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书籍/力学书籍/力学/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library)/auto/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library).md

@@ -796,7 +796,7 @@ f_{I}(t)+f_{D}(t)+f_{S}(t)=0
 $$  
 
 in which the damping and elastic forces can be expressed as in Eqs. (2-2). However, the inertial force in this case is given by  
-
+其中阻尼力和弹性力可以表示为方程 (2-2)。然而，在这种情况下，惯性力由下式给出
 $$
 f_{I}(t)=m\;\ddot{v}^{t}(t)
 $$  
@@ -811,37 +811,38 @@ m~{\ddot{v}}^{t}(t)+c~{\dot{v}}(t)+k~v(t)=0
 $$  
 
 Before this equation can be solved, all forces must be expressed in terms of a single variable, which can be accomplished by noting that the total motion of the mass can be expressed as the sum of the ground motion and that due to column distortion, i.e.,  
-
+在求解此方程之前，所有力都必须用一个单一变量来表示，这可以通过注意到质量的总运动可以表示为地面运动和由柱变形引起的运动之和来实现，即
 $$
 v^{t}(t)=v(t)+v_{g}(t)
 $$  
 
 Expressing the inertial force in terms of the two acceleration components obtained by double differentiation of Eq. (2-15) and substituting the result into Eq. (2-14) yields  
-
+将惯性力表示为通过对式 (2-15) 进行二次微分得到的两个加速度分量，并将结果代入式 (2-14)，可得
 $$
 m\;{\ddot{v}}(t)+m\;{\ddot{v}}_{g}(t)+c\;{\dot{v}}(t)+k\;v(t)=0
 $$  
 
 or, since the ground acceleration represents the specified dynamic input to the structure, the same equation of motion can more conveniently be written  
-
+或者，由于地面加速度代表了结构所受的规定动力输入，因此相同的运动方程可以更方便地写成
 $$
 m\;\ddot{v}(t)+c\;\dot{v}(t)+k\;v(t)=-m\;\ddot{v}_{g}(t)\equiv p_{\mathrm{eff}}(t)
 $$  
 
 In this equation, $p_{\mathrm{eff}}(t)$ denotes the effective support excitation loading; in other words, the structural deformations caused by ground acceleration ${\ddot{v}}_{g}(t)$ are exactly the same as those which would be produced by an external load $p(t)$ equal to $-m\ i_{g}(t)$ . The negative sign in this effective load definition indicates that the effective force opposes the sense of the ground acceleration. In practice this has little significance inasmuch as the engineer is usually only interested in the maximum absolute value of $v(t)$ ; in this case, the minus sign can be removed from the effective loading term.  
+在此方程中，$p_{\mathrm{eff}}(t)$ 表示有效支座激励荷载；换句话说，由地面加速度 ${\ddot{v}}_{g}(t)$ 引起的结构变形与由等于 $-m\ i_{g}(t)$ 的外部荷载 $p(t)$ 产生的变形完全相同。此有效荷载定义中的负号表明有效力与地面加速度的方向相反。实际上，这意义不大，因为工程师通常只关注 $v(t)$ 的最大绝对值；在这种情况下，可以从有效荷载项中移除负号。
 
 An alternative form of the equation of motion can be obtained by using Eq. (2- 15) and expressing Eq. (2-14) in terms of $v^{t}(t)$ and its derivatives, rather than in terms of $v(t)$ and its derivatives, giving  
-
+运动方程的另一种形式可以通过使用公式 (2-15) 并用 $v^{t}(t)$ 及其导数，而不是用 $v(t)$ 及其导数来表示公式 (2-14) 获得，从而得到
 $$
 m\:\ddot{v}^{t}(t)+c\:\dot{v}^{t}(t)+k\:v^{t}(t)=c\:\dot{v}_{g}(t)+k\:v_{g}(t)
 $$  
 
 In this formulation, the effective loading shown on the right hand side of the equation depends on the velocity and displacement of the earthquake motion, and the response obtained by solving the equation is total displacement of the mass from a fixed reference rather than displacement relative to the moving base. Solutions are seldom obtained in this manner, however, because the earthquake motion generally is measured in terms of accelerations and the seismic record would have to be integrated once and twice to evaluate the effective loading contributions due to the velocity and displacement of the ground.  
-
+在此公式中，方程右侧所示的有效载荷取决于地震运动的速度和位移，通过求解该方程得到的响应是质量相对于固定参考点的总位移，而不是相对于移动基座的位移。然而，很少以这种方式获得解，因为地震运动通常以加速度来测量，并且需要对地震记录进行一次和两次积分，才能评估由地面速度和位移引起的有效载荷贡献。
 # 2-5 ANALYSIS OF UNDAMPED FREE VIBRATIONS  
 
 It has been shown in the preceding sections that the equation of motion of a simple spring-mass system with damping can be expressed as  
-
+在前面的章节中已经表明，一个带有阻尼的简单弹簧-质量系统的运动方程可以表示为
 $$
 m\;\ddot{v}(t)+c\;\dot{v}(t)+k\;v(t)=p(t)
 $$  
@@ -849,7 +850,9 @@ $$
 in which $v(t)$ represents the dynamic response (that is, the displacement from the static-equilibrium position) and $p(t)$ represents the effective load acting on the system, either applied directly or resulting from support motions.  
 
 The solution of Eq. (2-19) will be obtained by considering first the homogeneous form with the right side set equal to zero, i.e.,  
+其中，$v(t)$表示动态响应（即，相对于静平衡位置的位移），$p(t)$表示作用在系统上的有效载荷，该载荷可以是直接施加的，也可以是支撑运动引起的。
 
+方程（2-19）的解将通过首先考虑右侧设为零的齐次形式来获得，即，
 $$
 m\ {\ddot{v}}(t)+c\ {\dot{v}}(t)+k\ v(t)=0
 $$  
@@ -857,7 +860,9 @@ $$
 Motions taking place with no applied force are called free vibrations, and it is the free-vibration response of the system which now will be examined.  
 
 The free-vibration response that is obtained as the solution of Eq. (2-20) may be expressed in the following form:  
+没有施加外力而发生的运动称为自由振动，现在将研究系统的自由振动响应。
 
+作为方程（2-20）解而获得的自由振动响应可以表示为以下形式：
 $$
 v(t)=G\,\exp(s t)
 $$  
@@ -867,7 +872,11 @@ where $G$ is an arbitrary complex constant and $\exp(s t)\equiv e^{s t}$ denotes
 in expressing dynamic loadings and responses; therefore it is useful now to briefly review the complex number concept.  
 
 Considering first the complex constant $G$ , this may be represented as a vector plotted in the complex plane as shown in Fig. 2-4. This sketch demonstrates that the vector may be expressed in terms of its real and imaginary Cartesian components:  
+其中 $G$ 是一个任意复常数，$\exp(s t)\equiv e^{s t}$ 表示指数函数。在随后的讨论中，使用复数
 
+来表示动态载荷和响应通常会很方便；因此，现在简要回顾复数概念是很有用的。
+
+首先考虑复常数 $G$，它可以表示为在复平面中绘制的向量，如图 2-4 所示。该示意图表明，该向量可以用其实部和虚部笛卡尔分量来表示：
 $$
 G=G_{R}+i\;G_{I}
 $$