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---
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epoch: 1761696459225
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modelKey: gemini-2.5-pro|google
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tags:
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- copilot-conversation
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---
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**user**: [[Chap 9]]详细描述chap 9的内容
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[Timestamp: 2025/10/29 08:07:39]
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**ai**: 根据您提供的笔记 [[Chap 9]]，该章节的核心内容是**建立多自由度（MDOF）系统的运动方程**。以下是该章节的详细内容描述：
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### **引言**
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章节开篇首先回顾了单自由度（SDOF）系统。它指出，将复杂结构简化为SDOF系统是一种近似方法，其结果的可靠性难以评估，尤其是在荷载分布复杂或结构物理特性不适合单一变形模式的情况下。因此，为了更准确地描述结构包含位移形状和振幅随时间变化的动态响应，必须采用多自由度（MDOF）模型。
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---
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### **9-1 自由度的选择 (SELECTION OF THE DEGREES OF FREEDOM)**
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这部分内容阐述了如何为结构选择自由度以建立MDOF模型。
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* **MDOF的必要性**：通常结构的动力响应不能用SDOF模型充分描述，因为响应不仅包括振幅的变化，还包括变形形状随时间的变化。这种行为需要用多个位移坐标来描述，即多个自由度。
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||||
* **自由度的定义**：在离散参数系统中，自由度可以被定义为结构上选定点（节点）的位移幅值。本章采用这种方法，而不是广义坐标法（将在第16章讨论）。
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* **示例**：以图9-1中的简支梁为例，其运动可以通过梁上一系列离散点的竖向位移 $v_{1}(t), v_{2}(t), ..., v_{N}(t)$ 来定义。
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* **选择与数量**：节点的选择应与结构的物理特性（如质量、刚度变化处）相关联，并能良好地定义结构的变形形状。自由度的数量由分析者决定，数量越多，对真实动力行为的近似越好。
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||||
---
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### **9-2 动态平衡条件 (DYNAMIC-EQUILIBRIUM CONDITION)**
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本节通过建立每个自由度的动态力平衡来推导MDOF系统的运动方程。
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* **力的平衡**：在任意一个节点 $i$ 上，存在四种力：外荷载 $p_{i}(t)$，以及由运动产生的惯性力 $f_{I i}$、阻尼力 $f_{D i}$ 和弹性力 $f_{S i}$。所有自由度的动态平衡可以写成矩阵形式：
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||||
$$
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||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}=\mathbf{p}(t)
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$$
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* **各力分量的矩阵表示**：假设结构行为是线性的，各个力向量可以通过影响系数矩阵与位移、速度和加速度向量联系起来。
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* **弹性力 (Elastic Force)**: $\mathbf{f}_{S}=\mathbf{k}\ \mathbf{v}$
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* $\mathbf{k}$ 是**刚度矩阵 (stiffness matrix)**，其元素 $k_{ij}$ 是刚度影响系数，定义为“坐标 $j$ 产生单位位移时，在坐标 $i$ 处产生的力”。
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* **阻尼力 (Damping Force)**: $\mathbf f_{D}=\mathbf c\,\dot{\mathbf v}$
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||||
* $\mathbf{c}$ 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”。
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||||
* **惯性力 (Inertial Force)**: $\mathbf{f}_{I}=\mathbf{m}\;\ddot{\mathbf{v}}$
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||||
* $\mathbf{m}$ 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”。
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||||
* **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：
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)=\mathbf{p}(t)
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||||
$$
|
||||
这个方程是SDOF运动方程的MDOF等价形式，其中每个标量项都被相应的矩阵（质量、阻尼、刚度矩阵）和向量（加速度、速度、位移、荷载向量）所替代。
|
||||
|
||||
---
|
||||
|
||||
### **9-3 轴向力效应 (AXIAL-FORCE EFFECTS)**
|
||||
|
||||
本节讨论了轴向力对结构刚度的影响，并将其纳入运动方程。
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||||
|
||||
* **几何刚度**：轴向力（或任何可能引起结构屈曲的荷载）会影响结构的刚度。这种效应会产生一个额外的力分量 $\mathbf{f}_{G}$，它与位移相关，通常会“助长”变形而不是抵抗变形。
|
||||
* **包含轴向力效应的平衡方程**：
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||||
$$
|
||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}-\mathbf{f}_{G}=\mathbf{p}(t)
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||||
$$
|
||||
* **几何刚度矩阵**：力 $\mathbf{f}_{G}$ 可以通过**几何刚度矩阵 (geometric-stiffness matrix)** $\mathbf{k}_{G}$ 表示：
|
||||
$$
|
||||
\mathbf{f}_{G}=\mathbf{k}_{G}\,\mathbf{v}
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||||
$$
|
||||
* **修正后的运动方程**：将 $\mathbf{f}_{G}$ 的表达式代入，运动方程变为：
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)-\mathbf{k}_{G}\,\mathbf{v}(t)=\mathbf{p}(t)
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||||
$$
|
||||
* **组合刚度矩阵 (Combined Stiffness Matrix)**：可以将弹性和几何刚度效应合并，定义一个**组合刚度矩阵** $\overline{{\mathbf{k}}}$：
|
||||
$$
|
||||
\overline{{\mathbf{k}}}=\mathbf{k}-\mathbf{k}_{G}
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||||
$$
|
||||
最终，考虑轴向力效应的MDOF运动方程可以写为：
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\overline{{\mathbf{k}}}\,\mathbf{v}(t)=\mathbf{p}(t)
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||||
$$
|
||||
|
||||
### **总结**
|
||||
|
||||
[[Chap 9]] 详细阐述了从概念到数学公式，如何建立一个多自由度系统的动力学模型。它首先强调了从SDOF模型过渡到MDOF模型的必要性，然后定义了自由度，并基于动态平衡条件，系统地推导出了包含质量、阻尼和刚度矩阵的MDOF运动方程。最后，还引入了轴向力对刚度的影响，即几何刚度，并给出了考虑该效应的最终运动方程形式。
|
||||
[Timestamp: 2025/10/29 08:08:02]
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||||
@ -27,13 +27,13 @@ The motion of this structure will be assumed to be defined by the displacements
|
||||
The equation of motion of the system of Fig. 9-1 can be formulated by expressing the equilibrium of the effective forces associated with each of its degrees of freedom. In general four types of forces will be involved at any point $i$ : the externally applied load $p_{i}(t)$ and the forces resulting from the motion, that is, inertia $f_{I i}$ , damping $f_{D i}$ , and elastic $f_{S i}$ . Thus for each of the several degrees of freedom the dynamic equilibrium may be expressed as
|
||||
图9-1所示系统的运动方程可以通过表述与其每个自由度相关的有效力的平衡来建立。通常，在任意点 $i$ 将涉及四种类型的力：外部施加的载荷 $p_{i}(t)$ 以及由运动产生的力，即惯性力 $f_{I i}$、阻尼力 $f_{D i}$ 和弹性力 $f_{S i}$。因此，对于若干个自由度中的每一个，动力平衡可以表示为
|
||||
$$
|
||||
\begin{array}{c}{f_{I1}+f_{D1}+f_{S1}=p_{1}(t)}\\ {f_{I2}+f_{D2}+f_{S2}=p_{2}(t)}\\ {f_{I3}+f_{D3}+f_{S3}=p_{3}(t)}\end{array}
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||||
\begin{array}{c}{f_{I1}+f_{D1}+f_{S1}=p_{1}(t)}\\ {f_{I2}+f_{D2}+f_{S2}=p_{2}(t)}\\ {f_{I3}+f_{D3}+f_{S3}=p_{3}(t)}\end{array}\tag{9-1}
|
||||
$$
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||||
|
||||
or when the force vectors are represented in matrix form,
|
||||
或者当力矢量以矩阵形式表示时，
|
||||
$$
|
||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}=\mathbf{p}(t)
|
||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}=\mathbf{p}(t)\tag{9-2}
|
||||
$$
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||||
|
||||
which is the MDOF equivalent of the SDOF equation (2-1).
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||||
@ -42,25 +42,25 @@ which is the MDOF equivalent of the SDOF equation (2-1).
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||||
Each of the resisting forces is expressed most conveniently by means of an appropriate set of influence coefficients. Consider, for example, the elastic-force component developed at point 1; this depends in general upon the displacement components developed at all points of the structure:
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||||
每一个阻力都可以通过一组合适的**影响系数**最方便地表示。例如，考虑在点1处产生的弹性力分量；这通常取决于在结构所有点处产生的位移分量：
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||||
$$
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||||
f_{S1}=k_{11}v_{1}+k_{12}v_{2}+k_{13}v_{3}+\cdot\cdot\cdot+k_{1N}v_{N}
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||||
f_{S1}=k_{11}v_{1}+k_{12}v_{2}+k_{13}v_{3}+\cdot\cdot\cdot+k_{1N}v_{N}\tag{9-3a}
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||||
$$
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||||
|
||||
Similarly, the elastic force corresponding to the degree of freedom $v_{2}$ is
|
||||
类似地，对应于自由度 $v_{2}$ 的弹性力是
|
||||
$$
|
||||
f_{S2}=k_{21}v_{1}+k_{22}v_{2}+k_{23}v_{3}+\cdot\cdot\cdot+k_{2N}v_{N}
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||||
f_{S2}=k_{21}v_{1}+k_{22}v_{2}+k_{23}v_{3}+\cdot\cdot\cdot+k_{2N}v_{N}\tag{9-3b}
|
||||
$$
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||||
|
||||
and, in general,
|
||||
|
||||
$$
|
||||
f_{S i}=k_{i1}v_{1}+k_{i2}v_{2}+k_{i3}v_{3}+\cdot\cdot\cdot+k_{i N}v_{N}
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||||
f_{S i}=k_{i1}v_{1}+k_{i2}v_{2}+k_{i3}v_{3}+\cdot\cdot\cdot+k_{i N}v_{N}\tag{9-3c}
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||||
$$
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||||
|
||||
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:
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||||
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：
|
||||
$$
|
||||
\begin{array}{r}{k_{i j}=\mathrm{~force~corresponding~to~coordinate~}i\mathrm{~due}\mathrm{~to}}\\ {\mathrm{~a~unit~}d i s p l a c e m e n t\;\mathrm{of~coordinate~}j\quad\quad}\end{array}
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||||
\begin{array}{r}{k_{i j}=\mathrm{~force~corresponding~to~coordinate~}i\mathrm{~due}\mathrm{~to}}\\ {\mathrm{~a~unit~}d i s p l a c e m e n t\;\mathrm{of~coordinate~}j\quad\quad}\end{array}\tag{9-4}
|
||||
$$
|
||||
|
||||
In matrix form, the complete set of elastic-force relationships may be written
|
||||
@ -94,7 +94,7 @@ $$
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||||
or, symbolically,
|
||||
|
||||
$$
|
||||
\mathbf{f}_{S}=\mathbf{k}\ \mathbf{v}
|
||||
\mathbf{f}_{S}=\mathbf{k}\ \mathbf{v}\tag{9-6}
|
||||
$$
|
||||
|
||||
in which the matrix of stiffness coefficients $\mathbf{k}$ is called the stiffness matrix of the structure (for the specified set of displacement coordinates) and $\mathbf{v}$ is the displacement vector representing the displaced shape of the structure.
|
||||
@ -131,35 +131,59 @@ $$
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||||
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):
|
||||
其中 $\dot{v}_{i}$ 表示第 $i$ 个位移坐标的时间变化率（速度），且系数 $c_{i j}$ 称为阻尼影响系数。这些系数的定义与式 (9-4) 完全平行。
|
||||
$$
|
||||
\begin{array}{c}{{c_{i j}=\mathrm{force\;corresponding\;to\;coordinate\;}i\mathrm{\;due\;to\;unit}}}\\ {{{\nu e l o c i t y\;\mathrm{of\;coordinate\;}j}}}\end{array}
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||||
\begin{array}{c}{{c_{i j}=\mathrm{force\;corresponding\;to\;coordinate\;}i\mathrm{\;due\;to\;unit}}}\\ {{{v e l o c i t y\;\mathrm{of\;coordinate\;}j}}}\end{array}\tag{9-8}
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||||
$$
|
||||
|
||||
Symbolically, Eq. (9-7) may be written
|
||||
|
||||
$$
|
||||
\mathbf f_{D}=\mathbf c\,\dot{\mathbf v}
|
||||
\mathbf f_{D}=\mathbf c\,\dot{\mathbf v}\tag{9-9}
|
||||
$$
|
||||
|
||||
in which the matrix of damping coefficients $\mathbf{c}$ is called the damping matrix of the structure (for the specified degrees of freedom) and $\dot{\mathbf{v}}$ is the velocity vector.
|
||||
其中阻尼系数矩阵 $\mathbf{c}$ 称为结构（对于指定的自由度）的阻尼矩阵，且 $\dot{\mathbf{v}}$ 是速度向量。
|
||||
|
||||
The inertial forces may be expressed similarly by a set of influence coefficients called the mass coefficients. These represent the relationship between the accelerations of the degrees of freedom and the resulting inertial forces; by analogy with Eq. (9-5), the inertial forces may be expressed as
|
||||
其中阻尼系数矩阵 $\mathbf{c}$ 称为结构（对于指定的自由度）的阻尼矩阵，且 $\dot{\mathbf{v}}$ 是速度向量。
|
||||
|
||||
惯性力可以类似地通过一组称为质量系数的影响系数来表示。这些表示自由度加速度与由此产生的惯性力之间的关系；参照式 (9-5)，惯性力可以表示为
|
||||
$$
|
||||
\left\{\begin{array}{l}{f_{I1}}\\ {f_{I2}}\\ {\cdot}\\ {\cdot}\\ {f_{I i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{l l l l l l l}{m_{11}}&{m_{12}}&{m_{13}}&{\cdots}&{m_{1i}}&{\cdots}&{m_{1N}}\\ {m_{21}}&{m_{22}}&{m_{23}}&{\cdots}&{m_{2i}}&{\cdots}&{m_{2N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {m_{i1}}&{m_{i2}}&{m_{i3}}&{\cdots}&{m_{i i}}&{\cdots}&{m_{i N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}\\ {m_{i1}}&{m_{i2}}&{m_{i3}}&{\cdots}&{\cdots}&{m_{i i}}&{\cdots}&{\cdots}\\ {\cdots}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]=\left\{\begin{array}{l}{\ddot{v_{1}}}\\ {\ddot{v_{2}}}\\ {\ddot{v_{2}}}\\ {\cdot}\\ {\dot{v_{i}}}\\ {\dot{v_{i}}}\\ {\cdot}\end{array}\right\}
|
||||
\begin{Bmatrix}
|
||||
f_{I1} \\
|
||||
f_{I2} \\
|
||||
\vdots \\
|
||||
f_{Ii} \\
|
||||
\vdots
|
||||
\end{Bmatrix}
|
||||
=
|
||||
\begin{bmatrix}
|
||||
m_{11} & m_{12} & m_{13} & \cdots & m_{1i} & \cdots & m_{1N} \\
|
||||
m_{21} & m_{22} & m_{23} & \cdots & m_{2i} & \cdots & m_{2N} \\
|
||||
\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
|
||||
m_{i1} & m_{i2} & m_{i3} & \cdots & m_{ii} & \cdots & m_{iN} \\
|
||||
\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots
|
||||
\end{bmatrix}
|
||||
=
|
||||
\begin{Bmatrix}
|
||||
\ddot{v}_1 \\
|
||||
\ddot{v}_2 \\
|
||||
\vdots \\
|
||||
\ddot{v}_i \\
|
||||
\vdots
|
||||
\end{Bmatrix}
|
||||
\tag{9-10}
|
||||
$$
|
||||
|
||||
where $\ddot{v}_{i}$ is the acceleration of the $i$ displacement coordinate and the coefficients $m_{i j}$ are the mass influence coefficients, defined as follows:
|
||||
其中 $\ddot{v}_{i}$ 是第 $i$ 个位移坐标的加速度，且系数 $m_{i j}$ 是质量影响系数，定义如下：
|
||||
$$
|
||||
m_{i j}=\mathrm{force\corresponding\to\coordinate\}i\ \mathrm{due}
|
||||
m_{ij} = \text{force corresponding to coordinate } i \text{ due to unit acceleration of coordinate } j
|
||||
\tag{9-11}
|
||||
$$
|
||||
|
||||
Symbolically, Eq. (9-10) may be written
|
||||
|
||||
$$
|
||||
\mathbf{f}_{I}=\mathbf{m}\;\ddot{\mathbf{v}}
|
||||
\mathbf{f}_{I}=\mathbf{m}\;\ddot{\mathbf{v}}\tag{9-12}
|
||||
$$
|
||||
|
||||
in which the matrix of mass coefficients $\mathbf{m}$ is called the mass matrix of the structure and $\ddot{\mathbf{v}}$ is its acceleration vector, both defined for the specified set of displacement coordinates.
|
||||
@ -169,7 +193,7 @@ Substituting Eqs. (9-6), (9-9), and (9-12) into Eq. (9-2) gives the complete dyn
|
||||
|
||||
将式 (9-6)、(9-9) 和 (9-12) 代入式 (9-2)，得到考虑所有自由度的结构完整动力平衡方程：
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)=\mathbf{p}(t)
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)=\mathbf{p}(t)\tag{9-13}
|
||||
$$
|
||||
|
||||
This equation is the MDOF equivalent of Eq. (2-3); each term of the SDOF equation is represented by a matrix in Eq. (9-13), the order of the matrix corresponding to the number of degrees of freedom used in describing the displacements of the structure. Thus, Eq. (9-13) expresses the $N$ equations of motion which serve to define the response of the MDOF system.
|
||||
@ -178,9 +202,9 @@ This equation is the MDOF equivalent of Eq. (2-3); each term of the SDOF equatio
|
||||
## 9-3 AXIAL-FORCE EFFECTS
|
||||
|
||||
It was observed in the discussion of SDOF systems that axial forces or any load which may tend to cause buckling of a structure may have a significant effect on the stiffness of the structure. Similar effects may be observed in MDOF systems; the force component acting parallel to the original axis of the members leads to additional load components which act in the direction (and sense) of the nodal displacements and which will be denoted by $\mathbf{f}_{G}$ . When these forces are included, the dynamic-equilibrium expression, Eq. (9-2), becomes
|
||||
在单自由度（SDOF）系统讨论中观察到，轴向力或任何可能导致结构屈曲的载荷可能会对结构的刚度产生显著影响。在多自由度（MDOF）系统中也可能观察到类似效应；作用于构件原始轴线平行的力分量会产生额外的载荷分量，这些分量作用于节点位移的方向（和指向），并用 $\mathbf{f}_{G}$ 表示。当包含这些力时，动力平衡表达式，即式（9-2），变为
|
||||
在单自由度（SDOF）系统讨论中观察到，**轴向力或任何可能导致结构屈曲的载荷可能会对结构的刚度产生显著影响**。在多自由度（MDOF）系统中也可能观察到类似效应；**作用于构件原始轴线平行的力分量会产生额外的载荷分量，这些分量作用于节点位移的方向（和指向）**，并用 $\mathbf{f}_{G}$ 表示。当包含这些力时，动力平衡表达式，即式（9-2），变为
|
||||
$$
|
||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}-\mathbf{f}_{G}=\mathbf{p}(t)
|
||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}-\mathbf{f}_{G}=\mathbf{p}(t)\tag{9-14}
|
||||
$$
|
||||
|
||||
in which the negative sign results from the fact that the forces $\mathbf{f}_{G}$ are assumed to contribute to the deflection rather than oppose it.
|
||||
@ -190,17 +214,43 @@ These forces resulting from axial loads depend on the displacements of the struc
|
||||
|
||||
这些由轴向载荷产生的力取决于结构的位移，并且可以通过影响系数（称为几何刚度系数）表示，如下所示：
|
||||
$$
|
||||
\left\{\begin{array}{l}{f_{G1}}\\ {f_{G2}}\\ {\cdot}\\ {\cdot}\\ {f_{G i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{c c c c c c c}{k_{G11}}&{k_{G12}}&{k_{G13}}&{\cdot\cdot}&{k_{G1i}}&{\cdot\cdot}&{k_{G1N}}\\ {k_{G21}}&{k_{G22}}&{k_{G23}}&{\cdot\cdot}&{k_{G2i}}&{\cdot\cdot}&{k_{G2N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {\vdots}&{k_{G i1}}&{k_{G i2}}&{k_{G i3}}&{\cdot\cdot}&{k_{G i i}}&{\cdot\cdot}&{k_{G i N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]\left\{\begin{array}{l}{v_{1}}\\ {v_{2}}\\ {v_{2}}\\ {\cdot}\\ {v_{i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}
|
||||
\begin{Bmatrix}
|
||||
f_{G1} \\
|
||||
f_{G2} \\
|
||||
\vdots \\
|
||||
f_{Gi} \\
|
||||
\vdots
|
||||
\end{Bmatrix}
|
||||
=
|
||||
\begin{bmatrix}
|
||||
k_{G11} & k_{G12} & k_{G13} & \cdots & k_{G1i} & \cdots & k_{G1N} \\
|
||||
k_{G21} & k_{G22} & k_{G23} & \cdots & k_{G2i} & \cdots & k_{G2N} \\
|
||||
\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
|
||||
k_{Gi1} & k_{Gi2} & k_{Gi3} & \cdots & k_{Gii} & \cdots & k_{GiN} \\
|
||||
\vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots
|
||||
\end{bmatrix}
|
||||
\begin{Bmatrix}
|
||||
v_1 \\
|
||||
v_2 \\
|
||||
\vdots \\
|
||||
v_i \\
|
||||
\vdots
|
||||
\end{Bmatrix}
|
||||
\tag{9-15}
|
||||
$$
|
||||
|
||||
in which the geometric-stiffness influence coefficients $k_{G_{i j}}$ have the following definition:
|
||||
|
||||
$k_{G_{i j}}=\mathrm{force\;corresponding\;to\;coordinate\;}i\mathrm{~due\;to\;unit}$ displacement of coordinate $j$ and resulting from axial-force components in the structure
|
||||
$$
|
||||
k_{G_{ij}} = \text{force corresponding to coordinate } i \text{ due to unit} \\
|
||||
\text{displacement of coordinate } j \text{ and resulting from} \\
|
||||
\text{axial-force components in the structure}
|
||||
\tag{9-16}
|
||||
$$
|
||||
|
||||
Symbolically Eq. (9-15) may be written
|
||||
|
||||
$$
|
||||
\mathbf{f}_{G}=\mathbf{k}_{G}\,\mathbf{v}
|
||||
\mathbf{f}_{G}=\mathbf{k}_{G}\,\mathbf{v}\tag{9-17}
|
||||
$$
|
||||
|
||||
where $\mathbf{k}_{G}$ is called the geometric-stiffness matrix of the structure.
|
||||
@ -210,19 +260,19 @@ When this expression is introduced, the equation of dynamic equilibrium of the s
|
||||
|
||||
当引入此表达式时，结构的动力平衡方程[由不考虑轴向力效应的式 (9-13) 给出]变为
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)-\mathbf{k}_{G}\,\mathbf{v}(t)=\mathbf{p}(t)
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)-\mathbf{k}_{G}\,\mathbf{v}(t)=\mathbf{p}(t)\tag{9-18}
|
||||
$$
|
||||
|
||||
or when it is noted that both the elastic stiffness and the geometric stiffness are multiplied by the displacement vector, the combined stiffness effect can be expressed by a single symbol and Eq. (9-18) written
|
||||
或者当注意到弹性刚度和几何刚度都乘以位移向量时，组合刚度效应可以用一个符号表示，并且式 (9-18) 写成
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\overline{{\mathbf{k}}}\,\mathbf{v}(t)=\mathbf{p}(t)
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\overline{{\mathbf{k}}}\,\mathbf{v}(t)=\mathbf{p}(t)\tag{9-19}
|
||||
$$
|
||||
|
||||
in which
|
||||
|
||||
$$
|
||||
\overline{{\mathbf{k}}}=\mathbf{k}-\mathbf{k}_{G}
|
||||
\overline{{\mathbf{k}}}=\mathbf{k}-\mathbf{k}_{G}\tag{9-20}
|
||||
$$
|
||||
|
||||
is called the combined stiffness matrix, which includes both elastic and geometric effects. The dynamic properties of the structure are expressed completely by the four influence-coefficient matrices of Eq. (9-18), while the dynamic loading is fully defined by the load vector. The evaluation of these physical-property matrices and the evaluation of the load vector resulting from externally applied forces will be discussed in detail in the following chapter. The effective-load vector resulting from support excitation will be discussed in connection with earthquake-response analysis in Chapter 26.
|
||||
|
||||
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Reference in New Issue
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