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@ -12,19 +12,28 @@ Roy R. Craig, Jr.\* The University of Texas at Austin Austin TX 78712
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Since the 1960s, substructuring, or component mode synthesis (CMS), has been used to model complex structures. Substructuring involves dividing the structure into a number of substructures, or components (Fig. 1), obtaining reduced-order models of the components, and then assembling a reduced-order model of the entire structure. This paper defines and illustrates many of the terms that are found in the literature on substructuring, (fixed-interface modes, free-interface modes, constraint modes, residual fexibility, etc.), discusses a general procedure for coupling substructures, and compares two widely used methods. The focus of this paper is on the presentation of figures that illustrate the physical meaning of various component mode transformations.
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Since the 1960s, substructuring, or component mode synthesis (CMS), has been used to model complex structures. Substructuring involves dividing the structure into a number of substructures, or components (Fig. 1), obtaining reduced-order models of the components, and then assembling a reduced-order model of the entire structure. This paper defines and illustrates many of the terms that are found in the literature on substructuring, (fixed-interface modes, free-interface modes, constraint modes, residual fexibility, etc.), discusses a general procedure for coupling substructures, and compares two widely used methods. The focus of this paper is on the presentation of figures that illustrate the physical meaning of various component mode transformations.
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自20世纪60年代以来，次结构法，或称部件模态综合法（CMS），已被用于模拟复杂结构。次结构法涉及将结构划分为若干个次结构或部件（图1），获得这些部件的降阶模型，然后组装整个结构的降阶模型。本文定义并阐释了文献中常见的许多术语，（固定界面模态、自由界面模态、约束模态、残余柔度等），讨论了一种耦合次结构的通用程序，并比较了两种常用的方法。本文的重点是展示图示各种部件模态变换的物理意义。
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Figure 1: Typical Airplane Substructures.
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Figure 1: Typical Airplane Substructures.
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When a large, complex structural system must be analyzed for its response to dynamic excitation, some form of substructure coupling method, or component mode synthesis (CMS) method, is usually employed. The term component modes is used to signify Ritz vectors, or assumed modes[1], that are used as basis vectors in describing the displacement of points within a substructure, or component. Component normal modes, or eigenvectors, are just one class of assumed modes. In the mid- $\cdot1960^{\circ}\mathrm{s}$ Hurty published several reports and papers on substructure coupling (e.g., [2, 3]). In collaboration with Hurty, Bamford created a CMS computer program that employed normal modes, rigid-body modes, constraint modes, and attachment modes[4]. A simplification of Hurty's method was presented by the author in 1968[5], and in the early 1970's MacNeal and Rubin introduced important alternatives to Hurty's CMS method[6, 7]. A number of CMS methods are described and compared in Refs. [8-10] and in at least three textbooks[11-13]. Although CMS methods have been developed for damped systems as well as for undamped systems, methods for damped systems are not discussed in the present paper.
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When a large, complex structural system must be analyzed for its response to dynamic excitation, some form of substructure coupling method, or component mode synthesis (CMS) method, is usually employed. The term component modes is used to signify Ritz vectors, or assumed modes[1], that are used as basis vectors in describing the displacement of points within a substructure, or component. Component normal modes, or eigenvectors, are just one class of assumed modes. In the mid- $\cdot1960^{\circ}\mathrm{s}$ Hurty published several reports and papers on substructure coupling (e.g., [2, 3]). In collaboration with Hurty, Bamford created a CMS computer program that employed normal modes, rigid-body modes, constraint modes, and attachment modes[4]. A simplification of Hurty's method was presented by the author in 1968[5], and in the early 1970's MacNeal and Rubin introduced important alternatives to Hurty's CMS method[6, 7]. A number of CMS methods are described and compared in Refs. [8-10] and in at least three textbooks[11-13]. Although CMS methods have been developed for damped systems as well as for undamped systems, methods for damped systems are not discussed in the present paper.
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当需要分析大型、复杂的结构系统对动态激励的响应时，通常会采用子结构耦合法，或称部件模态综合法（CMS）。**部件模态**一词用于表示 Ritz 矢量，或假定模态[1]，它们被用作描述子结构或部件内各点位移的基向量。简正模态，或特征向量，只是假定模态的一种类型。在 20 世纪 60 年代中期，Hurty 发表了多篇关于子结构耦合的报告和论文（例如 [2, 3]）。在 Hurty 的合作下，Bamford 创建了一个 CMS 计算机程序，该程序采用了简正模态、刚体模态、约束模态和**连接模态**[4]。作者于 1968 年提出了 Hurty 方法的简化版本[5]，在 20 世纪 70 年代初，MacNeal 和 Rubin 引入了 Hurty CMS 方法的重要替代方案[6, 7]。在参考文献 [8-10] 以及至少三本教科书[11-13] 中描述并比较了多种 CMS 方法。虽然 CMS 方法已被开发用于阻尼系统和未阻尼系统，但本文不讨论阻尼系统的方法。
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Component mode synthesis involves three basic steps: division of a structure into components, definition of sets of component modes, and coupling of the component mode models to form a reduced-order system model. The primary uses of dynamic substructuring are: (1) to couple reduced-order models of moderately complex structures (e.g., airplane components, as in Fig. 1, or systems of automotive components), (2) in test-verification of finite element models of components, or (3) to implement computation of the dynamics of very large finite element models (e.g., multi-million-DOF models). This paper addresses primarily applications of the first type; Refs. [14--20] illustrate the relationship of substructure analysis to substructure testing, and Refs. [21, 22] are representative of the third application of substructuring.
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Component mode synthesis involves three basic steps: division of a structure into components, definition of sets of component modes, and coupling of the component mode models to form a reduced-order system model. The primary uses of dynamic substructuring are: (1) to couple reduced-order models of moderately complex structures (e.g., airplane components, as in Fig. 1, or systems of automotive components), (2) in test-verification of finite element models of components, or (3) to implement computation of the dynamics of very large finite element models (e.g., multi-million-DOF models). This paper addresses primarily applications of the first type; Refs. [14--20] illustrate the relationship of substructure analysis to substructure testing, and Refs. [21, 22] are representative of the third application of substructuring.
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组件模态综合涉及三个基本步骤：将结构划分为组件、定义组件模态集合，以及耦合组件模态模型以形成降阶系统模型。动态次结构的 主要用途包括：（1）耦合复杂程度适中的结构降阶模型（例如，飞机组件，如图1所示，或汽车组件系统），（2）用于验证组件有限元模型的试验，或（3）实现计算非常大的有限元模型的动力学（例如，百万自由度模型）。本文主要关注第一种类型的应用；参考文献[14--20]阐述了次结构分析与次结构试验的关系，参考文献[21, 22]代表了次结构的第三种应用。
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In Section 2, the systematic procedures used to generate FE-based component modes are described, including discussions of inertia relief and residual fexibility. Section 3 is a review of a Lagrangemultiplier-based generalized substructure coupling procedure. This is followed, in Section 4, by a discussion of coupling analyses based on two widely used CMS methods, and, in Section 5, by conclusions.
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In Section 2, the systematic procedures used to generate FE-based component modes are described, including discussions of inertia relief and residual fexibility. Section 3 is a review of a Lagrangemultiplier-based generalized substructure coupling procedure. This is followed, in Section 4, by a discussion of coupling analyses based on two widely used CMS methods, and, in Section 5, by conclusions.
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在第2节中，描述了用于生成基于有限元（FE）的组件模态的系统化流程，包括惯性释放和残余挠曲的讨论。第3节是对基于拉格朗日乘子（Lagrange multiplier）的广义次结构耦合流程的回顾。随后，在第4节中，讨论了基于两种常用的CMS方法（组件模态方法）的耦合分析，并在第5节中，总结结论。
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# 2. Component Modes
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# 2. Component Modes
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The most general type of component, or substructure, is one that is connected to one or more adjacent components by redundant interfaces. Figure 2 illustrates a simple cantilever beam that is divided into three components; the middle one is a typical component with redundant interface (boundary) coordinates.
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The most general type of component, or substructure, is one that is connected to one or more adjacent components by redundant interfaces. Figure 2 illustrates a simple cantilever beam that is divided into three components; the middle one is a typical component with redundant interface (boundary) coordinates.
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最通用的构件或子结构，是指那些通过冗余接口连接到一个或多个相邻构件的结构。图2展示了一个简单的悬臂梁，它被划分为三个构件；中间的构件是一个典型的具有冗余接口（边界）坐标的构件。
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a. Components and coupled system.
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a. Components and coupled system.
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@ -34,24 +43,31 @@ b. Typical component with redundant boundary.
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As noted in Fig. 2, the coordinate sets $\tau,\mathcal{R},\mathcal{E}$ and $\pmb{{\cal B}}$ denote interior coordinates (i.e., not shared with an adjacent component), rigid-body coordinates, excess coordinates (i.e., redundant boundary coordinates), and boundary coordinates (i.e., shared with adjacent components). The numbers of coordinates in these sets are $N_{i},\,N_{r},\,N_{e},$ and $N_{b}$ , respectively, with $N_{b}=N_{r}+N_{e}$ and $N=N_{i}+N_{b}$
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As noted in Fig. 2, the coordinate sets $\tau,\mathcal{R},\mathcal{E}$ and $\pmb{{\cal B}}$ denote interior coordinates (i.e., not shared with an adjacent component), rigid-body coordinates, excess coordinates (i.e., redundant boundary coordinates), and boundary coordinates (i.e., shared with adjacent components). The numbers of coordinates in these sets are $N_{i},\,N_{r},\,N_{e},$ and $N_{b}$ , respectively, with $N_{b}=N_{r}+N_{e}$ and $N=N_{i}+N_{b}$
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The equation of motion of $\mathbf{a}$ typical undamped component, labeled $^c$ may be written as
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如图2所示，坐标系 $\tau,\mathcal{R},\mathcal{E}$ 和 $\pmb{{\cal B}}$ 分别表示内坐标（即与相邻部件不共享）、刚体坐标、过余坐标（即冗余边界坐标）和边界坐标（即与相邻部件共享）。这些坐标系中坐标的数量分别为 $N_{i},\,N_{r},\,N_{e},$ 和 $N_{b}$ ，其中 $N_{b}=N_{r}+N_{e}$ 且 $N=N_{i}+N_{b}$。
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The equation of motion of $\mathbf{a}$ typical undamped component, labeled $^c$ may be written as
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典型无阻尼构件 $\mathbf{a}$，标记为$c$，其运动方程可写为：
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$$
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$$
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M^{c}\ddot{\boldsymbol{u}}^{c}+K^{c}\boldsymbol{u}^{c}=\boldsymbol{f}^{c}
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M^{c}\ddot{\boldsymbol{u}}^{c}+K^{c}\boldsymbol{u}^{c}=\boldsymbol{f}^{c}
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$$
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$$
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where the superscript $c$ is the label of the particular component, and where $M^{c},\,K^{c}$ , and $\pmb{u}^{c}$ , are the component's mass matrix, stiffness matrix, and displacement vector, respectively. The force vector, $\pmb{f}^{c}$ includes both the externally applied forces and the forces on the component due to its connection to adjacent components at boundary degrees of freedom.
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where the superscript $c$ is the label of the particular component, and where $M^{c},\,K^{c}$ , and $\pmb{u}^{c}$ , are the component's mass matrix, stiffness matrix, and displacement vector, respectively. The force vector, $\pmb{f}^{c}$ includes both the externally applied forces and the forces on the component due to its connection to adjacent components at boundary degrees of freedom.
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In component mode synthesis, the component's physical displacement coordinates $\pmb{\mathit{u}}$ are represented in terms of component generalized coordinates $\pmb{p}$ by the Rayleigh-Ritz coordinate transformation
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其中，$c$ 上标代表特定部件的标签，而 $M^{c}$、 $K^{c}$ 和 $\pmb{u}^{c}$ 分别代表该部件的质量矩阵、刚度矩阵和位移矢量。力矢量 $\pmb{f}^{c}$ 包括外部施加的力和由于其与相邻部件在边界自由度处连接而产生的力。
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In component mode synthesis, the component's physical displacement coordinates $\pmb{\mathit{u}}$ are represented in terms of component generalized coordinates $\pmb{p}$ by the Rayleigh-Ritz coordinate transformation
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在部件模态综合法中，部件的物理位移坐标 $\pmb{\mathit{u}}$ 通过瑞利-里兹坐标变换，用部件广义坐标 $\pmb{p}$ 来表示。
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$$
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$$
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\pmb{u}^{c}=\pmb{\Psi}^{c}\pmb{p}^{c}
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\pmb{u}^{c}=\pmb{\Psi}^{c}\pmb{p}^{c}
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$$
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$$
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where the component mode matrir $\Psi^{c}$ is a coordinate transformation matrix of preselected component (assumed) modes, including the following types: rigid-body modes, normal modes of free vibration (i.e., eigenvectors), constraint modes, and attachment modes. In collaboration with Hurty, Bamford defined all four of these types of modes in Ref. [4]; they are also defined in Refs. [9-11]. Other types of assumed modes (e.g., Krylov vectors [23]) may also be employed as component modes.
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where the component mode matrir $\Psi^{c}$ is a coordinate transformation matrix of preselected component (assumed) modes, including the following types: rigid-body modes, normal modes of free vibration (i.e., eigenvectors), constraint modes, and attachment modes. In collaboration with Hurty, Bamford defined all four of these types of modes in Ref. [4]; they are also defined in Refs. [9-11]. Other types of assumed modes (e.g., Krylov vectors [23]) may also be employed as component modes.
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其中，组件模态矩阵 $\Psi^{c}$ 是预选组件（假设）模态的坐标变换矩阵，包括以下类型：刚体模态、简正模态（即特征向量）、约束模态和连接模态。**Hurty 和 Bamford 在文献 [4] 中定义了这四种类型的模态；这些定义也见于文献 [9-11]**。其他类型的假设模态（例如 Krylov 向量 [23]）也可以用作组件模态。
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The coordinate transformation relating component physical coordinates ${\pmb u}^{c}$ to component generalized coordinates $\pmb{p}^{c}$ is given by Eq. (2). This equation, together with the equation of motion in generalized coordinates, forms the component modal model. From Eqs. (1) and (2), the component equation of motion in generalized coordinates is
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The coordinate transformation relating component physical coordinates ${\pmb u}^{c}$ to component generalized coordinates $\pmb{p}^{c}$ is given by Eq. (2). This equation, together with the equation of motion in generalized coordinates, forms the component modal model. From Eqs. (1) and (2), the component equation of motion in generalized coordinates is
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与部件物理坐标 ${\pmb u}^{c}$ 与部件广义坐标 $\pmb{p}^{c}$ 之间的坐标变换由公式(2)给出。 结合广义坐标下的运动方程，该公式构成部件模态模型。 根据公式(1)和(2)，在广义坐标下，部件的运动方程为：
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$$
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$$
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\bar{M}^{\mathrm{c}}\,\ddot{\pmb{p}}^{\mathrm{c}}+\bar{K}^{\mathrm{c}}\,\pmb{p}^{\mathrm{c}}=\bar{\pmb{f}}^{c}
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\bar{M}^{\mathrm{c}}\,\ddot{\pmb{p}}^{\mathrm{c}}+\bar{K}^{\mathrm{c}}\,\pmb{p}^{\mathrm{c}}=\bar{\pmb{f}}^{c}
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$$
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$$
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$$
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$$
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The following partitioned forms of Eq. (1) will be useful in the derivation of component modes:
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The following partitioned forms of Eq. (1) will be useful in the derivation of component modes:
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以下划分形式的方程 (1) 将在分量模态推导中有所助益：
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$$
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$$
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\begin{array}{r l r}&{}&{\left[\begin{array}{c c}{M_{i i}}&{M_{i b}}\\ {M_{b i}}&{M_{b b}}\end{array}\right]\left\{\begin{array}{c}{\ddot{u}_{i}}\\ {\ddot{u}_{b}}\end{array}\right\}+\left[\begin{array}{c c}{K_{i i}}&{K_{i b}}\\ {K_{b i}}&{K_{b b}}\end{array}\right]\left\{\begin{array}{c}{{u_{i}}}\\ {{u_{b}}}\end{array}\right\}}\\ &{}&{=\left\{\begin{array}{c}{{{f}_{i}}}\\ {{{f}_{b}}}\end{array}\right\}}\end{array}
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\begin{array}{r l r}&{}&{\left[\begin{array}{c c}{M_{i i}}&{M_{i b}}\\ {M_{b i}}&{M_{b b}}\end{array}\right]\left\{\begin{array}{c}{\ddot{u}_{i}}\\ {\ddot{u}_{b}}\end{array}\right\}+\left[\begin{array}{c c}{K_{i i}}&{K_{i b}}\\ {K_{b i}}&{K_{b b}}\end{array}\right]\left\{\begin{array}{c}{{u_{i}}}\\ {{u_{b}}}\end{array}\right\}}\\ &{}&{=\left\{\begin{array}{c}{{{f}_{i}}}\\ {{{f}_{b}}}\end{array}\right\}}\end{array}
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@ -73,55 +91,57 @@ $$
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$$
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$$
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The superscript $c$ ,which was used above to designate a component, will be omitted from component matrices and vectors in the remainder of this section.
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The superscript $c$ ,which was used above to designate a component, will be omitted from component matrices and vectors in the remainder of this section.
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在本文的其余部分，用于标识构件的上标 $c$ 将在构件矩阵和向量中省略。
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# 2.1. Normal Modes
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## 2.1. Normal Modes
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Component normal modes are eigenvectors, and may be classified according to the interface boundary conditions specified for the component - fixed-interface normal modes, free-interface normal modes, hybrid-interface normal modes, or loadedinterface normal modes[11]. Component ficedinterface normal modes are obtained by restraining all boundary DOFs and solving the following eigenproblem:
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Component normal modes are eigenvectors, and may be classified according to the interface boundary conditions specified for the component - fixed-interface normal modes, free-interface normal modes, hybrid-interface normal modes, or loadedinterface normal modes[11]. Component ficedinterface normal modes are obtained by restraining all boundary DOFs and solving the following eigenproblem:
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组件简正模态是特征向量，可以根据为组件指定的界面边界条件进行分类——固定界面简正模态、自由界面简正模态、混合界面简正模态或载荷界面简正模态[11]。组件固定界面简正模态是通过约束所有边界自由度并求解以下特征问题得到的：
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$$
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$$
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\left[K_{i i}-\omega_{j}^{2}M_{i i}\right]\left\{\phi_{i}\right\}_{j}=0,\;j=1,2,...,N_{i}
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\left[K_{i i}-\omega_{j}^{2}M_{i i}\right]\left\{\phi_{i}\right\}_{j}=0,\;j=1,2,...,N_{i}
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$$
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$$
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The complete set of $N_{i}$ fixed-interface normal modes is labeled $\Phi_{n}$ and assembled according to the partitioning of Eq. (5) as columns of the modal matrix
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The complete set of $N_{i}$ fixed-interface normal modes is labeled $\Phi_{n}$ and assembled according to the partitioning of Eq. (5) as columns of the modal matrix
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完整的 $N_{i}$ 个固定界面简正模态标记为 $\Phi_{n}$，并根据公式(5)的分区方式，作为模态矩阵的列进行组装。
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$$
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$$
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\boldsymbol{\Phi}_{n}\equiv\left[\begin{array}{l}{\boldsymbol{\Phi}_{i n}}\\ {\boldsymbol{0}_{b n}}\end{array}\right]
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\boldsymbol{\Phi}_{n} ( N \times N_i)\equiv\left[\begin{array}{l}{\boldsymbol{\Phi}_{i n}}\\ {\boldsymbol{0}_{b n}}\end{array}\right]
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$$
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$$
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When normalized with respect to the mass matrix $M_{i i}$ , the fixed-interfacc modes satisfy
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When normalized with respect to the mass matrix $M_{i i}$ , the fixed-interfacc modcs satisfy
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当以质量矩阵 $M_{i i}$ 进行归一化时，固定界面模态满足
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$$
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$$
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\Phi_{i n}^{T}M_{i i}\Phi_{i n}=I_{n n},\ \Phi_{i n}^{T}K_{i i}\Phi_{i n}=\Lambda_{n n}=\mathrm{diag}(\omega_{j}^{2})
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\Phi_{i n}^{T}M_{i i}\Phi_{i n}=I_{n n},\ \Phi_{i n}^{T}K_{i i}\Phi_{i n}=\Lambda_{n n}=\mathrm{diag}(\omega_{j}^{2})
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$$
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$$
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Figure 3 shows the $N_{i}$ (four) fixed-interface normal modes for the 8DOF free-free beam component in Fig. 2b.
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Figure 3 shows the $N_{i}$ (four) fixed-interface normal modes for the 8DOF free-free beam component in Fig. 2b.
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图3显示了图2b中8DOF自由-自由梁部件的$N_{i}$ (四) 简正模态。
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Figure 3: Component Fixed-Interface Modes
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Figure 3: Component Fixed-Interface Modes
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A second type of component normal modes used in CMS is free-interface normal modes. These normal modes are defined by the equation
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A second type of component normal modes used in CMS is free-interface normal modes. These normal modes are defined by the equation
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第二种在CMS中使用的组件简正模态是自由边界简正模态。这些简正模态由以下方程定义：
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$$
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$$
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\left[K-\omega_{j}^{2}M\right]\{\phi\}_{j}=0,\;j=1,2,...,(N_{f}=N-N_{r}),
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\left[K-\omega_{j}^{2}M\right]\{\phi\}_{j}=0,\;j=1,2,...,(N_{f}=N-N_{r}),
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$$
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$$
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The assembled set of $N_{f}$ flexible (i.e., non rigidbody) free-interface normal modes is
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The assembled set of $N_{f}$ flexible (i.e., non rigidbody) free-interface normal modes is
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装配好的 $N_{f}$ 个柔性（即非刚体）自由边界简正模态是：
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$$
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$$
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\boldsymbol{\Phi}_{n}\equiv\left[\begin{array}{l}{\boldsymbol{\Phi}_{i n}}\\ {\boldsymbol{\Phi}_{b n}}\end{array}\right]
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\boldsymbol{\Phi}_{n}( N \times N_f)\equiv\left[\begin{array}{l}{\boldsymbol{\Phi}_{i n}}\\ {\boldsymbol{\Phi}_{b n}}\end{array}\right]
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$$
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$$
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Figure 4 shows the first four of the six free-free fex modes of the 8DOF component.
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Figure 4 shows the first four of the six free-free flex modes of the 8DOF component.
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图4显示了8自由度组件的六个简正模态中的前四个模态。
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Figure 4: Flex Modes for the Free-Free Component.
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Figure 4: Flex Modes for the Free-Free Component.
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A third important type of component normal modes is loaded-interface normal modes. This includes lumped-mass loaded-interface normal modes, commonly referred to as mass-additive normal modes[16, 19]. Benfield and Hruda described CMS methods based on “consistent" mass- and stiffnessadditive normal modes[24], but these methods require reduced-order models of all adjacent substructures, so they are generally of limited practical value.
|
A third important type of component normal modes is loaded-interface normal modes. This includes lumped-mass loaded-interface normal modes, commonly referred to as mass-additive normal modes[16, 19]. Benfield and Hruda described CMS methods based on “consistent" mass- and stiffnessadditive normal modes[24], but these methods require reduced-order models of all adjacent substructures, so they are generally of limited practical value.
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第三种重要的简正模态类型是载荷界面简正模态。这包括集质量载荷界面简正模态，通常被称为质量累加简正模态[16, 19]。Benfield 和 Hruda 描述了基于“一致”质量累加和刚度累加简正模态的 CMS 方法[24]，但这些方法需要所有相邻次结构的降阶模型，因此通常实用价值有限。
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# 2.2. Constraint Modes; Rigid-body Modes
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## 2.2. Constraint Modes; Rigid-body Modes
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A constraint mode is defined as the static deformation of a structure when a unit displacement is applied to one coordinate of a specified set of “constraint" coordinates, $c$ , while the remaining coordinates of that set are restrained, and the remaining degrees of freedom of the structure are force-free. The set of interface constraint modes based on unit displacement of the boundary coordinates $\pmb{u}_{b}$ is a very useful CMS set, because of the ease of enforcing inter-component compatibility when these constraint modes are employed, as will be explained in Section 4.1. This set, with ${\mathcal{C}}={\boldsymbol{B}}$ , is given by
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A constraint mode is defined as the static deformation of a structure when a unit displacement is applied to one coordinate of a specified set of “constraint" coordinates, $c$ , while the remaining coordinates of that set are restrained, and the remaining degrees of freedom of the structure are force-free. The set of interface constraint modes based on unit displacement of the boundary coordinates $\pmb{u}_{b}$ is a very useful CMS set, because of the ease of enforcing inter-component compatibility when these constraint modes are employed, as will be explained in Section 4.1. This set, with ${\mathcal{C}}={\boldsymbol{B}}$ , is given by
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一种约束模态定义为，当单位位移施加到一组指定的“约束”坐标 $c$ 的一个坐标上，而该组的其他坐标被约束，且结构的其余自由度无力时，结构的静态变形。基于边界坐标 $\pmb{u}_{b}$ 单位位移的界面约束模态集合是一个非常有用的 CMS 集合，因为在采用这些约束模态时，可以轻松地强制执行组件间的兼容性，如将在第 4.1 节中解释。该集合，其中 ${\mathcal{C}}={\boldsymbol{B}}$ ，表示为
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$$
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$$
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\left[\begin{array}{c c}{{K_{i i}}}&{{K_{i b}}}\\ {{K_{b i}}}&{{K_{b b}}}\end{array}\right]\left[\begin{array}{c}{{\Psi_{i b}}}\\ {{I_{b b}}}\end{array}\right]=\left[\begin{array}{c}{{0_{i b}}}\\ {{R_{b b}}}\end{array}\right]
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\left[\begin{array}{c c}{{K_{i i}}}&{{K_{i b}}}\\ {{K_{b i}}}&{{K_{b b}}}\end{array}\right]\left[\begin{array}{c}{{\Psi_{i b}}}\\ {{I_{b b}}}\end{array}\right]=\left[\begin{array}{c}{{0_{i b}}}\\ {{R_{b b}}}\end{array}\right]
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$$
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$$
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@ -133,7 +153,7 @@ $$
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$$
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$$
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From Eqs. (8) and (12) it can easily be shown that these constraint modes are stiffness-orthogonal to all of the fixed-interface normal modes, that is,
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From Eqs. (8) and (12) it can easily be shown that these constraint modes are stiffness-orthogonal to all of the fixed-interface normal modes, that is,
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从公式(8)和(12)可以看出，这些约束模态与所有固定界面简正模态正交，即：
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$$
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$$
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\Phi_{n}^{T}K\bar{\Psi}_{c}=0
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\Phi_{n}^{T}K\bar{\Psi}_{c}=0
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$$
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$$
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@ -145,6 +165,7 @@ Figure 5: Component Constraint Modes.
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Although they are often considered to be normal modes, rigid-body modes are actually a special case of constraint modes. They can be defined relative to anysetof $N_{r}$ coordinates that is just sufficient to restrain rigid-body motion of the component. For purposes of substructure coupling, rigid-body modes will be defined relative to a set $\mathcal{R}$ of boundary coordinates. Then,
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Although they are often considered to be normal modes, rigid-body modes are actually a special case of constraint modes. They can be defined relative to anysetof $N_{r}$ coordinates that is just sufficient to restrain rigid-body motion of the component. For purposes of substructure coupling, rigid-body modes will be defined relative to a set $\mathcal{R}$ of boundary coordinates. Then,
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虽然它们通常被认为是简正模态，刚体模态实际上是一种约束模态的特殊情况。 它们可以相对于任何一组 $N_{r}$ 坐标来定义，这组坐标仅足够约束组件的刚体运动。 为了子结构耦合的目的，刚体模态将相对于一组边界坐标 $\mathcal{R}$ 来定义。 然后，
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$$
|
$$
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\left[\begin{array}{l l l}{K_{i i}}&{K_{i e}}&{K_{i r}}\\ {K_{e i}}&{K_{e e}}&{K_{e r}}\\ {K_{r i}}&{K_{r e}}&{K_{r r}}\end{array}\right]\left[\begin{array}{l}{\Psi_{i r}}\\ {\Psi_{e r}}\\ {I_{r r}}\end{array}\right]=\left[\begin{array}{l}{0_{i r}}\\ {0_{e r}}\\ {0_{r r}}\end{array}\right]
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\left[\begin{array}{l l l}{K_{i i}}&{K_{i e}}&{K_{i r}}\\ {K_{e i}}&{K_{e e}}&{K_{e r}}\\ {K_{r i}}&{K_{r e}}&{K_{r r}}\end{array}\right]\left[\begin{array}{l}{\Psi_{i r}}\\ {\Psi_{e r}}\\ {I_{r r}}\end{array}\right]=\left[\begin{array}{l}{0_{i r}}\\ {0_{e r}}\\ {0_{r r}}\end{array}\right]
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$$
|
$$
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@ -179,7 +200,7 @@ $$
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Either the set of interface constraint modes $\Psi_{c}$ defined by Eq. (13), or the combined set $\left[\Psi_{r}\ \Psi_{e}\right]$ defined by Eqs. (16) and (19), spans the static response of the substructure to interface loading and allows for arbitrary interface displacements $\pmb{u}_{b}$ . Along with the interface displacement, there is accompanying displacement of the interior of the substructure, as determined by Eqs. (13), (16), and (19). Additional interior fexibility can be incorporated by including fixed-interface normal modes, fixed-interface Krylov vectors, or other fixed-interface assumed modes in the component mode matrix $\Psi[3,\,5,\,23]$
|
Either the set of interface constraint modes $\Psi_{c}$ defined by Eq. (13), or the combined set $\left[\Psi_{r}\ \Psi_{e}\right]$ defined by Eqs. (16) and (19), spans the static response of the substructure to interface loading and allows for arbitrary interface displacements $\pmb{u}_{b}$ . Along with the interface displacement, there is accompanying displacement of the interior of the substructure, as determined by Eqs. (13), (16), and (19). Additional interior fexibility can be incorporated by including fixed-interface normal modes, fixed-interface Krylov vectors, or other fixed-interface assumed modes in the component mode matrix $\Psi[3,\,5,\,23]$
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# 2.3. Attachment Modes
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## 2.3. Attachment Modes
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An attachment mode is defined as the component displacement vector due to a single unit force applied at one of the coordinates of a given set $\pmb{A}$ Consequently, attachment modes are just columns of the associated fexibility matrix. Attachment modes were defined by Bamford[4], and they get their name from their usefulness in representating the deformation of a structure to loading (e.g., an external force, an attached mass, or an attached fexible component) at the point where the attachment mode's unit force is applied. In this paper we are interested in defining attachment modes to represent the response of a component to forces at its interface with adjoining components. One diffculty encountered in using attachment modes is that many components have one to six rigid-body degrees of freedom, making it impossible to apply directly to the unrestrained component the necessary unit forces in order to compute the resulting attachment mode shapes. However, one option in this case is to select a set $\mathcal{R}$ of boundary rigid-body degrees of freedom, (mathematically) restrain the component at these DOFs, and then form cantilever attachment modes by applying unit loads at the redundant boundary coordinates, that is, for $\boldsymbol{A}=\boldsymbol{\mathcal{E}}$ .Then,
|
An attachment mode is defined as the component displacement vector due to a single unit force applied at one of the coordinates of a given set $\pmb{A}$ Consequently, attachment modes are just columns of the associated fexibility matrix. Attachment modes were defined by Bamford[4], and they get their name from their usefulness in representating the deformation of a structure to loading (e.g., an external force, an attached mass, or an attached fexible component) at the point where the attachment mode's unit force is applied. In this paper we are interested in defining attachment modes to represent the response of a component to forces at its interface with adjoining components. One diffculty encountered in using attachment modes is that many components have one to six rigid-body degrees of freedom, making it impossible to apply directly to the unrestrained component the necessary unit forces in order to compute the resulting attachment mode shapes. However, one option in this case is to select a set $\mathcal{R}$ of boundary rigid-body degrees of freedom, (mathematically) restrain the component at these DOFs, and then form cantilever attachment modes by applying unit loads at the redundant boundary coordinates, that is, for $\boldsymbol{A}=\boldsymbol{\mathcal{E}}$ .Then,
|
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|
|
||||||
@ -195,7 +216,7 @@ $$
|
|||||||
|
|
||||||
Two important topics that arise when freeinterface normal modes are to be employed to represent the fexible behavior of unrestrained components are inertia relief and residual fezibility, both of which were discussed by MacNeal[6] and Rubin[7]. Sections 2.4 and 2.5 treat these two topics, and the related forms of attachment modes are defined.
|
Two important topics that arise when freeinterface normal modes are to be employed to represent the fexible behavior of unrestrained components are inertia relief and residual fezibility, both of which were discussed by MacNeal[6] and Rubin[7]. Sections 2.4 and 2.5 treat these two topics, and the related forms of attachment modes are defined.
|
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|
|
||||||
# 2.4. Inertia-Relief Attachment Modes
|
## 2.4. Inertia-Relief Attachment Modes
|
||||||
|
|
||||||
When a component has rigid-body freedom, it is appropriate to employ inertia-relief attachment modes[6, 7, 11]. The term inertia relief refers to the process of applying to the component an equilibrated load system $\pmb{f}_{f}$ , which consists of the original force vector $\pmb{f}$ equilibrated by the rigid-body d'Alembert force vector $M\ddot{\boldsymbol{u}}_{r}$ , where $\pmb{u}_{r}$ is the rigid-body motion due to $\pmb{f}$ .Starting with Eq. (1), let the displacement vector be separated into rigid-body displacement and fexible-body displacement, that is, let
|
When a component has rigid-body freedom, it is appropriate to employ inertia-relief attachment modes[6, 7, 11]. The term inertia relief refers to the process of applying to the component an equilibrated load system $\pmb{f}_{f}$ , which consists of the original force vector $\pmb{f}$ equilibrated by the rigid-body d'Alembert force vector $M\ddot{\boldsymbol{u}}_{r}$ , where $\pmb{u}_{r}$ is the rigid-body motion due to $\pmb{f}$ .Starting with Eq. (1), let the displacement vector be separated into rigid-body displacement and fexible-body displacement, that is, let
|
||||||
|
|
||||||
@ -272,7 +293,7 @@ The top two plots in Fig. 6 are the shapes that correspond to the two columns of
|
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|
|
||||||
Figure 6: Elastic Flexibility Shapes
|
Figure 6: Elastic Flexibility Shapes
|
||||||
|
|
||||||
# 2.5. Flexibility Matrices and Residual Flexibility Attachment Modes
|
## 2.5. Flexibility Matrices and Residual Flexibility Attachment Modes
|
||||||
|
|
||||||
The complete set of component normal modes $\Phi_{n}$ and the corresponding set of eigenvalues $\Lambda_{n n}$ are identified by the subscript $n$ , whether these are the $N_{i}$ fixed-interface modes, the $N_{f}$ free-free flexible (fex) modes, or some other form of component normal modes.
|
The complete set of component normal modes $\Phi_{n}$ and the corresponding set of eigenvalues $\Lambda_{n n}$ are identified by the subscript $n$ , whether these are the $N_{i}$ fixed-interface modes, the $N_{f}$ free-free flexible (fex) modes, or some other form of component normal modes.
|
||||||
|
|
||||||
@ -446,7 +467,7 @@ Equations (42) through (56) describe a single level of substructuring; however,
|
|||||||
|
|
||||||
Most applications of component mode synthesis employ one of two approaches, called constraint-mode methods and attachment-mode methods. The former employ constraint modes and fixed-interface normal modes, as represented by Hurty's method[3] and the Craig-Bampton variant of Hurty's method[5]. The latter employ attachment modes and freeinterface normal modes, as represented by MacNeal's method[6] and Rubin's method[7]. It is possible to cite here only a small number of the significant papers dealing with the use of component modes in structural dynamics.
|
Most applications of component mode synthesis employ one of two approaches, called constraint-mode methods and attachment-mode methods. The former employ constraint modes and fixed-interface normal modes, as represented by Hurty's method[3] and the Craig-Bampton variant of Hurty's method[5]. The latter employ attachment modes and freeinterface normal modes, as represented by MacNeal's method[6] and Rubin's method[7]. It is possible to cite here only a small number of the significant papers dealing with the use of component modes in structural dynamics.
|
||||||
|
|
||||||
# 4.1 Constraint-Mode Methods
|
## 4.1 Constraint-Mode Methods
|
||||||
|
|
||||||
Although there had been previous applications of component modes, Hurty's 1965 paper[3] provided the first comprehensive development of a finite element oriented CMS method based on constraint modes and fixed-interface modes. Craig and Bampton[5] simplified Hurty's method by treating all interface degrees of freedom together, rather than requiring the interface degrees of freedom to be separated into rigid-body freedoms and redundant interface freedoms. The displacement transformation for this method employs a combination of fixed-interface normal modes (Eq. (7) and Fig. 3) and interface constraint modes (Eq. (13) and Fig. 5), and takes the form
|
Although there had been previous applications of component modes, Hurty's 1965 paper[3] provided the first comprehensive development of a finite element oriented CMS method based on constraint modes and fixed-interface modes. Craig and Bampton[5] simplified Hurty's method by treating all interface degrees of freedom together, rather than requiring the interface degrees of freedom to be separated into rigid-body freedoms and redundant interface freedoms. The displacement transformation for this method employs a combination of fixed-interface normal modes (Eq. (7) and Fig. 3) and interface constraint modes (Eq. (13) and Fig. 5), and takes the form
|
||||||
|
|
||||||
@ -490,7 +511,7 @@ $$
|
|||||||
|
|
||||||
Thus, component models based on the use of fixed-interface modes plus interface constraint modes are essentially superelements - all physical boundary coordinates are retained as independent generalized coordinates, greatly facilitating component coupling. Because of the simple, straightforward procedures for formulating the component modes employed by this method, because of the straightforward way in which components are coupled to form the component mode system model and the sparsity patterns of the resulting system matrices, and because this method also produces highly accurate models with relatively few component modes[8], this method has been widely used and is available in a number of commercial finite element codes (e.g., MSC/NASTRAN[27]).
|
Thus, component models based on the use of fixed-interface modes plus interface constraint modes are essentially superelements - all physical boundary coordinates are retained as independent generalized coordinates, greatly facilitating component coupling. Because of the simple, straightforward procedures for formulating the component modes employed by this method, because of the straightforward way in which components are coupled to form the component mode system model and the sparsity patterns of the resulting system matrices, and because this method also produces highly accurate models with relatively few component modes[8], this method has been widely used and is available in a number of commercial finite element codes (e.g., MSC/NASTRAN[27]).
|
||||||
|
|
||||||
# 4.2 Attachment-Mode Methods
|
## 4.2 Attachment-Mode Methods
|
||||||
|
|
||||||
If a reduced set of component normal modes is used without including a complete set of either interface constraint modes or interface attachment modes, the component mode set is not statically complete, as indicated in Section 2.5. This is true of the “classical" CMS method of using only a set of free-interface normal modes[8]. However, methods that employ free-interface normal modes together with attachment modes (including residualfexibility attachment modes and/or inertia-relief attachment modes) are widely used, especially MacNeal's method and Rubin's methodl6, 7, 26], and especially in context of experimental verification of finite element models[14-20]. Martinez, et al.[14, 28], simplified the formulation of this class of CMS methods, casting the component mode matrix $\Psi$ inthe same format as that of the constraint-mode method of Craig and Bampton. The variant of Rubin's method proposed by Martinez, et al., is described below.
|
If a reduced set of component normal modes is used without including a complete set of either interface constraint modes or interface attachment modes, the component mode set is not statically complete, as indicated in Section 2.5. This is true of the “classical" CMS method of using only a set of free-interface normal modes[8]. However, methods that employ free-interface normal modes together with attachment modes (including residualfexibility attachment modes and/or inertia-relief attachment modes) are widely used, especially MacNeal's method and Rubin's methodl6, 7, 26], and especially in context of experimental verification of finite element models[14-20]. Martinez, et al.[14, 28], simplified the formulation of this class of CMS methods, casting the component mode matrix $\Psi$ inthe same format as that of the constraint-mode method of Craig and Bampton. The variant of Rubin's method proposed by Martinez, et al., is described below.
|
||||||
|
|
||||||
|
|||||||
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