From eb4134f04f9772e310aa808ab6c656eeea78a0b6 Mon Sep 17 00:00:00 2001
From: aGYZ <5722745+agyz@user.noreply.gitee.com>
Date: Tue, 28 Oct 2025 08:17:40 +0800
Subject: [PATCH] vault backup: 2025-10-28 08:17:40

---
 .../auto/Chap 9.md                            | 56 +++++++++++++++++--
 1 file changed, 50 insertions(+), 6 deletions(-)

diff --git a/书籍/力学书籍/力学/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library)/auto/Chap 9.md b/书籍/力学书籍/力学/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library)/auto/Chap 9.md
index 21dcecc..1789306 100644
--- a/书籍/力学书籍/力学/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library)/auto/Chap 9.md	
+++ b/书籍/力学书籍/力学/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library)/auto/Chap 9.md	
@@ -60,14 +60,36 @@ $$
 In these expressions it has been tacitly assumed that the structural behavior is linear, so that the principle of superposition applies. The coefficients $k_{i j}$ are called stiffness influence coefficients, defined as follows:  
 在这些表达式中，已经默认假设结构行为是线性的，从而叠加原理适用。系数 $k_{i j}$ 被称为刚度影响系数，定义如下：
 $$
-\begin{array}{r}{k_{i j}=\mathrm{~force~corresponding~to~coordinate~}i\mathrm{~d}\mathrm{t}}\\ {\mathrm{~a~unit~}d i s p l a c e m e n t\;\mathrm{of~coordinate~}j\quad\quad}\end{array}
+\begin{array}{r}{k_{i j}=\mathrm{~force~corresponding~to~coordinate~}i\mathrm{~due}\mathrm{~to}}\\ {\mathrm{~a~unit~}d i s p l a c e m e n t\;\mathrm{of~coordinate~}j\quad\quad}\end{array}
 $$  
 
-In matrix form, the complete set of elastic-force relationships may be written  
+In matrix form, the complete set of elastic-force relationships may be written 
 以矩阵形式，完整的弹性力关系可以写成
-$$
-\left\{\begin{array}{l}{f_{S1}}\\ {f_{S2}}\\ {\cdot}\\ {\cdot}\\ {f_{S i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{l l l l l l l}{k_{11}}&{k_{12}}&{k_{13}}&{\cdots}&{k_{1i}}&{\cdots}&{k_{1N}}\\ {k_{21}}&{k_{22}}&{k_{23}}&{\cdots}&{k_{2i}}&{\cdots}&{k_{2N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {k_{i1}}&{k_{i2}}&{k_{i3}}&{\cdots}&{k_{i i}}&{\cdots}&{k_{i N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]\left\{\begin{array}{l}{v_{1}}\\ {v_{2}}\\ {v_{3}}\\ {\cdot}\\ {v_{i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}
-$$  
+ $$
+\begin{Bmatrix}
+f_{S1} \\
+f_{S2} \\
+\vdots \\
+f_{Si} \\
+\vdots
+\end{Bmatrix}
+=
+\begin{bmatrix}
+k_{11} & k_{12} & k_{13} & \cdots & k_{1i} & \cdots & k_{1N} \\
+k_{21} & k_{22} & k_{23} & \cdots & k_{2i} & \cdots & k_{2N} \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
+k_{i1} & k_{i2} & k_{i3} & \cdots & k_{ii} & \cdots & k_{iN} \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots
+\end{bmatrix}
+\begin{Bmatrix}
+v_1 \\
+v_2 \\
+\vdots \\
+v_i \\
+\vdots
+\end{Bmatrix}
+\tag{9-5}
+$$ 
 
 or, symbolically,  
 
@@ -81,7 +103,29 @@ If it is assumed that the damping depends on the velocity, that is, the viscous
 其中刚度系数矩阵 $\mathbf{k}$ 称为结构的刚度矩阵 (对于指定的一组位移坐标)，且 $\mathbf{v}$ 是表示结构变形形状的位移向量。
 如果假设阻尼取决于速度，即黏性类型，则对应于所选自由度的阻尼力可以类似地通过阻尼影响系数来表示。参照式 (9-5)，完整的阻尼力集由下式给出
 $$
-\left\{\begin{array}{c}{f_{D1}}\\ {f_{D2}}\\ {\cdot}\\ {\cdot}\\ {f_{D i}}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{c c c c c c c}{c_{11}}&{c_{12}}&{c_{13}}&{\cdots}&{c_{1i}}&{\cdots}&{c_{1N}}\\ {c_{21}}&{c_{22}}&{c_{23}}&{\cdots}&{c_{2i}}&{\cdots}&{c_{2N}}\\ {\cdots}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {c_{i1}}&{c_{i2}}&{c_{i3}}&{\cdots}&{c_{i i}}&{\cdots}&{c_{i N}}\\ {\cdots}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]\,\left\{\begin{array}{c}{\dot{v}_{1}}\\ {\dot{v}_{2}}\\ {\dot{v}_{3}}\\ {\cdot}\\ {\dot{v}_{i}}\\ {\dot{v}_{i}}\\ {\cdot}\end{array}\right\}\,\,
+\begin{Bmatrix}
+f_{D1} \\
+f_{D2} \\
+\vdots \\
+f_{Di} \\
+\vdots
+\end{Bmatrix}
+=
+\begin{bmatrix}
+c_{11} & c_{12} & c_{13} & \cdots & c_{1i} & \cdots & c_{1N} \\
+c_{21} & c_{22} & c_{23} & \cdots & c_{2i} & \cdots & c_{2N} \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
+c_{i1} & c_{i2} & c_{i3} & \cdots & c_{ii} & \cdots & c_{iN} \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots
+\end{bmatrix}
+\begin{Bmatrix}
+\dot{v}_1 \\
+\dot{v}_2 \\
+\vdots \\
+\dot{v}_i \\
+\vdots
+\end{Bmatrix}
+\tag{9-7}
 $$  
 
 in which $\dot{v}_{i}$ represents the time rate of change (velocity) of the $i$ displacement coordinate and the coefficients $c_{i j}$ are called damping influence coefficients. The definition of these coefficients is exactly parallel to Eq. (9-4):