Method Of Small Parameters Research Articles

Synchronization of a vibration system excited by two eccentric rotors on different rigid frames

This paper is focused on synchronization of a dynamic model with three rigid frames (RFs) driven by two reverse rotating eccentric rotors (ERs) distributed on different RFs. The differential equations of motion are established based on Lagrange equation. Applying the average method of small parameters, the condition is derived for the vibration system to meet synchronous motion. The stability condition of the vibration system is obtained by using Lyapunov indirect method. To investigate the selected motion characteristic on synchronous state in various frequency regions, theoretical results are discussed with frequency ratio numerically. Coupled motion of the vibration system, phase difference between two ERs, motion phase between RFs, and amplitude–frequency characteristics of RFs are discussed qualitatively to reveal the selected motion characteristic. A series of experiments are done to further illustrate the selected motion characteristic in sub-resonant state and super-resonant state of the vibration system with vertical direction. Finally, the experimental results validate the predication of the theoretical method.

Double synchronization states of two exciters with horizontal asymmetric structure in a vibrating system

Synchronization of two exciters with asymmetric structure is more widely used in engineering, such as vibrating conveyer, dewatering screen, vibrating screen, however, the synchronization state of two exciters with asymmetric structure in the sub-resonant and super-resonant states is less considered, and it is a valuable research topic. Firstly, we establish a kind of weakly nonlinear vibrating system driven by two exciters with opposite rotation directions and asymmetrical structure. Using Lagrange equations, the differential equations of the vibrating system are obtained. Secondly, we apply Fourier series expansion to obtain the nonlinear elastic force of the nonlinear stiffness. Thirdly, we introduce the modified averaging method of small parameters and Hamilton’s principle to get the criterions of synchronization synchronization stability of two exciters. The theoretical analysis shows that the phase differences are stabilized in the vicinity of Pi phase difference in a super-resonant state when l0 approaches to le, otherwise, in the neighborhood of zero phase difference in a sub-resonant state. Then, the above theory is verified by experiments. The stable phase difference, amplitudes responses and energy consumption are also discussed. This research provides the foundation and guidance for the synchronization of two exciters with horizontal asymmetrical structure and engineering design.

Synchronization and stability of an elastically coupled tri-rotor vibration system

A new mechanism, an elastically coupled tri-rotor system, is proposed to implement synchronization. It is composed of a rigid body, three induction motors, coupling unit and springs. According to the Lagrange equation, the model of the system is established. The average method of small parameters is applied to study the synchronization characteristics of the system, therefore, the balance equation and stability criterion of the system can be obtained. Obviously, many parameters affect the synchronous state of the rotors, especially the spring stiffness, the stiffness of the coupling unit and the installation location of the system. Finally, computer simulations are used to verify the correctness of theoretical analysis.

Existence of Equilibrium Points in the R3BP with Variable Mass When the Smaller Primary is an Oblate Spheroid

The paper deals with the existence of equilibrium points in the restricted three-body problem when the smaller primary is an oblate spheroid and the infinitesimal body is of variable mass. Following the method of small parameters; the co-ordinates of collinear equilibrium points have been calculated, whereas the co-ordinates of triangular equilibrium points are established by classical method. On studying the surface of zero-velocity curves, it is found that the mass reduction factor has very minor effect on the location of the equilibrium points; whereas the oblateness parameter of the smaller primary has a significant role on the existence of equilibrium points.

Improved Perturbative Solution of Yaroshevskii's Planetary Entry Equation

An improved approximate analytical solution is developed for Yaroshevskii's classical planetary entry equation for the ballistic entry of a spacecraft into planetary atmospheres at circular speed. Poincaré's method of small parameters is used to solve for the altitude and flight path angle as a function of the spacecraft's speed. From this solution, other important expressions are developed including deceleration, stagnation-point heat rate, and stagnation-point integrated heat load. The accuracy of the solution is assessed via numerical integration of the exact equations of motion. The solution is also compared to the classical solutions of Yaroshevskii and Allen and Eggers. The new second-order analytical solution is more accurate than Yaroshevskii's fifth-order solution for a range of shallow (−3 deg) to steep (up to −90 deg) entry flight path angles, thereby extending the range of applicability of the solution as compared to the classical Yaroshevskii solution, which is restricted to an entry flight path of approximately −40 deg.

Elastoplastic problem for noncircular openings under Coulomb’s criterion

An elastoplastic problem is considered for elliptical, arched, square and polygonal cross-section openings on the assumption of Coulomb’s limit equilibrium and varied lateral earth pressure coefficients in intact rock mass. The problem is solved using the method of small parameters and the finite element method. The elastoplastic solutions are obtained for mutually influencing underground openings. The mechanisms of concentration of vertical stresses in a pillar established between mutually influencing openings are determined.

On the Synchronization of Two Eccentric Rotors with Common Rotational Axis: Theory and Experiment

We study synchronization of two eccentric rotors (ERs) with common rotational axis in the vibration system of the far-resonant spatial motion. We deduce the dimensionless coupling equation of two ERs with applying the average method of small parameters. We convert the synchronization problem into the existence and stability of solving the zero solutions for the dimensionless coupling equations. By introducing the synchronization torque and the difference between the residual torques of two motors, we obtain the synchronization condition that two ERs achieve the synchronized motion. We derive the stability condition of the synchronized motion, which satisfies Routh-Hurwitz criterion. We discuss numerically the choosing motion feature of the vibration system, which indicates that the vibration system has two steady motion modes. The synchronization torque forces the phase difference to approachπwhen the structural parameters of the vibration system satisfy the condition of the spatial cone motion, and the synchronization torque drives the phase difference to approach zero when those satisfy the condition of the spatial circle motion. Finally, through the comparison and analysis of experimental data, the structural parameters of the vibration system satisfying the above two conditions can guarantee the synchronization stability for two ERs.

Vibratory synchronization transmission of two exciters in a super-resonant vibrating system

In this study, vibratory synchronization transmission (VST) of two exciters is investigated in a super-resonant vibrating system to save energy. First, the frequency capture equation of two exciters (FCETE) is deduced by the average method of small parameters. The criterion of VST being the equivalent effective residual torque of motor with power supply should be smaller or equal to the frequency capture torque of the system by setting the disturbance parameters of angular velocities in FCETE to zero. The stability criterion of synchronous states is derived by complying with the generalized Lyapunov equations. Lastly, experiment and computing simulation on VST are conducted. The VST regime of two exciters is discussed in detail from coupling characteristic, current change, visual picture, phase relation, and direct verification of parameters, respectively by comparison and analyses of corresponding data. Results indicate that the key factor in VST implementation is the torque of frequency capture, which serves as the ‘vehicle’ in transferring energy. Thus, the distance between rotational centre of each exciter and mass centre of the rigid frame should be far enough to enhance the ability of VST and ensure energy saving. The theory, experiment, and computing simulation prove that the theoretical approach used in this study is useful and feasible.

Vibratory synchronization and coupling dynamic characteristics of multiple unbalanced rotors on a mass-spring rigid base

The investigation of a generalization of the model on vibratory synchronization for multiple unbalanced rotors (URs) is considered in present work. Based on the previous publications and using the average method of small parameters, the dimensionless coupling equation of multiple URs is constructed, and the criterions of synchronization and stability in the simplified form for multiple URs are given. Taking three counting-rotating URs for example, the coupling dynamic characteristics and synchronization regime of the system are discussed numerically. Then a vibrating synchronization bedstand corresponding to the dynamical model used in numerical discussion is set up, a more detailed discussion and parameter study in experiment are provided, through the analyses on experimental data such as phase differences, rotational velocities and synchronization quality, it is shown that the experimental results are in approximate or good agreement with the numerical/theoretical results, which validates the validity of the model and approach.

Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame

This paper investigates synchronization of two self-synchronous vibrating machines on an isolation rigid frame. Using the modified average method of small parameters, we ded uce the non-dimensional coupling differential equations of the disturbance parameters for the angular velocities of the fo ur unbalanced rotors. Then the stability problem of synchronization for the four unbalanced rotors is converted into the stabili ty problems of two generalized systems. One is the generalized system of the angular velocity disturbance parameters for the four unbalanced rotors, and the other is the generalized system of three phase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix is definite positive, and that all the eigenvalues for the generalized s ystem of three phase disturbance parameters have negative real parts. Computer simulations are carried out to verify the results of the theoretical investigation.

Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame

This paper investigates synchronization of two self-synchronous vibrating machines on an isolation rigid frame. Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of the disturbance parameters for the angular velocities of the four unbalanced rotors. Then the stability problem of synchronization for the four unbalanced rotors is converted into the stability problems of two generalized systems. One is the generalized system of the angular velocity disturbance parameters for the four unbalanced rotors, and the other is the generalized system of three phase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix is definite positive, and that all the eigenvalues for the generalized system of three phase disturbance parameters have negative real parts. Computer simulations are carried out to verify the results of the theoretical investigation.

Synchronization of Two Asymmetric Exciters in a Vibrating System

We investigate synchronization of two asymmetric exciters in a vibrating system. Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of the two exciters (NDDETE). By using the condition of existence for the zero solutions of the NDDETE, the condition of implementing synchronization is deduced: the torque of frequency capture is equal to or greater than the difference in the output electromagnetic torque between the two motors. Using the Routh-Hurwitz criterion, we deduce the condition of stability of synchronization that the inertia coupling matrix of the two exciters is positive definite. A numeric result shows that the structural parameters can meet the need of synchronization stability.

Synchronization of two asymmetric exciters in a vibrating system

We investigate synchronization of two asymmetric exciters in a vibrating system. Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of the two exciters (NDDETE). By using the condition of existence for the zero solutions of the NDDETE, the condition of implementing synchronization is deduced: the torque of frequency capture is equal to or greater than the di fference in the output electromagnetic torque between the two motors. Using the Routh-Hurwitz criterion, we deduce the condition of stability of synchronization that the inertia coupling m atrix of the two exciters is positive definite. A numeric result shows that the structural parameters can meet the need of synchronization stability.

Relative Periodic Motion of a Rigid Body Pendulum on an Ellipse

In this paper, the properties of the relative periodic motions of a rigid body suspended on an elastic string in a vertical plane are considered. The supported point of this pendulum moves on an elliptic path with the rigid body suspended at the end point of the pendulum. Applying Lagrange’s equation, the equations of motion are obtained in terms of small parameter (ε). These equations represent a quasi-linear system of second order, which can be solved in terms of generalized coordinates Θ, Φ, and β using the method of small parameters. At the end, a discussion of the motion and a conclusion of the results are considered to show the orientation of the body and the geometric interpretation of this motion at any instant of time. Also, computerized data with graphical representations of the solutions are given for describing the behavior of the body in some periods of time.

Analysis of vibrations in large flexible hybrid systems

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible.

Periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in the Kovalevskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincare the existence of a periodic solution is established.

The general problem of regular motion of an axisymmetrical satellite in a central field

In this paper, the stationary motions of an axisymmetrical satellite in a central field of attraction are investigated. Both the orbital, “translational”, and the self-rotational motions of the satellite are taken into consideration. In the decomposition of the Newtonian force function, all the harmonics up to the third order are taken into account. The exact solutions are deduced by Poincare's method of small parameters. The results show the existence of two solutions, which satisfy all the necessary and sufficient conditions. One of the solutions belongs to the group of stationary motions of a triaxial satellite obtained before by the author. The second solution, in which the axis of symmetry can keep an arbitrary constant inclination to the orbital plane, is a new and rather a more general solution. These solutions generalize the conical and hyperbolic precessions of the satellite, suggested by Sarichev [ Sci. Technol. J., Moscow, 1978] and the stationary solutions of the limited problem suggested by the author.

A new class of periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in Kovaleveskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincare, the existence of a periodic solution is established.

Method of successive approximations for solution of optimal control problems

AbstractThe paper contains a survey of results devoted to one of the numerical methods of optimal control—the method of successive approximations. This method is based on Pontryagin's maximum principle and is known in the English literature as the min‐H method. Various modifications of the method and some theoretical results on its convergence are presented. Examples of applications of the method for the calculation of optimal trajectories are given. The method of small parameters which is close to the method of successive approximations, is also described.

Mathematical models of high temperature flow in gas ducts of rocket engines

Results are given on certain dynamic properties of a high temperature and pressure gas flow. The investigation concerns the effect of sonic processes on the transfer of external signals by the flow and of the gas nonidealities on the properties of a linear and a nonlinear mathematical model of the jet nozzle. The investigation was conducted by the method of small parameters for differential equations of the dynamics of the gas flow, with the help of invariants of these equations, and estimates of solutions with arbitrary signals of a specific class. The results lead to estimates of errors due to usual simplifications of gas flow models such as linearization of jet nozzle equations and neglect of sonic processes. The errors of the latter kind are shown to depend on the Mach number and smoothness of signals. Accurate and approximate models of nozzle flow are obtained useful for controllability analysis of rocket engines. Some computation examples are given.