- Email: [email protected]
Structural Analysis Method of Investigation of Linear Complex Conservative Systems
Copyright © IFAC System Structure and Control, Nantes, France, 1995
STRUCTURAL ANALYSIS METHOD OF INVESTIGATION OF LINEAR COMPLEX CONSERVATIVE SYSTEMS A.V. PESTEREV and G.A. TAVRIZOV
State Institute of Physics and Technology, 13/7, Prechistenka str. i\;!oscow, 119034, Russia. E-mail: [email protected]
Abstract. The structural analysis method developed at the State Institute of Physics and Technology is intended for investigation of complex mechanical systems. The method is well formalised and can be applied to solution of many problems in the field of vibrations of complex systems both passive and active. It allows one to investigate efficiently different complex systems consisting of distributed and finite-dimensional subsystems interacting at a finite number of points by using a unified formalism . In the paper the general formalism of the technique as applied to spectral problems for conservative systems is discussed . The method involves use of the subsystem Green 's operators . Natural frequencies of the system are obtained by investigating some symmetric characteristic matrix whose order is equal to the number of linearly independent forces and moments of the interaction . Abstract. La methode de l'analyse structurale developpee dans l'Institut d ' Etat des Problemes Physiques et Technologies est destinee pour l'investigation des systemes mechaniques complexes. La methode est bien formalisee et peut et re utilisee pour la solution de bon nombre de problemes dans le domaine des vibrations des systemes actives comme passives. Elle permit d ' utiliser une formalisme uniforme pour etudier efficacement les systemes differents se composant des sous-systemes aux parametres distribues et concentres. L'article expose le formalisme general de la methode appliquee aux problemes spectrals des systemes conservatives . La methode utilise l'operateurs de Green des sous-systemes. Les frequences propres du systeme sont obtenues comme le result at de l'investigation de quelque mat rice symetriquee caracteristiquee dont l'ordre est egal au nombre des forces et des moments lineairement independants d 'interaction parmi les sous-systemes .
Key Words. Structural analysis method ; distributed parameter systems ; finite-dimensional interaction : Green 's operators ; characteristic matrix.
1. INTRODUCTION
systems , e.g. , if some components of the system are considered as distributed ones (rods , beams , plates and so on ) whereas other components are approximated by finite-dimensional models . The " interaction space" is assumed to be " poor", i.e. distributed subsystems interact with each other via a finite number of points , the number of degrees of freedom (DOFs) of a finite-dimensional subsystem (FDS) taking part in the interaction between subsystems is small as compared to the total number of DOFs .
The structural analysis method (or fac torized perturbations method ) is intended for investigation of complex physical (controllable) systems . The first results related to the method discussed were published in 1974 (Azarov et al. , 1974) with a more detailed discussion being found in the book by Azarov et al. (1975) . Since then the method was developed successfully and applied to investigation of mathematical models of some real space structures.
The most related to the method discussed are the Green 's function method (Bergman and McFarland , 1988 ; McFarland and Bergman ,1990) and the modal for ce technique (Yee and Tsuei , 1989) . whi ch reduce the problem of finding natural fr equencies of a combined system to investigation of
The aim of this paper is to outline the current structural analysis technique as applied to solving spectral problems for conservative systems consisting of subsystems interacting at a finite number of points . The method is most convenient when the system is composed of different sub-
61
some characteristic matrix. The authors of the former method consider uniform distributed subsystems connected to discrete masses and/or supported at a finite number of points. To obtain the characteristic equation governing natural frequencies of the combined system they use explicit formulas for the Green's functions of the distributed subsystems. The latter method is formulated as applied to FOSs. The authors use the boundary coordinate FRF matrices of subsystems to obtain the modal force matrix. The order of both the characteristic equation and the modal force matrix is equal to the number of physical boundary coordinates between the subsystems. In both cases the procedure of obtaining the characteristic (modal force) equation is not formalized (at least in the cited papers) but only described by means of various examples.
is the dynamic stiffness operator of the jth subsystem depending on spectral parameter A = w" , w is a circular frequency, Fj is a vector-function of amplitudes of external harmonic generalized forces acting on the jth subsystem , Yj is an amplitude response vector-function of the subsystem . AIj and k j are mass and stiffness operators of the jth subsystem. In the case of a one-dimensional distributed subsystem k j is a differential operator , 111j is an operator of multiplication by the mass distribution function mj (x). In the case of a FDS k j and lvlj are operators of multiplication by symmetric stiffness and mass matrices respectively. It is evident that vibrations of the aggregate of the non-interacting subsystems satisfy the equation
A(A)Y
In the context of the structural analysis method the procedure of calculating the characteristic matrix is well formalized. As soon as the system under investigation is divided into subsystems, such that the interaction between the subsystems is of finite rank. one can introduce notations described in the next section and obtain the characteristic matrix following a specified formal procedure. The characteristic matrix obtained for the case of conservative systems is always symmetric. This , as well as a minimal order of the matrix , is achieved by the use of appropriate factorization of the operator governing the interaction between the subsystems. Factorization of the interaction is a simple but very useful expedient , which, as far as the authors know. is not used in other related methods of investigating complex structures.
=F
,
where A(A) is a diagonal operator matrix •
A(A)
=
. ,
J( -
•
'If
AM = diag [Aj(A)];=l ,
y and F are the N -component vector-functions of responses and external forces , jth components of which are vector-functions Yj and Fj respectively. Let now the subsystems interact elastically at a finite number of points (e.g .. the subsystems are connected to each other by means of a finite number of conservative translational and rotational springs) . The equation governing vibrations of the system can be obtained from (3) by adding forces and moments of the interaction between the subsystems to the right side of the equation . Clearly, the interaction forces and moments depend on only kinematic displacements of those points that take part in the interaction . i.e. the interaction forces and moments depend on the finite-dimensional vector Y S rather than on the distributed response Y. Denote by S' an operator transforming vector-function Y into ys ys = SY. To illustrate this note that if a subsystem is a space beam the operator 5 acting on its response vector-function Y transforms it into an aggregate of Eulerian variables of those points of the beam at which springs are attached. Acting on a response vector of a FDS it " chooses" responses of those DO Fs (or linear combinations of them) that take part in the interaction.
2. THEORY 2. 1. lv!athematlcal Formulation of the Problem
Let us consider a linear complex conservative system consisting of a finite number N interacting subsystems with a finite number of interaction points. As the equations governing vibrations of distributed and finite-dimensional subsystems are different it is reasonable to use operator terminology to unify the discussion. The hat' will mark operators, adjoint operators are denoted by the * superscript. Assuming stationary vibrations under action of external harmonic forces and eliminating the time dependent part of the motion one can write the equation governing vibrations of the jth isolated subsystem in the following operator form ( 1)
Let the interaction between the subsystems be given by a stiffness matrix [{S , i.e. , an amplitude vector of interaction forces and moments FS be given by the equation
FS
= _f{sy s
Note that such a description of the interaction is the most convenient . On the one hand . if the interaction is described in terms of translational and/or torsional springs one can easily obtain
where linear self-adjoint operator (2)
62
the corresponding stiffness matrix. On the other hand. the description of the interaction in terms of the stiffness matrix allows one to formalize the mathematical treatment. To transform F' into the right side of equation (3) the adjoint operator S· is used . As a result one obtains
.4.(A)Y
ht're) and
(7)
is invertible , where j is an identity operator. If is invertible then
= F - S· f{' SY .
=
+ 5* f{S 5)
Y
=0 .
the perturbed op-
erator f = [+ &d is invertible if, and only if, the characteristic operator
0-1
Assuming F 0 one arrives at the following operator equation governing free vibrations of the system
(.4.(A)
r is invertible then
r
.
= r- I -
_
f
__
r- 1 &X- 1,e r- 1
.
(8)
Otherwise , dimensions of null spaces of rand :\ are equal each other and there exists one-to-one relationship between vectors from the null spaces
(4)
The discrete spectrum of the system is defined to be the set of numbers A such that equation (4) has non-trivial solutions. An example of formulation of the spectral problem in form (4) for a space beam with oscillators is given in Pesterev and Tavrizov (1994a) .
~
= [-1&7], 7] = -iJf"
f, E Ker
f , 7] E Ker X·
(9)
Formula (8) is well-known in matrix analysis it is used to calculate the inverse of an invertible matrix perturbed by a matrix of small rank (Horn and Johnson , 1985) . If operator [ is perturbed by an operator of finite rank , as in the problem discussed in the paper , the characteristic operator is a matrix.
2.2. Factori=ation of the Interaction As the interaction between the subsystems is conservative. matrix f{s is positive semidefinite and , hence , can be factorized in the form (Horn and J ohnson, 1985)
Denote vt--P) = A-l(,\). Applying formula (7) to the operator of the left side of equation (6) one arrives at the formula for the characteristic matrix
(5) (10)
where B is the matrix of order n x dim f{s , BT is its transpose. The number of rows n of matrix B is equal to the rank of f{S and is also equal to the number of linearly independent interaction forces. By using formula (5) equation (4) can be written in the explicit self-adjoint form
where I is the identity matrix of order n . Operator W(A) is referred to as the Green's (dynamic flexibility , receptance) operator of the aggregate of the non-interacting subsystems. Clearly. it is (as well as .4.( A») a diagonal matrix of operators : W(A) = diag [W-k (A)]f=I ' where vt'dA) is the Green 's operator of the kth isolated subsystem. In the case of a one-dimensional distributed subsystem , such as a rod or a beam , Vt--k(A) is an integral operator , the kernel of which is the Green 's function of the subsystem Gk(x , f,: A). For a FDS Wk(A) is the operator of multiplication by the receptance matrix Gk(A) (I{k - Mvh)- I It is well known that the Green 's operators can be calculated by means of the modal series
Factorization (5) . which is sometimes referred to as the skeleton factorization , is not necessary but very useful and advantageous. As will be seen further to obtain the characteristic matrix one does not need the factorization. But with the factorization the characteristic matrix obtained will have two important properties. First, the characteristic matri...x will be always symmetric. Second , it will have a minimal order . Factorization (.5) can be obtained either analytically (for simple stiffness matrices) or numerically (for complicated matrices obtained both experimentally and by means of a spring model of the interaction) .
=
~t-
k
(A)
= '" 'Pki'Pki W'(A) = '" 'Pj'pj . ~A-A' ~A'-A i
kz
j
( 11)
J
where Aki and Cflci ' i = 1, 2, ... , are eigenvalues (nat ural frequencies squared) and J,J-orthonormal (normalized to unitary mass) eigenfunctions of the kth subsystem: ('Pki , tVh'Pkj) = 8ij . 8iJ is the Kronecker delta, the inner product in the formula depends on the subsystem under consideration and is defined in either functional or finite-dimensional space ; the set of values Aj is a union of eigenvalues Aki , eigenfunction Pj is the N -component vector-function . such that if Aj
2.3. Characteristic Iv! atrix and the Green 's Operators According to the theorem on perturbation of an invertible operator (Azarov et al. , 1974, 1975) , if & and iJ are linear operators (to save room the necessary conditions of the theorem are omitted
t,
63
is equal to ~ki then the kth component of
Let ~I; be some new (nondegenerate) eigenvalue of the combined system of multiplicity rk , which is to say that the equation X(~dhl; 0 has rk non-trivial linearly independent solutions h~ E Ker X('\k), i = 1, .. . ,7\. Then, as it follows from formulas (6) and (9). the corresponding linearly independent eigenfunctions of the system can be calculated by the formula
:2.4. Solving Spectral Problem in Intervals of Regularity
2.5. Degenerate Eigenfrequencies and Corresponding Eigenfunctions
According to the aforementioned theorem, if A(A) is invertible problem (6) has non-trivial solutions if and only if characteristic matrix (10) is singular. As.4.( A) is invertible for all values of A not belonging to the spectra of the isolated subsystems , "new" or "nondegenerate" (McFarland and Bergman, 1990) eigenvalues of the combined system (i .e., other than eigenvalues of the isolated subsystems) satisfy the determinant equation: det X(A) = O. Note that in practice it is more convenient not to solve the determinant equation but use one of the two following ways.
Some eigenfrequencies of the isolated subsystems may also belong to the spectrum of the combined system. As the Green's operator does not exist at these frequencies the theorem on perturbation of an invertible operator is not applicable to this case and such frequencies have to be analyzed separately.
=
Let Aj be some eigenvalue of the aggregate of the non interacting subsystems of multiplicity rj. Rewrite formula (ll) for vt-(A) in a neighbourhood of Aj separating those terms that have pole Aj
As established by Pesterev (1992) eigenvalues of the characteristic matrix: are monotonically nondecreasing continuous functions of A in any interval [AI, '\:d not containing eigenfrequencies of the isolated subsystems . As a consequence, the number of non-negative eigenvalues of the characteristic matrix N+ (A) is mono tonic non-decreasing function of A in the interval and the number r of new eigenfrequencies in the interval can be calculated by the formula
vt--('\)
= ~-reg (A) + pJA' (:pJf _ A J
(14)
'
where (functional) matrix :pJ has rj columns , and define matrix X reg (,\) by the formula
Xreg('\)
- -. T = 1+ BSHreg(>.)S· B
X(A) -
==
BS ,~O (BS:pO)T YJ
J
Aj - A
(15)
Clearly, Vt-'reg(A) and Xreg(A) exist at Aj. One way to calculate new eigenfrequencies is to solve the spectral problem for the characteristic matrix and to look for the values of ,\ at which the characteristic matrix eigenvalues (those that change sign in the interval) equal zero. Formula (12) and the aforementioned properties of monotonicity and continuity of the characteristic matrix eigenvalues make solving the problem not difficult and guarantee that all new eigenfrequencies of the combined system will be found.
It has been shown that to determine whether Aj is also the eigenvalue of the combined system one should investigate the following (written in the partitioned form) matrix of order n + rj (Gould , 1966 ; Azarov et al. , 1974, 19(5) (16)
Eigenvalue Aj belongs to the spectrum of the combined system if, and only if Xj is singular and its multiplicity rj is equal to the dimension of the null space of Xj.
Another way is to transform the matrix into a triangular form by means of some inertia-preserving (congruent) transformation, e.g ., by Gaussian elimination without row or column interchanges , (Bergman and McFarland. 1988; McFarland and Bergman, 1990) and to look for those values of A at which some diagonal elements change signs . Clearly, the number of non-negative diagonal elements of the triangular matrix at some value of A is equal to N+(A).
The different method , based upon investigation of limit behaviour of the characteristic matrix eigenvalues in a neighbourhood of Aj, was proposed by Pesterev (1992) . The multiplicity rj of Aj in the spectrum of the combined system can be defined by calculating inertia of the characteristic matrix
64
on the left and on the right of Aj, and is given by the formula
rank of matrix BT is equal to the number of its columns . It is not difficult to show that vector F' is orthogonal to the corresponding eigenfunction of the resonating subsystem when isolated , and therefore the interaction forces do not cause resonance of the subsystem. In terms of limit behaviour of the characteristic matrix eigenvalues this case means that there exists an eigenvalut' of X(A) such that its limit is equal to zero wht'n ,\ tends to Aj (Pesterev, 1992).
If rj = 0, then Aj does not belong to the combined system spectrum. Eigenfrequencies belonging both to the spectra of the isolated subsystems and to the spectrum of the combined system are called "degenerate" (McFarland and Bergman , 1990) or ., conservative" (Gould , 1966) eigenfrequenCles.
Eigenfunctions of the combined system corresponding to the first and to the second case are called by McFarland and Bergman (1990) " nondegenerate" and" degenerate" modes , respectively. It is possible for both degenerate and nondegener ate modes to exist at the same degenerate frequency (that is consistent with formula (19». one relevant example of such a phenomenon is given by McFarland and Bergman (1990).
Let Aj belong to the spectrum of the combined system and let its multiplicity be rj, then there exist rj linearly independent vectors rri satisfying the equation
( 17) Write
g;
in the partitioned form
2. 6. Orthogonalization and Normalization of Eigenfunctions
(18)
If some eigenvalue of the combined system is multiple then corresponding eigenfunctions in the general case may be not orthogonal. To orthogonalize (and normalize) them one has to calculate inner product in the space which the eigenfunctions belong to. That is, the orthogonality relations and the formulas for norms are given either in a functional space (in the case of distributed subsystems) or in R-dimensional space (in the case of finite-dimensional subsystems) , where R is the number of DOFs of the combined system. It is established by Pesterev (1992) that in the context of the structural analysis method the problems of orthogonalizing eigenfunctions and calculating their norms can be solved in a finite-dimensional space even if the subsystems are distributed.
According to Pesterev (1992) rj linearly independent eigenfunctions of the combined system corresponding to Aj are given by the formula - 'Pi -
T,T.(' VY reg Aj
)S'*BThji
+ 'Pj0 Zij
(19)
It can be seen from the formula that in the general case any system eigenfunction is the sum of some linear combination of the eigenfunctions of the resonating subsystems (the second term) and some " particular" solution. To illustrate formula (19) take , for simplicity, rj = 1 (only one resonating subsystem at A = Aj) and consider, without going into details. two cases which can take place: 1) hi = O. The case takes place when the eigenfunction of the resonating subsystem corresponding to Aj has nodes at the points of connections to the other subsystems. The corresponding eigenvibration of the combined system is as follows. The resonating subsystem vibrates as if it were isolated , the forces of interaction between the subsystems are equal to zero . Actually, acting on the both sides of (19) by operator S' and premyltiplying by matrix f{s , using (5) and (16)-(18) , one obtains
For each eigenvalue ~j of the combined system introduce inner product < . , . > in a finitedimensional space as follows. If Aj is nondegenerate define the inner product in the n-dimensional space according to the formula
If Aj is degenerate and rj is its multiplicity in the spectra of the subsystems the inner product in (n + rj )-dimensional space of vectors 9 of form (18) is given by the formula
where (as rj = 1) S''PJ is one-column matrix and is a scalar . In terms of limit behaviour of eigenvalues of X( A) the case means that all the eigenvalues have removable singularities at Aj and none of them equals zero (Pesterev, 1992). 2) hi -::P O. In this case the resonating subsystem exchanges forces with other subsystems. Analogously to (20) one can obtain F S = _BT hi, the right side of the equation is non-zero as the
< gi , gk >= (hi , X~eg(~j)hk) + (z; . Zk).
z;
(22 )
The round brackets in the right sides of formulas (21) and (22) denote usual (Euclidean) scalar product of vectors. Derivatives of the characteristic matrix X'(~j) = B.S'vtl ' ( \ )S" BT and its regular component X~e9 (~j) = BSvV: eg (~j ).5" BT can be easily obtained by differentiating the series in (11). It is evident that matrices X' (~j) and 65
X~eg(~'j) are positive definite and, thus, formulas (21) and (22) define the inner products correctly.
(1975). Mathematical
1\-[ethods for Investigation of Complex Physical Systems. Nauka,
Moscow (in Russian). Bergman, L.A. , and D.M . McFarland (1988). On the vibration of a point supported linear distributed structure. J. Vib.! Acous. ! Stress and Reliab. Design, 110 , 485-492.
To obtain orthonormal eigenfunctions of the combined system one should set the norms of vectors hi E Ker X().j) (or g{ E Ker Xj) to unity and orthogonalize the vectors corresponding to the same frequency in the space with inner product (21) ( (22) ) and then take advantage of formulas (13) ( (19) ). It is shown that vectors hi orthogonal in the space with the metrics defined by formula (21) are also orthogonal in the ordinary sense (Pesterev, 1992) , and that orthogonality of vectors gi in the space with the metrics defined by (22) results in Euclidean orthogonality of vectors hi (Pesterev and Tavrizov , 1994c).
Gould , S.H . (1966).
Variational Methods fur Eigenvalue Problems. Oxford Dniv. Press.
London. Hirai, 1., T. Yoshimura, and K. Takamura (1913). On a direct eigenvalue analysis for locally modified structures. Int . 1. Numer. M eth. Engin. , 6,441-442. Matrix Horn, R.A. , and C.R. Johnson (1985). Analysis. Cambridge University Press, Cambridge . McFarland, D.M. , and L.A . Bergman (1990) . Analysis of passive and active discretedistributed linear dynamical systems using Green's function methods. Technical Report
3. CONCLUSION The general formalism of the structural analysis method as applied to spectral problems for complex conservative systems has been presented. The method is well suited to investigation of systems consisting of distributed and finitedimensional subsystems interacting at a finite number of points. It reduces the problem of finding eigenfrequencies of the combined system to the problem of investigating some characteristic matrix. The order of the matrix is equal to the rank of the interaction (the number of linearly independent interaction forces and moments) . The ad vanced formalism of the method allows one to obtain the characteristic matrix in a predetermined way. The use of factorization of the interaction results in a minimal order of the characteristic matrix and makes it symmetric . An efficient algorithm of calculating multiplicity of degenerate eigenfrequencies and established properties of the characteristic matrix eigenvalues guarantee finding all eigenfrequencies and orthonormal eigenfunctions of the combined system in a given frequency range. The problem of calculating both nondegenerate and degenerate modes corresponding to a degenerate eigenfrequency is solved successfully. It is shown how one can solve problems of orthogonalizing eigenfunctions and calculating their norms in the space , dimension of which IS equal to the rank of the interaction .
AAE 90-4 , UIL U ENG 90-0504 , University of Illinois, Urbana, Illinois .
Pesterev , A.V. (1992) . The method of guaranteed finding complex conservative systems discrete spectra. In: Computatwnal and Applied Mathematics I - Algorithms and Theory, Selected and revised papers from the 13th 11V[ACS World Congo (C. Brezinski and C.
Kulish , Eds), pp. 399-408. Elsevier. Pesterev, A.V. , and G .A. Tavrizov (1994a) . Vibrations of beams with oscillators, I: Structural analysis method for solving the spectral problem. J. of Sound and Vibr ., 170 , 521536. Pesterev , A .V., and G .A. Tavrizov (1994b). Vibrations of beams with oscillators, II : Generalized static Green's operator . 1. of Sound and Vibr., 170, 537-544. Pesterev, A.V. , and G.A . Tavrizov (l994c) . On inversion of some meromorphic matrices. Linear Algebra and its Applicatzons. 212/213 , 505-.5 2l. Yee , E.K.L., and Y.G . Tsuei (1989). Direct component modal synthesis technique for system dynamic analysis. AIAA 1.,27, 1083-1088.
4. REFERENCES
Azarov , V .L., L.N . Lupichev , and G.A. Tavrizov (1974). Factorized perturbations method in study of complex physical systems, Proc. of VI IFAC Cong. , Sec. !II, Yerevan , USSR, pp. 63-77 . Azarov , V.L. , L.N . Lupichev , and G .A. Tavrizov
66
STRUCTURAL ANALYSIS METHOD OF INVESTIGATION OF LINEAR COMPLEX CONSERVATIVE SYSTEMS A.V. PESTEREV and G.A. TAVRIZOV
State Institute of Physics and Technology, 13/7, Prechistenka str. i\;!oscow, 119034, Russia. E-mail: [email protected]
Abstract. The structural analysis method developed at the State Institute of Physics and Technology is intended for investigation of complex mechanical systems. The method is well formalised and can be applied to solution of many problems in the field of vibrations of complex systems both passive and active. It allows one to investigate efficiently different complex systems consisting of distributed and finite-dimensional subsystems interacting at a finite number of points by using a unified formalism . In the paper the general formalism of the technique as applied to spectral problems for conservative systems is discussed . The method involves use of the subsystem Green 's operators . Natural frequencies of the system are obtained by investigating some symmetric characteristic matrix whose order is equal to the number of linearly independent forces and moments of the interaction . Abstract. La methode de l'analyse structurale developpee dans l'Institut d ' Etat des Problemes Physiques et Technologies est destinee pour l'investigation des systemes mechaniques complexes. La methode est bien formalisee et peut et re utilisee pour la solution de bon nombre de problemes dans le domaine des vibrations des systemes actives comme passives. Elle permit d ' utiliser une formalisme uniforme pour etudier efficacement les systemes differents se composant des sous-systemes aux parametres distribues et concentres. L'article expose le formalisme general de la methode appliquee aux problemes spectrals des systemes conservatives . La methode utilise l'operateurs de Green des sous-systemes. Les frequences propres du systeme sont obtenues comme le result at de l'investigation de quelque mat rice symetriquee caracteristiquee dont l'ordre est egal au nombre des forces et des moments lineairement independants d 'interaction parmi les sous-systemes .
Key Words. Structural analysis method ; distributed parameter systems ; finite-dimensional interaction : Green 's operators ; characteristic matrix.
1. INTRODUCTION
systems , e.g. , if some components of the system are considered as distributed ones (rods , beams , plates and so on ) whereas other components are approximated by finite-dimensional models . The " interaction space" is assumed to be " poor", i.e. distributed subsystems interact with each other via a finite number of points , the number of degrees of freedom (DOFs) of a finite-dimensional subsystem (FDS) taking part in the interaction between subsystems is small as compared to the total number of DOFs .
The structural analysis method (or fac torized perturbations method ) is intended for investigation of complex physical (controllable) systems . The first results related to the method discussed were published in 1974 (Azarov et al. , 1974) with a more detailed discussion being found in the book by Azarov et al. (1975) . Since then the method was developed successfully and applied to investigation of mathematical models of some real space structures.
The most related to the method discussed are the Green 's function method (Bergman and McFarland , 1988 ; McFarland and Bergman ,1990) and the modal for ce technique (Yee and Tsuei , 1989) . whi ch reduce the problem of finding natural fr equencies of a combined system to investigation of
The aim of this paper is to outline the current structural analysis technique as applied to solving spectral problems for conservative systems consisting of subsystems interacting at a finite number of points . The method is most convenient when the system is composed of different sub-
61
some characteristic matrix. The authors of the former method consider uniform distributed subsystems connected to discrete masses and/or supported at a finite number of points. To obtain the characteristic equation governing natural frequencies of the combined system they use explicit formulas for the Green's functions of the distributed subsystems. The latter method is formulated as applied to FOSs. The authors use the boundary coordinate FRF matrices of subsystems to obtain the modal force matrix. The order of both the characteristic equation and the modal force matrix is equal to the number of physical boundary coordinates between the subsystems. In both cases the procedure of obtaining the characteristic (modal force) equation is not formalized (at least in the cited papers) but only described by means of various examples.
is the dynamic stiffness operator of the jth subsystem depending on spectral parameter A = w" , w is a circular frequency, Fj is a vector-function of amplitudes of external harmonic generalized forces acting on the jth subsystem , Yj is an amplitude response vector-function of the subsystem . AIj and k j are mass and stiffness operators of the jth subsystem. In the case of a one-dimensional distributed subsystem k j is a differential operator , 111j is an operator of multiplication by the mass distribution function mj (x). In the case of a FDS k j and lvlj are operators of multiplication by symmetric stiffness and mass matrices respectively. It is evident that vibrations of the aggregate of the non-interacting subsystems satisfy the equation
A(A)Y
In the context of the structural analysis method the procedure of calculating the characteristic matrix is well formalized. As soon as the system under investigation is divided into subsystems, such that the interaction between the subsystems is of finite rank. one can introduce notations described in the next section and obtain the characteristic matrix following a specified formal procedure. The characteristic matrix obtained for the case of conservative systems is always symmetric. This , as well as a minimal order of the matrix , is achieved by the use of appropriate factorization of the operator governing the interaction between the subsystems. Factorization of the interaction is a simple but very useful expedient , which, as far as the authors know. is not used in other related methods of investigating complex structures.
=F
,
where A(A) is a diagonal operator matrix •
A(A)
=
. ,
J( -
•
'If
AM = diag [Aj(A)];=l ,
y and F are the N -component vector-functions of responses and external forces , jth components of which are vector-functions Yj and Fj respectively. Let now the subsystems interact elastically at a finite number of points (e.g .. the subsystems are connected to each other by means of a finite number of conservative translational and rotational springs) . The equation governing vibrations of the system can be obtained from (3) by adding forces and moments of the interaction between the subsystems to the right side of the equation . Clearly, the interaction forces and moments depend on only kinematic displacements of those points that take part in the interaction . i.e. the interaction forces and moments depend on the finite-dimensional vector Y S rather than on the distributed response Y. Denote by S' an operator transforming vector-function Y into ys ys = SY. To illustrate this note that if a subsystem is a space beam the operator 5 acting on its response vector-function Y transforms it into an aggregate of Eulerian variables of those points of the beam at which springs are attached. Acting on a response vector of a FDS it " chooses" responses of those DO Fs (or linear combinations of them) that take part in the interaction.
2. THEORY 2. 1. lv!athematlcal Formulation of the Problem
Let us consider a linear complex conservative system consisting of a finite number N interacting subsystems with a finite number of interaction points. As the equations governing vibrations of distributed and finite-dimensional subsystems are different it is reasonable to use operator terminology to unify the discussion. The hat' will mark operators, adjoint operators are denoted by the * superscript. Assuming stationary vibrations under action of external harmonic forces and eliminating the time dependent part of the motion one can write the equation governing vibrations of the jth isolated subsystem in the following operator form ( 1)
Let the interaction between the subsystems be given by a stiffness matrix [{S , i.e. , an amplitude vector of interaction forces and moments FS be given by the equation
FS
= _f{sy s
Note that such a description of the interaction is the most convenient . On the one hand . if the interaction is described in terms of translational and/or torsional springs one can easily obtain
where linear self-adjoint operator (2)
62
the corresponding stiffness matrix. On the other hand. the description of the interaction in terms of the stiffness matrix allows one to formalize the mathematical treatment. To transform F' into the right side of equation (3) the adjoint operator S· is used . As a result one obtains
.4.(A)Y
ht're) and
(7)
is invertible , where j is an identity operator. If is invertible then
= F - S· f{' SY .
=
+ 5* f{S 5)
Y
=0 .
the perturbed op-
erator f = [+ &d is invertible if, and only if, the characteristic operator
0-1
Assuming F 0 one arrives at the following operator equation governing free vibrations of the system
(.4.(A)
r is invertible then
r
.
= r- I -
_
f
__
r- 1 &X- 1,e r- 1
.
(8)
Otherwise , dimensions of null spaces of rand :\ are equal each other and there exists one-to-one relationship between vectors from the null spaces
(4)
The discrete spectrum of the system is defined to be the set of numbers A such that equation (4) has non-trivial solutions. An example of formulation of the spectral problem in form (4) for a space beam with oscillators is given in Pesterev and Tavrizov (1994a) .
~
= [-1&7], 7] = -iJf"
f, E Ker
f , 7] E Ker X·
(9)
Formula (8) is well-known in matrix analysis it is used to calculate the inverse of an invertible matrix perturbed by a matrix of small rank (Horn and Johnson , 1985) . If operator [ is perturbed by an operator of finite rank , as in the problem discussed in the paper , the characteristic operator is a matrix.
2.2. Factori=ation of the Interaction As the interaction between the subsystems is conservative. matrix f{s is positive semidefinite and , hence , can be factorized in the form (Horn and J ohnson, 1985)
Denote vt--P) = A-l(,\). Applying formula (7) to the operator of the left side of equation (6) one arrives at the formula for the characteristic matrix
(5) (10)
where B is the matrix of order n x dim f{s , BT is its transpose. The number of rows n of matrix B is equal to the rank of f{S and is also equal to the number of linearly independent interaction forces. By using formula (5) equation (4) can be written in the explicit self-adjoint form
where I is the identity matrix of order n . Operator W(A) is referred to as the Green's (dynamic flexibility , receptance) operator of the aggregate of the non-interacting subsystems. Clearly. it is (as well as .4.( A») a diagonal matrix of operators : W(A) = diag [W-k (A)]f=I ' where vt'dA) is the Green 's operator of the kth isolated subsystem. In the case of a one-dimensional distributed subsystem , such as a rod or a beam , Vt--k(A) is an integral operator , the kernel of which is the Green 's function of the subsystem Gk(x , f,: A). For a FDS Wk(A) is the operator of multiplication by the receptance matrix Gk(A) (I{k - Mvh)- I It is well known that the Green 's operators can be calculated by means of the modal series
Factorization (5) . which is sometimes referred to as the skeleton factorization , is not necessary but very useful and advantageous. As will be seen further to obtain the characteristic matrix one does not need the factorization. But with the factorization the characteristic matrix obtained will have two important properties. First, the characteristic matri...x will be always symmetric. Second , it will have a minimal order . Factorization (.5) can be obtained either analytically (for simple stiffness matrices) or numerically (for complicated matrices obtained both experimentally and by means of a spring model of the interaction) .
=
~t-
k
(A)
= '" 'Pki'Pki W'(A) = '" 'Pj'pj . ~A-A' ~A'-A i
kz
j
( 11)
J
where Aki and Cflci ' i = 1, 2, ... , are eigenvalues (nat ural frequencies squared) and J,J-orthonormal (normalized to unitary mass) eigenfunctions of the kth subsystem: ('Pki , tVh'Pkj) = 8ij . 8iJ is the Kronecker delta, the inner product in the formula depends on the subsystem under consideration and is defined in either functional or finite-dimensional space ; the set of values Aj is a union of eigenvalues Aki , eigenfunction Pj is the N -component vector-function . such that if Aj
2.3. Characteristic Iv! atrix and the Green 's Operators According to the theorem on perturbation of an invertible operator (Azarov et al. , 1974, 1975) , if & and iJ are linear operators (to save room the necessary conditions of the theorem are omitted
t,
63
is equal to ~ki then the kth component of
Let ~I; be some new (nondegenerate) eigenvalue of the combined system of multiplicity rk , which is to say that the equation X(~dhl; 0 has rk non-trivial linearly independent solutions h~ E Ker X('\k), i = 1, .. . ,7\. Then, as it follows from formulas (6) and (9). the corresponding linearly independent eigenfunctions of the system can be calculated by the formula
:2.4. Solving Spectral Problem in Intervals of Regularity
2.5. Degenerate Eigenfrequencies and Corresponding Eigenfunctions
According to the aforementioned theorem, if A(A) is invertible problem (6) has non-trivial solutions if and only if characteristic matrix (10) is singular. As.4.( A) is invertible for all values of A not belonging to the spectra of the isolated subsystems , "new" or "nondegenerate" (McFarland and Bergman, 1990) eigenvalues of the combined system (i .e., other than eigenvalues of the isolated subsystems) satisfy the determinant equation: det X(A) = O. Note that in practice it is more convenient not to solve the determinant equation but use one of the two following ways.
Some eigenfrequencies of the isolated subsystems may also belong to the spectrum of the combined system. As the Green's operator does not exist at these frequencies the theorem on perturbation of an invertible operator is not applicable to this case and such frequencies have to be analyzed separately.
=
Let Aj be some eigenvalue of the aggregate of the non interacting subsystems of multiplicity rj. Rewrite formula (ll) for vt-(A) in a neighbourhood of Aj separating those terms that have pole Aj
As established by Pesterev (1992) eigenvalues of the characteristic matrix: are monotonically nondecreasing continuous functions of A in any interval [AI, '\:d not containing eigenfrequencies of the isolated subsystems . As a consequence, the number of non-negative eigenvalues of the characteristic matrix N+ (A) is mono tonic non-decreasing function of A in the interval and the number r of new eigenfrequencies in the interval can be calculated by the formula
vt--('\)
= ~-reg (A) + pJA' (:pJf _ A J
(14)
'
where (functional) matrix :pJ has rj columns , and define matrix X reg (,\) by the formula
Xreg('\)
- -. T = 1+ BSHreg(>.)S· B
X(A) -
==
BS ,~O (BS:pO)T YJ
J
Aj - A
(15)
Clearly, Vt-'reg(A) and Xreg(A) exist at Aj. One way to calculate new eigenfrequencies is to solve the spectral problem for the characteristic matrix and to look for the values of ,\ at which the characteristic matrix eigenvalues (those that change sign in the interval) equal zero. Formula (12) and the aforementioned properties of monotonicity and continuity of the characteristic matrix eigenvalues make solving the problem not difficult and guarantee that all new eigenfrequencies of the combined system will be found.
It has been shown that to determine whether Aj is also the eigenvalue of the combined system one should investigate the following (written in the partitioned form) matrix of order n + rj (Gould , 1966 ; Azarov et al. , 1974, 19(5) (16)
Eigenvalue Aj belongs to the spectrum of the combined system if, and only if Xj is singular and its multiplicity rj is equal to the dimension of the null space of Xj.
Another way is to transform the matrix into a triangular form by means of some inertia-preserving (congruent) transformation, e.g ., by Gaussian elimination without row or column interchanges , (Bergman and McFarland. 1988; McFarland and Bergman, 1990) and to look for those values of A at which some diagonal elements change signs . Clearly, the number of non-negative diagonal elements of the triangular matrix at some value of A is equal to N+(A).
The different method , based upon investigation of limit behaviour of the characteristic matrix eigenvalues in a neighbourhood of Aj, was proposed by Pesterev (1992) . The multiplicity rj of Aj in the spectrum of the combined system can be defined by calculating inertia of the characteristic matrix
64
on the left and on the right of Aj, and is given by the formula
rank of matrix BT is equal to the number of its columns . It is not difficult to show that vector F' is orthogonal to the corresponding eigenfunction of the resonating subsystem when isolated , and therefore the interaction forces do not cause resonance of the subsystem. In terms of limit behaviour of the characteristic matrix eigenvalues this case means that there exists an eigenvalut' of X(A) such that its limit is equal to zero wht'n ,\ tends to Aj (Pesterev, 1992).
If rj = 0, then Aj does not belong to the combined system spectrum. Eigenfrequencies belonging both to the spectra of the isolated subsystems and to the spectrum of the combined system are called "degenerate" (McFarland and Bergman , 1990) or ., conservative" (Gould , 1966) eigenfrequenCles.
Eigenfunctions of the combined system corresponding to the first and to the second case are called by McFarland and Bergman (1990) " nondegenerate" and" degenerate" modes , respectively. It is possible for both degenerate and nondegener ate modes to exist at the same degenerate frequency (that is consistent with formula (19». one relevant example of such a phenomenon is given by McFarland and Bergman (1990).
Let Aj belong to the spectrum of the combined system and let its multiplicity be rj, then there exist rj linearly independent vectors rri satisfying the equation
( 17) Write
g;
in the partitioned form
2. 6. Orthogonalization and Normalization of Eigenfunctions
(18)
If some eigenvalue of the combined system is multiple then corresponding eigenfunctions in the general case may be not orthogonal. To orthogonalize (and normalize) them one has to calculate inner product in the space which the eigenfunctions belong to. That is, the orthogonality relations and the formulas for norms are given either in a functional space (in the case of distributed subsystems) or in R-dimensional space (in the case of finite-dimensional subsystems) , where R is the number of DOFs of the combined system. It is established by Pesterev (1992) that in the context of the structural analysis method the problems of orthogonalizing eigenfunctions and calculating their norms can be solved in a finite-dimensional space even if the subsystems are distributed.
According to Pesterev (1992) rj linearly independent eigenfunctions of the combined system corresponding to Aj are given by the formula - 'Pi -
T,T.(' VY reg Aj
)S'*BThji
+ 'Pj0 Zij
(19)
It can be seen from the formula that in the general case any system eigenfunction is the sum of some linear combination of the eigenfunctions of the resonating subsystems (the second term) and some " particular" solution. To illustrate formula (19) take , for simplicity, rj = 1 (only one resonating subsystem at A = Aj) and consider, without going into details. two cases which can take place: 1) hi = O. The case takes place when the eigenfunction of the resonating subsystem corresponding to Aj has nodes at the points of connections to the other subsystems. The corresponding eigenvibration of the combined system is as follows. The resonating subsystem vibrates as if it were isolated , the forces of interaction between the subsystems are equal to zero . Actually, acting on the both sides of (19) by operator S' and premyltiplying by matrix f{s , using (5) and (16)-(18) , one obtains
For each eigenvalue ~j of the combined system introduce inner product < . , . > in a finitedimensional space as follows. If Aj is nondegenerate define the inner product in the n-dimensional space according to the formula
If Aj is degenerate and rj is its multiplicity in the spectra of the subsystems the inner product in (n + rj )-dimensional space of vectors 9 of form (18) is given by the formula
where (as rj = 1) S''PJ is one-column matrix and is a scalar . In terms of limit behaviour of eigenvalues of X( A) the case means that all the eigenvalues have removable singularities at Aj and none of them equals zero (Pesterev, 1992). 2) hi -::P O. In this case the resonating subsystem exchanges forces with other subsystems. Analogously to (20) one can obtain F S = _BT hi, the right side of the equation is non-zero as the
< gi , gk >= (hi , X~eg(~j)hk) + (z; . Zk).
z;
(22 )
The round brackets in the right sides of formulas (21) and (22) denote usual (Euclidean) scalar product of vectors. Derivatives of the characteristic matrix X'(~j) = B.S'vtl ' ( \ )S" BT and its regular component X~e9 (~j) = BSvV: eg (~j ).5" BT can be easily obtained by differentiating the series in (11). It is evident that matrices X' (~j) and 65
X~eg(~'j) are positive definite and, thus, formulas (21) and (22) define the inner products correctly.
(1975). Mathematical
1\-[ethods for Investigation of Complex Physical Systems. Nauka,
Moscow (in Russian). Bergman, L.A. , and D.M . McFarland (1988). On the vibration of a point supported linear distributed structure. J. Vib.! Acous. ! Stress and Reliab. Design, 110 , 485-492.
To obtain orthonormal eigenfunctions of the combined system one should set the norms of vectors hi E Ker X().j) (or g{ E Ker Xj) to unity and orthogonalize the vectors corresponding to the same frequency in the space with inner product (21) ( (22) ) and then take advantage of formulas (13) ( (19) ). It is shown that vectors hi orthogonal in the space with the metrics defined by formula (21) are also orthogonal in the ordinary sense (Pesterev, 1992) , and that orthogonality of vectors gi in the space with the metrics defined by (22) results in Euclidean orthogonality of vectors hi (Pesterev and Tavrizov , 1994c).
Gould , S.H . (1966).
Variational Methods fur Eigenvalue Problems. Oxford Dniv. Press.
London. Hirai, 1., T. Yoshimura, and K. Takamura (1913). On a direct eigenvalue analysis for locally modified structures. Int . 1. Numer. M eth. Engin. , 6,441-442. Matrix Horn, R.A. , and C.R. Johnson (1985). Analysis. Cambridge University Press, Cambridge . McFarland, D.M. , and L.A . Bergman (1990) . Analysis of passive and active discretedistributed linear dynamical systems using Green's function methods. Technical Report
3. CONCLUSION The general formalism of the structural analysis method as applied to spectral problems for complex conservative systems has been presented. The method is well suited to investigation of systems consisting of distributed and finitedimensional subsystems interacting at a finite number of points. It reduces the problem of finding eigenfrequencies of the combined system to the problem of investigating some characteristic matrix. The order of the matrix is equal to the rank of the interaction (the number of linearly independent interaction forces and moments) . The ad vanced formalism of the method allows one to obtain the characteristic matrix in a predetermined way. The use of factorization of the interaction results in a minimal order of the characteristic matrix and makes it symmetric . An efficient algorithm of calculating multiplicity of degenerate eigenfrequencies and established properties of the characteristic matrix eigenvalues guarantee finding all eigenfrequencies and orthonormal eigenfunctions of the combined system in a given frequency range. The problem of calculating both nondegenerate and degenerate modes corresponding to a degenerate eigenfrequency is solved successfully. It is shown how one can solve problems of orthogonalizing eigenfunctions and calculating their norms in the space , dimension of which IS equal to the rank of the interaction .
AAE 90-4 , UIL U ENG 90-0504 , University of Illinois, Urbana, Illinois .
Pesterev , A.V. (1992) . The method of guaranteed finding complex conservative systems discrete spectra. In: Computatwnal and Applied Mathematics I - Algorithms and Theory, Selected and revised papers from the 13th 11V[ACS World Congo (C. Brezinski and C.
Kulish , Eds), pp. 399-408. Elsevier. Pesterev, A.V. , and G .A. Tavrizov (1994a) . Vibrations of beams with oscillators, I: Structural analysis method for solving the spectral problem. J. of Sound and Vibr ., 170 , 521536. Pesterev , A .V., and G .A. Tavrizov (1994b). Vibrations of beams with oscillators, II : Generalized static Green's operator . 1. of Sound and Vibr., 170, 537-544. Pesterev, A.V. , and G.A . Tavrizov (l994c) . On inversion of some meromorphic matrices. Linear Algebra and its Applicatzons. 212/213 , 505-.5 2l. Yee , E.K.L., and Y.G . Tsuei (1989). Direct component modal synthesis technique for system dynamic analysis. AIAA 1.,27, 1083-1088.
4. REFERENCES
Azarov , V .L., L.N . Lupichev , and G.A. Tavrizov (1974). Factorized perturbations method in study of complex physical systems, Proc. of VI IFAC Cong. , Sec. !II, Yerevan , USSR, pp. 63-77 . Azarov , V.L. , L.N . Lupichev , and G .A. Tavrizov
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