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ROY CRAIG, ENGINEERING EDUCATOR AND PIONEER CONTRIBUTOR TO COMPONENT MODE SYNTHESIS

François M. Hemez(a) Los Alamos National Laboratory Jeff Bennighof The University of Texas at Austin Daniel Kammer University of Wisconsin Roy R. Craig, Jr. The University of Texas at Austin Yung-Tseng Chung Boeing Huntington Beach Charlie Pickrel Boeing Commercial Airplanes

ABSTRACT

The contribution to Structural Dynamics of Professor Roy R. Craig, Jr. is recognized during a special honorary session of the 22nd IMAC. Professor Craig has pioneered the development of the component mode synthesis technology. These techniques are used to condense the information represented by dynamic finite element matrices on a subset of master degrees of freedom. The condensed representation can be analyzed at a fraction of the cost it would take to solve the full system of equations. Although originally developed for performing load analyses with linear models, the idea of component mode synthesis has lead to new developments and applications, for example, in domain decomposition and parallel processing, experimental modal analysis, re-analysis, and design optimization. In this paper, the concept of component mode synthesis is briefly reviewed, and recent developments are discussed. While originally developed to enable finite element-based coupled loads analysis, the concept of CMS has demonstrated a remarkable ability to evolve among the years to address new technological challenges, and adapt to novel computational environments. Examples are the synthesis of experimentally coupled models, multilevel sub-structuring, parallel computing, and design optimization.

1. INTRODUCTION

Component Mode Synthesis (CMS) is alive and well. CMS was developed in the 1960's, with a key contribution by Professor Roy R. Craig, Jr. in 1966 while he was spending a summer at Boeing Commercial Aircraft in Seattle. Today the so-called "Craig-Bampton" method is one of the most widely used techniques for reducing the size of large finite element models while retaining the modal dynamics of interest. Over the past forty years numerous free-interface, fixed-interface, and hybrid methods have been studied to establish CMS as one of the standard tools of modal analysis and finite element modeling.

(a)

Figure 1. Professor Roy R. Craig, Jr. The 22nd International Modal Analysis Conference (IMAC) recognizes the outstanding contribution of Professor Craig to Structural Dynamics in a special honorary session dedicated to his work. His contributions include two books and numerous research papers in the

Corresponding author; Mailing address: Engineering Sciences and Applications (ESA-WR), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A.; Phone: 505-663-5204; E-mail: [email protected]

fields of modal testing and analysis; frequencydomain system identification; use of modal testing to validate finite element models; and flexible multi-body dynamics and control. In the remainder, various aspects of Dr. Craig's contribution to Structural Dynamics are illustrated. A short biography is provided first in Section 2. Section 3 discusses the background and history of CMS in non-technical terms. Section 4 overviews briefly the concept of CMS. In Section 5, the discussion is made specific to the Craig-Bampton method, the applicability of which is discussed in Section 6. Bibliographical references are provided in Section 7.

Distinguished Advisor awards in 1979, 1980, and 1981, and Faculty Leadership Awards in 1987, 1990, 1994, and 1995. He was the 1996 recipient of the John Leland Atwood Award, bestowed jointly by the American Society for Engineering Education and the American Institute for Aeronautics and Astronautics upon an "outstanding aerospace engineering educator in recognition of the educator's contribution to the profession." In 2001 he received the D.J. DeMichele Award from the Society for Experimental Mechanics for "exemplary service and support of promoting the science and educational aspects of modal analysis technology." Dr. Craig's thirty-one refereed publications have been principally in the areas of Structural Dynamics analysis and testing; structural optimization; control of flexible structures; and the use of computers in engineering education. He has published two textbooks entitled Structural Dynamics -- An Introduction to Computer Methods (1981), and Mechanics of Materials (1996 and 2000). The latter has been translated into Chinese, Korean, and Portuguese. In addition to authoring textbooks, he has contributed chapters to six other books; made over fifty presentations at technical meetings; been invited Worldwide to lecture to various audiences; and published about thirtyfive technical reports. Some of the publications of Dr. Craig are listed in Section 7. Dr. Craig's industrial experience has been with the Boeing Company; the U.S. Naval Civil Engineering Laboratory; Lockheed Palo Alto Research Laboratory; Exxon Production Research Corporation; NASA Johnson Space Center; and IBM. He has been a regular consultant to NASA and the aerospace industry on Space Shuttle payload coupled loads analysis and on test-verification of payload finite element models. His latest research on system identification and model updating for space structures was supported by NASA Johnson Space Center and NASA Marshall Space Flight Center. Finally, Dr. Craig has received two citations for his contributions to the aerospace structures technology, including a NASA citation for contributions to the U.S. manned space flight program.

2. SHORT BIOGRAPHY

Roy Craig holds a B.S. degree (1956) in Civil Engineering from the University of Oklahoma, and M.S. (1958) and Ph.D. (1960) degrees in Theoretical and Applied Mechanics from the University of Illinois at UrbanaChampaign. He taught one year at the University of Illinois, and from 1961 to 2001 he taught at The University of Texas at Austin. He is currently the John J. McKetta Energy Professor Emeritus in Engineering in the Department of Aerospace Engineering and Engineering Mechanics. In addition to teaching undergraduate and graduate lecture courses, Dr. Craig taught a graduate course in Experimental Methods in Structural Dynamics, and he completed a major revision of the required ASE undergraduate Measurements and Instrumentation course. He has so far supervised twelve Ph.D. dissertations and nineteen M.S. theses. He has participated in continuing engineering studies as course coordinator and principal instructor for a four-day short course entitled "Introduction to Computer Methods in Structural Dynamics," and as an instructor in short courses entitled "Finite Elements" and "Design of Offshore Structures." Dr. Craig has received numerous teaching awards, distinguished advisor awards, and faculty leadership awards. His teaching awards include the General Dynamics Teaching Excellence Award in the College of Engineering (1966), a University of Texas Students' Association Teaching Excellence Award (1966), the ASE-EM Departmental Teaching Award in 1989, and a Tau-Beta-Pi Faculty Award in 1990. Dr. Craig received the College of Engineering

3. BACKGROUND AND HISTORY

Roy Craig has had a profound impact on the Structural Dynamics communities in both academia and industry. He has made many contributions in structural analysis, testing, and modal analysis. However, the contribution that truly sticks out is the formulation of the CraigBampton substructure representation for use in component mode synthesis or CMS. CMS is the process by which modal analysis is performed for a large finite element model by dividing the structure up into smaller substructures that are analyzed individually. Dr. Craig and his collaborators did not invent the CMS technique. That honor goes to earlier work by researchers such as Hurty. However, Craig and Bampton came up with an idea that transformed Hurty's representation into a form that was both easy to implement and apply. In Hurty's approach, degrees of freedom connecting adjacent substructures had to be divided into two sets, those that would just eliminate rigid body motion of the substructure if they were constrained, and all the others that could be considered redundant. Craig and Bampton recognized that all of the interface degrees of freedom could be treated equally. There was no need to divide up the response of the substructure due to motion of the interface into rigid and elastic components. All the degrees of freedom within a substructure could then be divided into two sets, those interior to the interface, and those at the interface to adjacent substructures. The response of the substructure could then be easily described as the sum of the elastic response of normal mode shapes computed with the interface degrees of freedom fixed, and the static response of constraint modes. Constraint modes represent the static response of the substructure to a unit deflection of one of the interface degrees of freedom with all the others held fixed. There are many different types of substructure representations that have been developed over the years. There are far too many certainly to discuss here. Several represent significant contributions to research in the field of analytical Structural Dynamics. However, the true importance of a contribution can be measured by how much it is actually used by practicing engineers on a daily basis. CMS using the Craig-Bampton substructure representation is the state-of-the-practice in

Structural Dynamic analysis all over the World. It is the default substructure representation in workhorse finite element codes such as MSC/NastranTM. It is the standard representation for U.S. Space Shuttle payloads.

4. A BRIEF OVERVIEW OF CMS

The development of the model reduction technology in the late 1960's and 1970's was motivated to a great extent by the ability to perform coupled load analysis with increasingly large and complicated numerical models. Coupled load analysis combines the dynamic models of a spacecraft and launch vehicle to form a system model that represents the coupled interaction of the two components. The system model is subjected to critical loading events including lift-off, air loads, engine ignition, engine shutdown, and pyrotechnic stage separation in order to estimate the maximum loads. Satellite manufacturers, for example, need to verify that the peak stress and acceleration loads do not damage critical components such as electronics packages. Launch integration engineers, on the other hand, need to ensure that the payload does not adversely affect the dynamics of the vehicle. Performing such analyses with complete structural models of the launch and payload systems is prohibitively expensive. Full models are therefore replaced by reduced models that provide an estimation of the loads at the interface between the launch vehicle and its payloads. By condensing the low-frequency dynamics of the entire launch vehicle to a few junction degrees of freedom, coupled load analysis can be repeated efficiently because the bulk of the computation is eliminated without sacrificing critical information about the dynamics of the coupled system. Over the past twenty years, the modeling of increasingly complex industrial structures has lead to the development of finite elements models of very large sizes. For example, aerospace structures represented by more than a million degrees of freedom are commonly analyzed. Nevertheless, many details of such large models are not critical when it comes to capturing the input-output relationship of interest between design parameters of the numerical model (inputs) and features of the predicted response (outputs). Model size is equally critical in the context of numerical optimization or finite

element model updating. It can also be a seriously limiting factor when it comes to introducing sources of non-linearity such as nonlinear materials, contact and friction between components. Reduction methods can be categorized in two families: direct reduction and component mode synthesis. To extract low-frequency eigen-solutions, it can be advantageous to reduce the eigen-value problem to a smaller size problem. Direct model reduction techniques apply to this situation and, by extension, to any situation where the size of the numerical quantities (vectors, matrices) is reduced down to a subset of generalized degrees of freedom to gain compactness and computational efficiency. The generalized degrees of freedom can be defined as physical coordinates or non-physical ones, such as modal participation factors. In the context of large projects, complicated systems such as airplanes and launchers are frequency integrated from numerous subsystems. It is increasingly common that teams from different geographical regions or countries are put in charge of developing the sub-systems. Different teams therefore develop the numerical models for each sub-system. Just like the fully integrated system is assembled from the individual sub-systems, the final numerical model must interface all the individual submodels. This could be a tedious endeavor because of mesh compatibility, software integration, and database management issues. Instead, the sub-models can be condensed on their interfaces with each other, and the final assembly only requires interfacing the junction coordinates. One then speaks of CMS, also referred to as sub-structuring. Sub-structuring currently plays a considerable role in the analysis of large finite element models. As previously mentioned, it consists in subdividing a structure in components called sub-structures or superelements that are analyzed and condensed separately while preserving junction degrees of freedom between the sub-structures. The advantage of sub-structuring for Structural Dynamics is two-fold. First, the individual substructures can be condensed and analyzed independently of each other. This is precisely the property exploited by parallel processing. Domain decomposition methods are similar in principle to sub-structuring methods, with

implementation differences depending on whether the domain decomposition adopts a primal or dual formulation. Second, substructuring methods can be implemented to analyze problems of quasi-unlimited size, given the current data processing resources. For example, the structural modification of a substructure makes it possible to re-analyze the full model at low computational cost. This is a significant advantage for the study of complex systems whose designs are entrusted to various equipment suppliers. 4.1 Direct Condensation The basic philosophy behind the creation of a reduced model from the global stiffness and mass matrices of a structural component is briefly described below in the context of a linear, conservative, elasto-dynamic sub-structure. The homogeneous equation of motion for such a model is given by: (K - M ) y = 0 (1) where M and K represent the discrete mass and stiffness matrices of the system and (; y ) is a given pair of eigen-solutions. The generic model reduction is performed via a transformation matrix T such that: y = Tc (2) where c is a vector of generalized coordinates representing the contribution of each column of matrix T. Note that some of these coordinates may correspond to physical coordinates. Others correspond to non-physical modal coordinates. Substituting equation (2) into equation (1) and pre-multiplying by the transposed of matrix T yields the reduced system of equations: ^ ^ (3) K - M c = 0

(

)

^ ^ where K and M are reduced-order stiffness and mass matrices defined as:

^ K = T T KT,

^ M = T T MT

(4)

The eigen-solutions obtained from solving equation (3) provide an approximation of the low-frequency modes that would be extracted from the full-size equation (1). These estimates are accurate to the extent where the subspace generated by the columns of matrix T represents the fundamental modes correctly. It is interesting to note that the various methods of reduction and sub-structuring differ only by the choice of the condensation matrix T.

In model reduction, a general rule of thumb is to include in matrix T the low-frequency eigenmodes of the structure. This typically provides an accurate representation of the low-frequency dynamics. To improve accuracy and increase the frequency range within which the dynamics are approximated with accuracy, static residual modes can be added in the columns of matrix T. Such static contributions are defined below. In the context of sub-structuring, the condensation matrix T replaces the junction degrees of freedom by appropriately selected coordinates of the model. Direct condensation can therefore be viewed as a particular case of CMS with only one sub-structure. 4.2 Component Mode Synthesis The objective of CMS is to calculate the dynamic behavior of a structure, starting from the knowledge of the dynamic behavior of its sub-structures. Numerical models of reduced sizes are used to describe the sub-structures. The general principle of sub-structuring consists of approximating the dynamic behavior of a substructure by a linear combination of q vectors that form a Ritz basis denoted, as before, by the matrix Ts. The subscript of matrix Ts refers to the sth sub-structure. The size of the model is reduced from N physical degrees of freedom to q "privileged" or generalized coordinates. The CMS matrix Ts discussed in this section may or may not be identical to the condensation matrix of a direct reduction. Differences between the sub-structuring methods are based upon: · The choice of reduction basis Ts and generalized coordinate vector cI (in the remainder, cI denotes the vector of generalized degrees of freedom defined across the interface common to the NS sub-structures); The technique of assembly of the substructures.

unresolved out-of-balance of forces over the interface. In this Section, we proceed, first, by introducing the notations commonly encountered in CMS. Then, three types of modes are defined: constraint modes, attachment modes, and normal modes. Finally, the overall procedure of sub-structuring is described. Figure 1 illustrates a finite element mesh whose degrees of freedom can be partitioned in two subsets. An internal domain is composed of all nodes that belong to the mesh with the exception of the right edge shown in red. The nodes of the right edge define the junction subset, also referred to as the interface nodes.

Internal coordinates denoted by the subscript i

Junction coordinates denoted by the subscript j

Figure 1. Illustration of internal (i) and junction (j) degrees of freedom or coordinates. The coordinates of the displacement vector ys of the sub-structure are partitioned into subsets of junction degrees of freedom yj and interior degrees of freedom yi:

·

When implementing a method of substructuring to approximate a structural response, the principal sources of error come from: · · The use of incomplete Ritz bases; Errors in the description and discretization of the geometry of an interface between two components or the contour of a connection; The inequality of displacements over an interface and/or the presence of

{ys } =

K jj Ks = K ij K ji , K ii

y j yi

M jj Ms = M ij M ji M ii

(5)

and the partitioned stiffness and mass matrices of the sub-structure inherit the following forms: (6)

·

Note that such partitioning is purely symbolic. Finite element matrices are never re-

arranged according to equation (6) because this would destroy their sparsity pattern. Instead, the partitioning is handled by defining the lists of nodes and degrees of freedom that belong to the internal and junction subsets and by treating them accordingly. 4.3 The Constraint Modes The constraint modes correspond to the static responses of the sub-structure obtained when unit displacements are imposed on one interface degree of freedom while holding the remaining interface degrees of freedom fixed. The number of constraint modes is therefore equal to the number of junction coordinates, denoted by Nj in the following. A constraint mode for the jth interface degree of freedom is solution to the problem:

K jj K ij

K ji I jj Fj = K ii c 0

(7)

Figure 2. Constraint mode deformation of the sub-structure. The concept of constraint modes has been used extensively by Guyan, Hurty, and Craig and Bampton to construct sub-structured models or super-elements. They are sometimes claimed to be appropriate when the junction between sub-structures, that is, the right edge shown in red in Figures 1 and 2, is significantly stiffer than the sub-structures that it connects. 4.4 The Attachment Modes The attachment modes are defined as the static deformation shapes obtained by applying a unit force or moment to one degree of freedom of the interface while all other degrees of freedom on the interface are free (no force or moment applied). The number of attachment modes is again equal to the number of junction coordinates (Nj). An attachment mode for the jth interface coordinate is solution to:

where the vector Ijj represents the unit vector at coordinate j, that is, a vector populated with zero entries except at the jth degree of freedom where it is equal to one. It represents a unit displacement imposed on the jth coordinate of the junction. The vector c represents the static displacements due to zero forces at the interior degrees of freedom. In the right hand side, the vector Fj denotes the reaction forces due to the known (applied) unit displacements. Solving equation (7) yields:

c = - K ii -1K ij I jj

Fj = K jj - K ji K ii K ij I jj

Note that solving equation (8) requires that the partition of the stiffness matrix Kii be nonsingular. This may not be the case with "floating" sub-structures, that is, subsets of internal degrees of freedom that are not attached to anything (except to each other) once the junction coordinates have been eliminated from the subset. In the event of a singular matrix, special numerical solvers can be implemented that overcome the difficulty of obtaining zero Gauss pivots during the factorization. Figure 2 illustrates the static displacements of the interior degrees of freedom due to the unit displacement applied to a rotational junction degree of freedom. All other junction coordinates are kept fixed and equal to zero.

(

(

)

-1

)

(8)

K jj K ij

K ji I jj a = K ii 0

(9)

where the vector Ijj represents, as before, the unit vector at degree of freedom j. Note that, to solve equation (9), the stiffness matrix Ks must not be singular, which physically means that the sub-structure must not possess any rigid body mode or mechanism. Figure 3 illustrates the static displacements of the interior degrees of freedom due to a unit moment applied to the junction node located at

the corner between the two plates. All other force and moment degrees of freedom on the junction are kept fixed and equal to zero.

associated with the retained mode shapes in . Rigid body motions are clearly not included in the truncated basis. The eigen-pairs (; ) denote modes that would already be included in the columns of the reduction matrix Ts. They could either be fixed interface modes (where the junction degrees of freedom are grounded, yj=0), free interface modes (where no internal or junction degree of freedom is constrained), or attachment modes. 4.5 The Normal Modes The normal modes are simply the mode shapes of the sub-structure. They are extracted from the eigen-problem of the undamped model (Ms;Ks) with a specific boundary condition applied to the interface degrees of freedom. Three types of boundary condition at the junction are possible: · Fixed interface: Zero displacements and rotations are imposed on the interface coordinates. These modes are generally used in supplement of the static modes in the sub-structuring methods of Hurty and Craig-Bampton. Free interface: The interface degrees of freedom are not constrained. These modes are generally used in the substructuring methods suggested by MacNeal and Rubin. Loaded interface: The mass and stiffness properties of the interface are modified by adding mass and/or stiffness perturbations. Masses and rotational inertias can be added as point values at specific degrees of freedom, in specific directions. Translation and rotational stiffness values can be added as springs between specific coordinates of the interface, or between a junction degree of freedom and a virtual attachment point (grounded point). The modal analysis is performed on the modified sub-structure. The sub-structuring method that uses Gladwell branch modes and the method suggested by Benfield and Hruda both employ loaded interface normal modes.

Figure 3. Attachment mode deformation of the sub-structure. The attachment mode obtained by inverting equation (9) is the solution to a linear system of equations that includes contributions from the low-frequency modes of the sub-structure. In fact, it includes contributions from all the eigenmodes of the sub-domain. It may be important to remove from the attachment mode (9) the contributions of low-frequency mode shapes . The reason is that the modes may already be included in the columns of the reduction matrix Ts, and it would be undesirable to include their contribution twice. This leads to the notion of residual attachment modes. Residual attachment modes are similar to attachment modes, except that the static contributions of eigen-modes already retained in the representation of the sub-structure are removed. These modes are introduced in the sub-structuring techniques of MacNeal and Rubin who propose to compute them as: ·

·

R j = a - -1 T I jj

(

)

(10)

In equation (10), the matrix is the truncated modal matrix composed of the retained eigen-modes of the sub-structure. The diagonal matrix is the corresponding spectral matrix whose entries are the eigen-values

Figure 4 illustrates the third mode of vibration obtained with a fixed interface. In this configuration, the translation and rotational degrees of freedom of the junction (right edge, shown in red in Figure 4) are constrained to

zero, hence removing them from the eigen-value problem.

1. Define the interface coordinate s, y ys = j yi 2. Define the interface coordinate s c I and reduction matrices Ts such that y s = Ts c I 3. Reduce the sub- domain matrices, ^ ^ M = TTM T , K = TT K T

s s s s s s s s

(11)

4. Assemble the interface matrices, ^ ^ ^ ^ M = M , K = K

I s =1... N s

I

s

I

s =1... N s

s

5. Solve the interface problem, ^ ^ K - M c = 0

(

I

)

I

6. Transform back into physical

Figure 4. Third mode shape with fixed interface degrees of freedom. Whether fixed interface, free interface or loaded interface modes are best appropriate for a particular application depends, to a great extent, on two factors. The first one is the analyst's experience with these different types of modes, and the analyst's judgment that one type of interface modes best complements the modes already included in the sub-structure's reduction matrix Ts. The other factor is the role that the interface dynamics play in the overall behavior of the system. In cases where the junction is expected to be stiffer than the sub-structures, fixed interface modes might be more appropriate because they constraint the motion of the interface degrees of freedom. In cases where it is important to account for the flexibility of the junction, free interface modes might be more relevant. 4.6 General Procedure for Sub-structuring Equation (11) below details the six-step method of sub-structuring, and the associated computations. The six steps are the same no matter which modes are chosen to build the reduction matrix Ts of each sub-structure. Steps 1, 4, and 5 must be performed for the entire structure. Steps 2, 3, and 6 can be performed for each sub-structure independently of the others. These steps are therefore naturally suited to parallel processing. Sub-structuring is generally implemented as follows:

coordinate s, y s = Ts c I

In equation (11), the mass and stiffness matrices are denoted by Ms and Ks for s=1...Ns sub-structures. The information is condensed in ^ ^ the reduced matrices M s and K s . Note that the reduction matrix Ts can be different for each sub-structure, provided that the generalized degrees of freedom on the interface are compatible with those from the other substructures. Once all sub-structures have been reduced, one is left with an interface problem whose mass and stiffness properties are ^ ^ represented by the matrices M I and K I . Once the interface solution has been transformed back into the physical coordinates, the final solution of the original eigen-problem is given by the eigen-values and the mode shapes y that collect the interface degrees of freedom cI and the internal degrees of freedom yi for all sub-structures. As mentioned previously, the overall substructuring procedure outlined in equation (11) is similar to the techniques of domain decomposition for parallel processing. More precisely, it is similar to primal domain decomposition techniques, that is, procedures where the equations of motion are formulated in terms of generalized displacements. Dual domain decomposition methods start from the same overall equations of motion but then

proceed by formulating an interface problem in terms of generalized forces. In any case, two constraints are used to assemble the substructures into an interface problem. They are the continuity of junction coordinates from each sub-domain across the interface:

T K sc c 0 T M - c s c T f M sc

(s) 0 y j q (s) f

y (1) = y (2) = ... = y (N S ) j j j

(s)

(12)

(s) T M s f y j Fj(s) (16) c (s) = I q f 0

where yj denotes the junction coordinates contributed by the sth sub-structure, and the equilibrium of forces contributed by the junction degrees of freedom from each sub-domain across the interface:

FI =

s =1... N s

F

where is the spectral matrix of the fixed interface modes. It is a diagonal matrix that stores the eigen-values of the sub-structure when its junction degrees of freedom are grounded (that is, yj=0). Note that writing equation (16) requires the resolution of several static systems to calculate the constraint modes. There are as many constraint modes as there are junction degrees of freedom yj of the sth sub-structure. In addition, writing equation (16) requires the resolution of an eigen-problem to calculate the fixed interface modes (; f ) . The size of the fixed interface eigen-value problem is equal to the number of internal coordinates yi of the sth sub-structure. What governs the computational cost is the fixed interface eigen-problem, especially when the number of internal degrees of freedom is large. As with Guyan sub-structuring, the reduced equations of motion (16) for each sub-structure are assembled into an interface problem by enforcing the continuity of displacements across the interface:

(s) j

(13)

where Fj(s) denotes the forces at junction degrees of freedom contributed by the sth substructure. When these two conditions cannot be enforced exactly, iterations are performed between the sub-domain problems and the interface problem until some convergence threshold is satisfied.

5. THE CRAIG-BAMPTON METHOD

The Ritz vectors of the sub-structuring methods of Hurty and Craig-Bampton include the constraint modes c and the fixed interface modes f . The fixed interface modes are the normal modes of the sub-structure with grounded junction degrees of freedom (yj=0, zero displacements and rotations). The generalized coordinates of the interface cI include two contributions, the modal coordinates qc of constraint modes (which are identical to the junction degrees of freedom, qc=yj) and the modal coordinates qf of fixed interface modes:

y (1) = y (2) = ... = y (N S ) = c I j j j

(17)

y j c I = (s) q f

(s)

The resulting system of equations depends only on the common interface degrees of freedom denoted by cI. Both Guyan and Craig-Bampton reductions therefore belong to the family of primal sub-structuring methods. The sub-structuring of Craig-Bampton is the technique used most intensively in the industry, especially in the automotive and aeronautics industries. Its popularity comes from the wellestablished accuracy of the approximation, and the ease of numerical implementation in general-purpose finite element packages. Also, equation (16) shows that this particular choice of Ritz basis "decouples" the stiffness matrix, meaning that there is no stiffness coupling term between coordinates yj and qf. It has been proposed to obtain a CraigBampton reduced model from experimental data. The fixed interface modes are obtained

(14)

The reduction matrix Ts of each substructure is composed of constraint and fixed interface modes. The value of junction degrees of freedom for these modes are, respectively, unit displacements and zero displacements:

I Ts = jj c

0 f

(15)

The resulting equation of motion of the sth sub-structure takes the form:

through modal testing of the sub-structure with a fixed interface. The constraint modes are extracted from the columns of the full flexibility matrix than can also be approximated from experimental data. The practical limitation is that the flexibility matrix is known only on the locations where driving point measurements are available. Drawbacks of the Craig-Bampton substructuring include the fact that the junction coordinates yj of all sub-structures must be represented in the final, condensed model. It may lead to a large-size interface problem, therefore, reducing the computational efficiency. The cost of solving the fixed interface eigenproblem for each sub-structure may also be significant even if only a few low-frequency modes are extracted. These potential limitations have, however, been addressed in the literature, for example, by developing multi-level reduction techniques.

later. One example is to view primal domain decomposition algorithms as nothing but CraigBampton condensed models. The reason for the extraordinary capacity of the concept of CMS to evolve and contribute to novel applications is simple. The Craig-Bampton form is a very useful representation for a structure as a whole. As a result, it has been applied in many ways that have no direct connection to CMS. It has been used, for example, in connection with placing sensors in an optimal fashion for sequentially assembled large space structures. It has been applied to help derive a measure of dynamic importance for fixed interface normal mode shapes. The Craig-Bampton condensation was also used as a basis for structural control. It turns out that the Craig-Bampton condensation is a special form of an inverse system in which the inputs (force-like) and outputs (displacement-like) are reversed from what we normally expect. These inverse systems can be employed to estimate applied forces based on measured structural responses. Inverse systems can be used to estimate response at locations on a structure where there are no sensors available. Finally, inverse system-based techniques have been developed for output-only modal identification. While the contributions by Dr. Craig form a cornerstone for structural component synthesis, their utility in the field of Structural Dynamics far exceeds these few applications.

6. CMS AND BEYOND

Most of us first got exposed to Dr. Craig's work in the early-to-late 1980's. CMS was then becoming a very popular and important method for structural analysis in the aerospace community. For example, SDRC in California was performing research to study and understand the relative merits of various substructure representations for use in aerospace structures. In general, aerospace structures are comprised of flexible components that are rigidly connected. It would be expected that the CraigBampton representation would give better results than sub-structure representations that use free interface normal modes. It has since been demonstrated that this is indeed the case. As the techniques were maturing, the IMAC became a forum for the, sometimes heated, discussion of the latest sub-structuring methods, their advantages, and limitations. Dr. Craig has always been and continues to be supportive of the conference. In the late 1980's and early 1990's, the concept of CMS started to be applied to many other problems that included, for example, the formulation of domain decomposition algorithms for parallel processing, and experimental modal synthesis. Often, the original investigations of these issues appeared disconnected from substructuring. Strong connections were discovered

7. REFERENCES

[1] Craig, R.R., Jr., "Rotating Beam With Tip Mass Analyzed by a Variational Method," Journal of Acoustical Society of America, Vol. 35, No. 7, pp. 990-993, Jul. 1963. Craig, R.R., Jr., Plass, H., "Vibration of Hub-Pin Plates," AIAA Journal, Vol. 3, No. 6, pp. 1177-1178, Jun. 1965. Craig, R.R., Jr., Kana, D.D., "Parametric Oscillations of a Longitudinally Excited Cylindrical Shell Containing Liquid," Journal of Spacecraft and Rockets, Vol. 5, No. 1, pp. 13-21, Jan. 1968. Craig, R.R., Jr., Bampton, M.C.C., "Coupling of Substructures for Dynamic Analyses," AIAA Journal, Vol. 6, No. 7, pp. 1313-1319, Jul. 1968.

[2]

[3]

[4]

[5]

Craig, R.R., Jr., Chang, T.-C., "Normal Modes of Uniform Beams," Journal of the EM Division, Proceedings of the ASCE, Vol. 95, No. EM4, pp. 1027-1031, Aug. 1969. Craig, R.R., Jr., Bampton, M.C.C., "On the Iterative Solution of Semi-definite Eigenvalue Problems," The Aeronautical Journal, Vol. 75, pp. 287-290, Apr. 1971. Craig, R.R., Jr., "Optimization of a Supersonic Panel Subject to a Flutter Constraint ­ A Finite Element Solution," AIAA Journal, Vol. 11, No. 3, pp. 404-405, Mar. 1973. Craig, R.R., Jr., Chang, T.-C., "Computation of Upper and Lower Bounds to the Frequencies of Elastic Systems by the Method of Lehmann and Maehly," International Journal for Numerical Methods in Engineering, Vol. 6, pp. 323-332, 1973. Craig, R.R., Jr., Su, Y.-W.T., "On MultipleShaker Resonance Testing," AIAA Journal, Vol. 12, No. 7, pp. 924-931, Jul. 1974.

Engineering Problems," Earthquake Engineering and Structural Dynamics, Vol. 11, No. 5, pp. 679-688, Sept.-Oct. 1983. [16] Craig, R.R., Jr., Engles, R.C., Hacrow, H.W., "A Survey of Payload Integration Methods," Journal of Spacecraft and Rockets, Vol. 21, No. 5, pp. 417-424, Sept.Oct. 1984. [17] Craig, R.R., Jr., Blair, M.A., "A Generalized Multiple-Input Multiple-Output Modal Parameter Estimation Algorithm," AIAA Journal, Vol. 23, No. 6, pp. 931-937, Jun. 1985. [18] Craig, R.R., Jr., "A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods," International Journal for Analytical and Experimental Modal Analysis, Vol. 2, No. 2, pp. 59-72, Apr. 1987. [19] Craig, R.R., Jr., Cheu, T.C., Johnson, C.P., "Efficient Initial Vectors for Subspace Iteration Method," International Journal for Numerical Methods in Engineering, Vol. 24, No. 10, pp. 1841-1848, Oct. 1987. [20] Craig, R.R., Jr., Kim, H.-M., "Structural Dynamics Analysis Using an Un-symmetric Block Lanczos Algorithm," International Journal for Numerical Methods in Engineering, Vol. 26, No. 10, pp. 23052318, Oct. 1988. [21] Craig, R.R., Jr., Hale, A.L., "The BlockKrylov Component Synthesis Method for Structural Model Reduction," AIAA Journal of Guidance, Control, and Dynamics, Vol. 11, No. 6, pp. 562-570, Nov.-Dec. 1988. [22] Craig, R.R., Jr., Ni, Z., "Component Mode Synthesis for Model Order Reduction of Non-Classically-Damped Systems," AIAA Journal of Guidance, Control, and Dynamics, Vol. 12, No. 4, pp. 577-584, Jul.Aug. 1989. [23] Craig, R.R., Jr., Su, T.-J., Ni, Z., "StateVariable Models of Structures Having RigidBody Modes," AIAA Journal of Guidance, Control, and Dynamics, Vol. 13, No. 6, pp. 1157-1160, Nov.-Dec. 1990. [24] Craig, R.R., Jr., Kurdila, A.J., Kim, H.-M., "State-Space Formulation of Multi-Shaker Modal Analysis," International Journal of Analytical and Experimental Modal

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[10] Craig, R.R., Jr., Erbug, I.O., "Application of a Gradient-Projection Method to Minimum Weight Design of a Delta Wing with Static and Aero-Elastic Constraints," Computers and Structures, Vol. 6, No. 6, pp. 529-538, Dec. 1976. [11] Craig, R.R., Jr., Chang, C.-J., "FreeInterface Methods of Substructure Coupling for Dynamic Analysis" AIAA Journal, Vol. 14, No. 11, pp. 1633-1635, Nov. 1976. [12] Craig, R.R., Jr., "Methods of Component Mode Synthesis," The Shock and Vibration Digest, Vol. 9, No. 11, pp. 3-10, Nov. 1977. [13] Craig, R.R., Jr., Johnson, C.P., "Quadratic Reduction for the Eigen-problem," International Journal for Numerical Methods in Engineering, Vol. 15, No. 6, pp. 911-923, Jun. 1980. [14] Craig, R.R., Jr., Chung, Y.-T., "A Generalized Substructure Coupling Procedure for Damped Systems," AIAA Journal, Vol. 20, No. 3, pp. 442-444, Mar. 1982. [15] Craig, R.R., Jr., Cornwell, R.E., Johnson, C.P., "On the Application of the Mode Acceleration Method to Structural

Analysis, Vol. 5, No. 3, pp. 169-183, Jul. 1990. [25] Craig, R.R., Jr., Kim, H.-M., "Computational Enhancement of an Un-symmetric Block Lanczos Algorithm," International Journal for Numerical Methods in Engineering, Vol. 30, No. 5, pp. 1083-1089, Oct. 1990. [26] Craig, R.R., Jr., Su, T.-J., "Model Reduction and Control of Flexible Structures Using Krylov Vectors," AIAA Journal of Guidance, Control, and Dynamics, Vol. 14, No. 2, pp. 260-267, Mar.-Apr. 1991. [27] Craig, R.R., Jr., Su, T.-J., "Krylov Model Reduction for Undamped Structural Dynamics Systems," AIAA Journal of Guidance, Control, and Dynamics, Vol. 14, No. 6, pp. 1311-1313, Nov.-Dec. 1991. [28] Craig, R.R., Jr., Su, T.-J., "Control Design Based on Linear State Function Observer," International Journal of Systems Science, Vol. 23, no. 7, pp. 1179-1190, Jul. 1992. [29] Kim, H.-M., Craig, R.R., Jr., "Application of Un-symmetric Block Lanczos Vectors in System Identification," International Journal of Analytical and Experimental Modal Analysis, Vol. 7, No. 4, pp. 227-241, Oct. 1992. [30] Craig, R.R., Jr., "Substructure Methods in Vibration," Transactions of the ASME, Vol. 117(B), pp. 207-213, Jun. 1995. [31] Su, T.-J., Babuska, V., Craig, R.R., Jr., "A Substructure-Based Controller Design Method for Flexible Structures," AIAA Journal of Guidance, Control, and Dynamics, Vol. 18, No. 5, pp. 1053-1061, Sept.-Oct. 1995. [32] Craig, R.R., Jr., Structural Dynamics ­ An Introduction to Computer Methods, John Wiley and Sons, 1981. [33] Craig, R.R., Jr., Solutions Manual for Structural Dynamics ­ An Introduction to Computer Methods, John Wiley and Sons, 1982. [34] Craig, R.R., Jr., Mechanics of Materials, John Wiley and Sons, 1996. [35] Craig, R.R., Jr., Solution Manual for Mechanics of Materials, John Wiley and Sons, Vol. 1, 1996, Vol. 2, 1997.

[36] Craig, R.R., Jr., Mechanics of Materials, 2nd Edition, John Wiley and Sons, 2000. [37] Craig, R.R., Jr., Solution Manual for Mechanics of Materials, 2nd Edition, John Wiley and Sons, 2000. [38] Craig, R.R., Jr., "Frames," in Shock and Vibration Computer Programs ­ Reviews and Summaries, Pilkey, W., Pilkey, B., Editors, The Shock and Vibration Information Center, pp. 129-150, 1975. [39] Craig, R.R., Jr., "A Review of Time-Domain and Frequency-Domain Component Mode Synthesis Methods," in Combined Experimental / Analytical Modeling of Dynamic Structural Systems, American Society of Mechanical Engineers, Vol. 67, pp. 1-30. [40] Craig, R.R., Jr., "Recent Literature on Structural Modeling, Identification, and Analysis," in Mechanics and Control of Large Flexible Structures, Junkins, J.L., Editor, Vol. 129 of AIAA Progress in Astronautics and Aeronautics, pp. 3-29, 1990. [41] Craig, R.R., Jr., "Component Modeling Techniques," in Flight Vehicle Materials, Structures, and Dynamics Technology ­ Assessment and Future Direction, Venneri, S.L., Noor, A.K., Editors, Published jointly by NASA and ASME (in press). [42] Su, T.-J., Craig, R.R., Jr., "Krylov Vector Methods for Model Reduction and Control of Flexible Structures," in Advances in Control and Dynamic Systems, Leondes, C., Editor, Academic Press, pp. 449-481, 1993. [43] Craig, R.R., Jr., "Component Modeling Techniques," in Flight Vehicle Materials, Structures, and Dynamics, Vol. 5., Structural Dynamics and Aeroelasticity, American Society of Mechanical Engineers, pp. 19-28, 1993. [44] Craig, R.R., Jr., "Computational Methods: Krylov-Lanczos Methods," in Encyclopedia of Vibration, Academic Press (to appear).

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