Nonlinear Dynamic Analysis of a Trochoid Cam Gear with the Tooth Profile Modification
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International Journal of Precision Engineering and Manufacturing https://doi.org/10.1007/s12541-020-00417-6
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Nonlinear Dynamic Analysis of a Trochoid Cam Gear with the Tooth Profile Modification Ronggang Yang1 · Bo Han2 · Jiawei Xiang1 Received: 11 February 2020 / Revised: 22 July 2020 / Accepted: 13 September 2020 © Korean Society for Precision Engineering 2020
Abstract A trochoid cam gear (TCG) is a kind of precise transmission device with high performance, such as non-backlash, high precision, low noise, etc. Generally, the tooth profile modification technique is essential to influence the dynamic performance of TCG, which has not been studied. In the present, the equations of tooth profile modification are constructed to analysis the nonlinear dynamics of the TCG. Firstly, the tooth profile modification equation is deduced according to the requirement of meshing impact reduction. Secondly, considering the time-varying principal curvature radius of tooth profile and the nonlinear relationship between forces and deformations, a translation-torsion nonlinear dynamic model is established to further construct the nonlinear model of TCG. Finally, the dynamic characteristics of the TCG with three key parameters (the damping c, the radius dr of roller and the short amplitude coefficient K) are investigated. The tooth profile modification technique proposed herein can improve the system stability. Moreover, by increasing c and K, or decreasing dr, the performance of TCG can be improved to maintain stable motion state. Keywords Trochoid cam gear · Tooth profile modification · Nonlinear dynamics · Chaos and bifurcation · System stability List of symbols α Rotation angle of the rolling circle r1 Radius of short amplitude circle R0 Radius of rolling circle O2 Center of roller Or Center of rolling circle rq Radius of the roller K Short amplitude coefficient, K = r1/R0 φ Angle at which the gear rotates φ2 Angle corresponding to the tooth profile n Number of the rollers a1, a2, a3, a4 Polynomial coefficients rp Maximum of modification rf Normal distance Δφ Angle interval corresponding to the tooth profile modification rh Radius of the tooth top arc k Meshing stiffness [1, 2]
d3, d4 Deformations of the 3rd roller and the 4th rollers respectively l10, l20, l30, l40 Distances from normals of the 1st, 2nd, 3rd and 4th rollers to center of rotation F3 Symbolic function F Force T Torque u Rotation freedom of gear ξ Translation freedom of gear c Damping coefficient Tc Clockwise torque Tm Magnitude of torque change φm, φf Angle functions of the 1st and 2nd rollers respectively lm, lf Distance functions of the 2nd and 3rd rollers to center of rotation γ Static elastic angle f1 Rotation frequency of the gear, f1 = ω/(2π) f2 Meshing frequency, f2 = nω/(2π)
* Jiawei Xiang [email protected] 1
College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou 325035, China
College of Mechanical Engineering, Yanshan University, Qin Huangdao 066004, China
2
Vol.:(0123456789
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