740 lines
78 KiB
Markdown
740 lines
78 KiB
Markdown
|
# DYNAMIC ANALYSIS OF LARGE STRUCTURES BY MODAL SYNTHESIS TECHNIQUES
|
|||
|
|
|
|||
|
|
WALTER C. HURTY,t JoN D. CoLLINS and GARY C. HARTf J. H. Wiggins Co., Palos Verdes Estates, California, U.S.A.
|
|||
|
|
|
|||
|
|
Abstract-The past decade has seen the development of several techniques for the dynamic analysis of large structures that involve division into substructures or components. These techniques make use of component displacement modes to synthesize global systems of generalized coordinates and, for that reason, they have come to be known as modal synthesis or component mode techniques.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
# 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.
|
|||
|
|
|
|||
|
|
The calculation of the natural frequencies and mode shapes of a structure, as well as dynamic response predictions, have long been of interest and concern to engineers. In the pre-computer era, the primary emphasis was on minimizing the number of hand calculations necessary in predicting these dynamical parameters. This emphasis usually resulted in approximate techniques for predicting the fundamental natural frequency and mode shapes. With the increased availability and capability of the digital computer, we can now develop a more realistic analytical model of a structure by using a large number of generalized coordinates. While the use of many coordinates improves the definition of the deformed shape of the structure, it also necessitates the computational handling of large matrices. Therefore, because of the size of the matrices involved, a direct cigenvalue solution using all of the generalized coordinates may exceed the size limitation of the computer and/or lead to errors in numerical calculations and long computer run times.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
# COMPONENT MODE METHODS, SURVEY, AND EVALUATION
|
|||
|
|
|
|||
|
|
Criteria useful in comparing methods of analysis
|
|||
|
|
|
|||
|
|
(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.
|
|||
|
|
|
|||
|
|
(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 mostclearlypermits isolation of the contractual components of the total systemwouldbesought.
|
|||
|
|
|
|||
|
|
(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.
|
|||
|
|
|
|||
|
|
(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 repetitivetestforconvergencemaybemade.
|
|||
|
|
|
|||
|
|
(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 ininterconnections.
|
|||
|
|
|
|||
|
|
(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 wellsuited 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.
|
|||
|
|
|
|||
|
|
(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.
|
|||
|
|
|
|||
|
|
(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 experimentallywhereverpossible.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
# 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.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
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'.
|
|||
|
|
|
|||
|
|
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
|
|||
|
|
|
|||
|
|
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
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
Component modes are defined in three categories :
|
|||
|
|
|
|||
|
|
(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.
|
|||
|
|
|
|||
|
|
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:
|
|||
|
|
|
|||
|
|
(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.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
The main purpose in using this variation in the original coordinate scheme is to simplify the task of programming because it is no longer necessary to identify statically determinate and redundant constraints. This identification requires judgement on the part of the analyst and thus requires a decision-making function which is not easily programmed.
|
|||
|
|
|
|||
|
|
However, experience has shown that there are structural systems for which the stiffness matrix is ill-conditioned when related to a coordinate system composed entirely of deformation modes as provided under the Craig and Bampton scheme. By use of rigid body modes, this matrix conditioning is much improved in some cases.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
The essential feature of the branch mode method is the grouping of components into subsets called branches. No rules are established for defining these branches; hence, their construction is left to the judgement and experience of the analyst. A system of branch modes must be complete in the sense that it must be possible to express any motion of the system as a sum of motions in the various branch modes.
|
|||
|
|
|
|||
|
|
Normally, although not necessarily, the coordinate system for this structure consists of the normal modes of vibration of the various separate branches. To determine these vibration modes, the boundary conditions on the branch must be specified. This means that the opposite ends of the components joined (i.e., the connections not included in the interface between the components joined in the branch) must be subjected to specified boundary conditions. Gladwell suggests that these boundaries may be either fixed or free depending upon the nature of the system and the judgement of the analyst. In fact, Gladwell suggests that branch modes may be defined by letting one of the components in the branch be either fixed or to vibrate as a rigid body. Hence, there are a number of options available to the analyst.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
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.
|
|||
|
|
|
|||
|
|
# 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.
|
|||
|
|
|
|||
|
|
The criteria previously discussed are considered in sequence :
|
|||
|
|
|
|||
|
|
(1) Natural Frequency and Mode Shape Convergence Method A. Numerous small problems have been solved by this technique with no difficulties experienced in convergence. The largest known problem on which data on convergence are available is a plane frame with up to 56-degrees of freedom [13]. Frequencies and modal vector directions are shown to converge uniformly and rapidly with added degrees of freedom. A tentative rule to be drawn from this example may be expressed in the following form :
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\scriptstyle n=C r
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$\pmb{n}\!=\!$ mode number up to which satisfactory accuracies are attained, $r\!=$ number of dynamic degrees of freedom, $c{=}\mathbf{a}$ constant varying generally in the range 0.65 to 0.80.
|
|||
|
|
|
|||
|
|
Method $\pmb{B}$ . To this point, only one small problem is available [9]. This is insuffcient to completely judge convergence. One can note only that in the example treated involving four-degrees of freedom, the first three mode frequencies were very accurate.
|
|||
|
|
|
|||
|
|
Method C. Two examples are treated in [10] in which mode frequencies are obtained with varying numbers of degrees of freedom. Each of the problems involves simple systems with only two components, but solutions are carried out up to 32-degrees offreedom.
|
|||
|
|
|
|||
|
|
With interface loading included in the analysis, convergence is good, and methods A and C are comparable in this respect. Both exhibit the desirable characteristic of producing accurate modes up to some critical mode number beyond which the results become abruptly very bad. Therefore, there is no problem in differentiating 'good' and ‘bad' modes.
|
|||
|
|
|
|||
|
|
Method $D$ . The use of free-free component modes with interface loading was studied for two problems in [1o]. While the convergence was poor when only a few free-free modes were used, their examples show that if a “large' number of modes are used, system natural frequencies up to perhaps one-half the total number of freefree modes were accurate within two percent. While some questions as to what is a large’ enough number of free-free modes need further study ,this method appears to be slightly less desirable than A and C.
|
|||
|
|
|
|||
|
|
Hou [12] shows no noticeable convergence problems. However, he uses and presents guides for selecting desirable free-free modes.
|
|||
|
|
|
|||
|
|
(2) Isolation of contractual components
|
|||
|
|
|
|||
|
|
Method A. This method involves the development of mass and stiffness matrices, as well as applied force vectors (if they exist) for each component independently. A single stage operation brings these together simultaneously to synthesize the system matrices and forces. This synthesis is completely mathematical, and its completion does not require information concerning physical nature or properties of thecomponents.
|
|||
|
|
|
|||
|
|
The information from component analyses to be transferred for system analysis is for each component:
|
|||
|
|
|
|||
|
|
$\bullet$ Mass matrix,
|
|||
|
|
$\bullet$ Stiffness matrix,
|
|||
|
|
$\bullet$ Matrix $\beta$ which relates component interface displacements to system coordinate displacements.
|
|||
|
|
|
|||
|
|
The isolation of components is not infuenced by the system topology.
|
|||
|
|
|
|||
|
|
Method $B$ This method requires bringing together pairs, or triplets, of components to form branch modes. For each branch mode, normally an eigenvalue analysis is required. To this branch would be added another component and so on in sequential manner if the components are topologically arranged in a chain. The topological character of the system has a considerable influence on the transfer of information required.
|
|||
|
|
|
|||
|
|
The branch mode method requires, by its very nature, quite a high degree of information exchange among components.
|
|||
|
|
|
|||
|
|
Method C. Since this method is basically a branch mode method, the comments underB above apply.
|
|||
|
|
|
|||
|
|
Method $D$ . The method of [13] and [12] is comparable to method A with respect to component isolation. This component mode concept does not require any information about other components. The free-free mode method of [1o], does require some small amount of information about the other components (stiffness and mass matrices).
|
|||
|
|
|
|||
|
|
# (3) Convergence of local stresses
|
|||
|
|
|
|||
|
|
Method A. No information is given in [4] with regard to the convergence of stresses, or forces, at the interfaces. Unpublished work using this method has shown that for the example treated in the above reference, force equilibrium at the interfaces is satisfied with but small errors in those conditions under which the frequencies have converged. This tentatively suggests that stress convergence follows the same course as frequency convergence.
|
|||
|
|
|
|||
|
|
Method B. No information is given in the technical literature. There is no reason to believe that any difficulty would be experienced in this regard provided the branch mode coordinates are suitably chosen.
|
|||
|
|
|
|||
|
|
Method C. Again, no specific information on stresses is contained in any published material. The comments under B above also apply to this method.
|
|||
|
|
|
|||
|
|
Method D. Convergence to accurate stresses in all free-free methods is expected to be slow and, in fact, stresses may not converge within satisfactory limits at the interface between components. The reason for this is that many free-free modes may have to be combined to in any way create ‘near-continuous’ displacement gradients at component interfaces.
|
|||
|
|
|
|||
|
|
(4) Repetitive test for convergence
|
|||
|
|
|
|||
|
|
Method A. In this method, it is relatively easy to add degrees of freedom to an already existing solution. The information to be added to the system mass and stiffness matrices is simply additive, and it is not necessary to alter the system in any way or to recompute the existing matrices. The added coordinates are found in the residual supply (normally, this is available) of component normal modes. Augmentation of the mass matrix along its diagonal consists of adding normalized component masses (usually these are normalized to unit values). The added terms to the principal diagonal of the system stiffness matrix are, then, the squares of the component modal frequencies, Coupling elements in the mass matrix must be also added, and these are the corresponding elements in the component mass matrices. A significant point to be made is the fact that new component normal modes can be added without affecting the interface continuity conditions. Hence, the transformation from component to system coordinates is unchanged.
|
|||
|
|
|
|||
|
|
Method B. In this method, convergence could be treated by adding previously deleted branch modes which normally would be available from the original branch eigensolutions. The eigensolutions would be repeated for each added component if these are in a topological chain with more component modes added in each case. For each new solution with added degrees of freedom, all the eigensolutions present in the original solution would be repeated.
|
|||
|
|
|
|||
|
|
Method C. This method closely parallels method B with respect to computations to test convergence. Hence, the comments under B apply also in this case.
|
|||
|
|
|
|||
|
|
Method $\pmb{D}$ . In both methods, augmentation of the original analysis would also use previously deleted component normal modes which normally would be available. However, since all the free-free modes define coordinate displacements at the interfaces, the addition of new modes alter the continuity equations. Therefore, the system eigenvalue equation is newly altered, but the computations do not appear complex.
|
|||
|
|
|
|||
|
|
# (5) Redundant connections
|
|||
|
|
|
|||
|
|
Method A. Redundant connections are handled routinely by this method of analysis However, when a very large number of connection points exist between components, the proportion of the matrices committed to the attachment degrees of freedom becomes very large and results in a reduction in the accuracy of the analysis.
|
|||
|
|
|
|||
|
|
Method B. The question of redundancy in the interconnection system is not con? sidered by Gladwell (1964). No redundant constraints exist in any of the examples treated.
|
|||
|
|
|
|||
|
|
Method C. Redundant connections are handled routinely by this method. The number of degrees of freedom to be used is independent of the number of connections among components.
|
|||
|
|
|
|||
|
|
Method $\boldsymbol{D}$ . In a formal sense, no difficulties are seen in handling redundancies in the interconnected system by these methods but increased interconnections may increase the possibility of ill-conditioning in the compatibility relationships.
|
|||
|
|
|
|||
|
|
# (6) Ability to treat dynamic response
|
|||
|
|
|
|||
|
|
Method A. In this method, the analysis of static response is a special case of the general procedure. Use of a part of the synthesized stiffness matrix of the system that relates to the interface coordinates leads to a response analysis identical with that normally employed for static analysis by the stiffness method. Therefore, accurate predictions to low frequency response are obtained.
|
|||
|
|
|
|||
|
|
It is appropriate here to point out that if large, concentrated quasi-static forces are applied to a component, then special modes representing the static response of the component to those forces relative to its constraints may be added to the coordinate system in order to enhance accuracy. However, this must be done with caution because such modes can make the stiffness and mass matrices ill-conditioned.
|
|||
|
|
|
|||
|
|
Method B. No information exists on this point relative to branch mode methods. It is conjectured that much would depend on how the branches are defined relative to the disposition of the applied forces.
|
|||
|
|
|
|||
|
|
Method $C_{\mathrm{~\,~}}$ . Although this is, in effect, a branch mode method, it is believed that the provision for dealing with interface loading should improve the accuracy of response to quasi-static forces.
|
|||
|
|
|
|||
|
|
Method $\textbf{\emph{D}}$ .For reasons already stated that relate to the stresses at interfaces, the methods are likely to be unsuitable in dealing with quasi-static responses where the distribution of applied force is entirely arbitrary.
|
|||
|
|
|
|||
|
|
# (7) Matrix conditioning
|
|||
|
|
|
|||
|
|
Method A. Under conditions in which a stiffness matrix used in static analysis becomes ill-conditioned the stiffness matrix used in this method will be subject to the same problem. However, examples can be cited in which the use of component rigid body modes eliminate the difficulty.
|
|||
|
|
|
|||
|
|
Method B. No study has been made of this method.
|
|||
|
|
|
|||
|
|
Method C. As mentioned with respect to highly redundant connections, this method can avoid the use of large matrices and perhaps ill-conditioning. However, because the rigid body modes are not separated out in the stiffness matrix, problems may existhere.
|
|||
|
|
|
|||
|
|
Method D. Hou [12] and Hart, et al [13] note that the number of free-free normal coordinates must always exceed the number of connection coordinates. This can be very troublesome for some systems because a matrix composed of eigenvectors which may be large must be inverted.
|
|||
|
|
|
|||
|
|
Benfield and Hruda [10] do not use the same condensation scheme as above and, hence, do not need to invert the reduced eigenvector matrix.
|
|||
|
|
|
|||
|
|
# (8) Use of experimental analysis
|
|||
|
|
|
|||
|
|
Method A. The matrices and component modes used in this method can be verified, for the most part, by tests which are confined to each component independently. The component stiffness matrix is verified by a combination of static tests in which responses to displacements at the interconnections are determined and by fixed constraint modal tests which verify the component natural frequencies. The stiffness matrix is uncoupled with respect to the combined static-dynamic elements.
|
|||
|
|
|
|||
|
|
The mass matrix can be partially verified experimentally as follows. The rigid body mass inertia matrix is checked by: (1) weighing the component; (2) finding its center of gravity; and (3) determining its principal moments of inertia by tests such as the pendulum test. The modal masses are verified by the fixed constraint modaltests.
|
|||
|
|
|
|||
|
|
Since the mass matrix is coupled, no procedure for verifying the coupled mass terms is easily found.
|
|||
|
|
|
|||
|
|
Methods B and C. The branch modes in these methods can be verified experimentally by standard modal survey tests. In all cases, appropriate boundary conditions, which may be either fixed or free depending on how the branch is defined, must be observed.
|
|||
|
|
|
|||
|
|
These methods require that all of the components that form the branch be brought together for the tests.
|
|||
|
|
|
|||
|
|
Method $\boldsymbol{D}$ . In this method [12], experimental verifcation of the unconstrained, non-interface loaded free-free normal modes would be made by standard modal survey tests on each component. However, a procedure for the experimental verification of attachment loaded normal modes [10] is not known.
|
|||
|
|
|
|||
|
|
From the foregoing evaluation, it appears that two methods are particularly useful when all factors are weighed. These are:
|
|||
|
|
|
|||
|
|
Method A. Component Mode Synthesis,
|
|||
|
|
|
|||
|
|
Method D. Coupled-Free-Free Component Modes with Interface Loading.
|
|||
|
|
|
|||
|
|
In the following sections these two methods are treated in detail and general flow charts are presented for guidance in programming. These methods are now identified as follows:
|
|||
|
|
|
|||
|
|
Method A--CM I Modal Synthesis with Fixed Attachment Modes.
|
|||
|
|
|
|||
|
|
Method D-CM II Modal Synthesis with Free Attachment Modes.
|
|||
|
|
|
|||
|
|
# THE MODAL SYNTHESIS METHOD WITH FIXED ATTACHMENT MODES
|
|||
|
|
|
|||
|
|
# Introduction
|
|||
|
|
|
|||
|
|
The logic behind the development of the modal synthesis method with fixed attachment modes (component mode method) can best be explained by the use of an example structure as shown in Fig. 1. The simplest analysis of this frame would put the node points at each intersection. The stifness matrix would then be developed by the standard method of imposing a unit deflection at each nodal coordinate point and determining the forces required to maintain zero deflections at all the other nodal coordinates. The resulting mass and stifness matrices using these nodal coordinates are adequate to be used in a simple dynamic analysis of the frame.
|
|||
|
|
|
|||
|
|
|
|||
|
|
FIG. 1. Dynamic analysis of a simple frame.
|
|||
|
|
|
|||
|
|
However, if more sophistication of the mathematical model is desired, the simple model can be improved by either increasing the number of node points on the structure or by making some assumption as to the dynamic behavior of the interior of the components and adding to the model by the inclusion of these interior characteristics. This means that if uniform beams connect the attachment points, theoretically derived mode shapes for the continuous beams could be used to describe the dynamic behavior of the interior portions of the structure. Moreover, the number of these mode shapes used could be based on the limitations of the computer or on the basis of expected participation in the fundamental modes of vibration of the composite frame. Nevertheless, it is easy to conclude that the introduction of, say, three mode shapes theoretically derived for a continuous beam should provide better convergence than the addition of one three-d.o.f. node point at the center of thebeam.
|
|||
|
|
|
|||
|
|
If the frame has rigid attachment points, the end points of the interior modes can show no deflection or rotation; hence, the selection of fixed-fixed interior modes. Also, if the proper coordinate systems are selected, the stiffness matrix for the composite structure will show uncoupling between the attachment stiffnesses and the internal stiffnesses. In fact, the internal stiffnesses can be expressed as generalized stiffnesses for each of the internal component modes, and the result is a series of diagonal matrices made up of the frequencies squared (generalized stiffnesses) of each of the internal modes of the components.
|
|||
|
|
|
|||
|
|
The dynamic equation for the system is shown below. This equation will be derived in the following Sections.
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
$[M^{B B}]$ and $[K^{B B}]$ are the mass and stiffness matrices for the simple dynamic model shown in Fig. 1. These never change, but the system model is improved with the progressive addition of $\lbrack M^{B1}],$ $[\omega^{2}{}_{N1}],$ etc., to better describe the internal dynamic behavior of the components. It is evident that the method of addition of the internal component characteristics (modes) will have a direct bearing on the accuracy of the results. Hence, a modal selection criteria is also of great importance to the development of an adequate component mode model.
|
|||
|
|
|
|||
|
|
# Substructure development
|
|||
|
|
|
|||
|
|
The dynamic equations for lumped-mass model of a single component without damping can be written as
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\lbrack m]\{\ddot{u}\}+[k]\{u\}=\{F\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
If the rows and columns of the matrices and vectors are rearranged such that the attachment (boundary) coordinates of the component are first, followed by the interior coordinates. Then,
|
|||
|
|
|
|||
|
|
$$\left[\stackrel{\lbrack m^{B B}]}{\lbrack0]}\ \stackrel{[0]}{\lbrack}m^{I I}]\right]\left\{\{\ddot{u}^{B}\}\right\}+\left[\stackrel{\lbrack k^{B B}]}{\lbrack{k^{I B}}][k^{I I}]}\right]\left\{\{u^{B}\}\right\}=\left\{\{F^{B}\}\right\}$$
|
|||
|
|
|
|||
|
|
A set of interior modes for the substructure can be obtained from (3) when the attachment coordinates are held fixed; i.e., (2) is reduced to
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{m^{I I}\}\{\ddot{u}^{I}\}+[k^{I I}]\{u^{I}\}=\{0\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Not all of the eigenvectors obtainable from (4) are necessary. High frequency modes will not contribute materially to the sought after mode shapes of the combined structure. Therefore, a cutoff, on the basis of frequency, will permit the use of only the first r normal modes. Let $[\Phi^{N}]$ represent the truncated set of interior normal modes. $\{\Phi^{N}\}$ will be a matrix having $\scriptstyle{n_{r}}$ columns and $n_{N}$ represents the number of generalized interior degrees of freedom in the substructure.
|
|||
|
|
|
|||
|
|
The objective of this analysis is to be able to reduce the number of degrees of freedom being treated in the substructure so that several substructures can be combined without completely over-running the capacity of the computer. Hence, if a Rayleigh-Ritz method is employed, the mass and stiffness matrices of the substructure can be written in a reduced form. The kinetic and strain energies of the substructure can be written as follows
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\begin{array}{r l}&{{\sf K}.\ {\sf E}.\ {=}\,{\xi}\{{\dot{u}}\}^{T}[m]\{{\dot{u}}\}}\\ &{{\sf S}.\ {\sf E}.\ {=}\frac{1}{2}\{u\}^{T}[k]\{{u}\}}\end{array}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Let $[\Phi]$ be a selected group of mode shapes. Then,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u\}\!=\![\phi]\{p\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where $\{p\}$ is a set of generalized coordinates related to the selected mode shapes. To establish $[\Phi]$ first requires that the attachment (boundary) coordinates of $\{p\}$ be equivalent to the attachment coordinates of $\{u\}$ .Thus,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\left\{\stackrel{\{u^{B}\}}{\{\hat{u}^{I}\}}\right\}=\left[\left[\phi^{B}\right]\stackrel{\cdot}{\vdots}[\phi^{N}]\right]\left\{\stackrel{\{p^{B}\}}{\{\bar{p^{N}}\}}\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Note that the dimension of $\{u^{I}\}$ is that of all of the interior coordinates of the substructure; whereas, the dimension of $\{p^{N}\}$ is that of the selected normal modes of the interior of the substructure.
|
|||
|
|
|
|||
|
|
$[\Phi^{B}],$ . the matrix of attachment modes of the substructure, is obtained by setting all interior forces of the substructure in equation (3) equal to zero. Then,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[k^{I B}]\{u^{B}\}+[k^{I I}]^{-1}\,\{u^{I}\}=\{0\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\bigl\{u^{I}\bigr\}=-[k^{I I}]^{-1}[k^{I B}]\bigl\{u^{B}\bigr\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
If displacements in the two coordinate systems $\{u\}$ and $\{p\}$ are equivalent at the boundary, then
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[\phi^{B}]\!=\!\!\left[\!\!\begin{array}{c}{{[l]}}\\ {{\bar{[\Phi}^{\bar{c}}\bar{]}}}\end{array}\!\!\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[\Phi^{C}]\!=-[k^{I I}]^{-1}[k^{I B}]\dag
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Since the interior modes have no displacements at the attachment points
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[\phi^{N}]\!=\!\!\left[\!\!\begin{array}{c}{{{[0]}}}\\ {{{\bar{[\phi^{N}]}}}}\end{array}\!\!\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
combining,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{{\pmb u}\}\!=\![\phi]\{{\pmb p}\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\left\{\stackrel{u^{B}}{u^{i}}\right\}=\left[\stackrel{[I]}{\dot{\Omega}^{\bar{c}}}\right]\stackrel{\dagger}{\vdots}\frac{[0]}{[\bar{\Phi}^{\bar{N}}]}\right]\left\{\stackrel{p^{B}}{p^{\bar{N}}}\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Substitute (14) into (5) and (6) to obtain the new expressions for kinetic and potential energy. After application of Lagrange's equations, the new mass and stiffness matrices in the $\{p\}$ coordinateswillbe
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\begin{array}{c}{{{[\overline{{{m}}}]\!=\!\!\left[\!\!\begin{array}{c c}{{\overline{{{m}}}^{B B}}}&{{\overline{{{m}}}^{B N}}}\\ {{\overline{{{m}}}^{N B}}}&{{\overline{{{m}}}^{N N}}}\end{array}\!\!\right]\!=\!\!\left[\!\!\begin{array}{c c}{{\left[m^{B B}\!+\!(\Phi^{C})^{T}m^{I I}\Phi^{C}\right]\!\!\!}}&{{\left[(\Phi^{C})^{T}m^{I I}\Phi^{N}\right]\!\!\!}}\\ {{\left[(\Phi^{N})^{T}m^{I I}\Phi^{C}\right]\!\!\!}}&{{\left[I\right]\!\!\!}}\end{array}\!\!\!\right]}}\\ {{[\!\!\!}}\\ {{[\!\!\!}}\\ {{\bar{k}}]\!\!=\!\!\left[\!\!\begin{array}{c c}{{\overline{{{k}}}^{B B}}}&{{\bar{k}^{B N}}}\\ {{\bar{k}^{N B}}}&{{\bar{k}^{N N}}}\end{array}\!\!\!\right]\!=\!\!\left[\!\!\begin{array}{c c}{{\left[k^{B B}\!+\!k^{B I}\Phi^{C}\right]\!\!\!}}&{{\left[0\right]\!\!\!}}\\ {{\left[0\right]\!\!\!}}&{{\left[\omega_{n}^{\ 2}\right]\!\!\!}}\end{array}\!\!\!\right]}}\end{array}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Two characteristics should be noted about the component coordinate system used to develop the mass and stiffness matrices [m] and $[\bar{k}]$ : (1) no modification has been made to the attachment coordinates, and the attachment coordinates of the substructure therefore remain compatible with all other substructures at the attachment points; and (2) the generalized coordinates describing the interior modes of the substructure are unique to the substructure and do not require compatibility with other substructures.
|
|||
|
|
|
|||
|
|
Synthesis of the system mass and stiffness matrices The partitioning of the system stiffness matrix is shown below:
|
|||
|
|
|
|||
|
|
There are no boundary and interior coupling terms in $[\pmb{K}]$ because these coupling terms are zero in the modified stiffness matrix of the substructure. The matrices $[\omega_{N1}^{2}]$ and $[\omega_{N2}^{2}]$ represent the frequencies of the selected interior normal modes of substructures 1 and 2 and are equivalent to $[\omega_{N}{}^{2}]$ in [16]. $[K^{B||}]$ is the stiffness matrix for the attachment nodes and is built-up from the attachment stiffness of the substructures, equation (16).
|
|||
|
|
|
|||
|
|
The system mass matrix has coupling between the attachment coordinates and the interior coordinates:
|
|||
|
|
|
|||
|
|
Both $[M^{B B}]$ and $[K^{B B}]$ should be developed with a sequential numbering of the attachment nodes with each node having, initially, six-degrees of freedom.
|
|||
|
|
|
|||
|
|
$[M^{B B}]$ is symmetric and is constructed in the same manner as $[K^{B B}]$ $[M_{\,\,\,1}^{B N}],\,[M_{\,\,\,2}^{B N}]$ etc.
|
|||
|
|
are not symmetric.
|
|||
|
|
|
|||
|
|
Removal of rigid body modes
|
|||
|
|
|
|||
|
|
$[M]$ and $[K]$ as defined in the previous section can both be singular because:
|
|||
|
|
|
|||
|
|
(1) constraints at the attachment nodes have not been treated;
|
|||
|
|
|
|||
|
|
(2) the stiffness matrix $[K^{B B}]$ for the attachments has singularities due to rigid body modes; and
|
|||
|
|
|
|||
|
|
(3) the mass matrix may have singularities because the analyst, in developing the substructure mass matrix, may have chosen to neglect moments of inertia. Zero masses (moments of inertia) in the substructure mass matrix do not necessarily lead to zero mass elements in the system mass matrix (but the singularities can be carried over) leading to serious problems when an attempt is made to invert the system mass matrix.
|
|||
|
|
|
|||
|
|
The removal of rows and columns to treat constraints is standard, but both matrices may remain singular because of the problems of zero mass or rigid body modes. The eigenvalue problem requires either the product [M- 1 K] or [K- 1 M] to obtain the modal characteristics. If masses are required to be non-zero or if a transformation is made to identify and remove mass singularities, the [M-1 K] approach can be used. However, [M} is a full matrix and, therefore, its inversion does not offer the advantage that it does when [M] is diagonal (as is the case in most structure problems).
|
|||
|
|
|
|||
|
|
On the other hand, rigid body modes can be systematically removed from $[K^{B B}]$ thus enabling a reduction of coordinates and inversion of [K]. This enables the use of zero inertias in the substructure mass matrices and has the added benefit of inverting the simpler (in this case) system stiffness matrix.
|
|||
|
|
|
|||
|
|
As examination of the system stiffness matrix $[K]$ will show that the rigid body modes (i.e., singularities) are in [KeB] but not in the series of diagonal matrices representing the internal fixed attachment modes. Moreover, there is no coupling between [K''] and the interior which would complicate the removal of the rigid body modes.
|
|||
|
|
|
|||
|
|
Since $[K^{B B}]$ is built-up from a systematic treatment of the attachment nodes, each having six-degrees of freedom, a standard matrix can be developed which will sweep out the rigid body modes. From kinematics, recall that the velocity of a point on a translating rotating rigid structure can be written in terms of the velocity of another point and the angular velocity. This is also true for displacements as long as the angular displacement is restricted to being very small. Therefore,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{U\}\!=\!\{U_{0}\}\!+\!\{\omega_{0}\}\times\{{r}\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{\omega\}\!=\!\{\omega_{0}\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$\{U_{0}\}$ displacement in global coordinates of reference point zerof,
|
|||
|
|
|
|||
|
|
$\{\omega_{0}\}$ rotation about point zero, $\{r\}$ vector distance between the two locations on the structure.
|
|||
|
|
|
|||
|
|
This can be rewritten in matrix form as follows:
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\left\{U_{i}\right\}=\left\{\begin{array}{l}{U_{x}}\\ {U_{y}}\\ {U_{z}}\\ {\cdots}\\ {\omega_{x}}\\ {\omega_{y}}\\ {\omega_{z}}\end{array}\right\}=\left[\begin{array}{l l l l l l}{1}&{0}&{0}&{\left\lfloor\begin{array}{l l l l l}{1}&{0}&{z}&{-y}\\ {0}&{1}&{0}&{\left\lfloor\begin{array}{l l l l l}{0}&{0}&{z}&{-y}\\ {0}&{0}&{1}&{\left\lfloor\begin{array}{l l l l l}{0}&{0}&{x}\\ {1}&{y}&{-x}&{0}\\ {-\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {0}&{0}&{\left\lfloor\begin{array}{l l l l l}{1}&{0}&{0}&{0}\\ {\vdots}&{1}&{0}&{1}&{0}\\ {0}&{0}&{0};}&{0}&{1}&{0}\\ {0}&{0}&{0};}&{0}&{0}&{1}\end{array}\right.}\end{array}\right]\right.\left[\begin{array}{l}{U_{x_{0}}}\\ {U_{y_{0}}}\\ {U_{z_{0}}}\\ {\omega_{x_{0}}}\\ {\omega_{y_{0}}}\\ {\omega_{y_{0}}}\\ {\omega_{z^{0}}}\end{array}\right]=\left[D_{i}\right]\left\{U_{0}\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$\{U_{i}\}$ is the vector describing the three translations and rotations at node i.
|
|||
|
|
|
|||
|
|
$\{U_{0}\}$ is the vector describing the three translations and rotations of a reference point, 0. $[D_{i}]$ is the transformation matrix including distances $x_{i},\,y_{i}$ and $z_{i}$ from point 0 to node i.
|
|||
|
|
|
|||
|
|
Using $[D]_{:}$ . a coordinate transformation can be constructed which will identify six rigid body modes:
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\left\{\begin{array}{c}{{\{U_{1}\}}}\\ {{\{U_{2}\}}}\\ {{\{U_{3}\}}}\\ {{\cdot}}\\ {{\cdot}}\end{array}\right\}=\left[\begin{array}{c c c c}{{[D_{1}]}}&{{[0]}}&{{[0]}}&{{.}}\\ {{[D_{2}]}}&{{[I]}}&{{[0]}}&{{.}}\\ {{[D_{3}]}}&{{[0]}}&{{[I]}}&{{.}}\\ {{.}}&{{.}}&{{.}}&{{.}}\end{array}\right]\left(\begin{array}{c}{{\{U_{0}\}}}\\ {{\{U_{2}\}}}\\ {{\{U_{3}\}}}\\ {{.}}\\ {{.}}\end{array}\right){=}[D]\{U\},
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
The vectors $\{U_{1}\},\,\{U_{2}\}$ , etc., on the left side of equation (22) represent total displacements of the nodes relative to a fixed frame of reference. The vectors $\{U_{o}\},\{U_{2}\}$ , etc., on the right side represent displacements relative to a moving frame of reference embedded in the rigid body. Vector $\{U_{0}\}$ represents the three rigid body translations and the three rigid body rotations (about point O). The dimension of this transformation will be the same as $[K^{B B}]$ since only the attachment nodes are involved. Pre and post multiplication of $[K^{B B}]$ by $[D]^{\tau}$ and $[D]$ , respectively, will produce zeros in the first six rows and six columns of the product (if $[K^{B B}]$ contains six rigid body degrees of freedom).
|
|||
|
|
|
|||
|
|
Assume at this point in the development that $[K^{B B}]$ does have six rigid body degrees of freedom (completely unconstrained) and carry out the multiplications:
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[M_{D}]\!=\!\!\left[\!\!\begin{array}{c c}{{\!\!\!\left[D^{T}\right]\!\!}}&{{\!\!\!\left[0\right]\!\!}}\\ {{\!\!\!\left[0\right]\!\!}}&{{\!\!\!\left[I\right]\!\!}}\end{array}\!\!\!\right]\!\!\left[\!\!\begin{array}{c c}{{\!\!\!\left[M^{B B}\right]\!\!}}&{{\!\!\!\left[M^{B N}\right]\!\!}}\\ {{\!\!\!\left[M^{B N}\right]\!\!}}&{{\!\!\!\left[I\right]\!\!}}\end{array}\!\!\!\right]\!\!\left[\!\!\begin{array}{c c}{{\!\!\!\left[D\right]\!\!}}&{{\!\!\!\left[0\right]\!\!}}\\ {{\!\!\!\left[0\right]\!\!}}&{{\!\!\!\left[I\right]\!\!}}\end{array}\!\!\!\right]\!\!=\!\!\!\left[\!\!\begin{array}{c c}{{\!\!\!\left[M^{B B}\right]\!\!}}&{{\!\!\!\left[M^{B N}\right]\!\!\!}}\\ {{\!\!\!\left[M^{N B}\!\!\!}}&{{\!\!\!\left[I\right]\!\!\!}}\end{array}\!\!\!\right]\!\!\right]}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[K_{D}]\!=\!\!\left[\!\!\begin{array}{c c c}{{\!\!\Gamma\!\!}[D^{T}]}&{{\!\!\![0]\!\!}}\\ {{\!\![0]\!\!}}&{{\!\!\![I]\!\!}}\end{array}\!\!\!\right]\!\!\!\left[\!\!\begin{array}{c c c}{{\!\![K^{B B}]}}&{{\!\!\![0]\!\!}}\\ {{\!\![0]\!\!}}&{{\!\!\![\omega_{N}^{2}]\!\!\!}}\end{array}\!\!\!\right]\!\!\!\left[\!\!\begin{array}{c c c}{{\!\![D]\!\!}}&{{\!\!\![0]\!\!}}\\ {{\!\![0]\!\!}}&{{\!\!\![I]\!\!}}\end{array}\!\!\!\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Repartition $[K_{D}]$ and $\left[{M_{p}}\right]$ to separate the six rigid body modes from the elastic degrees of freedom:
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\left[\!\!\left[M_{d11}\!\!\right]\!\!\right]\;\;\;\left[M_{d12}\!\!\right]\!\!\right]\!\!\left\{\{\dot{U}_{d1}\}\right\}\!\!\!\right\}+\!\!\left[\!\!\left[0\right]\!\!\!\right]\;\;\;\left[0\right]\!\!\!\right]\!\!\left\{\{U_{d1}\}\right\}\!\!\!\right=\!\!\!\left\{\{0\}\right\}\!\!\!\!
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
The coordinates, $\{U_{d2}\}$ , represent all of the elastic degrees of freedom of the composite structural system.
|
|||
|
|
|
|||
|
|
Separating the first equation from (26), $\{U_{d}\}$ can be solved in terms of $\{U_{d2}\}$ resulting in the equation
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\lbrack M_{d22}-M_{d21}M_{d11}^{-1}M_{d12}]\{\dot{U}_{d2}\}+[K_{d22}]\{U_{d2}\}=0.
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
The only inversion involved in this process is that of $[M_{d11}]$ which is always positive definite. $[M_{d1\,1}],$ a $\phantom{0}{6\times6}$ matrix in this case, represents the masses or inertias associated with the six rigid body degrees of freedom. Hence, the first three diagonal elements each equal the total mass of the system, and the next three equal the three system moments of inertia about the origin. Off-diagonal terms represent products of inertia and coupling due to the origin not necessarily being at the center of mass. The characteristics of $[M_{d11}]$ thereforeoffera check on the accuracy of the development of the model to this point.
|
|||
|
|
|
|||
|
|
The eigenvalues of the composite structure are computed from the mass and stiffness matrices in (27). $[K_{d22}]$ is non-singular and contains off-diagonal terms only in the $[K(_{\_d}^{B B})]$ portion. The inversion of $[K_{22}]$ is, therefore, straightforward, and eigenvalues can be obtained even though singularities may exist in the mass matrix.
|
|||
|
|
|
|||
|
|
When the system is constrained and has fewer than six rigid body degrees of freedom, the procedure leading to the reduced, non-singular stiffness matrix must be modified. In the developmentof $[K^{B B}],[M^{B B}],[M^{B N}],$ . etc., six-degrees of freedom were permitted at each node even though motion of any of the degrees of freedom may have been constrained. This resulted in zero rows and columns being included to represent these constrained coordinates These zeros introduce singularities, of course, but are not removed first because rigid body mode removal is more systematic if the $D_{i}^{\,\circ}\mathbf{s}$ are maintained as $6\times6$ matrices. The procedure, therefore, when there is a mix of rigid body modes and constraints is to:
|
|||
|
|
|
|||
|
|
(1) remove the rigid body modes from the mass and stiffness matrices by pre and post multiplication by $\boldsymbol{[D]^{T}}$ and $\{D\},$ respectively;
|
|||
|
|
(2) reduced the size of the mass and stiffness matrices by removing rows and columns associated with constrained coordinates; and
|
|||
|
|
(3) follow the procedure of equations (26) and (27) to obtain a reduced dynamic equation with a nonsingular stiffness matrix.
|
|||
|
|
|
|||
|
|
The only differences between the procedure for six rigid body d.o.f. and fewer rigid body d.o.f. are the dimensions of $[M_{d22}-M_{d21}M_{d11}^{-1}M_{d12}]$ and the formation of $[\pmb{D}]$ .The modified matrix will be symmetrical and have the same dimension as the number of rigid body degrees of freedom.
|
|||
|
|
|
|||
|
|
# MODAL SYNTHESIS METHOD WITH FREE ATTACHMENT MODES
|
|||
|
|
|
|||
|
|
This section presents the technical background for a component mode synthesis procedure which is based upon Rayleigh-Ritz at the system level and utilizes deflection functions which are free-free component modes. In addition to the system level Rayleigh-Ritz the method differs from standard free-free component mode analyses [11, 12, 13] in that the deflection functions obtained from the free-free eigenvalue analysis of each component include approximate effects of all attaching component stiffness and inertial characteristics. The procedure presented parallls in many respects the development presented by Benfield and Hruda [1o].
|
|||
|
|
|
|||
|
|
First we shall describe how to obtain the displacement functions associated with a set of substructure (component) generalized coordinates. The displacement functions are obtained by performing a free-free eigenvalue analysis of a component's modifed mass and stiffness matrix. Therefore, we may visualize each displacement function as a substructure free-free mode shape and each substructure generalized coordinate as a free-free normal mode coordinate. It should be kept in mind that the free-free modes associated with this substructure are not free-free modes in the classical sense. Instead, they take into account, in an approximate manner, the stiffness and inertia forces applied at the attachments due to connecting substructures. Figure 3 shows a general structural system composed of three substructures. Let us first consider the formulation of displacement functions for substructure 1. Note that this substructure is attached to two other substructures, 2 and 3. Therefore, interface loads will exist corresponding to the attachment points between substructure 1 and each of these substructures.
|
|||
|
|
|
|||
|
|
|
|||
|
|
FiG. 2. Schematic flow chart of component modes program I (fixed attachment modes).
|
|||
|
|
|
|||
|
|
|
|||
|
|
FIG. 3. General structural system.
|
|||
|
|
|
|||
|
|
Define the coordinate displacement vector for substructure 1 in the following way
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u\}_{1}\!=\!\left\{\!\!\begin{array}{l}{{\{u_{1,\,2}\}}}\\ {{\bar{\{u_{1,\,3}\}}}}\\ {{\bar{\{u_{1}^{I}\}}}}\end{array}\!\!\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$\left\{u_{1,\,2}\right\}$ attachment displacements between substructures 1 and 2
|
|||
|
|
|
|||
|
|
$\{{u_{1}},{B}_{3}\}$ attachment displacements between substructures 1 and 3
|
|||
|
|
$\left\{u_{1}^{I}\right\}$ internal displacements for substructure 1
|
|||
|
|
|
|||
|
|
Similarly, we define the associated force vector as
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{f\}_{1}=\left\{\begin{array}{l l}{\{f_{1,\,2}\}}\\ {\{\bar{f}_{1,\,3}\}}\\ {\{\bar{f}_{1,\,3}\}}\end{array}\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
The component stiffness matrix relating these forces and displacements is
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[k]_{1}=\left[\!\!\begin{array}{l l l}{\!\Gamma_{1}k_{2,\,2}^{B B}]}&{\left[\!\!1k_{2,\,3}^{B B}\!\!\right]}&{\left[\!\!1k_{2}^{B I}\!\!\right]}\\ {\!\!\left[\!\!1k_{3,\,2}^{B B}\!\!\right]}&{\left[\!\!1k_{3,\,3}^{B B}\!\!\right]}&{\left[\!\!1k_{3}^{B I}\!\!\right]}\\ {\!\!\left[\!\!1k_{2}^{I B}\!\!\right]}&{\left[\!\!1k_{3}^{I B}\!\!\right]}&{\left[\!\!k_{1}^{I I}\!\!\right]}\end{array}\!\!\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$[k_{m,\,n}^{B B}]$ stiffness matrix for substructure $"l"$ relating forces at attachments between $\psi$ and $\bullet m^{\bullet}$ due to unit displacements at attachments between $\ast l^{\star}$ and $\pmb{n}^{\star}$
|
|||
|
|
|
|||
|
|
$[\![k_{m}^{B I}]\!]\!=\![\![k_{m}^{B I}]^{T}$ stiffness matrix for substructure $\bullet\,\pmb{\gamma}$ relating forces at attachments between $\vartheta_{\ell}^{\bullet}$ and $\bullet\,\pmb{m}^{\bullet}$ due to unit internal displacements in substructure $"$
|
|||
|
|
|
|||
|
|
$[k_{l}^{t r}]$ stiffness rmatrix for substructure $\boldsymbol{\cdot}$ relating internal forces due to unit internal displacements.
|
|||
|
|
|
|||
|
|
The final force displacement stiffness equation for substructure 1 is
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{f\}_{1}{=}[k]_{1}\{u\}_{1}.
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Now, a classical free-free analysis of component 1 could use the stiffness matrix in (30) together with the corresponding mass matrix in a standard eigenvalue analysis. Such an analysis would provide mode shapes which could be used as displacement functions in the Rayleigh-Ritz procedure. However, instead of using these functions, we shall now modify (30) and its corresponding mass matrix in an approximate way to account for the elastic and inertial effects of the attaching substructures 2 and 3. In particular, we will add matrices to $[_{1}k_{2,\,2}^{B B}]$ and $[_{1}k_{3,\;3}^{B B}]$ which represent the forces at these attachment points due to the stiffness of substructures 2 and 3, respectively. These stiffness matrices which will be added are referred to as stiffness interface loads. The modified mass and stiffness matrices will then be substituted into an eigenvalue algorithm, and the resulting mode shapes will be used as displacement functions in the Rayleigh-Ritz solution of the composite structural system eigenvalueproblem.
|
|||
|
|
|
|||
|
|
Let us first consider the stiffness interface loads from substructure 2 on attachment coordinates between components 1 and 2. Using the previously defined notation (Fig. 3), wecanwrite.
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{f\}_{2}\!=\![k]_{2}\{u\}_{2}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u\}_{2}\!=\!\left\{\!\!\begin{array}{l}{{\{u_{2,\,1}^{B}\}}}\\ {{\{\bar{u}_{2,\,3}^{\overline{{B}}^{\overline{{B}}^{\overline{{-}}}}\}}}}\\ {{\{\bar{u}_{2}^{\overline{{l}}^{\overline{{l}}}}\}}}\end{array}\!\!\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{f\}_{2}\!=\!\left\{\begin{array}{l l}{{\{f_{2,\,1}^{B}\}}}\\ {{\{\bar{f}_{2,\,3}^{\bar{B}^{-}}\}}}\\ {{\{\bar{f}_{2,\,3}^{\bar{I}}\}}}\end{array}\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[k]_{2}=\left[\begin{array}{l l l}{\left[\l_{2}k_{1,\,1}^{B B}\right]}&{\left[\l_{2}k_{1,\,3}^{B B}\right]}&{\left[\l_{2}k_{1}^{B I}\right]}\\ {\left[\l_{2}k_{3,\,1}^{B B}\right]}&{\left[\l_{2}k_{3,\,3}^{B B}\right]}&{\left[\l_{2}k_{3}^{B I}\right]}\\ {\left[\l_{2}k_{1}^{I B}\right]}&{\left[\l_{2}k_{3}^{I B}\right]}&{\left[k_{2}^{I I}\right]}\end{array}\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Now, if we restrain all attachment generalized displacements of substructure 2, except those connected with substructure 1 (the one for which we are seeking attachment loads), in effect we take out the second row and column of submatrices in (33) and obtain
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[\hat{k}]_{2}\equiv\left[\stackrel{\left[_{2}k_{1,\ 1}^{B B}\right]}{\left[_{2}k_{1}^{I B}\right]}\quad\left[_{2}k_{1}^{B I}\right]\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Therefore, the corresponding force deflection equation becomes
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\left\{\stackrel{\{f_{2,1}^{B}\}}{\{\bar{f}_{2}^{\bar{I}}\}}\right\}=\left[\stackrel{\lceil_{2}k_{1,1}^{B B}\rceil}{\bigl[_{2}k_{1}^{I B}\bigr]}\quad\bigl[_{2}k_{1}^{B I}\bigr]\biggr]\left\{\stackrel{\{u_{2,1}^{B}\}}{\{\bar{u}_{2}^{\bar{I}}\}}\right\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Now, if we set $\{f_{2}^{t}\}\!=\!0$ we can obtain an expression relating $\{f_{2,\,1}^{B}\}$ and attachment boundary displacements $\left\{u_{2,\mathrm{~1~}}^{B}\right\}$ . In so doing, we arrive at
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\left\{f_{2,\,\,1}^{B}\right\}{=}[k_{1,\,\,2}^{*}]\{u_{2,\,\,1}^{B}\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[k_{1,\,2}^{*}]\!\equiv\![_{2}k_{1,\,1}^{B B}]\!-\![_{2}k_{1}^{B I}][k_{2}^{I I}]^{-1}[_{2}k_{1}^{I B}]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
In equation (36), the matrix $[k_{1}^{*},2]$ represents the attachment forces corresponding to the attachment generalized displacements between substructures 1 and 2 due to unit attachment displacements at the interface between substructures 1 and 2 from the stiffness ofsubstructure2.
|
|||
|
|
|
|||
|
|
Since $\{u_{1^{\prime}2}^{B}\}\,{=}\,\{u_{2^{\prime}1}^{B}\}$ by definition, we add $[k_{1,2}^{*}]$ to $[_{1}k_{2}^{B},_{2}]$ to obtain the final attachment interface forces due to unit attachment displacements at the interface between substructures 1 and 2. Therefore, $[k_{1^{\prime}2}^{*}]$ is the stiffness interface load between substructures 1 and 2.
|
|||
|
|
|
|||
|
|
Following the same procedure, we can obtain the stiffness interface load between substructures 1 and 3, denoted $[k_{1,3}^{*}]$
|
|||
|
|
|
|||
|
|
In so doing, we obtain
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[k_{1}^{*},3]\!=\![{_3k_{1,\,1}^{B}}]\!-\![{_3k_{1}^{B I}}][k_{3}^{I I}]^{-1}[{_3k_{1}^{I B}}]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
This stiffness interface load is then added to the stiffness matrix $[_{1}k_{3}^{B B}{}_{3}]$
|
|||
|
|
|
|||
|
|
Upon adding (37) and (38) to (30) as noted, we obtain the force deflection equation which corresponds to (31) but has interface stifness loading. Thus,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{f\}_{1}\!=\![k^{*}]_{1}\{u\}_{1}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[k^{*}]_{1}=\left[\left(\underset{\left[1\;k_{2}^{B B}\;\right]}{\left[\left(\underset{1\;k_{2}^{B B}\;\right]}{\left[1\;k_{2}^{B B}\;\right]}+\left[k_{1}^{*}\;,2\right]\right)}\;\left[1\;k_{2}^{B}\;,3\right]\;\;\;\;\;\;\;\;\;\;\;\left[1\;k_{2}^{B B}\;\right]\right)\;\left[1\;k_{3}^{B B}\;\right]\;,
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
The development of the interface mass loading proceeds in a similar manner. However, there are some important observations to be made. First, it is assumed that all the distributed masses of the system are replaced by equivalent concentrated masses; that is, the translation inertia terms are obtained by lumping concentrated masses at nodal points. Also, rotational inertia terms are obtained by calculating appropriate rotational inertia terms corresponding to nodal rotational coordinates. If the masses are handled in this way, the substructure mass matrix is diagonal. The mass matrix for substructure 1 is :
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[m]_{1}=\left[\!\!\begin{array}{c c c}{{\left[\!\!\begin{array}{c}{{{}_{1}m_{2}^{B B\!}}}{{{}_{2}}}\!\!\!}\end{array}\!\!\right]}}&{{[0]}}&{{[\!\!\begin{array}{c}{{{}[\!\!\begin{array}{c}{{0}}\!\!\!}\\ {{{}_{1}m_{3}^{B B\!}}}\!\!\!}\end{array}\!\!\!}}\\ {{\left[\!\!\begin{array}{c c c}{{{}[\!\!\begin{array}{c}{{0}}\!\!\!}}&{{{{}[\!\!\begin{array}{c}{{0}}\!\!\!}&{{{}}\!\!\!}}\end{array}\!\!\!}}}&{{{[m_{1}^{I\!}]}}}\end{array}\!\!\!\right]}}\end{array}\!\!\!\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$[m_{l}^{I I}]{=}\mathbf{m}\mathbf{a}\mathbf{s}\mathbf{s}$ matrix corresponding to internal coordinates of substructure $\bullet_{\bar{\ell}}$ $[\!_{l}m_{m}^{B B}]\!=\!\mathbf{mass}$ matrix corresponding to attachment coordinates between substructures $"l"$ and $\bullet_{m}\cdot$
|
|||
|
|
|
|||
|
|
Drawing upon the previous discussion leading up to (36), we repeat that when calculating attachment stiffness loading from substructure 2 upon substructure 1, we can express, using (35), the internal displacements of component 2 in terms of its attachment displacements; that is, we can show that
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u_{2}^{I}\}=-\[k_{2}^{I I}]^{-1}[_{2}k_{1}^{I B}]\{u_{2,\;1}^{B}\}\!=\![A]\{u_{2,\;1}^{B}\}\,.
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Now, noting that the kinetic energy of component 2 can be written as
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\mathrm{Kinetic~energy}\!=\!\mathbf{K}.\mathbf{E}.\!=\!\frac{1}{2}\!\left\{\!\hat{u}\!\right\}_{2}\![\hat{m}]_{2}\!\left\{\!\hat{\ddot{u}}\!\right\}_{2}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where the mass matrix corresponding to (34) is
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\begin{array}{c}{{{[\hat{m}]_{2}\equiv\left[\stackrel{\left[\displaystyle{\sum}\right.\left.\left.\left[\displaystyle{}\sum\right.\!m_{1}^{B B}\right]}\right.\right.\ \ \ \left.\left[\boldsymbol{0}\right]\right.\right]}}}\\ {{{[\hat{m}]\nonumber\left.\left.\left[m_{2}^{I I}\right]\right]}}}\\ {{{\{\hat{u}\}_{2}\equiv\left\{\left\{\dot{u}_{2,\ 1}^{B}\right\}\right\}\right\}}}\end{array}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
Substituting (42) into (45) and thus eliminating $\left\{\dot{u}_{2}^{I}\right\}$ , it can be shown that from (43) and (45)
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\mathbf{K}.\mathbf{E}.{=}\dag\{\dot{u}_{2,\;1}^{B}\}\bigl[m_{1,\;2}^{*}\bigr]\bigl\{\dot{u}_{2,\;1}^{B}\bigr\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[m_{1}^{*},_{2}]{=}[_{2}m_{1}^{B B}]\,{+}[A]^{T}[m_{2}^{I I}][A]\,.
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Recognizing that $[{m_{1}},^{\ast}_{2}]$ in (46) represents the inertia interface loading, we can add it to $[_{1}m_{2}^{B B}]$ in (41) and hence have our interface loading mass matrix for substructure 1 as modified by substructure 2.
|
|||
|
|
|
|||
|
|
The same procedure as outlined above for the inertia interface loading of substructure 2 upon substructure 1 is followed through for substructure 3; that is,
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\begin{array}{c}{{[m_{1}{}^{\ast},\stackrel{\ast}{3}]=[\L_{3}m_{1}^{B B}]+[B]^{T}[m_{3}^{I I}][B]}}\\ {{{}}}\\ {{[B]\equiv-[k_{3}^{I I}]^{-\L_{1}}[\L_{3}k_{1}^{I B}].}}\end{array}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Therefore, when (47) and (48) are added to (41), we obtain the modified mass matrix for substructure 1. This matrix is,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[m^{*}]_{1}=\left[\stackrel{\displaystyle(\sum_{\scriptscriptstyle1}m_{2}^{B B}]+\big[m_{1}^{\phantom{*},*},\big])}{\displaystyle[0]}\;(\big[_{\scriptscriptstyle1}m_{3}^{B B}\big]+\big[m_{1}^{\phantom{*},*},\big])\;\;\;\;[0]\atop[m_{1}^{I I}]\right]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Combining (40) and (49), we now can solve the eigenvalue problem to determine the free-- free mode shapes of substructure 1; that is,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
(-\omega_{j}^{2}[m^{*}]_{1}+[k^{*}]_{1})\{\Phi_{1}\}_{j}\!=\!\{0\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where, $\{\Phi_{1}\}_{j}$ is the jth mode shape which corresponds to the jth free-free mode of vibration. Also, $\{\Phi_{1}\}_{j}$ represents one displacement function which is to be used in the system Rayleigh-- Ritz solution.
|
|||
|
|
|
|||
|
|
The $n_{1}^{r}$ eigenvectors calculated can be written in the expanded form
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u\}_{1}\!=\!\left\{\!\!\begin{array}{l l}{\!\!\left\{u_{1,\,\,2}^{B}\right\}\!\!}\\ {\!\!\left\{u_{1,\,\,3}^{B}\right\}\!\!}\\ {\!\!\left\{\!\!\frac{\{u_{1}^{B}\}}{\{u_{1}^{I}\}}\!\!}\!}\end{array}\!\!\right\}\!=\![\{\Phi_{1}\}_{1};\!\!\{\Phi_{1}\}_{2}\!\!,\!\bar{\!\!\!\!\!\!\cdot\!\!\!\!\cdot\!\!\!\!\cdot\!\!\!\!,\!\!\!\cdot\!\!\!\{\{\Phi_{1}\}_{n_{1}^{r}}\!\!\!\}\!\!\{\{p\}\!\!\!\}}\!\!\!}\end{array}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$\{p\}_{1}=$ vector of length $n_{1}^{r}$ of distributed coordinates
|
|||
|
|
|
|||
|
|
It is convenient for later discussion to partition each free-free mode of vibration into vectors corresponding to attachment and internal coordinates; that is,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\begin{array}{r}{\{\Phi_{1}\}_{j}\!=\!\left\{\!\!\begin{array}{l}{\{\Phi_{1,\,2}^{B}\}_{j}}\\ {\bar{\{\Phi}_{1,\,3}^{B}}\bar{\{\}}_{j}}\\ {\bar{\{\Phi}_{1}^{\bar{I}}}\bar{\{\}}_{j}}\end{array}\!\!\right\}.}\end{array}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
Equation (52) is now expanded consistent with (51) and we can therefore write,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u\}_{1}\!=\!\left\{\!\!\begin{array}{c}{{\{u_{1}^{B},\,_{2}\}}}\\ {{\{u_{1}^{B},\,_{3}\}}}\\ {{\{u_{1}^{I}\}}}\end{array}\!\!\right\}\!=\!\left[\!\!\begin{array}{c}{{\!\!\left[\Phi_{1}^{B},\,_{2}\right]\!\!}}\\ {{\!\!\left[\Phi_{1}^{B},\,_{3}\right]\!\!}}\\ {{\!\!\left[\Phi_{1}^{I}\right]\!\!}}\end{array}\!\!\right]\{p\}_{1}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where $[\Phi_{1}^{B},2]$ is composed of the vectors defined in (52).
|
|||
|
|
|
|||
|
|
This completes the formulation of a set of displacement functions which describe the free vibration deformed shape of substructure I ; that is, we now have a set of $\pmb{n}_{1}^{r}$ distributed normal coordinates for component 1. The exact same procedure is now carried out for each substructure of our system. For the structure shown in Fig. 3, we therefore have three sets of displacement functions--one for each substructure.
|
|||
|
|
|
|||
|
|
Now we shall describe how the direct stiffness and mass matrices of the structure are assembled from individual substructure mass and stiffness matrices. Then, utilizing the Rayleigh-Ritz concept, the reduced system mass and stiffness matrices are formulated with the direct stiffness and mass matrices of the composite structure and the displacement functions obtained as described in the previous section. The system natural frequencies and mode shapes are then calculated from an eigenvalue analysis using these reduced matrices.
|
|||
|
|
|
|||
|
|
Equation (30) gives the stiffness matrix of substructure 1. Similar stiffness matrices are also formulated and arranged for the other substructures in the system. Now, the displacement coordinates of the system are defined and these include the internal displacement coordinates in each component plus the displacement coordinates associated with the attachment nodes. For the structure shown in Fig. 3, this would be $(n_{1}^{I}+n_{2}^{I}+n_{3}^{I}+n^{B})\!=\!n$
|
|||
|
|
|
|||
|
|
Now, the direct stiffness and mass matrices for the composite structure are formed by overlaying (or combining) the individual original non-interface loaded substructure mass and stiffness matrices such that compatibility of displacements at attachment nodes is satisfied. The result of this operation is $(n\times n)$ stiffness and mass matrices which satisfy compatibility conditions. The strain energy expression in this generalized coordinate system is
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\mathbf{S.E.}\!=\!\underset{(1\times n)}{\downarrow}\!\!\left\{u\right\}^{T}\!\underset{(n\times n)}{\subset}\!\underset{(n\times1)}{\subset}\!\!\left\{u\right\}\!}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where,
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u\}=\left\{\begin{array}{c}{\{u_{1}^{B},2\}}\\ {\{u_{1}^{-\frac{\imath}{2}},3\}}\\ {\{u_{2}^{-\frac{\imath}{2}},3\}}\\ {\{u_{1}^{\overline{{\imath}}}\}}\\ {\{\bar{u}_{2}^{\overline{{\imath}}}\}^{-\overline{{\imath}}}}\\ {\{\bar{u}_{2}^{\overline{{\imath}}}\}^{-\overline{{\imath}}}}\\ {\{\bar{u}_{3}^{-\overline{{\imath}}}\}^{-\overline{{\imath}}}}\end{array}\right\}\left\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
and $[K]$ is the direct stiffness matrix in these coordinates. Similarly, the kinetic ene rgy of the system can be written as
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\begin{array}{c}{{{\bf K.E.=}\frac{1}{2}\{\dot{u}\}^{T}\ \{M\}\ \{\dot{u}\}}}\\ {{(1\times n)(n\times n)(n\times1)}}\end{array}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
where $\{\dot{u}\}$ is the time derivative of (55), and $[M]$ is the direct mass matrix of the composite structure.
|
|||
|
|
|
|||
|
|
We will now use the displacement functions calculated in the previous section to reduce the mass and stiffness in order from $\pmb{n}$ to $\left(n_{1}^{r}\!+\!n_{2}^{r}\!+\!n_{3}^{r}\right)$ . This is accomplished by using (53) and (55); that is, we may express each term in (55) in terms of the free-free mode shapes (shape functions) of each component and the corresponding distributed coordinates. For the structure shown in Fig. 3 it follows, using the notation of (53), that
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\{u\}=\left\{\begin{array}{c}{\left\{u_{1,\,2}^{B}\right\}}\\ {\left\{u_{1,\,3}^{B}\right\}}\\ {\left\{u_{2,\,3}^{B}\right\}}\\ {\left\{u_{1}^{I}\right\}}\\ {\left\{u_{2}^{I}\right\}}\\ {\left\{u_{3}^{I}\right\}}\end{array}\right\}=\left[\begin{array}{c}{\left[\Phi_{1,\,2}^{~B}\right]^{\frac{1}{2}}\left\{\Phi_{2,\,1}^{~B}\right\}^{\frac{1}{2}}\left[\Phi_{2}^{1}\right]}\\ {\left[\Phi_{1,\,3}^{~B}\right]^{\frac{1}{2}}\left[\Phi_{2,\,3}^{~}\right]^{\frac{1}{2}}\left[\Phi_{3,\,1}^{~B}\right]}\\ {\left[\Phi_{1,\,1}^{~B}\right]^{\frac{1}{2}}\left[\Phi_{2,\,3}^{~B}\right]^{\frac{1}{2}}\left[\Phi_{3,\,2}^{~B}\right]}\\ {\left[\Phi_{1}^{I}\right]^{\frac{1}{2}}}\\ {\left[\Theta_{1}^{2}\right]^{\frac{1}{2}}}\\ {\left[0\right]^{\frac{1}{2}}}\end{array}\right]\left[\begin{array}{c}{\left[0\right]}\\ {\Phi_{3}^{I}}\\ {\left[\Phi_{3}^{1}\right]}\\ {\left[\Phi_{3}^{1}\right]}\end{array}\right]\left[\begin{array}{c}{\left\{p\right\}_{1}}\\ {\left\{p\right\}_{2}}\\ {\left\{p\right\}_{3}^{-}}\\ {\left\{p\right\}_{3}^{-}}\\ {\left\{p\right\}_{3}^{-}}\end{array}\right]\equiv[\Phi]\{p\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
If we now substitute (57) and its time derivative into (54) and (56), we obtain
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
{\bf S.E.}\!=\!\underline{{\mathfrak{k}}}\{p\}^{T}[\Phi]^{T}[K][\Phi]\{p\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
\mathbf{K.E.}\!=\!\pm\{\dot{p}\}^{T}\![\Phi]^{T}\![M][\Phi]\{\dot{p}\}
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
Hence, the problem is now transformed into generalized coordinates $\{p\}$ and thecorresponding reduced stiffness and mass matrices have been developed. The matrices are now denoted
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[K R]\!\equiv\![\Phi]^{T}[K][\Phi]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
$$
|
|||
|
|
[M R]\!\equiv\![\Phi]^{T}[M]\![\Phi]
|
|||
|
|
$$
|
|||
|
|
|
|||
|
|
and have the dimensions $(n_{1}^{r}+n_{2}^{r}+n_{3}^{r})\times(n_{1}^{r}+n_{2}^{r}+n_{3}^{r}).$
|
|||
|
|
|
|||
|
|
An eigenvalue solution for the composite structural problem is now possible using (60) and (61). The size of the eigenvalue problem has been reduced in size from $n\!=\!n^{B}\!+\!n_{1}^{\bar{I}}+n_{2}^{\bar{I}}$ $+n_{3}^{I}$ to $\widetilde{n}\!=\!n_{1}^{r}\!+\!n_{2}^{r}\!+\!n_{3}^{r}$ . The order of the reduced matrices are directly dependent upon the number of free-free displacement functions selected for each component.
|
|||
|
|
|
|||
|
|
Once the eigenvalues and eigenvectors are obtained, they are transformed back to the generalized coordinate space using (57).
|
|||
|
|
|
|||
|
|
It must be noted that the matrix operations in (60) and (61) are the largest, from a dimension standpoint, in the entire process. Since these operations involve multiplication only, the matrices can be partitioned; and the products can be developed section by section without ever having the entire mass, stiffness or deflection function matrices in core at one time. In fact, at no time in the entire development do the entire system mass or stiffness matrices have to be in core.
|
|||
|
|
|
|||
|
|
Figure 4 shows a schematic flow chart of the modal synthesis method with free attachment modes.
|
|||
|
|
|
|||
|
|
It is noted that the set of displacement functions used in (57) will not, in general exactly reproduce zero frequency system modes. Therefore the method should be used with caution with free-free systems.
|
|||
|
|
|
|||
|
|
Acknowledgment—This paper contains resuits of research performed for the National Aeronautics and Space Administration George C. Marshali Space Flight Center, under Contract NAS8-26192 and under thedirectionofDr.JohnR.Admire.
|
|||
|
|
|
|||
|
|
# REFERENCES
|
|||
|
|
|
|||
|
|
[1] B. A. HuNN, A method of calculating the space free resonant modes of an aircraft. J. Royal Aeronaut. Soc. 57, 420-422 (1953).
|
|||
|
|
[2] R. H. MAcNEAL, Vibrations of composite systems, AFOSR-TN-55-120, Office of Scientifc Research Air Research and Development Command, Technical Report No. 4, California Institute of Technology, Pasadena, California, Oct. (1954).
|
|||
|
|
[3] W. C. HuRTy, Vibrations of structural systems by component mode synthesis: J. Engng. Mech. Div. Proc. ASCE, 86, # M4, 51-69 (1960).
|
|||
|
|
[4] W. C. HuRTy, Dynamic analysis of structural systems by component mode synthesis. Technical Memorandum No. 32-530, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, January 15, 1964. Also published in part as Dynamic Analysis of Structural Systems using Component Modes, AIAA J., 3, 4, 678-685 (1965).
|
|||
|
|
[5] R. M. BAMForD, A modal combination program for dynamic analysis of structures. Technical Memorandum 33-290, Jet Propulsion Laboratory, July (1967).
|
|||
|
|
[6] R. R. CRAIG and M. C. C. BAMPToN, Coupling of substructures for dynamic analysis, AIAA J., 6, 7, 1313-1319 (1968).
|
|||
|
|
[7] R. L. BAJAN and C. C. FENG, Free vibration analysis by the modal substitution method. AAS Symposium Paper No. 68-8-1, Space Projections from the Rocky Mountain Region, Denver, Colorado, July (1968).
|
|||
|
|
[8] R. L. BAJAN, C. C. FENG and I. J. JAszLics, Vibration analysis of complex structural systems by modal substitution. Shock and Vibration Bulletin 39(3), 99-106 (1969).
|
|||
|
|
[9] G. M. L. GLADwELL, Branch mode analysis of vibrating systems. J. Sound Vibration, 1, 41-59 (1900).
|
|||
|
|
[10] W. A. BENFIELD and R. F. HRUDA, Vibration analysis of structures by component mode substitution, presented at the AIAA/ASME 11th Structures, Structural Dynamics. and Materials Conference, Denver, Colorado, April 22-24 (1970).
|
|||
|
|
[11] R. L. GoLDMAN, Vibration analysis by dynamic partitioning, AIAA J. 7, 6, 1152-1154 (1969).
|
|||
|
|
[12] S. N. Hou, Review of modal synthesis techniques and a new approach. Shock Vibration Bull. 40(4), 25-30(1969).
|
|||
|
|
[13] G. C. HART, W. C. HuRTY and J. D. CoLLINs, A Survey of Modal Synthesis Methods, presented at the SAE National Aeronautic and Space Engineering Meeting, Los Angeles, California, September 28- 30 1971.Paper No. 710783.
|