Vibration Of Gear System Research Articles

Vibration attenuation characteristics of squeeze film dampers in spiral bevel gear systems

A central drive bevel gear system of aero-engines extracts power from a high-pressure rotor. The high speed of the high-pressure rotor aggravates the gear system's vibration. This work adopts squeeze film dampers (SFD) as damping devices to suppress excessive vibration of the spiral bevel gear rotor system. This paper establishes the general Reynolds equation of SFD and calculates the nonlinear oil-film force of SFD in real-time based on the finite element method. Besides, the finite element method and Timoshenko beam element are employed to model the flexible gear shaft. Then, the dynamic model of the spiral bevel gear system supported by SFDs is established, including the nonlinear oil-film force of SFD and time-varying meshing stiffness of the gear system. The dynamic response of the spiral bevel gear system is analyzed theoretically and experimentally to study the vibration attenuation characteristics of SFD. The findings indicate that SFD can effectively suppress the vibration amplitude of the system at high speeds. The theoretical analysis and experimental amplitude are within an order of magnitude, and the changing trend is consistent.

Analytical Model of Gear Thermal Stiffness Based on the Potential Energy Method

Stiffness is the main internal excitation of gear system vibration and is the basis for studying the static and dynamic characteristics of the system. Based on the relationship between thermal stress and strain in plane strain theory and combined with the potential energy method, a novel gear thermal stiffness model is proposed in this paper. The model deduces more general analytical formulas for gear thermal mesh stiffness and time-varying thermal mesh stiffness. The results show that the gear thermal stiffness decreases as a whole compared with the elastic stiffness. The decrease in stiffness is nonuniform along the line of action. The effects of temperatures, temperature distributions and thermal stress on gear thermal stiffness are investigated. The accuracy of the model is verified by comparisons with existing methods and references.

Self-Excited Vibration Analysis of Gear-Bearing System with Multipoint Mesh and Variable Bearing Dynamic Coefficients

The gear-bearing system is the most important part of integrally centrifugal compressors. According to statistics, the majority of integrally geared compressor accidents are caused by excessive vibration of the geared rotor. However, its complicated dynamic characteristics and inevitable vibration faults in actual operation present significant challenges throughout the analysis and design stages. In this paper, the coupled self-excited vibration of the gear system characterized by multipoint meshing and oil film bearing supporting is investigated. Firstly, the structure of the gear system in an integrally geared compressor is used as a research object. The modeling approach of meshing excitation, including time-varying mesh stiffness, gear meshing error, and tooth backlash are introduced. However, the variable stiffness and damping coefficient equations of journal bearing and oil film thrust bearing are modeled and utilized to approximate the variable bearing force and simplify the vibration computation under the assumption of Newtonian fluid. Then, a dimensionless modeling method of the gear system considering gyroscopic moment of gear disk, variable meshing force, as well as variable stiffness and damping coefficient is proposed. Based on the dynamic model, the influence of the bearing dynamic coefficients and load on the vibration of the entire gear system is studied. Among which, the vibration displacement and meshing force are examined using frequency-domain and time-domain analysis methods. The results suggest that the flexible support can restrain the system’s nonlinear motion, whilst increasing load on the gear system can improve gear operation stability and reduce load fluctuation.

Theoretical analysis of nonlinear vibration characteristics of gear pair with shafts

The circumferential vibration of a gear pair is a parametric excitation caused by nonlinear tooth stiffness, which fluctuates with meshing. In addition, the vibration characteristics of the gear pair become complicated owing to the tooth profile error and backlash. It is considered that the circumferential vibration of the gear pair is affected by the torsional vibration of the shafts. It is important to understand quantitatively the vibration characteristics of the gear system considering the shafts. Therefore, the purpose of this research was to clarify the nonlinear vibration characteristics of a gear pair considering the influence of the shafts using theoretical methods. To achieve this objective, calculations were performed using equations of motion in which the circumferential vibration of the gear pair and the torsional vibration of the shafts were coupled. The nonlinear tooth stiffness was represented by a sine wave. The influence of tooth separation was considered by defining a nonlinear function using backlash and the tooth profile error. For the numerical calculations, both stable and unstable periodic solutions were obtained by using the shooting method. The effect of the shafts on the gear system vibration were clarified by comparing the results in the cases in which the shaft was not considered, one shaft was considered, and both shafts were considered.

Analytical investigation of profile shifts on the mesh stiffness and dynamic characteristics of spur gears

Nowadays, profile shift is one of the most powerful techniques to achieve high-performance spur gear transmission system. Although this technique has been widely used in industry, its influence on time-varying meshing stiffness (TVMS) and dynamic characteristics is still unclear. This paper presents a novel analytical model of TVMS for profile-shifted spur gears. The modification coefficient and transition curve revise the proposed model. Afterward, numerical simulations are carried out to analyze individual and compound shifts based on a six-degree of freedom gear model. For individual shifts, the positive shift can decrease the gear tooth's stiffness and TVMS so that it causes stronger vibrations in the gear system. On the contrary, the negative shift can improve the gear tooth's stiffness and TVMS, thereby mitigating the gear system's vibration. For compound shift, S0-gearing poses a slight impact on TVMS, but an unneglectable effect on dynamic characteristics. Besides, the S-gearing with the positive compound shift can decrease TVMS and increase vibration. In contrast, the S-gearing with the negative compound shift leads to TVMS rising and significantly reducing vibration. The insightful findings derived from the study can help engineers and manufacturers better understand the operation performance and dynamic features of profile-shifted spur gears.

Vibration characteristics of spur gear system with non-linear damping under no lubrication condition

Gear rattle induced by backlash between meshing gears results in excessive vibration and noise in many gear systems. A dynamic model of a light-loaded spur gear pair with non-linear damping under no lubrication condition is presented in this paper. Unlike previous models, the effect of backlash on damping is considered, so the damping is not linear. In order to illustrate the effect of non-linear damping on the dynamic response, the dynamic behaviors of gear system with non-linear damping (GSND) and linear damping (GSLD) under different impact status are compared by using the numerical integration method. The results show that the behavior of gear pair with non-linear damping is different from that with linear damping, and the impact vibration of GSND is much greater than GSLD under some conditions. The influences of parameters such as load ratio, load value, meshing stiffness, and damping ratio on the dynamic responses of GSND are also discussed, and some suggestions to reduce impact vibration of gear system are proposed. The results provide a theoretical basis for the design and manufacture of gear system.

Bifurcation and Erosion of Safe Basin for a Spur Gear System

Tooth surface contact deformation is part of the main causes for gear system vibration. The safety region of the single-stage spur system vibration is established based on the tooth surface contact deformation. Safe basin and its erosion are calculated numerically according to the safety region as the system control parameters are varied. The basin of attraction in the safe basin is computed by combining the simple cell-to-cell method. Vibration safety and global dynamics of the gear system are investigated. Some bifurcations and mechanisms of the erosion of safe basin are studied by using phase portraits, Poincaré maps, bifurcation diagrams under multi-initial values and top Lyapunov exponents (TLE). The sensitivity of system behavior and bifurcation to initial values are discussed as well. Hidden bifurcating points and attractors are revealed. To get a better understanding of the sensitivity of the system behavior to initial values, a bifurcation dendrogram under multi-initial values is designed. It is found that there is the erosion of safe basin existing in the examining area. Both vibration amplitude changing of coexisting attractors and appearing or disappearing of coexisting attractors are the main causes for the erosion of the safe basin. Bifurcation of the system behavior is selective to initial values. With varying frequency, backlash and comprehensive transmission error, the period-1 response gradually bifurcates under multi-initial conditions and evolves into other new periodic responses that coexist with the period-1 response. The new coexisting response has a key effect on the erosion of the safe basin. The study is helpful to the optimization design of the gear system and the control of the system behavior.

Bifurcation and chaos analysis of a gear pair system with multi-clearance

In order to investigate the characteristics of bifurcation and chaos for a spur gear pair system, a three-degree-of-freedom nonlinear dynamic model with multi-clearance is established, in which time-varying meshing stiffness, static transmission error, gear backlash and bearing clearance are comprehensively taken into account. Through introducing a relative generalized coordinate, the dimensionless dynamic equations of motion of system are derived and then solved by using Runge-Kutta numerical integration method. And the bifurcation and chaos features of gear pair are systematically analyzed and discussed from bifurcation diagrams with meshing frequency, gear backlash, bearing clearance and damping ratio as control parameters under different loaded conditions. Meantime, with the help of Poincaré map and phase diagram, the motion forms of system are accurately identified. The analysis results reveal that as meshing frequency increases, the system shows various types of motion states which contain periodic motion, quasi-periodic motion and chaotic motion. Similarly, with the increasing of gear backlash, the system undergoes complex motion forms under lightly loaded condition, whereas it is only in period-one motion state under heavily loaded condition. Furthermore, the system motion state is gradually switched from chaos to periodic or quasi-periodic motion under lightly loaded condition when bearing clearance changes. However, under heavily loaded condition, the bearing clearance has a weak effect on dynamic behavior of the gear system. Apparently, the system tends to be more stable under heavily loaded condition than that under lightly loaded condition. In addition, the growing damping ratio can effectively suppress the chaotic behavior and control nonlinear vibration of gear system. The research results provide useful guidance for dynamic design and vibration control for gear set.

Fault mode analysis and detection for gear tooth crack during its propagating process based on dynamic simulation method

Gearbox is one of the most important parts of rotating machinery, therefore, it is vital to carry out health monitoring for gearboxes. However, it is still an unsolved problem to disclose the impact of gear tooth crack fault on gear system vibration features during the crack propagating process, besides effective crack fault mode detection methods are lacked. In this study, an analytical model is proposed to calculate the time varying mesh stiffness of the meshing gear pair, and in this model the tooth bending stiffness, shear stiffness, axial compressive stiffness, Hertzian contact stiffness and fillet-foundation stiffness are taken into consideration. Afterwards, the vibration mechanism and effects of different levels of gear tooth crack on the gear system dynamics are investigated based on a 6 DOF dynamic model. Then, the crack fault vibration mode is studied, and a parametrical-Laplace wavelet method is presented to describe the crack fault mode. Furthermore, based on the maximum correlation coefficient (MCC) criterion, the optimized Laplace wavelet base is determined, which is then designed as a health indicator to detect the crack fault. The results show that the proposed method is effective in fault diagnosis of severe tooth crack as well as the early stage tooth crack.

Determination of Composite Meshing Errors and Its Influence on the Vibration of Gear System

Determination of Composite Meshing Errors and Its Influence on the Vibration of Gear System

Non-linear dynamics of gear pair with dynamic backlash subjected to combined internal and external periodic excitations

In this paper, we theoretically investigate the nonlinear dynamic characteristics of a gear pair system with dynamic backlash subjected to internal and external periodic excitations. The dynamic model of a viscously damped single degree of freedom gear pair model with periodic time-varying stiffness, dynamic backlash, static transmission error and external periodic excitation is established. The incremental harmonic balance method (IHBM) is applied to analyze the frequency response characteristics as well as the effects of dynamic backlash, time-varying stiffness, excitation force amplitude and damping ratio on the dynamic characteristics of the gear pair system. The analysis results are compared with the numerical integration results to show accuracy and authority of the IHBM. The difference between the constant backlash and the dynamic backlash models is also discussed. The results show that the nonlinear dynamic characteristics such as parametric resonance, multi-valued properties and jump phenomena are strongly attributed to the interaction between time-varying stiffness and backlash. Significantly, under combined internal and external periodic excitations, the multi-valued properties and jump phenomena occur not only in primary resonance frequency but also in super harmonic frequency. The increase of the dynamic backlash and damping ratio could effectively control the nonlinear vibration of gear system; whereas the excitation force amplitude has less influence on the nonlinear dynamic characteristics and the increase of the excitation force amplitude could no longer control the nonlinear vibration of gear system, which is different from the dynamic characteristics under internal periodic excitations.

The fault feature extraction and classification of gear using principal component analysis and kernel principal component analysis based on the wavelet packet transform

The vibration signal of a gear system is selected as the original information of fault diagnosis and the gear system vibration equipment is established. The vibration acceleration signals of the normal gear, gear with tooth root crack fault, gear with pitch crack fault, gear with tooth wear fault and gear with multi-fault (tooth root crack & tooth wear fault) is collected in four kinds of speed conditions such as 300rpm, 900rpm, 1200rpm and 1500rpm. Using the method of wavelet threshold de-noising to denoise the original signal and decomposing the denoising signal utilizing the wavelet packet transform, then 16 frequency bands of decomposed signal are got. After restructuring the decomposing signal and obtaining the signal energy in each frequency band, the signal energy of the 16 bands is as the shortlisted fault characteristic data. Based on this, using the methods of principal component analysis (short for PCA) and kernel principal component analysis (short for KPCA) to extract the feature from the fault features of shortlisted 16-dimensional data feature, then the effect of reducing dimension analysis are compared. The fault classifications are displayed through the information that got from the first and the second principal component and kernel principal component, and these demonstrate they have a different and good effect of classification. Meanwhile, the article discusses the effect of feature extraction and classification that caused by the kernel function and the different options of its parameters. These provide a new method for a gear system fault feature extraction and classification.

Vibration signal models for fault diagnosis of planetary gearboxes

A thorough understanding of the spectral structure of planetary gear system vibration signals is helpful to fault diagnosis of planetary gearboxes. Considering both the amplitude modulation and the frequency modulation effects due to gear damage and periodically time variant working condition, as well as the effect of vibration transfer path, signal models of gear damage for fault diagnosis of planetary gearboxes are given and the spectral characteristics are summarized in closed form. Meanwhile, explicit equations for calculating the characteristic frequency of local and distributed gear fault are deduced. The theoretical derivations are validated using both experimental and industrial signals. According to the theoretical basis derived, manually created local gear damage of different levels and naturally developed gear damage in a planetary gearbox can be detected and located.

Study on the Passive Vibration Control of Gear System

The passive vibration control is a very effective measure to gear system. But there exist some limitations. This paper viewed the development of the passive vibration control of gear system, and researched the damping characteristics of the friction dampers and viscoelastic dampers intensively. Firstly, the limitations of the existence passive damping technology are pointed out. Secondly, the feasibility and validity of piezoelectric shunt damping is guaranteed through experimentation on the gear system vibration control. Thirdly, a new hybrid damping, piezoelectric- viscoplastic material damping, is proposed, which extracts the advantages of piezoelectric shunt damping and viscoplastic damping.

Frequency response calculation of non-linear torsional vibration in gear systems

The current paper focuses on the non-linear torsional vibration of a one-stage transmission gear system. Four different methods have been applied for solution of the equation of motion; the discretization method, the perturbation method, the Ritz method, and the step-wise time integration of the equation of motion. The time and frequency results from these analyses have been compared with each other, as well as those reported in literatures. Although all of these methods are accurate and computationally effective for finding the main spectral contribution, however, only the discretization method and the step-wise time-integration model are able to identify the other frequency components.

Effect of manufacturing and assembly defects on two-stage gear system vibration

Effect of manufacturing and assembly defects on two-stage gear system vibration

Evaluation Method of Gear Dynamic Performance Using Information in Time-Frequency Domain.

The wavelet transform (WT) is a method for the time-frequency analysis of signals. It is possible to detect the change in each frequency through a sampling period by the WT. In this study, in order to discuss an evaluation method for the gear dynamic performance using the time-frequency analysis, the dynamic performance of both sintered and steel gears were measured using a power circulating gear testing machine in the range of the rotational speeds of 1600 to 10000 rpm. The measured signals of dynamic performance, which are tooth root strain, vibration and sound of gear box, were analyzed in the time-frequency domain by the continuous WT based on the Gabor function, and the signals of the dynamic performance were decomposed and reconstructed by the discrete WT based on the cardinal B-spline function. The validity on the new evaluation method using both continuous and discrete WTs was discussed. The dynamic performance can be divided between two frequency regions over and under the tooth mesh frequency fz. Consequently, it was clarified that the dynamic performance over the fz indicated the condition of the torsional vibration of gear system, and the dynamic performance under fz depended on the rotational speed of gear.

Vibration in gear systems

This paper simulates the non-linear vibration of a gear pair including shaft flexibility. Backlash is included because of the clearance between the teeth and the periodic variation in the stiffness of the mesh produces a parametric excitation. This paper shows the different behaviour that may be observed when the additional degree of freedom is added to the single degree of freedom model previously analysed.

Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems

Mesh stiffness variation, the change in stiffness of meshing teeth as the number of teeth in contact changes, causes parametric instabilities and severe vibration in gear systems. The operating conditions leading to parametric instability are investigated for two-stage gear chains, including idler gear and countershaft configurations. Interactions between the stiffness variations at the two meshes are examined. Primary, secondary, and combination instabilities are studied. The effects of mesh stiffness parameters, including stiffness variation amplitudes, mesh frequencies, contact ratios, and mesh phasing, on these instabilities are analytically identified. For mesh stiffness variation with rectangular waveforms, simple design formulas are derived to control the instability regions by adjusting the contact ratios and mesh phasing. The analytical results are compared to numerical solutions.

Low frequency vibration generated by gear tooth impacts

Excitation of low frequency vibration in gear systems by impacts arising from gear tooth defects is examined. It is demonstrated that in favourable conditions the low frequency vibration can be isolated and used to monitor the condition of the gear.