vault backup: 2025-11-03 08:12:26

copilot-conversations/20251029_080739__[[Chap_9]]详细描述chap_9的内容.md

@@ -41,7 +41,7 @@ tags:
     *   **阻尼力 (Damping Force)**: $\mathbf f_{D}=\mathbf c\,\dot{\mathbf v}$
         *   $\mathbf{c}$ 是**阻尼矩阵 (damping matrix)**，其元素 $c_{ij}$ 是阻尼影响系数，定义为“坐标 $j$ 产生单位速度时，在坐标 $i$ 处产生的力”。
     *   **惯性力 (Inertial Force)**: $\mathbf{f}_{I}=\mathbf{m}\;\ddot{\mathbf{v}}$
-        *   $\mathbf{m}$ 是**质量矩阵 (mass matrix)**，其元素 $m_{ij}$ 是质量影响系数，定义为“坐标 $j$ 产生单位加速度时，在坐标 $i$ 处产生的力”。
+        *   $\mathbf{m}$ 是**质量矩阵 (mass matrix)**，其元素 $m_{ij}$ 是质量z影响系数，定义为“坐标 $j$ 产生单位加速度时，在坐标 $i$ 处产生的力”。
 *   **MDOF运动方程**：将上述三个力向量的表达式代入动态平衡方程，得到MDOF系统的标准运动方程：
     $$
     \mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)=\mathbf{p}(t)

书籍/力学书籍/力学/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library)/auto/Chap 10 EVALUATION OF STRUCTURALPROPERTY MATRICES.md

@@ -6,7 +6,9 @@
 Before discussing the elastic-stiffness matrix expressed in Eq. (9-5), it will be useful to define the inverse flexibility relationship. The definition of a flexibility influence coefficient $\widetilde{f}_{i j}$ is  
 在讨论由式 (9-5) 表示的弹性刚度矩阵之前，定义逆柔度关系将是有益的。柔度影响系数 $\widetilde{f}_{i j}$ 的定义是
 $$
-\begin{array}{r l}{\widetilde{f}_{i j}=\ }&{\mathrm{deflection\;of\;coordinate\}i\mathrm{~due\;to\;unit\;load}}\\ &{\mathrm{applied\;to\;coordinate\}j}\end{array}
+\tilde{f}_{ij} = \text{deflection of coordinate } i \text{ due to unit load} \\
+\text{applied to coordinate } j
+\tag{10-1}
 $$  
 
 For the simple beam shown in Fig. 10-1, the physical significance of some of the flexibility influence coefficients associated with a set of vertical-displacement degrees of freedom is illustrated. Horizontal or rotational degrees of freedom might also have been considered, in which case it would have been necessary to use the corresponding horizontal or rotational unit loads in defining the complete set of influence coefficients; however, it will be convenient to restrict the present discussion to the vertical motions.  
@@ -23,7 +25,29 @@ $$
 Since similar expressions can be written for each displacement component, the complete set of displacements is expressed  
 由于可以为每个位移分量编写类似的表达式，因此完整的位移集表示为
 $$
-\left\{\begin{array}{l}{v_{1}}\\ {v_{2}}\\ {\cdot}\\ {v_{i}}\\ {v_{i}}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{l l l l l l}{\widetilde{f}_{11}}&{\widetilde{f}_{12}}&{\widetilde{f}_{13}}&{\cdots}&{\widetilde{f}_{1i}}&{\cdots}&{\widetilde{f}_{1N}}\\ {\widetilde{f}_{21}}&{\widetilde{f}_{22}}&{\widetilde{f}_{23}}&{\cdots}&{\widetilde{f}_{2i}}&{\cdots}&{\widetilde{f}_{2N}}\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&&\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&&\\ {\widetilde{f}_{i1}}&{\widetilde{f}_{i2}}&{\widetilde{f}_{i3}}&{\cdots}&{\widetilde{f}_{i i}}&{\cdots}&{\widetilde{f}_{i N}}\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&&{.}\end{array}\right]\left\}\begin{array}{l}{p_{1}}\\ {p_{2}}\\ {\cdot}\\ {p_{i}}\\ {p_{i}}\\ {\cdot}\end{array}\right\}
+\begin{Bmatrix}
+v_1 \\
+v_2 \\
+\vdots \\
+v_i \\
+\vdots
+\end{Bmatrix}
+=
+\begin{bmatrix}
+\tilde{f}_{11} & \tilde{f}_{12} & \tilde{f}_{13} & \cdots & \tilde{f}_{1i} & \cdots & \tilde{f}_{1N} \\
+\tilde{f}_{21} & \tilde{f}_{22} & \tilde{f}_{23} & \cdots & \tilde{f}_{2i} & \cdots & \tilde{f}_{2N} \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
+\tilde{f}_{i1} & \tilde{f}_{i2} & \tilde{f}_{i3} & \cdots & \tilde{f}_{ii} & \cdots & \tilde{f}_{iN} \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots
+\end{bmatrix}
+\begin{Bmatrix}
+p_1 \\
+p_2 \\
+\vdots \\
+p_i \\
+\vdots
+\end{Bmatrix}
+\tag{10-3}
 $$  
 
 or symbolically  
@@ -67,6 +91,9 @@ $$
 Alternatively, transposing Eq. (10-6) and substituting Eq. (9-6) leads to the second strain-energy expression (note that $\mathbf{p}=\mathbf{f}_{S}$ ):  
 此外，对式(10-6)进行移项并代入式(9-6)，可得到第二个应变能表达式（注意 $\mathbf{p}=\mathbf{f}_{S}$ ）：
 $$
+\mathbf{f}_{S}=\mathbf{k}\ \mathbf{v}\tag{9-6}
+$$
+$$
 U=\frac{1}{2}\mathbf{}\mathbf{v}^{T}\,\mathbf{k}\,\mathbf{v}
 $$