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FEM.asv
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FEM.asv
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Matlab ???????
%???????
%1. ???????????
%2. ?????????
% ??Euler-Bernoulli Beam ????
% ????????????????????????????????
syms gMaterial gNode L gElement gEigValue gEigVector
%?????
fprintf('??Euler-Bernoulli Beam ???????????2\n')
fprintf('??Timoshenko?????????3\n???????????????\n')
%?????????
O=input('??????(<6?: ');
Nel=input('?????? ');
L=input('?????m?: ');
E=input('??????????10^11pa??');
E=E*10^11;
S=input('?????????1?????2?????3??????');
%?????
if S==1
D=input('???????m?:');
I=D^4/64;
A=pi*D^24;
end
if S==2
a=input('???????m?: ');
b=input('???????m?: ');
I=a^3*b/12;
A=a*b;
end
if S==3
D=input('???????m?:');
d=input('???????m?:');
I=(D^4-d^4)/64;
A=pi*(D^2-d^2)/4;
end
%???????????????
Nnode=Nel+1; %???????
node=(1:Nnode)'; %?????
x=0:L/Nel:L;
xx=x'; %??x????
yy=zeros(Nnode,1);
gNode=[ node, xx, yy];
%????????
gMaterial=[ E I A 7800 ];
gElement=[(1:Nel)', (1:Nel)', (2:Nnode)'];
gBco=[1, 1, 0
1, 2, 0
Nnode, 1, 0
Nnode, 2, 0];
%???? ??? ?
gNF=[5, 2, -8e3];
ie=1;
E = gMaterial( 1, 1 ) ;
I = gMaterial( 1, 2 ) ;
A = gMaterial( 1, 3 ) ;
xi = gNode( gElement( ie, 2), 2 ) ;
yi = gNode( gElement( ie, 2), 3 ) ;
xj = gNode( gElement( ie, 3), 2 ) ;
yj = gNode( gElement( ie, 3), 3 ) ;
l =( (xj-xi)^2 + (yj-yi)^2 )^(1/2) ;
c=E*I/(l^3);
k=[12 6*l -12 6*l;
6*l 4*l^2 -6*l 2*l^2;
-12 -6*l 12 -6*l;
6*l 2*l^2 -6*l 4*l^2];
ke=c*k;
mass=gMaterial(1,4)*A*l;
m=[ 156 22*l 54 -13*l;
22*l 4*l^2 13*l -3*l^2;
54 13*l 156 -22*l;
-13*l -3*l^2 -22*l 4*l^2];
me=mass/420*m;
gK=zeros(Nnode*2);
gM=zeros(Nnode*2);
for ie=1:Nel
for i=1:2
for j=1:2
for p=1:2
for q=1:2
m=(i-1)*2+p;
n=(j-1)*2+q;
M=(gElement(ie,i)-1)*2+m;
N=(gElement(ie,j)-1)*2+n;
gK(M,N)=gK(M,N)+ke(m,n);
gM(M,N)=gM(M,N)+me(m,n);
end
end
end
end
end
[bc1_number,~]=size(gBco);
w2max = max( diag(gK)./diag(gM) );
for ibc=1:1:bc1_number
n = gBco(ibc, 1 ) ;
d = gBco(ibc, 2 );
m = (n-1)*2 + d ;
gK(:,m) = zeros( Nnode*2, 1 ) ;
gK(m,:) = zeros( 1, Nnode*2 ) ;
gK(m,m) = 1;
end
for ibc=1:1:bc1_number
n = gBco(ibc, 1 ) ;
d = gBco(ibc, 2 ) ;
m = (n-1)*2 + d ;
gM(:,m) = zeros( Nnode*2, 1 ) ;
gM(m,:) = zeros( 1, Nnode*2 ) ;
gM(m,m) = gK(m,m)/w2max/1e10 ;
end
for i=1:Nnode*2
for j=i:Nnode*2
gK(j,i) = gK(i,j) ;
gM(j,i) = gM(i,j) ;
end
end
[gEigVector, gEigValue] = eigs(gK, gM, O, 'SM' );
for ibc=1:1:bc1_number
n = gBco(ibc, 1 ) ;
d = gBco(ibc, 2 ) ;
m = (n-1)*2 + d ;
gEigVector(m,:) = gBco(ibc,3) ;
end
fre_number=length(diag(gEigValue));
fprintf( '\n\n ?? ????(??) \n' ) ;
for i=1:fre_number
fprintf( '----------------') ;
end
fprintf( '\n' ) ;
for i=1:fre_number
fprintf( ' %6d ', i ) ;
end
fprintf( '\n' ) ;
for i=1:fre_number
fprintf( '----------------') ;
end
fprintf( '\n' ) ;
[dof,~]=size(gEigVector) ;
for i=1:dof
for j=fre_number:-1:1
fprintf( '%15.7e ', gEigVector(i,j) ) ;
end
fprintf( '\n' ) ;
end
for i=1:fre_number
fprintf( '----------------') ;
end
fprintf( '\n' ) ;
% ?????
fprintf( '\n\n\n\n ?? ???(??)?? \n' ) ;
fprintf( '----------------------------------------------------------\n') ;
fprintf( ' ?? ??? ??????(Hz) ???(rad/s)\n') ;
fprintf( '----------------------------------------------------------\n') ;
for i=fre_number:-1:1
fprintf( '%6d %15.7e %15.7e %15.7e\n', fre_number-i+1, ...
gEigValue(i,i), sqrt(gEigValue(i,i))/2/pi, sqrt(gEigValue(i,i)) ) ;
end
fprintf( '----------------------------------------------------------\n') ;
for j=fre_number:-1:1
figure
x = gNode(:,2) ;
y = gNode(:,3) ;
dx = gEigVector(1:2:length(x)*2, j ) ;
dy = gEigVector(2:2:length(x)*2, j );
factor = max( [max(abs(x))/max(abs(dx)), max(abs(y))/max(abs(dy))] )* 0.05;
plot(x,y,'-', x+factor*dx, y+factor*dy,'r:');
hold on
title( sprintf( '?%d???: %.3f Hz', fre_number-j+1, sqrt(gEigValue(j,j))/2/pi ) ) ;
grid on
end
%---------------------------------------------------------------------
%?????????
% ????????
% ??????????????????
% gNode -------- ????
% gElement ----- ????
% gMaterial ---- ?????????????????????????
% gBC1 --------- ????
% gDeltaT ------ ????
% gTimeEnd ----- ??????
% gDisp -------- ??????
% gVelo -------- ??????
% gAcce -------- ???????
%??????
global gDeltaT gTimeEnd gDisp gVelo gAcce
%??????(Newmark?)
gDeltaT=0.01;
gTimeEnd = 4890*gDeltaT ;
timestep = floor(gTimeEnd/gDeltaT) ;
% ???????????
gDisp = zeros( Nnode*2, timestep ) ;
gVelo = zeros( Nnode*2, timestep ) ;
gAcce = zeros( Nnode*2, timestep ) ;
gForc = zeros( Nnode*2, timestep ) ;
% ????
gDisp(:,1) = zeros(Nnode*2, 1 ) ;
gVelo(:,1) = ones (Nnode*2, 1) ;
%gForce(:,1)
gama = 0.5 ;
beta = 0.25 ;
C=zeros(size(gK));
[~,N]=size(gK);
alpha0 = 1/beta/gDeltaT^2;
alpha1 = gama/beta/gDeltaT;
alpha2 = 1/beta/gDeltaT;
alpha3 = 1/2/beta - 1;
alpha4 = gama/beta - 1;
alpha5 = gDeltaT/2*(gama/beta-2);
alpha6 = gDeltaT*(1-gama);
alpha7 = gama*gDeltaT;
K1 = gK + alpha0*gM + alpha1*C;
[bc1_number,dummy] = size(gBco) ;
K1im = zeros(N,bc1_number) ;
for ibc=1:1:bc1_number
n=gBco(ibc,1);
d=gBco(ibc,2);
m=(n-1)*2+d;
K1im(:,ibc)=K1(:,m);
K1(:,m) = zeros( Nnode*2, 1 );
K1(m,:) = zeros( 1, Nnode*2);
K1(m,m) = 1.0;
end
[KL,KU] = lu(K1); % ????????????????
%???????
gAcce(:,1) =gM\(-gK*gDisp(:,1)-C*gVelo(:,1));
%?????????
for i=2:1:timestep
if mod(i,100) == 0
fprintf( '??????%d\n', i ); % ??????
end
f1 =gM*(alpha0*gDisp(:,i-1)+alpha2*gVelo(:,i-1)+alpha3*gAcce(:,i-1)) ...
+ C*(alpha1*gDisp(:,i-1)+alpha4*gVelo(:,i-1)+alpha5*gAcce(:,i-1)) ;
% ?f1????????
[bc1_number,dummy] = size( gBco ) ;
for ibc=1:1:bc1_number
n = gBco(ibc, 1 ) ;
d = gBco(ibc, 2 ) ;
m = (n-1)*2 + d ;
f1 = f1 - gBco(ibc,3) * K1im(:,ibc) ;
f1(m) = gBco(ibc,3) ;
end
y = KL\f1 ;
gDisp(:,i) = KU\y ;
gAcce(:,i) = alpha0*(gDisp(:,i)-gDisp(:,i-1)) - alpha2*gVelo(:,i-1) - alpha3*gAcce(:,i-1) ;
gVelo(:,i) = gVelo(:,i-1) + alpha6*gAcce(:,i-1) + alpha7*gAcce(:,i) ;
end
return
global gElement gMaterial gBC1 gDisp gVelo gAcce gDeltaT gTimeEnd
t = 0:gDeltaT:gTimeEnd-gDeltaT;
d = gDisp((floor(Nnode/2)*2)+1,:);
subplot(2,1,1);
plot( t, d );
title( 'L/4???????');
xlabel( '??(s)');
ylabel( '??(cm)' );
grid on
fd = fft( d ) ;
df = 1/gTimeEnd ;
f = (0:length(d)-1)*df ;
subplot(2,1,2);
plot(f,abs(fd)) ;
set(gca,'xlim',[0,100]) ;
title( 'L/4???????' ) ;
xlabel( '??(Hz)') ;
ylabel( '??' ) ;
fifi1 = diff(abs(fd));
n = length(fifi1) ;
d1 = fifi1(1:n-1);
d2 = fifi1(2:n) ;
indmax = find( d1.*d2<0 & d1>0 )+1;
for i=1:length(indmax)
if f(indmax(i)) > 10
break ;
end
text( f(indmax(i)+2), abs(fd(indmax(i)))*0.9, sprintf('f=%.3f',f(indmax(i))));
end