A survey of Nonlinear Vibration Research on the ho

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Overview of gear nonlinear vibration research

1 significance of studying gear nonlinear vibration

curved bevel gears and hypoid gears are the most complex gear drives. It is widely used in various industrial sectors such as aviation, aerospace, transportation, machinery and instrument manufacturing. The second type of gear mechanism is mainly used in ordinary precision experimental machines, and is developing in the direction of high speed, heavy load and low weight. Due to the reduction of structural mass and the increase of transmission power, the excitation force of the system increases, and even the dynamic system contains high-energy power sources that cause strong vibration. So far, the trend of decreasing mass and damping and increasing exciting force is still expanding. The vibration of the gear system will not only produce noise and lead to the instability of the transmission system, but also make the transmission system invalid and have serious consequences. Therefore, the research on the dynamic performance of the gear device has attracted much attention

the research on the dynamics of curved bevel gears and hypoid gears can not only have a comprehensive understanding of the vibration mechanism of the gear system, so as to design a gear system with good stability and low noise, but also the reduction of dynamic stress level will improve the reliability and service life of the gear system. In fault diagnosis, because cracks are closely related to gear stiffness, transmission error and tooth surface damage, dynamics research will be of great significance to the analysis of its spectrum mechanism. 2 research hotspots and existing problems of gear nonlinear vibration 2.1 geometry and mechanics

wildhaber and Baxter made outstanding contributions to the development of curved bevel gears and hypoid gears in the 1940s. Since then, Gleason's machine tool adjustment card, tooth surface contact analysis, and the loading contact analysis, edge contact analysis and finite element analysis proposed by krenzer have attracted the attention and exploration of scholars from all over the world. This aspect has become the main content of geometric and mechanical research on curved bevel gears and hypoid gears. While studying Gleason technology, it promotes the development of gear meshing theory. Litvin broke away from Gleason's calculation principle and proposed a local synthesis method. Wu Xutang and Wang Xiaochun got a similar method from another way. The most important contribution to the design and machining theory of curved bevel gears and hypoid gears is the third-order contact theory of Chinese scholar Wang Xiaochun. Liang Guiming put forward the idea of "non-zero displacement" in the tooth shape design of curved bevel gears. It breaks through the limitation that the traditional design can only carry out height displacement when selecting the displacement coefficient, and the displacement coefficient can be optimized according to a certain meshing performance. This provides a relatively free space for the selection of modification coefficients of curved bevel gears and hypoid gears. For example, in order to reduce vibration and noise, negative displacement can be selected to increase the flexibility and coincidence of gear teeth. In terms of mechanics, Zheng Changqi and Litvin presented a loading contact analysis method respectively. In addition, it must be mentioned that Litvin proposed a concept of pre parabolic error function to absorb vibration from geometry, and summarized it into a methodology with the local synthesis method. 2.2 dynamics

there have been more in-depth studies on the structural dynamics of curved bevel gears and hypoid gears, especially on the traveling wave resonance of curved bevel gears, but there are few literatures on the dynamics of curved bevel gears and hypoid gear systems. Fang Zongde gave a method to calculate the dynamic load of hypoid gear transmission in 1994. There are many literatures on the dynamics of spur and helical gear systems, and this aspect has been a research hotspot in recent years. Although there are great differences between curved bevel gears and hypoid gears and cylindrical gears, the dynamics of cylindrical gears still has great reference significance in terms of methodology. Therefore, we mainly comment on spur gears and helical gears

(1) the vibration equation containing time-varying stiffness and transmission error

the research on gear load can be traced back to two centuries ago, but the first systematic research on gear dynamics is marked by the article "high speed gear" published by Ross in the American Gear manufacturing journal in the 1920s. The first gear dynamic vibration model was proposed by tuplin in 1950. Subsequently, several vibration models appeared, in which parametric excitation such as tooth profile error and time-varying stiffness were considered. In 1959, attia presented the dynamic load test results of a group of spur gears. After the 1960s, a large number of achievements have been made in the theory and experiment of gear dynamics. Theoretical research mainly focuses on modeling, excitation terms and solving skills

(2) gear dynamics with clearance

under the assumption of rigid support and ignoring the input and output inertia, the renewable plastic granulator is also a major energy consumer in China, which is accurately simplified to a single degree of freedom vibration equation. The vibration equation includes time-varying stiffness, transmission error and clearance function. The existence of gap makes the vibration phenomenon more complex, and produces great noise and dynamic load, especially light load noise. In 1989, comparin and Singh proposed a mathematical model of single degree of freedom constant stiffness (equivalent stiffness) with gap vibration, which was solved by harmonic balance method. Although this equation takes only one term of harmonic, its solution can distinguish between unilateral impact, bilateral impact and no impact, and how do we choose that? The following is the selection criteria: the stability of the numerical solution of the differential equation is analyzed, and the possible multivalued response of the frequency response curve at a given frequency is pointed out. Kahraman and Singh solved a vibration equation with clearance and excitation of transmission error parameters in 1991. Although this model is still a single degree of freedom model with constant stiffness, they solved it with the harmonic balance method and found the transition frequency, subharmonic resonance and chaos. These phenomena are related to the average load, alternating load, damping and gap, and it is pointed out that various impact zones are related to the average load. They also give a complete initial value diagram of numerical integration, which solves the difficulties in the solution method of numerical integration. In the same year, Kahraman and Singh established a three degree of freedom equation considering gear clearance and bearing clearance at the same time. Many regular enterprises have appeared loss degree equation. They further changed the constant stiffness in the above single degree of freedom model and three degree of freedom model to time-varying stiffness, and solved it with multi-scale method. From 1995 to 1997, they developed time-varying stiffness, clearance function, transmission error and external excitation force into multiple harmonics, which were solved by harmonic balance method. Padmanabhan and Singh proposed a parameter continuous method for solving nonlinear differential equations based on the shooting method in 1995. This method changes the solution of periodic problems of differential equations into the solution of nonlinear eigenvalue problems by shooting method, and then solves this group of parameters by continuous method

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