Non-ideal Energy Source Research Articles

Bending-torsional coupling vibration characteristics of asynchronous motorized spindle considering electromechanical coupling and multi-excitation effect

For the asynchronous motorized spindle directly driven by the induction motor, the load-dependent slip ratio determines that its transient characteristics are sensitive to load. For the accurate evaluation of transient characteristics, it is necessary to deeply analyze the bending-torsional coupling vibration characteristics of the asynchronous motorized spindle with the non-ideal energy source. In this work, starting from energy conservation and clarifying the multi-physical field coupling mechanism, the bending-torsional coupling dynamic model is developed. Considering the speed fluctuation, the mass eccentricity excitation is deconstructed to explore its contribution to the coupling vibration. From the load-displacement relationship, the unbalanced excitation experienced by the journal from the bearing is modeled, considering the friction. According to the T-type equivalent circuit and mechanical characteristics of the induction motor, the electromagnetic torque considering power conservation is obtained. Furthermore, combined with Maxwell's stress formula, the electromagnetic excitation from electromagnetic torque and magnetic pull is obtained, considering the tangential and normal magnetic stress. Based on the projection and synthesis of forces, the semi-empirical mechanical cutting force model is employed to calculate the milling load. And a numerical algorithm for the fast acquisition of state-dependent delay is proposed. In particular, to reasonably characterize the energy characteristics in the system, the influence of structural vibration as an energy sink on the multi-physical field composed of mechanical, electromagnetic, and thermal fields is explored from the energy conservation. The proposed model was validated experimentally. The effects of electrical and mechanical parameters on the response are numerically discussed.

Extending the Laws of Thermodynamics for Arbitrary Autonomous Quantum Systems

Energy exchanges between arbitrary quantum systems are shown to obey thermodynamic laws, opening the way for the characterization of autonomous thermal devices at the microscale fueled by quantum, nonideal energy sources.

Dynamic modeling of a non-ideal gyroscopic rotor system with nonlinear damping and nonlinear rigidity of an elastic support

The article considers the effect of joint linear and nonlinear cubic damping on dynamics of a gyroscopic rigid rotor interacting with an electric motor with a rectilinear characteristic, taking into account the nonlinear rigidity of the support material. The method of regulating the control parameter (voltage on the motor), the amplitude of vibration, and the angular velocity of the shaft in the frequency equation, depending on the value of the coefficient of nonlinear cubic damping of the support, offers the most effective options for controlling resonant oscillations of large amplitudes. It is shown that the greater the value of the coefficient of nonlinear cubic damping, the easier it is to control these oscillations. Moreover, it is proved that the Sommerfeld effect (of the first kind) can also be weakened and eliminate with the help of joint linear and nonlinear damping. To do this, in the case of a rigid characteristic of the nonlinear elasticity of the support material, in a rotor system with a nonideal energy source to eliminate the bistability region, that is, jumping effects, more nonlinear damping of support or energy from a nonideal energy source will be required than in the case of an ideal rotor system.

Instability attenuation and bifurcation studies of a non-ideal rotor involving time-delayed feedback

In non-ideal vibratory system, the excitation is a nonlinear function of system response. The dynamic behavior of such system is often characterized by an energy source with limited power. The study of instability phenomena in non-ideal rotor driven through a non-ideal energy source is of considerable current interest. The non-ideal rotor system often gets destabilized on exceeding a critical input power near the resonance. This kind of instability is termed as Sommerfeld effect marked with nonlinear jump phenomena. This paper investigates the attenuation of nonlinear jump phenomena and numerical study of bifurcations of a non-ideal unbalanced rotor system with internal damping using time-delayed feedback via active magnetic bearings. The results show that the time delay indeed plays a critical role on the suppression of the jump phenomena. Following, some new insights are also revealed through a numerical study of saddle node, Hopf and trans-critical bifurcations with time delay as a bifurcation parameter. The transient analysis confirms the results obtained analytically through the steady-state consideration.

Resonance vibrations of a gyroscopic rotor with linear and nonlinear damping and nonlinear stiffness of the elastic support in interaction with a non-ideal energy source

The article examines the effect of linear damping and combined linear and nonlinear cubic damping of an elastic support on the dynamics of a gyroscopic rigid rotor with a non-ideal energy source, taking into account cubic nonlinear stiffness of the support material. Analysis of the research results shows that both linear damping and combined linear and nonlinear cubic damping can significantly suppress the resonance peak of the fundamental harmonic, reduce the amplitude of vibration frequency variation and stabilize the shaft rotation speed, but the effect of combined damping is more significant. In non-resonant regions, where the speed is higher than the natural frequency of the rotor system, both types of damping shorten the distance between jumps in nonlinear resonance curves and eliminate them. If linear damping mainly affects the boundaries of the instability region close to the resonant frequency, then nonlinear cubic damping significantly narrows the width of the instability region throughout the entire range beyond the resonant rotation speed. These results can be successfully used for the development of passive vibration isolators used to damp vibrations generated by rotary machines

Mixed forced, parametric, and self-oscillations with nonideal energy source and lagging forces

The purpose of this study is to determine the effect of retarded forces in elasticity and damping on the dynamics of mixed forced, parametric, and self-oscillations in a system with limited excitation. A mechanical frictional self-oscillating system driven by a limited-power engine was used as a model. Methods. In this work, to solve the nonlinear differential equations of motion of the system under consideration, the method of direct linearization is used, which differs from the known methods of nonlinear mechanics in ease of use and very low labor and time costs. This is especially important from the point of view of calculations when designing real devices. Results. The characteristic of the friction force that causes self-oscillations, represented by a general polynomial function, is linearized using the method of direct linearization of nonlinearities. Using the same method, solutions of the differential equations of motion of the system are constructed, equations are obtained for determining the nonstationary values of the amplitude, phase of oscillations and the speed of the energy source. Stationary motions are considered, as well as their stability by means of the Routh–Hurwitz criteria. Performed calculations obtained information about the effect of delays on the dynamics of the system. Conclusion. Calculations have shown that delays shift the amplitude curves to the right and left, up and down on the amplitude–frequency plane, change their shape, and affect the stability of motion.

A Tutorial Review on Time-Frequency Analysis of Non-Stationary Vibration Signals with Nonlinear Dynamics Applications

Time-frequency analysis (TFA) for mechanical vibrations in non-stationary operations is the main subject of this article, concisely written to be an introducing tutorial comparing different time-frequency techniques for non-stationary signals. The theory was carefully exposed and complemented with sample applications on mechanical vibrations and nonlinear dynamics. A particular phenomenon that is also observed in non-stationary systems is the Sommerfeld effect, which occurs due to the interaction between a non-ideal energy source and a mechanical system. An application through TFA for the characterization of the Sommerfeld effect is presented. The techniques presented in this article are applied in synthetic and experimental signals of mechanical systems, but the techniques presented can also be used in the most diverse applications and also in the numerical solution of differential equations.

Numerical and experimental investigation of the dynamic behavior of a cantilever beam driven by two non-ideal sources

Non-ideal energy sources are present in several engineering applications where the power source interacts with the dynamics of the structure to which it is coupled. The systems driven by this kind of source present a rich dynamic behavior and some phenomena that can only be observed under non-ideal consideration. The modeling of such systems becomes much more complex due to the interaction between the system and the energy source, requiring an additional equation. The present work proposes a continuous vibration model of a cantilever beam with two unbalanced DC motors. The motors are considered non-ideal energy sources. The advantage of the continuous approach is that the position of the motors can be easily varied. In addition, time–frequency methods are performed in order to characterize and identify the occurrence of the Sommerfeld effect on the non-stationary operation. In order to validate the model, experimental measurements were performed and, subsequently, compared with the numerical results.

Calculation by the method of direct linearization of mixed parametric oscillations and self-oscillations with a non-ideal energy source

Calculation by direct linearization of mixed parametric and self-oscillations with a non-ideal energy source and polynomial characteristics of elastic and friction forces of a sufficiently general form is considered. Equations of nonstationary and stationary motions are derived and stability conditions for steady-state oscillations are determined. Calculations are performed to obtain information on the amplitude-frequency dependence and stability of oscillations. Keywords mixed oscillations, parametric oscillations, self-oscillations, non-ideal energy source, method, direct linearization. a.alifov@yandex.ru

Coupled bending torsional vibrations of non-ideal energy source rotors under non-stationary operating conditions

In the automotive industry, high-speed elecrical engines will be used in a very large range of speeds, leading to non-stationary operating conditions. Thus more critical speeds may be crossed a very large number of times during the whole life-time of the engine. Therefore, estimating accurately the non-stationary loads and deformations during these transient regimes is of first importance for a correct design. In this paper, we propose a novel dynamic model for unbalanced high-speed rotors with less restrictive assumptions. The finite element model accounts for flexion, torsion and traction-compression leading to six degrees of freedom on each node. The non-ideal energy source is considered and the rotor is running under non-stationary operating conditions and crossing supercritical speeds. The angular displacement is defined in such a way that it combines simultaneously the intrinsic nominal rotation and the torsional deformation. The comparison of the proposed model with other models under different assumptions on the energy source and on the bending-torsion coupling shows that our model offers more accurate prediction for both lateral and torsional vibrations when crossing critical speeds. The ability of the proposed model to account for the mutual influence between lateral and torsional behavior is highlighted and is exhibited through time-frequency analyses.

Attenuation of Sommerfeld effect in an internally damped eccentric shaft-disk system via active magnetic bearings

Eccentric shaft-disk system with internal damping driven by a non-ideal power source exhibits Sommerfeld effect characterized by nonlinear jump phenomena of amplitude and rotor speed upon exceeding a critical power input around the critical speed. This effect causes instability in high speed rotors. So the diminution of such effect is extremely important in order to smooth running of the rotors at high speeds. The aim of this paper is to attenuate the Sommerfeld effect of an internally damped unbalanced flexible shaft-disk system via linearized active magnetic bearings. The shaft-disk system is excited through a brushed DC motor which acts as a non-ideal energy source. The characteristic equation of fifth order polynomial in rotor speed is obtained through energy balance of supplied power and the mechanical power dissipated at steady-state condition. Using MATLAB simulations, amplitude frequency responses are obtained close to system resonance for several values of bias current of active magnetic bearings. Thus the Sommerfeld effect is found to be attenuated as bias current increases gradually. The complete disappearance of Sommerfeld effect is also reported when the bias current reaches a specific value under certain conditions. A few numerical results are validated with established results when bias current is made to zero.

The nonlinear vibrations of orthogonal mechanism of vibrating table in view of friction

Abstract. In dynamics of the orthogonal mechanism of vibrating table main attention is paid to the effect of friction on vibrational process when it is interacting with non-ideal energy source. As a result of analysis of numerical solutions of nonlinear equation of motion, it is established that in general, for the period of rotation, both sliding friction and rotational friction increase the difference between the maximum and minimum angular velocity values of the driving link (engine shaft) when its average value is constant; both increase the coefficient of non-uniformity of rotation. When there is a cargo of significant mass in the working link, the effect of sliding friction impacts more on the amplitude and frequency of the angular velocity oscillations of the driving link (engine shaft) and on the frequency of the coordinate changes, as well as on the frequencies and maximum and minimum values of the projections of the velocity and acceleration of the hinge linking the work platform to coupler. Similarly, the larger of the load mass, the stronger the effect of rotational friction on the amplitude and frequency of angular velocity oscillations of the driving link (engine shaft).

Remarks on the Sommerfeld effect characterization in the wavelet domain

In many applications in engineering, a mechanical system operates in the nonstationary regime, either partially of fully, creating the possibility to generate nonlinearities in them. Great efforts have been made to better understand and characterize these phenomena. One of several methods that are being used for the processing of signals of a nonstationary nature, as well as for the characterization of nonlinearities in mechanical systems, is the wavelet transform. A particular phenomenon that is seen in systems operating in the nonstationary regime is the Sommerfeld effect, which occurs due to the nonlinear interaction between a nonideal energy source and a mechanical system. This phenomenon can lead to high amplitudes of vibration for the system that in turn can cause damage in it. Therefore, this work presents an application of the continuous wavelet transform and the wavelet packet transform for the characterization of the Sommerfeld effect in mechanical systems where only the time response is at hand. Experimental procedures were performed where a nonideal energy source (an unbalanced DC motor) was used to excite (a) a portal frame and (b) a three-story shear-building. The results showed the effectiveness and the potential of the methods proposed.

Dynamic characterization of a flexible internally damped spinning shaft with constant eccentricity

In the present study, the dynamic characterization of a flexible spinning shaft with constant eccentricity driven by a non-ideal energy source (DC motor) with external and internal damping has been focused. It is well established that the structural response of a vibratory system to which a non-ideal drive is connected may act as an energy sink under certain conditions such that a part of the energy supplied by the source is spent to vibrate the structure rather than to increase the drive speed. This phenomenon is formally known as the Sommerfeld effect. The Sommerfeld effect characterized by jump phenomena is studied through the steady-state amplitude obtained by instantaneous power balance method and further verified through numerical simulation. Finally, power balance equation is transformed into a characteristic equation through which jump phenomena and Sommerfeld effect are predicted numerically for the first and third modes using root loci method. The break-in and breakaway points of the root loci represent the values of the supply voltage at which jumps in shaft speed and flexural vibration amplitudes take place during coast-up and coast-down operation.

Non-linear vibrating systems excited by a nonideal energy source with a large slope characteristic

This paper revisits the problem of an unbalanced motor attached to a fixed frame by means of a nonlinear spring and a linear damper. The excitation provided by the motor is, in general, nonideal, which means it is affected by the vibratory response. Since the system behaviour is highly dependent on the order of magnitude of the motor characteristic slope, the case of large slope is considered herein. Some Perturbation Methods are applied to the system of equations, which allows transforming the original 4D system into a much simpler 2D system. The fixed points of this reduced system and their stability are carefully studied. We find the existence of a Hopf bifurcation which, to the authors’ knowledge, has not been addressed before in the literature. These analytical results are supported by numerical simulations. We also compare our approach and results with those published by other authors.

Nonstationary analysis of nonlinear rotating shafts passing through critical speed excited by a nonideal energy source

In this paper, the effect of nonlinearity on vibration of a rotating shaft passing through critical speed excited by nonideal energy source is investigated. Here, the interaction between a nonlinear gyroscopic continuous system (i.e. rotating shaft) and the energy source is considered. In the shaft model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. Firstly, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with nonconstant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the nonstationary vibration of the nonideal system, multiple-scale method is directly applied to the equations expressed in complex coordinates. Three analytical expressions that describe variation of amplitude, phase, and angular acceleration during passage through critical speed are derived. It is shown that Sommerfeld effect in specific range of driving torque occurs. Finally, effect of damping and nonlinearity on occurrence of Sommerfeld effect is investigated. It is shown that the linear model predicts the range of Sommerfeld effect occurrence inaccurately and, therefore, nonlinear analysis is necessary in the present problem.

Limit cycles in nonlinear vibrating systems excited by a nonideal energy source with a large slope characteristic

This study is the natural continuation of a previous paper of the authors Gonzalez-Carbajal and Dominguez in J. Sound Vib. (Under revision), where the possibility of finding Hopf bifurcations in vibrating systems excited by a nonideal power source was addressed. Herein, some analytical tools are used to characterize these Hopf bifurcations, deriving a simple rule to classify them as supercritical or subcritical. Moreover, we find conditions under which the averaged system can be proved to be always attracted by a limit cycle, irrespective of the initial conditions. These limit cycle oscillations in the averaged system correspond to quasiperiodic motions of the original system. To the authors’ knowledge, limit cycle oscillations have not been addressed before in the literature about nonideal excitations. Through supporting numerical simulations, we also investigate the global bifurcations destroying the limit cycles. The analytical results are verified numerically.

Nonlinear Dynamics of Electro-mechanical System Composed of Two Pendulums and Rotating Hub

Dynamics of a structure composed of two pendulums attached to a rotating rigid hub is analysed in the paper. The system is rotating in a horizontal plane so the gravity force does not influence its motion. Both pendulums are treated as lumped masses fixed at the ends of stiff and massless rods connected to the hub by flapping hinges. Three possible variants of the excitation are considered: (a) an ideal energy source, where torque is given by a defined harmonic function (b) a simplified limited power supply, where torque is given by an arbitrary defined function or (c) a non-ideal energy source, where DC output characteristic of limited power is taken into account.

On nonlinear dynamics behavior and control of a new model of a magnetically levitated vibrating system, excited by an unbalanced DC motor of limited power supply

We analyze new results on a magnetically levitated body (a block including a magnet whose bottom pole is set in such a way as to repel the upper pole of a magnetic base) excited by a non-ideal energy source (an unbalanced electric motor of limited power supply). These new results are related to the jump phenomena and increase of power required of such sources near resonance are manifestations of a non-ideal system and they are referred as the Sommerfeld effect, which emulates an energy sink. In this work, we also discuss control strategies to be applied to this system, in resonance conditions, in order to decrease its vibration amplitude and avoiding this apparent energy sink.

Non-ideal mechanical system with an oscillator with rational nonlinearity

In this paper the generalized form of a non-ideal system which contains a pure nonlinear oscillator and a non-ideal energy source is studied. In the non-ideal oscillator-motor system there is an interaction between the motions of the oscillator and those of the motor, as the motor has an influence on the oscillator and vice versa. The mathematical model of the system is represented with two coupled nonlinear differential equations. The averaging method for solving these differential equations is based on the application of the Ateb function, which is the exact solution of the pure nonlinear oscillator. Using the obtained approximate solution, the resonant motion of the system is considered. Significant attention is paid to the steady-state motion and to the Sommerfeld effect. The influence of the order of nonlinearity on the dynamics of the nonideal system is evident. In the paper the procedure for determination of the parameters for suppression of the Sommerfeld effect of the non-ideal system is also given.