#### 期刊菜单

Nonlinear Dynamic Analysis of Hysteresis Model of Magnetorheological Damper
DOI: 10.12677/AEPE.2021.95030, PDF, HTML, XML, 下载: 434  浏览: 1,055

Abstract: Magneto-Rheological Damper (MRD) has strong nonlinear characteristics such as yield and hystere-sis saturation, which makes the magneto-rheological (MR) suspension system produce complex nonlinear dynamic behavior under uneven road excitation, which is one of the important con-straints affecting the practical application of MR suspension system. Based on the 2-DoF suspension model with improved skyhook control, this paper analyzes the influence of Sigmoid model, modified Bouc-Wen model and S model on the chaotic response of MR suspension system under harmonic road excitation. The results show that the nonlinear dynamic behavior of the suspension system based on different MRD hysteresis models is different under the same road excitation. The chaotic suppression ability of the suspension system based on Sigmoid model is better than that of the modified Bouc-Wen suspension system and the S suspension system when the road excitation changes.

1. 前言

MRD作为智能控制的执行器或被控对象应用于车辆半主动悬架系统，具有可控性强、频率响应高、阻尼力可调范围广、低成本低能耗等特点 [1]，在动态特性上更能满足减振防护的要求，因此MR悬架系统具有良好的发展前景。

2. MRD滞环模型

${F}_{\text{d}}={f}_{\text{i}}\left(i\right){F}_{\text{h}}\left(x,\stackrel{˙}{x},\stackrel{¨}{x}\right)$ (1)

${f}_{\text{i}}\left(i\right)=1+\frac{{k}_{2}}{1+\mathrm{exp}\left(-{a}_{2}\left(i+{I}_{0}\right)\right)}-\frac{{k}_{2}}{1+\mathrm{exp}\left(-{a}_{2}{I}_{0}\right)}$ (2)

2.1. Sigmoid模型

${F}_{\text{h}}\left(v\right)={f}_{0}\left(1+\mathrm{exp}\left({a}_{1}{v}_{\text{m}}\right)\right)\frac{1-\mathrm{exp}\left(-\sigma \left(v+{v}_{\text{h}}\right)\right)}{1+\mathrm{exp}\left(-\sigma \left(v+{v}_{\text{h}}\right)\right)}\left(1+{k}_{\text{v}}|v|\right)$ (3)

${v}_{\text{h}}=\mathrm{sgn}\left(\stackrel{¨}{x}\right){k}_{4}{v}_{\text{m}}\left(1+\frac{{k}_{3}}{1+\mathrm{exp}\left(-{a}_{3}\left(i+{I}_{1}\right)\right)}-\frac{{k}_{3}}{1+\mathrm{exp}\left(-{a}_{3}{I}_{1}\right)}\right)$ (4)

$\sigma =\frac{{a}_{0}}{1+{k}_{0}{v}_{\text{m}}}$ (5)

${k}_{\text{v}}={k}_{1}\mathrm{exp}\left(-{a}_{4}{v}_{\text{m}}\right)$ (6)

${v}_{\text{m}}=\sqrt{{\stackrel{˙}{x}}^{2}-\stackrel{¨}{x}x}$ (7)

2.2. 修正Bouc-Wen模型

Figure 1. Sigmoid model

Figure 2. Modified Bouc-Wen model

${F}_{\text{h}}={c}_{1}\stackrel{˙}{y}+{k}_{1}\left(x-{x}_{0}\right)$ (8)

$\stackrel{˙}{y}=\frac{1}{{c}_{0}+{c}_{1}}\left[\alpha z+{c}_{0}\stackrel{˙}{x}+{k}_{0}\left(x-y\right)\right]$ (9)

$\stackrel{˙}{z}=-\gamma |\stackrel{˙}{x}-\stackrel{˙}{y}|z{|z|}^{n-1}-\beta \left(\stackrel{˙}{x}-\stackrel{˙}{y}\right){|z|}^{n}+A\left(\stackrel{˙}{x}-\stackrel{˙}{y}\right)$ (10)

2.3. S型滞环模型

${F}_{\text{h}}={k}_{\text{v}}\mathrm{tan}\mathrm{sgn}\left(\sigma \left(\stackrel{˙}{x}+{\stackrel{˙}{x}}_{\text{h}}\right)\right)+{c}_{\text{p}}\stackrel{˙}{x}+kx$ (11)

$\sigma =\frac{{a}_{0}}{1+{k}_{0}{v}_{\text{m}}}$ (12)

${k}_{\text{v}}={k}_{1}\mathrm{exp}\left(-{a}_{4}{v}_{\text{m}}\right)$ (13)

${\stackrel{˙}{x}}_{\text{h}}=\mathrm{sgn}\left(\stackrel{¨}{x}\right){k}_{4}{v}_{\text{m}}$ (14)

3. 基于MRD的车辆悬架模型

3.1. 二自由度MR悬架系统模型

Figure 3. 2-DoF MR suspension system dynamics model

${m}_{\text{s}}{\stackrel{¨}{x}}_{\text{s}}+{k}_{\text{s}}\left({x}_{\text{s}}-{x}_{\text{u}}\right)+{F}_{\text{d}}=0$ (15)

${m}_{\text{s}}{\stackrel{¨}{x}}_{\text{u}}-{k}_{\text{s}}\left({x}_{\text{s}}-{x}_{\text{u}}\right)+{k}_{\text{t}}\left({x}_{\text{u}}-{x}_{\text{in}}\right)+{c}_{\text{t}}\left({\stackrel{˙}{x}}_{\text{u}}-{\stackrel{˙}{x}}_{\text{in}}\right)-{F}_{\text{d}}=0$ (16)

3.2. 基于改进Skyhook的半主动控制策略

“天棚”(Skyhook)阻尼控制策略是由Karnopp [20] 提出的一种经典车辆悬架阻尼控制方法，目前在车辆半主动悬架系统振动控制中被广泛采用。理想模型如图4(a)所示，假设将系统中的阻尼器位于簧载质量与某固定的“天棚”之间，使其作用力方向与簧载质量相反，从而抑制车身振动。理想的阻尼力为：

${f}_{\text{sky}}=-{c}_{\text{sky}}{\stackrel{˙}{x}}_{\text{s}}$ (17)

(a) (b)

Figure 4. Skyhook control. (a) Ideal skyhook damping system model; (b) Improved skyhook damping system model

${i}_{\text{d}}=\left\{\begin{array}{l}{c}_{\text{sky}}{\stackrel{˙}{x}}^{2}{}_{\text{s}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{x}}_{\text{s}}{\stackrel{˙}{x}}_{u}>0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{x}}_{\text{s}}{\stackrel{˙}{x}}_{u}\le 0\end{array}$ (18)

Figure 5. System framework of improved skyhook damping controller

4. MR悬架系统非线性动力学分析

4.1. MRD滞环模型验证

Table 1. Three hysteresis model parameters

(a) (b) (c)

Figure 6. F-v characteristic curve at different control currents. (a) Sigmoid; (b) Modified Bouc-Wen; (c) S

(a) (b) (c)

Figure 7. F-v characteristic curve with different excitation amplitudes. (a) Sigmoid; (b) Modified Bouc-Wen; (c) S

(a) (b) (c)

Figure 8. F-v characteristic curve with different excitation frequency. (a) Sigmoid; (b) Modified Bouc-Wen; (c) S

4.2. MR悬架系统非线性动力学仿真结果

1) 激励频率f变化时MR悬架系统的非线性特性

(a) (b) (c)

Figure 9. Bifurcation diagram of suspension system sprung mass displacement at amp=2cm. (a)Sigmoid; (b) Modified Bouc-Wen; (c) S

(a) (b) (c)

Figure 10. Bifurcation diagram of suspension system sprung mass displacement at amp = 8 cm. (a) Sigmoid; (b) Modified Bouc-Wen; (c) S

2) 激励幅值amp变时MR悬架系统的非线性特性

(a) (b) (c)

Figure 11. Bifurcation diagram of suspension system sprung mass displacement at f = 3.2 Hz. (a) Sigmoid; (b) Modified Bouc-Wen; (c) S

(a) (b) (c)

Figure 12. Bifurcation diagram of suspension system sprung mass displacement at f = 11.5 Hz. (a) Sigmoid; (b) Modified Bouc-Wen; (c) S

5. 结论

1) 路况较好时，MR悬架系统对频率变化较为敏感。当路面激励信号频率较小时，基于三种模型的悬架系统均能长久保持周期运动；随着路面激励信号频率增大，悬架系统会多次出现混沌振动，基于Sigmoid模型的悬架系统最先进入混沌振动，但振动幅值最小。

2) 路况较差时，基于修正Bouc-Wen模型的悬架系统提前进入混沌振动，而对Sigmoid模型悬架系统和S模型悬架系统影响较小。

3) 路面激励信号频率较小时，Sigmoid模型悬架系统能长久保持周期运动，仅在amp大于0.092 m即路况极其糟糕的情况下，才会进入混沌振动；S模型悬架系统在0.05 m~0.065 m出现混沌振动，修正Bouc-Wen悬架系统则在大于0.04 m情况下长期处于混沌振动。

4) 路面激励信号频率较大时，Sigmoid模型悬架系统在0.03 m~0.048 m区间内，S模型悬架系统在0.016 m~0.020 m、0.029 m~0.042 m、0.072 m~0.0105 m区间，修正Bouc-Wen模型悬架系统在0.020 m~0.055 m区间内均出现不同程度的混沌振动。

NOTES

*通讯作者。

 [1] Pang, H., Pei, L., Sun, C. and Gong, X. (2018) Normal Stress in Magneto-Rheological Polymer Gel under Large Am-plitude Oscillatory Shear. Journal of Rheology, 62, 1409-1418. https://doi.org/10.1122/1.5030952 [2] Zhang, H.L., Wang, E.R., Min, F.H. and Zhang, N. (2016) Hysteresis-Induced Bifurcation and Chaos in a Magneto-Rheological Suspension System under External Excitation. Chinese Physics B, 25, Article No. 030503. https://doi.org/10.1088/1674-1056/25/3/030503 [3] 付一博. 磁流变阻尼器时滞效应研究[J]. 科技创新导报, 2020, 17(9): 75-77. [4] Yang, M.G., Li, C.Y. and Chen, Z.Q. (2013) A New Simple Non-Linear Hysteretic Model for MR Damper and Verification of Seismic Response Reduction Experiment. Engineering Structures, 52, 434-445. https://doi.org/10.1016/j.engstruct.2013.03.006 [5] 王璐, 于海龙, 江民, 朱洪涛. 装甲车辆悬挂系统模糊PID控制仿真研究[J]. 噪声与振动控制, 2020, 40(5): 152-158. [6] Zhang, G., Li, Y., Yu, Y., Wang, H. and Wang, J. (2020) Modeling the Nonlinear Rheological Behavior of Magnetorheological Gel Using a Computationally Efficient Model. Smart Materials and Structures, 29, Article ID: 105021. https://doi.org/10.1088/1361-665X/aba809 [7] 李秀领, 李宏男. 磁流变阻尼器的双sigmoid模型及试验验证[J]. 振动工程学报, 2006, 19(2): 168-172. [8] 徐赵东, 李爱群, 程文瀼, 叶继红. 磁流变阻尼器带质量元素的温度唯象模型[J]. 工程力学, 2005, 22(2): 144-148. [9] 臧传相, 侯保林, 谈乐斌. 磁流变阻尼器S型滞环模型的改进及辨识[J]. 燕山大学学报, 2010, 34(5): 401-404. [10] 陶柯免. 磁流变阻尼器参数化模型的研究综述[J]. 山西建筑, 2020, 46(23): 46-48. [11] 黄苗玉, 王恩荣, 闵富红. 磁流变车辆悬架系统的混沌振动分析[J]. 振动与冲击, 2015, 34(24): 128-134. [12] 张海龙, 闵富红, 王恩荣. 磁流变阻尼器的车辆悬架系统混沌分析与控制[C]//第三十一届中国控制会议论文集. 中国合肥: 中国自动化学会控制理论专业委员会, 2012: 425-428. [13] 吴参, 王维锐, 徐博侯, 李兴林, 江伟光. 路面激励下车辆悬架滞回模型的混沌研究[J]. 浙江大学学报(工学版), 2011, 45(7): 1259-1264+1287. [14] 王皖君, 应亮, 王恩荣. 可控磁流变阻尼器滞环模型的比较[J]. 机械工程学报, 2009, 45(9): 100-108. [15] Wang, E.R., Ma, X.Q. and Su, C.Y. (2004) Generalized Asymmetric Hysteresis Model of Con-trollable Magnetorheological Damper for Vehicle Suspension Attenuation. Chinese Journal of Mechanical Engineering, 17, 301-305. https://doi.org/10.3901/CJME.2004.02.301 [16] Wen, Y.K. (1976) Method for Random Vibration of Hysteretic Systems. Journal of the Engineering Mechanics Division, 102, 249-263. https://doi.org/10.1061/JMCEA3.0002106 [17] Spencer, B.F., Dyke, S.J., Sain, M.K. and and Carlson, J.D. (1997) Phenomenological Model for Magnetorheological Dampers. Journal of Engineering Mechanics, 123, 230-238. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:3(230) [18] 杨礼康. 基于磁流变技术的车辆半主动悬挂系统理论与试验研究[D]: [博士学位论文]. 杭州: 浙江大学, 2004. [19] 宁东红, 贾志娟, 董明明, 杜海平. 车辆座椅悬架减振系统研究进展[J]. 中国计量大学学报, 2018, 29(2): 113-120. [20] Karnopp, D., Crosby, M.J. and Harwood, R.A. (1974) Vibration Control Using Semi-Active Force Generators. Journal of Engineering for Industry, 5, 619-629. https://doi.org/10.1115/1.3438373