Modified Bouc?Wen model based on a fractional derivative for describing the hysteretic characteristics of magnetorheological elastomers
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摘要: 為了準確表征大范圍應變幅值、激勵頻率和磁場下磁流變彈性體(Magnetorheological elastomer, MRE)的力學行為，本文引入黏彈性分數階導數，提出一種描述磁流變彈性體滯回特性的分數階導數改進Bouc?Wen模型。分析了各向同性與異性MRE的微觀形貌特征，對MRE進行了性能試驗，研究發現，MRE的儲能和損耗模量隨著應變幅值(0～100%)增大先不變后減小，隨著頻率(0～100 Hz)增大而增大，隨著磁場(0～545 mT)增大而增大。在此基礎上，基于分數階導數提出改進Bouc?Wen模型，在Simulink軟件中建立仿真模型，利用Oustaloup濾波器算法對分數階導數項近似計算，對比分析驗證了改進模型的有效性，各工況下仿真數據和試驗數據的吻合度均高于98%。結果表明：改進Bouc?Wen模型能準確地模擬MRE應力應變滯回曲線，擬合精度較Bouc?Wen模型明顯提升，改進模型在較寬的應變幅值、頻率和磁場范圍內是準確有效的，為實現MRE的工程應用打下基礎。Abstract: As a new type of magnetic sensitivity smart material, magnetorheological elastomers showing a good magnetorheological effect have been broadly applied in the field of intelligent structures and devices. A viscoelastic fractional derivative element was introduced into the stress?strain relationship of magnetorheological elastomers based on the Bouc?Wen model to accurately characterize the mechanical behavior of magnetorheological elastomers under a wide range of strain amplitude, excitation frequency, and magnetic field and to make it better applied in engineering practice. Further, a modified Bouc?Wen model based on a fractional derivative was proposed to describe the hysteresis characteristics of magnetorheological elastomers. The Bouc?Wen model has good universality and can accurately describe the hysteretic characteristics of the magnetorheological elastomer’s nonlinear viscoelastic region, but it cannot accurately simulate magneto-viscoelasticity and frequency dependence. The fractional derivative can express this characteristic with fewer parameters and higher accuracy. The micromorphology characteristics of isotropic and anisotropic magnetorheological elastomers were analyzed, and the performance tests of the magnetorheological elastomers were conducted. The storage and loss modulus of the magnetorheological elastomers initially remain unchanged and then decrease with an increase in strain amplitude (0–100%). Moreover, the storage and loss modulus of the magnetorheological elastomers increase with an increase in frequency (0–100 Hz) and magnetic flux density (0–545 mT). On this basis, a modified Bouc?Wen model was proposed based on the fractional derivative. The simulation model was established using the Simulink software, and the fractional derivative part of the modified model was approximately calculated using the Oustaloup filter algorithm. The effectiveness of the modified model was verified through a comparative analysis. The fitness values of simulation and experimental data under different loading conditions are higher than 98%. Results show that the modified Bouc?Wen model can accurately simulate the stress?strain hysteresis loops of the magnetorheological elastomers, and the fitting accuracy is significantly improved compared with that of the Bouc?Wen model. The modified model is accurate and effective in a wide range of strain amplitudes, frequencies, and magnetic fields, which can lay a foundation for the engineering application of magnetorheological elastomers.
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圖 2 MRE的截面形貌。（a）CIP60%各向同性；（b）CIP80%各向同性；（c）CIP60%各向異性；（d）CIP80%各向異性
Figure 2. Cross section of MRE: (a) CIP60% isotropic; (b) CIP80% isotropic; (c) CIP60% anisotropic; (d) CIP80% anisotropy
圖 3 （a，b）旋轉流變儀；（c）MRE試件；（d）測試示意圖
Figure 3. (a, b) Rotary rheometer; (c) MRE specimen; (d) schematic of the measuring system
圖 4 MRE的儲能和損耗模量隨應變幅值的變化曲線。（a）儲能模量；（b）損耗模量
Figure 4. Storage modulus (a) and loss modulus (b) curves of MRE with strain amplitude
圖 5 MRE的儲能和損耗模量隨應變頻率的變化曲線。（a）CIP80%各向同性；（b）CIP80%各向異性；（c）CIP60%各向同性；（d）CIP60%各向異性
Figure 5. Storage and loss modulus curves of MRE with strain frequency: (a) CIP80% isotropic; (b) CIP80% anisotropic; (c) CIP60% isotropic; (d) CIP60% anisotropic
圖 6 MRE儲能和損耗模量隨磁感應強度的變化曲線。（a）儲能模量；（b）損耗模量
Figure 6. Storage modulus (a) and loss modulus (b) of MRE with magnetic flux density
圖 8 頻率響應曲線。（a）幅頻；（b）相頻
Figure 8. Frequency response curves: (a) amplitude-frequency; (b) phase-frequency
圖 11 不同幅值下的仿真與試驗數據對比。（a, b）Bouc?Wen模型；（c, d） 改進Bouc?Wen模型
Figure 11. Comparison of simulation and experimental data under different amplitudes: (a, b) Bouc?Wen model; (c, d) modified Bouc?Wen model
圖 12 不同頻率下的仿真與試驗數據對比。（a）Bouc?Wen模型；（b）改進Bouc?Wen模型
Figure 12. Comparison of simulation and test data under different frequencies: (a) Bouc?Wen model; (b) modified Bouc?Wen model
圖 13 不同磁場下的仿真與試驗數據對比。（a）Bouc?Wen模型；（b）改進Bouc?Wen模型
Figure 13. Comparison of simulation and test data under different magnetic flux densities: (a) Bouc?Wen model; (b) modified Bouc?Wen model
圖 14 各參數的變化趨勢。（a～h）不同應變幅值下；（i～p）不同磁感應強度下
Figure 14. Change trends of parameters under different strain amplitudes (a?h) and under different magnetic flux densities (i?p)
表 1 MRE的成分配比
Table 1. Composition of MRE
g Coumarone resin SA ZnO CZ RD 4010NA S NR CIP 12 1 5 0.5 3 2 3 100 190 12 1 5 0.5 3 2 3 100 506 
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表 2 Bouc?Wen模型識別結果及吻合率
Table 2. Identified values and fitness values of the Bouc?Wen model
B/ mT f/Hz xm/% fitness values k c α A γ β n 0 1 1.36 0.9730 6.1106 5.9804 0.5310 39.9773 11.3493 2.3570 0.7731 0 1 2.51 0.9681 4.4045 3.5773 0.6137 54.4966 17.0611 3.0967 0.5451 0 1 4.64 0.9565 2.6280 3.0000 0.6595 74.9881 12.5262 2.2414 0.9174 0 1 8.57 0.9467 1.8639 2.6686 0.4699 45.5637 11.5298 1.3959 0.5629 0 1 15.8 0.9397 2.7776 1.0000 0.6278 54.0000 12.8755 4.7350 0.8254 0 1 29.3 0.9430 1.0000 0.1139 0.3029 72.0000 8.0525 0.5736 1.1378 405 0.5 0.40 0.9554 9.8393 17.0504 0.1054 57.1985 15.1242 1.7420 0.1000 405 1 0.40 0.9419 11.2673 15.0000 0.4228 73.6573 14.6013 2.6526 0.5289 405 3 0.40 0.9765 7.9062 8.3759 0.0038 73.8195 9.9851 4.6420 0.8466 405 5 0.40 0.9741 11.7603 4.8326 0.0864 59.9666 11.5887 1.5288 0.3563 115 1 2.51 0.9461 4.4521 5.9309 0.4151 46.3326 12.5141 2.0746 1.3497 234 1 2.51 0.9409 4.7443 9.0000 0.4838 70.0000 13.4964 1.8649 1.3741 405 1 2.51 0.9386 6.4304 13.7396 0.6064 80.0000 6.6630 1.2085 1.3839 456 1 2.51 0.9344 6.2437 9.1639 0.3505 70.0000 18.7092 5.6384 0.6457 545 1 2.51 0.9380 6.0084 10.3366 0.2640 66.8783 16.8824 1.0650 1.2103
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表 3 改進Bouc?Wen模型識別結果及吻合率
Table 3. Identified values and fitness values of the modified Bouc?Wen model
B/ mT f/Hz xm/% fitness values k c α A γ β n p 0 1 1.36 0.9831 6.7864 4.9731 0.4624 32.5858 11.8852 1.7276 0.4847 1.0000 0 1 2.51 0.9904 9.2082 22.8462 0.6561 19.2403 20.6769 1.0023 0.5000 0.3334 0 1 4.64 0.9928 13.0716 18.9248 0.6167 10.2625 18.8049 2.7732 0.5684 0.3270 0 1 8.57 0.9884 9.8591 12.7818 0.3759 8.1462 11.4349 5.1525 0.5000 0.3984 0 1 15.8 0.9887 8.3377 4.0287 0.7254 18.0359 19.7456 1.8512 1.1993 0.6228 0 1 29.3 0.9802 2.5102 1.3608 0.3440 27.3639 8.9086 2.0680 1.3468 0.6513 405 0.5 0.40 0.9924 19.0175 34.0650 0.6236 61.4151 21.0888 1.0865 0.7105 1.0000 405 1 0.40 0.9918 19.8181 22.6808 0.5380 49.3512 19.723 2.5815 0.6600 0.9000 405 3 0.40 0.9955 18.2508 24.4390 0.5066 52.4992 2.3542 2.6633 1.2512 0.7043 405 5 0.40 0.9957 11.8541 22.7924 0.3706 69.4789 4.5770 3.4270 1.4916 0.6600 115 1 2.51 0.9881 6.2890 17.2922 0.5086 24.0852 23.4419 1.2659 0.4984 0.3998 234 1 2.51 0.9901 5.7381 22.5996 0.3092 26.5923 19.7345 3.1640 0.3708 0.4425 405 1 2.51 0.9855 8.0516 19.9349 0.3287 47.7665 20.9086 5.1609 0.4234 0.6927 456 1 2.51 0.9941 8.7869 21.4934 0.5697 31.3416 19.0964 0.5867 0.4529 0.6205 545 1 2.51 0.9938 8.1853 21.4785 0.5817 42.2406 19.0661 3.8937 0.4125 0.6129
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