Nonlinear dynamic behaviors of a rotor-labyrinth seal system

The nonlinear model of rotor-labyrinth seal system is established using Muszynska’s nonlinear seal forces. We deal with dynamic behaviors of the unbalanced rotor-seal system with sliding bearing based on the adopted model and Newmark integration method. The influence of the labyrinth seal one the nonlinear characteristics of the rotor system is analyzed by the bifurcation diagrams and Poincare’ maps. Various phenomena in the rotor-seal system, such as periodic motion, double-periodic motion, quasi-periodic motion and Hopf bifurcation are investigated and the stability is judged by Floquet theory and bifurcation theorem. The influence of parameters on the critical instability speed of the rotor-seal system is also included.     

Non-Linear Dynamic Response of a Single Wall Carbon Nanotube Subjected to Radial Impulse

This paper investigates non-linear dynamic response of a single wall carbon nanotube (CNT) based on a thin-walled shell mode. Following the subsequent motion of a single wall (CNT) under radial impulsive pressure, a nonlinear dynamic response may occur through interaction of circumferential membrane force with flexural curvature. The results carried out show that the initial deformation energy in the single wall CNT will be transferred, over a number of cycles, from the breathing mode to one or two high flexural modes, so that the nonlinear flexural stress of the single wall CNT is significantly higher than that given by linear theory. The principal point of interest are the conditions for significant interaction to occur and the increased flexural stress associated with the altered radial motion of a single wall CNT. An erratum to this article can be found at     

Non-Linear Self-Excited Random Vibrations and Stability of an SDOF System with Parametric Noises

Abstract. The paper presents the solution to the properties of stochastic response of a system with random parametric noises, which is prone to the loss of aerodynamical stability. The system is described by an equation of van der Pol type with the negative linear, and with the positive cubic dampings. The coefficients of the linear damping and of the stiffness include the multiplicative random perturbations, the external excitation being given as a sum of a deterministic function and of an additive perturbation. All three input random processes are supposed to be Gaussian and centered, with the non-zero mutual stochastic parameters, as it corresponds to the properties of real systems. The solution has been based on the method of stochastic linearisation and of the subsequent solution of the Fokker–Planck–Kolmogorov equation in the sense of the first and second stochastic moments for the transient and stationary states. There have been demonstrated several effects, which are typical for systems with parametric noises, differentiating them from the systems with constant coefficients. The principal attention has been devoted to the properties of the spectral density of the response, the character of which changes abruptly with the degree of non-linearity of the damping and of the level of random perturbations.Sommario. La presente memoria studia le proprietà della risposta stocastica di un sistema con eccitazione casuale parametrica, che tende alla perdita della stabilità aerodinamica. Il sistema è descritto mediante un'equazione del tipo di van der Pole con il termine lineare dello smorzamento negativo e il termine cubico positivo. Poichá l'eccitazione esterna è la somma di una funzione deterministica e di una perturbazione additiva, i coefficienti dello smorzamento lineare e della rigidezza comprendono le perturbazioni casuali moltiplicative. I tre processi stocastici di eccitazione sono assunti gaussiani e a media nulla con parametri stocastici incrociati diversi da zero, come si verifica per le proprietà dei sistemi reali. La soluzione è basata sul metodo della linearizzazione stocastica e della successiva soluzione dell'equazione di Fokker-Planck-Kolmogorov studiando i primi e i secondi momenti statistici per gli stati transitori e stazionari. Vengono mostrati diversi effetti, tipici dei sistemi con eccitazione parametrica, differenziandoli dai sistemi a coefficienti costanti. Particolare attenzione è rivolta alle proprietà della densità spettrale della risposta le cui caratteristiche cambiano bruscamente con il grado di non linearità dello smorzamento e del livello di casualità delle perturbazioni.     

Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system

This research studies the effects of axial preload on nonlinear dynamic characteristics of a flexible rotor supported by angular contact ball bearings. A dynamic model of ball bearings is improved for modeling a five-degree-of-freedom rotor bearing system. The predicted results are in good agreement with prior experimental data, thus validating the proposed model. With or without considering unbalanced forces, the Floquet theory is employed to investigate the bifurcation and stability of system periodic solution. With the aid of Poincarè maps and frequency response, the unstable motion of system is analyzed in detail. Results show that the effects of axial preload applied to ball bearings on system dynamic characteristics are significant. The unstable periodic solution of a balanced rotor bearing system can be avoided when the applied axial preload is sufficient. The bifurcation margins of an unbalanced rotor bearing system enhance markedly as the axial preload increases and relates to system resonance speed.     

Global Existence for a System of Non-Linear and Non-Local Transport Equations Describing the Dynamics of Dislocation Densities

In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system.     

Nonlinear Dynamics and Stability of a Two D.O.F. Elastic/Elasto-Plastic Model System

The local and global nonlinear dynamics of a two-degree-of-freedom model system is studied. The undeflected model consists of an inverted T formed by three rigid bars, with the tips of the two horizontal bars supported on springs. The springs exhibit an elasto-plastic response, including the Bauschinger effect. The vertical rigid bar is subjected to a conservative (dead) or non-conservative (follower) force having static and periodic components. First, the method of multiple scales is used for the analysis of the local dynamics of the system with elastic springs. The attention is focused at modal interaction phenomena in weak excitation at primary resonance and in hard sub-harmonic excitation. Three different asymptotic expansions are utilised to get a structural response for typical ranges of excitation parameters. Numerical integration of the governing equations is then performed to validate results of asymptotic analysis in each case. A full global nonlinear dynamics analysis of the elasto-plastic system is performed to reveal the role of plastic deformations in the stability of this system. Static 'force-displacement' curves are plotted and the role of plastic deformations in the destabilisation of the system is discussed. Large-amplitude non-linear oscillations of the elasto-plastic system are studied, including the influence of material hardening and of static and sinusoidal components of the applied force. A practical method is proposed for the study of a non-conservative elasto-plastic system as a non-conservative elastic system with an 'equivalent' viscous damping. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions

The effect of non-linear magnetic forces on the non-linear response of the shaft is examined for the case of superharmonic resonance in this paper. It is shown that the steady-state superharmonic periodic solutions lose their stability by either saddle-node or Hopf bifurcations. The system exhibits many typical characteristics of the behavior of non-linear dynamical systems such as multiple coexisting solutions, jump phenomenon, and sensitive dependence on initial conditions. The effects of the feedback gains and imbalance eccentricity on the non-linear response of the system are studied. Finally, numerical simulations are performed to verify the analytical predictions.     

Gyroscopic Stabilization of Passive Magnetic Levitation

A nonlinear mathematical model able to describe the motion of a passive magnetic levitation device, known as Levitron, is presented in this paper. Using the standard approach usually applied in rotordynamics and without introducing any preliminary assumptions, the equations of motion for all six degrees of freedom of the magnetic spinning top are obtained. By computing the four natural frequencies characterizing the horizontal translational vibrations of the rotor and the whirling motion of its axis, the conditions for stable levitation in terms of the spin speed are obtained. Some results coming from the numerical integration of the equations of motion are also presented and compared with those obtained using the simplified model based upon the fast top assumption.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis

The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass. In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates. However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates. Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the -film, short bearing model. After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present. Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part II: Experimental Analysis

In the first part of the present investigation [9], the dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. In the present paper an experimental confirmation of the theoretical results is sought. The steel rotor of the experimental rig was given a constant circular cross section in order to fix in an easy way the two distances between supports corresponding, respectively, to the values of the parameter assigned in [9]. Two steel rings, each one with a series of holes and a clamping screw, were mounted onto the rotor with a small clearance. This arrangement made it possible to fix the positions of the rings and their holes respect to the rotor, so as to realize a pre-estabilished unbalance. The two bronze journal bearings were characterised by a relatively low length/diameter ratio, and a relatively high value of the radial clearance and were lubricated with oil delivered from a thermostatic tank. In this way, despite the relative lightness of the rotor, the dimensionless static eccentricity _s was given the high values that were apt to realize the operating conditions assumed in the theoretical analysis. The rotor was driven by means of a d.c. motor connected to a toothed belt-drive. Varying the rotor speed in the range 1000 ÷ 10000 r.p.m., made it possible to assign the values of the modified Sommerfeld number assumed in the theoretical analysis. Three pairs of eddy-current probes were mounted in order to detect the trajectories of three points (C₁, C and C₂) suitably fixed along the rotor axis. These orbits were finally put in comparison with the corresponding ones previously obtained through numerical analysis. The comparison pointed out that the experimental data were in good agreement with the theoretical predictions, despite the approximations that characterise the theoretical model and the unavoidable errors affecting measures in the course of the experimental test.     

Non-Linear response of a long,discontinuous fiber/melt system in elongational flows

The authors investigated the transient elongational behavior of a highly-aligned 600% volume fraction long, discontinuous fiber filled poly-ether-ketone-ketone melt with a computer-controlled extensional rheometer at 370°C. Prior experiments at controlled strain rate and stress produced _E ⁺ (t, ) and ^– (t, _E) similar to a shear dominated flow of a non-linear viscoelastic fluid. Stress relaxation following steady extension showed nonlinear effects in the change in stress decay rate with increasing strain rate. Continuous relaxation spectra showed a shift in the spectral peak to smaller values of with increasing strain rate. The Giesekus nonlinear constitutive relation modeled the elongation and stress relaxation with shearing rate at the fiber surface set by a strain rate magnification factor. Suitable for elongation, the model produced insufficient shift in the stress relaxation spectrum to account for the large change in stress decay rate exhibited in the experiments.English alphabet a _r aspect ratio of the fibers or l/d - A ₀ initial uniform cross-section area of the specimen - d fiber diameter - f fiber volume fraction - H() relaxation spectrum found by the method of Ferry and William l length of the fiber - L(t) time function specimen length - L ₀ initial specimen length - r radial coordinate across the shear cell - R _i fiber radius and inner cell dimension - R _o outer cell radius - t time in s - t _max duration of the extension - T _g glass transition temperature of the polymer - v velocity of the moving end of the test specimen - x axial position where is calculated Greek alphabet nonlinearity parameter in the Giesekus relation - axial mass distribution along the specimen major axis - shear strain rate - strain tensor - ₍₁₎ first convected derivative of the strain tensor - ₍₂₎ second convected derivative of the strain tensor - average strain at the end of extension as determined from - extension strain rate - average extension strain rate determined from - ^– transient strain rate under controlled stress, creep, test - _E elongational viscosity - _Eapp apparent elongational viscosity determined from - _E ⁺ transient elongational viscosity - ₀ zero shear rate viscosity - relaxation parameter - ₁ relaxation parameter in either Jeffrey's or Giesekus fluid - ₂ retardation parameter in either Jeffrey's or Giesekus fluid - _max relaxation value at which 99.9% of the H spectrum had occurred - _p relaxation value at which H reaches a maximum - volumetric composite density - _E elongational stress - _E ⁺ transient elongational stress - _E ^– controlled elongational stress, creep stress - _E ^y peak elongational stress in controlled experiment - shear stress at surface of the fiber in a shear cell - _yx simple shear component of the strain rate tensor - stress tensor - ₁ first convected derivative of the stress tensor     

Modelling of Ground Moling Dynamics by an Impact Oscillator with a Frictional Slider

This paper describes current research into the mathematical modelling of a vibro-impact ground moling system. Due to the structural complexity of such systems, in the first instance the dynamic response of an idealised impact oscillator is investigated. The model is comprised of an harmonically excited mass simulating the penetrating part of the mole and a visco-elastic slider, which represents the soil resistance. The model has been mathematically formulated and the equations of motion have been developed. A typical nonlinear dynamic analysis reveals a complex behaviour ranging from periodic to chaotic motion. It was found out that the maximum progression coincides with the end of the periodic regime.     

Effect of Stress-softening on the Dynamics of a Load Supported by a Rubber String

The Mullins effect in the oscillatory motion of a load under gravity and attached to a stress-softening, neo-Hookean rubber string is investigated. Equations for the small amplitude vertical oscillations of the load superimposed on the finite static stretch of both the virgin and stress-softened cords, the latter subjected to varying degrees of preconditioning, are derived. The vibrational frequency of the small motion exhibits behavior similar to that observed in experiments by others on postmortem, human aortic tissue for which no stress-softening is reported. Standard numerical methods are applied to study the finite amplitude motion of the load in the stress-softened case. The resultant motions and their various physical aspects under free-fall and general initial conditions are described in several examples. Oscillations that engage all three phases of motion consisting of the suspension, the free-flight, and the retraction of the load in its general vertical motion are illustrated. Effects due to the degree of stress-softening are discussed; and the motion response for two values of the model softening parameter is compared in several examples. All results are illustrated graphically and numerous tabulated numerical results are provided.      

Nonlinear magneto-electric response of a giant magnetostrictive/piezoelectric composite cylinder

In this study,we investigate the nonlinear coupling magneto-electric(ME) effect of a giant magnetostrictive/piezoelectric composite cylinder.The nonlinear constitutive relations of the ME material are taken into account,and the influences of the nonlinear material properties on the ME effect are investigated for the static and dynamic cases,respectively.The influences of different constraint conditions on the ME effect are discussed.In the dynamic case considering nonlinear material properties,the double frequency ME response(The response frequency is twice the applied magnetic frequency) is obtained and discussed,which can be used to explain the experiment phenomenon in which the input signal with frequency f is converted to the output signal with 2 f in ME laminated structures.Some calculations on nonlinear ME effect are conducted.The obtained results indicate that the nonlinear material properties affect not only the magnitude of the ME effect in the static case but also the ME response frequency in the dynamic case.     

Dynamics of a kicked oscillator with a delay in its parametric feedback loop: An analytical study

This paper deals with the behavior of a parametrically-forced analytically-solvable oscillator in the presence of a delay in the feedback loop. In spite of the delay, it is here shown that, when the system is in the strong relaxation regime, a one-dimensional approximation to the Poincaré (stroboscopic) map may be constructed. The influence of both the delay and the amplitude of the impulses on the dynamics is explained in terms of the geometric properties of either the isochrone portrait or the PTC. Interest has recently grown on this kind of excitation because in real-life feedback systems a delay is always present.     

Nonlinear resonances and self-excited oscillations of a rotor caused by radial clearance and collision

This paper investigates oscillations in a flexible rotor system with radial clearance between an outer ring of the bearing and a casing by experiments and numerical simulations. The mathematical model considers the collisions of the bearing with the casing. The following phenomena are found: (1) Nonlinear resonances of subharmonic, super-subharmonic and combination oscillation occur. (2) Self-excited oscillation of a forward whirling mode occurs in a wide range above the major critical speed. (3) Entrainment phenomena from self-excited oscillation to nonlinear forced oscillation occur at these nonlinear resonance ranges. Moreover, this study analyzes periodic solutions of the mathematical model by the Harmonic Balance Method (HBM). As the results, the nonlinear resonances of subharmonic oscillation and its entrainment phenomenon can be explained theoretically by investigating the stability of the periodic solutions. The influence of the static force and the bearing damping on these oscillation are also clarified.     

Vibration of a Non-Linear Self-Excited System with Two Degrees of Freedom under External and Parametric Excitation

Vibration analysis of a non-linear parametrically self-excited system with two degrees of freedom under harmonic external excitation was carried out in the present paper. External excitation in the main parametric resonance area was assumed in the form of standard force excitation or inertial excitation. Close to the first and second free vibrations frequency, the amplitudes of the system vibrations and the width of synchronization areas were determined. Stability of obtained periodic solutions was investigated. The analytical results were verified and supplemented with the effects of digital and analog simulations.     

Dynamics analysis of slender vehicle influenced by random factors

In this paper, the aeroelastic problems of slender vehicles under the influence of random factors and thrust are studied. An aeroelastic dynamic model of a free-free Euler–Bernoulli beam considering thrust and aerodynamic forces is established based on Hamilton’s principle of nonconservative systems. On this basis, considering the influence of random factors, the elastic modulus and viscous drag are regarded as one-dimensional continuous stationary random fields and discretized. The stochastic finite element method is used to solve the dynamic model, and the results are compared with the Monte Carlo simulation results. Then, the influence of the correlation of the random field on the elastic displacement is further analyzed. The following simulation results are obtained: (1) the stochastic factor analysis model established in this paper can reflect the statistical characteristics of aeroelastic response well; (2) the stronger the correlation of the random field is, the greater the expectation of elastic displacement, but as the correlation increases, the expectation tends to be constant; and (3) it is necessary to choose the discrete length of the random field reasonably, and the discrete length depends on the correlation characteristics of the random field studied.     

Dynamics of repelling soliton collisions in coupled Schrödinger equations

The system of Coupled Nonlinear Schrödinger's Equations (CNLSE) is solved numerically by means of a conservative difference scheme. Values of the cross-modulation parameter, α₂, are chosen to induce repelling collisions. The resulting waveforms are fitted to soliton profiles and studied via internal parameters including phase velocity to gain insight into the dynamics of the repelling collisions. A decaying oscillation of the post-collision profiles is observed and investigated in terms of varying α₂ and relative initial velocity of colliding profiles.