# COUPLING OF SUBSTRUCTURES FOR DYNAMIC ANALYSES: AN OVERVIEW
Roy R. Craig, Jr.\* The University of Texas at Austin Austin TX 78712
# Abstract
# 1. Introduction
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.
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.
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.
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.
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.
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 $N=N_{i}+N_{b}$
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.
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
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.
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
\begin{array}{r l}&{\left[\begin{array}{c c c}{M_{\mathrm{ii}}}&{M_{\mathrm{ie}}}&{M_{\mathrm{ir}}}\\ {M_{\mathrm{ei}}}&{M_{\mathrm{ee}}}&{M_{\mathrm{er}}}\\ {M_{\mathrm{ri}}}&{M_{\mathrm{re}}}&{M_{\mathrm{rr}}}\end{array}\right]\left\{\begin{array}{c}{\dot{u}_{i}}\\ {\dot{u}_{e}}\\ {\dot{u}_{r}}\end{array}\right\}}\\ &{+\left[\begin{array}{c c c}{K_{\mathrm{ii}}}&{K_{\mathrm{ie}}}&{K_{\mathrm{ir}}}\\ {K_{\mathrm{ei}}}&{K_{\mathrm{ee}}}&{K_{\mathrm{er}}}\\ {K_{\mathrm{ri}}}&{K_{\mathrm{re}}}&{K_{\mathrm{rr}}}\end{array}\right]\left\{\begin{array}{c}{u_{i}}\\ {u_{e}}\\ {u_{r}}\end{array}\right\}}\\ &{=\left\{\begin{array}{c}{f_{i}}\\ {f_{e}}\\ {f_{r}}\end{array}\right\}}\end{array}
$$
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.
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:
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
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 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
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,
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,
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]$
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,
It can be seen that these attachment modes are just an expanded form of the columns of the righthand partition of the Hexibility matrix $\bar{G}_{c}$ of Eq. (17) with $A=\mathcal{E}$ . That is,
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.
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
are the appropriate orthogonality equation and the definition of the rigid-body modal mass matrix, respectively. (It is not assumed that the rigid-body modes are orthonormalized.） Since $K\Psi_{r}~\,=\,\,0$ Eqs. (1) and (22) can be combined to give
where $P_{r}$ is the inertia-relief projection matriz, de fined by
$$
P_{r}=I-M\Psi_{r}\bar{M}_{r r}^{-1}\Psi_{r}^{T}
$$
When any force vector is premultiplied by this inertia-relief projection matrix, the corresponding
force system is self-equilibrated. Also, from Eq. (26) it can easily be verified that $P_{r}^{T}$ is mass-orthogonal to the rigid-body modes, that is,
$$
\Psi_{r}{}^{T}M P_{r}^{T}=0
$$
Inertia-relief attachment modes are staticdeformation shapes defined by applying unit forces at the all interface coordinates $\quad A=B!$ , that is, by applying the force
pre-multiplied by the inertia-relief projection matrix, $P_{r}$ . Since the unit-force column vectors in $F_{b}$ are self-equilibrated by the inertia-relief projection matrix, no reaction forces are required, such as there are in Eq. (20). Deformation of the component due to this equilibrated force system is given by
$$
\hat{\Psi}_{b}=G_{c}P_{r}F_{b}
$$
where $G_{c}$ is the constrained feribility matriz, a spe cial expanded (singular) form of the cantilever fexibility matrix $\dot{G}_{c}$ in Eq. (17), given by
The attachment-mode set defined by Eq. (29) is made orthogonal to the rigid-body modes, and the resulting inertia-relief attachment modes are given by[11]
is the elastic flezibility matrir in inertia-relief format. In Eq. (31), the $\boldsymbol{I_{b b}}$ matrix in $F_{b}$ picks out the columns of the fexibility matrix $G_{f}$ that correspond to unit forces applied at the boundary. From Eqs. (27) and (32), it can be shown that the columns of $G_{f}$ are mass-orthogonal to the rigid-body modes $\Psi_{r}$ . Therefore, $G_{f}$ spans the same subspace as do the free-interface fex modes of Eq. (11).
The top two plots in Fig. 6 are the shapes that correspond to the two columns of the elastic flexibility matrix $G_{f}$ for unit forces at the transverse DOFs at the two ends of the 8DOF free-free beam in Fig. 2. It is clear that these two fexibility shapes (and the remaining six as well) are dominated by the contribution of the fundamental free-free fex mode (Fig. 4). This can be seen clearly by the bottom two figures, which represent symmetric and antisymmetric loading by unit forces at the two ends of the component.
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.
Let the (diagonal) modal mass matrix and modal stiffness matrix for modes $\Phi_{n}$ be
Note that each column of the $j$ th mode's contribution to the elastic fexibility matrix has the shape of mode $\phi_{j}$ .Although the elastic fexibility matrices $G$ of Eq. (34) and matrix $G_{f}$ of Eq. (32) and illustrated in Fig. 6 are formed in different ways, they are numerically the same. In this section we will be concerned with components that have rigid-body freedom, in which case $G$ is singular, with rank $N_{f}$ Regardless of whether the elastic fexibility matrix
$G$ is singular or not, from Eqs. (33b) and (34) it can be shown that
$$
G^{T}K G=G
$$
Since model reduction is one of the major objectives in CMS, the normal mode set is usually reduced to a smaller set of kept normal modes, denoted by $\Phi_{k}$ ，where $\Phi_{n}~\equiv~\left[\Phi_{k}~\Phi_{d}\right]$ .\*The deleted normal modes, $\Phi_{d}$ , are generally all of the modes above some specified cutof frequency. The portion of the fexibility matrix contributed by modes $\Phi_{d}$ is called the residual feribility matrir. It is given by
where $G$ is the total fexibility matrix. Since it is not usually feasible to compute or measure the $\Phi_{d}$ modes, Eq. (36) is useful only because Eq. (32) exists as an alternative to Eq. (34) for determining the elastic fexibility matrix $G$
The matrix $G_{d}$ will always be a singular matrix because of the modes deleted in Eq. (36). Also, because of the mass- and stiffness-orthogonality of the kept modes to the deleted modes,
Residual-fleribility attachment modes maybe de fined for forces applied at the interface coordinates, that is, for $\boldsymbol{A}=\boldsymbol{B}$ , by the following equation:
Figure 7 shows the attachment mode shape for the component with a unit force at DOF7; the top figure includes all six flex modes, the middle figure is the contribution of two “kept" modes, and the bottom figure is the corresponding residual fexibility attachment mode shape. It is clear that the order of magnitude of the residual fexibility is smaller than that of the fexibility of the kept modes. Figure 8 shows the residual-Hexibility attachmentmode shapes $({\bf k}{=}2)$ for the component with unit forces at DOFs 7 and 5 (left and right ends). The top two figure are the attachment-mode shapes for the individual unit forces; the middle figure is the shape produces by symmetric loading by two unit forces, and the bottom figure is the corresponding residual-fexibility attachment-mode shape for antisymmetric loading. It can easily be seen that these residual-fexibility shapes are free of the first two (kept) normal-mode contributions.
Incorporation of $\Psi_{d}$ into the component mode set ensures complete representation of static defection of the component due to forces applied at interface DOFs. In this sense, it is closely related to the mode-acceleration method for incorporating static completeness in dynamic-response computations[1, 7, 11]. Hintz has given an extensive discussion of the need for statically complete component mode sets in Ref. [25]. We will return to the topic of residual flexibility later in Section 4.2.
# 3. A Generalized Component Coupling Procedure for Undamped Structures
In this section a generalized substructure coupling procedure that employs Lagrange multipliers to enforce inter-component displacement compatibility equations (and other constraint equations, if applicable) is presented. Let the system be composed of two components, labeled $_\alpha$ and $\beta$ , that have a common (generally redundant) interface. The physical displacements at the interface are constrained by the displacement compatibility equation
$$
\pmb{u}_{b}^{\alpha}=\pmb{u}_{b}^{\beta}
$$
and the mutually reactive interface forces (i.e., not including external forces applied at the interface) are related by
Constraint equations, such as Eq. (40) and any other constraint equations that are to be imposed (say $N_{C}$ equations in all), can be written in terms of the generalized coordinates $\pmb{p}$ and combined to form a matrix constraint equation of the form
$$
C{\pmb p}={\bf0}
$$
For example, Eqs. (2) and (40) can be combined to give the constraint equation
The synthesis of the system equation of motion is based on Lagrange's equation of motion with undetermined multipliers[11, 26]. The Lagrangian for the system of two coupled substructures can be written
$$
\mathcal{L}=\mathcal{T}-\mathcal{V}+\lambda^{T}C p
$$
where $\tau$ is the system kinetic energy and $\nu$ is the system potential energy, given by
where $p_{j}$ refers to the $j$ th element of the merged displacement vector $\pmb{p}$ , and ${\bar{f}}_{j}$ refers to the corre sponding (externally applied) force. As required by Eq. (41), the mutually reactive interface forces cancel out and do not appear on the right-hand side of Eq. (48). In matrix form, the $\left(N_{\alpha}+N_{\beta}\right)$ equations of Eq. (48) can be written as
Since, due to the constraint equation, Eq. (42), thecoordinates $\pmb{p}$ are not linearly independent, practically all substructure coupling methods solve the coupled set of equations, Eqs. (42) and (49), by introducing a linear transformation of the form
$$
\pmb{p}=S\pmb{q}
$$
where $\pmb q$ is the vector of independent system generalized coordinates.
Let $\pmb{p}$ be rearranged, if necessary, and partitioned into $N_{C}$ dependent coordinates $\pmb{p}_{D}$ , and $(N_{\alpha}\!+\!N_{\beta}-$ $N_{C}\,.$ ) linearly independent coordinates $\pmb{p}_{I}$ , and let Eq. (42) be partitioned accordingly, giving
where $C_{D D}$ is a nonsingular square matrix. Then, the equation
$$
p\equiv\left\{\begin{array}{c}{{p_{n}}}\\ {{p_{I}}}\end{array}\right\}=\left[\begin{array}{c}{{-C_{\scriptscriptstyle D D}^{-1}C_{\scriptscriptstyle D I}}}\\ {{I_{\scriptscriptstyle I I}}}\end{array}\right]p_{I}\equiv S q
$$
defines both $S$ and $\pmb q$ . Then, the vector of independent system generalized coordinates is $\pmb q\equiv\pmb{p}_{I}$ ,and the coupling transformation matrix $S$ is given by
From Eqs. (51) and (52), it is seen that $C S\,=\,0$ Therefore, the system equation of motion, Eq. (54), becomes simply
$$
M_{q}{\ddot{q}}+K_{q}q=f_{q}
$$
Although Eq.(55) defines $M_{q},~K_{q}$ ,and $\pmb{f}_{q}$ in terms of matrix operations, the system matrices and force vector can usually be assembled from the substructure matrices by the “direct stiffness" assembly procedure, as is illustrated in Section 4.1.
Equations (42) through (56) describe a single level of substructuring; however, essentially the same procedure can be employed when a structure is partitioned into several levels of substructures (e.g.. Ref.[21]).
# 4. Component Mode Synthesis Methods
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.
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
where $\Phi_{i k}$ is the interior partition of the fixedinterface modal matrix, and $\Psi_{i b}$ is the interior partition of the constraint-mode matrix.
With component fixed-interface normal modes normalized according to Eq. (9), the reduced mass and stiffness matrices, Eqs. (4), have the special forms
The zeros in the $k b$ and $b k$ partitions of $\bar{K}_{C B}^{c}$ arethe result of orthogonality equation Eq. (14).
Equation (57) implies that $\pmb{p}_{b}^{c}=\pmb{u}_{b}^{c}$ .Therefore, in terms of component generalized coordinates, the interface compatibility equation, Eq. (40), becomes
and the component coupling matrix $S$ is justthe "direct-stiffness assembly” matrix. For the twocomponent system, component mass and stiffness matrices (Eq. (59)) are assembled to form the following system reduced-order mass and stiffness matrices, respectively:
$$
\begin{array}{r l}&{M_{C B}=\left[\begin{array}{l l l}{I_{k_{a}k_{o}}^{\alpha}}&{0_{k_{o}k_{i}\beta}^{\alpha}}&{\hat{M}_{k_{o}k\beta}^{\alpha}}\\ {0_{k_{i}k_{o}}^{\beta}}&{I_{k_{i}k_{i}\beta}^{\beta}}&{\hat{M}_{k_{i}\beta}^{\beta}}\\ {\bar{M}_{b k_{o}}^{\alpha}}&{\bar{M}_{b k_{o}}^{\beta}}&{\bar{M}_{b k}^{\alpha}+\bar{M}_{b k}^{\beta}}\end{array}\right]}\\ &{K_{C B}=\left[\begin{array}{l l l}{\Lambda_{k_{c}k_{o}}^{\alpha}}&{0_{k_{c}k_{i}}^{\alpha}}&{0_{k_{c}b}^{\alpha}}\\ {0_{k_{i}k_{o}}^{\beta}}&{\Lambda_{k_{i}k_{\beta}}^{\beta}}&{0_{k_{i}b}^{\beta}}\\ {0_{k_{k_{o}}}^{\alpha}}&{0_{k_{k_{\beta}}}^{\beta}}&{\bar{K}_{b k}^{\alpha}+\bar{K}_{b k}^{\beta}}\end{array}\right]}\end{array}
$$
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]).
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.
The initial form of the displacement transformation for this method employs a combination of rigidbody modes (from Eq. (16)), kept free-free normal modes (from Eq. (11)), and residual-fexibility attachment modes (from Eq. (39)). To simplify the following derivation, the rigid-body modes can be written in the following two-partition form:
When two of the six free-interface modes for the 8DOF beam component are kept, the resulting eight columns of the Rubin transformation matrix have the shapes shown in Fig. 9 and 10.
With mass-normalized free-interface normal modes $\Phi_{k}$ , from Eqs. (4) the component generalized mass matrix and stiffness matrix based on the Rubin transformation are
The zeros in these matrices are the result of orthogonality (e.g., Eqs. (37) and (38)), and the residual flexibility term $\Psi_{b b}$ of $\bar{K}_{R}^{c}$ is due to Eq. (35).s The above formulation is a consistent Ritz transformation; residual effects are included in both the stiffness and mass matrices. If the residual term $\bar{M}_{b b}$ in the mass matrix is omitted, the resulting nonconsistent transformation is referred to as the MacNeal method[6].
As suggested by Martinez, et al., let the lower partition of Eq. (65) be solved for the generalized coordinate vector $\pmb{p_{b}}$ in terms of $\mathbf{\deltau}_{b}$ , and the result incorporated back into the upper partition of Eq. (65). Finally, Eq. (65) can be re-cast in terms of the modal generalized coordinate vector $\pmb{p}_{k}$ and the interface displacement vector $\pmb{u}_{b}$ . This produces the following coordinate-transformation equation:
When two of the six free-interface modes for the 8DOF beam component are kept, the resulting eight columns of the Rubin-Martinez transformation matrix have the shapes shown in Figs. 11 and 12. Note the similarities and differences between the shapes in Figs. 11 and 3, and the similarities and differences between the shapes in Figs. 12 and 5.
When the Rubin-Martinez transformation is used in Eqs. (4) to form the generalized mass matrix and stiffness matrix, the resulting matrices no longer have the sparsity that is exhibited by the Rubin mass matrix and stiffness matrix in Eqs. (67).
Procedures used to formulate component modes for substructures and to assemble substructure models to form reduced-order models of the original system have been reviewed. The physical meaning of many CMS terms has been illustrated. The constraint-mode method described in Section 4.1 is widely used in reducing finite element models for dynamic analysis because it is very straightforward, and it leads to accurate reduced-order models. On the other hand, the attachment-mode method described in Section 4.2 also produces accurate reduced-order models, and is currently widely used in test-verifying finite element models. There is a pressing need for the development of efficient computational structural dynamics algorithms based on constraint-mode substructuring methods, and there is still a substantial need for a more thorough understanding of attachment-mode methods and their use in test verification of finite element models. Research is also needed on CMS methods for damped structural systems.
# 6. References
[1] Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., Aeroelasticity, Addison-Wesley Publishing Co., Inc., Reading, Ma, 1955.
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[17] Admire, J. R., Tinker, M. L., and Ivey, E. W., "Residual Flexibility Test Methods for Verification of Constrained Structural Models," AIAA Journal, Vol. 32, No. 1, Jan. 1994, pp. 170-175.
[18] Duarte, M. L. M., and Ewins, D. J., “Improved Experimental Component Mode Synthesis (IECMS) with Residual Compensation Based Purely on Experimental Results," Proc. 14th International Modal Analysis Conference, Dearborn, MI, Feb. 1996, pp. 641-647.
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