Multi stage gear meshing excitation modeling

As an elastic mechanical system, gear system produces vibration and noise problems under dynamic excitation. The dynamic excitation is the input of the system, so it is the primary problem to determine the dynamic excitation characteristics of the gear meshing process. The dynamic excitation of gear system can be divided into external excitation and internal excitation: external excitation mainly refers to the load fluctuation of prime mover and actuator, and the determination method is the same as that of general mechanical system; internal excitation mainly refers to meshing excitation, which is caused by the time-varying number of meshing teeth, gear machining or installation error. At present, the main methods of meshing excitation modeling are: time-varying meshing stiffness, static transfer error, slice method.

In the process of transmitting power and motion, the number of teeth participating in meshing changes periodically. Under the same load, the number of teeth participating in meshing is small, the total deformation is large, and the comprehensive meshing stiffness is small, on the contrary, the comprehensive meshing stiffness is large. The time-varying meshing stiffness describes the changing law of the comprehensive meshing stiffness of the gear pair with time (or rotation angle), which makes the dynamic model of the gear system contain time-varying parameters and is a kind of parametric excitation. The key problem of meshing stiffness calculation is to determine the elastic deformation of gear teeth. The early research mainly used the method of material mechanics or elastic mechanics. In 1949, Weber regarded the gear tooth as a cantilever beam with variable cross-section. Based on the method of mechanics of materials, a calculation method of gear tooth deformation was proposed, which comprehensively considered the bending deformation, shear deformation and contact deformation. In 1981, Cornell, on the basis of Weber, added the influence of tooth root fillet geometry and tooth base elasticity on tooth deformation. In the same year, Nagaya and uematsu increased the influence of the moving speed of the load on the tooth surface on the tooth deformation. The main representative of elasticity method is Terauchi, a Japanese scholar. The basic idea of this method is to map the curvilinear boundary of the gear to a straight line boundary by conformal mapping transformation. The displacement field of the half plane can be obtained from the complex function of the concentrated force acting on the half plane, and then the displacement field of the gear can be obtained. However, Terauchi uses trial method to construct mapping function, which is a very empirical work, and it is often time-consuming and difficult to achieve high accuracy. Therefore, Chinese scholar Cheng Naishi and others have compiled a computer program to solve conformal mapping function, which makes up for the disadvantage of time-consuming and laborious trial method, and can be used to calculate the tooth deformation of various tooth profile curves. In the 1970s, people began to use the finite element method to calculate the elastic deformation and root stress of gear teeth. Compared with mechanics of materials and elasticity, finite element method can simulate more complex geometry and boundary conditions. In the finite element analysis, the meshing force is usually treated as the concentrated force acting on the node, which will produce large errors at the node. Therefore, vijayakar uses the matching surface to divide the tooth contact area into two parts: the inner part and the outer part. The finite element method and the contact mechanics method are used to calculate the overall and local deformation of the tooth respectively. When a pair of teeth produce unit deformation, the load transmitted is the meshing stiffness of a single pair of teeth. When multiple pairs of teeth mesh at the same time, the comprehensive meshing stiffness (the comprehensive meshing stiffness of gear pair) is generally obtained by directly superimposing the meshing stiffness of a single pair of teeth. The results show that the meshing stiffness of a single pair of teeth increases with the increase of load, and the meshing in and out parts are smaller in the meshing area, and the meshing in and out parts are larger in the middle part of the meshing area. However, in order to simplify the calculation, some scholars treat the meshing stiffness of a single pair of gear teeth as a constant, so the comprehensive meshing stiffness of gear pair is simplified as a square wave function. The related research includes the work of Kahraman and Blankenship, Yang et al., Lin and Parker et al. At the same time, for helical gear or herringbone gear, some scholars proposed to approximate the comprehensive meshing stiffness of gear pair by using the product of the time-varying contact line length and meshing stiffness per unit length of the gear pair.

Transfer error is caused by gear elastic deformation and machining / installation error. It is defined as the deviation between the actual position and the theoretical position of the driven gear. It is generally expressed as a periodic function along the meshing line. It is the displacement excitation in the dynamic model of the gear system. According to the system load, the transfer error can be divided into load transfer error and no-load transfer error; according to the system speed, the transfer error can be divided into static transfer error and dynamic transfer error. Velex et al. [36] constructed the gear pair meshing excitation model with the static transmission error under load and the static transmission error without load, established the dynamic model of a pair of gear pairs based on the static transmission error, and verified the effectiveness of the method under the non resonance condition by comparing with the finite element simulation results. On this basis, velex et al. Extended the research to the multi-stage gear transmission system, and theoretically deduced the tooth profile modification criterion to minimize the dynamic load of the system. Using static transmission error to construct gear meshing excitation, the accurate internal excitation of complete gear system can be obtained by full finite element method and experimental method. Tang Jinyuan put forward the conceptual model and mechanical model of static transmission error calculation, and deduced the relationship between static transmission error and gear manufacturing error, loaded deformation, dynamic load and gear geometric parameters. Tesfahunegn uses the general contact algorithm of commercial software to solve the tooth surface contact problem, and accurately calculates the static transmission error under the joint influence of time-varying meshing stiffness and tooth profile modification (error excitation). Sainte Marie uses beam element to represent gear shaft, lumped mass to represent gear, and static transmission error as meshing excitation to establish a three-dimensional dynamic model of a pair of gears. The relationship between dynamic transmission error and gear dynamic load is analyzed by using this model. The results show that when the gear is dominated by torsional vibration, the two are linearly correlated, and when the gear has obvious yaw vibration The linear relationship disappears.

Slicing method is a numerical method to simulate gear meshing excitation in three-dimensional space. In this method, the tooth is divided into several independent slices along the direction of tooth width. The width and stiffness of each slice are the same, and the shear stress of each slice is not considered. The tooth load calculated by slice method is the superposition of the micro element load vectors of each slice gear pair participating in meshing at the current moment, so it depends on the instantaneous distribution of the load on the tooth. This method can simulate any machining error and installation error of gear pair, and is suitable for spur gear and helical gear at the same time, and can simulate meshing excitation in any direction in space, so it is widely used in academic circles. Velex and maatar used slicing method to calculate gear meshing excitation, and established a full degree of freedom model of a pair of gear pairs (6 degrees of freedom for driving and driven gears respectively). The modeling ability of the model considering tooth surface machining error and gear assembly error, as well as the calculation ability of outputting static / dynamic transmission error, time-varying meshing stiffness and tooth surface load distribution were demonstrated by numerical examples. Gu et al., Liu Yanfang et al., Wei Jing et al., Liu Wen et al. Applied the slicing method to the dynamic modeling of helical fixed axis gears, and compared the static transfer error calculated by the slicing method with the results calculated by the finite element method, which verified the effectiveness of the slicing method applied to helical gears. Chen Zaigang and Shao Yimin [26, 46, 47] established the gear dynamic model with root space crack fault by slicing method, studied the dynamic response of gear under different fault degrees, and provided theoretical basis for root crack fault monitoring. Beta य़ EB et al. Put the teeth of each sheet on the same Pasternak elastic foundation, and considered the interaction between the deformation of each sheet, so that the model is closer to the actual situation. The research of gear meshing excitation calculation by slicing method also includes the work of Bruy è re and velex, pears, Wang, sunghoon, Guo Dong, Wang Qibin, Zhang Yimin and Chang Lehao.

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