Nonlinear Dynamic Analysis of a Trochoid Cam Gear with the Tooth Profile Modification

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 d_r 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 d_r, the performance of TCG can be improved to maintain stable motion state.

Appendix

The nonlinear coupling dynamic model of translation-torsion of the TCG is established as

$$ \left\{ {\begin{array}{*{20}l} {J\ddot{u} + cf(\dot{\delta }_{1} )l_{10} + cf(\dot{\delta }_{m} )l_{m} + cf(\dot{\delta }_{f} )l_{f} + cf(\dot{\delta }_{4} )l_{40} + k(\Phi_{1} )(f(\delta_{1} ))^{10/9} l_{10} } \hfill \\ {\quad + k(\Phi_{2} )(f(\delta_{m} ))^{10/9} l_{m} + k(\Phi_{3} )(f(\delta_{f} ))^{10/9} l_{f} + k(\Phi_{4} )(f(\delta_{4} ))^{10/9} l_{40} = T_{c} + T_{m} \cos (\omega t)} \hfill \\ {m(\ddot{\xi } - \omega^{2} \xi ) + cf(\dot{\delta }_{1} )\cos \Phi_{11} + cf(\dot{\delta }_{m} )\cos \Phi_{m} - cf(\dot{\delta }_{f} )\cos \Phi_{f} - cf(\dot{\delta }_{4} )\cos \Phi_{41} } \hfill \\ {\quad + k(\Phi_{1} )(f(\delta_{1} ))^{10/9} \cos \Phi_{11} + k(\Phi_{2} )(f(\delta_{m} ))^{10/9} \cos \Phi_{m} } \hfill \\ {\quad - k(\Phi_{3} )(f(\delta_{f} ))^{10/9} \cos \Phi_{f} - k(\Phi_{4} )(f(\delta_{4} ))^{10/9} \cos \Phi_{41} = - F} \hfill \\ \end{array} } \right. , $$

(20)

where $l_{20} = - R_{0} \sin (\theta (\alpha + \Phi ))$, $l_{10} = - R_{0} \sin (\theta (\Phi ))$, $\Phi_{t} = \alpha + \Phi - (\pi /2 - \arctan y^{\prime } (x_{2} ))$, $l_{3x} = KR_{0} \cos (\Phi_{s} )$, $l_{2x} = - KR_{0} \cos (\Phi_{t} )$, $\Phi_{s} = 2\alpha - \Phi - (\pi /2 + \arctan y^{\prime } (x_{3} ))$, $f_{m} = \left\{ {\begin{array}{*{20}l} {f(\delta_{2x} ),} \hfill & {\Phi_{d} < \Phi_{2} < 2\alpha } \hfill \\ {f(\delta_{2} ),} \hfill & {\alpha < \Phi_{2} \le \Phi_{d} } \hfill \\ \end{array} } \right.$,$\Phi_{m} = \left\{ {\begin{array}{*{20}l} {\Phi_{22} ,} \hfill & {\Phi_{d} < \Phi_{2} < 2\alpha } \hfill \\ {\Phi_{21} ,} \hfill & {\alpha < \Phi_{2} \le \Phi_{d} } \hfill \\ \end{array} } \right.$, $l_{m} = \left\{ {\begin{array}{*{20}l} {l_{2x} ,} \hfill & {\Phi_{d} < \Phi_{2} < 2\alpha } \hfill \\ {l_{20} ,} \hfill & {\alpha < \Phi_{2} \le \Phi_{d} } \hfill \\ \end{array} } \right.$, $f_{f} = \left\{ {\begin{array}{*{20}l} {f(\delta_{3x} ),} \hfill & {\Phi_{d} < \Phi_{3} < 2\alpha } \hfill \\ {f(\delta_{3} ),} \hfill & {\alpha < \Phi_{3} \le \Phi_{d} } \hfill \\ \end{array} } \right.$, $\Phi_{f} = \left\{ {\begin{array}{*{20}l} {\Phi_{32} ,} \hfill & {\Phi_{d} < \Phi_{3} < 2\alpha } \hfill \\ {\Phi_{31} ,} \hfill & {\alpha < \Phi_{3} \le \Phi_{d} } \hfill \\ \end{array} } \right.$, $l_{f} = \left\{ {\begin{array}{*{20}l} {l_{3x} ,} \hfill & {\Phi_{d} < \Phi_{3} < 2\alpha } \hfill \\ {l_{30} ,} \hfill & {\alpha < \Phi_{3} \le \Phi_{d} } \hfill \\ \end{array} } \right.$, $l_{3} = KR_{0} \cos [2\alpha - \Phi - (\pi /2 + \theta_{3} )]$, $l_{30} = - R_{0} \sin (\theta_{3} )$, $l_{40} = - R_{0} \sin (\theta_{4} )$, $l_{4} = KR_{0} \cos [\alpha - \Phi - (\pi /2 + \theta_{4} )]$, $l_{3x} = KR_{0} \cos (\Phi_{s} )$, $f{ = }r_{q} - l_{\Delta }$, $F_{3} = \left\{ {\begin{array}{*{20}l} {f_{3} } \hfill & {\Phi_{d} < \Phi_{3} < 2\alpha } \hfill \\ 1 \hfill & {\alpha < \Phi_{3} \le \Phi_{d} } \hfill \\ \end{array} } \right.$, $f_{3} = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {f \le 0} \hfill \\ {1,} \hfill & {f > 0} \hfill \\ \end{array} } \right.$, $l_{f} = \left\{ {\begin{array}{*{20}l} {l_{3x} ,} \hfill & {\Phi_{d} < \Phi_{3} < 2\alpha } \hfill \\ {l_{30} ,} \hfill & {\alpha < \Phi_{3} \le \Phi_{d} } \hfill \\ \end{array} } \right.$, $\Phi_{s} = 2\alpha - \Phi - (\pi /2 + \arctan f^{\prime } (X_{1} ))$, N = 2, $\theta_{3} = \theta (\Phi - 2\alpha )$, $\theta_{4} = \theta (\Phi - \alpha )$.

According to the nonlinear system in Ref. [39], when the damping changes, the bifurcation characteristics of the system are shown in Fig. 21.

Fig. 21

Bifurcation with c

When c is 0.01 N s/mm, the system response is shown in Fig. 22, which contains high frequency vibration signal. The phase trajectory line does not form a closed curve after multi-turn winding clockwise, and the phase trajectory line is filled with part of the phase space. When the rollers begin to engage and end the engagement, the system produces a strange attractor. The Poincáre map is composed of points that are irregularly distributed in a similar trapezoidal region. The spectrogram appears as a continuous line and exhibits background noise. The above indicates that the system is in a chaotic motion state.

Fig. 22

The motion state at c = 0.01 N s/mm. a Time domain response, b phase diagram, c Poincáre map, d FFT, e partial response and f partial FFT

When c is 0.5 N s/mm, the system response is shown in Fig. 23. The time domain response curve exhibits a rule similar to periodic motion. The Poincáre map is composed of points that do not form a closed curve. The spectrogram exhibits background noise. The above indicates that the system is in a quasi-periodic motion state. Points a and b represent the points at that rollers participating in energy transfer engage in meshing and end meshing in each cycle, respectively. Assume that the phase trajectory starts from point b. After winding one circle clockwise, phase trajectory passes through point a. Then, the phase trajectory continues to wind in the clockwise direction from the outside to the inside. Finally, the phase track returns to the area near point b.

Fig. 23

The motion state at c = 0.5 N s/mm. a time domain response, b phase diagram, c Poincáre map, d FFT, e partial response, f partial phase diagram, g partial phase diagram and h partial FFT

When c is 2 N s/mm and 10 N s/mm, the system responses are shown in Figs. 24 and 25, respectively. The system time domain responses show the rule of periodic motion, and the Poincáre maps consist of finite number of points. The phase trajectory winds in a clockwise to form a closed curve. Spectrogram are the discrete spectrum without background noise. The above indicates that the system is in a stable periodic motion state.

Fig. 24

The motion state at c = 2 N s/mm. a Time domain response, b phase diagram, c Poincáre map, d FFT, e partial phase diagram and f partial FFT

Fig. 25

The motion state at c = 10 N s/mm. a Time domain response, b phase diagram, c Poincáre map, d FFT, e partial response and f partial phase diagram

When c is 10 N s/mm, the phase trajectory begins at point b and passes through point a after one circle of clockwise winding. Then after one circle of clockwise winding back to point b, phase trajectory forms one closed curve.

From the above analysis and Fig. 21, with the increase of c, after period doubling bifurcation, the system moves from chaotic motion state to stable periodic motion state, and the ability of the system to maintain stable operation is gradually enhanced. From chaotic vibration theory, the main way to chaotic vibration is paroxysmal.