On the problem of vibrations of non-linear-elastic electroconductive plates in transverse and longitudinal magnetic fields

Non-linear cylindrical-form vibrations of an electroconductive plate-strip in longitudinal and transverse magnetic fields are considered. For the plate material a non-linear relationship between stress and deformation is accepted. On the basis of the known hypotheses and assumptions the general system is reduced to one equation for the normal displacement of the plate. Using the Bubnov-Galerkin method and the method of asymptotic integration, the amplitude-frequency characteristics of vibrations are defined. On the basis of numerical analysis different degrees of the influence of various factors (non-linearity, magnetic field, conductivity, etc.) on the nature of vibrations are compared. Qualitative and quantitative differences between the nature of vibrations in longitudinal and transverse magnetic fields are revealed.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

水轮发电机转子偏心引起的非线性电磁振动

由于机械和电磁相互耦合,水轮发电机的电磁振动具有强非线性特征。根据不平衡磁拉力与转子偏心的非线性函数关系,通过简化的各向同性的单圆盘转子系统,建立了水轮发电机转子电磁振动的非线性系统。利用非线性振动理论的多尺度方法,从理论上分析了该系统强迫振动的稳态响应,进而研究了水轮发电机转子偏心引起的电磁振动。研究发现不平衡磁拉力使系统的涡动频率下降,使运动的中心发生变化;并且会引起两倍转频的振动。最后用模型转子仿真试验的结果验证了这些理论分析的结论。     

Dynamic Analysis of the Non-Linear Behavior of a Composite Sandwich Beam with a Magnetorheological Elastomer Core

This work deals with the active control of the vibrations of mechanical structures incorporating magnetorheological elastomer. The damping coefficient and shear modulus of the elastomer increase when exposed to a magnetic field. Compared with the vibration control where the elastomer is permanently exposed to a magnetic field, the control of this process through time reduces vibrations more effectively. The experimental study for the vibrations of a sandwich beam filled with an elastomer is conducted, followed by a numerical study using the Abaqus code.The vibration damping is found to be dependent on the loading rate of micro-size ferromagnetic particles in the elastomer.     

Effect of magnetic field and non-homogeneity on the radial vibrations in hollow rotating elastic cylinder

In this paper, the natural frequencies of the radial vibrations of a hollow cylinder with different boundary conditions under influences of magnetic field, rotation and non-homogeneity have been studied. The solution of the problem is obtained by using technique of variables separation. In the present paper three different boundary conditions are considered, namely the free, fixed and mixed boundary conditions. The displacement and stresses components have been obtained in analytical form involving Bessel function of first and second kind and of order n. The determination is concerned with the eigenvalues of the natural frequencies of the radial vibrations for different boundary conditions in the case of harmonic vibrations. Numerical results are given and illustrated graphically for each case considered. Comparisons are made with the results in the absence of magnetic field, rotation and non-homogeneity. The results indicate that the effect of magnetic field, rotation and non-homogeneity are very pronounced.     

Reduction of subsynchronous vibrations in a single-disk rotor using an active magnetic damper

Rotor instabilities in turbomachinery often manifest themselves as a re-excitation of the first rotor critical speed resulting in lateral rotor vibrations at a frequency below the rotor operating frequency. Considerable work exists in the literature involving the analysis of destabilizing mechanisms and passive solutions for reducing subsynchronous vibrations. The authors propose here a novel active control solution utilizing active magnetic bearing (AMB) technology in conjunction with conventional support bearings. The AMB is utilized as an active magnetic damper (AMD) at rotor locations inboard of conventional support bearings. Presented here are initial proof-of-concept experimental results using an AMD for vibration control of subsynchronous rotor vibrations in a high-speed single-disk laboratory rotor. The study shows that subsynchronous vibrations are reducible with an AMD and up to a 93% reduction in the amplitude of subsynchronous vibrations is demonstrated. The study also shows that the AMD can significantly increase synchronous vibration response (up to 218% in one case) by increasing system stiffness and pushing a critical speed closer to an operating speed. The overall results from this work demonstrate that reduction in subsynchronous response is feasible and that full rotor dynamic analysis and design is critical for the successful application of this approach.     

Influence of magnetic field on free vibrations in elastodynamic problem of orthotropic hollow sphere

The effect of magnetic field on the plane vibrations for an elastodynamic orthotropic sphere is studied. Equations of elastodynamic problems of the orthotropic hollow sphere in terms of displacement are solved. The numerical results of the frequency equations in the presence of magnetic field are discussed and shown graphically. Compar-isons are made with the result in the presence and absence of magnetic field in the case of orthotropic sphere. The results show that the effect of magnetic field is very pronounced.     

Analytical and numerical studies of free vibration frequencies of ferromagnetic plates with arbitrary electric conductance in a transverse magnetic field in the 3D setting

The problem on the vibrations of a magnetoelastic ferromagnetic plate was studied in [1–5] from the viewpoint of the averaging approach, i.e., on the basis of the classical Kirchhoff hypothesis. In [6–10], a new approach proposed for elastic plates in [11] was used to derive dispersion relations for magnetoelastic plates. In [10], the 3D approach was used to obtain the ferromagnetic plate vibration frequencies; in the case of a transverse magnetic field, the equations of the perturbed motion of the plates were written out with the initial stresses taken into account [5, 12] but without considering the initial strains. In [13, 14], the problems on the vibrations of conducting plates in a magnetic field were studied.In the present paper, we derive the dispersion equations, which are asymptotic equations for small magnetic fields and exact equations for the initial stresses and strains related by Hooke’s law. The corresponding numerical computations are also performed.     

Surface waves in an elastic medium in the presence of a magnetic field

Waves in a perfectly conducting elastic medium are considered in two cases: 1) at the free surface of a medium occupying an infinite halfspace and located in a homogeneous constant magnetic field; 2) at the interface of two media one of which is located in a magnetic field. At a certain relative velocity instability develops. A similar problem was investigated in [1] for zero magnetic field and a critical velocity was obtained. This paper examines the instability due to vibrations propagating at right angles to the magnetic field.The author thanks M. I. Kiselev for his helpful suggestions.     

Dispersion of radial vibrations of an orthotropic cylindrical tube placed in a magnetic field

The present paper studies the dispersion relation of the radial vibrations of an orthotropic cylindrical tube. The effects of the magnetoelastic interaction on the problem are investigated. The problem is represented by the equations of elasticity taking into account the effect of the magnetic field as given by Maxwell's equations in the quasi-static approximation. The stress free conditions on the inner and outer surfaces of the hollow cylindrical cube are satisfied to form a dispersion relation in terms of the wavelength, the cylinder radii and the material constants. This study shows that waves in a solid body propagating under the influence of a superimposed magnetic field can differ significantly from those propagating in the absence of a magnetic field. The results have been verified numerically and represented graphically.     

Bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings

Auxiliary bearings are utilized in practical installations of magnetically suspended rotating machines with the main functions to provide support to the rotating machines during their non-operational period, and to protect the magnetic bearings and the rotating assembly from being damaged due to power loss during operation. The relatively small clearances of these bearings, which are typically half of that of the magnetic bearings, may at time cause contact between the rotor and these bearings to occur even during normal operation of the rotating machines. The work presented herein examines the bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings, which occurs during contact between the rotor and the auxiliary bearings. Numerical results revealed the occurrence of period-doubling bifurcation resulting in vibrations of period-2, -4, -8, -16 and -32, as well as quasi-periodic and chaotic vibrations. The results further showed that for a relatively small rotor imbalance magnitude, which is within the prescribed level of certain classes of practical rotating machinery, such nonlinear dynamical phenomena would not have been discovered had the auxiliary bearings forces been omitted in the model of the rotor-bearing system. As these bearings are essential elements in practical installations of magnetically supported rotating machines, failure to include them in the rotor-bearing model may result in incorrect prediction of the rotor’s vibration response.     

Modelling of the Dynamic Behaviour of Electromechanical Relays for the Analysis of Sensitivity to Shocks and Vibrations

During their operating life, electromechanical relays (EMRs) are subjected to external mechanical vibrations that can induce undesirable vibratory responses in the movable part of the relay and thus produce an unwanted temporary break in electric connections. This paper presents a simplified multiphysics model dedicated to the prediction of the maximum vibration levels that these relays can undergo before a loss of contact occurs. Our methodology considers the magnetic aspects as well as the mechanical aspects. The dynamic behaviour of the movable part is modelled as a cantilever beam subjected at its extremity to an elastic force. The dynamic parameters were updated from the identification of the first natural modes of the movable part of the relay. The magnetic force acting on the movable part was computed using a one-dimensional approach with an equivalent magnetic circuit and was corroborated thanks to experimental measurements and two-dimensional Finite Element simulations. The interaction between the electromagnetic and mechanical phenomena was taken into account using a parameterization coupled approach. Our methodology has been applied to the study of the PED PXC-1203 relay. The numerical predictions were validated using experimental data measured in the frequency range [2–8 kHz]. A parametric analysis of our model was performed and shows the influence of some factors, like the air gap and the rated voltage, that affect the performance of the relay under external mechanical vibrations.     

Nonlinear vibration of a rotating cantilever beam in a surrounding magnetic field

The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam.     

Basic vibrations and anharmonic analysis of a vibration

We prove that, besides the simple harmonic vibrations, some anharmonic vibrations are basic as well, because a general vibration can be considered as a superposition of such vibrations with different frequencies. The results in this paper are a generalization of Fourier analysis and a new theory of vibration analysis.     

The effect of friction reduction in the presence of in-plane vibrations

A reduction of friction by vibrations has been observed in various experiments. This effect can be applied to actively control frictional forces by modulating vibrations. Moreover, common methods of controlling friction rely on lubricants and suitable material combinations. The superimposition of vibrations can further reduce the friction force. This study presents a theoretical approach based on the Dahl friction model that describes the friction reduction observed in the presence of the tangential vibrations at an arbitrary angle. Analysis results indicated that the tangential compliance should be considered in modeling the effect of vibrations in reducing friction. At any vibration angle, the tangential compliance of the contacts reduces the friction reduction effect. The vibrations parallel to the macroscopic velocity are most effective for friction reduction.     

Stabilization of a cantilever pipe conveying fluid using electromagnetic actuators of the transformer type

The article presents an investigation of the stabilization of a cantilever pipe discharging fluid using electromagnetic actuators of the transformer type. With the flow velocity reaching a critical value, the straight equilibrium position of the pipe becomes unstable, and self-excited lateral vibrations arise. Supplying voltage to the actuators yields two opposite effects. First, each of the actuators attracts the pipe, thus introduces the effect of negative stiffness which destabilizes the middle equilibrium. Second, lateral vibrations change the gap in magnetic circuits of the actuators, which leads to oscillations of magnetic field in the cores and the electromagnetic phenomena of induction and hysteresis that impede the motion of the pipe. The combination of these two non-linear effects is ambiguous, so the problem is explored both theoretically and experimentally. First, a mathematical model of the system in form of a partial differential equation governing the dynamics of the pipe coupled with two ordinary differential equations of electro-magnetodynamics of the actuators is presented. Then, the equation of the pipe’s dynamics is discretized using the Galerkin procedure, and the resultant set of ordinary equations is solved numerically. It has been shown that the overall effect of actuators action is positive: the critical flow velocity has been increased and the amplitude of post-critical vibrations reduced. These results have been validated experimentally on a test stand.

     

The transverse and torsional vibrations of A.C. motors in the starting process

According to the electromagnetic field theory, a set of differential equations is derived for coupling of the transient process for an electrical machine and transverse and torsional vibrations for the rotor by a unified method. A non-dimensional constant coefficient state equation is obtained by transformation. Computation is done. Rules of the change are obtained for transverse and torsional vibration, unilateral magnetic pull, electrical torque, and so on in the starting process. The theoretical results are shown to be in good agreement with the experimental results. The mathematical model verified by the experiment is used to analyse the effect of electromagnetic and mechanical parameters on electromagnetic force and vibration.     

Propagation of elastic waves in a rod in a longitudinal magnetic field

The propagation of harmonic waves in discussed for an ideally conducting continuous elastic cylindrical rod within an ideally conducting cylindrical rube. The annulus contains a steady homogeneous longitudinal magnetic field. The dispersion equation is derived. The case of bending vibrations is considered.     

Effect of Vibrations on Pore Fluid Distribution in Porous Media

Understanding the role of shuttle vibrations in pore fluid distribution is an essential task in the exploration of plant growth in root modules aboard space flights. Results from experimental investigations are reported in this paper on the distribution of immiscible fluid phases in glass beads under vibrations. Hexadecane, a petroleum compound immiscible with and lighter than water, was used in the experiments. The higher freezing point of Hexadecane (18 °C) allowed the solidification of the entrapped blobs in the presence of water in porous media, so that their size distribution can be obtained. van Genuchten function, commonly used to express moisture retention curves, is found to be an adequate fit for blob size distribution at residual saturation. The effect of vibrations on the fate (mobilization, stranding, or breakup) of a solitary ganglion in porous media was studied using a network model. A mobility criterion considering viscous, gravity, and capillary forces was developed to determine the fate of a solitary ganglion in a porous medium. It is concluded that the effect of vibrations is to increase the likelihood of breakup and mobilization of blobs entrapped in porous media at residual saturation. The pore fluid distributions after vibrations are less uniform than those before vibrations.     

Some features of the dynamics of ferroelectric (ferromagnetic) heterostructures

This paper considers the dynamic coupled problem of harmonic vibrations of a layeredinhomogeneous electromagnetoelastic medium subjected to an oscillating mechanical or electrical load under various electric and magnetic field conditions specified on the surface and internal boundaries of this medium. Green’s function of the medium is constructed. Dispersion curves and phase velocities for different boundary conditions and materials are obtained.