| The using webbed helical gears can effectively reduce the weight of the transmission structure and avoid the large impact load in high-speed transmission of large structures.But deformation of thin web and rim can make tooth and gear body large deformation to bring the tooth contact line deviating from the ideal position that may lead to the large fluctuation of mesh stiffness in gear transmission.Thus,the vibration and noise problems appear in a transmission system.So this paper proposes a helical gear coupled with piezoelectric layer on the rim to adjust the vibration of the gear transmission.The excitation voltage of piezoelectric layer will be changed according to system demand to control the deformation of thin rim which can achieve to change the contact line length of the teeth in meshing area.Then the reasonable variation of the mesh stiffness could be guaranteed which improves reliability and stability of the helical gear transmission.The main research contents of the paper are summarized as follows:(1)For the characteristics of the helical cylindrical gear transmission,a kind of dynamic model with seven degrees of freedom is established,which includes the time-varying mesh stiffness and backlash.In order to analyze the variation of circumferential dynamic transmission error,the helical gear transmission system is simplified into a single degree of freedom system.The system equation of the backlash is smoothed by using the polynomial fitting.The multi-scale method is used to solve equation of system to obtain the steady-state frequency response and stability domain of primary resonance and sub-resonance.Thus the mechanism of vibration and noise of helical gear transmission system is revealed.And the fourth order Runge-Kutta method is applied to solve the system equations.The two-dimensional bifurcation diagram obtained shows that the given the fluctuation of meshing stiffness and service load can make gear transmission system appear period-doubling bifurcation,quasi periodic motion and chaotic behavior within a certain range of the excitation frequency.(2)In order to accurately calculate the helical gear meshing stiffness,a calculation method of tooth contact line is put forward based on helical gear meshing mechanism,which is proved by comparing with the existing calculation method.This method is combined with the ISO calculation rules of biggest meshing stiffness and change rule of single tooth meshing stiffness in unit length to calculate meshing stiffness of helical gear.The influence of structure parameter on mesh stiffness and load distribution coefficient is studied.The results show that the method can be used to accurately calculate the mesh stiffness of helical gear and get the influence law of transverse contact ratio and face contact ratio on fluctuation of mesh stiffness.But the method does not include the change of the load variation along the tooth contact line so that it could not be used to accurately obtain the meshing stiffness of helical gear with web.(3)Due to the inaccuracy of the gear contact line caused by the deformation of the webbed helical cylindrical gear,the complex variation of the meshing stiffness is caused.For accurate calculation of the meshing stiffness,the paper puts forward a kind of calculation method of mesh stiffness by combining finite element method with strain energy method.The finite element model of webbed helical gear transmission system is established by use of Ansys software.The change rules of meshing stiffness with the addendum coefficient and the helical angle are obtained by the method and compared with the tooth contact line analysis method.The results show that the method can get changes of the line of contact due to the deformation of the tooth and gear body and can accurately extract strain energy and load of a pair of gear to ensure the accuracy of the mesh stiffness calculation.At the same time,this method can be used to calculate the influence of the applied load,the rim and the thickness of the web on the meshing stiffness and load distribution of tooth.An accurate change rule of the meshing stiffness of helical gear with web could be obtained by using the method.(4)In order to control the meshing stiffness of webbed helical gear effectively,the paper proposes a model whose rim is layered with piezoelectric materials to realize the active control of mesh stiffness.Based on the first order shear deformation theory of thin shell,the finite element method and Lagrange equation,a dynamical model of helical gear coupled with piezoelectric layer is established.The influence of the width and thickness of the piezoelectric ring on the rim deformation and coupling force of the rim is calculated.The influence of the control voltage on the contact line and load distribution of tooth surface in the zone of action is analyzed.It is shown that the change of control voltage can cause a large change in the tooth contact line,the meshing stiffness and the bearing behavior of the gear teeth.And the meshing stiffness of the helical gear transmission system is optimized by using a kind of piecewise linear quadratic programming adaptive control method.The results show that this control method can effectively reduce the fluctuation of meshing stiffness and improve the load distribution characteristics of gear teeth under the condition of ensuring the gear transmission application condition.In this way,the application of webbed helical gear is extended.(5)According to the dynamic model of the web helical gear coupled with piezoelectric layer,considering the time delay of piezoelectric materials,a time delayed speed and displacement feedback control model of the gear transmission dynamic system is established.The steady-state amplitude response and stable region of the control system about primary resonant and sub-resonance are obtained by multi-scale method.The results show that suited control parameters can be used to effectively suppress the steady-state amplitude of primary resonance and sub harmonic resonance and increase the stability of the web helical gear transmission.But combination of the certain control parameters can also lead to augment of steady-state amplitude and cause the system unstable.According to the results of multi-scale analysis,the time domain response,phase diagram,Poincaré phase diagram and spectrum of the system equation under different control parameters are analyzed by numerical method.It reveals the proper control parameters can make the amplitude of the gear system rapidly converge to periodic solution.The unreasonable parameters will be able to lead to multiply periodic motion of steady state or even aggravating vibration. |
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