基于机器学习的磁流变弹性体磁致储能模量的快速准确表征

任航; 赵丹; 董立强; 刘少刚; 杨金水

doi:10.7498/aps.73.20240482

摘要

磁流变弹性体在振动控制领域展现出巨大的潜力, 但其磁致力学性能的测量过程往往需投入较高的人工与时间成本. 本研究旨在利用机器学习方法在小样本试验数据驱动下实现磁流变弹性体磁致力学性能的快速准确预测. 基于加装可控磁场的剪切流变仪测试了磁流变弹性体 (9种配比, 4种加载频率)的磁致储能模量. 每种样品取5个测试点作为训练集并搭建支持向量回归机器学习模型, 从而表征磁流变弹性体的磁致储能模量. 结果表明, 相较于典型的理论模型, SVR模型仅使用5个样本点即可更准确表征磁流变弹性体磁致储能模量, 相关系数高达0.998. 另外, SVR模型训练时间仅为0.02 s, 可显著加速磁流变弹性体表征的进程. 更重要的是, SVR模型具有良好的泛化性, 对于不同硅油配比和不同加载频率的磁流变弹性体预测结果的相关系数仍可达 0.998 以上. 因此, 机器学习模型可实现磁流变弹性体磁致储能模量的快速准确表征, 为新型磁流变材料的研发提供参考.

关键词:

Abstract

Magnetorheological elastomers (MREs) are smart materials with a wide range of applications, particularly in reducing vibrations and noise. Traditional methods of testing their magnetically-induced properties, although thorough, are labor-intensive and time-consuming. In this work, we introduce an innovative method that harnesses machine learning to rapidly characterize MREs by using a smallest dataset, thus simplifying the characterization process. Initially, 12 types of MREs are prepared and tested on a shear rheometer with a controllable magnetic field. From these data, we strategically select five representative data points from each sample to form a training dataset. Using this dataset, we develop a support vector regression (SVR) model to characterize the magnetically-induced storage modulus of the MRE. The SVR model exhibits remarkable accuracy, with a correlation coefficient (R²) of 0.998 or higher, exceeding the precision of traditional models. The training time of this model is very brief, only 0.02 seconds, thus greatly accelerating the characterization speed of MRE. Moreover, the SVR model demonstrates strong generalization ability, maintaining a high correlation coefficient of 0.998 or greater even when silicone oil is added to the MREs or tested under various loading frequencies. In a word, the machine learning model not only accelerates the evaluation process but also provides a valuable reference for developing innovative MREs, marking a significant advancement in the field of smart materials research.

Keywords:

作者及机构信息

1.
哈尔滨工程大学机电工程学院, 哈尔滨　150000

2.
哈尔滨工程大学, 青岛创新发展基地, 青岛　266000

通信作者: 董立强, dongliqiang@hrbeu.edu.cn

基金项目: 国家自然科学基金(批准号: 52275098, 52075111, 51675111)资助的课题.

Authors and contacts

1.
School of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150000, China

2.
Qingdao Innovation and Development Base, Harbin Engineering University, Qingdao 266000, China

Corresponding author: Dong Li-Qiang, dongliqiang@hrbeu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 52275098, 52075111, 51675111).

文章全文 : translate this paragraph

参考文献

[1]	Vatandoost H, Hemmatian M, Sedaghati R, Rakheja S 2020 Compos. Part B Eng. 182 107648 Google Scholar
[2]	Nam T H, Petríková I, Marvalová B 2020 Polym. Test. 81 106272 Google Scholar
[3]	Kukla M, Warguła Ł, Talaśka K, Wojtkowiak D 2020 Materials 13 4795 Google Scholar
[4]	Agirre-Olabide I, Elejabarrieta M J 2018 Polym. Test. 66 114 Google Scholar
[5]	Zainudin A A, Yunus N A, Mazlan S A, Shabdin M K, Abdul Aziz S A, Nordin N A, Nazmi N, Abdul Rahman M A 2020 Appl. Sci. 10 1638 Google Scholar
[6]	Jaafar M F, Mustapha F, Mustapha M 2021 J. Mater. Res. Technol. 15 5010 Google Scholar
[7]	Leng D X, Zhu Z H, Liu G J, Li Y C 2022 Ocean Eng. 253 111293 Google Scholar
[8]	Jin T H, Liu Z M, Sun S S, Ren Z S, Deng L, Yang B, Christie M D, Li W H 2020 Mech. Syst. Signal Process. 135 106338 Google Scholar
[9]	刘少刚, 赵跃超, 赵丹 2019 物理学报 68 234301 Google Scholar Liu S G, Zhao Y C, Zhao D 2019 Acta Phys. Sin. 68 234301 Google Scholar
[10]	Wang Q, Chen Z X, Wang Y H, Gong N, Yang J, Li W H, Sun S S 2024 Mech. Syst. Signal Process. 208 111029 Google Scholar
[11]	胡红生, 王炅, 钱苏翔, 李延成, 沈娜, 严拱标 2011 机械工程学报 47 84 Hu H S, Wang J, Qian S X, Li Y C, Shen N, Yan G B 2011 Chin. J. Mech. Eng. 47 84
[12]	文永蓬, 孙倩, 周伟浩, 尚慧琳, 郭林生 2018 机械工程学报 54 114 Wen Y P, Sun Q, Zhou W H, Shang H L, Guo L S 2018 Chin. J. Mech. Eng. 54 114
[13]	Jolly M R, Carlson J D, Muñoz B C 1996 Smart Mater. Struct. 5 607 Google Scholar
[14]	Zhu Y S, Gong X L, Dang H, Zhang X Z, Zhang P Q 2006 Chin. J. Chem. Phys. 19 126 Google Scholar
[15]	Li W H, Zhang X Z 2010 Smart Mater. Struct. 19 035002 Google Scholar
[16]	Ivaneyko D, Toshchevikov V, Saphiannikova M, Heinrich G 2011 Macromol. Theory Simul. 20 411 Google Scholar
[17]	Ivaneyko D, Toshchevikov V, Borin D, Saphiannikova M, Heinrich G 2014 Macromol. Symp. 338 96 Google Scholar
[18]	Li W H, Zhou Y, Tian T F 2010 Rheol. Acta 49 733 Google Scholar
[19]	Gu Z R, Luo Y P, Su Z B, Zhang L Y, Ren H J, Wang Y, Luo J 2023 J. Magn. Magn. Mater. 580 170795 Google Scholar
[20]	Feng Y Y, Yang X J, Liu J G, Chen Z Q 2023 Phys. Stat. Mech. Its Appl. 621 128789 Google Scholar
[21]	Chen L, Gong X L, Li W H 2008 Polym. Test. 27 340 Google Scholar
[22]	Li Y C, Li J C, Li W H, Samali B 2013 Smart Mater. Struct. 22 035005 Google Scholar
[23]	Ahmad Khairi M H, Abd Fatah A Y, Mazlan S A, Ubaidillah U, Nordin N A, Nik Ismail N I, Choi S B, Abdul Aziz S A 2019 Int. J. Mol. Sci. 20 4085 Google Scholar
[24]	Gowda D K, Odenbach S 2023 J. Magn. Magn. Mater. 579 170856 Google Scholar
[25]	刘浩, 须颖, 罗杨泉, 肖海善 2022 机械工程学报 58 328 Google Scholar Liu H, Xu Y, Luo Y Q, Xiao S H 2022 Chin. J. Mech. Eng. 58 328 Google Scholar
[26]	孙涛, 袁健美 2023 物理学报 72 218901 Google Scholar Sun T, Yuan J M 2023 Acta Phys. Sin. 72 218901 Google Scholar
[27]	寇雯博, 董灏, 邹岷强, 韩均言, 贾西西 2021 物理学报 70 030701 Google Scholar Kou W B, Dong H, Zou M Q, Han J Y, Jia X X 2021 Acta Phys. Sin. 70 030701 Google Scholar
[28]	Goodall R E A, Lee A A 2020 Nat. Commun. 11 6280 Google Scholar
[29]	Goodall R E A, Parackal A S, Faber F A, Armiento R, Lee A A 2022 Sci. Adv. 8 eabn4117 Google Scholar
[30]	Bessa M A, Bostanabad R, Liu Z, Hu A, Apley D W, Brinson C, Chen W, Liu W K 2017 Comput. Methods Appl. Mech. Eng. 320 633 Google Scholar
[31]	Clément A, Soize C, Yvonnet J 2012 Int. J. Numer. Methods Eng. 91 799 Google Scholar
[32]	Le B A, Yvonnet J, He Q C 2015 Int. J. Numer. Methods Eng. 104 1061 Google Scholar
[33]	Shen L, Qian Q 2022 Comput. Mater. Sci. 211 111475 Google Scholar
[34]	Jung J, Kim Y, Park J, Ryu S 2022 Compos. Struct. 285 115210 Google Scholar
[35]	Huang D Z, Xu K, Farhat C, Darve E 2020 J. Comput. Phys. 416 109491 Google Scholar
[36]	Li X, Liu Z, Cui S, Luo C, Li C, Zhuang Z 2019 Comput. Methods Appl. Mech. Eng. 347 735 Google Scholar
[37]	Liu X, Yan Z, Zhong Z 2021 Int. J. Hydrog. Energy 46 22079 Google Scholar
[38]	El Said B 2023 Int. J. Solids Struct. 276 112334 Google Scholar
[39]	Li Z Q, Li X, Chen Y, Zhang C 2023 Compos. Struct. 323 117473 Google Scholar
[40]	Nguyen X B, Komatsuzaki T, Iwata Y, Asanuma H 2018 Mech. Syst. Signal Process. 101 449 Google Scholar
[41]	Wang L Z, Chen Z B, Jiang L K, Cheng L 2023 J. Magn. Magn. Mater. 570 170441 Google Scholar
[42]	Kumbhar S B, Chavan S P, Gawade S S 2018 Mech. Syst. Signal Process. 100 208 Google Scholar
[43]	Eem S H, Jung H J, Koo J H 2012 IEEE Trans. Magn. 48 3080 Google Scholar
[44]	Norouzi M, Sajjadi Alehashem S M, Vatandoost H, Ni Y Q, Shahmardan M M 2016 J. Intell. Mater. Syst. Struct. 27 1121 Google Scholar
[45]	Nguyen X B, Komatsuzaki T, Zhang N 2020 Mech. Syst. Signal Process. 141 106438 Google Scholar
[46]	Yang S, Wang P, Liu Y, Dong X, Tong Y, Zhao Y 2021 Front. Mater. 8 743716 Google Scholar
[47]	Wang Q, Dong X F, Li L Y, Ou J P 2017 Smart Mater. Struct. 26 065010 Google Scholar

施引文献

图 1 MRE制备流程

Fig. 1. Preparation process of MRE.

下载: 全尺寸图片幻灯片

图 2 MRE的磁致储能模量测试

Fig. 2. Magnetic storage modulus test of MRE.

下载: 全尺寸图片幻灯片

图 3 不同MRE样品的测试结果　(a)不同铁颗粒含量; (b) 不同硅油含量; (c) 不同加载频率

Fig. 3. Test results of different MRE samples: (a) Different iron particle content; (b) different silicon oil content; (c) different loading frequencies.

下载: 全尺寸图片幻灯片

图 4 典型的MRE力学模型　(a) 微观磁偶极子模型; (b) 宏观黏弹性模型

Fig. 4. Typical mechanical model of MRE: (a) Microscopic magnetic dipole model; (b) macroscopic viscoelastic model.

下载: 全尺寸图片幻灯片

图 5 SVR 模型机制

Fig. 5. Mechanism of SVR model.

下载: 全尺寸图片幻灯片

图 6 训练集采样过程

Fig. 6. Sampling process of training set.

下载: 全尺寸图片幻灯片

图 7 不同模型对MRE磁致储能模量的表征　(a) 磁偶极子模型与黏弹性模型; (b) SVR模型与对数模型

Fig. 7. Characterization of MRE magnetic induced storage modulus by different models: (a) Magnetic dipole model and viscoelastic model; (b) SVR model and logarithmic model.

下载: 全尺寸图片幻灯片

图 8 不同模型对不同样品预测的RMSE

Fig. 8. RMSE predicted by different models for different samples.

下载: 全尺寸图片幻灯片

图 9 不同模型对不同样品R²的对比　(a)不同铁颗粒含量; (b)不同硅油含量以及不同加载频率

Fig. 9. Comparison of different models on R² of different samples: (a) Different iron particle contents; (b) different silicon oil contents and loading frequencies.

下载: 全尺寸图片幻灯片

图 10 SVR模型对不同MRE样品的预测结果　(a)不同铁颗粒含量; (b) 不同硅油含量; (c) 不同加载频率

Fig. 10. Prediction results of SVR model for different MRE samples: (a) Different iron particle content; (b) different silicone oil content; (c) different loading frequencies.

下载: 全尺寸图片幻灯片

表 1 MER样品配比及测试工况

Table 1. Ratio and testing conditions of MRE.

	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
铁磁颗粒/%	12	15	18	21	24	27	27	27	27	27	27	27
硅油/%	5	5	5	5	5	5	0	10	15	5	5	5
加载频率/Hz	75	75	75	75	75	75	75	75	75	30	60	90

下载: 导出CSV

表 2 不同模型对S6样品的预测结果

Table 2. Prediction results of different models on S6 sample.

训练样本数量	4	5	10	20
train R²	0.999	0.999	0.999	0.999
test R²	0.878	0.998	0.998	0.998
RSME	0.112	0.0125	0.0133	0.0111

下载: 导出CSV

表 3 基于SVR模型预测不同MRE样品的RMSE与R²

Table 3. Prediction of RMSE and R² for different MRE samples based on SVR model.

样品	RMSE	R²
S1	3378	0.999
S2	5006	0.999
S3	4246	0.999
S4	6671	0.998
S5	8579	0.999
S6	8669	0.998
S7	2275	0.999
S8	6547	0.999
S9	3630	0.999
S10	17122	0.998
S11	12642	0.998
S12	10409	0.998

下载: 导出CSV

表 4 不同模型对S6样品的预测结果

Table 4. Prediction results of different models on S6 samples.

模型	磁场范围/mT	R²
磁偶极子模型	0—1000	0.836
动态黏弹性模型	0—326	0.93
四参数分数阶导数黏弹性模型	0—150	0.97
动态磁力学模型	90—178	0.99
三参数本构模型(Maxwell形式)	125—540	0.958
渗透模型	0—375	0.9
Ramberg-Osgood模型	0—500	0.9
修正Kelvin–Voigt黏弹模型	0—272	0.93
自适应光滑库仑摩擦模型	—	0.92
修正Bouc-Wen模型	0—545	0.9
非线性流变模型	0—330	0.98
SVR模型	0—1000	0.998

下载: 导出CSV

[1]	Vatandoost H, Hemmatian M, Sedaghati R, Rakheja S 2020 Compos. Part B Eng. 182 107648 Google Scholar
[2]	Nam T H, Petríková I, Marvalová B 2020 Polym. Test. 81 106272 Google Scholar
[3]	Kukla M, Warguła Ł, Talaśka K, Wojtkowiak D 2020 Materials 13 4795 Google Scholar
[4]	Agirre-Olabide I, Elejabarrieta M J 2018 Polym. Test. 66 114 Google Scholar
[5]	Zainudin A A, Yunus N A, Mazlan S A, Shabdin M K, Abdul Aziz S A, Nordin N A, Nazmi N, Abdul Rahman M A 2020 Appl. Sci. 10 1638 Google Scholar
[6]	Jaafar M F, Mustapha F, Mustapha M 2021 J. Mater. Res. Technol. 15 5010 Google Scholar
[7]	Leng D X, Zhu Z H, Liu G J, Li Y C 2022 Ocean Eng. 253 111293 Google Scholar
[8]	Jin T H, Liu Z M, Sun S S, Ren Z S, Deng L, Yang B, Christie M D, Li W H 2020 Mech. Syst. Signal Process. 135 106338 Google Scholar
[9]	刘少刚, 赵跃超, 赵丹 2019 物理学报 68 234301 Google Scholar Liu S G, Zhao Y C, Zhao D 2019 Acta Phys. Sin. 68 234301 Google Scholar
[10]	Wang Q, Chen Z X, Wang Y H, Gong N, Yang J, Li W H, Sun S S 2024 Mech. Syst. Signal Process. 208 111029 Google Scholar
[11]	胡红生, 王炅, 钱苏翔, 李延成, 沈娜, 严拱标 2011 机械工程学报 47 84 Hu H S, Wang J, Qian S X, Li Y C, Shen N, Yan G B 2011 Chin. J. Mech. Eng. 47 84
[12]	文永蓬, 孙倩, 周伟浩, 尚慧琳, 郭林生 2018 机械工程学报 54 114 Wen Y P, Sun Q, Zhou W H, Shang H L, Guo L S 2018 Chin. J. Mech. Eng. 54 114
[13]	Jolly M R, Carlson J D, Muñoz B C 1996 Smart Mater. Struct. 5 607 Google Scholar
[14]	Zhu Y S, Gong X L, Dang H, Zhang X Z, Zhang P Q 2006 Chin. J. Chem. Phys. 19 126 Google Scholar
[15]	Li W H, Zhang X Z 2010 Smart Mater. Struct. 19 035002 Google Scholar
[16]	Ivaneyko D, Toshchevikov V, Saphiannikova M, Heinrich G 2011 Macromol. Theory Simul. 20 411 Google Scholar
[17]	Ivaneyko D, Toshchevikov V, Borin D, Saphiannikova M, Heinrich G 2014 Macromol. Symp. 338 96 Google Scholar
[18]	Li W H, Zhou Y, Tian T F 2010 Rheol. Acta 49 733 Google Scholar
[19]	Gu Z R, Luo Y P, Su Z B, Zhang L Y, Ren H J, Wang Y, Luo J 2023 J. Magn. Magn. Mater. 580 170795 Google Scholar
[20]	Feng Y Y, Yang X J, Liu J G, Chen Z Q 2023 Phys. Stat. Mech. Its Appl. 621 128789 Google Scholar
[21]	Chen L, Gong X L, Li W H 2008 Polym. Test. 27 340 Google Scholar
[22]	Li Y C, Li J C, Li W H, Samali B 2013 Smart Mater. Struct. 22 035005 Google Scholar
[23]	Ahmad Khairi M H, Abd Fatah A Y, Mazlan S A, Ubaidillah U, Nordin N A, Nik Ismail N I, Choi S B, Abdul Aziz S A 2019 Int. J. Mol. Sci. 20 4085 Google Scholar
[24]	Gowda D K, Odenbach S 2023 J. Magn. Magn. Mater. 579 170856 Google Scholar
[25]	刘浩, 须颖, 罗杨泉, 肖海善 2022 机械工程学报 58 328 Google Scholar Liu H, Xu Y, Luo Y Q, Xiao S H 2022 Chin. J. Mech. Eng. 58 328 Google Scholar
[26]	孙涛, 袁健美 2023 物理学报 72 218901 Google Scholar Sun T, Yuan J M 2023 Acta Phys. Sin. 72 218901 Google Scholar
[27]	寇雯博, 董灏, 邹岷强, 韩均言, 贾西西 2021 物理学报 70 030701 Google Scholar Kou W B, Dong H, Zou M Q, Han J Y, Jia X X 2021 Acta Phys. Sin. 70 030701 Google Scholar
[28]	Goodall R E A, Lee A A 2020 Nat. Commun. 11 6280 Google Scholar
[29]	Goodall R E A, Parackal A S, Faber F A, Armiento R, Lee A A 2022 Sci. Adv. 8 eabn4117 Google Scholar
[30]	Bessa M A, Bostanabad R, Liu Z, Hu A, Apley D W, Brinson C, Chen W, Liu W K 2017 Comput. Methods Appl. Mech. Eng. 320 633 Google Scholar
[31]	Clément A, Soize C, Yvonnet J 2012 Int. J. Numer. Methods Eng. 91 799 Google Scholar
[32]	Le B A, Yvonnet J, He Q C 2015 Int. J. Numer. Methods Eng. 104 1061 Google Scholar
[33]	Shen L, Qian Q 2022 Comput. Mater. Sci. 211 111475 Google Scholar
[34]	Jung J, Kim Y, Park J, Ryu S 2022 Compos. Struct. 285 115210 Google Scholar
[35]	Huang D Z, Xu K, Farhat C, Darve E 2020 J. Comput. Phys. 416 109491 Google Scholar
[36]	Li X, Liu Z, Cui S, Luo C, Li C, Zhuang Z 2019 Comput. Methods Appl. Mech. Eng. 347 735 Google Scholar
[37]	Liu X, Yan Z, Zhong Z 2021 Int. J. Hydrog. Energy 46 22079 Google Scholar
[38]	El Said B 2023 Int. J. Solids Struct. 276 112334 Google Scholar
[39]	Li Z Q, Li X, Chen Y, Zhang C 2023 Compos. Struct. 323 117473 Google Scholar
[40]	Nguyen X B, Komatsuzaki T, Iwata Y, Asanuma H 2018 Mech. Syst. Signal Process. 101 449 Google Scholar
[41]	Wang L Z, Chen Z B, Jiang L K, Cheng L 2023 J. Magn. Magn. Mater. 570 170441 Google Scholar
[42]	Kumbhar S B, Chavan S P, Gawade S S 2018 Mech. Syst. Signal Process. 100 208 Google Scholar
[43]	Eem S H, Jung H J, Koo J H 2012 IEEE Trans. Magn. 48 3080 Google Scholar
[44]	Norouzi M, Sajjadi Alehashem S M, Vatandoost H, Ni Y Q, Shahmardan M M 2016 J. Intell. Mater. Syst. Struct. 27 1121 Google Scholar
[45]	Nguyen X B, Komatsuzaki T, Zhang N 2020 Mech. Syst. Signal Process. 141 106438 Google Scholar
[46]	Yang S, Wang P, Liu Y, Dong X, Tong Y, Zhao Y 2021 Front. Mater. 8 743716 Google Scholar
[47]	Wang Q, Dong X F, Li L Y, Ou J P 2017 Smart Mater. Struct. 26 065010 Google Scholar

[1]	王前进, 孙鹏帅, 张志荣, 张乐文, 杨曦, 吴边, 庞涛, 夏滑, 李启勇. 混合气体测量中重叠吸收谱线交叉干扰的分离解析方法. 物理学报, 2021, 70(14): 144203. doi: 10.7498/aps.70.20210286
[2]	赵丹, 王帅虎, 刘少刚, 崔进, 董立强. 磁流变液构成的类梯度结构振动传递特性. 物理学报, 2020, 69(9): 098301. doi: 10.7498/aps.69.20200326
[3]	刘少刚, 赵跃超, 赵丹. 基于磁流变弹性体多包覆层声学超材料带隙及传输谱特性. 物理学报, 2019, 68(23): 234301. doi: 10.7498/aps.68.20191334
[4]	敬玉梅, 黄少云, 吴金雄, 彭海琳, 徐洪起. 三维拓扑绝缘体antidot阵列结构中的磁致输运研究. 物理学报, 2018, 67(4): 047301. doi: 10.7498/aps.67.20172346
[5]	余洋, 米增强. 机械弹性储能机组储能过程非线性动力学模型与混沌特性. 物理学报, 2013, 62(3): 038403. doi: 10.7498/aps.62.038403
[6]	卞雷祥, 文玉梅, 李平. 磁致伸缩材料磁弹性内耗的场依赖特性及其用于磁场传感研究. 物理学报, 2010, 59(2): 883-892. doi: 10.7498/aps.59.883
[7]	张延芳, 文玉梅, 李平, 卞雷祥. 采用阶梯形弹性基底的磁致伸缩/压电复合结构磁电响应研究. 物理学报, 2009, 58(1): 546-553. doi: 10.7498/aps.58.546
[8]	蔡从中, 裴军芳, 温玉锋, 朱星键, 肖婷婷. 选择性激光烧结成型件密度的支持向量回归预测. 物理学报, 2009, 58(13): 8-S14. doi: 10.7498/aps.58.8
[9]	蔡从中, 庄魏萍, 温玉锋, 朱星键, 裴军芳, 肖婷婷. 基于拓扑结构的碱金属化合物摩尔磁化率的支持向量回归研究. 物理学报, 2009, 58(13): 272-S277. doi: 10.7498/aps.58.272
[10]	温玉锋, 蔡从中, 裴军芳, 朱星键, 肖婷婷, 王桂莲. AlON-TiN复相材料合成工艺参数的支持向量回归分析. 物理学报, 2009, 58(13): 15-S20. doi: 10.7498/aps.58.15
[11]	阳昌海, 文玉梅, 李平, 卞雷祥. 偏置磁场对磁致伸缩/弹性/压电层合材料磁电效应的影响. 物理学报, 2008, 57(11): 7292-7297. doi: 10.7498/aps.57.7292
[12]	张军峰, 胡寿松. 基于多重核学习支持向量回归的混沌时间序列预测. 物理学报, 2008, 57(5): 2708-2713. doi: 10.7498/aps.57.2708
[13]	任广军, 姚建铨, 王鹏, 张强, 张会云, 张玉萍. 液晶磁致旋光的研究. 物理学报, 2007, 56(2): 994-998. doi: 10.7498/aps.56.994
[14]	于振华, 蔡远利. 基于在线小波支持向量回归的混沌时间序列预测. 物理学报, 2006, 55(4): 1659-1665. doi: 10.7498/aps.55.1659
[15]	叶美盈, 汪晓东, 张浩然. 基于在线最小二乘支持向量机回归的混沌时间序列预测. 物理学报, 2005, 54(6): 2568-2573. doi: 10.7498/aps.54.2568
[16]	崔万照, 朱长纯, 保文星, 刘君华. 混沌时间序列的支持向量机预测. 物理学报, 2004, 53(10): 3303-3310. doi: 10.7498/aps.53.3303
[17]	王养璞, 金其淑. 热激活磁弹性弛豫的多体普适方法研究. 物理学报, 1988, 37(9): 1401-1405. doi: 10.7498/aps.37.1401
[18]	胡海昌. 橫觀各向同性弹性体的振动问题. 物理学报, 1955, 11(3): 231-238. doi: 10.7498/aps.11.231
[19]	胡海昌. 在体积力作用下橫觀各向同性弹性体的平衡问题. 物理学报, 1955, 11(3): 219-230. doi: 10.7498/aps.11.219
[20]	胡海昌. 横观各向同性的半无限弹性体的若干问题. 物理学报, 1954, 10(3): 239-258. doi: 10.7498/aps.10.239

计量

文章访问数: 1522
PDF下载量: 31
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板