Analyzing the Stability of a n-DOF System with Viscous Damping

In this paper we introduce a numerically stable method for determining the stability of n-DOF system without computing eigenvalues. In this sense, at first we reduce the second-order system to a standard eigenvalue problem with symmetric tridiagonal form. Then we compute the exact inertia by using an algorithm based on floating point arithmetic [1]. Numerical tests report the effectiveness of these methods.


Introduction
The matrix second-order system with real coefficient matrices , and M C K arises in a wide variety practical applications such as in the mechanical vibrations, and structural design analysis.Many important characteristic of physical and engineering systems, such as stability and inertia can often be determined only by knowing the nature and location of the eigenvalues.It is well known that the stability of a physical system modeled by a system of differential equation is determined just by knowing if the eigenvalues of the system matrix have all negative real parts.In many engineering applications, it may not be enough to determine if the system is stable .A problem more general than the stability problem is the inertia problem.The inertia of a matrix (denoted by ) is the triplet of the numbers of the eigenvalues of [2] In( ) A A with positive, negative and zero real parts.There are reliable algorithms to compute the inertia and stability of a n-DOF system with viscous damping.Some of these algorithms are numerically unstable and are primarily of theoretical interest .Another group of these methods are not practical for system of large numbers of degrees of freedom.The following are the usual computational approaches for determining the inertia of a nonsymmetric matrix [3] A . 1) Compute the eigenvalues of A explicitly. 2) Compute the characteristic polynomial of A and then apply the well-known Routh-Hurwitz criterion.
3) Solve the Lyapunov equation The second approach is usually discarded as a numerical approach [2 and the last approach is counterproductive.Thus, the only viable way, from a numerical viewpoint, of determining the inertia of a matrix, is to compute explicitly its eigenvalues.Carlson and Datta described a computational method for determining the inertia of a nonsymmetric matrix .The method is based on the implicit solution of a special Lyapunov equation.But this method is not practical for large and sparse matrices .The paper is organized as follows.In Section 2, we introduce some important theorems and applications of inertia ans stability problem.Then we describe two new methods for computing the inertia of a large sparse nonsymmetric matrix in Sections 3 and 4. Finally, the conclusion are given in the last section.] [4,5] [2]

Inertia and Stability
A are, respectively, the number of eigenvalues of A with positive, negative and zero real parts.
Not that , and A is a stable matrix if and only if Definition 2.4.The second order differential equations is asymptotically stable(that is, ), if and only if all the eigenvalues of the quadratic pencil have negative real Similarly, by the inertia of the quadratic pencil (2. is defined to be the triplet of the numbers of eigenvalues of 3)

 
p  with positive, negative, and zero real parts.
Remark 2.5.The effective numerical methods for the quadratic eigenvalue problem are still not well developed, especially for large and sparse problems that arise in practical applications.In case of general damping, assuming that the solutions are of the form = e t  y x , where is a constant vector, we will have the quadratic eigenvalue problem that we can rewrite (2.3) as where Thus we have reduced the quadratic eigenvalue problem to the standard eigenvalue problem , on the other hand, for determining the stability of , we can compute Example 2.

K
We would like to study the stability with determining the inertia of   p  or

 
In A where

 
In A can be routinely determine.However for system of large numbers of degrees of freedom the available methods can be costly.Specially that we would like to determine the In   A without computing the eigenvalues.

Shifted Lanczos Process
In this Section we provide a stable numerically method for determination of a nonsymmetric matrix.Our scheme is first to reduce a given matrix A to a symmetric tridiagonal form with a Lanczos process, and then compute the exact inertia of a symmetric matrix by a floating point algorithm .[5] Step 1. (Lanczos process): Given vectors 1 and 1 such that , this process provide a tridiagonal The following algorithm by using the Sylvester law of inertia describes a floating point process to compute the exact inertia of a symmetric tridiagonal matrix [5 .In this algorithm and  is a proper shift parameter.
, respectively, for the Krylov subspaces End.
Step 2. ( We apply Shifted Lanczos process to compute the exact inertia of A .This algorithm has been tested when the dimension of matrix A increases.The results are shown in Table 1.
In Table 1 the column of error is the precision of transforming the matrix A to a tri-diagonal matrix.Note that if the error is small, then the inertia of A can be computed correctly.But if the error is not small, this dose not mean that the inertia of A cannot be computed, in this case by choosing a proper shift, the inertia of A will be computed.Shift intervals are seen in Table 1.The best case is when the shifted parameter is zero.In that case the amount of computations is less, that is why we have a column called 1 to have more information.Also the results show that by increasing the dimension the matrix this method does not work very well.

Weighted Shifted Lanczos Method
According to the results shown in Table 1, we can see that the shifted Lanczos method computes the inertia accurately, when the matrix is not so large, but does not have an exact results when the dimension is large.The

W V I
method is an oblique projection method.In this section we have tried to decrease the error by making changes in the Lanczos algorithm to be able to develop an effective method for computing the exact inertia of a large sparse nonsymmetric matrices.Using (3.1) and (3.2) we have: 2) The right side of the above relations indicates the error of oblique projection in the Lanczos method.We multiply the both sides of and by a small scaler (4.1) (4.2) > 0


with the hope that to prevent the increasing error in the Lanczos process.Thus we obtain   According to the

Algorithm 1 .
(inertia of a symmetric tridiagonal matrix) Let A be the same matrix that used in Example 1 and we increase its dimension orderly.We apply alg rithm 3 to find the exact inertia of o A .The results for different values of n are shown in Ta e 2.

4.2. According to the results in Table 1 and ore timTable 2 ,
Table , we can see that by we see that the Weighted Shifted Lanczos in comparison with Shifted Lanczos method works better.Now consider A is the same matrix that used in Example 3.1.W apply our two computational methods e