Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 725616 | 10 pages | https://doi.org/10.1155/2009/725616

An Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems

Academic Editor: Jerzy Warminski
Received31 Mar 2009
Accepted29 Aug 2009
Published04 Nov 2009

Abstract

The inverse eigenvalue problem of constructing symmetric positive semidefinite matrix 𝐷 (written as 𝐷β‰₯0) and real-valued skew-symmetric matrix 𝐺 (i.e., 𝐺𝑇=βˆ’πΊ) of order 𝑛 for the quadratic pencil 𝑄(πœ†)∢=πœ†2π‘€π‘Ž+πœ†(𝐷+𝐺)+πΎπ‘Ž, where π‘€π‘Ž>0, πΎπ‘Žβ‰₯0 are given analytical mass and stiffness matrices, so that 𝑄(πœ†) has a prescribed subset of eigenvalues and eigenvectors, is considered. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are specified.

1. Introduction

Vibrating structures such as beams, buildings, bridges, highways, and large space structures, are distributed parameter systems. While it is desirable to obtain a solution of a vibration problem in its own natural setting of distributed parameter systems; due to the lack of appropriate computational methods, in practice, very often a distributed parameter system is first discretized to a matrix second-order model (referred to as an analytical model) using techniques of finite elements or finite differences and then an approximate solution is obtained from the solution of the problem in the analytical model. A matrix second-order model of the free motion of a vibrating system is a system of differential equations of the form

π‘€π‘Žξ€·π·Μˆπ‘₯(𝑑)+π‘Ž+πΊπ‘Žξ€ΈΜ‡π‘₯(𝑑)+πΎπ‘Žπ‘₯(𝑑)=0,(1.1) where π‘€π‘Ž,π·π‘Ž,πΊπ‘Ž, and πΎπ‘Ž are, respectively, analytical mass, damping, gyroscopic and stiffness matrices.

The system represented by (1.1) is called damped gyroscopic system. The gyroscopic matrix πΊπ‘Ž is always skew symmetric and, in general, the mass matrix π‘€π‘Ž is symmetric and positive definite and π·π‘Ž,πΎπ‘Ž are symmetric positive semidefinite; the system is called symmetric definite system. If the gyroscopic force is not present, then the system is called nongyroscopic.

It is well known that all solutions of the differential equation of (1.1) can be obtained via the algebraic equation

ξ€·πœ†2π‘€π‘Žξ€·π·+πœ†π‘Ž+πΊπ‘Žξ€Έ+πΎπ‘Žξ€Έπ‘₯=0.(1.2) Complex numbers πœ† and nonzero vectors π‘₯ for which this relation holds are, respectively, the eigenvalues and eigenvectors of the system. The β€œforward” problem is, of course, to find the eigenvalues and eigenvectors when the coefficient matrices are given. Many authors have devoted to this kind of problem and a series of good results have been made (see, e.g., [1–7]). Generally speaking, very often natural frequencies and mode shapes (eigenvalues and eigenvectors) of an analytical model described by (1.2) do not match very well with experimentally measured frequencies and mode shapes obtained from a real-life vibrating structure. Thus, a vibration engineer needs to update the theoretical analytical model to ensure its validity for future use. In view of in analytical model (1.1) for structure dynamics, the mass and stiffness are, in general, clearly defined by physical parameters. However, the effect of damping and Coriolis forces on structural dynamic systems is not well understood because it is purely dynamics property that cannot be measured statically. Our main interest in this paper is the corresponding inverse problem, given partially measured information about eigenvalues and eigenvectors, we reconstruct the damping and gyroscopic matrices to produce an adjusted analytical model with modal properties that closely match the experimental modal data. Recently, the quadratic inverse eigenvalue problems over the complex field have been well studied and there now exists a wealth of information. Many papers have been written (see, e.g., [8–15]), and a complete book [16] has been devoted to the subject. In the present paper we will consider an inverse problem related to damped gyroscopic second-order systems.

Problem P
Given a pair of matrices (Ξ›,𝑋) in the form ξ€½πœ†Ξ›=diag1,πœ†2,…,πœ†2π‘™βˆ’1,πœ†2𝑙,πœ†2𝑙+1,…,πœ†π‘ξ€Ύβˆˆπ‚π‘Γ—π‘ξ€Ίπ‘₯,(1.3)𝑋=1,π‘₯2,…,π‘₯2π‘™βˆ’1,π‘₯2𝑙,π‘₯2𝑙+1,…,π‘₯π‘ξ€»βˆˆπ‚π‘›Γ—π‘,(1.4) where Ξ› and 𝑋 are closed under complex conjugation in the sense that πœ†2𝑗=πœ†2π‘—βˆ’1βˆˆπ‚,π‘₯2𝑗=π‘₯2π‘—βˆ’1βˆˆπ‚π‘› for 𝑗=1,…,𝑙, and πœ†π‘˜βˆˆπ‘,π‘₯π‘˜βˆˆπ‘π‘› for π‘˜=2𝑙+1,…,𝑝, we find symmetric positive semidefinite matrix 𝐷 and real-valued skew-symmetric matrix 𝐺 that satisfy the following equation: π‘€π‘Žπ‘‹Ξ›2+(𝐷+𝐺)𝑋Λ+πΎπ‘Žπ‘‹=0.(1.5) In other words, each pair (πœ†π‘‘,π‘₯𝑑),𝑑=1,…,𝑝, is an eigenpair of the quadratic pencil 𝑄(πœ†)∢=πœ†2π‘€π‘Ž+πœ†(𝐷+𝐺)+πΎπ‘Ž,(1.6) where π‘€π‘Ž>0 and πΎπ‘Žβ‰₯0 are given analytical mass and stiffness matrices.

The goal of this paper is to derive the necessary and sufficient conditions on the spectral information under which the inverse problem is solvable. Our proof is constructive. As a byproduct, numerical algorithm can also be developed thence. A numerical example will be discussed in Section 3.

In this paper we will adopt the following notation. π‚π‘šΓ—π‘›,π‘π‘šΓ—π‘› denote the set of all π‘šΓ—π‘› complex and real matrices, respectively. πŽπ‘π‘›Γ—π‘› denotes the set of all orthogonal matrices in 𝐑𝑛×𝑛. Capital letters 𝐴,𝐡,𝐢,… denote matrices, lower case letters denote column vectors, Greek letters denote scalars, 𝛼 denotes the conjugate of the complex number 𝛼, 𝐴𝑇 denotes the transpose of the matrix 𝐴, 𝐼𝑛 denotes the 𝑛×𝑛 identity matrix, and 𝐴+ denotes the Moore-Penrose generalized inverse of 𝐴. We write 𝐴>0(𝐴β‰₯0) if 𝐴 is real symmetric positive definite (positive semi-definite).

2. Solvability Conditions for Problem P

Let 𝛼𝑖=Re(πœ†π‘–) (the real part of the complex number πœ†π‘–), 𝛽𝑖=Im(πœ†π‘–) (the imaginary part of the complex number πœ†π‘–), 𝑦𝑖=Re(π‘₯𝑖), 𝑧𝑖=Im(π‘₯𝑖) for 𝑖=1,3,…,2π‘™βˆ’1. Define

𝛼Λ=diagξ‚»ξ‚Έ1𝛽1βˆ’π›½1𝛼1𝛼,…,2π‘™βˆ’1𝛽2π‘™βˆ’1βˆ’π›½2π‘™βˆ’1𝛼2π‘™βˆ’1ξ‚Ή,πœ†2𝑙+1,…,πœ†π‘ξ‚Όβˆˆπ‘π‘Γ—π‘ξ‚ξ€Ίπ‘¦,(2.1)𝑋=1,𝑧1,…,𝑦2π‘™βˆ’1,𝑧2π‘™βˆ’1,π‘₯2𝑙+1,…,π‘₯π‘ξ€»βˆˆπ‘π‘›Γ—π‘,(2.2)𝐢=𝐷+𝐺.(2.3) Then the equation of (1.5) can be written equivalently as

π‘€π‘Žξ‚π‘‹ξ‚Ξ›2𝑋+𝐢Λ+πΎπ‘Žξ‚π‘‹=0,(2.4) and the relations of 𝐢,𝐷, and 𝐺 are

1𝐷=2𝐢+𝐢𝑇1,𝐺=2ξ€·πΆβˆ’πΆπ‘‡ξ€Έ.(2.5) In order to solve the equation of (2.4), we shall introduce some lemmas.

Lemma 2.1 (see [17]). If π΄βˆˆπ‘π‘šΓ—π‘™,πΉβˆˆπ‘π‘žΓ—π‘™, then 𝑍𝐴=𝐹 has a solution π‘βˆˆπ‘π‘žΓ—π‘š if and only if 𝐹𝐴+𝐴=𝐹. In this case, the general solution of the equation can be described as 𝑍=𝐹𝐴++𝐿(πΌπ‘šβˆ’π΄π΄+), where πΏβˆˆπ‘π‘žΓ—π‘š is an arbitrary matrix.

Lemma 2.2 (see [18, 19]). Let π΄βˆˆπ‘π‘šΓ—π‘š,π΅βˆˆπ‘π‘šΓ—π‘™, then 𝑍𝐡𝑇+𝐡𝑍𝑇=𝐴(2.6) has a solution π‘βˆˆπ‘π‘šΓ—π‘™ if and only if 𝐴=𝐴𝑇,ξ€·πΌπ‘šβˆ’π΅π΅+ξ€Έπ΄ξ€·πΌπ‘šβˆ’π΅π΅+ξ€Έ=0.(2.7) When condition (2.7) is satisfied, a particular solution of (2.6) is 𝑍0=12𝐴(𝐡+)𝑇+12ξ€·πΌπ‘šβˆ’π΅π΅+𝐴(𝐡+)𝑇,(2.8) and the general solution of (2.6) can be expressed as 𝑍=𝑍0+2π‘‰βˆ’π‘‰π΅+π΅βˆ’π΅π‘‰π‘‡(𝐡+)π‘‡βˆ’ξ€·πΌπ‘šβˆ’π΅π΅+𝑉𝐡+𝐡,(2.9) where π‘‰βˆˆπ‘π‘šΓ—π‘™ is an arbitrary matrix.

Lemma 2.3 (see [20]). Let 𝐻𝐻=[𝑖𝑗] be a real symmetric matrix partitioned into 2Γ—2 blocks, where 𝐻11 and 𝐻22 are square submatrices. Then 𝐻 is a symmetric positive semi-definite matrix if and only if 𝐻11𝐻β‰₯0,22βˆ’ξ‚π»21𝐻+11𝐻12𝐻β‰₯0,rank11𝐻=rank11,𝐻12.(2.10) Lemma 2.3 directly results in the following lemma.

Lemma 2.4. Let 𝐻𝐻=[𝑖𝑗]βˆˆπ‘π‘›Γ—π‘› be a real symmetric matrix partitioned into 2Γ—2 blocks, where 𝐻11βˆˆπ‘π‘ŸΓ—π‘Ÿ is the known symmetric submatrix, and 𝐻12,𝐻22 are two unknown submatrices. Then there exist matrices 𝐻12,𝐻22 such that 𝐻 is a symmetric positive semi-definite matrix if and only if 𝐻11β‰₯0. Furthermore, all submatrices 𝐻12,𝐻22 can be expressed as 𝐻12=𝐻11ξ‚π»π‘Œ,22=π‘Œπ‘‡ξ‚π»11π‘Œ+𝐻,(2.11) where π‘Œβˆˆπ‘π‘ŸΓ—(π‘›βˆ’π‘Ÿ) is an arbitrary matrix and π»βˆˆπ‘(π‘›βˆ’π‘Ÿ)Γ—(π‘›βˆ’π‘Ÿ) is an arbitrary symmetric positive semi-definite matrix.

By Lemma 2.1, the equation of (2.4) with respect to unknown matrix πΆβˆˆπ‘π‘›Γ—π‘› has a solution if and only if

ξ‚€π‘€π‘Žξ‚π‘‹ξ‚Ξ›2+πΎπ‘Žξ‚π‘‹ξ‚(𝑋Λ)+𝑋Λ=π‘€π‘Žξ‚π‘‹ξ‚Ξ›2+πΎπ‘Žξ‚π‘‹.(2.12) In this case, the general solution of (2.4) can be written as

𝐢=𝐢0𝐼+π‘Šπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Ά,(2.13) where π‘Šβˆˆπ‘π‘›Γ—π‘› is an arbitrary matrix and

𝐢0𝑀=βˆ’π‘Žξ‚π‘‹ξ‚Ξ›2+πΎπ‘Žξ‚π‘‹ξ‚(𝑋Λ)+.(2.14) From (2.5) and (2.13) we have

π‘Šξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Ά+ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άπ‘Šπ‘‡=2π·βˆ’πΆ0βˆ’πΆπ‘‡0.(2.15) For a fixed symmetric positive semi-definite matrix 𝐷, we know, from the lemma (2.2), that the equation of (2.15) has a solution π‘Šβˆˆπ‘π‘›Γ—π‘› if and only if

(𝑋Λ)𝑇𝐷𝑋1Ξ›=2𝑋Λ𝑇𝐢0+𝐢𝑇0𝑋Λ.(2.16) Let the singular value decomposition (SVD) of 𝑋Λ be

𝑋𝑃Λ=π‘ˆΞ£000𝑇=π‘ˆ1Σ𝑃𝑇1,(2.17) where π‘ˆ=[π‘ˆ1,π‘ˆ2]βˆˆπŽπ‘π‘›Γ—π‘›,𝑃=[𝑃1,𝑃2]βˆˆπŽπ‘π‘Γ—π‘,Ξ£=diag{𝜎1,…,πœŽπ‘Ÿ}>0, and define

π‘ˆπ‘‡ξ‚Έπ·π·π‘ˆ=11𝐷12𝐷𝑇12𝐷22ξ‚Ήwith𝐷11βˆˆπ‘π‘ŸΓ—π‘Ÿ.(2.18) Then (2.16) becomes

Σ𝐷111Ξ£=2Ξ£π‘ˆπ‘‡1𝐢0+𝐢𝑇0ξ€Έπ‘ˆ1Ξ£.(2.19) Clearly, 𝐷11β‰₯0 if and only if

π‘ˆπ‘‡1𝐢0+𝐢𝑇0ξ€Έπ‘ˆ1β‰₯0(2.20) or equivalently,

𝑋Λ𝑇𝐢0+𝐢𝑇0𝑋Λβ‰₯0.(2.21) According to Lemma 2.4, we know if condition (2.21) holds, then there are a family of symmetric positive semi-definite matrices

𝐷𝐷=π‘ˆ11𝐷11π‘Œπ‘Œπ‘‡π·11π‘Œπ‘‡π·11ξ‚Ήπ‘ˆπ‘Œ+𝐻𝑇,(2.22) where 𝐷11=(1/2)π‘ˆπ‘‡1(𝐢0+𝐢𝑇0)π‘ˆ1,π‘Œβˆˆπ‘π‘ŸΓ—(π‘›βˆ’π‘Ÿ) is an arbitrary matrix, and π»βˆˆπ‘(π‘›βˆ’π‘Ÿ)Γ—(π‘›βˆ’π‘Ÿ) is an arbitrary symmetric positive semi-definite matrix, satisfying the equation of (2.16).

Applying Lemma 2.2 again to the equation of (2.15) yields

π‘Š=π‘Š0𝐼+2π‘‰βˆ’π‘‰π‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άβˆ’ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άπ‘‰π‘‡ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άβˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+π‘‰ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Ά,(2.23) where

π‘Š0=12ξ€·2π·βˆ’πΆ0βˆ’πΆπ‘‡0ξ€Έξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Ά+12𝑋Λ𝑋Λ+ξ€·2π·βˆ’πΆ0βˆ’πΆπ‘‡0ξ€Έξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Ά(2.24) is a particular solution of (2.15) with 𝐷 the same as in (2.22), and π‘‰βˆˆπ‘π‘›Γ—π‘› is an arbitrary matrix.

Since 𝐢0(πΌπ‘›βˆ’ξ‚π‘‹ξ‚ξ‚π‘‹ξ‚Ξ›(Ξ›)+)=0, it follows from (2.13) and (2.23) that

1𝐺=2ξ€·πΆβˆ’πΆπ‘‡ξ€Έ=12𝐢0βˆ’πΆπ‘‡0ξ€Έ+12ξ‚΅π‘Š0ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άβˆ’ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άπ‘Šπ‘‡0ξ‚Ά+ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άξ€·π‘‰βˆ’π‘‰π‘‡ξ€Έξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Ά=12𝐢0βˆ’πΆπ‘‡0ξ€Έ+12ξ‚΅ξ€·2π·βˆ’πΆπ‘‡0ξ€Έξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άβˆ’ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άξ€·2π·βˆ’πΆ0ξ€Έξ‚Ά+ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άξ€·π‘‰βˆ’π‘‰π‘‡ξ€Έξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚ΆβˆΆ=𝐺0+ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άπ½ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Ά,(2.25) where

𝐺0=12𝐢0βˆ’πΆπ‘‡0ξ€Έ+12ξ€·2π·βˆ’πΆπ‘‡0ξ€Έξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άβˆ’12ξ‚΅πΌπ‘›βˆ’ξ‚π‘‹ξ‚Ξ›ξ‚€ξ‚π‘‹ξ‚Ξ›ξ‚+ξ‚Άξ€·2π·βˆ’πΆ0ξ€Έ,(2.26) and 𝐽 is an arbitrary skew-symmetric matrix.

By now, we have proved the following result.

Theorem 2.5. Let π‘€π‘Ž>0,πΎπ‘Žβ‰₯0, and let the matrix pair (𝑋,Ξ›)βˆˆπ‚π‘›Γ—π‘Γ—π‚π‘Γ—π‘ be given as in (1.3) and (1.4). Separate matrices Ξ› and 𝑋 into real parts and imaginary parts resulting Λ and 𝑋 expressed as in (2.1) and (2.2). Let the SVD of 𝑋Λ be (2.17). Then Problem P is solvable if and only if conditions (2.12) and (2.21) are satisfied, in which case, 𝐷 and 𝐺 are given, respectively, by (2.22) and (2.25).

Note that when rank(𝑋Λ)=𝑛, that is, 𝑋Λ is full row rank, then the arbitrary matrices π‘Œ and 𝐻 in the equation of (2.22) disappear, in this case, 𝐷 is uniquely determined, and so is 𝐺. Thus, we have the following corollary.

Corollary 2.6. Under the same assumptions as in Theorem 2.5, suppose that rank (𝑋Λ)=𝑛, if condition (2.12) and 𝐢0+𝐢𝑇0β‰₯0 are satisfied. Then there exist unique matrices 𝐷 and 𝐺 such that (1.5) holds. Furthermore, 𝐷 and 𝐺 can be expressed as 1𝐷=2𝐢0+𝐢𝑇0ξ€Έ1,𝐺=2𝐢0βˆ’πΆπ‘‡0ξ€Έ.(2.27)

3. A Numerical Example

Based on Theorem 2.5 we can state the following algorithm.

Algorithm 3.1. An algorithm for solving Problem P.(1)Input π‘€π‘Ž,πΎπ‘Ž,Ξ›,𝑋.(2)Separate matrices Ξ› and 𝑋 into real parts and imaginary parts resulting Λ and 𝑋 given as in (2.1) and (2.2).(3)Compute the SVD of 𝑋Λ according to (2.17).(4)If (2.12) and (2.21) hold, then continue, otherwise, go to (1).(5)Choose matrices π‘Œβˆˆπ‘π‘ŸΓ—(π‘›βˆ’π‘Ÿ), π»βˆˆπ‘(π‘›βˆ’π‘Ÿ)Γ—(π‘›βˆ’π‘Ÿ) with 𝐻β‰₯0, and π½βˆˆπ‘π‘›Γ—π‘› with 𝐽𝑇=βˆ’π½.(6)According to (2.22) and (2.25) calculate 𝐷 and 𝐺.

Example 3.2. Consider a five-DOF system modelled analytically with mass and stiffness matrices given by π‘€π‘ŽπΎ=diag{1,2,5,4,3},π‘Ž=⎑⎒⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.100βˆ’20000βˆ’20120βˆ’35000βˆ’3580βˆ’12000βˆ’1295βˆ’40000βˆ’40124(3.1) The measured eigenvalue and eigenvector matrices Ξ› and 𝑋 are given by ξ€½ξ€Ύ,⎑⎒⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.Ξ›=diagβˆ’1.7894+7.6421π‘–βˆ’1.7894βˆ’7.6421π‘–βˆ’1.6521+3.9178π‘–βˆ’1.6521βˆ’3.9178𝑖𝑋=0.1696+0.6869𝑖0.1696βˆ’0.6869𝑖0.0245βˆ’0.0615𝑖0.0245+0.0615𝑖0.3906+0.5733𝑖0.3906βˆ’0.5733π‘–βˆ’0.0820βˆ’0.2578π‘–βˆ’0.0820+0.2578𝑖0.0210βˆ’0.1166𝑖0.0210+0.1166π‘–βˆ’0.3025βˆ’0.5705π‘–βˆ’0.3025+0.5705π‘–βˆ’0.0389+0.0079π‘–βˆ’0.0389βˆ’0.0079𝑖0.5205+0.2681𝑖0.5205βˆ’0.2681π‘–βˆ’0.0486+0.0108π‘–βˆ’0.0486βˆ’0.0108𝑖0.1806+0.3605𝑖0.1806βˆ’0.3605𝑖(3.2) According to Algorithm 3.1, it is calculated that conditions (2.12) and (2.21) hold. Thus, by choosing ξ€Ίξ€»π‘Œ=0.37420.30620.37070.7067𝑇,⎑⎒⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐻=10,𝐽=0βˆ’0.45120.18790.0747βˆ’0.44680.451200.2956βˆ’0.03950.0506βˆ’0.1879βˆ’0.29560βˆ’0.60440.5844βˆ’0.07470.03950.604400.19740.4468βˆ’0.0506βˆ’0.5844βˆ’0.19740(3.3) we can figure out ⎑⎒⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.𝐷=10.8255βˆ’8.5715βˆ’4.68400.0327βˆ’7.7270βˆ’8.571515.90972.63321.223411.2417βˆ’4.68402.63329.2185βˆ’0.58370.14490.03271.2234βˆ’0.583713.82353.2361βˆ’7.727011.24170.14493.236126.5027𝐺=0.0000βˆ’1.0438βˆ’3.7921βˆ’0.5470βˆ’8.67401.0438βˆ’0.00006.6747βˆ’0.739110.32623.7921βˆ’6.6747βˆ’0.0000βˆ’7.0774βˆ’6.21010.54700.73917.0774βˆ’0.000012.64968.6740βˆ’10.32626.2101βˆ’12.6496βˆ’0.0000(3.4) We define the residual as ξ€·πœ†res𝑖,π‘₯𝑖=β€–β€–ξ€·πœ†2π‘–π‘€π‘Ž+πœ†π‘–(𝐷+𝐺)+πΎπ‘Žξ€Έπ‘₯𝑖‖‖,(3.5) where β€–β‹…β€– is the Frobenius norm, and the numerical results shown in Table 1.


( πœ† 𝑖 , π‘₯ 𝑖 ) ( πœ† 1 , π‘₯ 1 ) ( πœ† 2 , π‘₯ 2 ) ( πœ† 3 , π‘₯ 3 ) ( πœ† 4 , π‘₯ 4 )

r e s ( πœ† 𝑖 , π‘₯ 𝑖 ) 8.6580e-014 8.6580e-014 8.2462e-014 8.2462e-014

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Copyright © 2009 Yongxin Yuan. 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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