'Engineering Specialist, AdvanceVehicle Dynamics, Chelsea, Michigan 48118; E-Mail: jbf5chrysler.com, Phone: 734.475550I, FAX: 734.475.5540. 2
Professor, Department of Naval Architecture and MasineEngineering, The University of Michigan, Ann Arbor, Michigan 48109;
E-Mail: michaelb@umjch.edu, Phone: 734.764.9317, FAX: 734.936.8820.
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REDESIGNING THE DYNAMICS OF STRUCTURAL SYSTEMS
John B. Ferris'
Chiysler Corporation, Chelsea, Michigan Michael M. BeEnitsas2
The. University of Michigan, Ann Arbor, Michigan and
Jeffrey L. Stein3
The University of Michigan, Ann Arbor, Michigan
submitted for publication to the
AIAA Journczl
March 5, 1998
Rtmg the Diia cs o( StruiJ SI5
kim B Fa,u. Michael t Bcntiva5. kv L Siem
Submitted f publicazit to the AIM Joiirr,aL March 5. 1998
Abstract
A process is developed by which changes in the eigenvalues of a structural system are
prescribed and the required changes to the substructures are predicted. A structural system is comprised of an assembly of substructures. The structural system redesign process is based on the derivation of the first order perturbation relationship between the structural system eigenvalues and the eigerivalues of each independent substructure. It is assumed that the relationship between the eigenvalues and the tunable physical parameters of each substructure is known. Since each perturbation relationship is of first order, a transformation matrix is derived for each relationship. A series of these transformation matrices are combined, to form the ultimate structural transformationmatrix.
The structural transformation matrix is a set of constraint equations which form part of an optimization prOblem. The Objective function for the optimization is to minimize the
sum of the square of the fractional change
iii the substructure eigenvalues. This optimization problem may be satisfied, via the generalized inverse, when the problem is underconstrained. This generalized inverse matrix is used, to determine the predicted changes in the substructure eigenvalues. The result of the structural system redesign process is a predicted set of changes to the substructure eigenvalues that yield the objective changes to the structural system eigenvalues.This redesign process is demonstrated by applying it. to an isolated platform example. The changes in the substructure eigenvalues are predicted via the redesign process, then substituted back into the fill flonliiiear equations for the structural system. The errors
Redaiung the Dvtwnics of StniciuraJ Syacn
Jelm B. Fana. Michael M. BeryUtua. ieffieyL Stein Submitted for p icaiicn to the 4.1.4.4 Jo1rrdaL Mach 5. 1998 between the resulting structural system eigenvalues and the objective elgenvalues are
found. It is shown that the structural
system redesign process is successful, fOr this example. Care must be taken, however, to determine the allowable size of the prescribed changes in the structural system elgenvalues.
1. Nomenclature
= matrix of baseline structure = matrix of objective structure
matrix, change from the baselineto the objective structure, [X']
= [Xj + [XJ
transformation matrix from the {4Y} space to the {z.X} space
2 Introduction
Reducing cost and shortening the time required to bring a new product to,market, while improving quality, are three important objectives in the design of
any system. It is critical to be ab!e to evaluate design proposals quantitatively with respect to structural
Réagiung the Dwnc o(Structuril Sya Jo&m 8. Ferns. Nrdi t Baitsas. kv L Sum
Sübmined for pubbauion lo the AL4A Jow-,mol. Mreh 5. 1998
performance Specifications. Recommending design changes suchthat these performance specifications are achieved without trial and error or repeated analyses could provide a powerfiui tool to designers. Finding a feasible design - possibly optimal accotding to a selected criterion - early in the design process allows efficient implementation.
It is the premise of this work that the use of mathematical models to predict structural system performance, early in the design process, will allow high quality products to be designed quickly and. with reduced cost. Toward this end, the issues involved in
redesigning a structural system are addressed in this work. A structural system is comprised of an assemblage of substructures. It is assumed that the basic geometry and material propel-ties of each substructure has been established during the initial design phase. Once the baseline structural finite element model is established, it is desirable to redesign the structure such that performance specifications are achieved.
Currently structural performance specifications are satisfied on an ad hoc basis, using "computer aided trial-and-error." That is; when Structural systems are used, it is up tO the designer's experence and intuition to determine what substructures, and what specific properties of each substructure, will affect the system performance. Simulations and numerical sensitivity analyses are used as aides to determine how these substructure
parameters should be changed.
LargE Admissible Perturbation (LEAP) theory (iscussed in Section 3.1) can be applied to establish the relationship between the. parameters of a structure (e.g., plate thickness) and its eigenvalues. The process described by LEAP theory is appropriate when static and dynamic constraints are placed on the performance of a single structure.
Redung the D iimj C(Sb,jcwiJ
JQ1U%B. Ferns. Midiul iL 8a,U 3erv L Stein
Submitted foc pubIiuon to the iIL4A Joiir,,W.Msd 3.1998
The allowable complexity of a structural model that is not comprised of substructures is limited by the ability of nonlinear Qptirnization codes to solve large nOnlinear optimization
problems. Currently. it is not possible to apply LEAP theory to a structural
system comprised of an assemblage ofsubstructures. The relationship between changes in the dynamic response of substructures and the corresponding changes to the dynamic response of the structural system remains to be developed.
The objective of this work is to develop a process by which the structural system is redesigned to meet performance specifications placed on its eigenvalues. This objective is achieved by developing the expressions that relate changes in the eigenvalues Of the substructures to the corresponding changes in the structural system eigenvalues. The primary contribution of this work is the development of the first order perturbation relationship between changes in the structural system eigenvalues and changes in the substructure eigenvalues. The structural system redesign process, also developed in this work, is the result of deriving the first order perturbation relationship. That is, the
perturbation relationship is used in the redesign process to predict the minimal changes
in
the substructure eigenvalues that wil affect the objective changes in the structural
system eigenvalues.
The structural system may be represented by modal expansions and not necessarily represented by finite element models. In fact, the redesign process developed in this work
is applicable toany assemblage of substructures
or components that can berepresented by linear time-invariant sets of equations. In this work, however, the
description of the development of thisprocess will be n terms of structural
system redesign.
Rcdiping the Diazw o(SructwiJSyatcn
Jd Farit Michsel M. Banivas. kffi L. Steu
$unU f publication tothe AL4A Jc,gn,aL March 5 199S
The impetus for the structural system redësigñ rocess is that global changes -changes to the entire structural system (e.g., eigenvalucs) - should be addressed at the structural system level. The result of the redesign process is a predicted set of changes to the eigenvalues of each substrUcture. The predicted changes in the substructure eigenvaiues are then the objective eigenvaiue changes for each substructure redesign. Each substructure redesign can be handled currently by the implementation of LEAP theory. The objective changes in the substructure eigenvaluesare combined with objective changes of locali:ed performance, such as maximum allowable stresses or deflections at a point, during each substructure redesign.
3. Background
3.1. Large Admissible Perturbation Theory
The structural system redesign process is based on the theory of LargE Admissible. Perturbations (LEAP theory) (Bernitsas and Kang, 1991, Bernitsas and Taweka1 1991, and Bernitsas and Rim, 1994). The current state of LEAP theory, therefore, must be defined.
LEAP theory is applicable to structures described by
a finite element representation. These Structures are represented by a single model; using substructuring is not considered. Some of the redesign problems that are addressed are redesigning fOrspecified natural frequencies, mOde shapes, static displacements, and stresses. The models can be composed of bar, beam, plate, spring, mass, or solid finite elements.
Rignthg the D iwnics o(Suctu,jSv
a
.MiCtICI M. &nULsa. L SIeñi.Sunjued for bbcs*ioito the AL4.4 Journal. March 5. 1998
The redesignprocess by large admissible perturbations is nonlinear
in the. sense that it allows for large changes (on the order of 100%) between the baseline and the objective design. Thenew structure is completed without trial and error
or repeated finite element analyses. This is achieved by an iterative
predictor-coor algorithm.
In the predictor phase, an inadmissibleprediction is made of the
changes in the physical parameters of the model. The thickness of plates, and
moment of inertia and cross-sectional area of beams
are typically predicted. This inadmissible prediction
is based on the
first order perturbation relationship between these physical parameters and the objectiveresponse (e.g., natural frequencies and static defiections).
At the end of thepredictor phase in the first iteration, the cognate spaces are defined and the modes are grouped in
these spaces. For example, for a cantilevered beam, the torsion modes form a cognate space and' the bending modes form a separate cognate space. In the correction
phase, admissible nonlinear changes are made to define the changes in the physical
parameters of the mcdel. These changes are based on satisf,'ing the
nonlinear form of the equations for the objective response (e.g., natural frequencies) and the orthogonaljty conditiOns uirig the
uTpdted
mode information determined in the prediction
phase. The next increment is made baed on the results from the admissible
nonlinear correction phase. The prediction phase of LEAP theory is based on the matrix
perturbation techniques developed by Stetson (1975) and improved by Stetson and Harrison (1981), Sandstróm and Anderson (1982), and Kim and Anderson
(1984). These linear methods find an objective statevery close to the original
state. A new finite elementsolution is required to proceed to the next increment.
Riing thePWnIC of Structural Systems
ioma. Fems. MjdaeI M. B TTÜtSáLJccy L. Stcm
SUhued for pubbrat,on to the.4L4A Journal.March 5. 1998
the inadmissible linear predictioA step is not e4uired in the implementation of LEAP theory. The LEAP theory approach allows all incremental predictions to be made from
the results of the initial finite element
solution and the corresponding nonlinearcorrections.
3.L Component Mode Synthesis
Component Mode Synthesis (CMS) is used in this work to develop the structural system. This technique allows substructures (i.e., components) to be cOmbined, via the substructure connectivity constraints, and yet retain the aiaiytical relationship between the substructure degrees of freedom and the structural system degrees of freedom. This feature is consequential to the redesign process. Atnong the first to contribute to this technique include: Gladwell (1964), Hurty (1965), Craig and Bampton (1968), Goldman (1969), Hou (1969), Bajan et. al. (1969), MacNeal (1971) and Ben.fild and Hruda (1971). An outline of the technique is presented here because the first order perturbation ofthese relationships are developed in Section 4.
Assume each substructure is represented by second order equations in the form
[m(r)J{ii(r)} +[k0]{x) =
Eqn. I
where {c) is a vector which
represents the physical degrees of freedom of the rth substructure. Note. that damping can be included in Rayleigh form. The eigensolutionis
found based on a mixture of fixed and free geometric boundary conditionsyielding a set of
Pre-multiplying Eqn. 1 by [()(r)hl]T and usingEqn. 2 we have
['(r)J{P(r)} =
(0),
*qn.3
since the redesign process is dependent only on the eigensolution and where [A(F)] is the diagonal eigenvalue matrix of the rth substructure. These substructures are concatenated to form a diagonal [A,] matrix
Redithg the Dnam
C(StuJSà
i
a
Fañ3. Mi'haeI P4.Bcviñj. ée L.StemSubmittcd f th lion to the .4L4.4 Joi,. Muth 5 199g
modal matrix [()1 The modal
substructure degrees offreedom, (P(r)}, are related to the physical substçucture degrees of freedomby{X(rfll } =
[r)
]{P() I.:I2!IL. [o}
1::
[oJ 9 Egn. 2 Eqn. 4 or(P1 = [J{p}
Eqn. 5The first step in the synthesis process js to write the physical constraint equatiOns, corresponding to the substructure connectivity, in terms of the substructure physical degrees Of freedom,
[A.]{x)
(0).Eqn.6
where
[1
Redign1ng the D ,iamics of SUuctwaj Sya,s 1dm a Ferris. Michael M. Bernittas. Jeffiev L Stein
Submitted for publication to the AL4A Jow'nal. MarCh 5. 1998
The physical constraint equations are converted to modal degrees of freedom
[AJ{x}
= [. [ crsuu.DEqn. 8
and partitioned into [Ad] and [As] such that [Ad] is square and {p) is partitioned into {Pd} and {p8} to correspond with the partitioning of[Ad] and [A.J, that is
:i
Eqn. 9
Now, {p} represents the dependent degrees of freedom and
{P8} represents theindependent degrees of freedom.
The relationship between the dependent and
independent degrees of freedom is=
{:
II
= [t3J{pg } = [BJ{q),Eqn. 10
[ifbi]
Eqn. 11 The degrees of freedom of the structzra1system, { q }, are related to the modal degrees of freedom of the substructures, {p}, via the [f3] matrix.
11
Redesigning the Diiasnzcso(Stngswiii Sysiai
Jdm & FCiT%LMIChaeJ M. B mise&jetht L Steài
Subnüned f publicaziint to the A24A JOIP,,QL Marc 5. 1998
The [3] transformation matrix iS used to constrain the substructure equations t form the system equations.
{} =
Eqh. 12
where
[A}
=([]r[DJ)'[B]f[AJ
qn. 13
The eigenvalueproblem for the structural system is,
ffA}_
[ii){ (0),Eqn. 14
whch can be solved for the eigenvaites and eigenvectors of the system, X and and the eigenvalue and eigenvector matrces, {AS1 and [(V] {
} .J. Now
for computational convenience that will be demonstrated later in this work, the inverse
of
the eigenvector matrix is defined as[rrn]=fxçj' and[r4T=[y,}
Eqn.15
Writing the transpose of the inverse of the eigenvector matrix
as a concatenation of
vectors will help clari& the redesign process developed next.
4. Structural System Redesign Denoting properties of theobjective state byprimes we have
[iJ
=or
Eqn.17
The matrix of admixture coefficients for [C]
is [Ce], and the matrix of admixture coefficients for [ri,,] is [C1], that isC- C
ofO
Cand the diagonal terms are perturbation of Eqn. 20 becomes
[c
=
([çj+ [
[çJ= [rM}+[rj= [c4[c'JrM}.
Substituting Eqn. 18 and Eqn. 19 into Eqn. 17 yields
= ([ç,]
[c'!r])(JA]f]X[
Eqn20 Given that the perturbation of the substructure. eigeñvalues is on the order of an arbitrarily
small constant c, then if [cJ is on the order of ,
then [c'] is on the order
This is proven in Appendix A. Knowing this, the first order
Resàüng the Daxncsof SvciuraJ Svsten
Jo B.Fans.Michael,. Ben àsa.,. ieffiCy L Stein
Subnüued for pub'ication to the ALL4 Jow"aZ..Mj,rJi5.1998
Egn. 18
Eqri. 19
{x4=
=[.
13
Rcd'esiiig the Dianu O(Stnicturij Safts
kim B FanL Michael M. BaMaaa. JethEyL. Sieãi
Suimtiuedfr pubhcia to the AL4.4 Jovr,,aI, March 5.1998
4X =
Eqn. 22
Note that the diagonal terms Of [Cr
IA,,] = O(e) and [çcJT
= 0 and thereforeare not included in Eqn. 22.
Eqn. 22 is the basis for the redesign equation for the 1th
system eigenvalue; not all of the system eigenvalues need to have a prescribed change. For instance, if the first, third, and seventh eigenvaluesar to be changed, then
Eqfl. 23 In this case, the first, third,, and seventh system eigenvalues will define the redesign
constraint equations (not to be conâised.vith the physical constraint equations of Eqt. 6).
Eqn. 22 is rewritten in the form
of a trmsformation matrix forall of the system
eigenvalues that define the redesign constraint equations. This transformation fliatrixdescribes the first order pei-turbation relationships between changes in [A.,j
and changes in (XSV,), that is
Eq24 The relationship between changesin [Arn] and changes in [A]
and U3] is derived next The first order pertutbation relationship
between [iA] and [Ab] aiid
[j3],
derived from Eqn. 12, isRcdcsigning the Drnaimcs of Strucwr*l Syam
icha B. Fei Michael%f. Bemiisaa. Jeffity L Steifl
Submittedfor publicaticm totha.4L4AJOUrITO& March5. 1998
] = J[13] +([r3]r[13]) [JT[A J[t3J
([]T[3]
[13]T[A([P1TE]Y
E3]r[AJ[3j
Egn. 25
Introducing the term [Crn] = [131TU3] provides a means of dividing the task of finding the
first order perturbation relationship into several smaller steps. With the introduction of
this new matrx, the expression for the [EAS%S]relationship becomes
[L'AI} =
([c]1)]T[A,
][] +
[c41 [i3]T[A,
][f3]
[cJ' []T[
][f3] + [c] [13]T[A
][p],
or written in transformation matrix form is
) } +
[1'.,A
J{LASUb}Eqn.26
Eqn. 27
It is shown in Appendix A that the first order perturbation
of( [CJ') is
'4'
1_I) =[Cm
Eqn. 28
so the transformation matrix from the change in the [C1] space to the [CJ' space is
=
[TCIC]{CSVJ.
Eqn. 29
By definition, the [Crn] matrix is [p1T[p1 therefore the first order perturbation of the
and the transformation matrix is then where and I
{c,4 =[T](8f3}.
Eqn. 31 At this point, several first order perturbation relationships have been derived and the corresponding transformation matrices have been developed. Eqn. 24, Eqn. 27, Eqn. 29 and Eqn. 31 are combined to formJ[TACIITC_.
or written in transformation matrix form as
,,Jt4I3 +[T
J
[TJ
Eqn.34
Now, the first order perturbation relations are known between
{A) afld {43}
nd{iA,b). Next, {13} and {A}
will be related back to (AX}.The first order perturbationof Eqn. 11 is
[oJ
AJ'[LSAJ]
Eqn35
15
Rsiiizig the Dynamicz ó(StructurjSvstcn
Jaha ft Furà. Michael M. Berflita&s Je L Stein SulatUned f abhcthonto the 4L4A Jowno1 MarCh 5. 1998
+ [JT[ØJ.
Eqn. 30
ITC4+[TLJ[-r
Eqn.32
perturbation relation for [A.1[cD"] is
=
- C1a IN
Redesing the Diwnâcs of StnctunJ Sat
kim 8 Fum. Michael M. BcrTi13aLkffie LStgn
Submined forpublicatioti tothe AL-LA Journal. Mircti 5. 1998
{i13} =[TA }{A,
s.).
Eqn. 36
Since [A] is
a constant matrix, for a given substructure connectivity, the first Order1l
cI-a IN4 J,
Eqn. 39
..J.
Eqit 40
It is assumed that the sensitivity of the substructure eigenvectors with respect to the substructure eigenvalues is known- a priori. The sensitivity relationship is dependent on the specific substructure. It is beyond the scope of-the structural system redesign process to determine this sensitivity relationship. The relationship is represented, in transformation matrix form, as
n}
[t}{LX
Eqn. 41
where
[T}
O1TA.Next, and {Ab) will be related back
to {Xb).
Eqn. 37
and the transformation matrix form is
{ a,ttZ
} -
[i
°.:J{lEqn. -38
The transformations described by Eqn. 36 and Eqn. 38 are combined with Eqn. 32 to form
Rcdeimg the Diwni of Structurij Svai
Joim ft Fris. Michael M. &niitias.Je L Stein
Submitted for publication to the AL4A Journal.March 5, 998
It is clear from, Eqn.
4 and Eqn. 5 that
[ftib]
was created from the substructureeigenvalues. The transformation
matrix between {A) and
(Ab}, therefore, is
comprised of zeros and ones,where
Eqn 42
The transformations in Eqn. 41 and Eqn. 42 are combined with those found in
Eqn. 39 to
form the final transformation matrix
[i
J = [JT
1T I + {t ,,hr
JL A_bA_s 17 I) !TA__ ..4T.A_s] Eqn. 43 Eqn.44The final structural transformation matrix of Eqn. 44 is the result of derivingthe first order perturbation relationship between the
substructure eigenvalues and the Structural system elgenvalues. The primary objective of this work is to establish this relationship, now the structural system redesign process is developed.
The structural system redesign process is based on the solution to an optimization
problem. The structural transformation matrix is
a set of linear equality constraint
functions for the optimization process. The objective function used here is a minimum change criterion on the square of the fractional changes of the substtucture
Re&aigning the Dynamics of StxijcturaJ Svgana
Jolm B Fcrrii. Michael M. Banitsas. Je&cy L Steiii Sübmiñed f pubIicati to the AIM Jo.rnwl. March 5. 1998
This function allows relatively large changes in the structural system eigenvalues
compared to other objective functions.
The solution to this optimization problem is
{
&} [TAx
f
{x).
Eqn. 45
The form of the generalized inverse denoted by [ ], depends on whether the problem is underconstrajned, overconstrained or determinant. For the underconstrained problem
=
[TJ(Jt.J[T..]T)
Eqn. 46
5. Example
5.1. Structural System Description
Two simple types of substructures are used to demonstrate the structural system redesign process. These suspension and plate substructures are represented
schematically
by Figure 1. The dynamic equations for these substructures are in modal form, the plate was derived via a finite element analysis and the suspension was derived from first
19
RcdeWi the Ditantjcs o(StjturJ Svcn
iom & Fur,i. Mitháel M. Benitsa. Jcfficy L Stci
Suflined picatito the .tL4A Joirno1, Mardi 5. 1998
Figure I. Schematic of Suspension and Plate Suttnictures
The structural system is an isolated platforrn shown schematically
consisting of a plate and four suspension substructures. The plate is substructure #1 and includes 3 elastic modes. The rigid body modes must always have eigenvalues equal to zero. Only the eigenvalues in the plate finite element model that correspond to the elastic modes can be changed, so only these eigenvalues are induded in the transformation
matrices. The four suspensions are substructures #2 through #5.
Substructures #2 through #4 include modes while substructure #5 includes I mode.
The structure is synthesized via the process described in Section 3.2. The first five eigenvalues for this structure are: 18.36 rads/sec, 34.16 rads/sec,. 39.35 rads/sec, 72.67 rads/sec, and 183.01 rads/sec.
n Figure 2,
Figure 2. Schematic of an Isolated Platfonu
It is assumed that the first two structural system eigenvalues
must be changed.
[iJ=
A scaling technique is employed before the generalized inverse of the transformation matrix is found. The fractional changes in the eigenvalues are used instead of the absolule changes. The a notation is used to denote a fractional change, and the following substitutions are made in Eqn. 47
where
=[
In this example, the [T1 mitrix is,
I,
d{X,
{a } = [tci]{a },
Rcdgning the t)rnanüc5 o(Snciwii Sait
hIm B. Frà. Mi.±aeI M. Bernitsas. Jcffi'ë, L Stein
Submmed for pubhcabcm to the .4L4A ./o,jrnoL Mud S. 199$
the baseline structure, is all the inforrriätiofl needed to begin the redesign process. In this example, the selection of the first tWo eigenvalues is arbitrary; in practice this choice would be base4 on issues such as the input frequency spectrum and the desired output frcquency spectrum.
Now each transformation matrix in the structural system redesign process is formed from the corresponding first order perturbation relationship. Only the final structural
transformation matrix, [I;,b], and its generalized inverse, [I;b], will be
discussed in this section. The structural transformation matrix for this example is,k,
J[A].
Eqn. 48
Eqn. 49
Eqn. 50
1-326E -3 362E -4 137E -3 6.82E -2 -554E -2 7 43E -0 236E - I 22E -2 -2 16E -2 284E -2
[-.u2E_6 $.30E-6 1S3E_6 333E-i -2S3E-4 9.IOE-O -7J1E-9 LO6E-8 -962E-9 6.68E-I Egn. 47
b} =.[A,bJ[T
Eqns2
This transformation is used to predict the changes in the substructure. eigenvalues that will produce, in turn, the objective changes in the structural system eigenvalués.
Each set of objective changes in the structural system eigenvalues corresponds to a predicted set of changes in the substructure eigenvalues. For instance, when the objective change in the first eigenvalue of the structure is -2% and the objective change in the second eigenvalue is + 1%, then the corresponding changes in the substructure eigenválues are
21
RiUng the Dsiiamcs o(Suictu1 Svt
Jm 8. Pà. Michael M. Scnütsai. hthty L SLein
Submined f pubbcaiion to the .4L4A Jour,,0!, Matth5. I 99S
ETa J=
f2.72E+I 335E-o 731E+I U7E-j -I.IOE-o 6.92E-o 235E-o &SOE-2 -839E-I
5.29E-2
[-L9oE-2 -2.27E-2 -516E-2 333E-! -3.03E-3 435E-O -321E-8 2.12E-S -2.06E-7 668E-I Eqn. 51
It is important to note that the generalized inverse of [TJ implies that the objective
function of this optimization is a minimum change criterion on the square of the fractional changes of the substructure eigenvalues. This objective function is desirable for the structural system redesign process.
The burden of satising the redesign
constraint equations for the objective structural system eigenvalues is distributed among all the fractional changes in the substructure eigenvalues, independently of the magnitude of those substructure eigenvalues. Since the example problem isunderconstrained, the transformation for fractional changes is
=[iJ
Error = Error; -5.15E-4 -3.82E-4 '-468E-4 5 5.96E 3l-6.14E-5i
-
f4;48E-6 1152E -55531E-7
[-557E-6
{L19E_2Redesigning the Dynamics o(Stnwa1 Syai
Jcm B. Fa. Michael M.BanAssi. Jcth L Steut
Subenined for publication to the AMA JoiirrtLMuch 5. 199*
and -7.92E + 1 -6.88E + 1 -460E + 2. 12.04E-1 i-2.24E - 2
-
f7.65E 5 12.78E-3 I 3.76E -5 1 l-4.07E-{4.o8E
-
1) Eqn.53Introducing the predicted substructure eigenvalue changes in ihe ft1i nonlinear cquations, in general5 will not exactly produce the objective, changes in the structural system eigenvalues. This is true in this case. The actual eigenvalue changes, actual eigenvalues and objective eigenvalues of the structural system are, respectively
1 0.0218 1 117.971 117.99
= .
andR}
e%)actua1
1-0.0439J
"
3435JEqn. 54
The error associate with this discrepancy is definednext.
The error between the ith objective system eigenvalue and the ith
actual eigenvaiue is defined as
The error in the first and secondstnictural system eigenvalues are
Redesiiñg the Damics oIStn*ctEàjSvenis
Jov. S. Fans. Midiacl M. Bcnüi kyLSteüi
Submjtied f pubhciicn to the tL4.4 Joiin,aL Maidi 5.1998
This error is small with respect to the objective changes, as expected.
Note that the value of the actual structural system eigenvalues is dependent
on all the objective eigenvalue changes. That is, the actual structural system eigenvalues are derived from the predicted changes in the substructure eigenvalues. The predicted changes in he substructure eigenvalues are determined, in turn, by the objective
changes in all he structural system eigenvalues via
Eqn.. 41 The
error for each of the structuralsystm
eigenvalues, as defined by Eqn. 55, is therefore a function of theobjective changes in ll
of the structuralsystem eigenvalues. Presently, this fact
will become clear when a domain of structural system eigenvalue changes is introduced.
Changes in the first and second eigenvalues of the structural system produces a t'vyo dimensional domain. The error between the objective
eigenvalues and the resultilg eigenvalues is calculated over this domain via Eqn 55. The
result is an error stirface over the domain of the changes inobjective eigenvalues. Since there
are two eigenvalues beirg changed, two error surfhces will result. The firsterror surface corresponds to the error the first eigenvalue over the domain, and the second error surface corresponds to the
errcr in the second eigenvalue. The first and second elgenvalues are varied by +1- 5% and tle error surfaces resulting from this domain are shown in Figure 3 and Figure 4.
- - - -- S
-Figure 3. Error Surface for the First Elgenvalue Of the StructuralSystem
2rd E igeriekie Error
--p
RIgnmg the Dnasnics of SuJduriJ Svems Jct B Femi. Michael M. Btñ.u. Jeffrey L Stein
SuiUcd for publication to the AMA Journal. Maecb5. 1998
id EineueErmr
S Ch&e: Ianbda 1
4-5
S Oi&ge: ltbde 2 Figure 4. Error Surface for the Second Eigenvalueof the Structural System,5 -5 S chew lanbde 2
Redmignmg the Diwnà of S*nicturj Sygcm J0IE ft Fans. Michael M. Bernitsai. JeeyL. Stem
Submitted for publication to the A1.L4 Journal. MMdt. 1998
5.2. of Example
The results summarized in Figure 3 and Figure 4 demonstratetwo important issues in the structural system redesign process. First, the
accuracy of the process can be evaluated for this example. Recall that the transformation matrices are derived from first
Order
perturbat ion relationships Theerror surfaces, therefore, should exhibit
two properties of first order perturbation relationships. The error in eigenvalue. predictions should
be zerc
when the objective change in the eigenvalues is zero, and the slope at this 'zero-zéro' point should be zero. Both Figure 3 and Figure 4 exhibit these properties. The
applcaton
of this process, therefore, is yielding appropriate results. The second issue is
the domain of allowable changes for this example.
Appropriate application of this process is dependent on the magnitude of the objective changes in the structural system eigenvalues. In the example, the changes shown for the first and second eigenvalues re +1- 5%. Figure 4 shows that this domain
is appropriate according to the error in the second eigenvalue. The maximum error in the second
eigenvalue is -1 .6%; this
error occurs when the first eigenvalue is changed by -5% and the second eigenvalue is changed by +5%. Part of this domain, however, is 'lot
appropriate according to the error in the first eigenvalue. Figure 3 shows that the error in the first eigenvalue is large (i.e., on the order of the objective changes in the eigenvalues)when th change in the first eigenvalue is positive and the change in the
second eigenvalue i negative. In orderto achieve changes in this direction several smaller incremental
changà
must be made.
Redesignthg the Diwne of truceUraI Sen
km B F Michael M Beninsa. Je&ev L Stem
Submitted for pubhcation to the .4L4.4 Je1rnaj,. Mitch 3. 1998
6. Discussion
Several development and implementation issues exist with this process and need to be addressed. The first development issue is the use of multiple transformation matrices. This representation has two major advantages. Thefirst major advantage is that it reduces
an otherwise complex set of perturbation equations into several smaller sets of
perturbation equations. This ifiakesthe derivation and explanation of the
finaltransformation matrix easier. Perhaps more importantly, it provides a framework on which future perturbation relationships can be derived more readily.
The first order perturbation relationship derived in this work relates the changes in the structural system eigenvalues to the changes in the substructure eigenvalues. In future work, it may be desirable to find the perturbation relationships involving other changes. These changes may include global 'unforced' responses such as structural system modes, or 'forced' responses such as maximum displacement, accelerations, or stresses. If this is the case, the first order perturbation relationship need only be found with respect to a perturbation relationship that is already known. For instance, the relationship between changes in the structural modes and the substructure eigenvalues may be desired. Since
changes in the structural modes, [D], are thnctions of the changes in [A], then the
[T3] transformation matrix
would have to be derived. This would be the only new transformation matrix that would have to be derived.The [Tb] transforTnatjon
matrix is already presented in this work. In this way, less development work is required as various first order perturbation relationships are developed in the future.Rcdcsiung the D%lwnia O(SuiS
kim 8. Fayi. Michael M. BCmIIS*L Jeffiev L S*elh
Submitted for pubbcauan to the .4L4AJoiirra1..Madt 5.1998
A majority of the first order
perturbation relationships developed in this work are required due to the [13] transformation matrix. It
s tempting to assume. that first otder changes in the substructure eigenvectors only cause second order changes in the system
eigenvalues. The system elgenvalues are based on Ritz approximations
of the system eigenvectors, which are linear combinations of the
substructure elgenvectors, and therefore first order changes to the substructure eigenvectors should only cause second order changes to the system eigenvalues. This is not the case, however, due to the matrix inversions required in the formation of [AS] and
[13]. The Ritz argument is applicable
when [AJ is
constant and the Ritz functions approximating the system eigenvectorsare perturbed. This explicit second order relationship is accounted fOr in Eqn. (6) when the diagonal terms are eliminated. In this case, however, [An,] itself is being perturbed.
7. Acknowledgment
The authors gratefully acknowledge Chrysler Corporation for supporting John Fèrtis, the Offshore Naval Research Department (DOD..G-N00014-94-1-1192) for Supporting Prof Bernitsas, and the Army Tank Automotive Command
(ARC DAAEo7-94Q...B).
for supporting of Prof. Stein.Appendix A
The following definitionsare required for the ensuing proofs: e is an arbitrarily small number
[Xl and [Y] are arbitrary invertable baseline matrices
Proof 1
We wish to prove: ([ii+
([i]
-
[c9)
+ o().
Knowing that ([i] +[Cx})([I]_[CxJ) =[IJ+.[C'Ic1
=and ptemultiplying by (Er]+
[Cu' yields
([IJ_[cxj)
= ([I}+[C1i' °&)
or ([i]+
[cxJj'
= ([i}
-
[c
Proof 2
We wish to prove: C 0(e2),C =0(e),i
j.
Weknow
[rIx']
['b
expanding primed matrices
([J
+[wD(jx]
+
[D
=[iJ, orRdcsigning the Diwm of Surnjri Sa15
Jo BFcvnt li,±a I M. B ns.jcffiyL Stein Subniv.d foi pubbcaii to the.4LL4 Journal. Mardi 3.1998 [X'] and [Y'J are invertable objective matrices
[X] and [iW] are matrices of changes to the elements of[X} and [Y] respectively
[Cx]and [C'] are admixture coefficient matrices
[X]
[K]+ [AX] [X] + [X][Cx] [X]([I] + [Cx])C =0, C' = 0(e), for i j
[Y']
[Y] + [EY] [Y] + [Y1[C [Y]([I] +[C'])The diagonal terms, therefore, are
c:: =0(c)
and the off-diagonal terms are
C = O(c),i
j.
Proof 3
We wish to prove: =
+ O(c1.
We know[x'IY'l
=['L
expanding the prime terms ([xJ+
+ =
['L
or ([xJ
+ [D[] +
([x][xD[Y]
=[I]. Substituting [XIYJ= [I] [AXIYJ + ([x] + [XD[Y] = [0],
or
[xX[IJ + [c1)[w]
=29
Rcduiung the D.nazTi o(Structurai S'i*rn
JQi B. Ferris. MICh*CI Nt. BeniñzaLJé&CV L Stern Suheiñt for pubIii ion to the AL4A JOurnal. March3. 1998
subtracting [YIx] [I] = [o],
or
-E'Fx}
Substituting the matrix of admixturecoefficients for [X] and [Y]yields
[CY]([x][x{c1
-[Yx][c1,
but [YJ[XJ=(1], so [c 'J([ij + [çX]) = _[c<},
from ProOf 1 =
and since
[cxIc<]
= (c)
=_[cx]+cc).
Redeaigning the Damics of StncturiiSteim
Jclma FemL Midtail M. Berñflsa. Jeffrey LStem
SnIin r publicationtotheAL4A JOkrnal.March5.1998
and pre-multiplying by [X']' [Y]
= +[CxJ[l[xJ[AxIyJ.
Using Proof 1
[YJ
=_([I]_[cx]+
0c)xr'[xIyJ,
or
[YJ
= -1
Xr'[iXJYJ 0(c),
and substituting [YJ=[Xr
[xJ')
=[xrt[xxJ-'
+ 0(e). Q.E.D.Appendix B Plate Substructure
The plate Substructure is 600 mm long and. has 6 plate elements in the x-direction and
1000
mm long and has 10 plate elements in the y-direction. The thickness of the plate elements varies as indicated below, where the thickness valuesare in mm.
-direcuon 4 16 16 16 18 IS 18 16 16 16 18 IS IS 16 16 16 18 18 18 16 16 16 18 18 18 12 12 12 14 14 14 12 12 12 14 14 14 10 I0 10 8 10 I0 10 10 I0 10 10 10 10
Riug the Dw ot&nw,waj
Sm
kim & Far.Michael M. Buyiitsa kffievL Seth
Submitted k totheAL4.4 Jov.ri,äl. Macb 5. I9S
The modulus ofelasticity is 206800 N/(min)2,
Poisson's ratio is 0.3, and the density is 782Er9 kg/(rnm)3. The first 6 eigenvalues and eigenvectors (for the displacementsat the four corners) are listed below.
=X2=3o, X4=1.536362E+05,
X5 = 1.800375E±05, =9.82l3Q7E+05Suspension Substructu, -7.389 4.557 -8.941 3.005 Substructure Number ki 2 5 20 40 20 40 365.83 182.92 731.66 365.83 -0.049 0.309 -0.034 0.2 18 -0.049 0.309 0.034 0.218 References
Bajan, R. L., Feng, C. C, and Jaszlics, I. J., "Vibration Analysis of Complex Structural Systems by Modal Substitution," The
Shock and Vibration Bulletin, No. 39, Part 3, 1969, pp. 99-105. Benuield, W. A.., and Hruda, R. F., "Vibration
Analysis of Structures by Component Mode Substitution," AIAA Journal, Vol. 9, No. 7, 1971, pp. 1255-61.
Berrátsas, M. M., and
Kang, B., "Admissible Large Perturbations in Structural Redesign," AlA Journal, Vol. 29, NO. 1, 1991, pp. 104..i 13.
31 5075 7.799 13.177 18.338 -2o.45c -1.923 10.246 8.498 16.696 10.432 -2.404 -4.540 5.061 7.344 -10.034 -9.402 -2.093 5.932 5.594 9.140 10 1000 2500 34. 169 0.218 0.069 20 1000 2500 17.084 0.154 QQ49 10 2000 5000 68.338 0.218 0.069 20 2000 5000 34. 169 O. 154 0.049
Redcsiung the Dtamics o(SuiciünJ Snn
Jo&m B. Fenis MicIaeI M. &niitsaL kthev L Stcj
Submitted f- pub1icatio, to the.4L4A Journal.MarCh 5. 1998 Bernitsas, M. M., and Rim, C. W., "Redesign of Plates by Large Admissible Perturbations," AIAA
Journal, Vol. .32, 1994, pp. 1021-1028.
Bernitsas, M. M, and Tawekal, R. L., "Structural MOdel Correlation Using Large Admissible Perturbations in Cognate Space," AIAA Journal, Vol. 29, No. 12, 1991, pp. 2222-32. Craig, R.. R., and Bampton, M. C. C., "Coupling of Substructures for Dynamic
Analysis;" AJAA
Journal, Vol. 6, No. 7, 1968, pp. 1313-19.
Gladwell, G. M. L., "Branch Mode Analysis of Vibrating Systems," Journal
of Sound and
Vibration, Vol. 1, 1964, pp. 41-59.
Goldman, R. L., "Vibration Analysis by Dynamic Partitioning," AIAA Journal, Vol. 7, No. 6, l969,pp. 1152-4.
Hou, S. N., "Review of
a Modal Synthesis Technique and a New Approach," Shock and Vibration Bulletin, No. 40, Part 4, 1969, pp. 25-30.Hurty, W. C, "Dyflamic Analysis. of Structural Systems Using Component Modes," AMA Journal, Vol. 3, No. 4, 1965, pp. 678-85.
Kim, K. -0., and Anderson, W. J., "Generalized Dynamic Reduction in Finite Element Dynamic Optimization," AIAA Journal, Vol. 22 No. 1.1, 1984, pp. 1616-17.
MacNeal, R. 11.4 "A Hybrid Method of Component Mode Synthesis", computers & Structures, Vol. 1, 1971, pp. 581-601..
Sandstrorn, R. E., and Anderson, W. J., "Modal Perturbation Methods for Marine Structures," Transactions of the Society of Naval Architects and Marine Engineers, Vol. 90, 1982,
pp.
