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Journal of Sound and Vibration 267 (2003) 1047 ­ 1064

JOURNAL OF SOUND AND VIBRATION
www.elsevier.com/locate/jsvi

Interaction of eigenvalues in multi-parameter problems
A.P. Seyranian*, A.A. Mailybaev
Institute of Mechanics, Moscow State Lomonosov University, Michurinsky pr. 1, 119192 Moscow, Russia Received 8 July 2002; accepted 29 October 2002

Abstract The paper presents a new general theory of interaction of eigenvalues of matrix operators depending on parameters. Both complex and real eigenvalues are considered. Strong and weak interactions are distinguished, and their geometric interpretation on the complex plane is given. Mechanical examples are presented and discussed in detail. r 2003 Elsevier Ltd. All rights reserved.

1. Introduction Behaviour of eigenvalues with a change of parameters is a problem of general interest for applied mathematicians and natural scientists. This problem has many important applications in aerospace, mechanical, civil, and electrical engineering. Behaviour of eigenvalues depending on parameters is of a special value for vibration and stability problems, see, for example, the books by Bolotin [1], Panovko and Gubanova [2], Ziegler [3], Huseyin [4], Leipholz [5], Thompson [6], Thomsen [7], and Paidoussis [8]. The theory of interaction of eigenvalues, presented below, is based on the constructive perturbation methods, developed mostly by Vishik, Lyusternik, and Lidskii, see Refs. [9­11], and their extension to the multi-parameter case done by Seyranian [12,13], Mailybaev and Seyranian [14], and Seyranian and Kirillov [15]. In Refs. [12,13] the notion of strong and weak interactions of eigenvalues was introduced based on a Jordan form of the system matrix. If there are two eigenvectors corresponding to a double eigenvalue the interaction is called weak, and if there exists only one eigenvector with a double eigenvalue the interaction is called strong. It was shown that the strong interaction in one-parameter systems is characterized by two parabolae of equal curvature at the intersection point lying in perpendicular planes, while weak interaction occurs in
*Corresponding author. Tel.: +7095-939-5478; fax: +7095-939-0165. E-mail address: seyran@imec.mus.ru (A.P. Seyranian). 0022-460X/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0022-460X(03)00360-2


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p p

Re 0 Im Im 0

Re

Fig. 1. Strong and weak interactions of eigenvalues in one-parameter problem.

the same plane; see Fig. 1. It is remarkable that the speed dl=dp of strong interaction tends to infinity at the intersection point, while weak interaction is characterized by finite interaction speed. Some results on interaction of complex eigenvalues in vibrational systems were obtained by Seyranian [16], and Seyranian and Pedersen [17]. In these two papers the important mechanical effects like destabilization of a non-conservative system by infinitely small damping and transference of instability between eigenvalue branches were described and explained. Overlapping of frequency curves in circulatory systems was studied in Kirillov and Seyranian [18], and stability boundaries of Hamiltonian and gyroscopic systems were examined by Mailybaev and Seyranian [19] and Seyranian and Kliem [20]. In this paper, a general theory of interaction of two eigenvalues of matrix operators on the complex plane depending on multiple parameters is presented. Both complex and real eigenvalues are considered. Strong and weak interactions of eigenvalues, when one of the parameters is changed while increments of others remain constant, are distinguished based on a Jordan form of the system matrix. It is shown that the strong interaction of eigenvalues on the complex plane is described by hyperbolae with perpendicular asymptotes, if the double eigenvalue l0 is complex, and by a parabola and straight line, if l0 is real. Weak interaction of eigenvalues, characterized by two linearly independent eigenvectors at the point of coincidence, is studied. It is revealed that weak interaction is described by hyperbolae or a small elliptic bubble appearing from the point of the double eigenvalue l0 perpendicular to the plane of original interaction in the case of a real l0 : And if l0 is complex the interacting eigenvalues keep or interchange their main directions of motion on the complex plane before and after the weak interaction. It is emphasized that the presented theory of interactions gives not only qualitative, but also quantitative results on behaviour of eigenvalues based only on the information at the initial point in the parameter space. As mechanical examples, vibrational systems with coincident frequencies and stability of motion of a rigid panel in airflow are considered and discussed in detail.

2. Strong interaction Consider an eigenvalue problem Au ¼ lu; Ï1÷ where A is a real m á m non-symmetric matrix with the elements aij smoothly depending on a vector of real parameters p ¼ Ïp1 ; y; pn ÷; l is an eigenvalue, and u is a corresponding eigenvector. The eigenvalues l are determined from the characteristic equation detÏA þ lI÷ ¼ 0; where I is the identity matrix.


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Assume that at p0 ¼ Ïp0 ; y; p0 ÷ the matrix A0 ¼ AÏp0 ÷ possesses a double eigenvalue l0 ; as a n 1 root of the characteristic equation, with an eigenvector u0 and associated vector u1 satisfying the Jordan chain equations A0 u0 ¼ l0 u0 ; A0 u1 ¼ l0 u1 × u0 : Ï2÷

Eqs. (2) mean that there is only one linearly independent eigenvector u0 corresponding to the double eigenvalue l0 : Along with Eqs. (2), a left eigenvector v0 and associated vector v1 are defined by the left Jordan chain equations vT A0 ¼ l0 vT ; 0 0 vT A0 ¼ l0 vT × vT : 1 1 0 vT u0 ¼ 0; 0 vT u0 ¼ vT u1 a0: 1 0 Ï3÷ Ï4÷

The eigenvectors and associated vectors of problems (2) and (3) are related by the equalities [9] Note that the eigenvectors and associated vectors are not defined uniquely. Upon assuming that the vectors u0 and u1 are given, the following normalization conditions are introduced, vT u1 ¼ 1; 0 vT u1 ¼ 0; 1 Ï5÷ uniquely determining the vectors v0 and v1 : The aim of the paper is to study the behaviour of two eigenvalues l; which are coincident and equal to l0 at p0 ; with a change of the vector of parameters p in the vicinity of the initial point p0 : For this purpose, a variation p ¼ p0 × ee is assumed, where e ¼ Ïe1 ; y; en ÷ is a vector of variation with jjejj ¼ 1; and e > 0 is a small parameter. As a result, the matrix A takes the increment AÏp × ee÷ ¼ A0 × eA1 × e2 A2 × ?; where A1 ¼
n X @A ej ; @pj j ¼1 n 1 X @2 A ej ek ; 2 j;k¼1 @pj @pk

Ï6÷

A2 ¼

Ï7÷

with the derivatives taken at p0 : Due to these variations the eigenvalue l0 and the corresponding eigenvector u0 take increments, which can be given in the form of expansions [9] l ¼ l0 × e u ¼ u0 × e
1=2

l1 × el2 × e3=2 l3 × ?; Ï8÷

1=2

w1 × ew2 × e3=2 w3 × ?:

By substituting expressions (6) and (8) into eigenvalue problem (1), a chain of equations for the unknowns l1 ; l2 ; y and w1 ; w2 ; y is obtained. Solving these equations yields [9,14] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 ¼ 7 vT A1 u0 ; l2 ¼ ÏvT A1 u1 × vT A1 u0 ÷=2: Ï9÷ 0 1 0 For the sake of convenience, the following notation is introduced @A u0 ; aj × ibj ¼ vT 0 @pj


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1 T @A T @A cj × idj ¼ u1 × v1 u0 ; v 2 0 @pj @pj Dpj ¼ pj þ p0 ¼ eej ; j j ¼ 1; 2; y; n; Ï10÷

where aj ; bj ; cj ; dj are real constants, and i is the imaginary unit. Then, using expressions (8)­(10) gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n Xn l ¼ l0 7 Ïaj × ibj ÷Dpj × Ïcj × idj ÷Dpj × oÏe÷: Ï11÷ j ¼1
j ¼1

Eq. (11) describes the increments of two eigenvalues l; when the parameters p1 ; y; pn are changed under the assumption that e is small. Due to the condition jjejj ¼ 1 one has jjDpjj ¼ jjeejj ¼ e51: Ï12÷ Thus, inequality (12) implies that all the increments Dp1 ; y; Dpn are small for their absolute values. From expression (11) it is seen that when only one parameter, say the parameter p1 ; is changed while others remain unchanged Dpj ¼ 0; j ¼ 2; y; n; then the speed of interaction dl=dp1 is infinite at p1 ¼ p0 : Indeed, following Eq. (11) yields 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dl 1 a1 × ib1 × OÏ1÷: Ï13÷ ¼7 dp1 2 p1 þ p0 1 Since a1 × ib1 is a complex number, which is generally non-zero, dl=dp1 -N as p1 -p0 : 1 2.1. Real eigenvalue l
0

Consider the case of a real double eigenvalue l0 : In this case the eigenvectors u0 ; v0 and associated vectors u1 ; v1 can be chosen real and, hence, the constants bj ¼ dj ¼ 0; j ¼ 1; y; n; in Eq. (11). Let us fix the increments Dp2 ; y; Dpn and consider behaviour of the interacting eigenvalues depending on the increment Dp1 : Then, formula (11) can be written in the form l ¼ l0 × X × iY × oÏe÷; where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a c1 a X × iY ¼ 7 a1 Dp1 × × c1 Dp1 × þ × b; a1 a1 a1
n X j ¼2 n X j ¼2

Ï14÷

Ï15÷

and a and b are small real numbers a¼ aj Dpj ; b¼ cj Dpj : Ï16÷

Real quantities X and Y describe, respectively, the real and imaginary parts of the leading terms in eigenvalue perturbation (11). If a1 ÏDffiffiffiffiffi × a=a1 ÷ > 0; then Eq. (15) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 a a c1 a X ¼ 7 a1 Dp1 × × b; Y ¼ 0: Ï17÷ × c1 Dp1 × þ a1 a1 a1


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If a1 ÏDp1 × a=a1 ÷o0; then separating the real and imaginary parts in Eq. (15) gives a c1 a × b; þ X ¼ c1 Dp1 × a1 a1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a : Y ¼ 7 þa1 Dp1 × a1 Eliminating Dp1 × a=a1 from Eqs. (18) yields the parabola c1 ac1 X × Y2 ¼ b þ a1 a1

Ï18÷

Ï19÷

in the ÏX ; Y ÷ plane symmetric with respect to the X -axis. Since a and b are small numbers depending on Dpj ; j ¼ 2; y; n; parabola (19) gives trajectories of l on the complex plane with a change of the parameter p1 while the others remain fixed. Note that the constants a1 and c1 involved in Eq. (19) are taken at the initial point p0 in the parameter space. First, assume that a1 > 0 and Dp2 ¼ ? ¼ Dpn ¼ 0; which implies a ¼ b ¼ 0: It follows from Eqs. (17)­(19) that with the increase of Dp1 the eigenvalues come together along parabola (19), merge to l0 at Dp1 ¼ 0; and then diverge along the real axis in opposite directions. The general picture of strong interaction is shown in Fig. 2, where x ¼ 0 and the arrows indicate motion of the eigenvalues, when Dp1 increases. The case a1 o0 implies the change of direction of motion for the eigenvalues. If Dp2 ; y; Dpn are non-zero and fixed, then the constants a and b are generally non-zero. This means the shift of parabola (19) along the real axis by x ¼ b þ ac1 =a1 ; see Fig. 2. One can see that the double eigenvalue does not disappear. It changes to l0 × x × oÏe÷ and appears at p1 ¼ p0 þ a=a1 × oÏe÷: 1 2.2. Complex eigenvalue l
0

Consider a complex eigenvalue l0 : In this case the vectors u0 ; u1 ; v0 ; and v1 are complex and, hence, the constants bj and dj in expression (11) are generally non-zero. Keeping the lowest order term in Eq. (11) gives l ¼ l0 × X × iY × oÏe where
1=2

÷;

Ï20÷

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn X × iY ¼ 7 Ïaj × ibj ÷Dpj : j ¼1
a 1> 0 Im
0

Ï21÷

a 1< 0 Im Re
0

Re

Fig. 2. Strong interaction of eigenvalues (real l0 ).


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Taking the square of Eq. (21) and separating real and imaginary parts, one gets the equations X2 þ Y2 ¼
n X j ¼1 n X j ¼1

aj Dpj ;

2XY ¼

bj Dpj :

Ï22÷

Expressing the increment Dp1 from one of equations (22) and substituting it into the other equation yields X2 þ where g is a small real constant: g¼
n X j ¼2

2a1 XY þ Y 2 ¼ g; b1 a1 bj aj þ Dpj : b1

Ï23÷

Ï24÷

In Eq. (23) it is assumed that b1 a0; which is the non-degeneracy condition for the complex eigenvalue l0 : Eq. (23) describes trajectories of the eigenvalues l; when only the parameter Dp1 is changed and the increments Dp2 ; y; Dpn are fixed. If all Dpj ¼ 0; j ¼ 2; y; n; or if they are non-zero, but satisfy the equality g ¼ 0; then Eq. (23) yields two perpendicular lines sffiffiffiffiffiffiffiffiffiffiffiffiffi! a1 a2 × 1 × 1 Y ¼ 0; Xþ b1 b2 1 a1 þ b1 sffiffiffiffiffiffiffiffiffiffiffiffiffi! a2 1 × 1 Y ¼ 0; b2 1

Xþ

Ï25÷

which intersect at the origin. Two eigenvalues l ¼ l0 × X × iY × oÏe1=2 ÷ approach along one of lines (25), merge to l0 at Dp1 ¼ 0; and then diverge along another line (25), perpendicular to the line of approach; see Fig. 3, where the arrows show motion of l with increasing Dp1 : Strong interaction in the three-dimensional space Ïp1 ; Re l; Im l÷ is shown in Fig. 1 (left) with two identical parabolae lying in perpendicular planes.
<0 =0 >0

Im

Im
0

Im

0



0

Re

Re

Re

Fig. 3. Strong interaction of eigenvalues (complex l0 ).


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If ga0; then Eq. (23) defines two hyperbolae in the (X,Y) plane with asymptotes (25). When Dp1 increases, two eigenvalues come closer, turn, and diverge; see Fig. 3. When g changes the sign, the quadrants containing hyperbola branches are changed to the adjacent. Example 1. As an example, stability of vibrations of a rigid panel of infinite span in airflow is considered. It is assumed that the panel is maintained on two elastic supports with the stiffness coefficients k1 and k2 per unit span. The panel has two degrees of freedom: a vertical displacement y and a rotation angle j; see Fig. 4. It is supposed that the aerodynamic lift force L is proportional to the angle of attack j; the dynamic pressure of airflow, and the width b of the panel L¼c
y

rv2 bj: 2

Ï26÷

Here, cy is the aerodynamic coefficient, r and v are the density and speed of the flow respectively. It is assumed that m is the mass of the panel per unit surface. Damping forces are not involved in the model. Small vibrations of the panel in airflow are described by the differential equations [2] y × a11 y × a12 j ¼ 0; . j × a21 y × a22 j ¼ 0; . where a
11

Ï27÷

¼

k1 × k2 ; mb

a

12

¼

k1 þ k2 rv2 þ cy ; 2m 2m Ï28÷

a21 ¼

6Ïk1 þ k2 ÷ ; mb2

a22 ¼

3Ïk1 × k2 ÷ rv2 þ 3cy : mb 2mb rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 × k2 ; t¼t mb

It is convenient to introduce the non-dimensional variables k1 þ k2 k¼ ; 2Ïk1 × k2 ÷ cy rv2 q¼ ; 2Ïk1 × k2 ÷ y y¼ ; * b

Ï29÷

L ¾b v y k
1



k b

2

Fig. 4. A panel on elastic supports vibrating in airflow.


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where k is a relative stiffness parameter changing in the interval þ1=2pkp1=2; and qX0 is a load * parameter. Using these variables in Eqs. (27) and separating the time Ïy; j÷T ¼ u expÏint÷; one obtains the eigenvalue problem (1) with ! 1 kþq Ï30÷ A¼ ; l ¼ n2 : 12k 3 þ 3q The stability problem of motion of the panel depending on two parameters p ¼ Ïq; k÷ has been studied in Ref. [15]. The characteristic equation is l2 ×Ï3q þ 4÷l × 12kq þ 3q þ 12k2 × 3 ¼ 0: Ï31÷

Motion of the panel is stable if all eigenvalues l are positive and simple. The stability of the panel can be lost by divergence or by flutter, the boundaries of which are given by l ¼ 0 or double positive eigenvalues with a single eigenvector respectively. Setting the discriminant of Eq. (31) equal to zero gives pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qf ¼ Ï1 × 4k þ 2 kÏk × 2÷÷: Ï32÷ 3 This is the boundary between flutter and stability regions; see Fig. 5, where S and F denote the stability and flutter regions respectively. It follows from Eq. (32) that the flutter region belongs to the half-plane kX0: The other branch of the solution, with plus before the square root in Eq. (32), corresponds to the boundary between flutter and divergence regions [15], shown in Fig. 5 by dashed line. Consider a point Ïk; qf ÷ on the flutter boundary (32). At this point, the eigenvalue problem (1) with Eqs. (30) and (32) is solved and the double eigenvalue l0 ¼ 2 þ 3qf =2 with the corresponding eigenvectors and associated vectors satisfying normalization conditions (5) are found as 1 1 0 0 0 1 2Ïqf þ k÷ 0 12k C C B B u0 ¼ @ 3qf þ 2 A; u1 ¼ @ 2 A; v0 ¼ @ 2 þ 3qf A; 1 2 þ 3qf 2 0 1 24k B 3q þ 2 C v1 ¼ @ f A: 0

Fig. 5. The stability and flutter regions for the panel vibrating in airflow.


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Then, according to Eqs. (10) the quantities become 3 a1 ¼ þ Ï8k þ 3qf × 2÷; a2 ¼ 12Ï2k þ qf ÷; 2 b1 ¼ b2 ¼ 0; d1 ¼ d2 ¼ 0;

3 c1 ¼ þ ; 2

c 2 ¼ 0; Ï33÷

and the approximate formula (11) for the eigenvalues takes the form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 3qf 3 3 l¼ 7 þ Ï8k þ 3qf × 2÷Dq × 12Ï2k þ qf ÷Dk þ Dq: 2 2 2

Ï34÷

This formula coincides with that of obtained from characteristic equation (31) with first order Taylor expansion of the terms under and out of the square root. The equation of parabola (19) becomes Y 2 ×Ï8k þ 3qf × 2÷X ¼ 12Ïqf þ 2k÷Dk: Ï35÷ Due to expressions (32) and (33) the term a1 is negative for 0pkp1=2: This means that for Dk ¼ 0 with the increase of q in the vicinity of the flutter boundary two positive eigenvalues l approach each other, merge to l0 ¼ 2 þ 3qf =2; become complex conjugate (flutter) and diverge along parabola (35). If Dka0 is small and fixed, then there is a shift of the double eigenvalue by x ¼ þa2 c1 Dk=a1 ¼ 12Ïqf þ 2k÷Dk=Ï8k þ 3qf × 2÷ and a shift of the parameter q at which the eigenvalue becomes double (flutter boundary) 8Ï2k þ qf ÷ Dk : Ï36÷ qf Ïk × Dk÷Eqf Ïk÷þ a2 Dk=a1 ¼ qf Ïk÷× 8k þ 3qf × 2 Note that these ffiffiffi 2ffiffiffi p shifts are negative or positive depending on the sign ofpk þ qf ; which is negative pffiffiffi for 0pko2= 3 þ 1 and positive for 2= 3 þ 1okp1=2: At k ¼ 2= 3 þ 1 the function qf Ïk÷ takes the minimum; see Fig. 5.

3. Weak interaction Consider a double eigenvalue l0 of the matrix A0 with two linearly independent eigenvectors. Such an eigenvalue is called semi-simple. Two right eigenvectors u1 ; u2 and two left eigenvectors v1 ; v2 are determined by the equations A0 u i ¼ l 0 u i ; vT A0 ¼ l0 vT ; i i i; j ¼ 1; 2; i ¼ 1; 2; Ï37÷

vT uj ¼ dij ; i

where dij is the Kronecker delta. The last equation represents the normalization condition determining left eigenvectors uniquely for given right eigenvectors. Note that any linear combination of right (or left) eigenvectors is a right (or left) eigenvector. Consider a perturbation of the parameter vector p ¼ p0 × Dp; where Dp ¼ ee with a direction e in the parameter space Ïjjejj ¼ 1÷ and a small positive perturbation parameter e: Then, the eigenvalue l0 and corresponding eigenvector u0 take increments, which can be given in the form of


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expansions [9,13] l ¼ l0 × el1 × e2 l2 × ?; u ¼ u0 × ew1 × e2 w2 × ?: Ï38÷

By substituting expressions (6) and (38) into eigenvalue problem (1), a chain of equations for the unknowns l1 ; l2 ; y and u0 ; w1 ; w2 ; y is obtained. Solution of these equations yields the vector u0 ¼ g1 u1 × g2 u2 ; where g1 and g2 are unknown coefficients determined by (see [13]) ! ! ! g1 g1 vT A1 u1 vT A1 u2 1 1 ¼ l1 : vT A1 u1 vT A1 u2 g2 g2 2 2 A non-trivial solution g1 ; g2 of this equation matrix on the left side. Two eigenvalues l1 of determine leading terms in the expansions for (38), which appear due to bifurcation of the Introducing the notation Ï39÷

Ï40÷

exists if and only if l1 is an eigenvalue of the 2 á 2 this matrix and corresponding eigenvectors Ïg1 ; g2 ÷T two eigenvalues l and corresponding eigenvectors u double semi-simple eigenvalue l0 : Ï41÷

X × iY ¼ el1 ;

where X and Y are, respectively, real and imaginary parts of the term el1 ; expansion for eigenvalue (38) can be written in the form l ¼ l0 × X × iY × oÏe÷: According to relations (7) and (40), X × iY is an eigenvalue of the 2 á 2 matrix ! n X fj11 fj12 Dpj ; fj21 fj22 j ¼1 where fjkl ¼ vT k @A ul ; @pj Ï44÷ Ï42÷

Ï43÷

with the derivatives evaluated at p0 : Solving the characteristic equation for matrix (43) gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X Xn X × iY ¼ gj Dpj 7 h Dpj Dpk ; Ï45÷ j ;k¼1 jk
j ¼1

where gj ¼ Ïf
11 j

× fj22 ÷=2;

hjk ¼ Ïf

11 j

þ fj22 ÷Ïf

11 k

þ fk22 ÷=4 ×Ïfj12 fk21 × fj21 fk12 ÷=2:

Ï46÷

Note that hjk ¼ hkj for any j and k: Expression (45) determines approximation of eigenvalues (42), when the parameter vector Dp is changing under the assumption that it is small for the absolute value. The coefficients gj and hjk in this expression depend on the left and right eigenvectors, corresponding to the eigenvalue l0 ; and first derivatives of the matrix A with respect to parameters taken at p0 :


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3.1. Real eigenvalue l

0

Consider a real semi-simple eigenvalue l0 : In this case the eigenvectors u1 ; u2 ; v1 ; v2 can be chosen real and, hence, the coefficients fjkl ; gj ; and hjk in expressions (44) and (46) are real. By expressing the square root from equality (45) and taking the square of the obtained relation, two equations for real and imaginary parts are found as follows: !2 n n X X gj Dpj þY 2 ¼ hjk Dpj Dpk ; Xþ
j ¼1 n X j ¼1 j ;k¼1

! gj Dp
j

2 Xþ The second equation requires that X ¼ are obtained: Xþ and Y2 ¼ þ
n X j ;k¼1 n X j ¼1

Y ¼ 0:

Ï47÷

Pn

j ¼1 2

gj Dpj or Y ¼ 0: Therefore, two independent systems
n X j ;k ¼ 1

! gj Dpj

¼

hjk Dpj Dpk ;

Y ¼ 0;

Ï48÷

hjk Dpj Dpk ;

X¼

n X j ¼1

gj Dpj :

Ï49÷

Consider now the behaviour of eigenvalues depending on the parameter p1 ; when other parameters p2 ; y; pn are fixed. First, let us put the increments Dp2 ¼ ? ¼ Dpn ¼ 0: Then, Eqs. (48) and (49) take the form ÏX þ g1 Dp1 ÷2 ¼ h11 Dp2 ; 1 Y 2 ¼ þh11 Dp2 ; 1 Y ¼ 0; Ï50÷ Ï51÷

X ¼ g1 Dp1 :

Upon assuming that h11 a0 (the non-degenerate case), only one of systems (50) and (51) has nonzero solutions. If h11 > 0; then system (51) has only the zero solution X ¼ Y ¼ Dp1 ¼ 0; and system (50) yields pffiffiffiffiffiffi Ï52÷ X ¼ g1 Dp1 7 h11 Dp1 ; Y ¼ 0: Expression (52) describe two real eigenvalues (42), which cross each other at the point l0 on the complex plane as Dp1 changes from negative to positive values; see Fig. 6 Ïr0 ÷; where the arrows show motion of the eigenvalues with an increment of Dp1 : If h11 o0; then system (50) has only the zero solution, and system (51) yields pffiffiffiffiffiffiffiffiffiffi Ï53÷ X ¼ g1 Dp1 ; Y ¼ 7 þh11 Dp1 : These formulae describe two complex conjugate eigenvalues crossing at the point l0 on the real axis with a change of Dp1 ; see Fig. 6 Ïr00 ÷:


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Im

c

r Re 0 r

Fig. 6. Weak interaction of eigenvalues for Dp2 ¼ ? ¼ Dpn ¼ 0:

From expressions (52) and (53) one can see that in the three-dimensional space Ïp1 ; Re l; Im l÷; and Fig. 1 (right). Now, consider the case, when the increments Dp2 the notation n n X X gj Dpj ¼ g1 Dp1 × s; hjk
j ¼1 j ;k¼1

the weak interaction occurs at the same plane the speed of interaction dl=dp1 remains finite, ; y; Dpn are small and fixed. Then, upon using Dpj Dpk ¼ h11 ÏDp1 þ d÷2 × c; Ï54÷

where s; d; and c are small real constants depending on Dp2 ; y; Dpn ; n n n X X h1j X s¼ gj Dpj ; d ¼ þ Dpj ; c ¼ hjk Dpj Dpk þ h11 d2 ; h11 j ¼2 j ¼2 j ;k ¼ 2 Eqs. (48) and (49) take the form ÏX þ s þ g1 Dp1 ÷2 þ h11 ÏDp1 þ d÷2 ¼ c; Y 2 × h11 ÏDp1 þ d÷2 ¼ þc; Y ¼ 0;

Ï55÷

Ï56÷ Ï57÷

X ¼ s × g1 Dp1 :

Solutions of systems (56) and (57) depend qualitatively on the signs of the constants h11 and c: Under the non-degeneracy conditions h11 a0 and ca0 there are four possibilities. Case r0× Ïh11 > 0; c > 0÷: System (56) determines two hyperbolae in the ÏDp1 ; X ÷ plane; system (57) has no solutions; see Fig. 7. Two simple real eigenvalues approach, and then diverge as Dp1 is changed; a double eigenvalue does not appear; see Fig. 8. Case r0þ Ïh11 > 0; co0÷: System (56) determines two hyperbolae in the ÏDp1 ; X ÷ plane; system (57) defines an ellipse in the ÏDp1 ; Y ÷ plane; see Fig. 7. The hyperbolae and ellipse have two common points pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ï58÷ Dp7 ¼ d7 þc=h11 ; X 7 ¼ s × g1 Dp7 ; Y 7 ¼ 0: 1 1
þ With ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p increasing Dp1 two simple real eigenvalues approach, interact strongly at Dp1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × d þ þc=h11 ; become complex conjugate, interact strongly again at Dp1 ¼ d × þc=h11 ; and then diverge along the real axis; see Fig. 8. By eliminating Dp1 from Eq. (57), an ellipse on the


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X r+

Y

0

p

1

0

p

1

X r- p

Y

0

1

0

p

1

X r+ p

Y

0

1

p

1

X r- p

Y

0

1

p

1

Fig. 7. Weak interaction of eigenvalues for small Dp2 ; y; Dpn :
Im r+ 0 r+ r- Re

r-

Fig. 8. Weak interaction of eigenvalues on the complex plane for small Dp2 ; y; Dpn :

complex plane is obtained as Y 2 × h11 X þ s þ g1 d 2 ¼ þc ; g1 Ï59÷

see Fig. 8. If the eigenvalues are plotted in the Ïp1 ; Re l; Im l÷ space, one observes a small elliptic bubble appearing from the point Ïp0 ; l0 ; 0÷; see Fig. 9. This bubble is placed in the plane perpendicular to 1 the plane of original interaction. At points (58) the double real eigenvalues l
7

¼ l0 × X

7

× oÏe÷;

Ï60÷


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Re
0

r-

Re
0

r+

Im

0

0 p1

p

1

Im

0

0 p1

p

1

Fig. 9. Weak interaction of eigenvalues in the Ïp1 ; Re l; Im l÷ space for small Dp2 ; y; Dpn in the cases r0þ and r00 : ×

appear. It is easy to show that each of these eigenvalues has a single eigenvector. Indeed, if eigenvalues (60) were semi-simple, then X 7 have to be semi-simple eigenvalues of 2 á 2 matrix (43) at Dp1 ¼ Dp7 : Hence, matrix (43) becomes 1 X
7

× f111 ÏDp1 þ Dp7 ÷ 1 X

f112 ÏDp1 þ Dp7 ÷ 1
7

! : Ï61÷

f121 ÏDp1 þ Dp7 ÷ 1

× f122 ÏDp1 þ Dp7 ÷ 1

Using this matrix with expressions (46) and (54), gives h11 ÏDp1 þ d÷2 × c ¼
n X j ;k ¼ 1

hij Dpj Dpk þ f122 ÷2 =4 × f112 f121 ÷ÏDp1 þ Dp7 ÷2 ¼ h11 ÏDp1 þ Dp7 ÷2 1 1 Ï62÷

¼ÏÏf

11 1

and, hence, c ¼ 0: But this is a contradiction to the assumption that co0: Therefore, two interactions at points (58) are strong and follow the scenarios described in the previous section. Case r00 Ïh11 o0; c > 0÷: System (56) determines an ellipse in the ÏDp1 ; X ÷ plane; system (57) × defines two hyperbolae in the ÏDp1 ; Y ÷ plane; see Fig. 7. The hyperbolae and ellipse have two common points (58), where double real eigenvalues (60) appear and cause strong interactions of eigenvalues. Therefore, with a monotonous pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp1 two complex conjugate eigenvalues change of approach, interact strongly at Dp1þ ¼ d þ þc=h11 ; become real, interact again at Dp1× ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d × þc=h11 ; then become complex conjugate and diverge; see Fig. 8. For this case Eq. (59) gives hyperbolae on the complex plane. The behaviour of eigenvalues in the three-dimensional space Ïp1 ; Re l; Im l÷ is shown in Fig. 9, where one can see a small elliptic bubble appearing in the plane Im l ¼ 0 perpendicular to the plane of original interaction. Case r00 Ïh11 o0; co0÷: System (56) has no solutions; system (57) determines two hyperbolae þ in the ÏDp1 ; Y ÷ plane symmetric with respect to the Dp1 axis; see Fig. 7. Two complex conjugate eigenvalues approach, and then diverge with an increment of Dp1 ; a double eigenvalue does not appear; see Fig. 8. Note that hyperbolae (59) change the vertical angles, where they appear, compared to the case r00 : × Variations of the parameters Dp2 ; y; Dpn change a picture of weak interaction in two ways: either the double semi-simple real eigenvalue disappear and simple eigenvalues move along hyperbolae as Dp1 changes, or the double semi-simple eigenvalue splits in two double eigenvalues with single eigenvectors, which leads to a couple of successive strong interactions with the appearance of a small bubble in the Ïp1 ; Re l; Im l÷ space.


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3.2. Complex eigenvalue l

0

Finally, consider the case when a double semi-simple eigenvalue l0 is complex. In this case the eigenvalues u1 ; u2 ; v1 ; v2 and, hence, the coefficients fjkl ; gj ; and hjk are complex. If Dp2 ¼ ? ¼ Dpn ¼ 0; then expression (45) yields pffiffiffiffiffiffi X × iY ¼ Ïg1 7 h11 ÷Dp1 ; Ï63÷ where g1 and h11 are complex numbers. With a change of Dp1 two eigenvalues (42) cross each other at the point l0 on the complex plane; see Fig. 6 Ïc÷: Upon assuming that the increments Dp2 ; y; Dpn are small and fixed, Eq. (45) yields pffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X × iY ¼ s × g1 Dp1 7 h11 ÏDp1 þ d÷2 × c=h11 ; Ï64÷ where s; d; and c are small complex numbers defined by expressions (55). Upon assuming that the second term under the square root in Eq. (64) is much smaller than the first term, the following formula can be deduced: pffiffiffiffiffiffi c X × iY E s × g1 d × g1 ÏDp1 þ d÷7 h11 ÏDp1 þ d÷× 2h11 ÏDp1 þ d÷ pffiffiffiffiffiffi Ï65÷ ¼ s × g1 d ×Ïg1 7 h11 ÷ÏDp1 þ d÷× oÏDp1 þ d÷ showing that the main directions of eigenvalues on the complex plane before and after the weak interaction remain the same as for unperturbed case (63). The expression under the square root in Eq. (64) z ¼ ÏDp1 þ d÷2 × c=h11 ; Ï66÷

defines a parabola in the complex plane with a change of Dp1 ; see Fig. 10 (in the case Im d ¼ 0 the parabola degenerates to a ray). Computing points z1 and z2 of the parabola belonging to the imaginary axis, which is perpendicular to the axis of the parabola, results in c c2 Z ¼ z1 z2 ¼ 4ÏIm d÷4 þ 4ÏIm d÷2 Re þ Im AR : Ï67÷ h11 h11 Assume that Za0; which is a non-degenerate case. This means that za0 for all Dp1 and, hence, two values of X × iY given by expression (64) are different. As a result, eigenvalues (42) are different and the double eigenvalue disappears.

Imz z
2

0 z
1

Rez

Fig. 10. Image of the function zÏDp1 ÷ with monotonic change of Dp1 :


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If Z > 0; then two purely imaginary points z1 and z2 lie at different sides of the origin, i.e., the origin belongs to the interior of the parabola. In this case z makes a turn around the origin as Dp1 changes. This means that eigenvalues (42) approach, and then diverge without a change of direction as shown in Fig. 11 Ïc× ÷: If Zo0; then the origin lies outside the parabola (this condition remains valid, when the parabola does not intersect the imaginary axis). As a result, eigenvalues (42) approach, and then diverge with a change of direction as shown in Fig. 11 Ïcþ ÷: Variations of the parameters Dp2 ; y; Dpn destroy a double semi-simple complex eigenvalue. A picture of weak interaction can change in two ways: either eigenvalues follow the same directions after passing the neighbourhood of l0 ; or the eigenvalues interchange their directions. Behaviour of the eigenvalues in the neighbourhood of l0 can be rather complicated due to the square root of complex expression in formula (64). Example 2. Consider a linear conservative system Mx × Cx ¼ 0; . Ï68÷ where xARm is a vector of generalized coordinates; M and C are symmetric positive definite real matrices of size m á m smoothly depending on a vector of two real parameters p ¼ Ïp1 ; p2 ÷: Taking a solution of this system in the form x ¼ u expÏiot÷; the following eigenvalue problem is obtained, Cu ¼ o2 Mu; where oAR is a frequency and u is a mode of vibrations. Upon denoting A ¼ Mþ1 C; l ¼ o2 ; Ï70÷ Eq. (69) can be written in standard form (1). Consider now a point p0 in the parameter space, where the matrix A0 ¼ Mþ1 C0 has a double 0 eigenvalue l0 ¼ o2 : Since the matrices M0 and C0 are symmetric, the multiple eigenvalue l0 is 0 always semi-simple. Let u1 and u2 be right eigenvectors (modes) corresponding to the eigenvalue l0 : It can be shown that left eigenvectors are v1 ¼ M0 u1 ; Normalization conditions (37) take the form uT M0 u1 ¼ uT M0 u2 ¼ 1; 1 2 uT M0 u2 ¼ uT M0 u1 ¼ 0: 1 2 Ï72÷ Using expressions (70) and (71) in formula (44) gives kl T @C 2 @M fj ¼ uk ul ; þ o0 @pj @pj
Im c Im
+

Ï69÷

v2 ¼ M0 u2 :

Ï71÷

Ï73÷

c-
0



0

Re

Re

Fig. 11. Weak interaction of eigenvalues for small Dp2 ; y; Dpn (complex l0 ).


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where fj12 ¼ f obtains

21 j

due to the symmetry of the matrices M and C: With the use of relations (46), one g1 ¼ Ïf111 × f122 ÷=2; h ¼ Ïf
11 1

11

þ f122 ÷2 =4 ×Ïf112 ÷2 X0:

Ï74÷

Upon assuming that h11 a0; the bifurcation of the double eigenvalue l0 is given by expression (52) for the case Dp2 ¼ 0: This bifurcation is of the type r0 ; see Fig. 6, where l0 splits into two real eigenvalues for small increments Dp1 : This agrees with the general theory, which says that all frequencies of the conservative system under consideration are real. If Dp2 is non-zero and small, then expression (55) gives s ¼ Ïf211 × f222 ÷Dp2 =2; d ¼ þÏÏf111 þ f122 ÷Ïf211 þ f222 ÷=4 × f112 f212 ÷Dp2 =h11 ; c ¼ ÏÏf111 þ f122 ÷f212 þÏf211 þ f222 ÷f112 ÷2 ÏDp2 ÷2 =Ï4h11 ÷X0: Ï75÷

Hence, behaviour of the eigenvalues with a change of Dp1 is described by two hyperbolae (56); see Figs. 7 and 8 Ïr0× ÷: Two real eigenvalues approach, turn at some distance from each other, and diverge with a monotonic change of Dp1 : The frequencies pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X × iY × oÏe÷; Ï76÷ o ¼ l ¼ o2 × X × iY × oÏe÷ ¼ o0 × 0 2o0 have the same type of behaviour in the neighbourhood of o0 : A small perturbation of the second parameter Dp2 destroys a picture of weak interaction in such a way that the double frequency disappears. This agrees with the results of singularity theory [21].

4. Conclusion A new theory of interaction of eigenvalues in multi-parameter problems has been presented. This theory describes behaviour of eigenvalues with a change of parameters based on the information at the initial point, where the eigenvalues coincide. This information includes determination of the eigenvectors and associated vectors and first order derivatives of the system matrix with respect to parameters. The presented theory of interaction of eigenvalues in multiparameter problems has a very broad field of applications because any physical system contains parameters. The study of behaviour of eigenvalues in vibrational systems is especially important for dynamic stability problems, since even a small change of parameters can lead to instability and catastrophic response of the system. Interaction of two eigenvalues, which is obviously the most important case frequently taking place in vibrational problems, has been studied. The study of eigenvalues of higher multiplicity as well as investigation of some degenerate cases can be done using the methods developed in the present paper.


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References
[1] V.V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Pergamon, New York, 1963. [2] Ya.G. Panovko, I.I. Gubanova, Stability and Oscillations of Elastic Systems, Consultants Bureau, New York, 1965. [3] H. Ziegler, Principles of Structural Stability, Blaisdell, Waltham, MA, 1968. [4] K. Huseyin, Vibrations and Stability of Multiple Parameter Systems, Sijthoff & Nordhoff, Alphen aan den Rijn, 1978. [5] H. Leipholz, Stability Theory, B.G. Teubner, Stuttgart, 1987. [6] J.M.T. Thompson, Instabilities and Catastrophes in Science and Engineering, Wiley, London, 1982. [7] J.J. Thomsen, Vibrations and Stability. Order and Chaos, McGraw-Hill, London, 1997. [8] M.P. Paidoussis, Fluid­Structure Interactions: Slender Structures and Axial Flow, Vol. 1, Academic Press, London, 1998. [9] M.I. Vishik, L.A. Lyusternik, The solution of some perturbation problems for matrices and selfadjoint or nonselfadjoint differential equations, Russian Mathematical Surveys 15 (1960) 1­74. [10] V.B. Lidskii, Perturbation theory of non-conjugate operators, USSR Computational Mathematics and Mathematical Physics 6 (1) (1966) 73­85. [11] J. Moro, J.V. Burke, M.L. Overton, On the Lidskii­Vishik­Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure, SIAM Journal on Matrix Analysis and Applications 18 (4) (1997) 793­817. [12] A.P. Seyranian, Sensitivity analysis of eigenvalues and development of instability, Strojnicky Casopis 42 (3) (1991) 193­208. [13] A.P. Seyranian, Sensitivity analysis of multiple eigenvalues, Mechanics of Structures and Machines 21 (2) (1993) 261­284. [14] A.A. Mailybaev, A.P. Seyranian, On singularities of a boundary of the stability domain, SIAM Journal on Matrix Analysis and Applications 21 (1) (1999) 106­128. [15] A.P. Seyranian, O.N. Kirillov, Bifurcation diagrams and stability boundaries of circulatory systems, Theoretical and Applied Mechanics 26 (2001) 135­168. [16] A.P. Seyranian, Collisions of eigenvalues in linear oscillatory systems, Journal of Applied Mathematics and Mechanics 58 (5) (1994) 805­813. [17] A.P. Seyranian, P. Pedersen, On two effects in fluid/structure interaction theory, in: P.W. Bearman (Ed.), Flow Induced Vibration, A.A. Balkema, Rotterdam, 1995, pp. 565­576. [18] O.N. Kirillov, A.P. Seyranian, Overlapping of frequency curves in nonconservative systems, Doklady Physics 46 (3) (2001) 184­189. [19] A.A. Mailybaev, A.P. Seyranian, The stability domains of Hamiltonian systems, Journal of Applied Mathematics and Mechanics 63 (4) (1999) 545­555. [20] A.P. Seyranian, W. Kliem, Bifurcations of eigenvalues of gyroscopic systems with parameters near stability boundaries, ASME Journal of Applied Mechanics 68 (2) (2001) 199­205. [21] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978.