Quasi-harmonic self-oscillations in discrete time: analysis and synthesis of dynamic systems

For sampling of time in a differential equation of movement of Thomson type oscillator (generator) it is offered to use a combination of the numerical method of finite differences and an asymptotic method of the slowl-changing amplitudes. The difference approximations of temporal derivatives are selected so that, first, to save conservatism and natural frequency of the linear circuit of self-oscillatory system in the discrete time. Secondly, coincidence of the difference shortened equation for the complex amplitude of self-oscillations in the discrete time with Eulers approximation of the shortened equation for amplitude of self-oscillations in analog system prototype is required. It is shown that realization of such approach allows to create discrete mapping of the van der Pol oscillator and a number of mappings of Thomson type oscillators. The adequacy of discrete models to analog prototypes is confirmed with also numerical experiment.

MATHEMATICAL JUSTIFICATION OF STABILITY OF STEERING BRIDGE OF WHEELED TRACTOR

The article deals with the influence of wheel oscillation and the micro-profile of the road surface on the stability of the wheel tractor axle movement. The reasons for the oscillations of the steered wheels, the design diagram of the controlled axle of the tractor and the sequence for determining the oscillation frequency of the axle of the tractor are presented. The reasons for the oscillation of the steered wheels are collisions with bumps, imbalance of the wheels and a double connection with the tractor frame through the steering system and the fastening of the steering axle beam. The most common functions for describing road irregularities that affect the movement of a tractor are the mathematical expectation and the average value of the ordinates of the micro-profile, the variance or standard deviation of the ordinates, the correlation function characterizing the relationship of various implementations of the micro-profile functions along the length of the road section and spectral density. Oscillations of the steered wheels have a side effect on the stability of the tractor, which leads to oscillations of the steered axle due to the presence of an additional degree of freedom (turning around the pivot) in comparison with uncontrolled ones. In addition, the steered wheels are interconnected by a steering linkage, which is damped due to clearances. Oscillations of the wheels can also occur due to the fact that the radial (normal) stiffness of the tires around the circumference is not the same. When such a tire rolls, the wheel begins to oscillate in a vertical plane. Such oscillations, performed due to changes in the parameters of the oscillatory system, are called parametric. Self-oscillations of the steered wheels cause significant dynamic loads on the steering parts, intense tire wear and lead to a loss of tractor controllability and driving stability. One of the main reasons for the occurrence of oscillations of the steered wheels is the presence of a gyroscopic relationship between the angular oscillations of the beam of the steered bridge in the transverse plane and the rotation of the wheels of this bridge relative to the pins. The article also discusses the physical essence of the processes occurring during self-oscillations of the tractor's controlled wheels.

Method for Determining the Pressure of a Pneumatic Wheel on the Soil

A method for determining the pressure of a pneumatic wheel on the soil is proposed. The method consists in measuring the magnitude of the vertical acceleration of the axles of a self-propelled vehicle, its speed and air pressure in the tire. The research results allow the developers of vehicles fitted with pneumatic wheels to choose rational characteristics of tires and optimal parameters of the oscillatory system of wheeled tractors, which help to reduce soil compaction.

On problem of calculating longitudinal vibrations of piston engine crankshaft

One of the initial stages of calculating the crankshaft longitudinal vibrations is developing an oscillatory system model, which includes the determination of longitudinal pliability (rigidity) of elastic sections. If it is impossible to determine the pliability experimental, the empiric formulas or the final element method (FEM) are used. There are given the values of crank longitudinal pliability of the crankshafts of different marine engine types found by using the formulas of L. Gugliemotti – R. Machciotta, P. Draminsky, E. Y. Gorbunov, S. F. Dorey, N. S. Skorchev, V. S. Stoyanov, etc. It is shown that the calculation results obtained from these formulas for the same engine significantly differ; therefore, the choice of one or another empirical formula for practical calculations is difficult. The preference of using FEM for determining the longitudinal (axial) compliance of cranks and other areas with complex geometric shapes has been proven. The possibility of its application is also shown to determine the longitudinal disturbing force as the reaction of the crankshaft support against the action of the radial force exerted to the connecting rod journal. It is proposed to use, along with empirical formulas, regression equations connecting the longitudinal compliance of the cranks with a significantly larger number of their design dimensions.

Determination of ship roll damping coefficients by a differential evolution algorithm

The execution of the so-called extinction tests represents the classical experimental method used to estimate the damping of an oscillatory system. For the specific case of ship roll motion, the roll decay tests are carried out at model-scale in a hydrodynamic basin. During these tests, the vessel is posed in an imbalance condition by an external moment and, after the release, the motion decays to the equilibrium condition. When the damping is far below the critical one, the transient decay is oscillatory. Here a new methodology is presented to determine the damping coefficients by fitting the roll decay curves directly, using a least-square fitting through a differential evolution algorithm of global optimisation. The results obtained with this methodology are compared with the predictions from standard methods. This kind of approach seems to be very promising when the motion model of the system under investigation is established with any level of non-linearities included. The usage of the fitting procedure on the approximate analytic solution of the differential equation of motion demonstrates the flexibility of the method. As a benchmark example, two experimentally measured roll extinction curves have been considered and suitably fitted. The newly predicted results, compared with the ones obtained from standard roll decay analysis, show that the algorithm is capable to perform a good regression on the experimental data.

Self-oscillations that cause the destruction of intermolecular bonds in the water coolant of a nuclear power plant

The digital acoustic model of a nuclear reactor (DAMNR) is presented as an auto-oscillatory system belonging to a special class of nonlinear dissipative systems capable of generating undamped oscillations. It is established that a water-water power reactor with a turbulent flow of a coolant is an open system of high complexity with a large number of elements, the connections between which are not predetermined, but probabilistic. Elements of the coolant circuit with negative dissipation (negative friction) are identified. It is shown that they self-organize chaotic turbulent pulsations and vortices into ordered wave oscillations, the frequency of which is determined by the Thomson (Kelvin) formula. In radio engineering circuits, an electronic self-oscillating generator with transformer feedback has similar properties. The presence of negative resistance in nonlinear dynamical systems leads to self-organization of chaotic turbulent perturbations and generation of self-oscillations in the form of acoustic standing waves (ASW). On the basis of theoretical and experimental data, the reliability of a previously unknown property of a reactor with connected pipelines - the ability to generate several ASW simultaneously-was confirmed. The use of DAMNR in the design and operation of nuclear power plants allows to identify the sources of ASW occurring in the coolant, the conditions for their occurrence and frequency.

Calculation of the main parameters of the ultrasonic oscillatory system for the intensification of gas nitriding processes

The paper studies the matters of designing an ultrasonic oscillatory system taking into account the peculiarities of the impact on the gas environment. Calculations of the elements of an ultrasonic oscillatory system are presented. The types of membrane emitters for influencing the saturating medium during gas nitriding are analyzed.

A Bridge between the Breath and the Brain: Synchronization of Respiration, a Pupillometric Marker of the Locus Coeruleus, and an EEG Marker of Attentional Control State

Yogic and meditative traditions have long held that the fluctuations of the breath and the mind are intimately related. While respiratory modulation of cortical activity and attentional switching are established, the extent to which electrophysiological markers of attention exhibit synchronization with respiration is unknown. To this end, we examined (1) frontal midline theta-beta ratio (TBR), an indicator of attentional control state known to correlate with mind wandering episodes and functional connectivity of the executive control network; (2) pupil diameter (PD), a known proxy measure of locus coeruleus (LC) noradrenergic activity; and (3) respiration for evidence of phase synchronization and information transfer (multivariate Granger causality) during quiet restful breathing. Our results indicate that both TBR and PD are simultaneously synchronized with the breath, suggesting an underlying oscillation of an attentionally relevant electrophysiological index that is phase-locked to the respiratory cycle which could have the potential to bias the attentional system into switching states. We highlight the LC’s pivotal role as a coupling mechanism between respiration and TBR, and elaborate on its dual functions as both a chemosensitive respiratory nucleus and a pacemaker of the attentional system. We further suggest that an appreciation of the dynamics of this weakly coupled oscillatory system could help deepen our understanding of the traditional claim of a relationship between breathing and attention.

Study of a Stripper Header for Grain Harvesting as a Vibrating System

Introduction. Grain losses caused by stripping defects are the main problem to be solved in designing a stripper header. To reduce these losses, a design of a stripper header with a vibration drive is proposed. This device combines the processes of stripping grain crops and the vibration effect of the stripping fingers upon the ears of plants. The most important stage of the mathematical description of these processes is composing the differential equation of the stripping fingers motion. Materials and Methods. A computational-graphic diagram of an oscillatory system with one degree of freedom is proposed. To compose the differential equation of the stripping fingers motion, a method based on the application of the Lagrange equation was used. The oscillations of the system under studying arise from the motion of a point in the system according to a given law. The problem of kinematic excitation is reduced to the problem of force perturbation. This stage of the study was carried out without taking into account the resistance forces. Results. An equation for motion of stripping fingers making vibrational reciprocating movements is obtained. It is proposed to select the elastic element in the design scheme and consider a more general case of the stripping fingers movement. In this case, the movement of the stripping fingers is considered to be difficult. A characteristic feature of the mathematical description is the presence of a generalized force of potential forces. The differential equation of motion of a comb in the presence of an elastic element and the solution of this equation are composed. Discussion and Conclusion. Forced oscillations of a system without resistance, excited by a harmonic disturbing force, are harmonic oscillations with constant amplitude. On close values of the angular frequency of vibration of the drive output link and the root of the ratio of the stiffness coefficient of the elastic element to the stripping fingers mass, the case of resonance takes place. The system parameters must be selected so as to avoid this negative phenomenon.