PERTURBATION METHOD FOR THE EIGENVALUE PROBLEM OF SYSTEMS WITH NONCLASSICAL DAMPING

Moon K . Kwak*

Virginia Polytechnic Institute and State University Blacksburg, VA 24061

Abstract

An efficient method is developed for the determination of the eigenvalues and eigenvectors of systems with nonclas- sical damping by means of a perturbation technique. The motion of mechanical systems is described by a second-order matrix differential equation which consists of mass, stiffness and damping matrices. The free vibration problem of such systems leads to an eigenvalue problem which results in com- plex eigenvalues and eigenvectors. To cope with nonclassi- cal damping, the second-order ordinary differential equation needs to be transformed into a first-order state equation, where we lose the advantage of the symmetric property of mass, stiffness and damping matrices. Moreover, the eigen- value problem of a nonsymmetric coefficient matrix requires additional computation compared to the case of a symmet- ric coefficient matrix. In this paper, simple algebraic equa- tions based on the eigensolution of the undamped system are derived by means of a perturbation technique for the determination of the eigensolution of the same system with nonclassical damping. A numerical example demonstrates the effectiveness of t,he new approach.

1. Introduction

The motion of mechanical systems is in general described by second-order matrix differential equations consisting of symmetric mass, stiffness, and damping matrices. In the dy- namic analysis of structures, the eigenvalue problem of the system is to he solved a priori in order to avoid resonance or to define the natural vibration characteristics in which the effect of damping is often ignored. Thus, the eigenvalue prob- lem of the undamped system results in real eigenvalues and eigenvectors. Undamped eigenvectors form a modal matrix which is very useful in analyzing the response of the system because it can decouple the equations of motion by using the orthogonality property. To deal with the multi-degree- of freedom system with damping, the effect of damping is introduced in the decoupled equation to ease the analysis. As a result, the damping matrix is generally assumed to be a diagonalizable matrix for the calculation of the damped- free vibration characteristics and the response of the damped system under excitations. In this regard, the diagonalizable damping matrix has been defined as a classical damping ma- trix. The eigenvalue problem of systems involving classical damping results in complex eigenvalues and real eigenvec-

tors which are the same as eigenvectors of undamped sys- tems. However, it is generally known that even in the case of viscous damping, the resulting damping matrix cannot be diagonalized by the modal matrix except for the case for which the damping matrix is proportional to mass and stiff- ness matrices.

To take the non-diagonalizable property of the damp- ing matrix into consideration, we need to transform the second-order matrix differential equation into the so-called s tate equation which is a first-order matrix differential equa- tion, which will double the size of a coefficient matrix. The eigenvalue problem of undamped systems is characterized by symmetric mass and stiffness matrices, which result in real eigenvalues and real eigenvectors readily solvable by various algorithms. By contrast, the eigenvalue problem of the state equation transformed from the second-order matrix differen- tial equation involving the damping matrix, which is char- acterized by an unsymmetric coefficient matrix, results in complex eigenvalues and eigenvectors. It requires additional computations. If the effect of damping on the system is small, then such damping can be regarded as a slight change in the coefficient matrix of the state equation. Hence, the use of a perturbation technique seems a natural choice. The advantage of the perturbation method to an eigenvalue prob- lem is that the eigensolution of a new coefficient matrix can be obtained by algebraic equations. Such algebraic equations have been derived for various cases in Refs. 1 through 4 up to the second-order perturbation solution by using the state

form expression. Kwak5 proposed a new approach based on the second-order matrix equation rather than using the state form and obtained a simple expression for higher-order per- turbation solutions for the calculation of the eigensolution of systems having light damping. This paper is the extension of the approach developed in Ref. 5 to the case of nonclassical damping.

It has been observed that in most cases the modal damp- ing matrix obtained from the nonclassical damping matrix by pre-multiplying the modal matrix and post-multiplying the transpose of it becomes a diagonally dominant matrix. This implies that the modal damping matrix can be divided into a diagonal matrix which is zero-order in magnitude and an off-diagonal matrix which is first-order in magnitude. In this way, we can cope with overdamped systems. Based on this observation, an efficient algorithm is developed for the cal-

* Assistant Professor, Department of Engineering Science and Mechanics, Member AIAA

Copyright c 1992 by the m r i c a n I n s t i t u t e of Aeronautics and Astronautics, Inc. ALl r igh t s reserved. 2 2 5 3

culation of complex eigenvalues and eigenvectors by means of a perturbation technique without the aid of the state form. Once the eigenvalue problem of the undamped sys- tem is solved, a zero-order solution is obtained from decou- pled modal equations and the higher-order solutions which reflect the coupling effect of damping can be obtained from the matrix equations. The zero-order eigenvalue problem is different from the undamped eigenvalue problem in that the zero-order eigenvalue problem includes the diagonal compo- nents of the modal damping matrix. It is shown that those matrix equations can be further reduced to simple algebraic equations. A numerical example demonstrates the effective- ness of the new approach.

2. Eigenvalue P r o b l e m

Let us consider the free vibration problem of a multi- degree-of-freedom mechanical system as

where x represents the configuration vector, M, C and Ii are N x N symmetric mass, damping and stiffness matrices, respectively. M and I( are assumed to be positive definite matrices. To cope with the general damping matrix, Eq. (1) needs to be transformed into state form. By employing iden- tity, x = x, Eq. (1) can be transforrned into the first-order matrix differential equation1.

t = Az (2)

where 2 = [zTkTIT and

There are several ways of forming a s tate equation from the second-order matrix differential equation. However, only the simplest way is presented here. The eigenvalue problem for the above s tate equation can be written as

If there is a slight change in the design, then the eigenvalue problem for a new system can be written as

where E is a srnall coefficient and A1 represents the slight change in the coefficient matrix. Applying a perturbation technique, a new eigensolution can be expressed in terms of the original eigensolution in series.

where X i and z; (i 2 1) are obtained from the algebraic equations. This is the common approach used in Refs. 1 through 4. Because the coefficient matrix of the state equa- tion is unsymmetric as can be seen in Eq. (2), mathematical manipulation involved in the process is complicated, so that algebraic equations for the perturbation solutions are pre- sented up to the second order in Refs. 1 through 4. This difficulty is mainly clue to t,he unsy~nmetric coefficient ma- trix of the s t a k equation. Quite recently, it has been found5 that the perturbation method can be applied directly to the

second-order matrix equation and simple algebraic equations are derived for higher-order perturbation solutions of systems having light damping. Light damping implies that systems have complex conjugates of eigenvalues arid eigenvectors. In this paper, the same technique developed in Ref. 5 is applied to systems having nonclassical damping.

In structural dynamic analysis, natural vibration char- acteristics are generally obtained from the undamped case. Then, the eigenvalue problem of the undamped system can be written as

Ii-x = X1Clz ( 7 )

The above eigenvalue problem results in eigenvalues R, and the modal matrix U which satisfy the orthogonality re- lationships:

U ~ M U = I , uTliu = 0: (8% b)

where R, is a diagonal matrix whose diagonals are eigen- values and each column of U contains corresponding eigen- vector. All the undamped eigenvalues are assumed to be distinct. The modal matrix, U , has been used to decouple Eq. (1). To this end, the following transformation is intro- duced:

x = U q (9)

Inserting Eq. (9) into Eq. ( I ) , pre-multiplying by uT and recalling Eqs. (8), we obtain

where C = uTcu (11)

which is referred to as a modal damping matrix. Equation (10) are known as modal equations of motion. In the classical damping theory, C is assumed to he a diagonal matrix so that the coupling effect can be neglected in the analysis. Otherwise, the modal equations are uselcss. In addition, if the effect of damping is negligible, then this assumption seems reasonable to some extent. Another approach is the use of a proportional damping matrix, which is assumed to be proportional t o mass and stiffness matrices, i.e.,

where CY and p are arbitrary constants. Inserting Eq. (12) into Eq. (11) and using Eqs. (S), we can obtain

which is a diagonal matrix. It has been shown that the as- sumption of the diagonalizable damping matrix is introduced to ease the analysis. But in nature, there are many cases in which the aforementioned assumptions do not hold. Espe- . cially, this is true when we design active damping control. Yet, the most noteworthy property of modal damping ma- trices in many cases is that they are diagonally dominant. Based on this observation, the modal damping matrix, C, can be assumed to consist of a zcro-order diagonal matrix and a first-order off-diagonal matrix, i.e.,

By definition,

In other words, the damping induces only a small coupling effect in the modal equations of motion. The solution of Eq. ( 1 0 ) can be written in the form

Thus, we obtain

The matrix equation ( 1 7 ) conatltutes the eigenvalue problem for the system with the nonclassical damping matrix. Equa- tion ( 1 7 ) can be generalized further by introducing square matrices, R and V . Then, using Eq. ( 1 4 ) , we can rewrite the eigenvalue problem in matrix form

where R is an N x N diagonal matrix, whose diagonal terms represent the eigenvalues and V is an N x N modal matrix containing the eigenvectors in each column corresponding to each eigenvalue.

3. Perturbation Method

By means of a perturbation technique, the eigensolution of Eq. ( 1 8 ) can be expressed in terms of an infinite series. Thus, we can write

To ensure that V = I when C = 0 , K(1 > 1 ) should have zero diagonals. This is due to the fact that the scalar mul- tiple of any eigenvector can also be the eigenvector. Thus, in expressing Eqs. ( 1 9 ) , the matrices R , are assumed to be diagonal matrices and the matrices K(1 2 1) are assumed t o have zero diagonals, i.e.,

The modal vector of the original damped system can be ob- tained by using Eqs. ( 9 ) and ( 1 9 b )

Inserting Eqs. ( 1 9 ) into Eq. ( 1 8 ) , we obtain

Considering the n th order equation, the following equations for the n-th order perturbation solution can be derived:

Using the definitions, Eqs. ( 2 0 ) and ( 2 1 ) , Eq. ( 2 4 ) can be divided into:

(Pn), i ( o n ) ; ; =

( 2 v o n 0 + c'o),, ( 2 6 )

If P , is obtained from the summation formula, Eq. ( 2 5 ) , then higher-order solutions can be easily determined by using algebraic equations, Eqs. ( 2 6 ) and ( 2 7 ) .

4. Zero-Order Solution

As can be seen from Eqs. ( 2 4 ) , ( 2 6 ) and ( 2 7 ) , it is evident that the zero-order solution forms the basis for higher-order solutions. The zero-order solution is different from the un- damped solution in that it includes the diagonal of the damp- ing matrix. To fully investigate the zero-order problem, the diagonal matrix Co introduced in Eq. ( 1 4 ) is rewritten as

where Z = d iag ( ( ; ) in which ( , is a nondimensional quantity known as the viscous damping factor6. Thus, the zero-order problem can be defined by decoupled equations:

The solution of Eq. ( 2 9 ) has the general form

Introducing Eq. ( 3 0 ) into Eq. ( 2 9 ) and dividing through by aie3i1, we conclude that the exponent must satisfy the fol- lowing characteristic equation:

Three cases can be considered for the solution of Eq. ( 3 1 ) according t o the value of (;. Solutions for each case are as follows:

(a) Underdamped Case: 0 < (; < 1

(b) Critically Damped Case: (, = 1

(c) Overdamped Case: ( ; > 1

where where i = a. Based on Eqs. ( 3 2 ) , R o can be written as

In the underdamped case, eigenvalues have the form of com- plex conjugates, so that one step is enough for the calcula- tion of the pair of eigenvalues of lightly damped systems5. However, if any of the viscous damping factors introduced in Eq. (28) is larger than unity, two steps are necessary to obtain the solution. It can be readily seen from Eq. (29) that the eigenvector for the zero order problem is the same as the eigenvector of the undamped system, so that we may conclude

v, = I (34)

The zero-order problem amounts to the eigenvalue problem of systems including only the diagonal of the damping ma- trix, so that it results in complex or real eigenvalues but real eigenvectors.

6. Numer ica l E x a m ~ l e

As a numerical example, the spring-mass-damper system shown in Fig. 1 is adopted. Following are the numerical data:

where I4 is a 4 x 4 identity matrix. Eigenvalues and eigen- vectors of the undamped case are as follows:

Then, Co and CI are obtained by using Eqs. (11) and (15)

3.3346 0 0

c,, = 0 0.7205 0 I : 0 1.0000 0 0 0 7.1449

It can be seen that C1 is one order of magnitude smaller than C o . The damping factor matrix can be derived from C, by using Eq. (28)

As can be seen from the damping factor matrix, three modes of the zero-order problem are underdamped and one mode is overdamped, so that two sets of zero-order solution are necessary to obtain higher-order solutions:

With the first set of zero-order solutions, using Eq. (25), we obtain

0 0.0787 - 0.2364i

p!" = 0.3642 - 0.4084'1 0 0.7563 - 0.84823 -0.0175 + 0.05243

-0.2304 + 0.25843 0.1731 - 0.5200i 0.2268 - 0.7406i -0.3389

-0.0242 + 0.07913 1.1780 0 -0.6490

-0,1323 + 0.43213 0

Then, using Eqs. (26) and (27), we can obtain first-order solutions

fly = 0

With zero- and first-order solutions, we obtain

and second-order solutions

As can be seen above, the coupling term in the modal damp- ing matrix has a second-order effect in the eigenvalue. Eigen- values and eigenvectors are calculated for up to 5 terms in the series and the results are shown in Tables 1 through 4 in which the complex eigenvalues are presented in terms of the number of terms in the series. Table 5 shows the eigenvec- tors corresponding to the third eigenvalue versus the number of terms. As can be seen from the tables, the convergence is remarkable so that we can conclude that the new algo- rithm developed in this paper gives fast convergence on both eigenvalues and eigenvectors. It should be stressed here that the new algorithm developed in this paper is not compu- tationally cumbersome and can be implemented on micro- computers.

7. Discussion a n d Conclusions

Mechanical systems are in general described by second- order matrix differential equations which consist of symmet- ric mass, stiffness and damping matrices. In order to consider the damping effect on the natural vibration characteristics and responses, the damping matrix is often put into a diago- nalizable form, which results in complex eigenvalues and real eigenvectors. To deal with the general damping matrix, we need to transform the second-order matrix differential equa- tions into first-order state equations, which implies eigenso- lutions of a large unsymmetric coefficient matrix resulting in complex eigenvalues and eigenvectors. In this paper, an efficient algorithm is developed to cope with the nonclassi- cal damping matrix by means of a perturbation technique. Once the eigensolution of the undamped system is available, which can be easily obtained by various algorithms because of the symmetric property of mass and stiffness matrices, the eigensolution of the system involving a nonclassical damping matrix can be obtained by using simple algebraic equations. A numerical example demonstrates the effectiveness of the new approach.

8. References

2. Meirovitch, L. and Ryland, G. Il., "Response of Slightly Damped Gyroscopic Systems", Journal of Sound and Vi- bration, Vol. 67, 1979, pp. 1-19.

3. Chung, K. R. and Lee, C. W., "Dynamic Reanalysis of Weakly Non-Proportionally Damped System", Journal of Sound and Vibration, Vol. 117, 1986, pp. 37-50.

4. Cronin, D. L., "Perturbation Approach for Determin- ing Eigenvalues and Eigenvectors for Nonproportional Damped Systems", Proc. of the 6th International Modal Analysis Conference, Feb. 1988, Kissimmee, Florida, pp. 23-29.

5. Kwak, M. K., "Perturbation Method for the Eigenvalue Problem of Lightly Damped System", Journal of Sound and Vibration ( to appear).

6. Meirovitch, L., Elements of Vibration Analysis, 2nd ed., McGraw-Hill, Inc., 1986.

I Order I 0 1 I

Table 1. 1st Eigenvalue

3

4

5

Exact

I Order I w7 I

-0.36048673363583 f 1.10221099641070i

-0.36001811136315 & 1.102525756358673

-0.36005135022337 f 1.10256989501924i

-0.36003275664756 k 1.102556789909283

I Exact 1 -0.47482057665660 & 1.67083157946636i 1 Table 2. 2nd Eigenvalue

1. Meirovitch, L., Computational Method for Structural Dy- namics, Noordhoff Sitjhoff, 1980.

I Exact / -1.68752773864076 f 1.78495989755505i 1 Table 3. 3rd Eigenvalue Table 4. 4th Eigenvalue

4 -4.69274608635096

Order

0

r 4

5

Exact

Table 5. 3rd Eigenvector

4 -2.45219220617508

Fig. 1 A Spring-Mass-Damper System

-2.30924228750856

-2.30973060627833

-2.30828593436722

-4.84583586944626

-4.84548709031904

-4.84695192174191