Gyroscopic stabilization of nonconservative systems

The problem is considered of the stabilization of a mechanical system having only nonconservative positional forces by adding gyroscopic forces. The gyroscopic stabilization is proved to be always realizable in the case of a degenerate matrix of nonconservative forces and even number of coordinates. If the matrix of nonconservative forces is nonsingular then a possibility of the gyroscopic stabilization is established for all systems whose number of coordinates is divisible by four. For a nonautonomous systemwith nonconservative positional forces and dissipative forces with complete dissipation, some sufficient conditions are obtained for stabilization up to the exponential stability by addition of gyroscopic forces.     

Stabilizing and destabilizing effect of small velocity‐dependent forceson non‐conservative systems: new results

A theory of the destabilization paradox in general non‐conservative systems with small dissipative and gyroscopic forces is presented. The problem is investigated by the approach based on the sensitivity analysis of multiple eigenvalues. An explicit asymptotic expression for the critical load as a function of the dissipation and gyroscopic parameters allowing to calculate a jump in the critical load is derived. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the influence of gyroscopic forces on the stability of steady-state motion : PMM vol. 39, n 6, 1975, pp. 963–973

The question of the influence of gyroscopic forces on the stability of steady-state motion of a holonomic mechanical system when the forces depend upon the velocities of only the position coordinates was answered by the Kelvin-Chetaev theorems [1] on the influence of gyroscopic and dissipative forces on the stability of equilibrium. However, if the gyroscopic forces depend as well on the velocities of the ignorable coordinates, then their influence on the stability of steady-state motions can, as the two problems in [2] show, prove to be entirely different from the influence of gyroscopic forces depending only on the velocities of the position coordinates. In this paper we investigate the influence of gyroscopic forces depending linearly on the velocities of the generalized coordinates, including the ignorable ones, on the stability of the steady-state motion of a holonomic conservative system. We prove that when the gyroscopic forces applied with respect to the ignorable coordinates are given as total time derivatives of certain functions of the position coordinates, the gyroscopic forces can both stabilize as well as destabilize the steady-state motion. Under certain conditions, this influence is also preserved for the action of dissipative forces depending on the velocities of only the position coordinates. In the case of action of dissipative forces depending also on the velocities of the ignorable coordinates, we have indicated the stability and instability conditions of the steady-state motion. Examples are considered. In conclusion, we discuss the conditions under which the application of gyroscopic forces to the system is equivalent to adding terms depending linearly on the generalized velocities to the Lagrange function.     

The stability and stabilization of the motion of non-conservative mechanical systems

Two stability problems are solved. In the first, the stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces (systems of general form) act, is investigated. The stability is considered in the case when the potential energy has a maximum at equilibrium. The condition for asymptotic stability is obtained by constructing Lyapunov's function. In the second problem, the possibility of stabilizing a gyroscopic system with two degrees of freedom up to asymptotic stability using non-linear dissipative and positional non-conservative forces is investigated. Stability of the gyroscopic system is achieved by gyroscopic stabilization. The stability conditions are obtained in terms of the system parameters. Cases when the gyroscopic stabilization is disrupted by these non-linear forces are indicated.     

Mechanical systems,equivalent in Lyapunov's sense to systems not containing non-conservative positional forces

Developing results obtained previously (Refs. Koshlyakov VN. Structural transformations of the equations of perturbed motion of a certain class of dynamical systems. Ukr Mat Zh 1997; 49 (4): 535–539; Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. Prikl Mat Mekh 1997; 61 (5): 774–780; Koshlyakov VN, Makarov VL. The theory of gyroscopic systems with non-conservative forces. Prikl Mat Mekh 2001; 65 (4): 698–704; Koshlyakov VN, Makarov VL. The stability of non-conservative systems with degenerate matrices of dissipative forces. Prikl Mat Mekh 2004; 68 (6): 906–913), the general problem of eliminating non-conservative positional structures from the second-order differential equation with constant matrix coefficients, obtained when modelling many mechanical systems, is considered. It is assumed that the matrices of the dissipative and non-conservative positional structures may, in particular, be degenerate. Under fairly general assumptions, theorems containing the necessary and sufficient conditions for a Lyapunov transformation to exist are proved. This converts the initial matrix equation to an equivalent autonomous form (in Lyapunov's sense) with a symmetrical matrix of the positional forces. An illustrative example is considered.     

Stabilization and destabilization in non-conservative gyroscopic systems

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is described in detail and its role in the disc brake squeal problem is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilization of the nonautonomous potential systems with the forces of a dissimilar structure

The problem is considered of stabilizing a nonautonomous system given potential forces by adding some dissipative, gyroscopic, and nonconservative positional forces. The stabilizability domain is found for the relative equilibrium of a satellite in the circular orbit.     

Gyroscopic Stabilization of 2nd-Order-Systems with Indefinite Damping

We consider the gyroscopic stabilization of the unstable system ẍ + D ẋ + Kx = 0 with positive definite stiffness matrix K. The indefinite damping matrix D is responsible for the instability of the system. The modelling of sliding bearings can lead to negative damping, see [6]. A gyroscopic stabilization of an unstable mechanical system with indefinite damping matrix was investigated in [4] in the case of matrix order n = 2 using the Routh-Hurwitz criterion. The question was raised whether an unstable system can be stabilized by adding a gyroscopic term Gẋ with a suitable skew-symmetric matrix G = −G^T . (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces

The exponential stability of the unperturbed motion of a non-autonomous mechanical system with a complete set of forces, that is, dissipative, gyroscopic, potential and non-conservative positional forces, is investigated. The problem of stabilizing a non-autonomous system with specified non-conservative forces is considered with and without the use of potential forces. The problem of stabilizing a non-autonomous system with specified potential forces by the action of the forces of another structure is studied. The domain of stabilizability of the relative equilibrium position of a satellite in a circular orbit is found.     

Non‐self‐adjoint Bessel and Sturm–Liouville boundary value problems in limit‐circle case

It is shown in the limit‐circle case that system of root functions of the non‐self‐adjoint maximal dissipative (accumulative) Bessel operator and its perturbation Sturm–Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the maximal dissipative (accumulative) Bessel operators is investigated, and it is proved that system of root functions form a basis (Riesz and Bari bases) in the same Hilbert space. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability of nonconservative systems and applications

The paper studies the stability of mechanical systems subjected to dissipative, gyroscopic, and nonconservative positional forces.     

Gyroscopic control and stabilization

Summary In this paper, we consider the geometry of gyroscopic systems with symmetry, starting from an intrinsic Lagrangian viewpoint. We note that natural mechanical systems with exogenous forces can be transformed into gyroscopic systems, when the forces are determined by a suitable class of feedback laws. To assess the stability of relative equilibria in the resultant feedback systems, we extend the energy-momentum block-diagonalization theorem of Simo, Lewis, Posbergh, and Marsden to gyroscopic systems with symmetry. We illustrate the main ideas by a key example of two coupled rigid bodies with internal rotors. The energy-momentum method yields computationally tractable stability criteria in this and other examples.     

Estimation of the time a perturbed Lagrangian system stays in an assigned region

The problem of estimating the mean time a weakly perturbed dynamical system stays in a fixed region of the phase variables is investigated. The motion is described by Lagrange's equations with an attractive force potential and in the presence of additive dissipative forces. The corresponding Cauchy problem is obtained in Hamiltonian variables for a non-linear first-order partial differential equation. Its classical positive solution specifies the action functional and the estimate sought for the time interval. The structure of the equations that allows of an explicit solution in terms of expressions for the kinetic and potential energy, as well as dissipative and dispersion matrices for a random Wiener-type perturbation, is established. The phenomenon of the escape of a phase point from different parts of the boundary of the region is investigated. Interesting problems of estimating the time for the inversion of the inner gimbal of a gyroscope, the time taken to reach an assigned level or a potential barrier of a multidimensional oscillatory system that has central symmetry, and the time a non-linear system with two degrees of freedom takes to escape over a potential barrier for a Henon–Heiles potential are investigated as examples.     

Structure preserving eigenvalue embedding for undamped gyroscopic systems

This paper concerns the eigenvalue embedding problem of undamped gyroscopic systems. Based on a low-rank correction form, the approach moves the unwanted eigenvalues to desired values and the remaining large number eigenvalues and eigenvectors of the original system do not change. In addition, the symmetric structure of mass and stiffness matrices and the skew-symmetric structure of gyroscopic matrix are all preserved. By utilizing the freedom of the eigenvectors, an expression of parameterized solutions to the eigenvalue embedding problem is derived. Finally, a minimum modification algorithm is proposed to solve the eigenvalue embedding problem. Numerical examples are given to show the application of the proposed method.     

Reduction of the equations of dynamics of systems with constraints to a given structure

The problem of constructing systems of second-order ordinary differential equations, the solutions of which, with the appropriate initial conditions, satisfy given equations of the constraints, is considered. The conditions for representing the differential equations in the form of Lagrange equations of the second kind are determined. It is shown that, when the equations of the non-holonomic constraints are specified by polynomials of order no higher than two with respect to the generalized velocities, the generalized forces of a system with energy dissipation comprise the sum of the gyroscopic, potential and dissipative forces.     

The stability and stabilization of non-linear,non-stationary mechanical systems

Mechanical systems acted upon by extremely non-linear positional forces are considered. The decomposition method is used to determine the sufficient conditions for asymptotic stability of an equilibrium. Problems of stabilizing the equilibrium of non-linear, non-stationary systems with specified potential forces by adding forces of different structure are studied. For systems with a non-stationary, homogeneous, positive-definite potential, the possibility of stabilization by linear dissipative forces, uncharacteristic of linear systems, is established. For systems with an even number of coordinates n ≥ 4, in the presence of dissipative forces with complete dissipation, the possibility of vibrational stabilization by adding circular and gyroscopic forces with coefficients fluctuating about zero is demonstrated.     

Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry

Formation of energy bands in the system of rotation-vibration quantum states of molecules is described within semi-quantum models under the presence of a symmetry group characterizing the equilibrium molecular configuration. Effective rotation-vibration Hamiltonians are written in two-quantum state models with rotational variables treated as classical ones. Eigen-line bundles associated with eigenvalues of 2×2 Hermitian matrix defined on rotational classical phase space which is a two-dimensional sphere are characterized by the first Chern class. Explicit procedure for the calculation of Chern numbers are suggested and realized for a family of Hamiltonians depending on extra control parameters in the presence of symmetry. Effective Hamiltonians for two vibrational states transforming according to some representations of the cubic symmetry group are studied. Chern numbers are evaluated for respective model Hamiltonians. The iso-Chern diagrams are introduced which split the parameter space into regions with fixed Chern numbers.     

The stability of linear periodic Hamiltonian systems under non-Hamiltonian perturbations

A definition of strong stability and strong instability is proposed for a linear periodic Hamiltonian system of differential equations under a given non-Hamiltonian perturbation. Such a system is subject to the action of periodic perturbations: an arbitrary Hamiltonian perturbation and a given non-Hamiltonian one. Sufficient conditions for strong stability and strong instability are established. Using the linear periodic Lagrange equations of the second kind, the effect of gyroscopic forces and specified dissipative and non-conservative perturbing forces on strong stability and strong instability is investigated on the assumption that the critical relations of combined resonances are satisfied.     

On linear vibrational systems with one dimensional damping II

We are interested in the quadratic eigenvalue problem of damped oscillations where the damping matrix has dimension one. This describes systems with one point damper. A generic example is a linearn-mass oscillator fixed on one end and damped on the other end. We prove that in this case the system parameters (mass and spring constants) are uniquely (up to a multiplicative constant) determined by any given set of the eigenvalues in the left half plane. We also design an effective construction of the system parameters from the spectral data. We next propose an efficient method for solving the Ljapunov equation generated by arbitrary stiffness and mass matrices and a one dimensional damping matrix. The method is particularly efficient if the Ljapunov equation has to be solved many times where only the damping dyadic is varied. In particular, the method finds an optimal position of a damper in some 60n ³ operations. We apply this method to our generic example and show, at least numerically, that the damping is optimal (in the sense that the solution of a corresponding Ljapunov equation has a minimal trace) if all eigenvalues are brought together. We include some perturbation results concerning the damping factor as the varying parameter. The results are hoped to be of some help in studying damping matrices of the rank much smaller than the dimension of the problem.     

大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．