vault backup: 2025-09-25 08:18:04

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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

@@ -1303,66 +1303,69 @@ $(a)$ the lateral spring stiffness $k$ $(b)$ the damping ratio $\xi$ $(c)$ the d
 
 RESPONSE TO HARMONIC LOADING  
 
-# 3-1 UNDAMPED SYSTEM  
+# Chap 3 RESPONSE TO HARMONIC LOADING谐波载荷响应
 
-# Complementary Solution  
+## 3-1 UNDAMPED SYSTEM无阻尼系统 
+
+
+### Complementary Solution互补解决方案  
 
 Assume the system of Fig. 2-1 is subjected to a harmonically varying load $p(t)$ of sine-wave form having an amplitude $p_{o}$ and circular frequency $\overline{{\omega}}$ as shown by the equation of motion  
-
+假设图2-1所示系统受到一个正弦波形式的简谐载荷 $p(t)$，其幅值为 $p_{o}$，圆频率为 $\overline{{\omega}}$，如运动方程所示。
 $$
 m\;\ddot{v}(t)+c\;\dot{v}(t)+k\;v(t)=p_{_o}\;\sin\overline{{{\omega}}}t
 $$  
 
 Before considering this viscously damped case, it is instructive to examine the behavior of an undamped system as controlled by  
-
+在考虑这种粘性阻尼情况之前，研究一个由...控制的无阻尼系统的行为是很有启发性的。
 $$
 m\ \ddot{v}(t)+k\ v(t)=p_{o}\ \sin\overline{{\omega}}t
 $$  
 
-which has a complementary solution of the free-vibration form of Eq. (2-31)  
+which has a complementary solution of the free-vibration form of Eq. (2-31) 其互补解具有式 (2-31) 的自由振动形式 
 
 $$
 v_{c}(t)=A\,\cos\omega t+B\,\sin\omega t
 $$  
 
-# Particular Solution  
+### Particular Solution  
 
 The general solution must also include the particular solution which depends upon the form of dynamic loading. In this case of harmonic loading, it is reasonable to assume that the corresponding motion is harmonic and in phase with the loading; thus, the particular solution is  
-
+通解还必须包括特解，该特解取决于动载荷的形式。在简谐载荷的情况下，合理假设相应的运动是简谐的并与载荷同相；因此，特解为
 $$
 v_{p}(t)=C\,\sin{\overline{{\omega}}t}
 $$  
 
 in which the amplitude $C$ is to be evaluated.  
-
+其中振幅 $C$ 待确定。
 Substituting Eq. (3-4) into Eq. (3-2) gives  
-
+将式 (3-4) 代入式 (3-2) 可得
 $$
 -m\,\overline{{{\omega}}}^{2}\,C\,\sin\overline{{{\omega}}}t+k\,C\,\sin\overline{{{\omega}}}t=p_{o}\,\sin\overline{{{\omega}}}t
 $$  
 
 Dividing through by $\sin{\overline{{\omega}}}t$ (which is nonzero in general) and by $k$ and noting that $k/m=\omega^{2}$ , one obtains after some rearrangement  
-
+除以 $\sin{\overline{{\omega}}}t$ (通常不为零) 和 $k$，并注意到 $k/m=\omega^{2}$，经过一些重新整理后，得到
 $$
 C=\frac{p_{o}}{k}\left[\frac{1}{1-\beta^{2}}\right]
 $$  
 
 in which $\beta$ is defined as the ratio of the applied loading frequency to the natural free-vibration frequency, i.e.,  
-
+其中 $\beta$ 定义为施加的载荷频率与固有自由振动频率的比值，即
 $$
 \beta\equiv\overline{{\omega}}\mathrm{~/~}\omega
 $$  
 
-# General Solution  
+### General Solution通解  
 
 The general solution of Eq. (3-2) is now obtained by combining the complementary and particular solutions and making use of Eq. (3-6); thus, one obtains  
-
+方程(3-2)的通解现通过结合互补解和特解，并利用方程(3-6)得到；由此，可得
 $$
 v(t)=v_{c}(t)+v_{p}(t)=A\ \cos{\omega t}+B\ \sin{\omega t}+{\frac{p_{o}}{k}}\left[{\frac{1}{1-\beta^{2}}}\right]\ \sin{\overline{{\omega t}}}
 $$  
 
 In this equation, the values of $A$ and $B$ depend on the conditions with which the response was initiated. For the system starting from rest, i.e., $v(0)=\dot{v}(0)=0$ , it is easily shown that  
-
+在这个方程中，$A$ 和 $B$ 的值取决于响应启动的条件。对于从静止状态开始的系统，即 $v(0)=\dot{v}(0)=0$ ，可以很容易地证明
 $$
 A=0~~~~~~~~~~~~B=-{\frac{p_{o}\beta}{k}}\left[{\frac{1}{1-\beta^{2}}}\right]
 $$  
@@ -1374,6 +1377,7 @@ v(t)=\frac{p_{o}}{k}\,\left[\frac{1}{1-\beta^{2}}\right]\,\left(\sin\overline{{\
 $$  
 
 where $p_{o}/k\,=\,v_{\mathrm{st}}$ is the displacement which would be produced by the load $p_{o}$ applied statically and $1/(1-\beta^{2})$ is the magnification factor (MF) representing the amplification effect of the harmonically applied loading. In this equation, $\sin{\overline{{\omega}}}t$ represents the response component at the frequency of the applied loading; it is called the steady-state response and is directly related to the loading. Also $\beta\sin\omega t$ is the response component at the natural vibration frequency and is the free-vibration effect controlled by the initial conditions. Since in a practical case, damping will cause the last term to vanish eventually, it is termed the transient response. For this hypothetical undamped system, however, this term will not damp out but will continue indefinitely.  
+其中，$p_{o}/k\,=\,v_{\mathrm{st}}$ 是由静载荷 $p_{o}$ 产生的位移，而 $1/(1-\beta^{2})$ 是表示简谐载荷放大效应的放大系数 (MF)。在该方程中，$\sin{\overline{{\omega}}}t$ 表示施加载荷频率下的响应分量；它被称为稳态响应，并与载荷直接相关。此外，$\beta\sin\omega t$ 是固有振动频率下的响应分量，并且是由初始条件控制的自由振动效应。由于在实际情况中，阻尼最终会使最后一项消失，因此它被称为瞬态响应。然而，对于这个假设的无阻尼系统，这一项不会衰减，而是会无限期地持续下去。
 
 Response Ratio — A convenient measure of the influence of dynamic loading is provided by the ratio of the dynamic displacement response to the displacement produced by static application of load $p_{o}$ , i.e.,