The calculation procedure using the method of direct linearization of mixed self-oscillations, forced and parametric oscillations in a system with limited power-supply is considered. On the basis of this method, the equations of non-stationary and stationary motions are derived. Using these equations and the Routh-Hurwitz criteria, the conditions for the stability of steady-state oscillations are obtained. Calculations were carried out to obtain information on the amplitude-frequency dependence and stability of oscillations in order to compare with the results obtained by the known methods of nonlinear mechanics. These calculations show that the results based on the method of direct linearization are qualitatively completely similar to the results obtained using the well-known methods of nonlinear mechanics, there are only fairly small quantitative differences. At the same time, in contrast to the known methods of nonlinear mechanics, the use of the method of direct linearization is quite simple, it takes much less time and labor spent.

On the Calculation by the Method of Direct Linearization of Mixed Oscillations in a System with Limited Power-Supply

This paper investigates the dynamics of a pneumatic dot peen marker without a slide valve. Dot peen marking is a promising method of marking part at any stage of the manufacturing process. A mathematical model of the marker is proposed, where vibration and impact properties of the system were described using methods of stereo-mechanical impact theory; the loads were calculated by solving a coupled gas dynamics problem. The design model is comprised of a nonlinear oscillator having one degree of freedom. The problem was solved using numerical integration. The proposed approach allows to describe major characteristics of dot peen marker operation, such as boundaries of the effective self-excited mode, amplitude and frequency of peen oscillation, pressure and temperature in the working chamber. The influence of the marker’s characteristics on the operating parameters was investigated using the calculated time histories of the state vector elements.

Mathematical Model of Dot Peen Marker Operating in Self-exciting Vibration Mode

The method of analytical study of the influence of impulse moments on nonlinear torsional oscillations of a homogeneous constant cross-section of a body under classical boundary conditions of the first, second, and third types have been developed. When the flexible material properties meet the body close to the power law of flexibility, mathematical models of the process have been obtained. It is the boundary value problem for an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of pulse momentum on the oscillatory process. Under the effect of periodic pulse momentum on a flexible body, resonant and non-resonant processes are possible. Resonant processes occur when the amplitude of natural oscillations approaches a fixed value. The peculiarities of resonant oscillations are established. The amplitude of passing through the primary resonance is significant for larger values of the nonlinearity parameter and, in the case of the action of pulse momentum, closer to the middle of the body. If the initial perturbation amplitude is less than the amplitude at which the resonance occurs in the presence of only internal forces of viscous friction. So, the external periodic impulse moments of resonance processes have occurred. The extreme case confirms the reliability of the results obtained related to the dynamics of the respective objects under the continuous action of autonomous perturbation.

Impulse Moments Influence on Single-Frequency Torsional Oscillations of Nonlinear Flexible Bodies

Self-oscillations under delays in elasticity and damping in a system with an energy source of limited power are considered. The dynamics of the system is described by nonlinear differential equations. The method of direct linearization of non-linearity is used for their solution. It has a number of advantages over the known methods of analysis of nonlinear systems: labor and time costs are reduced by several orders of magnitude; regardless of the specific type and degree of nonlinearity, you can get the final calculated ratios; there are no labor-intensive and complex approximations of various orders inherent in known methods. On the basis of this method, the equations of non-stationary and stationary movements are obtained, and the conditions for stability of stationary oscillations are derived. An analysis was performed to obtain information about the influence of delays on the parameters of stationary oscillations. The influence of delays on the amplitude of oscillations and the load on the energy source is shown. It turned out that depending on the delay value, the amplitude and load curves shift in the speed range of the energy source.

Influence of Delays on Self-oscillations in System with Limited Power-Supply

The Ensuring of the Economic Security of Industrial Enterprises in the Context of Forming a Flexible Management Model: Prerequisites and Tools

The purpose of this paper is to study the numerical oscillations of Runge-Kutta methods for the solution of alternately advanced and retarded differential equations with piecewise constant arguments. The conditions of oscillations for the Runge-Kutta methods are obtained. It is proven that the Runge-Kutta methods preserve the oscillations of the analytic solution. In addition, the relationship between stability and oscillations are shown. Some numerical examples are given to confirm the theoretical results.

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Qi Wang

Papers

Numerical Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Alternately Advanced and Retarded Type