(1); mxi=nfuns(2); 133: eta=linspace(0,pi,neta)’; xi=linspace(0,R,nxi)’; 134: [Xi,Eta]=meshgrid(xi,eta); z=h*cosh(Xi+i*Eta); 135: x=real(z); y=imag(z); cptim=zeros(1,3); 136: 137: % Form the Mathieu equation for the circumferential 138: % direction as: A*E+alpha*E-lambda*B*E=0 139: tic; [Veta,A]=funceta(meta,type,eta); 140: A=Veta\[A,repmat(cos(2*eta),1,meta).*Veta]; 141: B=A(:,meta+1:end); A=A(:,1:meta); 142: 143: % Form the modified Mathieu equation for the radial 144: % direction as: P*F-alpha*F+lambda*Q*F=0 © 2003 by CRC Press LLC [...]... 7 .81 51e-002 -2.2675e-002 7. 288 1e-004 7.5555e-004 | 2.4 7 .81 51e-002 -1. 486 0e-002 -6. 989 8e-004 7.5597e-004 | 2.5 7 .81 51e-002 -7.0445e-003 -1.7447e-003 6. 286 5e-004 | 2.6 7 .81 51e-002 7.7058e-004 -2.0539e-003 4.3228e-004 | 2.7 7 .81 51e-002 8. 585 7e-003 -1.7105e-003 2.4008e-004 | 2 .8 7 .81 51e-002 1.6401e-002 -1. 084 0e-003 9.9549e-005 | 2.9 7 .81 51e-002 2.4216e-002 -4.7454e-004 2.2493e-005 | 3 7 .81 51e-002 3.2031e-002... 1.6 3.0918e-003 3.2643e-002 6.5694e-004 -1.4078e-003 | 1.7 -9.6908e-002 2.7953e-002 3.0625e-003 -1.2125e-003 | 1 .8 -1.9691e-001 1.3262e-002 4.1954e-003 -8. 3907e-004 | 1.9 -2.9691e-001 -1.1429e-002 4. 284 3e-003 -4. 086 0e-004 | 2 7 .81 51e-002 -4.6120e-002 3 .83 58e-003 -1.1102e-016 | 2.1 7 .81 51e-002 -3 .83 05e-002 3.1202e-003 3.5021e-004 | 2.2 7 .81 51e-002 -3.0490e-002 2. 080 1e-003 6.1308e-004 | 2.3 7 .81 51e-002... Beam width and Young’s modulus BeamWidth=[]; BeamE=[]; Depth=[]; DepthX=[]; end elseif Problem == 2 Title=[’From Timoshenko and Young,’, ’ p 434, haunch beam’]; Printout=12; NoSegs=144*4+1; BeamLength=144; © 2003 by CRC Press LLC 60: 61: 62: 63: 64: 65: 66: 67: 68: 69: 70: 71: 72: 73: 74: 75: 76: 77: 78: 79: 80 : 81 : 82 : 83 : 84 : 85 : 86 : 87 : 88 : ExtForce=[]; ExtForceX=[]; ExtRamp=[-1 -1 0 1 08] ; IntSupX=[36;... %============================================== 173: 174: 175: 176: 177: 1 78: 179: 180 : 181 : 182 : 183 : 184 : 185 : 186 : 187 : 188 : 189 : function [eigs,vecs,Amat,Bmat]=eigenrec(A,B,C,D) % [eigs,vecs,Amat,Bmat]=eigenrec(A,B,C,D) % Solve a rectangular eigenvalue problem of the % form: X*A+B*X=lambda*(X*C+D*X) % % A,B,C,D - square matrices defining the problem % A and C have the same size B and D % have the same size % eigs - vector of... NoExtRamp); if NoExtRamp > 0 fprintf(’\n | # X-start Load’); © 2003 by CRC Press LLC 150: 151: 152: 153: 154: 155: 156: 157: 1 58: 159: 160: 161: 162: 163: 164: 165: 166: 167: 1 68: 169: 170: 171: 172: 173: 174: 175: 176: 177: 1 78: 179: 180 : 181 : 182 : 183 : 184 : 185 : 186 : 187 : 188 : 189 : 190: 191: 192: fprintf(’ X-end Load’); fprintf(’\n | - ’); fprintf(’ ’); for i=1:NoExtRamp fprintf(’\n... 377: 3 78: 379: 380 : 381 : 382 : 383 : 384 : 385 : 386 : % are affected by end conditions, external % loads, and support reactions if ns>0 a(js4,1)=interp1(x,ymat(:,1),r); a(js4,2)=interp1(x,ymat(:,2),r); a(js4,3)=r; a(js4,4)=ones(ns,1); for j=1:ns-1 a(j+5:ns+4,j+4)= interp1(x,ymat(:,j+2),r(j+1:ns)); end end b(js4)=ysprt-interp1(x,ymat(:,ns+3),r); 387 : 388 : 389 : 390: % Solve for unknown reactions and end... 7.5273e+000 | 3 2.0202e-002 7. 084 8e+000 | 4 3.0303e-002 6.6 688 e+000 | 5 4.0404e-002 6.2776e+000 Material deleted for publication | | | | | 296 297 2 98 299 300 2.9596e+000 2.9697e+000 2.9798e+000 2. 989 9e+000 3.0000e+000 6.2776e+000 6.6 688 e+000 7. 084 8e+000 7.5273e+000 7.9976e+000 Solution time was 0.55 secs Reactions at Internal Supports: | X-location Reaction | -| 1 1.0 782 e+000 | 2 4.7506e-001... corresponding % cartesian coordinate matrices u(:,:j) % gives the modal surface for the j’th % natural frequency 2 78: 279: if nargin . indx=[indx;2*ones(length(frqso),1)]; 83 : [frqs,k]=sort(frqs); modes=modes(:,:,k); 84 : indx=indx(k); cptim=cptim+cpto; 85 : end 86 : 87 : if nargin==6, return, end 88 : 89 : % Plot a sequence of modal functions 90: neig=length(frqs); 91:. X*A+B*X=lambda*(X*C+D*X) 1 78: % 179: % A,B,C,D - square matrices defining the problem. 180 : % A and C have the same size. B and D 181 : % have the same size. 182 : % eigs - vector of eigenvalues 183 : % vecs -. vecs(:,:,j) 184 : % contains the rectangular eigenvector 185 : % for eigenvalue eigs(j) 186 : % Amat, 187 : % Bmat - matrices that express the eigenvalue 188 : % problem as Amat*V=lambda*Bmat*V 189 : % ©