Computing eigenpair derivatives of asymmetric damped system by generalized inverse

Pingxin Wang1 , Xibei Yang2 , Jusheng Mi3

1School of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, P. R. China

1, 3College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, P. R. China

2School of Computer Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, P. R. China

1Corresponding author

Journal of Vibroengineering, Vol. 19, Issue 7, 2017, p. 5290-5300. https://doi.org/10.21595/jve.2017.18232
Received 6 February 2017; received in revised form 5 May 2017; accepted 16 May 2017; published 15 November 2017

Copyright © 2017 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

Many existing approaches for asymmetric damped system are based on the assumption that the eigenvalues are simple or semisimple with separated derivatives. This paper presents a new algorithm for computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of asymmetric damped system. Compared with the existing methods, the algorithm can be applicable to problems whether the repeated eigenvalues have well separated derivatives. In the proposed method, the derivatives of eigenvectors are divided into a particular solution and a homogeneous solution, where the particular solution is constructed by using generalized inverse matrix. The effectiveness of the proposed algorithm is illustrated by one numerical example.

Keywords: asymmetric damped system, eigenvalues derivatives, eigenvectors derivatives, generalized inverse.

1. Introduction

In the past decades, damped system has become a topic of great interest to researchers because it arises frequently in many areas such as applied mechanics, electrical oscillation, vibro-acoustics, fluid dynamics, signal processing and so on. In this paper, we consider the equation of damped system with the following form [1]:

(1)
M q ¨ ( t ) + C q ˙ ( t ) + K q ( t ) = 0 ,

where the matrices M, C and K are asymmetric and damping is non-proportional.

Studies on the sensitivity analysis of damped system have widely used in damage detection [2], system identification [3], carbon nanotube vibration [4], model updating [5, 6] and many other disciplines. The fundamental objective of sensitivity analysis is to compute the derivatives of eigenvalues and eigenvectors. Research on this topic has received much attention and many different algorithms [7-18] have been developed in a recent treatise. Most of the existing algorithms can be divided into two categories. The first category is to convert the damp system to undamped system by using linearization [7-9]. The second category is to differentiate Eq. (1) and compute the derivatives of eigenvectors directly [10-18]. Although there are various algorithms for damped system, most of existing approaches are based on the assumption that the eigenvalues are simple or semisimple with separated derivatives. In order to fill this gap, Qian et al. [19] relaxed the restriction that the repeated eigenvalues must have well separated derivatives and presented a new algorithm for computing eigensensitivity of damped system by utilizing the method of [20]. By using generalized inverses, Wang and Dai [21] introduced a method for computing the first-order derivatives of eigenvalues and eigenvectors for symmetric damped system.

Different from symmetric damped system, eigenvalues of an asymmetric damped system have different right and left eigenvectors. Therefore, the eigenvectors may be not unique by one normalization. By adopting an additional normalization to guarantee the uniqueness of the eigenvectors, we extend the method of [21] to asymmetric damped system in this paper. we derive the first-order derivatives of right and left eigenvectors regardless of whether the eigenvalues have multiple derivatives.

The structure of this paper is divided into five parts. We begin in Section 2 with a brief description of algorithm for eigenvalue derivatives, and Section 3 gives a comprehensive discussion on the derivations of the eigenvector in two cases according to whether the derivatives of the eigenvalue have multiple roots. A numerical example is presented to illustrate effectiveness of the proposed algorithm in Section 4. Section 5 concludes the paper with some final remarks.

2. Calculation of eigenvalues derivatives

We consider the damped system with parameters p. Suppose that λi(p) is the eigenvalue and uRi(p), ULi(p) are the corresponding right and left eigenvectors respectively, i.e.:

(2)
( λ i 2 ( p ) M ( p ) + λ i ( p ) C ( p ) + K ( p ) ) u R i ( p ) = 0 ,
(3)
U L i T ( p ) ( λ i 2 ( p ) M ( p ) + λ i ( p ) C ( p ) + K ( p ) ) = 0 ,

where M(p), C(p) and K(p)Cn×n are asymmetric analytic matrices and M(p) is nonsingular in this paper.

We consider the algorithm for eigenpair derivatives of Eq. (2) with multiple eigenvalue. For this reason, we suppose that λ0 is the semisimple eigenvalue with multiplicity r>1 at p0 and the column vectors of ΦR, ΦLCrn×r are the right and left eigenvectors respectively. From lemma 3 in Ref. [22], ΦLT(2λ0M+C)ΦR is non-singular. Without loss of generality, let ΦL, ΦR satisfy:

Φ L T ( 2 λ 0 M + C ) Φ R = I r .

Suppose Λ(p)=diag(λ1(p),,λr(p)) and the eigenvector subsets:

U R p = u R 1 p , , u R r p ,       U L p = u L 1 p , , u L r p ,

are the eigenvalues functions and corresponding eigenvectors functions when p varies at the neighborhood of p0. Then:

(4)
M ( p ) U R ( p ) Λ 2 ( p ) + C ( p ) U R ( p ) Λ ( p ) + K ( p ) U R ( p ) = 0 ,
(5)
Λ 2 ( p ) U L T ( p ) M ( p ) + Λ 2 ( p ) U L T ( p ) C ( p ) + U L T ( p ) K ( p ) = 0 ,

where UR(p), UL(p) satisfy:

(6)
u L i T ( p ) ( 2 λ i ( p ) M ( p ) + C ( p ) ) u R i ( p ) = 1 ,         ( i = 1 , , r ) .

It is well known that the eigenvectors given by Eqs. (4), (5) and (6) are nonunique to the extent of a non-zero constant, and another normalized condition should imposed to result in unique eigenvectors and thereafter, for one to solve the corresponding eigenvectors derivatives. Because the mass matrix is not positive definite, the conventional mass normalization is not used. In this paper, the normalizing condition in Ref. [15] is adopted. Suppose:

u R i ( p 0 ) = ( u R i 1 ( p 0 ) , ,       u R i n ( p 0 ) ) T ,       q i = m i n { k | | u R i k ( p 0 ) | = u R i ( p 0 ) } ,
i = 1 , , r .

The normalizing condition of uRip (i=1,,r) is:

(7)
u R i q i ( p ) 1 .

Because the linear combinations of eigenvectors corresponding to eigenvalues λ0 are also eigenvectors of λ0, we can express ΦR, ΦL by the differentiable eigenvector subset UR(p0), UL(p0) in the following manner:

(8)
U R p 0 = Φ R Γ R ,       U L ( p 0 ) = Φ L Γ L .

For the sake of simplify notation, p0 is omitted for variables evaluated at p=p0. Using Eq. (6) at p=p0, we have:

(9)
U L T 2 λ 0 M + C U R = I r .

Then:

(10)
Γ L T Γ R = Γ L T Φ L T 2 λ 0 M + C Φ R Γ R = U L T 2 λ 0 M + C U R = I r ,

i.e., ΓR is the inverse of ΓLT.

In order to compute Λ' and ΓR, let W=λ02M+λ0C+K, W'=λ02M'+λ0C'+K', where ()' denotes the derivative of (). Differentiating both sides of Eqs. (2) and (3) with respect to p and letting pp0, one has:

(11)
W U ' R = F 1 ,
(12)
U ' L T W = H 1 ,

where F1=-(2λ0M+C)URΛ'-W'UR,H1=-Λ'ULT(2λ0M+C)-ULTW'. Premultiplying each side of Eq. (11) by ULT one obtains:

(13)
Λ ' = - U L T W ' U R .

Substituting Eq. (9) into Eq. (13), we have:

(14)
D Γ R = Γ R Λ ' ,

where D=-ΦLTW'ΦR. Eq. (14) shows that diagonal elements of matrix Λ' are n eigenvalues of D and matrix ΓR are corresponding right eigenvectors matrix. Hence, solving eigenproblem DΓR=ΓRΛ' yields eigenvalues derivative Λ' and matrix ΓR.

3. Calculation of eigenvectors derivatives

In this section, we will present a comprehensive discussion on the derivations of the eigenvectors according to whether the derivatives of the eigenvalues have multiple roots.

Case 1. The eigenvalues of Eq. (14) are distinct (λ'jλ'(k),jk,j,k=1,,r). In this case, by solving Eq. (14) one can obtain the eigenvalue derivative Λ' and the corresponding eigenvector matrix ΓR uniquely up to scalar multipliers, and then UR are determined by the relation UR=ΦRΓR and normalizing condition Eq. (7). In order to obtain UR', we introduce the generalized inverse matrix G of matrix W, i.e. G satisfies WGW=W. Then GF1 will be a particular solution of Eq. (11) and the general solution of Eq. (11) can be expressed as:

(15)
U R ' = G F 1 + Φ R a 1 ,

where a1 is a r×r constant matrix. Let H=2λ0M+C, H'=2λ0M'+C', we will get another particular solution UR(1) of Eq. (11) by taking a1=-ΦLTHGF1 in Eq. (15):

(16)
U R ( 1 ) = ( I n - Φ R Φ L T H ) G F 1 = P L T G F 1 ,

where PLT=In-ΦRΦLTH. Because:

F 1 = - ( 2 λ 0 M + C ) U R Λ ' - W ' U R = - ( 2 λ 0 M + C ) Φ R Γ R Λ ' - W ' U R
          = - W ' U R + ( 2 λ 0 M + C ) Φ R Φ L T W ' U R = - ( I n - H Φ R Φ L T ) W ' U R = - P R W ' U R ,

where PR=In-HΦRΦLT, then:

(17)
U R ( 1 ) = - P L T G P R W ' U R = - G e W ' U R ,

where Ge=PLTGPR is a restricted generalized inverse of W [10], and UR(1) satisfies:

(18)
Φ L T H U R ( 1 ) = 0 .

Therefore, the solution of Eq. (11) can be written as:

(19)
U R ' = U R ( 1 ) + U R d 1 .

The computation of matrix d1 need employ the second-order differential information of Eq. (2):

(20)
W U R ' ' = F 2 ,

where:

F 2 = - W ' ' U R - 2 W ' U R ' - H U R Λ ' ' - 2 H U R ' Λ ' - 2 H ' U R Λ ' - 2 M U R ( Λ ' ) 2 ,
W ' ' = λ 0 2 M ' ' + λ 0 C ' ' + K ' ' .

Premultiplying each side of Eq. (20) by TLH , yields:

(21)
Λ ' ' = U L T g 2 ,

where g2=F2+HURΛ'', then:

(22)
F 2 = g 2 - H U R Λ ' ' = g 2 - H U R U L T g 2 = ( I n - H U R U L T ) g 2 = P R g 2 .

Using the similar discussion about Eq. (11), one can express the general solution of Eq. (20) as:

(23)
U ' ' R = U R ( 2 ) + U R d 2 ,

where UR(2)=PLTGF2=PLTGPRF2=Geg2. By the relation g2=F2+HURΛ'', one has:

(24)
g 2 = - W ' ' U R - 2 W ' U R ' - 2 H U R ' Λ ' - 2 H ' U R Λ ' - 2 M U R ( Λ ' ) 2 = - W ' ' U R - 2 W ' U R d 1
              - 2 W ' U R 1 - 2 H U R 1 Λ ' - 2 H ' U R Λ ' - 2 M U R ( Λ ' ) 2 - 2 H U R d 1 Λ ' .

Then:

(25)
U R ( 2 ) = G e g 2 = 2 U R ( 1 ) d 1 + U R ( 2 ) ,

where UR(2)=-W''UR-2W'UR(1)-2HUR(1)Λ'-2H'URΛ'-2MUR(Λ')2. Substituting Eq. (24) into Eq. (21), rearranging the equation yields:

(26)
d 1 Λ ' - Λ ' d 1 + 0.5 Λ ' ' = - 0.5 U L T W ' ' U R - U L T H ' U R Λ '
            - U L T M U R ( Λ ' ) 2 - U L T W ' U R ( 1 ) = d e f ( u 1 i j ) .

Then we can get the off-diagonal elements of matrix d1:

(27)
d 1 i j = u 1 i j λ ' j - λ ' i .

The diagonal elements of d1 can be determined from Eq. (7). Let UR=(uRij), by Eq. (7) and Eq. (19) one has:

(28)
e q i T ( U R 1 + U R d 1 ) e i = 0 .

Then:

(29)
e q i T U R d 1 e i + e q i T U R ( 1 ) e i = 0 .

So:

(30)
d 1 i i = - e q i T U R ( 1 ) e i - k = 1 , k i n u R q i k d 1 k i .

Repeat the above process with Eq. (12), we have the derivatives of left eigenvectors:

(31)
U L ' = U L ' + U L b 1 ,

where UL'=-GeTW'TUL and the off-diagonal elements of b1 can be determined by:

(32)
b 1 i j = u 2 j i λ j ' - λ i ' ,

where:

u 2 i j = - 0.5 U L T W ' ' U R - Λ ( 1 ) U L T H ' U R - ( Λ ' ) 2 U L T M U R - U L ( 1 ) W ' T U R .

Differentiating both sides of Eq. (6) and letting pp0, we get:

(33)
b 1 i i = - u L i T H u R i - 2 λ i ' u L i T M u R i - u L i T H ' u R i - j = 1 , j i r b j i u L j T H u R i .

Then, we complete the process for computing derivatives of eigenvectors when the eigenvalues of Eq. (14) are distinct.

Case 2. Eq. (14) has repeated eigenvalues. Suppose that the eigenvalues of Eq. (14) are λs' with multiplicity is ms, (s=1,,h), then s=1hms=r. The right and left eigenvectors is defined:

(34)
Γ R = Γ R 1 , Γ R 2 , , Γ R h ,         Γ L = ( Γ L 1 , Γ L 2 , , Γ L h ) .

Because the eigenvectors are not uniquely determined when eigenvalues have repeated roots, ΓR and ΓL, and then UR and UL , are not uniquely defined. Let:

(35)
U R = U R 1 , U R 2 , , U R h ,         U L = U L 1 , U L 2 , , U L h ,
(36)
U R s = Φ R Γ R s Q R s ,           U L s = Φ L Γ L s Q L s ,

where QRs and QLs are coefficient matrices. Dividing Eq. (26) into ms×mt block forms and substituting Eqs. (34) and (35) into it, one obtains:

(37)
( ( λ t ' - λ s ' ) d 1 s t + 0.5 Λ ' ' δ s t ) = ( U R s T A 2 t U R t ) ,
(38)
A 2 t = - 0.5 W ' ' + W ' G e W ' - λ t ' H ' - ( λ t ' ) 2 M ,

where:

(39)
δ s t = d e f I m s , s = t , 0 m s m t , s t ,
Λ ' ' = d i a g ( Λ 1 ' '   , , Λ s ' '   , , Λ h ' '   ) .

Premulitiplying each side of Eq. (37) by HRs with s=t, yields a standard eigenvalue problem:

(40)
A 2 s Q R s = Q R s Λ s ' ' ,         A 2 s = d e f 2 Γ L s T Φ L T A 2 t Φ R Γ R s ,       s = 1,2 , , h .

Suppose the eigenvalues of Eq. (40) are distinct, Λ'' and QRs are uniquely determined to the extent of an non-zero constant multiplier by Eq. (40), then UR=(UR1,,URh), URs=ΦRΓRsQRs are uniquely determined by the normalized condition Eq. (7). By the relation QLsTQRs=Ims, QLs are also determined, and then UL, UR(1)=-GeW'UR are also uniquely identified by Eqs. (36) and (17). So, the result of UR'=UR(1)+URd1 also becomes determination of the off-diagonal elements of d1 because the relation Eq. (30) is also right. On both sides of Eq. (37), we take st:

(41)
d 1 s t = U L s T A 2 t U R t λ t ' - λ s ' ,           s t .

To extra the off-diagonal elements of d1ss and use the third derivative of Eq. (2), we obtain following relation by Eqs. (17), (26), (31):

(42)
d 2 Λ ' - Λ ' d 2 + d 1 Λ ' ' - Λ ' ' d 1 + 1 3 Λ ' ' ' = 2 J d 1 Λ ' - Λ ' d 1 - 2 d 1 Λ ' - Λ ' d 1 d 1 - T ,

where:

T = U L T 2 H ' U R 1 Λ ' + H ' ' U R Λ ' + W ' U R 2 + H ' U R Λ ' ' + 2 M U R 1 ( Λ ' ) 2      
            + 2 M ' U R ( Λ ' ) 2 + W ' ' U R 1 + 2 M U R Λ ' Λ ' ' + 1 3 W ' ' ' U R   ,
J = - U L T H ' U R ,           H ' ' = 2 λ 0 M ' ' + c ' ' ,             W ' ' ' = λ 0 2 M ' ' ' + λ 0 c ' ' ' + k ' ' ' .

Taking out the diagonal block equation, one has:

(43)
d 1 s s Λ s ' ' - Λ s ' ' d 1 s s + 1 3 Λ s ' ' ' = - T s s + 2 t s k λ t ' - λ s ' J s t + d 1 s t d 1 t s = d e f v s i j .

Then:

(44)
λ s j ' '   - λ s i ' '   d 1 i j s s + 1 3 λ s i ' ' '   δ i j = v s i j ,

where λsj'' is the jth diagonal element of Λs'' . From Eq. (44), one has:

(45)
λ ' ' ' = 3 v s i i ,           i = 1,2 , , m s ,             s = 1,2 , , k ,         d 1 i j s s = v s i j λ s j ' '   - λ s i ' '   ,         i j .

The derivatives of left eigenvectors can be computed by similar method. Then, we complete the process of computing the derivatives of eigenvectors in two cases. Formally, an algorithm based on the above procedure can be designed as following.

Algorithm procedure for computing the eigenpair derivatives:

Step 1. Input: Matrices M, C, K and the first to third order derivatives of M, C, K, the repeated eigenvalue λ0 and corresponding right and left normalized eigenvectors matrices ΦR, ΦLCn×r.

Step 2. Compute:

W = λ 0 2 M + λ 0 C + K ,          
W ' = λ 0 2 M ' + λ 0 C ' + K ' ,
H = 2 λ 0 M + C ,
D = - Φ L T W ' Φ R .

The generalized inverse G of W and Ge=(I-ΦRΦLTH)G(I-HΦRΦLT).

Step 3. Obtain the eigenvalue derivatives Λi'  by computing the eigenproblem DΓR=ΓRΛ'.

Step 4. When the eigenvalue derivatives λi'  are distinct, then obtain UR, UL and UR(1) by Eqs. (8) and (17), then get d1, by Eqs. (29) and (32).

Step 5. If the eigenvalues of DΓR=ΓRΛ' have repeated roots, calculate:

A 2 t = - 0.5 W ' ' + W ' G e W ' - λ t ' H ' - ( λ t ' ) 2 M ,
A 2 s = d e f 2 Γ L s T Φ L T A 2 t Φ R Γ R s ,     s = 1,2 , , h .

Solve the eigenproblem A2sQRs=QRsΛs'' , compute UR, UL and UR(1) by Eqs. (35) and (17) and d1, by Eqs. (41) and (45), (30).

Step 6. Calculate UR'=UR(1)+URd1.

Step 7. Output: Λi' , UR'.

The computational cost of the Algorithm is O(n3), where the generalized inverse G of matrix W is calculated by the singular value decomposition of W.

4. Numerical illustration

To illustrate the effectiveness of the proposed algorithm, we carried out one numerical example shown in Fig. 1. All the following calculations used MATLAB2015b.

In Fig. 1, there are five springs, four damping and three disks. In order to obtain asymmetric damped system, we suppose that the system is rotated with high speed in a container. From Fig. 1, we can get the mass, stiffness and damping matrices of the system:

(46)
M = m 1 0 0 0 m 2 0 0 0 m 3 ,       K = k 1 + k 2 - k 2 0 - k 2 k 2 + k 3 + k 5 - k 3 0 - k 3 k 3 + k 4 ,
D = c + c 2 + c 3 - c - c 2 - c 2 c + c 1 - c - c 2 - c 2 c + c 2 .

We assume that the system has the following gyroscopic matrix:

(47)
G = 0 - 2 c - c 2 2 c 0 - c c 2 - c 0 .

Fig. 1. An asymmetric damped system

 An asymmetric damped system

Then, we can get the damped matrix of the asymmetric system by the relation:

(48)
C = D + G = c + c 2 + c 3 - 3 c - 2 c 2 c 2 c + c 1 - 2 c 0 0 2 c + c 2 .

Setting m1=m2=m3= 1 kg, k1=k4=k5= 1000 N/m, k2=k3= 0 N/m, c1=c2=c3= 10 N s/m. We select c as the design parameter and consider c0= 0. Then:

(49)
M = 1 0 0 0 1 0 0 0 1 ,         K = 1000 0 0 0 1000 0 0 0 1000 ,         C = c + 20 - 3 c - 20 c 2 c + 10 - 2 c 0 0 2 c + 10 ,
M ' = K ' = 0 ,       C ' = 1 - 3 0 1 2 - 2 0 0 2 .

The eigenvalues of the damped system are λ1=λ2=531.225i at p= 0 and the corresponding eigenvectors are:

(50)
Φ R = 1 0 1 1 0.5 0 ,         Φ L = 0 0 0 0.0160 i 0.0320 i - 0.0320 i .

By Eq. (12) we have:

(51)
Λ ' = - 1.0000 + 0.1601 i 0 0 - 1.0000 + 0.1601 i ,       Γ R = Γ L = 1 0 0 1 .  

This means the derivatives of eigenvalues are equal. Using Eq. (40), we get:

(52)
Λ ' ' = 0 + 0.0328 i 0 0 - 0.3000 + 0.0809 i .

Then we derive the adjacent right and eigenvectors U, V and the right eigenvectors derivatives U'R as following:

(53)
U = 0 1 1 - 0.3333 0 0.5 ,       V = 0 0 0.1601 i 0 0.0107 i 0.0320 i ,         U R ' = 0.3000 0 0 0 0 0 .  

To examine the correctness of the above results, we give the parameter c a small perturbation Δc= 0.1. We compute the eigenpair of perturbed system at c=c0+Δc and compare the result with the linear approximation of perturbed system by Taylor series. Table 1 and Table 2 show the comparisons of the eigenvalues and eigenvectors.

Table 1. Comparison of eigenvalues

Original eigenvalue
Approximated eigenvalues
Perturbed eigenvalues
Errors
–5–31.225i
–5.1–31.2088i
–5.1–31.2088i
8.5e-8
–5.1015–31.2086i
–5.1015–31.2086i
1.5e-5

Table 2. Comparison of right eigenvectors

Original eigenvectors
Approximated eigenvectors
Perturbed eigenvectors
Errors
1.0000
1.0000
1.0000
0
–0.3333
–0.3333
–0.3333
8e-14
5000
0.5000
0.5000
0
0
0.0300
0.0303
3e-4
1.0000
1.0000
1.0000
0
0
0
0
3e-11

5. Conclusions

Derivatives of eigenvalues and eigenvectors of parameter-dependent matrix eigenproblems play a key role in the optimum design of structures in engineering. In this paper, we extend the method of [21] to asymmetric damped system and develop a numerical algorithm for computing the eigenpair derivatives by adopting a new normalization to guarantee the uniqueness of the eigenvectors. The proposed algorithm divides the eigenvectors derivatives into a particular solution and a homogeneous solution, where the particular solution to the governing equations is constructed by using the generalized inverse matrix. The present study is the first step for the research of eigensensitivity of damped systems. The higher order derivatives of eigenvalues and eigenvectors will be challenges for further research.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Nos. 61503160, 11571171 and 61572242) and Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJB110004).

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Mathematical Problems in Engineering
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2018