Stabilization and destabilization of a circulatory system by small velocity-dependent forces
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 velocitieswhere , and are constant real square matrices of order , corresponding to inertial, dissipative plus gyroscopic, and non-conservative positional forces, respectively, is a small parameter, is a vector of generalized coordinates, and dot indicates differentiation with respect to time . The matrix is assumed to be non-singular.
Separating time with we get the eigenvalue problemThe eigenvalues are solutions of the characteristic equation
Consider now system (1) in the absence of velocity-dependent forces . Such a system is called circulatory. In this case it follows from Eq. (3) that if is an eigenvalue, then are eigenvalues too. Therefore, a circulatory system is marginally stable if and only if all the eigenvalues 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 . If , then secular terms proportional to , (instability) appear in the general solution of Eq. (1). Thus, existence of a pair of algebraically double eigenvalues 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 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 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 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” matrices by the Lyapunov direct method. It should be noted that his theorems provided some sufficient but not necessary conditions for the matrix to be “non-destabilizing”. Another attempt to find stabilizing configurations of the matrix 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 of velocity-dependent forces but failed in constructing stabilizing matrices even for 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 of velocity-dependent forces must satisfy in order to stabilize the unperturbed circulatory system with arbitrary degrees of freedom for sufficiently small . 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 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 . 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 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 . 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 is found and the inequalities on its elements are derived.
Section snippets
Behavior of eigenvalues due to perturbation
Let at the spectrum of eigenvalue problem (2) contain a complex-conjugate pair of double purely imaginary eigenvalues with the Jordan chain of length 2. The left and right eigenvectors and associated vectors and corresponding to the double eigenvalue satisfy the equations [24], [25], [26], [27]In addition, these vectors are related by the following conditions:The vectors
Stabilization conditions
Stability of the circulatory system perturbed by the velocity-dependent forces with the matrix 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 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 explicitly by means of the coefficients of the matrices and of the unperturbed circulatory system.
It seems that Bolotin [4] was the first who in the early 1960s noticed that the matrix 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 in an explicit form, and therefore obtain full description of the set of stabilizing matrices.
We consider system (1) with . Multiplying Eq. (1) by from the left and introducing the notationwe get the equationFor 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 , which carry concentrated masses and . The generalized coordinates and 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 but different damping coefficients and . Introducing the dimensionless quantitieswhere 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.
References (31)
- et al.
Effects of damping on stability of elastic systems subjected to nonconservative forces
International Journal of Solids and Structures
(1969) On the paradox of the destabilizing effect of damping in nonconservative systems
International Journal of Nonlinear Mechanics
(1992)Some stability problems in applied mechanics
Applied Mathematics and Computation
(1995)- et al.
Effect of damping on the postcritical behaviour of autonomous non-conservative systems
International Journal of Nonlinear Mechanics
(2002) - et al.
On the stabilizing and destabilizing effects of damping in wood cutting machines
International Journal of Machine Tools and Manufacture
(2003) - et al.
Dissipation induced instabilities
Annales de l’Institut Henri Poincaré
(1994) Damping configurations that have a stabilizing influence on nonconservative systems
International Journal of Solids and Structures
(1973)A note on stabilizing damping configurations for linear non-conservative systems
International Journal of Solids and Structures
(1973)- et al.
Stabilizing and destabilizing effects in non-conservative systems
PMM U.S.S.R.
(1989) - et al.
Reversible dynamical systemsdissipation-induced destabilization and follower forces
Applied Mathematics and Computation
(1995)