Model Of Gear System Research Articles

Study on multi-state of meshing-impacting dynamic characteristics of spur gear systems considering friction under localized tooth breakage

Localized tooth breakage is one of the common failures of spur gears, which affects the smooth and safe operation of spur gear transmission systems. The tooth collision cannot be neglected, and it is especially important to reveal the meshing-impacting dynamic characteristics of the gear system under localized tooth breakage to improve the safe and stable operation of the gear system. Based on the gear meshing principle and the dissipative collision contact force model, the drive-side tooth meshing model and back-side tooth impacting model are established with considering the transient nature of tooth back-side contact. The tooth surface meshing model and tooth back collision model under partial breakage are established. According to the contact state and force environment of the gear pair, the multi-state meshing-impacting behaviour under local breakage is classified, and the discrete meshing-impact dynamics model of an involute spur gear system under local breakage is established to explore the influence of local breakage on the meshing stiffness and load distribution. The mechanism of contact force under partial tooth breakage is revealed, and the influence of load coefficient and meshing frequency on the nonlinear dynamics is studied by defining two Poincaré maps. It is found that local tooth breakage affects the contact force of single-and double-tooth meshes and reduces gear load carrying capacity. Larger loads inhibit back-side impact, and smaller loads induce the coexistence behaviour and back-side impact. Larger or smaller meshing frequency induce back-side impact behaviour. The coexistence of chaotic and periodic motions induces back-side impact behaviour, and localized tooth breakage affects the coexistence phenomenon and aggravates the complexity of the dynamic behaviour. The dynamic model of gear system considering energy dissipation and nonlinear vibration under the presence of local tooth breakage of the pinion is explored, and the conditions of back-side impact are studied. This research provides new methods and ideas for nonlinear dynamic modelling and analysis of faulty gear systems.

Dynamic characteristics of spur gear system with tooth root crack considering gearbox flexibility

Considering the calculation efficiency and accuracy of meshing characteristics of gear pair with tooth root crack fault, the parametric model of cracked spur gear is established by simplifying the crack propagation path. LTCA method is used to calculate the time-varying meshing stiffness (TVMS) and transmission error (TE) of the cracked gear pair, and is verified by finite element (FE) method. The crack area share index is proposed to measure the degree of crack fault and the application range of simplified crack propagation path is obtained. The gearbox is introduced into the calculation as the condensation model, and the dynamic model of gear system with root crack fault considering gearbox flexibility is established. The effects of gearbox flexibility, rotational speed, crack degree and gear torque on the vibration response fault characteristics of gear system are analyzed. Compared with the simulation results without considering gearbox flexibility, the simulation results considering gearbox flexibility are smaller in amplitude and closer to the experimental results. The fault shock degree in acceleration waveform and fault frequency modulation in FFT spectrum are more obvious considering gearbox flexibility. The fault characteristics of vibration response are more obviously affected by the reduction of gearbox flexibility and the increase of fault degree than the change of rotational speed and gear torque. The research results provide theoretical and technical support for fault diagnosis and health monitoring of gear systems.

Dynamical modeling of spur gear with pitting based on image processing tooth surface

Reliable dynamic model of gear system with pitting can be utilized to investigate the effects of surface pitting on vibration response and to identify its severity. In this study, a dynamic model with pitting based on image processing is proposed, and pitting is simulated through the discrete tooth surface. The contact points are modeled by a network of linear springs with different stiffness in parallel. The revised mesh stiffness is calculated and a new mesh stiffness calculation method with surface pitting is proposed, which combines pixel matrix with mesh stiffness to improve the accuracy and efficiency of calculation. The dynamic model is validated by comparison with experimental test signals under different rotations and pitting states. As the Hilbert–Huang transform energy spectrum method has been applied for the detection of the vibration signals with small pitting areas, the signal characteristics caused by pitting can be well represented based on the statistical indicators and energy spectrum.

Nonlinear dynamic modeling and analysis of spur gears considering uncertain interval shaft misalignment with multiple degrees of freedom

Shaft misalignment is regarded as a deterministic parameter in traditional dynamics analysis of gear system, but due to design tolerance and assembly accuracy, shaft misalignment is unavoidably uncertain and this kind of uncertainty exhibits interval attribute. To evaluate the dynamic characteristics of gear system more reasonably, a new nonlinear dynamic model of spur gears considering the uncertainty of shaft misalignment with multiple degrees of freedom (DOFs) is proposed and studied in this paper. First, the meshing analysis of spur gears under shaft misalignment is completed based on tooth contact theory. Then a nonlinear interval dynamic model of gear system is established and an interval analysis approach based on Chebyshev inclusion function and optimization method is used to obtain the upper and lower bounds of the dynamic response. Afterwards, the effectiveness of proposed dynamic model and interval algorithm is validated by comparison with the existing experiments and simulations respectively. Finally, parameter studies are carried out in detail and the result shows that the uncertainty of shaft misalignment will significantly change the internal excitation of gear system and thus enormously enhancing the system vibration, this will bring valuable guidance for the tolerance allocation in gearbox design.

Dynamic characteristics of single-loop gear system based on bond graph method

The complexity of single-loop gear system transmission structure makes it difficult for traditional modeling methods to establish precise dynamic model, which greatly affects the accuracy of its dynamic characteristics research. Firstly, a structure diagram is established by adopting modularization idea according to the structural properties of single-loop gear system. On this basis, a precise bond graph model of the single-loop gear system is obtained combining the modeling principle of bond graph method and the advantages of rich graphics library. Secondly, the dynamic state equation of single-loop gear system is obtained from bond graph model. The simulation model of gear system is established by numerical simulation method. Eventually, the dynamic characteristics of a single-loop gear system are acquired by calculating two dynamic indexes of the system under linear and weakly nonlinear states. The simulation results show that the bond graph method can accurately describe the mathematical model of single-loop gear train and master the dynamic characteristics of complex gear train. This will provide a reference for the structural design and dynamic characteristics of the transmission system.

Research on dynamic characteristic of planetary gear system based on bond graph method

The dynamic characteristics of planetary system directly affect the stable state of the whole transmission system. Firstly, an accurate dynamic model of planetary gear system is established by using bond graph method. Formerly, the state equation of the gear train dynamics is obtained based on the bond graph model. On this basis, the simulation model of the gear system was acquired by using the numerical simulation method, and four dynamic indexes including zero-pole diagram, Bode diagram, Nyquist diagram and Nichols diagram of the gear dynamics system were gained. Finally, the analysis results show that the accurate mathematical model is described by the bond graph method and the dynamic characteristics is comprehended of planetary gear system, while will provide a basis for the stability research of the transmission system.

헬리컬 기어의 3차원 진동 해석을 위한 동역학 모델 개발

In this study, a dynamic model of a parallel-axis helical gear system is developed. A linear time-invariant dynamic model of a gear system is proposed. This model is constructed by using a three-dimensional model of a helical gear pair as the beam element in combination with a finite element shaft structure. Bearing flexibility is included in the model as a linear spring, and the torsional, transverse (line of action), rotational, and axial vibratory motions of the gear pair are also considered. For the gear pair model, the static transmission error is regarded as the excitation of the gear mesh, under the assumption that the gear mesh stiffness is a constant value. To validate the proposed model in predicting gear vibratory motions, the experimental results of the helical gear system are presented and compared with the predicted results. The predicted results are found to agree well with the experimental results in terms of natural frequencies and forced vibrations.

Reliability modeling of gear system considering strength degradation

The traditional reliability models of gear system were usually developed without considering strength degradation of gear tooth. Although some analysis methods considered strength degradation, but they ignored the uncertainty of strength degradation that is due to the random load. In order to consider the statistically dependent failure between the gears and the uncertainty of gear strength in the strength degradation process, the reliability models of gear system are developed based on the theory of stress - strength interference. The failure relationships between teeth are expressed by the minimum order statistics. The uncertainty of the strength degradation is expressed by the distribution of material parameter. Using the Monte Carlo simulation, the proposed model is validated. Taking a planetary gear system for a case study, the system reliabilities with strength degradation are analyzed. The results show that the strength degradation has a little influence on the gear system reliability at the early stage and it has a great influence on the gear system reliability at the middle and late stages.

Multi-objective optimal design of modification for helical gear

Vibration and noise produced in the process of gear transmission and uneven distribution load on tooth surface seriously affect the transmission performance of gear system. Gear modification is a valid technology to solve this problem. Gear meshing contact analysis considering gear modification, assembling errors and processing errors can accurately obtain the meshing performance under the simulated real working condition. On the one hand, it can provide load distribution on tooth surface, on the other hand, it can provide data for dynamic characteristic analysis. Therefore, taken gear meshing contact analysis as the link, the connection between gear modification, load distribution on tooth surface and dynamic model of gear system is established. A multi-objective modification optimal design method for helical gear considering vibration/noise reduction and load even distribution on tooth surface is proposed. An example is provided to illustrate the proposed method. The results show that the root mean square value after optimization reduces nearly 4 times and the load difference between left and right tooth surfaces after optimization reduces nearly 320 times, the modification optimization effect is very obvious.

Investigation on the influence of heat on the dynamic characteristics of a gear transmission system

In order to study the influence of heat on dynamic characteristics of gear transmission system, the meshing thermal stiffness of gear was proposed, and it was introduced into the system dynamics equation. Meanwhile, the equivalent viscous damping of the system was studied by using the Coulomb friction damping model. Thus, the thermal dynamics model of system was established. In this article, the thermal dynamics model of gear system was established and solved. The comparative analysis of the thermal dynamic response and elastic dynamic response of the system was also provided. The results show that the maximum peak value, vibration velocity, and equilibrium position of thermal dynamic response are all bigger than those of the elastic dynamic response. The maximum dynamic load is found at the alternating point of double teeth region to single tooth region, and the maximum thermal dynamic load and thermal dynamic load coefficient are smaller than those of the elastic system because of the increases of the equivalent viscous damping. Additionally, the theoretical model is verified by finite element method and experimental method, respectively.

Dynamic Modeling of Gear System Based on 3D Finite Element Model and Its Application in Spalling Fault Analysis

A reduced-order dynamic model, based on three-dimensional (3D) finite element model (FEM) and component modal synthesis technique (CMS), was presented for simulating the dynamic behavior of the spur gear system. The gear shaft and gear body were established via 3D elements to simulate bending and torsion of the gear system. The CMS technique was used to generate a reduced-order model of a spur gear system. A pair of mating teeth was assimilated to two different foundations (one for the pinion tooth and the other one for the gear tooth) linked in series by some independent springs, which was used to simulate the contact stiffness. The validity of the proposed model was verified by static analysis, dynamic analysis, and experimental analysis. The results show that the proposed model is an effective model. In addition, the proposed model has also been applied to analyze spur gear spalling faults. The results show that the dynamic response of the gear system is periodic vibration shock response due to the alternate meshing of single and double teeth. When the spalling fault occurs, some shock responses with significantly enhanced amplitude will be generated as the result of contact loss.

Research on Meshing Stiffness and Vibration Response of Pitting Fault Gears with Different Degrees

In order to study the influence of pitting on meshing stiffness, the normal distribution function is used to simulate the pitting location of pitting gear, and the potential energy method is used to analyze the influence of pitting on meshing stiffness. At the same time, the meshing stiffness of pitting gears with different degrees is analyzed by finite element method, and the validity of the calculation results with potential energy method is verified. On the basis of meshing stiffness, the dynamic model of gear system is established, and the vibration response of pitting gear system with different degrees is analyzed. The results show that with the increase of pitting area, the meshing stiffness decreases; the closer the meshing area of the driving wheel is to the pitting line, the more the meshing stiffness decreases, resulting in the intensification of vibration response and periodic impact; and in the time history diagram, there is a small spurious frequencies near the meshing frequency; in the phase diagrams and the Poincare diagram, trajectory and discrete point aggregation area is gradually increased.

Automated requirements-driven design synthesis of gearboxes with graph-based design languages using state of the art tools

In order to best match individual customer requirements in gearbox applications, customer-tailored gearboxes are a desirable solution. That leads to the demand for a truly individualized product development. In order to keep costs down despite an increasing product variability, novel digital design methods which automate the design process and avoid redundant manual work need to be developed and deployed. Graph-based design languages in UML (Unified Modeling Language) use a central data model to guarantee model consistency and to generate automatically multi-disciplinary models and simulations thereof. The paper shows the use of graph-based design languages for automated gear synthesis and the automated three dimensional arrangement of gearset parts. The aim of this language is to generate the gearset including the gear calculation. The calculation automation is supported with the domain-specific synthesis tool GAP (‘Getriebe Auslegungs Programm’). The resulting gearset is then packaged into a given design space provided as one of the customer requirements. The package problem is the subject of several boundary conditions and for this reasons normally multiple solutions do exist. Therefore, optimization methods are used in the following to select the final three-dimensional arrangement. The current state of the gear system model also includes a fully parametric CAD housing. The approach shown in this paper can be extended to the entire product life cycle, also including domain-specific and already established software solutions. Finally, the opportunity is present to include further product assessments like a cost calculation and a first reliability check of the system. The authors are aware of the fact that this fully automated process is currently still restricted to a specific class of product design problems such as gears and gearbox design.

Reliability Modeling and Estimation of the Gear System

Traditional reliability models of the gear system either simplify the gear system unreasonably or are too complicated to be applicable for some gear systems. Therefore, a reliability model of the gear system considering statistically dependent failure based on the theory of the order statistics is developed in this paper. Firstly, the gear bending fatigue test of small sample is implemented to obtain the P-S-N curves that are fitted by using the least square method and the linear regression method of the statistical parameters under the test stress levels. Then, according to stress-strength interference theory and the theory of the order statistics, the reliability models considering the number of load cycles from a gear to gear system are built, which indicates clear modeling process of the gear system. Finally, the proposed reliability model of gear system is validated by using the Monte Carlo Simulation (MCS) and the system reliability under special load history is analyzed qualitatively.

Quasi-static analysis of the nonlinear behavior of a railway vehicle gear system considering time-varying and stochastic excitation

The gearboxes of machines generally operate under a time-varying state rather than under steady-state conditions. However, it is difficult to investigate the nonlinear dynamics of a time-varying gear system. A gear system model of a railway vehicle was proposed in consideration of its time-varying mesh stiffness, nonlinear backlash, transmission error, time-varying external excitation, and rail irregularity. To obtain the nonlinear behaviors of a time-varying stochastic gear system, a quasi-static analysis was performed to observe its doubling-periodic bifurcation, chaotic motion, and transition from a lower to a higher power periodic motion. Based on the energy comparison results, the time-varying stochastic gear system was degraded to a time-varying system to simplify the calculation. Furthermore, the nonlinear response of the time-varying system was computed using the Runge–Kutta method and was compared with the results of a quasi-static analysis that employed a short-time Fourier transform method. The results of the quasi-static analysis were consistent with the results of the time–frequency analysis for the time-varying gear system except for the result at 3180 r/min, which represented a short period wherein the process transitioned to chaos. Hence, the comparison demonstrates the applicability of the quasi-static analysis for the nonlinear behavior analysis of a time-varying stochastic system.

Nonlinear time-varying dynamic analysis of a spiral bevel geared system

In this paper, a nonlinear time-varying dynamic model of a drivetrain composed of a spiral bevel gear pair, shafts and bearings is developed. Gear shafts are modeled by utilizing Timoshenko beam finite elements, and the mesh model of a spiral bevel gear pair is used to couple them. The dynamic model includes the flexibilities of shaft bearings as well. Gear backlash and time variation of mesh stiffness are incorporated into the dynamic model. Clearance nonlinearity of bearings is assumed to be negligible, which is valid for preloaded rolling element bearings. Furthermore, stiffness fluctuations of bearings are disregarded. Multi-term harmonic balance method (HBM) is applied on the system of nonlinear differential equations in order to obtain a system of nonlinear algebraic equations. Utilizing receptance method, system of nonlinear algebraic equations is grouped in nonlinear and linear sets of algebraic equations where the nonlinear set can be solved alone decreasing the number of equations to be solved significantly. This reduces the computational effort drastically which makes it possible to use finite element models for gear shafts. In the calculation of Fourier coefficients, continuous-time Fourier transform as opposed to the gear dynamics studies that utilize discrete Fourier Transform is used. Thus, convergence problems that arise when the number of nonlinear DOFs is large are avoided. Moreover, analytical integration is employed for the calculation of Fourier coefficients rather than numerical integration in order to further reduce the computational time required. Nonlinear algebraic equations obtained are solved by utilizing Newton’s method with arc-length continuation. Direct numerical integration is employed to verify the solutions obtained by HBM. Several case studies are carried out, and the influence of backlash amount, fluctuation of gear mesh stiffness and variation of bearing stiffness are investigated. In addition to these, the response of the coupled gear system model is compared with that of gear torsional model in order to study the influence of the coupling on dynamics of the system.

Dynamic response of a single-mesh gear system with periodic mesh stiffness and backlash nonlinearity under uncertainty

Parametric uncertainties play a critical role in the response predictions of a gear system. However, accurately determining the effects of the uncertainty propagation in nonlinear time-varying models of gear systems is awkward and difficult. This paper improves the interval harmonic balance method (IHBM) to solve the dynamic problems of gear systems with backlash nonlinearity and time-varying mesh stiffness under uncertainties. To deal with the nonlinear problem including the fold points and uncertainties, the IHBM is improved by introducing the pseudo-arc length method in combination with the Chebyshev inclusion function. The proposed approach is demonstrated using a single-mesh gear system model, including the parametrically varying mesh stiffness and the gear backlash nonlinearity, excited by the transmission error. The results of the improved IHBM are compared with those obtained from the scanning method. Effects of parameter uncertainties on its dynamic behavior are also discussed in detail. From various numerical examples, it is shown that the results are consistent meanwhile the computational cost is significantly reduced. Furthermore, the proposed approach could be effectively applied for sensitivity analysis of the system response to parameter variations.

Tooth profile modification based on lateral- torsional-rocking coupled nonlinear dynamic model of gear system

In this paper, the influence of tooth profile modification on the vibration of a single gear is studied. Considering geometric eccentricity, gyroscopic moment, teeth to the partial load torque and backlash, a 10 degree of freedom (DOF) lateral-torsional-rocking coupled nonlinear dynamic model of a single gear system is proposed. Considering actual motion state and tooth profile modification, stiffness of the meshing model was studied using an analytical method. Results show influence of the modification amount with different modification types and different modification length on dynamic load. For the case of different modification lengths and modification types, with increasing modification amount, profile of dynamic load coefficient follows a ‘V’ type pattern. An optimal modification amount can be determined, where dynamic load of gear can be minimized. At the same time, when modification amount exceeds a certain critical value, dynamic load for modified gear was higher than the dynamic load for gear without modification. The optimum modification can effectively slow mutation of meshing stiffness and reduce the dynamic load of gear; the meshing force amplitude of modification gear is decreased significantly in the high order frequency, while the impact load caused by the alternation of single and double teeth is slowed down, and contact rate is basically unchanged.

Rigid-flex coupling modeling and simulation on multistage gear transmission system

As multistage gear transmission systems are complex and precise, the flexibility of shaft can influence the dynamic response of system. In order to study dynamic response of the system, we build the rigid model of gear system and the finite element model of the gear shaft. The rigid-flex coupling model is established with the virtual prototype technology, and a contrast between rigid model and rigid-flex coupling model is constructed. With these methods, the dynamic responses with different rotation speeds and different loading magnitudes are examined. We also analyze the influence of shaft flexibility, rotation speeds and loading magnitudes on the vibration characteristics of gear transmission systems.

볼 베어링의 강성과 스퍼기어 계의 모드 특성

Abstract : This study focuses on a detailed ball-bearing model for spur gears. The nonlinear ball-bearing stiffness with radial clearance is derived and calculated. The bearing stiffness is used to the 3-degree-of- freedom (DOF) spur gear system model. The mode characteristics of the gear system model are analyzed and verified by performing finite-element analysis (FEA). From the results, the bearing stiffness on the radial clearance was more sensitive under low-load conditions compared to high-load conditions. The bearing stiffness significantly affected the gear dynamics in the low-frequency region, while the mesh stiffness affected the high natural frequency. †Corresponding Author, pci@gwnu.ac.krⒸ 2016 The Korean Society of Mechanical Engineers - 기호설명 -  : 베어링 반경방향 틈새  : 기어 감쇠  : 축과 베어링의 조합 감쇠  : 볼지름  : 내륜지름  : 외륜지름  : 베어링 피치 지름e:정적 전달오차E:탄성계수EI : 굽힘정수  : 베어링에 작용하는 외력H:하중을 받는 베어링 구름요소수   : 베어링강성   : 내륜 접촉강성   : 외륜 접촉강성  : 내륜-전동체-외륜 접촉 유효 강성  : 기어 치강성  : 축과 베어링의 조합 강성n: 베어링 하중변위관계 지수