Relevant bibliographies by topics / Gear-train backlash

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Gear-train backlash'

Author: Grafiati
Published: 6 September 2023

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Journal articles on the topic "Gear-train backlash"

1

Sommer, Andrew, Jim Meagher, and Xi Wu. "Gear Defect Modeling of a Multiple-Stage Gear Train." Modelling and Simulation in Engineering 2011 (2011): 1–8. http://dx.doi.org/10.1155/2011/754257.

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This study demonstrates the transient and steady state dynamic loading on teeth within a two-stage gear transmission arising from backlash and geometric manufacturing errors by utilizing a nonlinear multibody dynamics software model. Backlash between gear teeth which is essential to provide better lubrication on tooth surfaces and to eliminate interference is included as a defect and a necessary part of transmission design. Torsional vibration is shown to cause teeth separation and double-sided impacts in unloaded and lightly loaded gearing drives. Vibration and impact force distinctions between backlash and combinations of transmission errors are demonstrated under different initial velocities and load conditions. The backlash and manufacturing errors in the first stage of the gear train are distinct from those of the second stage. By analyzing the signal at a location between the two stages, the mutually affected impact forces are observed from different gear pairs, a phenomenon not observed from single pair of gears. Frequency analysis shows the appearance of side band modulations as well as harmonics of the gear mesh frequency. A joint time-frequency response analysis during startup illustrates the manner in which contact forces increase during acceleration.
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Hsu, Y.-L., and S.-G. Wang. "Minimizing angular backlash of a multistage gear train." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 216, no. 4 (April 1, 2002): 565–69. http://dx.doi.org/10.1243/0954405021520256.

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An optimization model is constructed to select the optimum reduction ratios that minimize the total angular backlash of a gear train, under constraints on the total reduction ratio and available space. It is found that a proper layout of reduction ratios has a major effect on the total angular backlash of a gear train.
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Zhu, Zengbao, Longchao Cheng, Rui Xu, and Rupeng Zhu. "Impacts of Backlash on Nonlinear Dynamic Characteristic of Encased Differential Planetary Gear Train." Shock and Vibration 2019 (May 27, 2019): 1–15. http://dx.doi.org/10.1155/2019/9347925.

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A multifreedom tensional nonlinear dynamic equation of encased differential planetary gear train with multibacklash and time-varying mesh stiffness was developed in the present research. The nonlinear dynamic response was obtained by solving the formulated nonlinear dynamic equation, and the impacts of backlash on dynamic characteristics of the gear train were then analyzed by combining time process diagram, phase diagram, and Poincaré section. The results revealed that bilateral shock in meshing teeth was caused due to smaller backlash, thus causing dramatic changes in meshing force; hence, the gears were found to be in a chaotic state. Further, during stable motion state, no contact between intermeshing teeth with bigger backlash was noticed; thus, they were in a stable quasiperiodic motion state in the absence of teeth exciting force. Therefore, in order to avoid a bilateral shock in gears as well as to maintain gear teeth lubrication, a slightly bigger backlash is required. The backlash change in any transmission stage caused significant impacts on gear force and the motion state of its own stage; however, the impact on gear force of another stage was quite small, whereas the impact on the motion state of another stage was quite large.
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Litvin, F. L., D. Vecchiato, A. Demenego, E. Karedes, B. Hansen, and R. Handschuh. "Design of One Stage Planetary Gear Train With Improved Conditions of Load Distribution and Reduced Transmission Errors." Journal of Mechanical Design 124, no. 4 (November 26, 2002): 745–52. http://dx.doi.org/10.1115/1.1515797.

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The authors propose an approach for the design of one-stage planetary gear train with reduced transmission errors, localized bearing contact and improved conditions of distribution of load between the planetary gears. The planetary gear train is considered as a multi-body mechanical system of rigid bodies. The proposed approach is based: (i) on modification of geometry of gear tooth surfaces, and (ii) minimization and equalization of the backlash between the contacting gear tooth surfaces. The modification of geometry is accomplished: (i) by double-crowning of planetary gears, and (ii) by application of screw involute surfaces of the same direction of screws for the sun and the ring gears. The proposed geometry enables: (i) predesign of parabolic function of transmission errors for reduction of noise and vibration, and (ii) a simple method of regulation and equalization of the backlash between the gear tooth.
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Dankov, A. M. "PLANETARY CONTINUOUSLY ADJUSTABLE GEAR TRAIN WITH FORCE CLOSURE OF PLANET GEAR AND CENTRAL GEAR: FROM IDEA TO DESIGN." Science & Technique 17, no. 3 (May 31, 2018): 228–37. http://dx.doi.org/10.21122/2227-1031-2018-17-3-228-237.

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One of the reasons constraining use of a continuously adjustable gear train with compound poly-sector gear wheels in general and its most perspective version that is a planetary continuously adjustable gear train in particular is its design complexity. For example, a complex design of the mechanism for regulation of transmission gear ratio is specified by the use of involute gearing that requires a presence of backlash and bottom clearances for normal functioning and, therefore, autonomous but synchronous movement of a planet gear, as well as sectors of a central gear. In order to simplify the design of especially this mechanism for continuously adjustable gear train the paper justifies a transition to a backlash-free gearing and during this process either a planet gear or sectors of central gear are forcibly moving for regulation of the transmission ratio without losing contact with the mating element. Contact constancy of interacting elements under load is ensured by their force closure which is meant to overcome an action of gearing forces. The paper describes options for implementation of backlash-free gearing as a result of planet gear and central gear force closure, each variant is characterized by selection of active (controlled) and passive (execution of force closure) gear element. In the case of gear transmission with planet pinion coaxial tooth rims it is appropriate to implement a version with involute gearing, passive planet gear and active gear sectors of central gears. In the case of gear transmission with planet pinion opposite tooth rims the most acceptable option is with the cycloidal pin wheel gear, active planet gear and passive gear sectors of central gears. The paper proposes to carry out the force closure for teeth of gear components by means of an elastic element (a spring). A method for determination of parameters for an elastic element has been described depending on a planet gear design and the paper also shows simplification level in the gear design.
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6

Joshi, Yashodhan V., and Jordan E. Kelleher. "Gear Train Mesh Efficiency Study: The Effects of an Anti-Backlash Gear." SAE International Journal of Commercial Vehicles 7, no. 1 (April 1, 2014): 271–77. http://dx.doi.org/10.4271/2014-01-1769.

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Zhang, Lianchao, Hongbo Liao, Dapeng Fan, Shixun Fan, and Jigui Zheng. "Design optimization analysis of an anti-backlash geared servo system using a mechanical resonance simulation and experiment." Mechanical Sciences 12, no. 1 (March 17, 2021): 305–19. http://dx.doi.org/10.5194/ms-12-305-2021.

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Abstract. In many mechatronic systems, gear transmission chains are often used to transmit motion and power between motors and loads, especially for light, small but large torque output systems. Gear transmission chains will inevitably bring backlash as well as elasticity of shafts and meshing teeth. All of these nonlinear factors will affect the performance of mechatronic systems. Anti-backlash gear systems can reduce the transmission error, but elasticity has to be considered too. The aim of this paper is to find the key parameters affecting the resonance and anti-resonance frequencies of anti-backlash gear systems and then to give the design optimization methods of improving performance, both from element parameters and mechanical designing. The anti-backlash geared servo system is modeled using a two-inertia approximate model; a method of computing the equivalent stiffness of anti-backlash gear train is proposed, which comprehensively considers the total backlash of transmission chain, gear mesh stiffness, gear shaft stiffness and torsional spring stiffness. With the s-domain block diagram model of the anti-backlash geared servo system, the influences of four main factors on the resonance and anti-resonance frequencies of system are analyzed by simulation according to the frequency response, and the simulation analysis results dependent on torsional spring stiffness of anti-backlash gear pair and load moment of inertia variation are verified by the experiment. The errors between simulation and experimental results are less than 10 Hz. With these simulation and experiment results, the design optimization methods of improving the resonance and anti-resonance frequencies such as designing the center distance adjusting mechanism to reduce the initial total backlash, increasing the stiffness of torsional spring and lightweight design of load are proposed in engineering applications.
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Yang, Tianfu, Shaoze Yan, Wei Ma, and Zengyao Han. "Joint dynamic analysis of space manipulator with planetary gear train transmission." Robotica 34, no. 5 (July 30, 2014): 1042–58. http://dx.doi.org/10.1017/s0263574714002045.

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SUMMARYJoints of space manipulators are usually simplified as torsional springs in modeling motion equations, and the nonlinear behaviors of the reducer in the joints are generally neglected. In this study, a dynamic model of a space manipulator that considers the joints that are transmitted through a typical 2K-H planetary gear reducer is developed using the Lagrangian method. The backlash clearances, gear tooth profile error, and time-variant meshing stiffness are integrated into the process. The simulation results show that the backlash clearances lead to the accumulation of positioning errors in the space manipulator when the joints rotate back and forth. The tooth profile error is the main cause of severe acceleration fluctuations and meshing force impacts. These fluctuations influence torque instability, which may accelerate gear system failure.
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Blanche, J. G., and D. C. H. Yang. "Cycloid Drives With Machining Tolerances." Journal of Mechanisms, Transmissions, and Automation in Design 111, no. 3 (September 1, 1989): 337–44. http://dx.doi.org/10.1115/1.3259004.

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The cycloidal speed reducer, or cycloid drive, is an epicyclic gear train in which the profile of the planet gear is an epitrochoid and the annular sun gear has rollers as its teeth. The cycloid drive has very high efficiency and small size, in comparison with a conventional gear mechanism, making it an attractive candidate for limited space applications. On the other hand, in this type of transmissions there exist two major drawbacks, namely, backlash and torque ripple. Backlash, the angle through which the output shaft can rotate when the input shaft is held fixed, has a degrading effect on the output accuracy. Torque ripple, the variation in mechanical advantage as the input shaft rotates, causes vibrations and could lead to dynamic instability of the machinery. If the cycloid drive were manufactured to the ideal dimensions, there would be no backlash nor torque ripple. However, in reality, there will always be some machining tolerances. In this paper an analytical model is developed which models the cycloid drive with machining tolerances. Consequently, the effect of machining tolerances on backlash and torque ripple are investigated. It is found that both the backlash and the torque ripple are inherent periodic functions of the input crank angle.
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Donmez, Ata, and Ahmet Kahraman. "An experimental methodology to study engine gear rattle problems." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 263, no. 3 (August 1, 2021): 3335–43. http://dx.doi.org/10.3397/in-2021-2377.

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Gear induced noise represents a major part of overall automotive drivetrain noise. Gear rattle noise is caused by strongly nonlinear dynamic behavior of the gear pair, primarily due to external torque of speed fluctuations under lightly loaded conditions. Such loading conditions cannot be generated by using the conventional gear dynamics test set-ups that employ power recirculating gearbox arrangements or conventional electric motors. In this paper, a new test set-up is introduced to emulate the actual torque/velocity fluctuations of the input and/or output members of a gear train through three-phase synchronous servo-motors. In addition to establishing backlash boundaries, a pair of absolute encoders are used to measure the relative motions of the gears as well as their impacts along the drive and coast sides flanks or gears. Torsional vibratory behavior of a gear pair is presented at different backlash values under several input/output fluctuation conditions along with the companion sound pressure measurements.
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Dissertations / Theses on the topic "Gear-train backlash"

1

Sommer, Andrew Patrick. "VIBRATION-BASED HEALTH MONITORING OF MULTIPLE-STAGE GEAR TRAIN AND DIFFERENTIAL PLANETARY TRANSMISSION INVOLVING TEETH DAMAGE AND BACKLASH NONLINEARITY." DigitalCommons@CalPoly, 2011. https://digitalcommons.calpoly.edu/theses/631.

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The objective of this thesis is to develop vibration-based fault detection strategies for on-line condition monitoring of gear transmission systems. The study divides the thesis into three sections. First of all, the local stresses created by a root fatigue crack on a pinion spur gear are analyzed using a quasi-static finite element model and non-linear contact mechanics simulation. Backlash between gear teeth which is essential to provide better lubrication on tooth surfaces and to eliminate interference is included as a defect and a necessary part of transmission design. The second section is dedicated to fixed axis power trains. Torsional vibration is shown to cause teeth separation and double-sided impacts in unloaded and lightly loaded gearing drives. The transient and steady-state dynamic loading on teeth within a two stage crank-slider mechanism arising from backlash and geometric manufacturing errors is investigated by utilizing a non-linear multi-body dynamics software model. The multi-body model drastically reduces the computation time required by finite element methods to simulate realistic operation. The gears are considered rigid with elastic contact surfaces defined by a penalty based non-linear contact formulation. The third section examines a practical differential planetary transmission which combines two inputs and one output. Planetary gears with only backlash errors are compared to those containing both backlash and tooth defects under different kinematic and loading conditions. Fast Fourier Transform (FFT) analysis shows the appearance of side band modulations and harmonics of the gear mesh frequency. A joint time-frequency analysis (JTFA) during start-up reveals the unique vibration patterns for fixed axis gear train and differential planetary gear, respectively, when the contact forces increase during acceleration.
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Prajapat, Ganesh Prasad. "Advanced modeling and control of DFIG based wind turbine systems." Thesis, 2018. http://localhost:8080/xmlui/handle/12345678/7602.

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Book chapters on the topic "Gear-train backlash"

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Karba, Bahadır, Nihat Yıldırım, Mert Vardar, Fatih Karpat, and Milan Rackov. "The Self-Compensation Approach for Backlash on Gear Train." In Machine and Industrial Design in Mechanical Engineering, 85–96. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-88465-9_6.

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Shi, Chunyang, Yushu Bian, and Zhihui Gao. "Research on Coupling Analysis of Factors Influencing Backlash in Gear Train Based on Probability Method." In Advances in Mechanism, Machine Science and Engineering in China, 117–33. Singapore: Springer Nature Singapore, 2023. http://dx.doi.org/10.1007/978-981-19-9398-5_7.

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Conference papers on the topic "Gear-train backlash"

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Zhu, Rupeng, Dongping Sheng, Fengxia Lu, Miaomiao Li, and Heyun Bao. "Modeling and Bifurcation Characteristics of Double Stage Planetary Gear Train With Multiple Clearances." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46105.

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This paper proposes a new non-linear transverse-torsional coupled model for double stage planetary gear train, and gear’s geometric eccentricity error, synthetical transmission error, time-varying meshing stiffness, sun-planet and planet-ring gear pair’s backlashes and sun gear’s bearing clearance are taken into account. The differential governing equations of motion are derived and solved by applying variable step-size Runge-Kutta numerical integration method. The system motion state is investigated systematically and qualitatively, and exhibits diverse bifurcation and chaos characteristics under different bifurcation parameters including meshing frequency, sun-planet backlash and planet-ring backlash. Analysis results showed that the increasing damping could suppress the region of chaotic motion and improve the system’s stability significantly; the route of period-doubling to chaotic motion was observed for both first and second stage’s motion state under the bifurcation parameter of meshing frequency; The routes of period doubling and crisis to chaos were identified under the bifurcation parameter of sun-planet backlash; Besides, the increasing damping ratio could split the bifurcation diagram window into different sections and strong coupling effects are generated to second stage’s motion. Several different types of routes to chaos were observed under the bifurcation parameter of planet-ring backlash including period doubling and 3T-periodic channel; Besides, it concluded that planet-ring backlash could generate a strong coupling effect to both stage’s nonlinear behavior.
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Ghassabzadeh Saryazdi, M., and M. H. Ghaffari Saadat. "Detecting Backlash and Crack in Torsional Vibration of a Gear Train System." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-39154.

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Torsional vibration of a two-stage gear train system is investigated, considering backlash and crack. Two criteria are used to detect backlash and crack for the purpose of vibration monitoring. A discretized model is considered which consists of a servomotor, a two-stage gearbox and a brake. It is shown that second moment of acceleration increases, increasing backlash. Kurtosis number diagnoses the tooth crack. Abrupt change in Kurtosis number shows that crack has been developed.
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Zhang, Q. John, and Jerry Rescigno. "Study of Transient Response of a Gear Train Under Pulsating Torque." In ASME 2000 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2000. http://dx.doi.org/10.1115/detc2000/ptg-14439.

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Abstract This paper studied the transient response of a single reduction gear train under a pulsating torque. A 4-DOF model has been used to estimate the impact load on the gear teeth. The system equations are solved under different gear backlash, inertia, stiffness and gear ratio conditions. The sensitivities of impact load versus these parameters have been summarized. The nonlinear behavior of the impact force changing from period to random due to the torque drop is investigated. Angular velocity test verified the power input curve. Angular acceleration and lock-rotor tests show the agreement between the model simulation and experimental result.
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Taheri Kahnamouei, Jalal, and Jianming Yang. "Statistical Linearization of Nonlinear Stiffness Matrix of Planetary Gear Train." In ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/imece2017-70767.

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Stochastic dynamic analysis of the planetary gear train is complicated and it becomes more challenging when the nonlinear term is considered in the equation. A backlash between gears’ teeth is one of the nonlinearity sources in the gearbox which changes the equation of random vibration to nonlinear. In this paper, the linearization of the random vibration of multi-degrees of freedom (MDOF) with nonlinear stiffness is examined for planetary gear trains. The method used to treat nonlinearity is the statistical linearization method (SL). At first, to achieve adequate accuracy, the time domain is divided into very small time intervals then SL is utilized in each time step. For each time step an equivalent linear stiffness matrix is calculated and replacs the nonlinear stiffness matrix. The comparison between equivalent stiffness and linear stiffness at each step has shown a good agreement.
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Fridman, Alexander, Olga Privalova, and Ilya Piraner. "Method of Computation for Periodic Oscillations of a High Power Density Internal Combustion Engine Drive Train in the Presence of Backlash." In ASME 2008 Internal Combustion Engine Division Spring Technical Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/ices2008-1624.

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It is well known that presence of backlash in the timing gear train of modern internal combustion engines leads to significant increase in dynamic loads and noise. Accounting for the backlash introduces non-linearity in the numerical model. Traditionally such a system is solved as an initial value problem using multi-cycle numerical integration of the equation of motion in time domain. Present work offers an efficient method for analysis of periodic oscillations of multi-mass system of complex structure by converting it into a chainlike system. At first, the method is applied to a linear system. It is also shown that this method is applicable to a nonlinear system offering significant improvement in the simulation time as compared to the traditional approach. The developed method is applied to studying dynamics of timing gear train of a high power density diesel engine.
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Blanche, J. G., and D. C. H. Yang. "Cycloid Drives With Machining Tolerances: Part I — Kinematic Analysis." In ASME 1987 Design Technology Conferences. American Society of Mechanical Engineers, 1987. http://dx.doi.org/10.1115/detc1987-0076.

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Abstract The cycloidal speed reducer, or cycloid drive, is an epicyclic gear train in which the profile of the planet gear is an epitrochoid and the annular sun gear has rollers as its teeth. The cycloid drive has very high efficiency and small size, in comparison with a conventional gear mechanism, making it an attractive candidate for limited space applications. On the other hand, in this type of transmissions there exist two major drawbacks, namely, backlash and torque ripple Backlash, the angle through which the output shaft can rotate when the input shaft is held fixed, has a degrading effect on the output accuracy. Torque ripple, the variation in mechanical advantage as the input shaft rotates, causes vibrations and could lead to dynamic instability of the machinery. If the cycloid drive were manufactured to the ideal dimensions, there would be no backlash nor torque ripple. However, in reality, there will always be some machining tolerances. In this paper an analytical model is developed which models the cycloid drive with machining tolerances. This model is used in Part II of this investigation to determine the effect of machining tolerances on backlash and torque ripple. As a result, simple and practical equations for design synthesis of this type of drives are formulated.
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Oh, Sukil, Koo-Tae Kang, Kang-Young Soh, and Jung-Ho Kim. "Whine Noise Development of Engine Timing Gear System in Heavy Duty Vehicle." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47251.

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Gear train is the most applied in the heavy duty engine timing system on the benefit of reliability, durability, timing accuracy, maintenance, and high torque transmission. But the gear train is vulnerable to rattle and whine noise due to many serial gear connections, which lead to more possibility for gear backlash impact and gear transmission errors compared with chain and toothed meshed belt timing systems. Furthermore, normal heavy duty diesel vehicles like truck and bus are well vibration isolated from engine at gear noise frequency range. Therefore, noise source control is inevitable in the developing process in heavy duty diesel vehicles. The objective of this paper is to reduce timing gear whine noise in the engine developing process for heavy duty vehicles. Main focus is modification of gear tooth shape in the proto engine developing process considering vehicle driving modes. To investigate engine timing gear whine noise, transmission error analysis was executed in accordance with proto type gears and optimal gear tooth modification was induced by transmission error analysis, engine bench, and vehicle test.
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Giagopulos, D., C. Salpistis, and S. Natsiavas. "On Some Peculiarities Encountered in the Identification of Nonlinear Gear-Pair Systems." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85006.

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This study investigates issues related to parametric identification of a typical power train component with nonlinear characteristics. In particular, a gear-pair system supported on bearings with rolling elements is selected. This model accounts for gear backlash and bearing stiffness nonlinearities. A Bayesian statistical framework is first adopted in order to estimate the optimal values of the gear and bearing model parameters. This is achieved by combining experimental information from vibration measurements with theoretical information built into a parametric mathematical model of the system. Then, characteristic numerical results are presented. The emphasis is put on explaining some of the peculiar results obtained by applying classical gradient-based optimisation methodologies for the strongly nonlinear system examined. Some serious difficulties, associated with the existence of irregular response or the coexistence of multiple motions, are first pointed out. A solution to some of these problems, through the application of a suitable genetic algorithm, is then presented.
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Wasfy, Tamer M., and Michael Lee Stark. "Multibody Dynamics Model for Predicting the Vibration Response and Transient Tooth Loads for Planetary Gear Systems." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-48814.

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A high-fidelity multibody dynamics model for predicting the transient response of planetary gear trains is presented. The model supports an arbitrary number of gears, stages and arms. The model accurately accounts for the effects of gear tooth stiffness/damping/friction and tooth backlash. The multibody system representing the system is modeled using rigid bodies, revolute joints and rotational actuators. A penalty model is used to impose the joint and normal contact constraints. The normal contact penalty stiffness and damping are used to model the tooth stiffness and damping. The contact model detects contact between discrete points on the surface of a gear tooth (master contact surface) and a polygonal surface representation of the mating gear tooth (slave contact surface). A recursive bounding box/bounding sphere contact search algorithm is used to allow fast contact detection. An asperity friction model or an elasto-hydrodynamic lubrication model can be used for the contact friction forces. The governing equations of motion are solved along with joint/constraint equations using a time-accurate explicit solution procedure. The model is partially validated by comparing its predictions of the resonant frequencies of a planetary gear train to those of a previously published steady-state dynamic model. The model can help improve the design of planetary gear boxes including increasing the range of operating speeds, torque capacity and durability.
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Kaplan, Jason A., Roger L. Fittro, Alexandrina Untaroiu, and Houston G. Wood. "Non-Linear Time-Transient Rotor Dynamic Analyses of Geared Systems." In ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/gt2015-43481.

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The ability to accurately predict the response of rotating machinery to external forces and to assess system-level stability for different modes is crucial from a reliability and preventive maintenance perspective. Geared systems, in particular, contain many complexities, which may lead to instability and even chaotic vibration behavior. No methods for determining the effects of the dynamic meshing forces on the vibrations of complete shaft/bearing systems have been proposed in the literature. Several time-transient and steady-state models for analyzing gear forces and deflections have been proposed, but they focus primarily on the dynamics of the gearbox itself and neglect vibration transmission through the remainder of the drive-train. Models that do incorporate other components of the drive-train propose simplified lumped-parameter models for the shafts and bearings. Recent models have used the finite element method to couple the lateral, torsional, and axial degrees-of-freedom of geared shaft systems to the forces and moments exchanged between the gears via stiffness matrices. Other models in literature capture the backlash non-linearity and the state-varying mesh stiffness and observe the time-transient response of the gearbox and simplified shaft/bearing structure. A finite element formulation of complete geared systems, which couples the axial, lateral, and torsional degrees-of-freedom, is developed in which the shaft is modeled with Timoshenko beam elements and captures the forces and moments due to gyroscopic effects, and rotational accelerations due to start-up. It includes the capability of modeling non-linear contact loss due to backlash clearance and parametric excitations resulting from the state-varying mesh stiffness and solves the time-transient state equations for the displacements and velocities of the shafts using the direct Runge-Kutta method.
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