Typora测试

Abstract

格式ctrl+B,I,U

typora typora typora <u>typora</u>

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<center>图1. 测试图片</center>

插入表格ctrl+T

1	2	3
a	1	4
b	3	1
c	2	1

一段matlab代码

%find the Linear system's response by state-space method 
function Xt = findResponse(K,M,C,dt,U,X0)
nd = size(M,1); N = max(size(U));
B = eye(nd);              %外荷载作用的位置
Ac = [zeros(nd),eye(nd); -inv(M)*K, -inv(M)*C];
Bc = [zeros(nd);inv(M)*B];

1. Introduction

2. Conventional SOD method

2.1 sod theory

数学证明：

• 证：取（1）的逆 $\Psi^{-1} (XX^T）{-1} \Psi^{-T} = \Sigma_q^{-1}$

  并右乘（2）得$\Psi^{-1} (XX^T）{-1} （VV^T）\Psi = Sigma_q^{-1}\Sigma_\dot{q}=\Lambda_{xv}$

  移项得$（VV^T）\Psi =XX^T \Psi\Lambda_{xv}$

  两边除以$N-1$即$R_{v}\Psi =R_{x} \Psi\Lambda_{xv}$,证毕。

2.2 sod for damped and noise contaminated signals

2.3 sod for $m\lt n$

由于SOD倾向于提取出能量较大模态坐标振动，假设前m阶响应能量较大, 设$\Sigma_q$的前m行m列为$\hat{\Sigma}q$, 后n-m行n-m列为$\check{\Sigma}q$
$XX^T = \Phi{m \times m} \hat{\Sigma}q \Phi{m \times m}^T+\Phi{m \times (n-m)} \check{\Sigma}q \Phi{m \times (n-m)}^T$

2.4 sod for $m\gt n$

3. SOD with decomposed signals