A continuous vibration theory for rotors with an open edge crack

In this paper a new continuous model for flexural vibration of rotors with an open edge crack has been developed. The cracked rotor is considered in the rotating coordinate system attached to it. Therefore, the rotor bending can be decomposed in two perpendicular directions. Two quasi-linear displacement fields are assumed for these two directions and the strain and stress fields are calculated in each direction. Then the final displacement and stress fields are obtained by composing the displacement and stress fields in the two directions. The governing equation of motion for the rotor has been obtained using the Hamilton principle and solved using a modified Galerkin method. The free vibration has been analyzed and the critical speeds have been calculated. Results are compared with the finite element results and an excellent agreement is observed.     

Modal balancing of flexible rotors with bow and distributed unbalance

Unbalance and bow are found to be one of the most common causes of synchronous machinery vibrations in rotating systems. Concentrated lumped mass models are adopted in most of the finite element approach for modeling unbalances and subsequent balancing in rotating systems. But this assumption may not be appropriate for long slender rotors with unbalances distributed along the length of the rotor. A polynomial curve for eccentricity distribution with finite element modeling is used to identify the distributed unbalance. The unbalance eccentricity distributions are estimated using the measured vibration responses at a speed below the balancing speed. Modal correction mass required to balance a rotor at its first bending critical speed, having both distributed unbalance and bow is computed knowing the amplification factor at critical speed. The rotor is balanced at its first bending critical speed using modal balancing method in a single trial run and using a single balancing plane. The method thus avoids multiple trial runs required for modal balancing of flexible rotors. This method is verified on an experimental rotor having both bow and unbalance. The concept of quantifying the distributed unbalance using ‘Norm’ of eccentricity polynomial function is also introduced for the first time.     

New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis

The actual breathing mechanism of the transverse breathing crack in the cracked rotor system that appears due to the shaft weight is addressed here. As a result, the correct time-varying area moments of inertia for the cracked element cross-section during shaft rotation are also determined. Hence, two new breathing functions are identified to represent the actual breathing effect on the cracked element stiffness matrix. The new breathing functions are used in formulating the time-varying finite element stiffness matrix of the cracked element. The finite element equations of motion are then formulated for the cracked rotor system and solved via harmonic balance method for response, whirl orbits and the shift in the critical and subcritical speeds. The analytical results of this approach are compared with some previously published results obtained using approximate formulas for the breathing mechanism. The comparison shows that the previously used breathing function is a weak model for the breathing mechanism in the cracked rotor even for small crack depths. The new breathing functions give more accurate results for the dynamic behavior of the cracked rotor system for a wide range of the crack depths. The current approach is found to be efficient for crack detection since the critical and subcritical shaft speeds, the unique vibration signature in the neighborhood of the subcritical speeds and the sensitivity to the unbalance force direction all together can be utilized to detect the breathing crack before further damage occurs.     

VIBRATION CHARACTERISTICS OF SLOTTED SHAFTS

The dynamics of asymmetric rotors has been investigated in the literature. However, considering the importance in the practical fields, the vibration characteristics of slotted shafts are not studied in detail. It needs more attention particularly with the emerging of new materials, such as composites for shafts. The present study aims at the analysis of the slotted shafts including the compensatory inertia slots. Also, the slotted composite shaft has been modelled based on first order shear deformation theory using finite element method with shell elements. Different materials such as, Boron epoxy, Carbon epoxy and Graphite epoxy have been tried for various stacking sequences. The slot parameters, stacking sequences and material properties have found to influence to great extent on the vibrational characteristics of rotors. The results are compared with those of isotropic slotted shaft.     

DYNAMIC ANALYSIS OF A SPIRAL BEVEL-GEARED ROTOR-BEARING SYSTEM

Spiral bevel gears can transmit motion between two rotors, which are commonly perpendicular to each other. In this paper, the dynamic analysis of a spiral bevel-geared rotor-bearing system is studied. Firstly, the constraint equation describing the relationship between the generalized displacements of spiral bevel gear pairs is derived briefly. Then the modelling of coupled axial-lateral-torsional vibration of the rotor system geared by spiral bevel gears is discussed. Finally, the mechanism of coupled vibration of the spiral bevel-geared rotor system is analyzed theoretically and the dynamic behavior of the system is investigated numerically. The conclusions are characterized as follows. The influences of the critical speeds in rigid journal supports, stability threshold speed and unbalanced responses in hydrodynamic journal bearings are not remarkable in comparison with the spur bevel-geared system under the same conditions. However, the critical speeds and stability threshold speed are essentially affected by boundary conditions such as the torsional stiffness, and meanwhile the effect of the unbalanced responses is not prominent under the concerned rotating speeds except that around the resonance peaks. The steady state response due to torsional excitation is also analyzed, and the results show that it cannot be neglected either in the torsional direction or in the lateral and axial directions in the spiral bevel-geared rotor system.     

Rapporteur's report,session 3: Other sources of railway noise

A method of finding the flexural and torsional normal modes of beams which have straight stiffness axes is given. The Lanzcos method of minimized iterations, based on the integral equation of beam vibrations, is used to obtain intermediate modes with the distributions along the beam of its mass and rigidity as data. The inertia matrix appropriate to the intermediate modes is theoretically tridiagonal and the stiffness matrix is unit. The dominant eigenvalues and vectors of the inertia matrix give very good approximations to the frequencies and shapes of the graver normal modes. Results of test calculations are given.     

Experimental study of noncontact ultrasonic motor with non-symmetrical electrode

We have proposed a novel noncontact ultrasonic motor based on non-symmetrical electrode driving. The configuration of this electrode and the fabrication process of rotors are presented. Its vibration characteristics are computed and analysed by using the finite element method and studied experimentally. Good agreement between them is obtained. Moreover, it is also shown that this noncontact ultrasonic motor is operated in antisymmetric radial vibration mode of $B_{21}$ mode. The maximum revolution speed for three-blade and six-blade rotors are 5100 and 3700\,r/min at an input voltage of 20V, respectively. Also, the noncontact high-speed revolution of the rotors can be realized by the parts of I, III of the electrode or II, IV of the electrode. The levitation distance between the stator and rotor is about 140$\mu $m according to the theoretical calculation and the experimental measurement.     

A theoretical-experimental comparison of the synchronous response of a bowed rotor in five different sets of fluid film bearings

A comparison of theoretical and experimental synchronous unbalance responses of a bowed Jeffcott rotor in fluid film bearings has been completed. A transfer matrix method was used to predict theoretically the response of a 25·4 mm shaft in fluid film bearings and results were compared to data from a previous experimental study. Four bearing types were used: two axial groove, pressure dam, tilting pad and four-lobe. Very good agreement was found for all bearing types at the rotor critical speed (3000 rpm). Differences less than 15% in peak response were found and the theoretical and experimental peaks were found to occur within 200 rpm. Worst agreement was found for the preloaded four-lobe bearings and this disagreement was found for speeds other than the critical speed. Also, for equal bow and unbalance the tilting pad and four-lobe bearings were found to produce the least and most damping at the critical speed, respectively. Previous to this time a comparison of theoretical and experimental synchronous responses of a rotor system representing industrial turbomachines has not been published, nor has a comparative study of the different bearing types.     

Acceleration of unbalanced flexible rotors through the critical speeds

The bending vibrational behaviour of a flexible rotor with a continuous mass distribution passing its critical speeds under a driving torque is considered. It is shown that the (non-linear) equations of motion for an actual shaft can be formally traced back to those of a Laval rotor. In this way, the results for a Laval rotor, which, in an earlier publication by the authors [1], have been presented generally for constant load torque can be applied to actual rotors. The system parameters of the Laval rotor merely have to be replaced by the generalized parameters of the respective bending modes. A special study shows that the effect of the torsional flexibility of the shaft on the bending vibrational behaviour is negligible.     

Vibration of a rotating beam with tip mass

The vibration frequency of a rotating beam with tip mass is investigated. The finite element method is used, a third order polynomial being assumed for the variation of the lateral displacement. The effects of the root radius, the setting angle and the tip mass are incorporated into the finite element model. The results are compared with results from previous authors utilizing Myklestad and extended Galerkin methods. The results show that the setting angle has a significant effect on the first mode frequencies but not on the high frequencies. The tip mass tends to depress the frequencies at low speeds of rotation but it tends to increase the frequencies at high speeds of rotation. The results of this work have applications in wind turbine rotors, helicopter rotors, etc., and the method used here can be extended to investigate the vibration frequency of flexible blade auto cooling fans.     

Seismic response of underwater members of variable cross section

This paper is concerned with a method for solving seismic response problems of a pile of variable cross section with a tip inertia subjected to a sea bottom seismic displacement. The analysis developed here is based on an elastodynamic theory in which the effects of the continuously distributed mass and rigidity of the pile are included. The method includes use of Fourier series expansion, the Laplace transform, the transfer matrix method and the residue theorem in order to deal with the complex seismic displacement and arbitrarily shaped piles; considerable simplification of the calculations is thus achieved. The theoretical results given are applicable to seismic response problems for a pile of arbitrary shape with a tip inertia, excited by arbitrary displacements. As an application of the present theoretical results, the dynamic response has been calculated for hollow piles of curved conical shape with tip inertias and double taper beams subjected to seismic displacements.     

Structural models and undamped torsional natural frequencies of a superconducting turbogenerator rotor

Superconducting turbogenerators with a “double rotor structure” have a torsional natural frequency within the generator, the outer rotor moving in opposition to the inner rotor. For large machines this natural frequency may approach 100 Hz. In this paper a finite element model and simple lumped mass and spring models of the rotor, for the calculation of the undamped torsional natural frequencies, are described and compared. A method by which equivalent spring stiffnesses for both the inner and outer rotors can be derived is described, allowing one to use a rotor model with one lumped mass and equivalent spring stiffness for each of the inner and outer rotors. Such a rotor model can be readily used for studying electromagnetic interaction effects and assessing fault torques in the outer rotor and inner rotor torque tubes.     

Theoretical and experimental research on a disk-type non-contact ultrasonic motor

We developed a disk-type non-contact ultrasonic motor based on B22 vibration mode. The rotors of SU-8 photoresist are fabricated by the UV-LIGA process to control their shapes and thicknesses. So the structures of them are optimized by the experiments. It is found that the revolution speed of disk-type non-contact ultrasonic motor not only depends on the vibration amplitude of the stator, but also the weight and construction of the rotors. The maximum revolution speed of the optimal rotor is 3569 rpm at the input voltage of 20 V and the driving frequency of 45.6 kHz. The exciting principle of traveling wave is presented with theoretical equations. The electric signals applied to the piezoelectric ceramic are designed by the principle. The natural frequency and corresponding vibration mode are calculated and analyzed using finite element method. It is shown that experimental results are in good agreement with simulation, which verifies the effectiveness of the finite element model. Moreover, the levitation distance between the stator and rotor is measured by a CCD laser displacement transducer.     

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.     

Vibration response of misaligned rotors

Misalignment is one of the common faults observed in rotors. Effect of misalignment on vibration response of coupled rotors is investigated in the present study. The coupled rotor system is modelled using Timoshenko beam elements with all six dof. An experimental approach is proposed for the first time for determination of magnitude and harmonic nature of the misalignment excitation. Misalignment effect at coupling location of rotor FE model is simulated using nodal force vector. The force vector is found using misalignment coupling stiffness matrix, derived from experimental data and applied misalignment between the two rotors. Steady-state vibration response is studied for sub-critical speeds. Effect of the types of misalignment (parallel and angular) on the vibration behaviour of the coupled rotor is examined. Along with lateral vibrations, axial and torsional vibrations are also investigated and nature of the vibration response is also examined. It has been found that the misalignment couples vibrations in bending, longitudinal and torsional modes. Some diagnostic features in the fast Fourier transform (FFT) of torsional and longitudinal response related to parallel and angular misalignment have been revealed. Full spectra and orbit plots are effectively used to reveal the unique nature of misalignment fault leading to reliable misalignment diagnostic information, not clearly brought out by earlier studies.     

The acoustic spectrum of axial flow machines

Rotor spectra from a variety of axial flow machines have been compared and acoustic trends examined. Types of rotor considered in the investigation included aircraft propellers, helicopter rotors, jet engine compressor and by-pass fans and a range of domestic, automotive and industrial cooling fans. Rotor sizes varied from 15 in to 60 ft in diameter and tip speeds from 200 ft/s to transonic speeds.The study showed that the acoustic output from these various rotor devices can be described by a common characteristic spectrum. In particular, the minimum broad band noise generated by ducted and free field rotors corresponds to laminar and turbulent boundary layer shedding, and discrete excess noise is generated by various degrees of impulsive blade loading. Finally the study showed that aerodynamic details can be interpreted from the acoustic spectra, and as a result, new insight into the mechanisms of rotor noise has been formed.     

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.     

Response of infinite length rods and beams with periodically varying area

This paper develops a solution method for the longitudinal motion of a rod or the flexural motion of a beam of infinite length whose area varies periodically. The conventional rod or beam equation of motion is used with the area and moment of inertia expressed using analytical functions of the longitudinal (horizontal) spatial variable. The displacement field is written as a series expansion using a periodic form for the horizontal wavenumber. The area and moment of inertia expressions are each expanded into a Fourier series. These are inserted into the differential equations of motion and the resulting algebraic equations are orthogonalized to produce a matrix equation whose solution provides the unknown wave propagation coefficients, thus yielding the displacement of the system. An example problem of both a rod and beam are analyzed for three different geometrical shapes. The solutions to both problems are compared to results from finite element analysis for validation. Dispersion curves of the systems are shown graphically. Convergence of the series solutions is illustrated and discussed.     

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.     

Bending wave propagation of carbon nanotubes in a bi-parameter elastic matrix

This article studies transverse waves propagating in carbon nanotubes (CNTs) embedded in a surrounding medium. The CNTs are modeled as a nonlocal elastic beam, whereas the surrounding medium is modeled as a bi-parameter elastic medium. When taking into account the effect of rotary inertia of cross-section, a governing equation is acquired. A comparison of wave speeds using the Rayleigh and Euler-Bernoulli theories of beams with the results of molecular dynamics simulation indicates that the nonlocal Rayleigh beam model is more adequate to describe flexural waves in CNTs than the nonlocal Euler-Bernoulli model. The influences of the surrounding medium and rotary inertia on the phase speed for single-walled and double-walled CNTs are analyzed. Obtained results turn out that the surrounding medium plays a dominant role for lower wave numbers, while rotary inertia strongly affects the phase speed for higher wave numbers.