结构动力学上机二.docx
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结构动力学上机二.docx
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结构动力学上机二
结构动力学上机二
1解单自由度无阻尼自由振动方程:
,
,并画图像:
解:
X=sin(11*t)/11
2解单自由度无阻尼受迫振动方程:
,并画图像:
解:
X=cos(11*t)*(sin(21*t)/42-sin(t)/2)-sin(11*t)*(cos(21*t)/42-cos(t)/2)-(89*sin(11*t))/231
出现拍振的现象。
(外激励频率接近共振频率)
3解无阻尼受迫振动
,
,并画图像:
X=sin(11*t)/8-sin(33*t)/88-cos(11*t)*(t/2-sin(22*t)/44)
当外激励频率达到固有频率时,结构共振,振幅越来越大。
4解单自由度有阻尼自由振动:
X=(4*215^(1/2)*sin((3*215^(1/2)*t)/4))/(645*exp(t/4))
图像振荡衰减。
5解单自由度有阻尼受迫振动方程:
并画图像。
X=(2*cos((3*215^(1/2)*t)/4))/exp(t/4)-cos((3*215^(1/2)*t)/4)*(cos(11*t-(3*215^(1/2)*t)/4)+cos(11*t+(3*215^(1/2)*t)/4)+(44*215^(1/2)*cos(11*t-(3*215^(1/2)*t)/4))/645-(44*215^(1/2)*cos(11*t+(3*215^(1/2)*t)/4))/645+(215^(1/2)*sin(11*t-(3*215^(1/2)*t)/4))/645-(215^(1/2)*sin(11*t+(3*215^(1/2)*t)/4))/645)-(22*215^(1/2)*sin((3*215^(1/2)*t)/4)*((sin(t*((3*215^(1/2))/4-11))/4-cos(t*((3*215^(1/2))/4-11))*((3*215^(1/2))/4-11))/(((3*215^(1/2))/4-11)^2+1/16)-(sin(t*((3*215^(1/2))/4+11))/4-cos(t*((3*215^(1/2))/4+11))*((3*215^(1/2))/4+11))/(((3*215^(1/2))/4+11)^2+1/16)))/645-(15488*215^(1/2)*sin((3*215^(1/2)*t)/4))/(215*exp(t/4)*(264*215^(1/2)-3872)*(264*215^(1/2)+3872))
振幅先增大,后趋于稳定。
MATLAB源代码:
x=dsolve('D2x+121*x=0','x(0)=0,Dx(0)=1')
t=1:
0.01:
10;
plot(t,sin(11*t)/11)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x=dsolve('D2x+121*x=11*sin(10*t)','x(0)=0,Dx(0)=1')
y=eval(x);
t=0:
0.01:
50;
plot(t,y)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x=dsolve('D2x+121*x=11*sin(11*t)','x(0)=0,Dx(0)=1')
y=eval(x);
t=0:
0.01:
10;
plot(t,y)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x=dsolve('D2x+0.5*Dx+121*x=0','x(0)=0,Dx(0)=1')
y=eval(x);
t=0:
0.01:
15;
plot(t,y)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x=dsolve('D2x+0.5*Dx+121*x=11*sin(11*t)','x(0)=0,Dx(0)=1');
y=eval(x);
t=0:
0.01:
30;
plot(t,y)
多自由度:
对于多自由度系统的上机实习,我采用教材第三章课后题第十题练习:
一栋三层楼房,如图3-17,其刚度、质量矩阵如下:
(1)确定模态质量、模态刚度矩阵M,K;
(2)若
确定模态力
;(3)确定稳定响应
的表达式;
(4)用模态位移法确定各位移响应。
解:
归一化的模态质量阵:
归一化的刚度阵:
模态化的力向量:
各阶各阶广义位移向量:
q1=
(1000*cos((9*310^(1/2)*t)/10))/3597489-(1000*cos(600*t))/3597489
q2=
cos(20*3^(1/2)*t)/3588-cos(600*t)/3588
q3=
(1000*cos((254890^(1/2)*t)/10))/3574511-(1000*cos(600*t))/3574511
U1U2U3的位移响应为:
x1=
(8091669514322434166450125*cos(600*t))/57912976571624697202143657984-(384951200277470875*cos((9*310^(1/2)*t)/10))/2025206264983682285568+(810684587087756875*cos((1110212776293937*t)/21990232555520))/16098166407631739027456+(2^(1/2)*(cos(600*t)/3588-cos(20*3^(1/2)*t)/3588))/2
x2=
(40025366925079229938625*cos(600*t))/166416599343749129891217408-(352174228340508625*cos((9*310^(1/2)*t)/10))/2700275019978243047424-(30556388613671125*cos((1110212776293937*t)/21990232555520))/277554593235029983232-(2^(1/2)*(cos(600*t)/3588-cos(20*3^(1/2)*t)/3588))/4
x3=
(322869390835085125*cos((1110212776293937*t)/21990232555520))/2012270800953967378432-(34520151149168375*cos((9*310^(1/2)*t)/10))/578630361423909224448-(208469934843040588782125*cos(600*t))/2068320591843739185790844928-(2^(1/2)*(cos(600*t)/3588-cos(20*3^(1/2)*t)/3588))/4
U1的位移响应曲线:
U2的位移响应曲线:
U3的位移响应曲线:
MATLAB代码:
clear,clc
M=[100;020;002];
K=[800-8000;-8002400-1600;0-16004000];
[ddhh]=eig(K,M);
dd(:
1)=dd(:
1)/sqrt(dd(:
1)'*M*dd(:
1));
dd(:
2)=dd(:
2)/sqrt(dd(:
2)'*M*dd(:
2));
dd(:
3)=dd(:
3)/sqrt(dd(:
3)'*M*dd(:
3));
M1=dd'*M*dd;
K1=dd'*K*dd;
symst;
p=[100100100]'*cos(600*t);
M1=[100;010;001];
K1=[251.100;012000;002548.9];
q1=simple(dsolve('1*D2q+251.1*q=100*cos(600*t)','q(0)=0','Dq(0)=0'))
q2=simple(dsolve('1*D2q+1200*q=100*cos(600*t)','q(0)=0','Dq(0)=0'))
q3=simple(dsolve('1*D2q+2548.9*q=100*cos(600*t)','q(0)=0','Dq(0)=0'))
q1=(eval(q1));
q2=(eval(q2));
q3=(eval(q3));
x1=dd(1,1)*q1+dd(1,2)*q2+dd(1,3)*q3
x2=dd(2,1)*q1+dd(2,2)*q2+dd(2,3)*q3
x3=dd(3,1)*q1+dd(3,2)*q2+dd(3,3)*q3
t=0:
0.01:
10;
y=(8091669514322434166450125*cos(600*t))/57912976571624697202143657984-(384951200277470875*cos((9*310^(1/2)*t)/10))/2025206264983682285568+(810684587087756875*cos((1110212776293937*t)/21990232555520))/16098166407631739027456+(2^(1/2)*(cos(600*t)/3588-cos(20*3^(1/2)*t)/3588))/2
plot(t,y)
t=0:
0.01:
10;
y=(40025366925079229938625*cos(600*t))/166416599343749129891217408-(352174228340508625*cos((9*310^(1/2)*t)/10))/2700275019978243047424-(30556388613671125*cos((1110212776293937*t)/21990232555520))/277554593235029983232-(2^(1/2)*(cos(600*t)/3588-cos(20*3^(1/2)*t)/3588))/4;
plot(t,y)
t=0:
0.01:
10;
y=(322869390835085125*cos((1110212776293937*t)/21990232555520))/2012270800953967378432-(34520151149168375*cos((9*310^(1/2)*t)/10))/578630361423909224448-(208469934843040588782125*cos(600*t))/2068320591843739185790844928-(2^(1/2)*(cos(600*t)/3588-cos(20*3^(1/2)*t)/3588))/4
plot(t,y)
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- 结构 动力学 上机