Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Mitigation of edgewise vibrations in wind turbine blades by means of roller dampers

Edgewise vibrations in wind turbine blades are lightly damped, and large amplitude vibrations induced by the turbulence may significantly shorten the fatigue life of the blade. This paper investigates the performance of roller dampers for mitigation of edgewise vibrations in rotating wind turbine blades. Normally, the centrifugal acceleration of the rotating blade can reach to a magnitude of 7–8g, which makes it possible to use this kind of damper with a relatively small mass ratio for suppressing edgewise vibrations effectively. The parameters of the damper to be optimized are the mass ratio, the frequency ratio, the coefficient of rolling friction and the position of the damper in the blade. The optimization of these parameters has been carried out on a reduced 2-DOF nonlinear model of the rotating wind turbine blade equipped with a roller damper in terms of a ball or a cylinder, ignoring the coupling with other degrees of freedom of the wind turbine. The edgewise modal loading on the blade has been calculated from a more sophisticated 13-DOF aeroelastic wind turbine model with due consideration to the indicated couplings, the turbulence and the aerodynamic damping. Various turbulence intensities and mean wind speeds have been considered to evaluate the effectiveness of the roller damper in reducing edgewise vibrations when the working conditions of the wind turbine are changed. Further, the optimized roller damper is incorporated into the 13-DOF wind turbine model to verify the application of the decoupled optimization. The results indicate that the proposed damper can effectively improve the structural response of wind turbine blades.     

Generalised modal stability of inclined cables subjected to support excitations

Parametric excitation is of concern for cables such as on cable-stayed bridges, whereby small amplitude end motion can lead to large, potentially damaging, cable vibrations. Previous identification of the stability boundaries for the onset of such vibrations has considered only a single mode of the cable, ignoring non-linear coupling between modes, or has been limited to special cases. Here multiple cable modes in both planes are included, with support excitation close to any natural frequency. Cable inclination, sag, parametric and direct excitation and nonlinearities, including modal coupling, are included. The only significant limitation is that the sag is small. The method of scaling and averaging is used to find the steady-state amplitude of the directly excited mode and, in the presence of this response, to define stability boundaries of other modes excited parametrically or through nonlinear modal coupling. It is found that the directly excited response significantly modifies the stability boundaries compared to previous simplified solutions. The analysis is validated by a series of experimental tests, which also identified another nonlinear mechanism which caused significant cable vibrations at twice the excitation frequency in certain conditions. This new mechanism is explained through a refinement of the analysis.     

Dynamic stability of a slender beam under horizontal–vertical excitations

The dynamic stability of a vertically standing cantilevered beam simultaneously excited in both horizontal and vertical directions at its base is studied theoretically. The beam is assumed to be an inextensible Euler–Bernoulli beam. The governing equation of motion is derived using Hamilton's principle and has a nonlinear elastic term and a nonlinear inertia term. A forced horizontal external term is added to the parametrically excited system. Applying Galerkin's method for the first bending mode, the forced Mathieu–Duffing equation is derived. The frequency response is obtained by the harmonic balance method, and its stability is investigated using the phase plane method. Excitation frequencies in the horizontal and vertical directions are taken as 1:2, from which we can investigate the influence of the forced response under horizontal excitation on the parametric instability region under vertical excitation. Three criteria for the instability boundary are proposed. The influences of nonlinearities and damping of the beam on the frequency response and parametric instability region are also investigated. The present analytical results for instability boundaries are compared with those of experiments carried out by one of the authors.     

Three dimensional vibration generators with a single rotational input

This paper presents a novel device capable of generating three-dimensional vibrations with a single-axis of rotation. The device resembles a vibration motor with an eccentric weight, but the weight is allowed to move up and down in parallel to the axis of rotations. While spinning of the eccentric weight causes lateral vibrations, vertical vibrations are excited by superposing a periodic torque on the rotary shaft. The frequency and magnitude of vertical oscillations can be independently regulated. Since the vertical natural frequency is sensitive to rotational speeds, maximum vertical oscillations can be achieved by properly adjusting the rotational speed according to the excitation frequency. Relations between the excitation torque and the vertical shaking force are examined using the frequency response for the linearized system. Numerical simulations on the original nonlinear system are conducted to verify the performance of vertical oscillations.     

Convective instability of the thermovibrational flow of binary mixture in the presence of the Soret effect

The convective instability of the thermovibration flow in a plane horizontal layer filled with an incompressible binary gaseous mixture is investigated. The study takes into account the effect of thermal diffusion or the Ludwig-Soret effect. Several instability mechanisms are discussed. To determine the instability threshold with respect to cell and long-wave perturbations, the Floquet theory was applied to the linearized equations of convection formulated in the Boussinesq approximation. We found that regime parametric instability can occur owing to the finite frequency vibrations. The evolution of plane, spiral and three-dimensional disturbances is studied. We demonstrated that, because of the properties of the system, the subharmonic response of plane disturbances to the external periodic action cannot be observed. The instability can be associated only with synchronous or quasiperiodic modes. Depending on the vibration parameters, modulations can stabilize or destabilize the base state. For spiral perturbations the stability boundary does not depend on the amplitude and frequency of vibrations. In the case of long-wave instability we apply the regular perturbation approach with the wavenumber as a small parameter in power expansions. The stability boundaries are found.     

Dynamical parametric instability of carbon nanotubes under axial harmonic excitation by nonlocal continuum theory

Structures under parametric load can be induced to the parametric instability in which the excitation frequency is located the instability region. In the present work, the parametric instability of double-walled carbon nanotubes is studied. The axial harmonic excitation is considered and the nonlocal continuum theory is applied. The critical equation is derived as the Mathieu form by the Galerkin's theory and the instability condition is presented with the Bolotin's method. Numerical calculations are performed and it can be seen that the van der Waals interaction can enhance the stability of double-walled nanotubes under the parametric excitation. The parametric instability becomes more obvious with the matrix stiffness decreasing and small scale coefficient increasing. The parametric instability is going to be more significant for higher mode numbers. For the nanosystem with the soft matrix and higher mode number, the small scale coefficient and the ratio of the length to the diameter have obvious influences on the starting point of the instability region.     

Experimental investigations of the response of a cracked rotor to periodic axial excitation

In this paper, the coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Steady state unbalance response of a cracked rotor with a single centrally situated crack subjected to periodic axial impulses is investigated experimentally. The cracked rotor is excited axially using an electrodynamic exciter at a frequency equal to its bending natural frequency in both non-rotating and rotating conditions. The resulting time domain and frequency domain signals of the cracked rotor are studied. Spectral response of the cracked rotor with and without axial excitation is found to be distinctively different. When excited axially, it shows prominent presence of rotor bending natural frequency. However for an uncracked rotor, the response is similar with or without axial excitation. It is thus proposed that the response of the rotor to axial impulse excitation could be used for more reliable diagnosis of rotor cracks.     

ADDING ONE BIFURCATION IN A ROTOR SYSTEM WITH CLEARANCES

A mathematical model of a rotor system with clearances is analysed by the application of modern nonlinear dynamic theory. From the bifurcation diagrams, it is discovered that the rotor system alternates between periodic and chaotic motions at a supercritical rotational speed, and after undergoing a chaotic region the periodic number of the motion will increase by one. At the same time, the periodic number is equal correspondingly to the integral multiple of the critical rotational speed. At the subcritical rotational speed, it is discovered that the chaotic bands among successive orders of superharmonic responses return to the period one through a reversed period-doubling bifurcation, as a result of a period-doubling bifurcation. It is shown that the increase of damping may reduce the width of the chaotic bands and the amplitude of the periodic response; the increase of nonlinear degree also leads to the reduction of chaotic bandwidth, but makes the amplitude of the subharmonic response increase. So it is suggested that proper damping and correct material selection by considering the dynamic characteristics of the rotor system may reduce the proportion of faults and enhance the dynamic characteristics when designing the rotor system. The working speed should not be selected at N times its natural frequency and should not be set in the chaotic bands among the successive orders of periodic motion for the same purpose.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

Dynamic behaviours of a full floating ring bearing supported turbocharger rotor with engine excitation

The rotor dynamic behaviour of turbochargers (TC) has been paid significant attention because of its importance in their healthy operation. Commonly, the TC is firmly mounted on engines and they will definitely suffer from the vibrations originated from engines in operation. However, only a limited number of papers have been published with consideration of this phenomenon. In this paper, a finite element model of a TC rotor supported by nonlinear floating ring bearings has been established. The nonlinear bearing forces have been calculated by a newly proposed analytical method. An efficient numerical integration approach has been employed to conduct the investigation including the traditional unbalance and the considered engine excitation effects in question. The results show that the unbalance will place considerable influence on the rotor response at a low working speed. At high speeds, the effect will be prevented by the dominant sub-synchronous vibrations, which also prohibit the appearance of a chaotic state. The novel investigation with the proposed model considering engine excitation reveals that the engine induced vibration will greatly affect the TC rotor response at relatively lower rotor speeds as well. At higher speed range, the dominant effect of sub-synchronous vibrations is still capable of keeping the same orbit shapes as that without engine excitation from a relative viewpoint.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

Effect of unbalanced rotor whirl on blade vibrations

In this paper, vibration of a bladed unbalanced flexible rotor is studied. The blade that is attached to the disk is considered as a fixed-free Euler-Bernoulli beam. Position of the blade with respect to the eccentric mass is taken into consideration. Coupled equations of the motion of unbalanced rotor and the blades are obtained through Lagrange equations. The dynamic equations have time variant periodic coefficients. Transient vibration analysis showed that rotor acceleration excites the blade vibration with its own natural frequency. While the rotor passes through its own natural frequency (critical speed) the blade vibration is again excited but this time with the rotor natural frequency. Modal behavior of the blades are different for subcritical, supercritical and for critical speed of the rotor. In the subcritical run of the rotor, blades located from 0° to 180° with respect to the eccentric mass are deflected in the negative direction while the rest are deflected in the positive direction. For supercritical run of the rotor, modal behavior of the blades is just the opposite. For critical speed of the rotor, blades located 90° to 270° from the eccentric mass are deflected in the positive direction while the rest of the blades are deflected in the negative direction. Blades have also different deflections. When the deflections of the blades are plotted with respect to their position angle, distribution of the blade deflections has a sinusoidal shape.     

ANALYSIS OF THE RESPONSE OF A CRACKED JEFFCOTT ROTOR TO AXIAL EXCITATION

The coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. Steady state unbalance response of a Jeffcott rotor with a single centrally situated crack subjected to periodic axial impulses is studied. Partial opening of crack is considered and the stress intensity factor at the crack tip is used to decide the extent of crack opening. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Therefore, lateral vibration response of a cracked rotor to axial impulses is studied in detail. Spectral analysis of response to periodic multiple axial impulses shows the presence of rotor bending natural frequency as well as side bands around impulse excitation frequency and its harmonics due to modulations caused by rotor running frequency. It is concluded that the above approach can prove to be a useful tool in detecting cracks in rotors.     

Interpolation of aerodynamic damping of lifting rotors in forward flight from measured response variance

With the necessity of experimentally measuring the aerodynamic derivatives of lifting rotors in forward flight, it becomes desirable for reasons of parametric resonance to measure the variance data of the response to white noise inputs, and then to interpolate with the numerically computed variance values for different assumed values of the damping parameter or the inertia number. Accordingly, methods are developed to compute directly the blade response variance matrix up to high rotor advance ratios for a series of inertia numbers. Numerical results refer to a rigid blade flapping model with an elastically restrained flapping hinge at the rotor centre. Different combinations of the advance ratio, tip-loss factor, flapping restraint parameter and inertia number are included. The white noise excitation is treated without and with input modulation, the latter case including azimuthwise blade input variation. The mean square response study of the model shows that up to an advance ratio of the order of 0·3 and at conventional values of the inertia number the perturbation approach is satisfactory. The numerical scheme, in addition to solving directly the response variance equations which are similar in structure to the original blade dynamic equations, makes use of the fact that the variable part of system parameter functions are independent of the inertia number. Compared to earlier studies the methods therefore offer significant saving in machine time. Numerical results pertaining to previously proposed experimental and analytical models show excellent agreement.     

OPTIMAL DAMPING OF STAYS IN CABLE-STAYED BRIDGES FOR IN-PLANE VIBRATIONS

Significant vibrations have been reported in stays of recently constructed cable stayed bridges. The vibrations appear as in-plane vibrations that may be caused by rain-wind- induced aeroelastic interaction or by resonance excitation of the cables from the motion of the pylons. The stays of modern cable-stayed bridges are often designed as twin cables with a spacing of, say 1m. In such cases, it is suggested in the paper to suppress the mentioned in-plane types of vibrations by means of a tuned mass-damper (TMD) placed between the twin cables at their midpoints. The TMD divides the stay into four half-cables, and resonance may occur in each of the half-cables as well as in the entire stay. The optimal tuning of the TMD is investigated based on a mathematical model, where the motion of the support points on the pylons is considered to be the main cause of excitation. The indicated motion is modelled as a band-limited Gaussian white noise process. Three load scenarios are considered: narrow-banded excitations, with the central frequency of the autospectrum close to the lowest eigenfrequency of each of the two cables constituting the stay, and a broadbanded excitation which encompasses both of the mentioned frequencies. The spring and the damper constants of the TMD are optimized so that the variances of the displacement of the adjacent four half-cables, the support point of the TMD and the secondary mass are minimized. At optimal design, it is shown that the variances reduce below 14% of those of the unprotected stay.     

Torsional vibrations of non-uniform rotating blades with attachment flexibility

The torsional vibrations of non-uniform pretwisted rotating blades are studied by using finite element methods based on both the Rayleigh-Ritz and Galerkin formulations. The apparent differences between the matrices obtained from these formulations are explained and, as obtained by using three different orders of elements, results are presented for blades with flexibly attached roots and for a non-uniform blade representative of a bearingless rotor. A parametric study is carried out to resolve a controversy regarding the relative importance of certain terms in the equations of motion of pretwisted rotating blades. In Appendix I, an exact solution is presented for the torsional vibrations of flexibly attached rotating blades with piecewise constant inertia and elastic properties, which serves as a benchmark solution for the finite element results.     

Optimizing parametric oscillators with tunable boundary conditions

Parametric excitation or pumping is an effective method to create large oscillations by periodically altering a physical parameter of the governing dynamics. Precisely tuned pumping frequencies can lead to exponentially growing oscillations limited only by nonlinear effects like axial stretching of transversely vibrating string. It is demonstrated that a tuned passive dynamical system amplifies the otherwise limited transverse vibrations amplitudes of a nonlinear string considerably and thus increasing the selectivity of the system. This effect becomes more noticeable for shorter wavelengths where nonlinear stretching limits the obtainable vibration amplitudes severely. The present work analyses a passive dynamical system connected to one end of a taught string which parametrically couples its axial motion to transverse vibration. Analysis shows that a specific selection of parameters can reduce the limiting effect of nonlinear stretching thus allowing one to excite high-order modes with small external forces. The result can possibly affect other disciplines where effective parametric amplification is necessary.     

RANDOM VIBRATION OF A ROTATING BLADE WITH EXTERNAL AND INTERNAL DAMPING BY THE FINITE ELEMENT METHOD

The finite element model is employed to investigate the mean-square response of a rotating blade with external and internal damping under stationary or non-stationary random excitation. The blade is considered to be subjected to white-noise and earthquake excitations. The effects of rotational speed, external and internal damping on the mean-square response are studied. It is found that the mean-square response decreases quickly when the external and internal damping increases within some scope. Moreover, the increment of rotational speed will reduce the mean-square response of a rotating blade. It is also found that the mean-square response decreases when the low natural frequency of base decreases. Inversely, the mean-square response increases when the high natural frequency of base (natural frequencies of base are over the first natural frequency of blade) decreases. The reliability of a rotating blade subjected to stationary or non-stationary excitations is also obtained.