On stability in a case of oscillations of a pendulum with a mobile point mass

The problem of the orbital stability of periodic motions of a heavy rigid body with a fixed point is investigated. The periodic motions are described by a particular solution obtained by D. N. Bobylev and V. A. Steklov and lie on the zero level set of the area integral. The problem of nonlinear orbital stability is studied. It is shown that the domain of possible parameter values is separated into two regions: a region of orbital stability and a region of orbital instability. At the boundary of these regions, the orbital instability of the periodic motions takes place.

The pendulum-like motions of a rigid body in the Goryachev-Chaplygin case

In this paper the motion of a rigid body with a fixed point in a uniform gravity field is considered. It is assumed that the main moments of inertia of the body correspond to the case of Goryachev–Chaplygin, i.e., they are in the ratio of 1:1:4. In contrast to the integrable case of Goryachev–Chaplygin, there are no restrictions imposed on the position of the center of mass of the body. The problem of the orbital stability of periodic pendulum motions of a body (oscillations and rotations) is investigated. The equations of perturbed motion were obtained and the problem of orbital stability was reduced to the stability problem of the equilibrium position of a second-order linear system with 2π-periodic coefficients, the right-hand sides of which depend on two parameters. Based on the analysis of the linearized system, it has been established that the rotations are orbitally unstable for all possible values of the parameters. Moreover, a diagram of stability of pendulum oscillations has been constructed, on which the regions of orbital instability (parametric resonance) and regions of orbital stability in the linear approximation are indicated. At small values of oscillations amplitudes, a nonlinear analysis was performed. Based on the analysis of the coefficients of the normalized Hamilton function using theorems of the KAM theory, rigorous conclusions on the orbital stability of pendulum oscillations with small amplitudes were obtained.

This paper deals with the identification of a small mass point in a vibrating rod based on the knowledge of the variations induced in a pair of natural axial frequencies. The analysis is based on an explicit expression of the frequency sensitivity to point mass variations and allows consideration of non-uniform bars under general boundary conditions. The inverse problem is generally ill-posed, that is, even if the system is not symmetrical, mass points in different locations can still produce identical changes in a pair of natural frequencies. Despite this ill-posedness, it is found that there are certain situations concerning uniform rods in which the effects of the non-uniqueness of the solution may be considerably reduced by means of a careful choice of the data. Some of the results are also valid for beams in bending and the identification technique can be extended to include the case of two equal point masses. The theoretical results are confirmed by a comparison with dynamic measurements on steel beams with one and two point masses.

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

The direct variational method, developed for studying the asymptotic behavior of a wide class of nonlinear oscillation and wave problems, is extended to the study of the nonlinear stability problems. For systems which are unstable against small distrubances but stable against finite amplitude disturbances, the variational method can yield significant results in regimes even far away from the critical region. T The procedure of the variational method is illustrated by applications is various physical problems: the Duffing oscillation, a model wave equation for an unstable mechanical system, the two-stream instability in plasma, and the nonlinear Kelvin-Helmholtz stability problem.

In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.

The effect of ``promoters'' or ``inhibitors'' on the vibrational frequency of CO chemisorbed on transition-metal surfaces has been investigated by means of cluster models in which the coadsorbed electropositive or electronegative atoms have been replaced by point charges. Ab initio Hartree-Fock wave functions have been determined for CO chemisorbed on small Cu and Pd clusters in the presence of point charges as well as of uniform electric fields of different magnitude. It is shown that the simple electrostatic interaction between the point charges and the CO dipole makes an important contribution to the large vibrational shifts observed experimentally. Further contributions to the shifts occur because the cluster electron distribution polarizes in response to the field created by the point charges. These electronic effects, however, are much less important than the electrostatic contribution. The dependence of both the electrostatic, or Stark, effects and the electronic, or chemical, effects on field strength is studied by placing the cluster in uniform electric fields of different magnitude. As the field strength is increased, the electronic effect on the frequency shift becomes larger but, even for the largest fields considered here, it is always smaller than the Stark shift. The frequency shifts due to the point charges are shown to correspond to those which would be caused by a large uniform field.

The orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom in the case of 3:1 resonance

Understanding particle motion in force fields (PMFF), which encompasses the nature of forces and the relationship between force and motion, is fundamental to mastering mechanics and electromagnetism. Effectively solving PMFF-related problems requires advanced reasoning skills and the ability to apply knowledge across diverse contexts. Despite evidence that students often encounter significant challenges with these concepts, comprehensive assessment tools to reliably evaluate their understanding remain limited. In high school and introductory college physics courses, the most commonly studied force fields include those generated by a point mass or a point charge, as well as uniform gravitational, electric, and magnetic fields. The basic types of motion observed in these fields include uniform linear motion, uniformly accelerated rectilinear motion, uniformly accelerated curvilinear motion, and uniform circular motion. This study developed a nine-item multiple-choice test designed to assess students’ understanding of PMFF concepts. Data were collected from 34 college physics majors using test scores, interviews, and eye-tracking technology. The analysis revealed several alternative conceptions that varied depending on the specific force field context. Students performed relatively well in uniform force field scenarios but faced significant difficulties with fields generated by a point mass or a point charge, as well as uniform magnetic fields. In fields of a point mass or point charge, common misconceptions centered on uniformly accelerated motion. In the uniform magnetic field context, students often struggled to differentiate between uniform linear motion and uniformly accelerated curvilinear motion. Eye-tracking data provided additional insights, revealing attention patterns that corroborated the observed difficulties. These findings highlight critical areas where students face challenges in understanding PMFF. The results can guide future research efforts to develop targeted educational interventions aimed at addressing these specific learning difficulties, ultimately improving student comprehension of PMFF concepts.

This investigation studied the problem of the parametric instability of a beam-column with variable cross-section resting on an elastic foundation. The problem was analyzed both theoretically and experimentally. The equation of motion for the system was developed following the classical equilibrium approach. The terms arising from consideration of the distributed longitudinal inertia were incorporated into the equation. The equation of motion was reduced to a system of ordinary differential equations of the second order by means of a slightly modified Galerkin method developed by the author. Periodic solutions of these equations, which represent the boundaries of the regions of parametric instability, were sought in the form of a Fourier series with periods T and 2T. This approach reduced the problem to determination of the eigenvalues of a set of matrices. The QR transformation was used to compute the eigenvalues of these matrices. This transformation was preferred over other methods of extracting eigenvalues mainly because of its versatility, numerical stability and less computer time. A part of this investigation was devoted to the study of the steady-state amplitudes of parametric vibrations within the principal region of instability. The problem of determining the amplitudes was also reduced to an eigenvalue problem. An experimental investigation was conducted to verify the validity of the theoretical results. The experiment was designed so as to permit an independent variation of the constant and dynamic components of the load and the excitation frequency parameters. A comparison of the experimental and the theoretical results showed close agreement both for the boundaries of the principal and second region of instability and the steady-state parametric response within the principal region of instability. The experimental and theoretical findings revealed that the slope factor of a column with linearly variable cross-sectional depth has a pronounced effect upon the boundaries of the principal region of instability. The upper boundary, as compared to that of a uniform column, moved closer to the lower boundary as the dynamic load was increased. This in effect narrowed the width of the principal instability region. The addition of an elastic foundation also narrowed the width of the regions of instability to some extent, and altered their location as well. The amplitudes of the steady-state parametric response in the regions of instability were observed to be relatively small in presence of an elastic foundation. The modified Galerkin method used in this study greatly simplified the computations and can effectively be used to compute the natural frequencies and the critical loads of beam-columns with variable cross-sectional dimension.

On the interdependency of primary and initial secondary equilibrium paths in sensitivity analysis of elastic structures

The motion of a heavy rigid body with a fixed point in a uniform gravitational field is considered. It is assumed that the main moments of inertia of the body for the fixed point satisfy the condition of D.N. Goryachev–S.A. Chaplygin, i.e., they are in the ratio 1 : 4 : 1. In contrast to the integrable case of D.N. Goryachev–S.A. Chaplygin, no additional restrictions are imposed on the position of the center of mass of the body. The problem of orbital stability of pendulum periodic motions of the body is investigated. In the neighborhood of periodic motions, local variables are introduced and equations of perturbed motion are obtained. On the basis of a linear analysis of stability, the orbital instability of pendulum rotations for all values of the parameters has been concluded. It has been established that, depending on the values of the parameters, pendulum oscillations can be both orbitally unstable and orbitally stable in a linear approximation. For pendulum oscillations that are stable in the linear approximation, based on the methods of KAM theory, a nonlinear analysis is performed and rigorous conclusions about the orbital stability are obtained.

In this article, the problem of finite-time stabilization of two strings connected by point mass is discussed. We use the so-called Riemann coordinates to convert the study system into four transport equations coupled with the dynamic of the charge. We act by Bhat-Bernstein feedbacks in various positions (two extremities, the point mass and one of boundaries, only on the point mass, ...) and we show that in some cases the nature of the stability depends sensitively on the physical parameters of the system.

Topicality. When solving a number of problems associated with the design of complex control systems [1-3], it often becomes necessary to adjust the motion in the vicinity of the calculated trajectories set at the initial moment of time and belong to some dynamic sets of phase space [2, 3]. In this case, the correction requirements may relate to the individual coordinate of the calculated trajectories and their linear combination. These tasks belong to the class of tasks of stabilization of motion and reduce to the construction of regulatory influences that provide the original system a certain type of stability [1, 2, 4]. This approach is used, for example, in modeling charged particles in order to control them in dynamics in the presence of appropriate sustainability requirements. To construct constructive algorithms, the stability analysis is carried out on a finite time interval, and the initial conditions are given in the structural form [3, 5-7]. The latter allows us to obtain numerical estimates for the stabilization of motion to practical stability in the presence of dynamic constraints on phase coordinates. Analysis of recent research and publications. With the analysis of practical stability and sensitivity of dynamic systems, problems of optimal control of a beam of trajectories are closely related [2, 3, 6]. Thus, numerical algorithms for calculating the optimal areas of practical stability are used to estimate the area of particle capture in the acceleration process. In this case, the field of initial conditions can be in the given structures (sphere, ellipsoid), and the maximum in volume (optimal by inclusion). Then the question is raised about optimization of estimates of such sets due to the proper choice of system parameters [7]. The tasks of constructing regulatory influences that provide the desired stability requirements are an important part of a set of tasks that arise in the design of technical systems, in particular accelerating-focusing [1, 2]. Unlike the classical statements [1], the analysis of the stability of parametric systems [2-4, 6, 7] allows us to study the system's performance in real modes, taking into account the requirements for sensitivity, and to solve such problems numerically from the standpoint of stability [3]. The purpose of the research ─ development of constructive algorithms for solving stabilization problems for practical stability for linear parametric systems with disturbances. Materials and methods of research. The paper uses methods of stability theory, differential equations and control theory. Research results and their discussion. For linear parametric systems of differential equations with perturbations, the sequence of stabilization problems to practical stability in the given domains of dynamic constraints is investigated. Considered cases of known and limited perturbation norm. For structurally-defined areas of initial conditions, numerical estimates of the stabilization of motion to practical stability have been obtained. Conclusions and perspectives of further research. Formation of stabilization problems for practical stability of linear parametric systems with perturbations is formulated. On the basis of practical stability algorithms, numerical criteria for stabilizing the system motion to a certain type of stability were obtained. Such an approach can be extended to the solution of problems of stabilization of motion to asymptotic stability in regions of a more general structure. Keywords: mathematical model, stabilization, regulator, parameters, practical stability, parametric system, dynamic constraints