Pergamon Int. J. Non-Linear Mechanics. Vol. 31, No. 6, 907-913, 1996 pp. Copyright 0 1996 Elsetier Science Ltd

Printed-in &eat Britain. All rights reserved OOZO-7462/96 515.00 + 0.00

PIL: SOOZO-7462(%)00073-X

ON THE EQUATIONS OF NON-LINEAR VIBRATIONS

F. Ma and W. C. Lee

Department of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A.

Abstract-The purpose of this short paper is to show that equations of non-linear vibrations whose linear parts involve two coefficient matrices can be simplified so that the linear portions become uncoupled. Equivalence transformations are utilized in the simplification, which can substantially streamline any subsequent analysis. Copyright 0 1996 Elsevier Science Ltd.

Keywords: equation of motion, non-linear vibrations, equivalence transformation

1. INTRODUCTION

In the last few decades, the response of non-linear systems having a single degree of freedom has been studied extensively. Quantitative investigation of non-linear vibration problems with several degrees of freedom, by contrast, is still a relatively unstructured engineering discipline [l]. Interest in this field has grown somewhat in recent years, due in part to the advent of computers and in part to the need to design structures and machinery reliably. Owing to the complexity of non-linear analysis, engineering systems of practical signifi- cance are usually assumed to be linear or piecewise linear as a first approximation. This permits considerable simplification in the solution process and often generates sufficiently accurate data on system performance. Deviations from linear behaviour are subsequently accounted for by augmenting to the governing linear differential equations suitable non- linear terms. This procedural viewpoint will be adopted in the present paper. However, it must be pointed out that equations of non-linear vibrations arise theoretically in a rather different fashion.

In classical formulation of dynamics, a holonomic system specified by n generalized coordinates qi (i = 1,2, . . . , n) is governed by a set of Lagrange’s equations:

i = 1,2, . . . , n. (1)

In the above differential equations, the scalar L is the system Lagrangian, and Qi is the generalized non-conservative force associated with the coordinate qi. It must be kept in mind that conservative forces are derivable from a potential function, and as such they are accounted for in the Lagrangian, of which the potential energy is a part. Upon simplifica- tion [2,3], the equations of motion can be cast in the form

Mh, t)ii + g(q, 40 = Q, (2)

where q = h, q2,. . . , q,JT is the vector of generalized coordinates and Q = [Q1. Qz, . . . , QJT is the vector of generalized non-conservative forces. The coefficient matrix M is symmetric and positive definite of order n, and g(q, 4, t) is a non-linear vector function. Equation (2) constitutes the general equation of non-linear vibrations. This differential equation is necessarily linear in the acceleration terms.

In traditional applications, the excitation Q depends only on q, 4 and t. Assuming time-invariance, equation (2) can be linearized [3] to obtain

Mii + BQ + Cq = f(t),

907

(3)

908 F. Ma and VI. C. Lee

where M, B and C are constant square matrices of order n. These coefficient matrices are real. The mass matrix M is symmetric and positive definite, but B and C are arbitrary. However, a real square matrix can always be decomposed as the sum of a real symmetric matrix and a real skew-symmetric matrix. Therefore, the coefficient matrix of velocity can be decomposed as B = D + G, where D is a symmetric damping matrix and G is a skew- symmetric gyroscopic matrix. Likewise, the coefficient matrix of displacement can be decomposed as C = K + H, where K is a symmetric stiffness matrix and H is a skew- symmetric circulatory matrix. The role that each of these constituent matrices plays has been discussed, for example, by Miiller and Schiehlen [3] as well as Inman [4]. Throughout the previous development, the coefficient matrix of acceleration M remains symmetric. In certain modern applications, the excitation Q can also depend linearly on 4. Such depend- ence may arise, for instance, from the use of control devices in structures. Again assuming time-invariance, linearization of equation (2) now gives

Aij + BQ + Cq =f(t), (4)

where the coefficient matrix of acceleration A is no longer symmetric. Equation (4) contains equation (3) as a particular case, and for this reason it will be used as the representative equation of a linear system [S]. Specific engineering examples involving a non-symmetric coefficient matrix of acceleration A have been given by Schmitz [6] as well as Soom and Kim [7]. Thus, it is observed that non-linear equations precede linear equations theoret- ically, and equation (4) is a mere result of linearization of the original non-linear equation of motion.

In applications, engineering systems of practical significance are usually assumed to obey linear differential equations as a first approximation. Since the various constituent matrices in equation (4) possess physical meanings, the linear equations of motion can be set up for many systems in a relatively straightforward manner [S]. The assumption of linearity also permits any subsequent analysis to be streamlined. Deviations from linear behaviour are then accounted for by augmenting equation (4) with higher order terms. This leads to a non-linear differential equation of motion, which is analysed to produce a closer approxi- mation of reality. In this interpretation, non-linear equations of motion arise from augmen- tation of additional features to the linear equations. As an example, consider a linear non-circulatory structure for which C = K in equation (4). Should the force-deformation plots of materials of the structure be found to exhibit significant hysteresis loops under cyclic deformations, the structure must be considered inelastic. The linear resisting force Kq is then replaced by a force due to non-linear stiffness g,(q, a), which depends on the history of deformations and on whether the velocity deformation is positive or negative. The equation of motion then becomes

A4 + Bil + g&I, il) = f(r). (5)

The exact form of the non-linear resisting force g,(q, 4) may be specified by an additional assumption such as elastoplastic idealization [S], which permits further analysis to be conducted efficiently. In other applications, one may start with a linear non-gyroscopic structure, for which B = D in equation (4). If the damping force is found to be appreciably non-linear, then equation (4) may be replaced by

Aii + g&I, il) + Cq = f(r). (6)

Again, the exact form of non-linear damping gn(q, 4) may be specified by additional assumptions. A common characteristic of equations (5) and (6) is that both involve two coefficient matrices in their linear parts. This is not a mere coincidence. In fact, non-linear equations of motion that arise from augmentation of a single non-linear feature to the original linear equations frequently contain two coefficient matrices in their linear parts. Development of an efficient method for the simplification of such non-linear equations is thus warranted.

The purpose of this paper is to show that equations of non-linear vibrations whose linear parts involve two coefficient matrices can be simplified so that the linear portions become uncoupled. The simplification utilizes equivalence transformations, which are the most general non-singular linear transformations. The organization of this paper is as follows. In Section 2, it is shown that linear equations of motion containing only two coefficient

Equations of non-linear vibrations 909

matrices can be completely decoupled by equivalence transformation. Equivalence trans- formation is applied to equations of non-linear vibrations with two coefficient matrices in Section 3, in which practical implications of the method are also discussed. Two illustrative examples are given in Section 4, and a summary of findings is provided in Section 5.

2. DIAGONALIZATION OF TWO MATRICES

Consider a linear undamped non-gyroscopic system, for which B = 0. The equation of motion is

Aij + Cq =f(t). (7)

This equation can be decoupled if and only if there exist two non-singular matrices U and V such that VAU, VCU are diagonal. Two square matrices P and Q, related by P = VQU, are said to be connected by an equivalence transformation. An equivalence transformation between P and Q preserves the rank of the matrices. If V = U- ‘, the equivalence trans- formation is called a similarity transformation. In the event that V = UT, the equivalence transformation is a congruence transformation. The classical modal transformation is an example of congruence transformation. A congruence transformation is also a similarity transformation if U is an orthogonal matrix. Equivalence transformations that are neither similarity nor congruence transformations are rarely used in structural analysis. Neverthe- less, equivalence transformations are the most general non-singular linear transformation, and it will be shown that A and C can be diagonalized simultaneously by an equivalence transformation in practically every situation.

Let u be a column vector of order it and 2 be a scalar constant. Equation (7) gives rise to an algebraic eigenvalue problem of the form

Cu = 1Au. (8)

Generalized eigenvalue problems of this kind were traditionally addressed in the abstract theory of matrix pencils [9]. Emphasis was usually placed on symmetric and definite pencils, in which A and C are symmetric and definite matrices. As a consequence, results applicable to the above eigenvalue problem are scattered and rather incomplete. Most constructive methods for the eigenvalue problem (8) have been summarized by Golub and Van Loan [lo]. In what follows, an extension based upon the discussion by Ma and Caughey [11] will be presented. Associated with eigenvalue problem (8) is the adjoint eigenvalue problem

CTv = IATv. (9)

As equations (8) and (9) lead to the same characteristic determinant, the corresponding eigenvalues are identical. At this point, two assumptions are made to streamline the ensuing development. First, it is required that A be non-singular. Technically speaking, there is no practical loss of generality in accommodating this assumption. Should the matrix coefficient A be singular, at least one component of the acceleration can be removed from the formulation. If the excitation Q in equation (2) does not depend on 4, then A = M and therefore it is necessarily non-singular. An equation of motion is termed degenerate if its coefficient of acceleration A is singular. The second requirement presumes independence of the eigenvectors associated with eigenvalue problem (8). An eigenvalue problem is termed defective if it does not possess a full complement of independent eigenvectors. Experience indicates that an eigenvalue problem (8) possessing physical significance is invariably not defective. And the assumption that there is a full complement of independent eigenvectors can be made without fail in almost every application. A sufficient condition under which problem (8) is not defective is that the eigenvalues be distinct. However, this is only a sufficient and not a necessary condition. Henceforth, without practical loss of generality, it will be assumed that an equation of motion is not degenerate or defective. The above two assumptions can somehow be relaxed. For example, it is permissible for both A and C to be singular in some cases. However, the treatment of such cases requires the notion of ill-disposed systems [12,13]. While equations of this type are of great theoretical interest, they will not be pursued extensively in the present paper.

910 F. Ma and W. C. Lee

Corresponding to each eigenvalue ni (i = 1,2,. . . , n), eigenvectors ui and vi can be found satisfying, respectively, the eigenvalue problems (8) and (9). Let the n independent eigenvec- tors ui be collected as columns of a matrix U, and the n adjoint eigenvectors Vi be collected as rows of a matrix V. If equation (9) is not degenerate or defective, it can be shown [l l] that

VAU = I,

vcu = AC,

where Ac is a diagonal matrix of the eigenvalues. That is,

(10)

(11)

Ac = diag[I,, AZ,. . . , A.].

Let q = Up. Equation (7) may be simplified to

(12)

ii + ACP = Vf(0, (13)

which represents a completely decoupled system. As a result, the system can be regarded as composed of n independent single-degree-of-freedom systems. Needless to emphasize, the solution of a decoupled system is immediate. If both A and C are singular, VAU cannot be normalized to yield an identity matrix. Equation (10) must be modified so that its right side is replaced by a diagonal matrix with some other structure [ 121. An analogous development can be made for a system containing only the coefficient matrices A and B. The same can perhaps be stated for a system with only B and C, but in this case the dynamical problem reduces to one involving uniform motion. It has therefore been established that a linear equation with only two coe$ficient matrices can be decoupled by equivalence transformation in practically every situation.

The decoupling equivalence transformation associated with equation (7) is defined by two adjoint eigenvalue problems (8) and (9). It has been proven [l l] that any other equivalence transformation that decouples system (7) must be derivable from these eigenvalue problems. The decoupling transformation can be uniquely determined within arbitrary multiplicative constants in the eigenvectors Ui and vi if the eigenvalues are distinct. In the event that A and C are symmetric and definite, the eigenvalue problems (8) and (9) are identical, the eigenvectors and adjoint eigenvectors are equal and V = UT. Thus, the decoupling equiva- lence transformation reduces to classical modal transformation if the coefficient matrices possess symmetry and definiteness. The method of equivalence transformation represents a direct extension of classical modal analysis, and the lack of symmetry in the coefficient matrices only approximately doubles the computational effort.

3. NON-LINEAR VIBRATIONS

An equation of non-linear vibration whose linear part contains two coefficient matrices can be simplified so that the linear portion becomes uncoupled. Referring to a structure with non-linear damping governed by equation (6), an equivalence transformation can be constructed to diagonalize A and C. As before, let this equivalence transformation be defined by the two non-singular matrices U and V. Recall q = Up. Equation (6) may be expressed in the form

c + VMP, PI+ ACP = w-(t), (14)

where h,(p, p) = g,(q, 4). In an analogous fashion, an inelastic structure (5) may be cast in the form

ii + A,P + Vhk(P, P) = V/-(t), (15)

for which AB = VBU and h,(p, fi) = g,(q, 4). In the above expression, U and V define the diagonalizing transformation for A and B.

The above type of simplification can provide physical insight into the structural behav- iour, since the set of differential equations is now coupled through the non-linear term only. It can also offer substantial reduction in computational effort, particularly when the resulting non-linear term takes a simple form. As a concrete example, consider an inelastic

Equations of non-linear vibrations 911

structure of the form (15). Suppose the non-linear stiffness can be represented as a finite power series:

hK(p, P) = Krp + &p3 + P3fi3, (16)

where E, p are constant parameters and Kr, K3, B3 are constant matrices. Note that the power of a vector is defined through the Hadamard or element-wise multiplication [14], so that p3 = [p:,p& . . . , p,“]’ if p = [p1,p2,. . . ,pJT. Likewise, p3 = [d:,d:, . . . ,@,31T. Substitue equation (16) into (15) to obtain

ji + As@ + VKlp + &VK3p3 + pVB3P3 = I’&). (17)

A natural question at this stage is whether any of VK1, VK3, or VB3 would be diagonal. If the coefficient of acceleration A in the original non-linear equation (5) is non-singular, it has been proved by Ma and Caughey [l l] that an equivalence transformation that diagonalizes A and B also diagonalizes a third matrix S if and only if A-lB and A-% commute in multiplication. In other words, if an equivalence transformation defined by two non- singular matrices U and V is such that VAU, VBU are diagonal, then

BA-IS = SA-‘B (18)

is a necessary and sufficient condition for VSU to be diagonal as well. When the above assertion is applied to equation (17), it can be seen that

BA- ‘KIU-’ = KIU- ‘A- ‘B,

BA-‘K3U-l = K3U-‘A-‘B,

BA-‘B,U-’ =B3U-‘A-‘B

are the respective conditions for VK1, VK3 and VB3 to be conditions are indeed satisfied, equation (17) becomes

fi + Ani + A;p + &p3 + pA;fi3 =

(19)

(20)

(21)

diagonal. If the above three

Vf(t) (22)

which is completely decoupled. Subsequent analysis of equation (22) can be substantially streamlined. Clearly, conditions (19)-(21) are not satisfied in general, and equation (15) can only be simplified to the form (17). Even in this case inspection of VK1, VK, and VB, could point to the dominant non-linear coupling terms.

It has thus been shown that an equation of non-linear vibration whose linear part contains two coefficient matrices can be simplified so that the linear portion becomes uncoupled. If the resulting non-linear part can be represented as a finite power series in p and p, then each coefficient S in the power series is diagonal as long as BA-‘SU-l = SU’A-‘B. Inspection of the coefficients of the power series can reveal the dominant coupling effect.

4. EXAMPLES

Two illustrative examples will be presented. These examples also address some interest- ing features not previously mentioned. The first example is adapted from an earlier paper [ 151 by the same authors on linear controlled structures. The second example involves two singular coefficient matrices.

4.1. Example 1 An inelastic structure, whose equation of motion has the form (5), is defined by

7.0000 1 .oooo 2.0000

A= I -1.0000 7.0000 0.0000 I ) (23) 1.0000 - 1.0000 6.0000

B = i

2.2577 -0.7016 -0.0567

0.7305 4.4257 0.9294 1.4657 0.8812 3.9978

I , (24)

912 F. Ma and W. C. Lee

-(2.7240q1 + 1.4078q2 + 1.9061q,)3

gK(% 4) = &K3 -(8.2244q1 + 7.1019q2 + 5.5798q3)3 ,

I

(25) (5.6148q1 + 5.7533q2 + 4.7538q3)3

-4.6612

K3 = I 13.9460

- 13.4589

Solution of the two adjoint eigenvalue normalization,

-0.2136

u= 1 1 .ooOO

- 0.9580

-0.4705 -0.1516

-0.3241 0.0575

I

. (26) 1.2613 0.2580

problems of the form (8) and (9) yields, after

-0.5502 -0.7315

- 0.2893 0.0614 I , (27) 1.0000 1.0000

-0.1728 -0.1887

-0.9129 -0.5196 P-9 0.7751 0.5124

With this eigenvalence transformation, the linear part of equation (5) is decoupled and can be expressed in the form (15), where

AB = diagC0.4685, 0.5109, 0.57331, (29)

and hK(P, P) = gK(q, 4) = sK3p3. An examination of the transformed non-linear part reveals that

Vh,(p, P) = sVK3p3 = sdiag[L9333p:, 0.2198p:,O.O495~~] (30)

and therefore equation (15) is completely decoupled. As a result, the system in this example can be regarded as composed of three independent single-degree-of-freedom systems. According to the discussion of the last section, condition (20) must be satisfied. This can be verified by a simple calculation. Although the coefficient matrices A and B in this structure can be diagonalized by equivalence transformation, they cannot be directly diagonalized by similarity transformation. Among other things, A itself is not diagonalizable by similarity transformation because there is only one eigenvector [l, 1, -11’ associated with the repeated eigenvalue 6. From this discussion, it can be seen that the method of equivalence transformation is quite compact and can lead to substantial reduction in computational effort.

4.2. Example 2

An inelastic structure of the form (5) is governed by

[; ‘6]c:]+[;5 ~5][~:]+g,(q,i)=f(t). (31)

The coefficient matrices A and B are both singular in the present case. Solution of the two adjoint eigenvalue problems of the type (8) and (9) yields, after normalization,

u= [

1 .oooo 1.0000 1 [ v= - 0.6000 1.2000

-0.3333 -0.5000 ’ 0.9487 1 -0.3162 ’ (32)

Recall q = Up. An equivalence transformation defined by U and V reduces equation (31) to the form

0.0000 o.oooo jl

0.0000 -1.1859 I[ 1 fiZ + Vk(P, P) = Vf(t),

where the transformed coefficient matrix of acceleration is given by

&=VAU= 1 .oooo 0.0000

0.0000 1 0.0000 .

(33)

(34)

Equations of non-linear vibrations 913

As previously mentioned, VAU cannot be normalized to yield an identity matrix if A and B are singular. Nevertheless, the linear part of equation (31) can still be decoupled by equivalence transformation. The resulting equation can then be cast in a form similar to equation (15). Further treatment of the decoupling of singular matrices requires certain abstract concepts [12,16] and will be given elsewhere. Finally, it must be stated that system (31) cannot be simplified in any way by similarity transformation. From this discussion, the power and generality of equivalence transformation over similarity transformation are clear.

5. CONCLUSIONS

It has been established that equations of non-linear vibrations whose linear parts contain two coefficient matrices can be simplified so that the linear portions become uncoupled. If the resulting non-linear part can be expanded in a power series of its arguments, then each coefficient in the power series is diagonal as long as a certain commutativity condition is satisfied. In addition, examination of the coefficients of the power series may reveal the dominant coupling effect. Simplification of non-linear equations involving two coefficient matrices with the procedure described in this paper can provide physical insight and at the same time substantially streamline any subsequent analysis.

Acknowledgement-This research has been supported in part by the Computer Mechanics Laboratory of the Department of Mechanical Engineering, University of California at Berkeley.

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