Stabilization and destabilization of a circulatory system by small velocity-dependent forces

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Abstract

A linear autonomous mechanical system under non-conservative positional forces is considered. The influence of small forces proportional to generalized velocities on the stability of the system is studied. Necessary and sufficient conditions are obtained for the matrix of dissipative and gyroscopic forces to make the system asymptotically stable. A system with two degrees of freedom is studied in detail. Explicit formulae describing the structure of the stabilizing matrix and the stabilization domain in the space of the matrix elements are found and plotted. As a mechanical example, a problem of stability of the Ziegler–Herrmann–Jong pendulum is analyzed.

Introduction

Consider a linear mechanical system with non-conservative positional forces proportional to the vector of generalized coordinates and small forces proportional to the vector of generalized velocitiesMq¨+εDq˙+Aq=0,where M, D and A are constant real square matrices of order m, corresponding to inertial, dissipative plus gyroscopic, and non-conservative positional forces, respectively, ε0 is a small parameter, q is a vector of generalized coordinates, and dot indicates differentiation with respect to time t. The matrix M is assumed to be non-singular.

Separating time with q=ueλt we get the eigenvalue problem(Mλ2+εDλ+A)u=0.The eigenvalues λ1,,λ2m are solutions of the characteristic equationdet(Mλ2+εDλ+A)=0.

Consider now system (1) in the absence of velocity-dependent forces (ε=0). Such a system is called circulatory. In this case it follows from Eq. (3) that if λ is an eigenvalue, then -λ,λ¯,-λ¯ are eigenvalues too. Therefore, a circulatory system is marginally stable if and only if all the eigenvalues ±iωj,ωj0 are purely imaginary and semi-simple. The semi-simple eigenvalue means that the number r of linearly independent eigenvectors corresponding to that eigenvalue is equal to its algebraic multiplicity k. If r<k, then secular terms proportional to tαeωjt, αk-1 (instability) appear in the general solution of Eq. (1). Thus, existence of a pair of algebraically double eigenvalues ±iω0,ω0>0 with only one eigenvector, other eigenvalues being purely imaginary and simple, corresponds to the boundary between stability and flutter instability (i.e., oscillations with growing amplitude).

Perturbation of the circulatory system by small velocity-dependent forces εD destroys the symmetry of eigenvalues. The eigenvalues can move to the right or left half of the complex plane. As the result, the non-conservative system can become unstable or asymptotically stable depending on the behavior of perturbed eigenvalues. It is important and practical to know what kind of matrices of velocity-dependent forces D stabilize or destabilize the unperturbed circulatory system.

The dependence of stability of a linear autonomous mechanical system on the structure of forces acting on the system is a classical subject going back to the works by Thomson and Tait [1]. However, the general interest to the problem of influence of small velocity-dependent forces on the stability of a linear non-conservative system arose in the early 1950s due to the famous work by Ziegler [2].

The destabilizing effect of viscous damping in the specific linear systems with two degrees of freedom subjected to non-conservative forces was first recognized by Ziegler [2], [3] and Bolotin [4]. This effect was further explored by Herrmann and Jong [5], Nemat-Nasser and Herrmann [6], Bolotin and Zhinzher [7], Kounadis [8], Beletsky [9], Bolotin, Grishko and Panov [10], Gallina and Trevisani [11] and others. However, this effect was not placed into the framework of theorems of sufficient generality. Nor was it shown whether a more general system with many degrees of freedom can also exhibit such behavior. The destabilization paradox has attracted much attention in the world literature; see review papers by Herrmann [12], Seyranian [13], Bloch et al. [14], and a recent work by Kirillov [15].

However, already in 1960s Bolotin [4] and Herrmann and Jong [5] found that for some specific problems there exist damping configurations for which the destabilization paradox does not take place. Done [16] tried to find the stabilizing matrices in a general formulation, however, he did not succeed in getting results in an explicit form even for 22 matrices. Walker [17] formulated a stabilization problem: assuming that the unperturbed circulatory system is stable how to get the asymptotic stability due to small velocity-dependent forces. He found a class of “non-destabilizing” mm matrices εD by the Lyapunov direct method. It should be noted that his theorems provided some sufficient but not necessary conditions for the matrix D to be “non-destabilizing”. Another attempt to find stabilizing configurations of the matrix D was done in a cluster of works by Banichuk, Bratus and Myshkis published in the early 1990s; see for example Refs. [18], [19]. They succeeded in getting one of the necessary conditions for the matrix D of velocity-dependent forces but failed in constructing stabilizing matrices even for 22 case. The next clarifying step was taken by O’Reilly et al. [20], [21] who obtained in explicit form the domain of stabilization for general systems with two degrees of freedom assuming that the unperturbed system has only distinct purely imaginary eigenvalues. In important papers by Seyranian and Pedersen [22], and Seyranian [23], two-dimensional stabilization domains for the classical examples by Bolotin [4] and Herrmann-Jong [5], assuming that the unperturbed system is on the boundary between flutter and stability, were found.

The purpose of the present paper is to find the necessary and sufficient conditions that the matrix D of velocity-dependent forces must satisfy in order to stabilize the unperturbed circulatory system with arbitrary degrees of freedom for sufficiently small ε. The paper is organized in the following way:

In Section 2, we derive main formulae for perturbations of eigenvalues of a circulatory system due to small velocity-dependent forces. Both general and degenerate cases are considered.

Based on these results, Section 3 treats stabilization conditions for the matrix D of velocity-dependent forces of a system with arbitrary degrees of freedom. It is assumed either that the circulatory system is stable or it is taken at the boundary between stability and flutter domains. Two theorems for the necessary and sufficient stabilization conditions are formulated and proved. These conditions imply linear and quadratic constraints on the elements of the stabilizing matrix D. To compute the coefficients of the linear and quadratic forms, one only needs to know the spectrum of the circulatory system with the corresponding right and left eigenvectors and the so-called associated vectors.

Section 4 is devoted to synthesis of stabilizing matrices using Walker's results [17]. We formulate and prove Theorem 3 giving a class of matrices D making a circulatory system asymptotically stable.

Section 5 treats systems with two degrees of freedom. Here we find stabilizing matrices in explicit form with the inequalities implied on the elements of the matrix D. Both cases of non-symmetric and symmetric matrices are considered.

Finally, in Section 6, we discuss the Ziegler–Herrmann–Jong pendulum loaded by tangential follower force. The form of the stabilizing matrix D is found and the inequalities on its elements are derived.

Section snippets

Behavior of eigenvalues due to perturbation εD

Let at ε=0 the spectrum of eigenvalue problem (2) contain a complex-conjugate pair of double purely imaginary eigenvalues ±iω0 with the Jordan chain of length 2. The left and right eigenvectors and associated vectors u0,u1 and v0,v1 corresponding to the double eigenvalue λ0=iω0 satisfy the equations [24], [25], [26], [27](A-ω02M)u0=0,(A-ω02M)u1=-2iω0Mu0,v0T(A-ω02M)=0,v1T(A-ω02M)=-2iω0v0TM.In addition, these vectors are related by the following conditions:v0TMu0=0,v0TMu1=v1TMu00.The vectors u0,u

Stabilization conditions

Stability of the circulatory system perturbed by the velocity-dependent forces with the matrix εD depends on whether the eigenvalues shift to the left- or to the right-hand side of the complex plane. The explicit formulae describing splitting of eigenvalues derived in the previous section allow us to find constructive conditions of stability of the perturbed non-conservative system.

Consider first the case when the unperturbed (ε=0) circulatory system is stable and its spectrum consists of

Synthesis of the stabilizing matrix D

The necessary and sufficient conditions of stabilization of circulatory system (1) by the velocity-dependent forces given by Theorems 1 and 2 are constructive and can be used to check the stability of the system. However, in practice one often needs to synthesize the stabilizing matrix D explicitly by means of the coefficients of the matrices M and A of the unperturbed circulatory system.

It seems that Bolotin [4] was the first who in the early 1960s noticed that the matrix D proportional to the

Stabilization of a system with two degrees of freedom

In this section we will show that for systems with two degrees of freedom it is possible to find the structure of stabilizing matrices D in an explicit form, and therefore obtain full description of the set of stabilizing matrices.

We consider system (1) with m=2. Multiplying Eq. (1) by M-1 from the left and introducing the notationD˜=M-1D,A˜=M-1A,we get the equationq¨+εD˜q˙+A˜q=0.For ε=0 we have a circulatory system. Consider the case when the circulatory system is situated on the boundary

Mechanical example: Ziegler–Herrmann–Jong pendulum

Consider a double pendulum [2], [5] composed of two rigid weightless bars of equal length l, which carry concentrated masses m1=2m and m2=m. The generalized coordinates ϕ1 and ϕ2 are assumed to be small. A follower load Q is applied at the free end as shown in Fig. 4. The visco-elastic hinges are characterized by the same stiffness c but different damping coefficients εb1 and εb2. Introducing the dimensionless quantitiesq=Qlc,k1=b1cml2,k2=b2cml2,τ=tcml2,where τ is time, we write the equations

Conclusion

Stabilization and destabilization phenomena of circulatory systems due to small velocity-dependent forces have been attracting substantial interest from researchers for half a century since the work by Ziegler [2]. In the present paper three theorems on the necessary and sufficient conditions for the matrices of velocity-dependent forces to stabilize an unperturbed circulatory system are established. These results are of general nature and have an applicable form allowing one to find elements

Acknowledgements

The work is supported by the research grants RFBR-NSFC 02-01-39004, RFBR 03-01-00161, and CRDF-BRHE Y1-MP-06-19.

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