Error of small parameter methods in solving shortened equations of a synchronized oscillator

The paper considers the use of recently appeared analytical methods for solving shortened equations of a synchronized oscillator. These are a quasi-small parameter method and a combined small parameter method. Both methods use the classic small parameter method. A peculiarity of their application is that in this case they are used for solving nonlinear differential equations that do not contain a small parameter. The difference between the above methods is in obtaining the equations of the first approximation. In the quasi-small parameter method, they are linear differential equations obtained by linearizing the original nonlinear differential equations in the area of the zero frequency detuning. In the combined small parameter method, the equations of the first approximation are obtained by approximating the original nonlinear differential equations. Of course, a number of transformations of these equations were made for this. The approximation made it possible to obtain better representation of the original nonlinear differential equations by means of linear differential equations. This representation provided a smaller error, which in both cases was presented as a discrepancy. The discrepancy does not allow obtaining a relative error and investigating its peculiarity. A study of the relative error of the quasi-small parameter method shows that this error is a continuous function of the frequency detuning with a zero value for a zero frequency detuning. A function representing relative error has a gap at zero frequency detuning for the combined small parameter method. However, this kind of gap can be eliminated by additional function definition.

Investigation of bending of an elastically supported beam taking into account the inhomogeneity of the base

Рассмотрен изгиб шарнирно закрепленной балки на упругом основании. Начальный прогиб и неоднородность жесткости основания заданы с точность до малых параметров. Получено условие, определяющее границу области сходимости метода малого параметра. Найдена функция, характеризующая прогиб, с точностью до величин четвертого порядка малости. Проанализирован случай, когда малые параметры являются случайными величинами. The bending of a pivotally fixed beam on an elastic base is considered. The initial deflection and inhomogeneity of the base stiffness are set up to small parameters. A condition is obtained that defines the boundary of the convergence region of the small parameter method. A function describing the deflection is found up to the fourth order of smallness. The case when small parameters are random variables is analyzed.

An analysis of annular plate in curvilinear non-orthogonal coordinates with the help of equations of a shell theory

The complete system of equations of a linear theory of thin shells in curvilinear non-orthogonal coordinates proposed in the paper was taken as the basis of the investigation. Earlier, this system was used for static analysis of a long developable helicoid. In the article, this system is applied for the determination of stress-strain state of annular and circular plates under action of the external axisymmetric uniform load acting both in the plane of the plate and out-of-their plane. Presented results for annular plate given in the non-orthogonal coordinates ex-pand a number of problems that can be solved analytically. They can be used as the first terms of series of expansion of displacements of degrees of the small parameter if a small parameter method is applied for examining a long tangential developable helicoid.

Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.

Stabilization of coupled inverted pendula: From discrete to continuous case

This study is focused on the investigation of stabilization problem for the system of coupled inverted pendula. The corresponding algorithm of control is based on the feedback principles and demonstrates some features in the physical implementation of the stabilization process. In the discrete case, the wave-like motion appears and leads to an observable interference pattern. In the continuous case, the provided feedback algorithm allows to simulate the stabilization process for the initially unstable solid medium. In these case conditions, ensuring the stabilization process is presented in the form of corresponding physical restrictions on the wave motion. Also, within the small parameter method, we investigate the nonlinear oscillatory motion of the material (the solid medium is modeled by the proposed system with nonlinear coupling). Results of numerical simulation for the system under consideration are presented and discussed. Particularly, numerical results show that the nonlinear material exhibits greater stability than the linear one.