The theory of gyroscopic systems with non-conservative forces

Koshlyakov, V. N.; Макаров, В. Л.

doi:10.1016/s0021-8928(01)00072-7

Cited by 8 publications

(11 citation statements)

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“…Equation (2) describes the behavior of numerous mechanical systems under the action of dissipative, gyroscopic, potential, and, in particular, nonconservative positional forces. In systems containing gyroscopes, J is regarded as a positive-definite matrix of the moments of inertia of the system about the corresponding axes [6][7][8][9][10]. The equations of motion of a body suspended from a string studied in what follows represent a special case of the general equation (2).…”

Section: Statement Of the Problem Basic Matrix Equationmentioning

confidence: 99%

“…In [7][8][9][10], it is shown that, under certain conditions, the nonconservative positional structures can be eliminated from the equations by using the Lyapunov transformation. It is well known that this transformation does not affect the properties of stability.…”

Section: Equations Of Perturbed Motion Of a Symmetric Body Suspended mentioning

confidence: 99%

“…Due to the structure of matrices (5), the necessary and sufficient conditions (established in [8]) of reducibility of the equation for the vector y = colon(y 1 , y 2 ) to the form…”

Section: Equations Of Perturbed Motion Of a Symmetric Body Suspended mentioning

confidence: 99%

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Method of averaging in the problem of stability of motion of a body suspended from a string

Koshlyakov

2007

Ukr Math J

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On the basis of the Bogolyubov-Mitropol'skii method of averaging, we study the problem of stability of the vertical rotation of a body suspended from a string. Statement of the Problem. Basic Matrix EquationIn the collection of efficient procedures aimed at the approximate investigation of the problems of nonlinear mechanics (and, in particular, of the mathematical models of various oscillatory processes), an important place is occupied by the method of averaging based on the differential equations presented in a special (standard) Bogolyubov-Mitropol'skii form [1]:where x = colon(x 1 , . . . , x n ) and X = colon(X 1 , . . . , X n ) are n -dimensional vector columns and ε is a small real nonnegative parameter. The procedure of averaging is carried out on the right-hand side of Eq. (1). In some cases, the stability (instability) of the averaged system is accompanied by a similar state in the original system [not necessarily represented in the standard form (1)]. However, there are no general theorems substantiating the applicability of the method of averaging in these cases and the procedure of formal replacement of a given set of equations by a different set does not always give correct results, as indicated, e.g., by Chetaev [2]. In [3], Bogolyubov, for the first time, established a correspondence between the solutions of the exact and averaged equations and formulated theorems based on the standard form (1). Therefore, the reduction of the original equations of the analyzed problem to the standard form should be regarded as a necessary stage in the substantiation of the method of averaging [4].In what follows, we use the following basic matrix equation [5]:where x = colon(x 1 , . . . , x 2m ) is the required vector and J = J T , D = D T , H = −H T , Π = Π T , and P = −P T (the symbol (·) T denotes the operation of transposition) are given constant 2m × 2m matrices. Equation (2) describes the behavior of numerous mechanical systems under the action of dissipative, gyroscopic, potential, and, in particular, nonconservative positional forces. In systems containing gyroscopes, J is regarded as a positive-definite matrix of the moments of inertia of the system about the corresponding axes [6-10]. The equations of motion of a body suspended from a string studied in what follows represent a special case of the general equation (2).

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Section: Statement Of the Problem Basic Matrix Equationmentioning

confidence: 99%

Section: Equations Of Perturbed Motion Of a Symmetric Body Suspended mentioning

confidence: 99%

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Method of averaging in the problem of stability of motion of a body suspended from a string

Koshlyakov

2007

Ukr Math J

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“…An approach based on transformation of the state vector of the system that enables non-conservative positional forces to be eliminated from the equations of motion under certain additional conditions has been developed. [3][4][5] The conditions for the motion of autonomous mechanical systems to be stable under the action of dissipative, gyroscopic, potential and non-conservative positional forces have been obtained using a Lyapunov function of quadratic form. 7-9 Several theorems regarding the exponential stability and stabilization of the motions of non-autonomous mechanical systems with non-conservative forces have also been obtained by using a Lyapunov function of quadratic form.…”

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confidence: 99%

Stabilization of the motions of non-autonomous mechanical systems

Peregudova¹

2009

Journal of Applied Mathematics and Mechanics

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“…The authors of [5,13] proposed a method to eliminate the nonconservative positional terms from the equations for two-degree-of-freedom systems. The studies [7][8][9]18] are devoted to the problem of reducing general nth-order systems to a form convenient for applying the Thompson-Tait-Chetaev theorems. In the present paper, we will resolve this problem for systems of n coupled symmetric bodies with external and internal dissipation [10].…”

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confidence: 99%

Dissipation in systems of coupled Lagrange gyroscopes

Storozhenko

2004

Int Appl Mech

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A system of coupled symmetric solids (Lagrange gyroscopes) moving under the simultaneous action of dissipative and nonconservative positional forces is analyzed for stability. The equations of perturbed motion are transformed so that nonconservative terms are eliminated and the Thomson-Tait-Chetaev theorems can be applied.Keywords: system of coupled solids, Lagrange gyroscope, external and internal dissipation, nonconservative positional forcesOne of the most important problems arising in the stability analysis of mechanical systems is dissipative forces [6,[15][16][17]. Attempts to allow for these forces often cause new terms, called nonconservative positional forces, to appear in the equations of perturbed motion, which complicates the analysis, making impossible the direct use of the Thompson-Tait-Chetaev theorems [14]. Such systems were studied in [11,12]. The state of the art in this research area is outlined in [1]. The authors of [5,13] proposed a method to eliminate the nonconservative positional terms from the equations for two-degree-of-freedom systems. The studies [7][8][9]18] are devoted to the problem of reducing general nth-order systems to a form convenient for applying the Thompson-Tait-Chetaev theorems. In the present paper, we will resolve this problem for systems of n coupled symmetric bodies with external and internal dissipation [10].

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The theory of gyroscopic systems with non-conservative forces

Cited by 8 publications

References 1 publication

Method of averaging in the problem of stability of motion of a body suspended from a string

Method of averaging in the problem of stability of motion of a body suspended from a string

Stabilization of the motions of non-autonomous mechanical systems

Dissipation in systems of coupled Lagrange gyroscopes

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