A general purpose, multi-stage, component modal synthesis method

Finite Elements in Analysis and Design 1 (11985) 153-164 North-Holland

153

A GENERAL PURPOSE, MULTI-STAGE, COMPONENT MODAL SYNTHESIS METHOD D.N. HERTING The MacNeaI-Schwendler Corporation, 815 Colorado BoulevartL Los Angeles, CA 90041, U.S.A. Received June 1984 Revised December 1984 Abstract. A general purpose algorithm for the dynamic analysis of large-order finite element structural models using real modal coordinates is presented. Each component substructure may be formulated to include free or fixed boundary modes, inertia relief displacement shapes, and physical displacements as generalized degrees of freedom. Capabilities are developed for transforming interior loads, such as aerodynamic effects, to the reduced set of coordinates. Theory and tests prove that the method duplicates results from most of the other popular methods by selecting various optional terms. The conclusion contains practical recommendations based upon several applications to structural analysis.

Introduction Component modal synthesis is a technique for mathematically representing a finite element structural model by including truncated sets of normal mode generalized coordinates defined for two or more components of the structural model. The objective of the technique is to simulate the dynamic response of a large-order finite element model while reducing computational costs by greatly reducing the number of equations describing the structure's behavior. The.significance of this presentation is that it provides a more fundamental justification for the modal formulation than the typical Rayleigh-Ritz approach. The method starts from the 'exact' response of the complete system, develops the approximation system, and predicts the error of the truncated modes. The method has been implemented in several finite element programs including MSC/NASTRAN, initiated by The MacNeal-Schwendler Corporation, and NASTRAN ®distributed by COSMIC. t The following sections of the paper present background and summaries of ~'fferent modal synthesis approaches, a complete, theoretical development of the NASTRAN . ~proaches, a discussion of accuracy and numerical results obtained with 'classical' test problems, and a summary of recommended procedures.

ckpomt The use of structural modes as generalized degrees of freedom in dynamic models originated in the analog computer field where structural models were combined with aeroelastic and control system models. ,'Tne first appfications to digital computers were simple extensions of the analog techniques. However, this so-called classical approach has been found to be highl) restrictive and limited in accuracy. Many different approaches were developed in recent years t Registered Trademark of NASA. 0168.874X/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

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having more generality and accuracy for solving large-order structural dynamics problems. Current methods used in component modal synthesis vary considerably in both approach and application; however, they may be grouped into two distinct categories. The first category contains all methods using a Rayleigh-Ritz approach in which the component degrees of freedom represent the deflections of normal modes and static deflection shapes. The second category contains methods in which the component degrees of freedom are physical (boundary) point displacements and modal coordinates. Here, the classical method has been improved by adding flexibility coefficients to the matrices to account for the effects of a truncated set of modes. The two basic categories and their variations are described below. An early Rayleigh-Ritz component modal synthesis application to digital computers was devised by Hurry [1] in which 'constraint' point modes, obtained by applying unit static loads, were introduced to represent tlie static deformation shapes of the structure. Bamford [2,3] extended the method by using unit displacements to define the rigid body and flexible 'attach' modes. Several variations followed [4-7] in which the displacements at both the interior and the boundary grid points were defined using combinations of different shapes. Unfortunately, the processes used by these methods to connect the component structures together are unwieldy because each interconnected degree of freedom generates a constraint equation coupling all of the various degrees of freedom from several component structures. Thc~se constraint equations generate full matrices of order equal to all degrees of freedom in the combination structure. Other difficulties arise in the solution of the combined structure because the original physical displacements are not available for load or boundary constraint application. These conditions must be specified at the original substructure level for these methods. A popular variation of the Rayleigh-Ritz approach was developed by Craig and Bampton [8] in which boundary grid point displacements were retained as degrees of freedom, thereby simplifying the connection of component structures. Unfortunately, all boundary degrees of freedom must be constrained when obtaining the normal modes with the Craig-Bampton approach. These modes are often of little practical use to an analyst who wishes to verify the component modes using vibration test data from an unconstrained specimen. A different approach was developed from the classical modal formulation used in analog computer systems. MacNeal [9] published a method in which the 'residual flexibility' of the structure was calculated to correct for the missing static effects which occur when a truncated set of normal modes is used. This method has a distinct advantage in that actual displacements of the boundary points are retained as degrees of freedom, thereby simplifying the combination process. Rubin extended MacNeal's method to include the inertial effects of the truncated modes. Rubin's method also provides a superior method for calculating the resultant displacements of the interior points. Interestingly, the different methods showed very close results. The similarities of the different methods were discussed in [14,15]. In parallel with theoretical modal synthesis developments, the state-of-the-art in the related field of substructure analysis has also been expanding. In particular, substructuring and superelement methods have been incorporated into major programs. With substructure analysis, the task of analyzing large-order finite element structural models has been simplified by dividing the process into manageable steps. Capabilities include open-ended, multi-stage combination of structural components, Guyan reduction of stiffness and mass matrices, and loaded response solution capabilities. The implementation of component modal synthesis may be viewed as a logical extension of the substructuring process. This modal formulation provides an alternative to the Guyan reduction process in that matrices representing compone:R structures are reduced in size before they are combined with other structures' matrices. Modal generalized coordinates are added to the system to represent the additional response and loads occurring in dynamic analyses. Although the methods of component modal synthesis described above are adaptable to a

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general purpose substructuring analysis system, none satisfy all of the following criteria. (1) Physical displacements should be retained as generalized coordinates to simplify connections to other components. (2) Total inertia and flexibility for each component should be retained. (3) Modal boundaries should be independent of physical connection points with other structural components. (4) The method must be computationally efficient, both in memory and time. The equations developed in the next section define an approach to modal synthesis which does satisfy all the above criteria. The method has been used in both NASTRAN and MSC/NASTRAN analysis computer programs for more than five years, and has bec,~ proven with many applications.

Equation development The equations of motion are developed below for the normal mode modal reduction of substructure models. The modal reduction is applicable to undamped or lightly damped structures represented by symmetric mass and stiffness matrices. Each component substructure is defined by a stiffness matrix, [K], and a mass matrix, [M]. For the undamped case, the equations of motion are [M]{i~}+[K]{u}ffi {P}, (1) where { u} are the displacement coordinates and { P } the loads due to external forces or boundary reaction loads from other substructures. For the idealized case when all mode shape vectors, { ~bj}, of the unconstcained substructure have been extracted, the equations of motion are uncoupled, one equation for each mode, such that

+,%)

=

(2)

where ~j is the generalized displacement of mode j, and where

,~ j = { ¢~j}T [M ]{ c~j} ,

(3)

e2 _ (1/.,W'j) { ~i }'r[ K ]{ ~j },

(4)

ej= {,j}*{p}.

(5)

The physical displacements, in terms of the modal displacements, are N

u,- E ,j j

or

(6)

j--|

In this development, j denotes the mode, i denotes a displacement coordinate, and N is the total number of modes. The matrix [ok] represents the collection of all mode shape vectors, and [~j] represents the set of mode shape vectors to be retained in the analysis. Using the Laplace operator s where O/Ot- s, equations (2), (5), and (6) are combined to obtain the matrix equation for coordinate displacements so that

1"}-['1

(s2+

Eq. (7) is exact when a/i modes of the system are used. Normally, however, only a truncated set, ~1,-.-, ~,,, is used to reduce the number of variables.

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Effects of truncated modes The response of higher modes, {~l, }, where ~o~>> [s [ 2 is primarily static, and equation (7) becomes {u} =

[epj][~Zj(s)...][epj] T {P}

(8)

+ [,k ]['-Zk..,][4~,] T { P },

where 1

Zj(s)-'-jgj(s2+o~ )

for j~< m,

(9)

and Zk - - - -

1

=

1

for k > m,

(10)

and the number of retained modes is m. The form of the last term of equation (8) is that of a flexibihty matrix, [AZ], times a load, which produces a corrective displacement vector. This residual flexibility is obtained from the basic equations of motion, equation (1), as shown next. Eq. (8) may be written in terms of free-body modes [~o], retained flexibility modes [~j], and residual flexibility in the form {u}=

-~-~[,o][d~'0]-'[,0] T + [ , j ] [ ' Z j ( s ) . j [ , j l T + [ A Z ]

1

{e},

(11)

where the subscript o denotes free-body modes. If we examine the 'steady state' response components, only free-body acceleration exists, and

{~} = ~ { . } = [~o][~go]-'[,o]~{P} Substituting (11) and (12) into (1) with

s =

(~-. 0).

0, and noting that [K]{ ¢o }

02) =

0, we obtain

[KI[Az]{ P} -- { P} - [ Ml[,ol[~go]-l[,o] T { e } - [K ][~,j] [ Z(0)] [,I,~] { P }. (13) Multiplying (11) by [K] and substituting (13) into that result gives an approximation, below, which inclades she stad¢ effect of all modes and the dynamic effect of selected modes

[El{ u} = [Kl[,j] [ Z ( s ) - z(0)l [,~] { P } - [ Ml[~,ol[~go]-'[,~0] + { P } + { P }. (14) A direct solution of the above approximation would provide excellent results for an uncoupled structure. However, the matrix .K will gain contributions from other connected structures at the boundary points. For this reason the boundary degrees of freedom must be separated from the interior degrees of freedom. But, first, equation (14) will be simplified by introducing modal coordinates, { 8 }, as defined below. Eq. (14) becomes [K]{u}

= [K][rbj]{8}-[M][~po]{8o}

+{p},

(15)

where { 8 } = [ Z ( s ) - Z(0)] [~j] T { P }

(16)

is the relative displacements of the modes, and { 8o } - [ . ~ ' o ] - ' [ , 0 ]T { P }

(17)

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is the rigid body acceleration in terms of the free-body modes. Note that {8} represents the difference between the modal displacement and the static response. Substituting (9) into (16) we observe that

Note that 8j has units of displacement, but decreases as the fourth power of ~0j. The response magnitudes of the high frequency modes (i.e., Is/,ojl << 1) will rapidly disappear. Also note that 8o has units of acceleration and represents free-body motion.

Development of interior/boundary transformations For connection to other structures, equation (15) must be separated into boundary points, Ub, and interior points, ui (which will be replaced in the medal coordinates). The matrices are partitioned such that [ K ] --* l Kib ', Kii J and

{ u } -,

, etc.

(2o)

Although the full matrix [K] may be singular, the partition [Kii ] must be nonsingular. The interior displacements may be defined by the lower half of equation (15) which becomes

[ Kib ] { Ub } -!- [ Kfi ] { u i } ~-"[ Kib~b -!- Kii~i ] { 8 } -- [ Mib~b 0 4- Mii~i 0 ] { 80 } -t- { Pi }(21) The loads Pi are appliexl- only to the interior and are known functions of time and frequency. Solving equation (21) for the interior displacements~ { u i }, we obtain { ui} = [Gib] { Ub } + ['l~i- aibCb] { 8 }

-[ Kii]- l[,Mib¢b0 + ii¢i0 ] { 80 } +[ Kii]-I {/'i },

(22)

where [Gib] = --[ K,i]-'[ Kib]. (23) Also, note that {~i0 } = [Gib]{ ~b0 } for the rigid body modes and the matrix [Gib ] is identical to the transformation matrix used for the Guyan reduction technique. Without modal coordinates, the formulation degenerates to a static matrix condensation. A higher-order approximation, {~}, to the interior point displacements is obtained as follows. An expression for { fit } is obtained by differentiating equation (22). Eq. (1) is rewritten in partitioned form and { ui } is used instead of { ui } so that

[MLbj_Mb_i ~i }--'~ {~/) LMib : Mii {~iUb } ..}-[_~bb~_5~l/ub Pb .

[Kib ', Kii ][

(24)

The second derivative of equation (22) is substituted into a lower expression of (24), and the expression is solved for a second degree approximation, { ui },

{ ~i} ~ [Gib] { Ub } --[ Arii]-l[ Mib -t- MfiGib] { ~b } --[ Kii]-l[ Mfi ][~pi - Gib~b] { ~ } 4-[ Kii ]- i[ Mii] [ Kill- 1[ Mib ~" MiiGib] [ ~Pb0]{ ~0 } •+ [ K i , ] - ' [

P, M~K~'i',]. -

(25)

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The above expression for { ~ ' -~ay be s,,~bstitut~d into the upper partition of equation (24) as shown below to obtain "~,e ~yr.~,~,li,~ eq~.ations of motion for the boundary coordinates. It should be noted that ~6i is usuaiiy unm:portant and ignored. In a similar fashion, the equations of motion for the modal coordinates defined in equation (15) may also be developed resulting in a complete reduced system. These equations are complicated ~o show. However, it is fortunate for calculation purposes that the resulting equations of motion are identical to those obtained by application of the Rayleigh-Ritz method, with a particular set of generalized 'shape functions'. A set of shape functions are combined int'~ a coordinate transformation matrix, [H ghi, by using equation (22), such that

(0)

(26)

[~I ol [Hgh] = L~b~-/.~,O~ ~ i - Gib~b '

0 ]

(27a)

{u,} --

(27b)

(,,)=.,

/

where

(27c) [a,bl ffi -[K.]-'[K,b],

(2.8)

[S,o] = -[ K,,]-'([ M,,,] + [m,,][G,b])[,bo].

(29)

The columns of all [Hgh] correspond to the 'modes' of the generalized system. Free-body motion and redundant constraint information are contained in the [Gib] partition. Inertia relief deformation shapes are contained in the [Hio ] matrix. The generalized coordinates, { u h }, define the new coordinate set for modal-reduced structures. Note that, when fixed boundary modes are used and the inertia relief effects are ignored, the transformation matrix given in (27) is the same as in [8]. When unconstrained points exist, they are defined by dynamic motion relative to static deformations. ~u i is a relative static motion that is calculated in the recovery step. After applying the transformations defined by (22). the reduced mass matrix, [Mhh], iS typically coupled between the boundary and modal coordinates, where [Mhh]-

[Hgh]TIM][ Hsh].

(~.,

However, the stiffness matrix takes the f o ~

[

(31)

where [Kbb ] is the stiffness partition obtained from a Guyan reduction and the subscript p refers to the modal coordinates. Note that the boundary and modal coordinates are statically uncoupled from each other, which allows the connection of substructures by simply adding the 'b' matrices. The externally applied loads are also transformed to the modal system resulting in component loads defined as { P h } - [Hsh]'{ e }-

(32)

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159

It is significant to note that equations (30) and (31) are closely related to equation (24). In fact, both methods result in the same solution matrices since the upper half of (24) may be expanded using (25) to obtain

[Mbb]{ ~b } 4-[Mbi]{ ~i } 4"[Kbb]{ Ub } 4-[Kbi]{ ~i } T T -" [ Mbb "4"MbiGib 4- GibMib 4- GibMiiGbi] { fib } 4- [Mbi 4- GTMii] [~bi- Gib~b] { ~b } 4-[ Mbi 4- Gi~*J'ii] [/'/riO]{ ~0 } 4- [Kbb 4- gbiGib] { Mb } -- [Gib] { Pi } -~-[Mbh]{ ~h } 4-[ gbh]{ Uh } -- [aibl{ Pi }.

(33)

In other words, the solution equations actually provide accuracy equal to the second degree of approximation of (25) rather than the static shapes of (22). This provides the justification for calculating the second degree terms in the data recovery process described below. Examination of the preceding development will indicate the following major features of the method. (1) The interior degrees of freedom of the substructure may be replaced with a much smaller set of modal degrees of freedom. Eq. (18) indicates that the effects of the high frequency truncated modes decrease with the f3urth power of their natural frequencies. (2) Since the modal coordinates are uncoupled from the boundary grid points, as shown in (32), no approximation is used for static solutions. (3) The location of fLXedboundary conditions on the normal modes is completely arbitrary. Either interior modal deflections {~i } or boundary deflections {~b } may contain zero components. (4) The method will produce results equivalent to other methods when the correspon~g restrictions are applied. Eliminating inertia relief effects will reproduce the Hurty [1] results. In addition, fixing all boundary points will result in exactly the same matrices as those of Craig and Bampton [8]. The use of free-free modes with inertia relief effect will duplicate Rubin's results [10]. Solution vector recovery for reduced structures

The reduced model equation of motion

[ Mhh]{ ~h } 4-[ Bhh]{/~h } 4-[ Kith]{ Uh } = { eh }

(34)

is solved for system response in the form of normal modes or solution vectors as a function of time or frequency. After the solution vectors, { i~h }, {/ah }, and { Uh} are obtained for the system model, displacements in the original coordinates are recovered using equations previously developed, namely = [H

I CUh} +

(35)

and similarly for the velocities and ac.celeration~, =

(36)

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and

However, Rubin [10] has shown that a mode acceleration technique will result in improved accuracy. If this technique is selected, then equation (25) may be used in the following form for the alternative, 'improved', displacements of the interior points, { ~i }. The argument for mode acceleration is that the inertial forces convergefaster than the elastic forces. Therefore, we solve a statics problem for interior displacements. (The boundary points are dependent on the connecting structures and their displacements must not be changed.) The most direct method is to evaluate the lower half of equation (24) for ~i, using the known values of Ub, i~b, and the approximate values of /i i obtained from (37), wl-,ich contains the 'mode accelerations' along with additional effects of the boundary accelerations. The explicit result is given by equation (25). However, we may note that many of its parts have simpler forms for ease of calculation. We may cast equation (25) in the classical mode acceleration form: { ~i } ~- [Gib] { 14b} -["[ Kii]-I { e i - gib~b -- gbi~i },

(38)

where the right-hand side contains the modal inertial forces and the accelerations are obtained from (37). From (27) we note that the accelerations include 'static mode' accelerations. { ~i} ~-~[Gib] { ~b } "l"[ ~ i - Gib~b] { ~ } "~'[nio]{ Go}.

(39)

Comparison with other approaches In the preceding development, particular choices were made for the generalized coordinates and the form of the equations. Other researchers have taken direct approaches which have resulted in sindlar results. In the discussion below it is shown that these different approaches are actually closely related to special cases of the general method. Eq. (14) is significant because it may be used as a basis for developing other approaches to modal synthesis. Moreover, since it is developed using only the assumption of static approximation for the higher frequency modes, it eliminates the question of whether additional effects exist. The discussion below shows that several other modal synthesis methods are obtained when different coordinates are used to represent the right-hand terms of (14). For any modal synthesis formulation, the complete set of degrees of freedom, { u}, are replaced by a reduced set of generalized coordinates. The examples presented next illustrate two widely different modal synthesis methods which can be developed from (14). MacNeai-Rubin methods

These methods utilize the concept of a residual flexibility matrix, AZ. One form is obtained directly from (13): az-

K-'A

-

(40)

where

[A]=

(41)

Note that the matrix [A] removes all resultant free-body forces, thereby allowing the matrix [K] to be constrained before it is inverted. By similar arguments, the result may be premultiplied by

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161

[A T] tO preserve symmetry and remove arbitrary free-body displacements. The actual form of the residual flexibility matrix, [Gr], used by these methods is [G r ] = [ ATK-'A] -[~jZj(0)~bT] • From (11), the displacement: equation is { u } = [G']{ e } + [~p]{~j }, where

{~ } = [z~(~)][~1{e }.

(42)

(43)

(44)

Note that the modal coordinates, /~j,must contain the zero-frequency modes. In MacNeal's procedure, equation (43) is partitioned into connection { u c) and interior(u i) displacement sets, and, assuming that no interiorloads exist,the system is directlyreduced to the form of equation (34). Note, however, that the mass matrix only reflectsthe inertiaof the modal coordinates. Rubin, however, extends the procedure [10] to include the inertialeffects of the additional motion caused by the residual flexibility.Although Rubin develops his mass matrix using force derivatives, the same results are obtained using a Rayleigh-Ritz procedure with displacements defined from the reduced form of (43), namely

~i

=

~[bi(~bb I {Ub--C~b~}"l'[~]{~}"

(45)

Eq. (47) may be rearranged into a single transformation of the form

(Ub/

0

(46)

where

[HI= [ci][o~.]-'

(4v)

Note that (46) differs from (27) in the choice of the physical meaning of ~ (total displacement) versus B (relative displacement).

Transformation methods Several variations exist for this method. The most general is described by Hintz [6] in which he chooses static coordinates, qb, modal coordinates ~, and 'inertia relief' coordinates, qr, which correspond to the first three terms of the right-hand of (14). To obtain these results, the displacements of the boundary points, u b, are assumed to be { Ub} -- { qb + ¢~b~}(48) Partfioning (14) into boundary coordinates, %, and interior coordinates, u~, the lower set of partitions produce the equation

[ gib]{ qb + *b~ } + [ Kii]{ £/i } "~ [ ~ib~b] { ~ } -I-[ gill{ ~[Ji} ~ "]"[ Mi~o'~o I¢~] { P }" (49)

Solving (49) for ui, we obtain { ui } -- [Gib] { qb } + [@i] { ~ } +[ Hi]{ q0 },

(5o)

where

(51)

[ o,] = [ r. ]-'[ r~].

(52)

{qo}'- [ ~ ] {q}"

(53)

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D.N. Herting / Generalpurpose, multi- stage, component modal synthesis method •-

Boundary Points

,o

l

~

Superelement. !

I

Fig. 1. Hine cell truss-rod elements.

Note that the colunms of the matrix [H~] correspond to the deformed shapes of the structure which occur to freeobody inertial forces resulting in the use of the term inertia relief. Combining (48) and (50), we obtain Hintz's form

/ __

"'J

LC: " "

@'J[FJ

,

(54)

Note that except for a minor change in the definition of ~, equation (54) is"identical to (27a). Eq. (54) may be used to transform the stiffness and mass matrices directly by pre- and post-multiplication. Note, however, that the resulting degrees of freedom do not directly define modal displacements (unless ~ b -- 0). The process of connecting adjoining structures requires a solution to the simultaneous equations resulting from the connection equations.

Test results

Two-component truss model

The two-component truss model described next is a 'classical' problem used by several other inve~,tigators to study the accuracy of modal synthesis formulations. The structural model, shown in Fig. 1, is constructed from two substructu,:es. Each substructure is reduced to its normal modes plus other shape functions (depending on the modal synthesis approach). The substructures are combined at three common nodes, and free-body normal modes are computed for the complete model. Results are nearly identical to those obtained by Hintz [6]; the differences appear to be due to the differences in numerical precision. Other test cases have been shown to be identical to Rubin, and Craig and Chang [7], in previous reports [12,13]. The model was analyzed using the new modal synthesis method. A complete system model was also run to provide a baseline for assessment oi; frequency and mode shape errors produced by mode truncation. Four modal synthesis approaches were exercized: cantilevered component modes with inertia relief shapes, free-body modes with inertia relief shapes, cantilevered modes only, and free-body modes only. Summaries of the system frequency errors for these four approaches versus the complete model results ~ e presented in Table 1. The first three approaches produced frequency errors less than one percent up to the system frequency equal to component mode truncatiq~ frequency. However, the ~ase using free-body component modes without inertia relief shapes produced poor results. This is because the half-wave, free-body shapes do not approximx~e the total system mode shapes for this particular

D.N. Herring / General purpose, multi, stage, component modal synthesis method

163

Table 1 Percent frequency errors with 20 elastic degrees of freedom two-component truss problem Elastic

Analysis approach

mode number

Cantilevered c~mponent modes with inertia relief

1 2 3 4

5 6 7 8 9 10 11 12

0.00034 0 0.0061 0.00002 0 0.00060 0.268 0.021 0.54 0.40 0.98 12.3

13 14 15 16 17

Free-body moves with inertia relief

Cantilevered modes

0.00034 0.00902 0.0135 0.00023 0.00083 0.0020 0.080 0.0071 0.00098 0.0041 0.021

0.000438 0.0017 0.0098 0.0096 0.033 0.0098 0.947 0.122 0.59 0.36 0.33

8.92 1.21 7.67 1.08 6.00 0.85 0.61 1.58 0.084 0.030 0.90

0.428

0.49

3.30

5.35 7.87

0.16 0.77 2.37 12.15

4.01 0.244 1.10 -0.59 6.69

Free-body modes

Note: Dashed lines indicate limits of truncated mode frequencies.

model. Note, though, the use of inertia relief shapes with free-body component modes produces very accurate system frequencies.

Summary and recommendations The theoretical development and model analyses described herein illustrate how the new modal synthesis method is capable of achieving equivalent or better results and accuracies compared to other state-of-the-art approaches. The effects of static inertia relief deflections are important for both constrained mode and free mode systems. The effects on the free mode case is shown by the truss problem. The effects on fixed base problems are also dramatic, particularly in the cases where base reactions to low frequency excitation are important. In actual practice the cost is small for these six additional degrees of freedom. The choices of constrained attach points or free boundary points are highly dependent on the loads provided by the attached structures. For cases where the attached structure is large and heavy relative to the component, the attach points should be constrained during the mode extraction. An example for an aircraft is the attach point on a wing component which will be connected to the fuselage. Free attach points are recommended where the adjoining structure is fight or weakly attached. These points are also useful at points loaded by external force function, (even if no structure is attached), since they provide good static stress data in the region. An example of a free attach point would be the point on the aircraft where an antenna is attached. If the point was constrained in the modal solution, the normal modes would be poor approximations of the actual response. The free point option will provide mode shapes closer to the final value with

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the relative local deflections represented by the static shapes associated with the Guyan reduction. References [1] HURTY, U.C., "Dynamic analysis of structural systems using component modes", AIAA Journal 3 (4) pp. 678-685, 1965. [2] BAMFORD,R.M., "A modal combination program for dynamic analysis of structures", NASA TM 33-290, Jet Propulsion Laboratory, 1966. [3] BAMIFORD,R.M., B.K. WADAet al., "Dynamic analysis of large structural systems", Synthesis of Vibrating Systems, ASME Booklet, pp. 57-71, 1971. [4] BAJAN,R.L. and C.C. FFNG, "Free vibration analysis by the modal substitution method", American Astronautics Society Symposium, Paper No. 68-8-1, 1968. [5] BENFIELD,W.A. and F.R. HRUDA,"Vibration analysis of structures by component substitution", AIAA Journal 9, pp. 1255-1261, 1971. [6] Hnsrrz, R.M., "Analytical methods in component modal synthesis", AIAA Journal 13, pp. 1007-1016, 1975. [7] CRAI¢;, R.R. and C.J. Ct/~,N¢~, "On the use of attachment modes in substructure coupling for dynamic analysis", AIAA Paper 77-405, Presented at AIAA / A S M E / A S C E / A H S 18th Structures, Structural Dynamics, and Materials Conference, 1977. [8] CRAIG, R.R. and M.D.D. BAMPTON,"Coupling of substructures for dynamic analysis", AIAA Journal 6 (7) pp. 1313-1319, 1968. [9] MACNF.AL,R.H., "A hybrid method of component mode synthesis", Computers and Structures I, pp. 581-601, 1971. [10] Rtmttq, S., "Improved component-mode representation for structural dynamic analysis", AIAA Journal 12, pp. 995-1006, 1975. [11] HERTtNG, D.N. and R.L. HOF.SLX',"Development of an automated multi-stage modal synthesis system for NASTRAN", 6th NASTRAN User's Colloquium Papers, NASA Conference Publication 2018, 1977. [12] HERTXNO,D.N. and M.J. MORGAN, "Component mode synthesis", Section 4.7, NASTRAN Theoretical Manual, Level 17.5 Updates. [13] I-I~RTIN¢3, D.N., "Accuracy of results with NASTRAN model synthesis", 7th NASTRAN Users' Colloquium Papers, NASA CP.2062, 1978. [14] KOOSTEI~tAN,A.L. and W. MCCt.ELLAtqD,"Combined experimental and analytical techniques for dynamic system synthesis", Tokyo Seminar on Finite E!e~.nentAnalysis, 1983. [15] Ct~jG, R.R. and C. -J. CHtnqo, "A review of substructure coupling for dynamic analysis", Advances in Engineering Science 2, NASA CP-2001, pp. 393-408, 1976.