外加周期激励控制齿轮传动系统安全-吸引盆侵蚀与分岔

银奎帮; 石咏荷; 郭峰

doi:10.21656/1000-0887.460071

外加周期激励控制齿轮传动系统安全-吸引盆侵蚀与分岔

doi: 10.21656/1000-0887.460071

西安铁路职业技术学院机电工程学院，西安 710026

基金项目:

西安铁路职业技术学院2025年度立项课题 XTZY25K02

详细信息

通讯作者:
银奎帮(1994—)，男，硕士(通信作者. E-mail: 18153991602@163.com)

中图分类号: O322; TB533⁺.1
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- 文章访问数: 120
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- 被引次数: 0
出版历程
- 收稿日期: 2025-04-10
- 修回日期: 2025-06-17
- 刊出日期: 2026-04-01

External Periodic Excitation Control of Gear Transmission Systems for Safety-Attraction Basin Erosion and Bifurcation

School of Mechanical and Electrical Engineering, Xi'an Railway Vocational & Technical Institute, Xi'an 710026, P.R.China

摘要

摘要: 针对高速重载齿轮系统因时变啮合刚度与齿侧间隙等强非线性因素耦合引发的失稳问题，本研究引入外加周期激励控制策略，建立单自由度齿轮传动系统动力学模型并进行数值求解.采用胞映射法，定量分析控制参数对系统安全-吸引盆的侵蚀与分岔转迁过程及吸引域占比p演化规律的影响.基于Floquet乘子分析，建立倍频系数、激励幅值与分岔阈值的定量映射关系，结合系统相图和Poincaré映射图，揭示了外加激励通过重构相空间拓扑实现稳定控制的机理，定量阐明了关键控制参数对系统全局稳定性转变的调控机制.研究表明：低频激励易诱发高周期吸引子导致运动越界失稳；高频激励触发安全-吸引盆侵蚀与分岔，其中P3S吸引子稳定，P2S吸引子经逆倍化分岔向P1S单周期安全轨道转迁；反向激励幅值破坏稳定性，而增大正向激励幅值可加速稳定化进程，最终实现P1S吸引域全域覆盖.研究结果为齿轮传动系统的振动抑制、参数优化与安全设计提供了理论支撑.
- 齿轮传动系统 /
- 非线性动力学 /
- 振动控制 /
- 安全-吸引盆
Abstract: In response to the instability problem caused by the coupling of strong nonlinear factors such as time-varying mesh stiffness and backlash in high-speed heavy-duty gear systems, an external periodic excitation control strategy was introduced, to establish and numerically solve a dynamic model for single-degree-of-freedom gear transmission systems. With the cell mapping method, the effects of control parameters on the erosion and bifurcation transition process of the system safety-attraction basin, as well as the evolution law of the attraction domain proportion p, were quantitatively analyzed. Based on the Floquet multiplier analysis, a quantitative mapping relationship between the doubling coefficient, the excitation amplitude, and the bifurcation threshold, was established. Combined with the system phase diagram and the Poincaré mapping diagram, the stable control mechanism realized through reconstruction of the phase space topology under external excitation, was revealed, and the control mechanism of key control parameters on the global stability transition of the system was quantitatively elucidated. The results shows that, the low-frequency excitation can easily induce high period attractors, leading to motion boundary instability; the high frequency excitation can trigger safety-attraction basin erosion and bifurcation, where the P3S attractor is stable and the P2S attractor undergoes inverse doubling bifurcation to transition to the P1S single period safe orbit; and the reverse excitation amplitude will destroy the system stability, while increasing the forward excitation amplitude can accelerate the stabilization process, ultimately achieving full coverage of the P1S attraction domain. The research provides a theoretical support for vibration suppression, parameter optimization, and safety design of gear transmission systems.
- gear transmission system /
- nonlinear dynamics /
- vibration control /
- safety-attraction basin

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图 1 单自由度齿轮非线性动力学模型

Figure 1. Nonlinear dynamic model for the single-degree-of-freedom gear

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图 2 倍频系数b变化时安全-吸引盆的侵蚀与分岔过程

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 2. The erosion and bifurcation process of the safety-attraction basin with the change of doubling coefficient b

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图 3 倍频系数b变化时相图及Poincaré映射图

Figure 3. The phase diagram and the Poincaré mapping diagram with the change of doubling coefficient b

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图 4 激励幅值a变化时安全-吸引盆的侵蚀与分岔过程

Figure 4. The erosion and bifurcation process of the safety-attraction basin with the change of excitation amplitude a

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图 5 激励幅值a变化时相图及Poincaré映射图

Figure 5. The phase diagram and the Poincaré mapping diagram with the change of excitation amplitude a

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表 1 齿轮几何参数

Table 1. Geometric parameters of gears

gear	number of teeth	gear module/m	pressure angle/(°)	pitch radius/m	quality/kg	moment of inertia/(kg·m²)
drive gear	20	0.002 4	20	0.024	1.081	6.24×10^-4
driven gear	90	0.002 4	20	0.108	15.372	1.09×10^-1

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表 2 安全-吸引盆标注明细

Table 2. Safety-attraction basin label details

number of cycles	P1	P2	P3	P4	P5	P6	P7	P8	chaos
safety-attraction basin	black	orange	white	yellow	purple	blue	pink	green	grey
attractor	●	×	▲	+	★	■	*	◆	—

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表 3 安全-吸引盆吸引域占比p值分布

Table 3. The distribution of p-values for the proportion of attraction zones in the safety-attraction basin

b	p	safety-attraction basin	Floquet multiplier
π/9	p_3U=0.319 9，p_4S=0.642 1，p_NS=0.037 8	P3U，P4S，PNS	(-0.479 3+0.000 0i，-1.441 8+0.000 0i)
π/8	p_3U=0.333 3，p_2S=0.643 1，p_NS=0.023 6	P3U，P2S，PNS	(-0.482 1+0.000 0i，-1.433 1+0.000 0i)
π/7	p_3U=0.446 0，p_2S=0.512 9，p_NS=0.041 0	P3U，P2S，P8S	(-0.501 9+0.000 0i，-1.376 8+0.000 0i)
π/6	p_3S=0.408 7，p_2S=0.591 3	P3S，P2S	(-0.605 9+0.000 0i，-1.140 5+0.000 0i)
π/5	p_3S=0.435 5，p_1S=0.499 9，p_6S=0.064 5	P3S，P1S，P6S	(-0.549 1+0.624 1i，-0.549 1+0.624 1i)
π/4	p_3S=0.462 5，p_1S=0.537 3	P3S，P1S	(-0.728 9+0.399 6i，-0.728 9-0.399 6i)
π/π	p_3S=0.341 9，p_2S=0.658 1	P3S，P2S	(-0.530 6+0.000 0i，-1.302 4+0.000 0i)
π/3	p_3S=0.304 4，p_2S=0.695 6	P3S，P2S	(-0.524 1+0.000 0i，-1.318 4+0.000 0i)
π/2	p_3S=0.149 7，p_1S=0.850 3	P3S，P1S	(-0.828 0+0.074 2i，-0.828 0-0.074 2i)

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表 4 安全-吸引盆吸引域占比p值分布

Table 4. The distribution of p-values for the proportion of attraction zones in the safety-attraction basin

a	p	safety-attraction basin	Floquet multiplier
-1.0	p_NU=1.0	PNU	(-0.466 3+0.000 0i，-1.481 8+0.000 0i)
-0.2	p_3U=0.367 1，p_2S=0.629 3，p_NS=0.003 6	P3U，P2S，PNS	(-0.493 0+0.000 0i，-1.401 5+0.000 0i)
0.30	p_3S=0.149 7，p_1S=0.850 3	P3S，P1S	(-0.828 0+0.074 2i，-0.828 0-0.074 2i)
0.40	p_3S=0.030 7，p_1S=0.969 3	P3S，P1S	(1.000 0+0.000 0i，0.691 0+0.000 0i)
0.42	p_5S=0.010 9，p_1S=0.989 1	P5S，P1S	(-0.677 9+0.481 1i，-0.677 9-0.481 1i)
0.43	p_1S=1.0	P1S	(-0.653 6+0.513 5i，-0.653 6-0.513 5i)

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