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## 利用杜哈美积分求位移、速度和加速度

(2009-06-25 16:36:00)

### 杂谈

 %单自由度系统适用，初始条件任意，比happy提供的适应性强。对于多自由度系统，正 %则化之后适用，加循环即可（有点修改，for t=0:dTaT end之间的不用改）。 clc; clear n=1000; m=3; T=2; si=0.02; k=2700; w=30; dTao=T/n; wD=w*(1-si^2)^0.5; p2=[ones(1,333),zeros(1,668)]; % p2=[1000,zeros(1,1000)]; %%%%%%%%%%%%%%%杜哈美积分开始%%%%%%%%%%%%%%%% x0=0; v0=0; j=1; for t=0:dTaT sysN=0; sysP=0; %%%%%%%%%%%%位移\速度\加速度公用部分%%%%%%%%%%%%%% for i=0:t/dTao sysN=sysN+p2(i+1)*exp(si*i*dTao*w)*cos(wD*i*dTao)*dTao; sysP=sysP+p2(i+1)*exp(si*i*dTao*w)*sin(wD*i*dTao)*dTao; end %%%%%%%%%%%%位移\速度\加速度公用部分%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%求位移%%%%%%%%%%%%%%%%%%% sysA=exp(-si*w*t)*(x0*cos(wD*t)+(si*w*x0+v0)/wD*sin(wD*t)); sysB=exp(-si*w*t)/(m*wD)*(sin(wD*t)*sysN-cos(wD*t)*sysP); x(j)=sysA+sysB; %%%%%%%%%%%%%%%%%求位移%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%求速度%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%V=C+D+E+F%%%%%%%%%%%%%%%%%% sysC=exp(-si*w*t)*(-x0*wD*sin(wD*t)-(si*w)^2*x0*sin(wD*t)/wD); sysD=exp(-si*w*t)*(v0*cos(wD*t)-si*w*v0*sin(wD*t)/wD); %%%%%%%%%%%%%%%E=G+H%%%%%%%%%%%%%%%%%%%%%% sysG=sin(wD*t)*sysN; sysH=-cos(wD*t)*sysP; sysE=(sysN+sysP)*-si*w*exp(-si*w*t)/(m*wD); %%%%%%%%%%%%%%%F=I+J+K+L%%%%%%%%%%%%%%%%%% sysI=wD*cos(wD*t)*sysN; sysJ=sin(wD*t)*p2(j)*exp(si*w*t)*cos(wD*t); sysK=wD*sin(wD*t)*sysP; sysL=-cos(wD*t)*p2(j)*exp(si*w*t)*sin(wD*t); sysF=(sysI+sysJ+sysK+sysL)*exp(-si*w*t)/(m*wD); v(j)=sysC+sysD+sysE+sysF; %%%%%%%%%%%%%%%%%%%%%%%%%求速度%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%求加速度%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%sysdC%%%%%%%%%%%%%%% sysdC1=-x0*wD^2*cos(wD*t)-(si*w)^2*x0*cos(wD*t); sysdC=-si*w*sysC+exp(-si*w*t)*sysdC1; %%%%%%%%%%%%%%%%%%%sysdD%%%%%%%%%%%%%%% sysdD1=-v0*wD*sin(wD*t)-si*w*v0*cos(wD*t); sysdD=-si*w*sysD+exp(-si*w*t)*sysdD1; %%%%%%%%%%%%%%%%%%%sysdE%%%%%%%%%%%%%%% sysdG=wD*cos(wD*t)*sysN+sin(wD*t)*p2(j)*exp(si*w*t)*cos(wD*t); sysdH=wD*sin(wD*t)*sysP-cos(wD*t)*p2(j)*exp(si*w*t)*sin(wD*t); sysdE=(si*w)^2*exp(-si*w*t)*(sysG+sysH)/(m*wD)-si*w*exp(-si*w*t)*(sysdG+sysdH)/(m*wD); %%%%%%%%%%%%%%%%%%%sysdF%%%%%%%%%%%%%%% sysdI=-wD^2*sin(wD*t)*sysN+wD*cos(wD*t)*p2(j)*exp(si*w*t)*cos(wD*t); %%%%%%%%%%%%怎样考虑dP2,暂时取0%%%sysdJ%%%%%%%%%% sysdJ1=wD*cos(wD*t)^2*p2(j)*exp(si*w*t); sysdJ2=sin(wD*t)*0*exp(si*w*t)*cos(wD*t); sysdJ3=sin(wD*t)*p2(j)*si*w*exp(si*w*t)*cos(wD*t); sysdJ4=-wD*sin(wD*t)^2*p2(j)*exp(si*w*t); sysdJ=sysdJ1+sysdJ2+sysdJ3+sysdJ4; %%%%%%%%%%%%%%%%%%%sysdK%%%%%%%%%%%%%%% sysdK=wD^2*cos(wD*t)*sysP+wD*sin(wD*t)*p2(j)*exp(si*w*t)*sin(wD*t); %%%%%%%%%%%%怎样考虑dP2,暂时取0%%%sysdL%%%%%%%%%% sysdL1=wD*sin(wD*t)^2*p2(j)*exp(si*w*t); sysdL2=cos(wD*t)*0*exp(si*w*t)*sin(wD*t); sysdL3=-cos(wD*t)*p2(j)*si*w*exp(si*w*t)*sin(wD*t); sysdL4=-wD*cos(wD*t)^2*p2(j)*exp(si*w*t); sysdL=sysdL1+sysdL2+sysdL3+sysdL4; %%%%%%%%%%%%%%%%%sysdF%%%%%%%%%%%%%% sysdF=-si*w*exp(-si*w*t)*(sysI+sysJ+sysK+sysL)/(wD*m)+exp(-si*w*t)*(sysdI+sysdJ+sysdK+sysdL)/(wD*m); %%%%%%%%%%%%%%%%%acc%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% acc(j)=sysdC+sysdD+sysdE+sysdF; %%%%%%%%%%%%%%%%%%%%%%%%%求加速度%%%%%%%%%%%%%%%%%%%%%%%%% j=j+1; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%杜哈美积分完毕%%%%%%%%%%%%%%%%%%%%%%%%% figure; t=0:dTaT; subplot(3,1,1);plot(t,x);grid on; subplot(3,1,2);plot(t,v);grid on; subplot(3,1,3);plot(t,acc);grid on; 杜哈梅积分，求任意激励的响应，振动力学里的，采用matlab函数！ 这是我写的duhamel积分，完全是按照西安交大版《振动力学》上的公式写的，调试无误。 % duhamel        duhamel integral % y=duhamel(x,h,fs) % inputs:     x:analysed signal(vector) %             h:impulse response %             fs:frequency of sampling % outputs:    y:duhamel integral of analysed signal x function y=duhamel(x,h,fs)     n=length(x);     for i=1:n         xx=x(1:i);         hh=fliplr(h(1:i));         temp=xx.*hh;         y(i)=trapz(temp)*(1/fs);     end FROM CHINAVIB

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