Qinkai Han

Analytical model of bolted disk–drum joints and its application to dynamic analysis of jointed rotor

Bolted joints are widely used in aero-engines. One of the common applications is to connect the rotor disks and drums. An analytical model for the bending stiffness of the bolted disk–drum joints is developed. The joint stiffness calculated using the analytical model shows sound agreement with the calculation obtained based on finite element analyses. The joint stiffness model is then implemented into the dynamic model of a simple rotor connected through the bolted disk–drum joint. Finally, the whirling characteristics and steady-state response of the jointed rotor are investigated to evaluate the influence of the joint on the rotor dynamics, where the harmonic balance method is employed to calculate the steady-state response to unbalance force. The simulation results show that the joint influence on the whirling characteristics of the rotor system can be neglected; whereas, the presence of the bolted disk–drum joint may lead to a decrease in the rotor critical speeds due to the softening of the joint stiffness. The proposed analytical model for the bolted disk–drum joints can be adopted conveniently for different types of rotor systems connected by bolted disk–drum joints.

Vibration, Buckling and Dynamic Stability of a Cracked Cylindrical Shell with Time-Varying Rotating Speed

Vibration, linear elastic buckling and dynamic stability behaviors of a cracked cylindrical shell with time-varying rotating speed are analyzed. Finite element method is used to obtain the system mass and stiffness matrix, and Bolotins method is applied to explore the dynamic stability region. The effects of constant rotating speed, crack length and orientation, and length-diameter ratio of the cylindrical shell on the free-vibration and buckling behaviors are investigated. The stability characteristics of the cracked shell are also researched and the influences of crack length, crack orientation, rotating speed basic value, steady load factor, dynamic load factor and damping ratio are considered. Numerical examples show that crack length and orientation can affect the vibration, buckling and dynamic stability behavior of cylindrical shell significantly.

Fault diagnosis for wind turbine planetary ring gear via a meshing resonance based filtering algorithm.

Identifying the differences between the spectra or envelope spectra of a faulty signal and a healthy baseline signal is an efficient planetary gearbox local fault detection strategy. However, causes other than local faults can also generate the characteristic frequency of a ring gear fault; this may further affect the detection of a local fault. To address this issue, a new filtering algorithm based on the meshing resonance phenomenon is proposed. In detail, the raw signal is first decomposed into different frequency bands and levels. Then, a new meshing index and an MRgram are constructed to determine which bands belong to the meshing resonance frequency band. Furthermore, an optimal filter band is selected from this MRgram. Finally, the ring gear fault can be detected according to the envelope spectrum of the band-pass filtering result.

Frequency Response Characteristics of Parametrically Excited System

The frequency response characteristic of a general time-invariant system has been extensively analyzed in literature. However, it has not gained sufficient attentions in the parametrically excited system. In fact, due to the parametric excitation, the frequency response of time-periodic system differs distinctly from that of the time-invariant system. Utilizing Sylvester’s theorem and Fourier series expansion method, commonly used in the spectral decomposition for matrix, the frequency response functions (FRFs) of a single-degree-of-freedom (SDOF) parametrically excited system are derived briefly in the paper. The external resonant condition for the system is obtained by analyzing the specific expressions of FRFs. Then, a spur-gear-pair with periodically time-varying mesh stiffness is selected as an example to simulate the frequency response characteristics of parametric system. The effects of parametric stability, periodic mesh stiffness parameters (mesh frequency and contact ratio), and damping are considered in the simulation. It is shown from both theoretical and simulation results that the frequency response of parametric system has the following properties: there are multiple FRFs even for a SDOF periodic system as the forced response contains many frequency components and each FRF is corresponding to a certain response spectrum; the system has multiple external resonances. Besides the resonance caused by the external driving frequency equals to the natural frequency, the system will also be external resonant if external frequency meets the combination of natural frequency and parametric frequency. When the system is in external resonant state, the dominant frequency component in the response is the natural frequency; damping makes the peak values of FRFs drop evidently while it has almost no impact on the FRFs in nonresonant regions.

Detection and Localization of Tooth Breakage Fault on Wind Turbine Planetary Gear System considering Gear Manufacturing Errors

Sidebands of vibration spectrum are sensitive to the fault degree and have been proved to be useful for tooth fault detection and localization. However, the amplitude and frequency modulation due to manufacturing errors (which are inevitable in actual planetary gear system) lead to much more complex sidebands. Thus, in the paper, a lumped parameter model for a typical planetary gear system with various types of errors is established. In the model, the influences of tooth faults on time-varying mesh stiffness and tooth impact force are derived analytically. Numerical methods are then utilized to obtain the response spectra of the system with tooth faults with and without errors. Three system components (including sun, planet, and ring gears) with tooth faults are considered in the discussion, respectively. Through detailed comparisons of spectral sidebands, fault characteristic frequencies of the system are acquired. Dynamic experiments on a planetary gear-box test rig are carried out to verify the simulation results and these results are of great significances for the detection and localization of tooth faults in wind turbines.

The Fault Characteristics of Planetary Gear System with Tooth Breakage

Tooth breakage is a typical failure form of wind-turbine planetary gear transmission system, it is important to study the influence of tooth breakage on vibration characteristics of planetary gear transmission system. In this paper, considering the tooth breakage defect, a lumped parameter vibration model of a planetary gear system with time-periodic mesh stiffness is established. Effects of the length and width of tooth breakage on meshing stiffness and dynamic response are discussed in detail. The relation between characteristic frequency of the tooth breakage fault and rotating speeds is pointed out. Several statistical indicators are utilized to show the influence of two parameters (length of planet tooth breakage and input speed) on the dynamic response of the system. Experiments are carried out to verify the simulation results. These results would be useful for fault diagnosis of wind turbine transmission system at different operation conditions.

Finite element analysis of stiffness characteristics of an electromagnetic device

An electromagnetic device, consisting of a pair of permanent magnets and a non-magnetic coil, has shown its linear advantages for inducing parametric stiffness excitation in the laboratory. The electromagnetic device powered by a DC coil current acts like a mechanical spring with constant stiffness. The effects of design parameters upon the electromagnetic stiffness and its linear operating range (referred as the stiffness characteristics) are important and crucial for the design and usage of the device. Thus, a numerical method, based upon the commercial finite element method (FEM) package ANSYS/Emag, was presented for analyzing the stiffness characteristics of the device with DC coil current. The theory foundation for computing the electromagnetic force using FEM was derived and the program realization in ANSYS was introduced briefly. Two typical examples were chosen to verify the accuracy of the numerical analysis method. Based upon these, the restoring force characteristics of the electromagnetic device with various DC coil currents were obtained utilizing the numerical method. Two important indicators (proportional coefficient between linear stiffness and coil current and linear range of the stiffness) were defined for measuring the stiffness characteristics of the electromagnetic device. Then, the effects of the design parameters (both the electromagnetic and structural parameters) on the two indicators were investigated, respectively. The research results provide useful references for the stiffness design of the electromagnetic device in actual practice.

Theoretical analysis of the natural frequency of a geared system under the influence of a variable mesh stiffness

Abstract Owing to the change in the number of contact tooth pairs, the mesh stiffness is periodically time varying, which leads to additional time-variant coefficients in the corresponding governing equations of a geared dynamic system. The natural frequency of such a system differs distinctly from that of a time-invariant system. However, it has not gained sufficient attention in geared dynamic analysis, and the natural frequency of a geared system is usually assumed to be the average frequency obtained using the time-average mesh stiffness. Whether it is reasonable to obtain the natural frequency of a geared system in this way should be analysed seriously. Thus, the natural frequency of parametric vibration for a spur-gear-pair system will be researched using the Floquet theory in this paper. The influences of the periodically time-varying mesh stiffness parameters (including the mesh frequency and the contact ratio) on the natural frequency will be discussed in detail. Contrasts between the natural frequency and the average frequency for the system in the stable and unstable regions will be processed and the scope of application for taking the average frequency as a substitution for the natural frequency of a geared system will then be presented.

Investigation of Parametric Instability of the Planetary Gear under Speed Fluctuations

Planetary gear is widely used in engineering and usually has symmetrical structure. As the number of teeth in contact changes during rotation, the time-varying mesh stiffness parametrically excites the planetary gear and may cause severe vibrations and instabilities. Taking speed fluctuations into account, the time-varying mesh stiffness is frequency modulated, and therefore sideband instabilities may arise and original instabilities are significantly affected. Considering two different speed fluctuations, original and sideband instabilities are numerically and analytically investigated. A rotational lumped-parameter model of the planetary gear is developed, in which the time-varying mesh stiffness, input speed fluctuations, and damping are considered. Closed-form approximations of instability boundaries for primary and combination instabilities are obtained by perturbation analysis and verified by numerical analysis. The effects of speed fluctuations and damping on parametric instability are systematically examined. Because of the frequency modulation, whether a parametric instability occurs cannot be simply predicted by the planet meshing phase which is applicable to constant speed. Besides adjusting the planet meshing phase, speed fluctuation supplies a new thought to minimize certain instability by adjusting the amplitude or frequency of the speed fluctuation. Both original and sideband instabilities are shrunken by damping, and speed fluctuation further shrinks the original instability.

Unbalanced Response of Cracked Rotor-Bearing System Under Time-Dependent Base Movements

Unbalanced response of cracked rotor-bearing system under time-dependent base movements is studied in this paper. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Both the open and breathing transverse cracks are considered in the analysis. The finite element model is established for the base excited rotor-bearing system with open or breathing cracks. Considering the time-varying base movements and transverse cracks, the second order differential equations of the system will not only have time-periodic gyroscopic and stiffness coefficients, but also the multi-frequency external excitations. An improved harmonic balance method is introduced to obtain the steady-state response of the system under both base and unbalance excitations. The whirling frequencies of the equivalent time-invariant system, orbits of shaft center, response spectra and frequency response characteristics, are analyzed accordingly. The effects of various base angular motions, frequency and amplitude of base excitations, and crack depths on the system dynamic behaviors are considered in the discussions.Copyright