| Gears and gear transmission components are important basic components of mechanic equipments.They are widely applied to automobiles,machine tool,coal machinery,wind power equipments,aerospace,ships,railway industry and so on.With the development of industrial technology,gears need to meet high precision,miniaturization,high bearing capacity,high hardness,good surface quality,long service life and other technical indicators.To meet the requirements mentioned above,the hardened gear technique was applied to improve the hardness of gear tooth surface.Therefore,the development of gear manufacturing technology mainly concentrated on adopting new technology of hardened tooth face gear processing and improving the gears' processing efficiency.There is still a big gap between the domestic manufacturing technology of hardened tooth face gear and the foreign advanced technology at present.In order to improve domestic hard gear surface machining technology,it is essential to introduce ultrasonic machining into the manufacturing of hardened tooth face gear to form ultrasonic honing composite machining.To realize ultrasonic gear honing,one of the key problems need to be solved firstly is to design and analyze the vibration characteristics of the ultrasonic vibration system.The traditional "full-resonant" design and analysis theory requires that all parts of the ultrasonic vibration system have the same resonance frequency which is within the frequency regulation range of the system ultrasonic power supply.However,in the actual machining,the size of the gear to be machined cannot be determined by the resonance frequency of the system and the frequency adjustment range of the ultrasonic power supply is limited.Therefore,it is difficult to ensure that the resonance frequency of any size of the gear to be machined is within the frequency adjustment range of the transducer and the ultrasonic power supply.At this time,the ultrasonic honing system cannot be analyzed by using the full-resonant theory.Therefore,the theory of "non-resonant" design and analysis of ultrasonic transformer forms a complex ultrasonic vibration system with the gear whose structure size can not be determined arbitrarily and the horn whose structure size can be determined arbitrarily.By adjusting the structure size of the horn,the resonance frequency of the transformer composed of gear and horn is within the frequency regulation range of the system,so as to realize non-resonant ultrasonic machining of gears.However,non-resonant design and analysis theory still has two shortcomings:(1)The non-resonant theory uses one-dimensional longitudinal vibration theory to analyze the horn,while two-dimensional vibration plate theory to analyze the gear,which not only can not fully reflect the vibration characteristics of the horn and plate,but also limits the size of the two;(2)because the horn and plate are analyzed by one-dimensional and two-dimensional vibration theory respectively,this phenomenon also leads to the mismatch of coupling conditions at the joint of the horn and the gear,thus affecting the accuracy of the calculation results of the transformer.In this paper,to perfect the non-resonant theory,using Chebyshev-Ritz method based on three-dimensional elastic vibration theory was proposed for analyzing the vibration characteristics of the transformer in three-dimensional condition.The following researches have been studied:(1)Based on the Euler Bernoulli beam theory,the Adomian decomposition method is used to analyze the vibration characteristics of the linear tapered bar with circular cross-section.The analytical solution of the deflection function of the horn under the condition of complete freedom is derived,and then the natural frequency of any order of the horn is obtained.The method is more direct,simple,easy to program and easy to use in engineering.(2)Based on the three-dimensional elastic vibration theory,the vibration characteristics of tapered linear and nonlinear circular rod and tapered annular plate are analyzed by Chebyshev-Ritz method,and the analytical solutions of the elements of stiffness and mass matrix in the eigenvalue equation of rod and annular plate are derived.It is proved that the three-dimensional Chebyshev-Ritz method can obtain more accurate natural frequency of rod and annular plate in any vibration mode and has better versatility.(3)The theoretical model of the stepped annular plate is established,and the vibration characteristics of the stepped annular plate are analyzed by the three-dimensional Chebyshev-Ritz method,and the modal experiment platform is established.By comparing the calculation results of the Chebyshev-Ritz method with the experimental data and the finite element analysis data,it is proved that the three-dimensional Chebyshev Ritz method can be used to analyze the vibration characteristics of the stepped annular plate.(4)Based on the analysis results of(2)and(3),a three-dimensional coupling model of the transformer is established.At the joint of the horn and the plate,the three-dimensional displacement coupling is carried out.The eigenvalue equation of the transformer and the analytical solution of each element in the stiffness and mass matrix of the equation are derived.It is proved that the Chebyshev-Ritz method has excellent convergence to analyze the vibration characteristics of the transformer and establish the hammering method platform.At the same time,the finite element method is used to analyze the modal of the transformer.Because the calculation results of Chebyshev-Ritz method are similar to the experimental results and the finite element simulation results,the effectiveness of the method is proved,and the non-resonant theory is initially advanced to the full three-dimensional level. |
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