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多体+耦合求解器/分段模态叠加法/参考文献/Dynamic analysis of large structures by modal synthesis techniques/auto/Dynamic analysis of large structures by modal synthesis techniques.md
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@ -7,7 +7,11 @@ Abstract-The past decade has seen the development of several techniques for the
This paper discusses some of the approaches used to develop structural dynamic characteristics from substructure dynamic characteristics. Several criteria that may be used to evaluate the merits of the various methodsarediscussed.
Two methods classed as (1) fixed-attachment mode and (2) free-attachment mode methods are developed in detail. General flow charts are presented that can be used in preparing computer programs.
摘要-过去十年，出现了多种用于大型结构动态分析的技术，这些技术涉及将结构划分为子结构或组件。这些技术利用组件位移模态来合成全局广义坐标系统，因此被称为模态合成或组件模态技术。
本文讨论了一些从子结构动力学特性发展到结构动力学特性的途径，并讨论了可用于评估各种方法优缺点的若干准则。
本文详细阐述了两类方法：(1) 固定附件模式；(2) 自由附件模式。提供了可用于编写计算机程序的一般流程图。
# INTRODUCTION
DuRING the past decade, a body of technology has developed within the general field of structural dynamics that has come to be identified by the term modal synthesis. The basic idea is to treat the structure as an assembly of connected components, or substructures, each of which is analyzed separately to derive a set of modes or displacement shapes from which a set of generalized coordinates applicable to the complete structure is synthesized.
@ -17,64 +21,111 @@ The calculation of the natural frequencies and mode shapes of a structure, as we
Several researchers have formulated various modal synthesis procedures in an attempt to reduce computation errors and minimize computer costs. These procedures attempt to retain the accuracy of the modal characteristics deemed important for the problem under consideration. However, inherent in each is the calculation of inaccurate answers for items considered unimportant. All modal procedures are based upon the Rayleigh-Ritz method of modal estimation, and each has strong points and weaknesses for different types of problems.
In order to evaluate the merits of the various procedures, it is important to establish objectives to be achieved through their use. These serve as criteria against which the strong and weak points in the several techniques can be detected.
在过去十年中，结构动力学一般领域内发展出一套技术，已被称为模态合成。其基本思路是将结构视为由相互连接的组件或子结构组成，每个子结构单独分析，得到一组模态或位移形状，再从中合成一组适用于整个结构的广义坐标。
# COMPONENT MODE METHODS, SURVEY, AND EVALUATION
结构的固有频率和模态形状的计算，以及动态响应预测，一直是工程师关注的重点。在计算机出现之前，主要关注的是尽量减少预测这些动力参数所需的手工计算次数。这种强调通常导致使用近似方法来预测基本固有频率和模态形状。随着数字计算机的普及和性能提升，我们现在可以通过使用大量广义坐标来构建更逼真的结构分析模型。虽然使用大量坐标可以更准确地描述结构的变形形状，但也需要处理大规模矩阵的计算。因此，由于矩阵尺寸庞大，直接使用所有广义坐标求解特征值可能超出计算机的尺寸限制，或导致数值计算错误和长时间的计算运行。
Criteria useful in comparing methods of analysis
多位研究者制定了各种模态合成程序，旨在降低计算误差并最小化计算成本。这些程序试图保留对所考虑问题重要的模态特征的准确性。然而，每种方法本身都会对被视为不重要的项目产生不准确的答案。所有模态程序都基于Rayleigh-Ritz模态估计方法，并且在不同类型的问题上各有优缺点。
为了评估各程序的优劣，必须确立通过使用它们所要实现的目标。这些目标作为评判各技术强弱点的标准。
# COMPONENT MODE METHODS, SURVEY, AND EVALUATION 组件模态方法、调查与评估
## Criteria useful in comparing methods of analysis
比较分析方法的有用准则
(1) A basic reason for the use of modal synthesis methods is to maximize both the quantity and quality of analysis obtainable with a given computer facility. By isolating the separate components of the structure (except for the final synthesis operations), the number of equations that may be solved for each component is greater than the number that would be possible if all components were treated together. In other words, each component can be treated by a more accurate and refined model and coordinate system. For system synthesis, the coordinate system obtained by simply synthesizing the conponent coordinates must be truncated. Otherwise, no economy would be achieved. Therefore, the matter of accuracy of the solution of the truncated system is of great importance. This can be investigated by determining the rate at which solutions converge to some asymptotic value as the number of degrees of freedom is increased. Therefore, various methods must be compared in terms of natural frequency and mode shapeconvergence.
(1) 使用模态综合方法的基本原因是最大化在现有计算机设施下可获得的分析数量与质量。通过将结构的各个组成部分分离（仅不包括最终综合操作），每个组成部分可求解的方程数量大于将所有组成部分一起处理时可求解的方程数量。换句话说，每个组成部分可以采用更精确、更细致的模型和坐标系进行处理。对于系统综合，仅通过综合各组成部分坐标得到的坐标系必须截断。否则，无法实现经济性。因此，截断系统求解精度的问题非常重要。可以通过确定随着自由度增加，解收敛到某一渐近值的速率来研究这一点。因此，必须在固有频率和模态形状收敛方面比较各种方法。
(2) Another important reason for using modal synthesis methods relates to the geographic and/or organizational dispersion of engineering efforts in the design and analysis of the structural system. It is a matter of considerable convenience and efficiency to be able to separate the engineering effort at the same interfaces as those used in contractual stipulations. With this in mind, it is highly desirable to minimize the necessary flow of engineering information across these interfaces. Therefore, a method of analysis that most clearly permits isolation of the contractual components of the total system would be sought.
(2) 使用模态综合方法的另一个重要原因与结构系统设计与分析中工程工作在地理和/或组织上的分散有关。能够在与合同条款相同的接口上分离工程工作，既方便又高效。基于此，极力希望尽量减少必要的工程信息在这些接口间的流动。因此，最能清晰地实现对整个系统合同组成部分隔离的分析方法将被优先考虑。
(3) An important criterion in the selection of a method of analysis is the accuracy of local stresses. In connection with modal synthesis methods, this relates, in fact, to the normal mode truncation to which reference has already been made. It is quite possible for some methods to converge satisfactorily to correct frequencies but to converge much less rapidly, or fail to converge, to correct stresses. Therefore, an important criterion in evaluation is the convergence of local stresses.
(3) 在选择分析方法时，一个重要的标准是局部应力的准确性。就模态综合方法而言，这实际上与已经提到的简正模态截断有关。某些方法可能在收敛到正确频率时表现良好，但在收敛到正确应力时收敛速度较慢，甚至无法收敛。因此，在评估时，一个重要的标准是局部应力的收敛性。
(4) A reliable test for the convergence of a modal synthesis method involves one or more repetitions of the analysis with more highly refined models and/or refined coordinate systems with larger numbers of degrees of freedom. This process can be laborious, requiring a complete re-analysis, or it can be relatively easy, requiring only some additions to the original coordinate system with only a part of the analysis repeated. The ease with which this repetitive analysis can be performed depends very much on the method used. Hence, an important criteria in the evaluation of methods is the ease with which a repetitive test for convergence may be made.
(4) 对模态综合方法收敛性的可靠检验涉及一次或多次使用更高精度的模型和/或更精细的坐标系（具有更多自由度）的分析重复。该过程可能繁琐，需要完整的重新分析；也可能相对简单，只需在原始坐标系中添加一些内容并重复部分分析。重复分析的难易程度在很大程度上取决于所使用的方法。因此，在评估方法时，一个重要的标准是进行重复收敛性检验的便利程度。
(5) Another criterion that relates to the joining together of the components has to do with the multiplicity of load paths in the connection systems. In most practical structures, the components are joined by means of statically indeterminate connection systems. Therefore, the method of analysis used should be capable to dealing with redundancies in interconnections.
(5) 与组件连接相关的另一个标准涉及连接系统中负载路径的多重性。在大多数实际结构中，组件通过静态不确定的连接系统连接。因此，所使用的分析方法应能够处理互连中的冗余。
(6) The modal synthesis methods considered here have in common the use of generalized coordinates based, at least in part, on the vibration modes of the several components considered separately. Therefore, these methods are particularly well suited to the determination of the vibration modes of the entire system. Usually, this information is not in itself the end to be sought but is useful in the analysis of various responses of the structure such as response to environmental forces, response as part of a control or guidance system, etc. Although combination of the various methods may relate primarily to their efficiency in determining the system normal modes of vibration, it should be kept in mind that some may be more useful than others when dealing with these various response problems. Hence the ability of the method to treat dynamic response is very important to the analyst. One must also determine whether the method is compatible with a static analysis of the same structure.
(6) 这里考虑的模态综合方法共有一个共同点，即使用基于各个单独考虑的部件振动模态的广义坐标，至少部分基于此。因此，这些方法特别适合确定整个系统的振动模态。通常，这些信息本身并不是最终目标，但在分析结构的各种响应时非常有用，例如对环境力的响应、作为控制或导向系统的一部分的响应等。虽然将各种方法组合起来主要与它们在确定系统简正模态方面的效率有关，但应记住，在处理这些不同响应问题时，有些方法可能比其他方法更有用。因此，方法处理动态响应的能力对分析师非常重要。还必须确定该方法是否与同一结构的静态分析兼容。
(7) All structural analyses in which numerical solutions are performed are subject to errors due to roundoff in arithmetical operations. It is possible for a set of equations to be either well-conditioned or ill-conditioned depending on whether the inescapable roundoff errors affect the solution but slightly or significantly.
Sources of ill-conditioning lie, in part, in the way the structure is modeled and, more specifically, in the kind of coordinate system, or systems, adopted. This latter factor is within the control of the analyst and may depend somewhat on the method of analysis used. Therefore, the various methods to be evaluated should be studied with reference to their inherent properties as they affect the conditioning of the equations to be solved.
(7) 所有进行数值求解的结构分析都受到算术运算中舍入误差的影响。对于一组方程，是否为良条件或差条件取决于不可避免的舍入误差对解的影响是轻微还是显著。
差条件的来源部分在于结构的建模方式，更具体地说，在所采用的坐标系统或系统类型上。后者因素在分析师的控制之下，并且可能在一定程度上取决于所使用的分析方法。因此，应根据各方法的固有属性来研究它们对待求解方程条件性的影响。
(8) Many sources of error exist in the theoretical predictions of the properties of structures. This is particularly true in calculating the stiffness matrix. It is also true with respect to the prediction of damping properties. These errors lie in our imperfect knowledge of the mechanics of materials and structures, in the existence of regions of localized stresses and defections (such as joints), as well as in many other factors. In view of this, one may be tempted to derive these static and dynamic properties experimentally wherever possible.
(8) 在结构属性的理论预测中存在许多误差来源。尤其在计算刚度矩阵时尤为明显。对阻尼属性的预测同样如此。这些误差源于我们对材料和结构力学的不完善认识、局部应力和缺陷（如接头）区域的存在，以及许多其他因素。鉴于此，人们可能倾向于在可能的情况下通过实验来推导这些静态和动态属性。
The ease and extent to which analysis can be experimentally verified depends, in part, upon the method of analysis used. The method will dictate the kinds of tests to be performed ; hence, to some degree, the difficulty in carrying out the experimental program. Ease and usefulness of experimental analysis provides still another criterion by which the various methods may be evaluated.
分析能够被实验验证的容易程度和范围在一定程度上取决于所采用的分析方法。该方法将决定要执行的测试类型；因此，在某种程度上，执行实验程序的难度也随之而来。实验分析的易用性和实用性为评估各种方法提供了另一种标准。
## Background in the development of modal synthesis methods 模态合成方法发展背景
# Background in the development of modal synthesis methods
Several variations in the general method of modal synthesis have developed during the past few years. Despite early investigations [1--3l, not much attention was given to the subject prior to the work reported in [4]. In this report, a comprehensive development of the procedure called component mode synthesis was given. In the procedure, a general displacement within a component is defined by superimposing the displacement relative to the component interfaces upon the interface displacements. The first class of displacements are defined, in turn, by a superposition of a truncated set of “fixed interface' or “fixed constraint' normal modes of vibration of the separate components. Separation of the interface displacements into rigid body and so-called ‘constraint’ displacements is not essential to the procedure although such separation into these two sets of modes may be advantageous in some cases.
Several variations in the general method of modal synthesis have developed during the past few years. Despite early investigations 1--3, not much attention was given to the subject prior to the work reported in [4]. In this report, a comprehensive development of the procedure called component mode synthesis was given. In the procedure, a general displacement within a component is defined by superimposing the displacement relative to the component interfaces upon the interface displacements. The first class of displacements are defined, in turn, by a superposition of a truncated set of “fixed interface' or “fixed constraint' normal modes of vibration of the separate components. Separation of the interface displacements into rigid body and so-called ‘constraint’ displacements is not essential to the procedure although such separation into these two sets of modes may be advantageous in some cases.
过去几年中，模态综合的一般方法出现了几种变体。尽管早期的研究1--3，但在[4]报告的工作之前，关于该主题并未受到太多关注。在本报告中，对称为组件模态综合的程序进行了全面的发展。在该程序中，组件内部的一般位移是通过将相对于组件接口的位移叠加到接口位移上来定义的。第一类位移随后由对分离组件的“固定接口”或“固定约束”简正模态的截断集合的叠加来定义。将接口位移分离为刚体位移和所谓的“约束”位移并不是该程序的必要条件，尽管在某些情况下将其分为这两组模态可能是有利的。
In the subsequent discussion in this paper, the method described above is classed as a 'fixed constraint mode' method. Bamford [5] programmed this procedure and in the process added another class of displacement modes. Craig and Bampton [6] also programmed essentially the same procedure but without separating rigid body and “constraint' modes. Bajan et al. [I7, 8] reported, also, on a procedure which is essentially the same but with the addition of an algorithm designed to assist in optimizing the choice of the component modes.
在本文后续讨论中，上述方法被归类为“固定约束模态”方法。Bamford [5] 对此过程进行了编程，并在此过程中添加了另一类位移模态。Craig 和 Bampton [6] 也几乎相同地编程了该过程，但没有将刚体模态和“约束”模态分离。Bajan 等人 [17, 18] 也报告了一种本质上相同的程序，但增加了一个用于协助优化组件模态选择的算法。
The basic process of modal synthesis was recognized independently by Gladwell [9] who developed a method which he called a branch mode' analysis. Two or three components of the entire assembly are joined to form a branch whose principal modes of vibration are determined either with other boundary interfaces fixed or free. The boundaries are chosen so as to overlap, and their principal modes form a basis for a coordinate system which defines displacements at both the interfaces and the interiors of all components.
模态综合的基本过程是由Gladwell [9] 独立识别的，他开发了一种他称之为“分支模态”分析的方法。整个装配的两到三个组件被连接形成一个分支，其主要振动模态是通过将其他边界接口固定或自由来确定的。边界被选择为重叠，其主要模态构成了一个坐标系统的基底，该坐标系统定义了所有组件的接口和内部的位移。
Benfield and Hruda [10] developed a method which is essentially a branch mode method but added a comprehensive treatment offering several alternative procedures for constructing the branch matrices, including an approximation to the effects of loading by a component on its neighbors. This loading effect is called “interface loading'.
Benfield 和 Hruda [10] 开发了一种方法，本质上是一种分支模态方法，但增加了全面的处理，提供了几种构造分支矩阵的替代程序，包括对组件对其邻居施加负载效应的近似。该负载效应被称为“界面负载”。
Another alternative is to select as component modes the modes of vibration of each separate component with its interfaces free. This procedure is appropriately called a free interface mode’ method. In order to satisfy displacement compatibility at interfaces, rigid body modes must also be included. It is known that this procedure has been used by some aircraft companies for some years, but the first published report seems to be that of Goldman [11]. His method appears not to take advantage of the possibility of truncation so that it fails to meet a principal objective of modal synthesis. Other variations of the “free interface mode’ procedure were reported [12] [13], and the former procedure includes the possibility of truncation in which a most suitable choice of modes can be made using an error index based on convergence of the eigenvalues.
# Discussion of methods
另一种替代方案是将每个独立组件在其接口自由的情况下的振动模态作为组件模态来选择。该过程适当地称为“自由接口模态”方法。为了满足接口处的位移兼容性，还必须包含刚体模态。已知该程序已被一些航空公司使用多年，但最早的公开报告似乎是Goldman的[11]。他的办法似乎没有利用截断的可能性，因此未能满足模态综合的主要目标。其他“自由接口模态”程序的变体已被报道[12][13]，而前一种程序包含截断的可能性，在此情况下可以使用基于特征值收敛的误差指数来选择最合适的模态。
## Discussion of methods
The following four methods of modal synthesis are identified in the present literature. The basic concepts of each are described in this Section.
A. Component Mode Synthesis B. Branch Mode Analysis C. Component Mode Substitution D. Coupled Free-Free Conponent Modes
A. Component Mode Synthesis
B. Branch Mode Analysis
C. Component Mode Substitution
D. Coupled Free-Free Conponent Modes
Method A. Component mode synthesis. The structural system is considered to be a finite set of components connected together by a finite number of connections which serve as constraints on each of the components. The components may be hyperstatic substructures and the connection system may contain any finite number of redundancies. The topological arrangement of components and connections is, in principle, of no consequence in the analysis. The analysis uses the principle of superposition; hence, force displacement properties must be linear. The method is, in essence, a displacement (or stiffness) method with an arbitrary virtual displacement of the system (or of its components) described by a superposition of several categories of modal displacements.
以下四种模态合成方法在现有文献中已被识别。每种方法的基本概念在本节中进行了描述。
A. 组件模态合成
B. 分支模态分析
C. 组件模态替代
D. 耦合自由-自由组件模态
方法A. 组件模态合成。结构系统被视为由有限数量的组件通过有限数量的连接相互连接而成，这些连接对每个组件起约束作用。组件可以是超静子结构，连接系统可以包含任意有限数量的冗余。组件和连接的拓扑排列在分析中原则上并不重要。分析采用叠加原理；因此，力-位移特性必须是线性的。该方法本质上是一种位移（或刚度）方法，其系统（或其组件）的任意虚位移由若干类别的模态位移的叠加来描述。
Component modes are defined in three categories :
组件模态分为三类：
(1) ‘Rigid Body Modes'---which may number for a body in three space up to six if no external constraints exist.
(2) ‘Constraint Modes'or 'Attachment Modes'-which are defined by displacements (singly or in combinations) of the redundant interface constraints. As many such modes exist as there are redundant constraints in the interface connections of the component.
(3) ‘Fixed Constraint Normal Modes'which are defined by displacements of interior points in the component relative to its interface constraints. These may be normal modes of vibration of the component with all interface constraints fixed. In general, there will be a large number of such modes, and for a continuous body this number becomes infinity in principle. The set of these modes to be used in the coordinate system will be truncated so that a source of error is introduced which depends on the extent of truncation.
(1) “刚体模态”——如果没有外部约束，三维空间中一个物体的刚体模态最多可达六个。
Compatibility of displacements at the component interface connections is explicitly assured by a coordinate transformation relating component coordinates and system coordinates. This transformation involves only the rigid body and constraint modes. Each normal mode defined for a component becomes an added generalized coordinate in the system. The minimum possible number of system degrees of freedom is equal to the total number of rigid body and constraint modes for all components less the number of displacement compatibilityequations. A paper by Craig and Bampton [6] presents a variation on the method of component mode synthesis proposed by Hurty [4]. The coordinate system is comprised of two classes of component modes instead of three. These are:
(2) “约束模态”或“附着模态”——由冗余接口约束的位移（单独或组合）定义。此类模态的数量与组件接口连接中的冗余约束数量相同。
(3) “固定约束简正模态”——由相对于其接口约束的组件内部点的位移定义。若所有接口约束固定，这些模态可能是组件的简正振动模态。一般而言，这类模态数量很大，对于连续体而言，原则上该数量趋于无穷大。为在坐标系中使用这些模态时会截断该集合，从而引入取决于截断程度的误差源。
Compatibility of displacements at the component interface connections is explicitly assured by a coordinate transformation relating component coordinates and system coordinates. This transformation involves only the rigid body and constraint modes. Each normal mode defined for a component becomes an added generalized coordinate in the system.
The minimum possible number of system degrees of freedom is equal to the total number of rigid body and constraint modes for all components less the number of displacement compatibility equations. A paper by Craig and Bampton [6] presents a variation on the method of component mode synthesis proposed by Hurty [4]. The coordinate system is comprised of two classes of component modes instead of three. These are:
(1) “Constraint Modes? (later called “"Attachment Modes" in this paper) defined by displacements of all of the interface constraints; and
(2) ‘Fixed Constraint Normal Modes' as previously defined.
兼容性位移在组件接口连接处通过将组件坐标与系统坐标关联的坐标变换得到明确保证。该变换仅涉及刚体和约束模态。为每个组件定义的每个简正模态在系统中成为一个附加的广义坐标。
系统自由度的最小可能数等于所有组件的刚体和约束模态总数减去位移兼容性方程的数量。Craig 和 Bampton [6] 的一篇论文提出了对 Hurty [4] 提出的组件模态综合方法的变体。该坐标系统由两类组件模态组成，而非三类。它们是：
(1) “约束模态？（后文称为“Attachment Modes”）由所有接口约束的位移定义；
(2) “固定约束简正模态”，如前所述。
The difference lies in the fact that rigid body modes are not separately identified. Therefore, the number of coordinates established under (1) above is the same as the number of coordinates established under both (1) and (2) under Hurty's method. The coordinate systems under the two methods are the same within a simple linear transformation.
@ -84,6 +135,14 @@ However, experience has shown that there are structural systems for which the st
Bajan et al. [8] propose a technique which differs in some respects from the above. While they use rigid body, constraint and a truncated number of normal modes for each component they present a method for automatic selection and substitution of modes which are added to the above in order to improve the analysis.
差异在于刚体模态未被单独识别。因此，在上述 (1) 下建立的坐标数与在 Hurty 方法下同时满足 (1) 与 (2) 的坐标数相同。两种方法下的坐标系在简单线性变换内相同。
使用此原始坐标方案的变体的主要目的在于简化编程任务，因为不再需要识别静定与冗余约束。这种识别需要分析师的判断，因此需要一个不易编程的决策功能。
然而，经验表明，对于某些结构系统，当刚度矩阵与完全由变形模态组成的坐标系相关时，矩阵条件不良。通过使用刚体模态，在某些情况下可以显著改善矩阵条件。
Bajan 等人[8]提出了一种在某些方面与上述不同的技术。虽然他们为每个组件使用刚体、约束和截断数量的简正模态，但他们提出了一种自动选择和替换模态的方法，以改进上述分析。
Method B. Branch mode analysis. The structural system consists, as under A, of a finite set of components. In Gladwell's initial development [9], the components are conneted together in a chain-like configuration; i.e., each component is connected to not more than two of its neighbors in an end-to-end arrangement. In later examples treated by this method,the topological arrangements are somewhat more general. For example, in a plane frame example three components are connected together at a single point.
Gladwell does not treat the problem of redundancies in the interconnection systems although there is no basic reason for limiting the branch mode method to a statically determinate connection system.
@ -95,15 +154,37 @@ Normally, although not necessarily, the coordinate system for this structure con
Since the coordinate system is composed entirely of vibration modes, it is essential that branches be defined so that these modes will be generally characterized by non-zero displacements or nodes at the intercomponent connections. Otherwise, the coordinate system will not be kinematically complete.
The determination of the complete coordinate system by synthesis of the various branch modes is not discussed in detail by Gladwell. He points out that the sets of branch. modes will be truncated and that the matter of how many modes are to be used is one which must be decided on the basis of experience and judgement.
方法B：分支模态分析。结构系统与方法A相同，由有限数量的构件组成。
在Gladwell的初始研究[9]中，构件以链式配置相连；即每个构件与其相邻构件的连接数不超过两端，形成端到端的排列。
在后续使用该方法的例子中，拓扑排列更为一般化。例如，在平面框架例子中，三个构件在一个点相连。
Gladwell并未讨论互连系统中冗余的问题，尽管没有基本理由将分支模态方法限制在静定连接系统上。
分支模态方法的核心特征是将构件分组为称为分支的子集。尚未制定定义这些分支的规则；因此，其构造留给分析师的判断和经验。分支模态系统必须完整，即能够将系统的任何运动表示为各分支模态运动的叠加。
通常，尽管不一定如此，该结构的坐标系由各独立分支的简正模态组成。为确定这些振动模态，必须指定分支上的边界条件。这意味着连接的构件相对端（即不包含在分支内构件间接口中的连接）必须施加指定的边界条件。Gladwell认为这些边界可根据系统性质和分析师判断为固定或自由。实际上，Gladwell建议通过让分支中的某个构件固定或作为刚体振动来定义分支模态。因此，分析师有多种选择。
由于坐标系完全由振动模态构成，必须定义分支，使得这些模态在构件间连接处通常具有非零位移或节点。否则，坐标系将不具备运动学完整性。
Gladwell未详细讨论通过合成各分支模态来确定完整坐标系的方法。他指出分支模态集将被截断，使用多少模态的问题必须基于经验和判断来决定。
Method C. Component mode substitution. This method is similar to that of Gladwell with respect to the definition of branches. The conceptual model of the structure includes a main body, which is considered to be a free-free body, to which are attached one or more branches. Each branch may consist of two or more components which may be redundantly interconnected.
As described previously [10] a branch, 'ab', is constructed by attaching a component 'b' to a component \*a'. Attachment requires that the displacements at the interface be compatible. This condition establishes a transformation which permits the construction of the branch stiffness and mass matrices from those of the separate components. The matrices thus constructed are, in fact, those matrices for component ‘a' augmented by matrices which in effect add mass and stiffness interface loadings representing approximately the effect of component “b' acting on component 'a'. The matrices so formed can be used to express the potential and kinetic energies of the branch, and these energies do not account for the motion of component “b' relative to the interface between the two components. To account for this motion, the fixed interface normal modes of component 'b' are added to the coordinate system, and the above stiffness and mass matrices are extended accordingly. They then represent the matrices appropriate to branch “ab'. These matrices are used to formulate an equation of motion for the branch whose eigenvectors are the normal modes of the branch. These modes may be used to extend the solution to a larger branch obtained by adding another component, or to find the matrices for the complete system if branch ‘ab' is attached directly to the main body.
If component eigenmodes are used throughout for a branch of two components, three eigenvalue solutions will be performed. For each additional component added to form a chain-like branch, two additional eigenvalue solutions are indicated. Thus, for a chain-like branch of n components as many as $_{2n-1}$ eigenvalue solutions would be performed. It is not essential that eigenmodes be used throughout provided other sets of suitable, independent displacement shapes can be derived as coordinate bases. At each step in the process of adding components, the set of fixed constraint branch modes will be truncated in order to keep the coordinate systems within acceptable bounds. No rules are given [10] concerning truncation, but several examples are included which provide information on this matter.
方法 C. 组件模态替换。
该方法在分支定义方面类似于 Gladwell 的方法。结构的概念模型包括一个主体，视为自由自由体，并附有一个或多个分支。 每个分支可能由两个或更多组件组成，这些组件可能冗余地相互连接。
如前所述 [10]，分支 'ab' 是通过将组件 'b' 附加到组件 \*a' 来构造的。 附件要求接口处的位移相容。 此条件建立了一个变换，允许从各独立组件的刚度和质量矩阵构造分支的刚度和质量矩阵。 构造出的矩阵实际上是组件 'a' 的矩阵，增加了矩阵，这些矩阵实际上添加了质量和刚度接口负载，近似表示组件 'b' 对组件 'a' 的作用。 构造出的矩阵可用于表示分支的势能和动能，这些能量不考虑组件 'b' 相对于两组件之间接口的运动。 为考虑此运动，将组件 'b' 的固定接口简正模态添加到坐标系中，并相应扩展上述刚度和质量矩阵。 它们随后表示适用于分支 'ab' 的矩阵。 这些矩阵用于制定分支的运动方程，其特征向量为分支的简正模态。 这些模态可用于将解扩展到通过添加另一个组件获得的更大分支，或在分支 'ab' 直接附加到主体时找到完整系统的矩阵。
如果在两个组件的分支中始终使用组件特征模态，则将执行三次特征值求解。 每增加一个组件以形成链式分支，需额外进行两次特征值求解。 因此，对于 n 个组件的链式分支，最多会执行 $_{2n-1}$ 次特征值求解。 只要能导出其他合适、独立的位移形状作为坐标基，持续使用特征模态并非必要。 在添加组件的每一步中，将截断固定约束分支模态集合，以保持坐标系在可接受范围内。 文献 [10] 未给出截断规则，但包含了若干示例，提供了相关信息。
Method D. Coupled free-free component modes. There are two separate approaches based upon the concept of coupled free-frcee component modes. The first approach synthesizes the free-free normal mode shapes and natural frequencies of vibration of each component which are obtained from a solution of the eigenvalue problem using only the mass and stiffness matrices of each isolated component. Hart, et al [13] Hou [12] and Goldman [11] present this approach. A second approach is presented, in part, [10]. In this latter approach the analyst first formulates the discrete direct stiffness and mass model of the complete structural system. Then a coordinate reduction is performed by selection of deflection functions constructed from the component interface loaded free-free modes. An extension of this latter approach is presented in detail later in this paper.
方法 D. 耦合自由-自由组件模态。 基于耦合自由-自由组件模态的概念，有两种不同的方法。 第一种方法合成每个组件的自由-自由简正模态形状和固有频率，这些模态是通过仅使用每个孤立组件的质量矩阵和刚度矩阵求解特征值问题得到的。 Hart 等人[13]、Hou[12] 和 Goldman[11] 提出了此方法。 第二种方法在部分内容中提出[10]。 在此后一种方法中，分析师首先制定完整结构系统的离散直接刚度和质量模型。 随后通过选择由组件接口加载的自由-自由模态构造的变形函数来执行坐标约简。 此后一种方法的扩展将在本文后面详细介绍。
# Evaluation of methods
A carefully considered evaluation of the several methods of analysis with respect to all of the criteria discussed will require more knowledge and experience than is presently available. The following discussion presents an initial evaluation based on what is presently known or understood to be true. Where insufficient evidence is available to support even a conjecture, this fact will be noted for future reference.