# Bifurcation characteristics of two stage planetary gear system varying with excitation frequency

The input torque tin = 1 000 n · m, the output torque tout = – 100 N · M. here, the relative displacement δ I SP1 between the first stage sun gear s I and the first stage planetary gear P I 1 is taken as an example to analyze. It is verified that other tooth pairs are in the same condition. The variable step size gill method is used for numerical solution [16]. Taking the continuously changing dimensionless excitation frequency of the system Ω 1 = ω Ⅰ M / ω D; Ω 2 = ω Ⅱ M / ω D (where Ω 2 = Ω 1 (1 + Z Ⅱ R / Z Ⅱ s)) as the independent variable, the bifurcation data and the maximum Lyapunov exponent spectrum of the system are obtained. Taking Ω 1 as the abscissa, the bifurcation diagram and the maximum Lyapunov exponent spectrum of the system are drawn, as shown in Fig. 1 and Fig. 2 respectively.

According to the calculation, only one of the 54 Lyapunov exponents of the system is positive, so nine Lyapunov exponents of the system are selected in Figure 2. Because only one exponent of the system is positive, the system does not appear super chaotic state. It can be seen from Figure 1 that the system shows extremely rich nonlinear dynamic characteristics. When the dimensionless excitation frequency Ω of the system is in the range of [0.5, 0.528], the system is in a chaotic state. It can be seen from Figure 2 that λ Max switches between positive and negative, which indicates that the system switches between chaos and steady state, and periodic and quasi periodic motions will appear intermittently during this period. Because of the low rotation speed, the single side contact and double side contact between the meshing gears of the system exist at the same time, and the system motion is more complex. In order to analyze the motion state of the system in this interval, take Ω = 0.511, and the phase trajectory diagram, Poincare section and power spectrum of the system are shown in Figure 3

See Figure 3 It can be seen that the system is in a chaotic state, the phase trajectories are intertwined and crossed, disordered and unclosed, and tend to fill a part of the phase space in the phase diagram; the Poincare section is a strange attractor with fractal structure, which is stretched and folded, and has topological self similar structural characteristics, and the power spectrum is a continuous spectrum with obvious noise background At the same time, there are spikes at 1 / 16 and M / 16 fundamental frequencies, which indicates the strict periodicity of the system accessing 16 different regions of strange attractors. The maximum Lyapunov exponent of the system under this excitation frequency is 0.01844, and λ Max > 0, which verifies that the system is in chaos.

With the increase of excitation frequency, when the dimensionless excitation frequency Ω is in the range of [0.529, 0.595], the system enters into a periodic state, and λ Max is negative, indicating that the system is in a steady state. In the interval of [0.596, 0.648], the system goes into chaotic state, there is periodic motion in the interval, and there is paroxysmal chaos in the interval. The positive and negative switching in Lyapunov exponent spectrum indicates that the system switches between steady state and chaos. In the interval of [0.649, 0.74], the system enters the periodic state, λ Max < 0. In the interval of [0.741, 0.763], the system enters into quasi periodic state, and λ Max fluctuates near zero within the allowable error range. In the interval of [0.764, 0.829], the system goes into periodic state, λ Max < 0. In the interval [0.83, 0.857], the system goes into quasi periodic state. In the interval of [0.858, 0.946], the system goes into periodic state, λ Max < 0.

At the frequency of 0.946, the chaotic attractor suddenly appears and the system changes from periodic state to chaotic state. 946, 0. 947). The phase trajectory, Poincare cross section and power spectrum of periodic state in this region are shown in Figure 4. When Ω = 0.946, the system is in 16 periodic motion state, the Poincare cross section is 16 fixed points, the power spectrum is distributed on the discrete points of H ·Ω B / 16 (where h is a positive integer and Ω B is the fundamental frequency), and the maximum Lyapunov exponent of the system is λ max = – 0.003 117 < 0; when Ω = 0.947, the system is in chaotic state, and the system’s λ max = 0.010 82 > 0, which can verify the state of the system. The phase trajectory, Poincare section and power spectrum of the chaotic state (Ω = 0. 947) in this interval are shown in Figure 5.

The gear backlash of multi-stage planetary gear train is the main factor that causes the transition. The existence of backlash will make the meshing tooth pair lose contact in the process of gear transmission. Therefore, when the contact is restored, the collision will occur, causing severe vibration.

When the excitation frequency is in the range of [0.947, 1.01], the system goes into chaotic state, and λ Max switches between positive and negative, which indicates that the system switches between stable state and chaos, and there is a “window” with periodic behavior during this period. When Ω = 0.96, the system changes from chaos state to 32 periodic motion state (Ω = 0.97, as shown in Figure 6), and λ max = – 0.002 95 < 0. Then, when Ω = 0.971, the system enters chaos state through quasi periodic path (Ω = 0.976, as shown in Figure 7).

In the range of [1.011, 1.099], the system bifurcates from chaos to 16 periodic state, which indicates that [1.1, 1.154] system enters into quasi periodic state. [1. 155, 1. 489] the system enters the periodic state.

When Ω = 1.49, Hopf bifurcation occurs and the system bifurcates from periodic motion to quasi periodic motion. The phase trajectory diagram, Poincare cross section and power spectrum of quasi periodic state (Ω = 1. 496) are shown in Figure 8. The Poincare cross section consists of two closed curves composed of several discrete points. There are peaks in a series of irreducible frequency components and their linear combinations in the power spectrum.

When the excitation frequency is in the range of [1.52, 1.54], the system enters into periodic motion.

Through the qualitative analysis of the global bifurcation and Poincare cross section of the above excitation frequencies and the quantitative verification of Lyapunov exponent, we can obtain the excitation frequency ranges [0.5, 0.528], [0.947, 0.96] and [0.974, 1.01] of the chaotic motion, as well as the bifurcation point of the cataclysm of the system In order to avoid chaos and bifurcation point, the system can be in a stable working state.

When Ω = 0.511, 0.947 and 0.976, the system goes into chaos, and strange attractors with fractal structure appear in Poincare section. Correlation dimension can describe the degree of correlation between phase points, and it is easy to be realized on computer. The strange attractor has fractal structure, and all the points are stretched and folded. Therefore, the correlation dimension of the strange attractor in chaotic state is a fractal dimension. The GP algorithm proposed by Grassberger and Procaccia is used to calculate the LN cn ∼ ln ξ curve of the system attractor under the above two parameter settings. As shown in Figure 8, the slopes of the straight line are 1.435, 1.633 and 1.789 respectively, the correlation dimensions of the strange attractor are both fractional, and the correlation dimensions of the system in the quasi periodic state are 1 011。

Scroll to Top