Stability Of Non-conservative Systems Research Articles

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

This paper intends to summarize the scientific production of Angelo Luongo on the occasion of his Sixtieth Birthday, focusing on his main contributions in the field of Mechanics. The task will not be easy because of the breadth of his scientific production, only apparently attributable to a restricted number of keywords. In fact, even when the work seems purely algorithmic, speculation on the physical and mechanical aspects of the problem is always present, providing new interpretations and innovative openings to a careful reader. Similarly, also the works, which apparently seem to be high-level applications, always reserve methodological aspects that are not negligible. The editorial choice to divide his papers through a small number of keywords is certainly simplistic, but offers the possibility to better highlight all the connections among his variegated production. The most original contributions of Angelo Luongo in the context of perturbation methods, linear and nonlinear dynamics and control, elastic buckling and structural analysis, bifurcation and stability of non-conservative systems, are discussed in detail. Finally, the Angelo Luongo’s central role in the creation and development of activities of the international research center M&MoCS is pointed out.

P‐positive definite matrices and stability of nonconservative systems

AbstractThe bifurcation problem of constrained non‐conservative systems with non symmetric stiffness matrices is investigated. It leads to study the subset Dp,n of ℳ︁n(ℝ) of the so called p‐positive definite matrices (1 ≤ p ≤ n). The main result (D1,n ⊂ Dp,n) is proved, the reciprocal result is investigated and the consequences on the stability of elastic nonconservative systems are highlighted.

On the stability of nonconservative systems with small dissipation

In the present work, we study the paradoxical influence of small dissipative and gyroscopic forces on the stability of linear nonconservative systems consisting of the nonpredictable (at first glance) behavior of a critical nonconservative loading. By studying bifurcations of multiple roots of the characteristic polynomial of the nonconservative system considered, the analytical description of this effect is obtained. The model of a disk brake describing the appearance of a creak in the braking of a car is considered as a mechanical example.

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.

Stability of nonconservative systems and applications

Stability of nonconservative systems and applications

Stability of nonconservative systems and applications

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

Stability of elastic and viscoelastic plates in a gas flow taking into account shear strains

It is well known that the internal friction in a material can have a considerable destabilizing effect on the stability of non-conservative systems. Apart from the Voigt model, the viscoelastic body model is sometimes utilized to describe material damping. This relates the stability problem for non-conservative elastic systems with that for viscoelastic system. The Bubnov–Galerkin method is usually applied for solving the problems. In this case, the displacement functions are represented by series in terms of natural vibration modes ϕ i( x ) of the elastic system. To provide a high degree of accuracy for the solution, one should involve a fairly large number of modes. For a viscoelastic plate, the number of terms to be kept in the expansion of the deflection can be substantially more. One should bear in mind, however, that as the number of modes preserved in the expansion increases, the influence of shear strains and rotational inertia on the behavior of the solution becomes more pronounced. In view of this, it is important to study the stability of non-conservative viscoelastic systems with the shear strain and rotational inertia being taken into account. In the present paper this problem is solved for a viscoelastic plate in a supersonic gas flow.

The stability of non-conservative systems with singular matrices of dissipative forces

Earlier results 1. , 2. , 3. , 4. are developed in application to a certain special class of non-conservative mechanical systems in which the matrices of dissipative and non-conservative forces are singular. For this class of systems, necessary and sufficient conditions are formulated for reducing the initial matrix equation to a form that admits of direct application of the Kelvin-Chetayev theorems. An example is presented.

Effect of small dissipative and gyroscopic forces on the stability of nonconservative systems

We analyze the effect of small forces proportional to the generalized velocity vector on the stability of a linear autonomous mechanical system with nonconservative positional forces. It is known that arbitrarily small dissipation generally destabilizes a nonconservative system [1‐5]. Necessary and sufficient conditions on the matrix of dissipative and gyroscopic forces under which the system is asymptotically stable are obtained. The two-dimensional system is studied in detail. The problem of the stability of the Ziegler‐Herrmann‐Jong pendulum is considered as a mechanical example. 1. We consider a linear mechanical system with nonconservative positional forces and small forces proportional to the velocity vector:

The stability of non-conservative systems and an estimate of the domain of attraction

The stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces act, is investigated. The condition for asymptotic stability is obtained using the Lyapunov function and an estimate of the domain of attraction is also found in terms of the system being considered. A precessional system is also examined. It is shown that the condition for the asymptotic stability of a system is the condition of acceptability in the sense of the stability of a precessional system. The results obtained are applied to the problem of the stabilization, using external moments, of the steady motion of a balanced gyroscope in gimbals.

On the Stability of Nonconservative Systems with Estimation of the Attraction Domain

Mechanical systems subjected to dissipative, gyroscopic, conservative, and also nonconservative positional forces are considered. The question of the effect of dissipative, gyroscopic, and conservative forces on the motion stability of a mechanical systems is determined by classical Kelvin–Chetaev theorems [1]. The presence of nonconservative positional forces considerably complicates the situation and excludes direct application of these theorems. Applying Lyapunov's functions method the condition of asymptotic stability of a mechanical system under the action of all listed above forces is obtained. Moreover, the estimation of the attraction domain in phase space is found. The precession system which is used in the solution of some problems in the applied theory of the gyroscopic systems is also examined. The connection between the stability of origin and precession systems is detected. Theoretical results are applied to the stabilization problem of stationary motion of the balanced gimbal suspension gyro by means of external moments.

About the influence of damping forces randomly changing in time on the stability of nonconservative systems

Using the pendulum with one degree of freedom subjected to a follower longitudinal force as an example of nonconservative viscoelastic systems, the stability of the equilibrium state of such systems is investigated at the deterministic and stochastic setting. The stochastic treatment supposes the random variation in time of characteristics of the external damping and viscous properties of the material. In this case the conditions of the almost sure stability and stability in mean-square are obtained.

The effect of damping on the dynamic stability of nonconservative systems

A study of the effect of vanishing viscous damping on the stability of nonconservative linear elastic systems, with special reference to systems that lose stability statically through the emergence of new equilibrium configurations close to the principal one, i. e. by divergence. It is shown that in this case slight damping presents a wide range of effects on the stability of the system.

Bifurcation Criterion and Plastic Buckling of Plates and Columns

Bifurcation Criterion and Plastic Buckling of Plates and Columns

Some implications of work hardening and ideal plasticity

The purpose of this note is to point out the severe restriction imposed on possible stress-strain relations by a mathematical formulation of the concepts of work hardening and ideal plasticity. Using this condition, an algebraic derivation and extension is given of Prager’s extension of the Mises plastic potential function. A brief discussion is also given of the meaning of stability of plastic deformation as contrasted with stability of non-conservative systems in general.