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@ -559,6 +559,11 @@ The finite-element type of idealization is applicable to structures of all types
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The first step in the finite-element idealization of any structure, e.g., the beam shown in Fig. 1-5, involves dividing it into an appropriate number of segments, or elements, as shown. Their sizes are arbitrary; i.e., they may be all of the same size or all different. The ends of the segments, at which they are interconnected, are called nodal points. The displacements of these nodal points then become the generalized coordinates of the structure.
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现已普及一种将任意给定结构的位移表示为有限个离散位移坐标的第三种方法，该方法结合了集中质量法和广义坐标法的某些特点。这种方法是结构连续体有限元分析的基础，为系统提供了一种方便可靠的理想化方法，并且在数字计算机分析中特别有效。
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有限元类型的理想化适用于所有类型的结构：由一维构件（梁、柱等）组成的框架结构；由二维构件组成的平面应力、板壳结构；以及一般三维实体。为简单起见，本次讨论中只考虑一维结构构件，但将该概念扩展到二维和三维结构单元是直接的。
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任何结构的有限元理想化的第一步，例如图1-5所示的梁，涉及将其划分为适当数量的段或单元，如图所示。它们的尺寸是任意的；即，它们可以全部相同或全部不同。这些段的端点，即它们相互连接的地方，称为节点。这些节点的位移随后成为结构的广义坐标。
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FIGURE 1-5 Typical finite-element beam coordinates.
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@ -572,43 +577,57 @@ Because the interpolation functions used in this procedure satisfy the requireme
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In general, the finite-element approach provides the most efficient procedure for expressing the displacements of arbitrary structural configurations by means of a discrete set of coordinates.
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完整结构的变形形状现在可以通过一组适当的假定位移函数，使用类似于公式(1-2)的表达式，用这些广义坐标来表示。然而，在这种情况下，这些位移函数被称为插值函数，因为它们定义了由指定节点位移产生的形状。例如，图1-5显示了与节点3的两个自由度相关的插值函数，它们在图示平面内产生横向位移。原则上，每个插值函数可以是任何内部连续并满足由节点位移施加的几何位移条件的曲线。对于一维单元，方便使用这些相同的节点位移在均匀梁中产生的形状。稍后在第10章中将显示，这些插值函数是三次Hermite多项式。
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由于本程序中使用的插值函数满足前一节中提出的要求，因此应该很明显，有限元方法中使用的坐标只是广义坐标的特殊形式。这种特殊方法的优点如下：
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(1) 仅通过将结构划分为适当数量的段，即可引入所需数量的广义坐标。
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(2) 由于为每个段选择的插值函数可以是相同的，因此计算得到简化。
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(3) 通过这种方法建立的方程大部分是解耦的，因为每个节点位移只影响相邻单元；因此求解过程大大简化。
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总的来说，有限元方法提供了一种最有效的方法，通过一组离散的坐标来表示任意结构构型的位移。
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# 1-5 FORMULATION OF THE EQUATIONS OF MOTION
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As mentioned earlier, the primary objective of a deterministic structural-dynamic analysis is the evaluation of the displacement time-histories of a given structure subjected to a given time-varying loading. In most cases, an approximate analysis involving only a limited number of degrees of freedom will provide sufficient accuracy; thus, the problem can be reduced to the determination of the time-histories of these selected displacement components. The mathematical expressions defining the dynamic displacements are called the equations of motion of the structure, and the solution of these equations of motion provides the required displacement time-histories.
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The formulation of the equations of motion of a dynamic system is possibly the most important, and sometimes the most difficult, phase of the entire analysis procedure. In this text, three different methods will be employed for the formulation of these equations, each having advantages in the study of special classes of problems. The fundamental concepts associated with each of these methods are described in the following paragraphs.
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如前所述，确定性结构动力学分析的主要目标是评估给定结构在给定随时间变化的载荷作用下的位移时程。在大多数情况下，仅涉及有限数量自由度的近似分析将提供足够的精度；因此，问题可以简化为确定这些选定位移分量的时程。定义动力位移的数学表达式称为结构的运动方程，这些运动方程的解提供了所需的位移时程。
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# Direct Equilibration Using d’Alembert’s Principle
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动力系统运动方程的建立可能是整个分析过程中最重要的阶段，有时也是最困难的阶段。在本文中，将采用三种不同的方法来建立这些方程，每种方法在研究特殊类型问题时都具有优势。与这些方法中的每一种相关联的基本概念将在以下段落中描述。
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## Direct Equilibration Using d’Alembert’s Principle 基于达朗贝尔原理的直接均衡
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The equations of motion of any dynamic system represent expressions of Newton’s second law of motion, which states that the rate of change of momentum of any mass particle $m$ is equal to the force acting on it. This relationship can be expressed mathematically by the differential equation
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任何动力学系统的运动方程都代表了牛顿第二运动定律的表达式，该定律指出，任何质量为 $m$ 的质点的动量变化率等于作用在其上的力。这种关系可以用微分方程表示为
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$$
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\mathbf{p}(t)={\frac{d}{d t}}{\bigg(}m{\frac{d\mathbf{v}}{d t}}{\bigg)}
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$$
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where ${\bf p}(t)$ is the applied force vector and $\mathbf{v}(t)$ is the position vector of particle mass $m$ . For most problems in structural dynamics it may be assumed that mass does not vary with time, in which case Eq. (1-3) may be written
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其中 ${\bf p}(t)$ 是施加的力矢量，$\mathbf{v}(t)$ 是质量为 $m$ 的质点的**位置矢量**。对于结构动力学中的大多数问题，可以假设质量不随时间变化，在这种情况下，式 (1-3) 可以写成
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$$
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\mathbf{p}(t)=m{\frac{d^{2}\mathbf{v}}{d t^{2}}}\equiv m\,{\ddot{\mathbf{v}}}(t)
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$$
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where the dots represent differentiation with respect to time. Equation (1-3a), indicating that force is equal to the product of mass and acceleration, may also be written in the form
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其中点表示对时间求导。方程 (1-3a) 表明力等于质量和加速度的乘积，也可以写成以下形式
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$$
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\mathbf{p}(t)-m\,\ddot{\mathbf{v}}(t)=\mathbf{0}
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$$
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in which case, the second term $m{\ddot{\mathbf{v}}}(t)$ is called the inertial force resisting the acceleration of the mass.
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在这种情况下，第二项 $m{\ddot{\mathbf{v}}}(t)$ 被称为抵抗质量加速度的惯性力。
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The concept that a mass develops an inertial force proportional to its acceleration and opposing it is known as d’Alembert’s principle. It is a very convenient device in problems of structural dynamics because it permits the equations of motion to be expressed as equations of dynamic equilibrium. The force $\mathbf p(t)$ may be considered to include many types of force acting on the mass: elastic constraints which oppose displacements, viscous forces which resist velocities, and independently defined external loads. Thus if an inertial force which resists acceleration is introduced, the equation of motion is merely an expression of equilibration of all forces acting on the mass. In many simple problems, the most direct and convenient way of formulating the equations of motion is by means of such direct equilibrations.
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# Principle of Virtual Displacements
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一个质量产生一个与其加速度成正比且方向相反的惯性力的概念被称为达朗贝尔原理。在结构动力学问题中，它是一个非常方便的工具，因为它允许将运动方程表达为动力平衡方程。力 $\mathbf p(t)$ 可以被认为包括作用在质量上的多种类型的力：抵抗位移的弹性约束力、抵抗速度的粘性力以及独立定义的外载荷。因此，如果引入一个抵抗加速度的惯性力，运动方程就仅仅是作用在质量上的所有力的平衡表达式。在许多简单问题中，建立运动方程最直接和方便的方法就是通过这种直接的平衡。
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## Principle of Virtual Displacements
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However, if the structural system is reasonably complex involving a number of interconnected mass points or bodies of finite size, the direct equilibration of all the forces acting in the system may be difficult. Frequently, the various forces involved may readily be expressed in terms of the displacement degrees of freedom, but their equilibrium relationships may be obscure. In this case, the principle of virtual displacements can be used to formulate the equations of motion as a substitute for the direct equilibrium relationships.
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The principle of virtual displacements may be expressed as follows. If a system which is in equilibrium under the action of a set of externally applied forces is subjected to a virtual displacement, i.e., a displacement pattern compatible with the system’s constraints, the total work done by the set of forces will be zero. With this principle, it is clear that the vanishing of the work done during a virtual displacement is equivalent to a statement of equilibrium. Thus, the response equations of a dynamic system can be established by first identifying all the forces acting on the masses of the system, including inertial forces defined in accordance with d’Alembert’s principle. Then, the equations of motion are obtained by separately introducing a virtual displacement pattern corresponding to each degree of freedom and equating the work done to zero. A major advantage of this approach is that the virtual-work contributions are scalar quantities and can be added algebraically, whereas the forces acting on the structure are vectorial and can only be superposed vectorially.
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然而，如果结构系统相当复杂，涉及多个相互连接的质点或有限尺寸的物体，那么直接平衡系统中所有作用力可能会很困难。通常，所涉及的各种力可以很容易地用位移自由度来表示，但它们的平衡关系可能不明确。在这种情况下，虚位移原理可以用来建立运动方程，作为直接平衡关系的替代。
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# Variational Approach
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虚位移原理可以表述如下。如果一个系统在外部施加的一组力的作用下处于平衡状态，并受到一个虚位移（即，一个与系统约束条件相容的位移模式），那么这组力所做的总功将为零。依据此原理，显而易见，虚位移过程中所做功的消失等同于平衡的表述。因此，动力系统的响应方程可以通过首先识别作用在系统质量上的所有力（包括根据达朗贝尔原理定义的惯性力）来建立。然后，通过分别引入与每个自由度对应的虚位移模式并将所做功设为零来获得运动方程。这种方法的一个主要优点是，虚功贡献是标量，可以代数相加，而作用在结构上的力是矢量，只能进行矢量叠加。
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## Variational Approach变分方法
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Another means of avoiding the problems of establishing the vectorial equations of equilibrium is to make use of scalar quantities in a variational form known as Hamilton’s principle. Inertial and elastic forces are not explicitly involved in this principle; instead, variations of kinetic and potential energy terms are utilized. This formulation has the advantage of dealing only with purely scalar energy quantities, whereas the forces and displacements used to represent corresponding effects in the virtual-work procedure are all vectorial in character, even though the work terms themselves are scalars.
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@ -616,6 +635,11 @@ It is of interest to note that Hamilton’s principle can also be applied to sta
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It has been shown that the equation of motion of a dynamic system can be formulated by any one of three distinct procedures. The most straightforward approach is to establish directly the dynamic equilibrium of all forces acting in the system, taking account of inertial effects by means of d’Alembert’s principle. In more complex systems, however, especially those involving mass and elasticity distributed over finite regions, a direct vectorial equilibration may be difficult, and work or energy formulations which involve only scalar quantities may be more convenient. The most direct of these procedures is based on the principle of virtual displacements, in which the forces acting on the system are evaluated explicitly but the equations of motion are derived by consideration of the work done during appropriate virtual displacements. On the other hand, the alternative energy formulation, which is based on Hamilton’s principle, makes no direct use of the inertial or conservative forces acting in the system; the effects of these forces are represented instead by variations of the kinetic and potential energies of the system. It must be recognized that all three procedures are completely equivalent and lead to identical equations of motion. The method to be used in any given case is largely a matter of convenience and personal preference; the choice generally will depend on the nature of the dynamic system under consideration.
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避免建立矢量平衡方程问题的另一种方法是利用标量形式的变分法，即哈密顿原理。惯性力和弹性力不明确地涉及此原理；相反，利用动能和势能项的变分。**这种表述的优点是只处理纯标量能量，而在虚功法中用于表示相应效应的力和位移都具有矢量性质，即使功项本身是标量**。
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值得注意的是，哈密顿原理也可以应用于静力学问题。在这种情况下，它简化为在静力分析中广泛使用的众所周知的最小势能原理。
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已经表明，动力系统的运动方程可以通过三种不同的方法中的任何一种来建立。最直接的方法是直接建立系统中所有作用力的动平衡，通过达朗贝尔原理考虑惯性效应。然而，在更复杂的系统中，特别是那些涉及质量和弹性分布在有限区域的系统中，直接的矢量平衡可能很困难，而只涉及标量的功或能量表述可能更方便。这些方法中最直接的是基于虚位移原理，其中作用在系统上的力被明确地评估，但运动方程是通过考虑在适当虚位移期间所做的功来推导的。另一方面，基于哈密顿原理的替代能量表述不直接使用作用在系统中的惯性力或保守力；相反，这些力的效应由系统动能和势能的变分来表示。必须认识到，所有这三种方法是完全等效的，并导致相同的运动方程。在任何给定情况下使用的方法很大程度上是方便性和个人偏好的问题；选择通常取决于所考虑的动力系统的性质。
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# 1-6 ORGANIZATION OF THE TEXT
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This book, “Dynamics of Structures,” has been written in five parts. Part One presents an extensive treatment of the single-degree-of-freedom (SDOF) system having only one independent displacement coordinate. This system is studied in great detail for two reasons: (1) the dynamic behavior of many practical structures can be expressed in terms of a single coordinate, so that this SDOF treatment applies directly in those cases, and (2) the response of complex linear structures can be expressed as the sum of the responses of a series of SDOF systems so that this same treatment once again applies to each system in the series. Thus, the SDOF analysis techniques provide the basis for treating the vast majority of structural-dynamic problems.
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@ -628,6 +652,15 @@ Part Four covers the general topic of random vibrations of linear SDOF and MDOF
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Earthquake engineering, with special focus on structural response and performance, is the subject of Part Five. A very brief seismological background on the causes and characteristics of earthquakes is given, along with a discussion of the ground motions they produce. Methods are then given for evaluating the response of structures to these motions using both deterministic and nondeterministic procedures.
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本书《结构动力学》分为五个部分。第一部分详细阐述了仅具有一个独立位移坐标的单自由度（SDOF）系统。对该系统进行详细研究的原因有二：(1) 许多实际结构的动力行为可以用单个坐标来表示，因此这种SDOF处理方法可以直接应用于这些情况；(2) 复杂线性结构的响应可以表示为一系列SDOF系统响应之和，因此同样的分析方法再次适用于该系列中的每个系统。因此，SDOF分析技术为处理绝大多数结构动力学问题提供了基础。
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第二部分讨论了离散参数多自由度（MDOF）系统，即其动力响应可以用有限数量的位移坐标来表示的系统。对于线性弹性系统的分析，本部分介绍了评估其自由振动状态下特性的程序，即评估简正模态形状和相应频率的程序。然后，给出了计算这些系统在任意指定载荷作用下动力响应的两种通用方法：(1) 利用模态叠加法，其中总响应表示为各种简正振动模态中各个响应之和，每个响应都可以通过SDOF系统的分析程序确定；(2) 直接求解原始耦合形式的MDOF运动方程。最后，介绍了结构动力学问题的变分公式，并阐述了逐步数值积分技术，用于直接求解代表线性或非线性系统的SDOF和MDOF运动方程。
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第三部分考虑了具有连续分布特性的动力线性弹性系统。此类系统具有无限多个自由度，要求其运动方程以偏微分方程的形式书写。然而，本部分表明模态叠加法仍然适用于这些系统，并且通过仅考虑有限数量的低阶振动模态可以获得实际的解。
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第四部分涵盖了线性SDOF和MDOF系统随机振动的一般主题。由于所考虑的载荷只能在统计意义上进行表征，因此相应的响应也以类似方式进行表征。为了提供处理这些系统的基础，本部分介绍了概率论和随机过程。
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第五部分的主题是地震工程，特别关注结构响应和性能。本部分简要介绍了地震的成因和特征的地震学背景，并讨论了地震产生的地面运动。然后，给出了使用确定性和非确定性程序评估结构对这些运动响应的方法。
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PART I
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SINGLE-DEGREE-OF-
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