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Article(id=1225751356099707721, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1225751351125263080, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.202311064, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1701273600000, receivedDateStr=2023-11-30, revisedDate=1706284800000, revisedDateStr=2024-01-27, acceptedDate=null, acceptedDateStr=null, onlineDate=1770171496653, onlineDateStr=2026-02-04, pubDate=null, pubDateStr=null, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770171496653, onlineIssueDateStr=2026-02-04, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770171496653, creator=13701087609, updateTime=1770171496653, updator=13701087609, issue=Issue{id=1225751351125263080, tenantId=1146029695717560320, journalId=1225147924628267009, year='2025', volume='38', issue='10', pageStart='2205', pageEnd='2462', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1770171495466, creator=13701087609, updateTime=1774228911890, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1242769389133611807, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1225751351125263080, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1242769389133611808, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1225751351125263080, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=2255, endPage=2263, ext={EN=ArticleExt(id=1225751356355560277, articleId=1225751356099707721, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Optimization and analysis of artificial neural network‑based model for the prediction of the seat transmissibility, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Artificial neural network modelling has been preliminarily employed to investigate effects on the biodynamic responses. In order to evaluate the vibration transmission characteristics of the seat‑occupant system, further quantitative research is needed. Drawing from a low frequency experimental investigation into whole body vibration, this study is aimed to develop an ANN model with the response surface method optimization. The age, stature, sitting height, knee height, buttock‑to‑knee, weight, gender, BMI, cushion thickness and frequency are used as network input to explore that these how to predict transmissibility from the seat base to the seat pan. Based on the interaction between hyperparameters, the mapping relationship between model hyperparameters and prediction performance indexes was established, and the optimal combination of hyperparameters was optimized and obtained. The results show that the resonance frequencies in the vertical inline and the fore‑and‑aft cross‑axis transmissibilities from seat base to seat pan decreased with increasing thickness of foam at the seat pan. BP‑ANN model has good performance in establishing the nonlinear relationship between the anthropometric, seat structure characteristics and vibration transmission characteristics of seat‑occupant system. Compared with BP‑ANN model, the error of RSM‑BP‑ANN model is reduced by 25% and 18% respectively in predicting vertical in‑line transmissibility and fore‑and‑aft cross‑axis transmissibility from seat base to seat pan. And this also provides an idea for adjusting the parameters of neural network models to improve the prediction accuracy of seat transmissibility.

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人工神经网络（ANN）已初步应用于研究人体视在质量的响应预测，但在评估人‑椅耦合系统的振动传递特性方面尚需进一步量化研究。本文以低频振动激励下的人‑椅系统振动试验为基础，寻求构建一种基于响应面法优化的反向传播人工神经网络模型（RSM‑BP‑ANN），以人员年龄、身高、坐高、膝盖高度、臀膝长度、体重、性别、BMI，以及坐垫处泡沫厚度、频率作为输入特征，探究优化后的人工神经网络模型对座椅频响函数的预测性能。基于超参数之间的交互影响建立模型超参数与预测性能指标的映射关系，优化并获取最佳超参数组合。结果显示，随着坐垫处泡沫厚度的增加，垂向同轴和水平前后向正交轴座椅频响函数的共振频率显著降低。BP‑ANN模型在建立人体体征参数、座椅结构特征与人‑椅系统振动传递特性的非线性关系方面表现出良好性能。与BP‑ANN模型相比，经过超参数优化的RSM‑BP‑ANN模型在预测垂向同轴和水平前后向正交轴座椅频响函数时的误差分别降低了25%与18%。因此，经过响应面法优化后的反向传播人工神经网络模型可以更准确地预测座椅频响函数，为快速有效地分析人‑椅系统振动传递特性提供了思路。

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张筱璐（1984―），男，博士，副教授。E‑mail：
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林森（2000―），男，硕士研究生。E‑mail：

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林森（2000―），男，硕士研究生。E‑mail：

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林森（2000―），男，硕士研究生。E‑mail：

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Journal of Vibration Engineering，2023,36(5):1234-1243., articleTitle=An approach for compressive sensing of vibration signal using BP neural network optimization, refAbstract=null)], funds=[Fund(id=1225751368779088298, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, awardId=51605010, language=CN, fundingSource=国家自然科学基金资助项目(51605010), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1225751360566640663, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, xref=1., ext=[AuthorCompanyExt(id=1225751360579223575, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, companyId=1225751360566640663, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1.College of Mechanical & Energy Engineering, Beijing University of Technology, Beijing 100124, China), AuthorCompanyExt(id=1225751360591806489, 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figureFileBig=kDylAododP52DPQ+ThFAug==, tableContent=null), ArticleFig(id=1225751367302693188, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=CN, label=图8, caption=座椅频响函数预测结果与试验测得值的对比情况, figureFileSmall=2qksqof32TIvq5LxX7SN1w==, figureFileBig=kDylAododP52DPQ+ThFAug==, tableContent=null), ArticleFig(id=1225751367457882447, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=EN, label=Fig. 9, caption=The relative importance of input on the prediction of seat transmissibilities by the RSM‑BP‑ANN model, figureFileSmall=iVCF3F4S00VwvYdPHP6z6Q==, figureFileBig=TbJ38Nl7uYMA7gtXdBheGg==, tableContent=null), ArticleFig(id=1225751367596294484, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=CN, label=图9, caption=各输入特征对RSM‑BP‑ANN模型预测座椅频响函数的相对重要性, figureFileSmall=iVCF3F4S00VwvYdPHP6z6Q==, figureFileBig=TbJ38Nl7uYMA7gtXdBheGg==, tableContent=null), ArticleFig(id=1225751367843758435, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=EN, label=Tab. 1, caption=

Box‑Behnken experiment design factors and levels

, figureFileSmall=null, figureFileBig=null, tableContent=
因素学习率动量因子第一层隐藏层节点数第二层隐藏层节点数
-10.010.345
10.030.567
), ArticleFig(id=1225751367977976173, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=CN, label=表1, caption=

Box‑Behnken试验设计因素及水平

, figureFileSmall=null, figureFileBig=null, tableContent=
因素学习率动量因子第一层隐藏层节点数第二层隐藏层节点数
-10.010.345
10.030.567
), ArticleFig(id=1225751368070250870, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=EN, label=Tab. 2, caption=

Box‑Behnken experimental sequences

, figureFileSmall=null, figureFileBig=null, tableContent=
序号ABCDY1Y2
10.010.3560.9130.871
20.030.3560.8830.854
30.010.5560.9110.907
40.030.5560.8560.853
50.020.4450.8620.852
…………………
250.020.4560.9010.879
260.020.4560.9200.893
270.020.4560.9280.912
280.020.4560.8770.912
290.020.4560.9190.891
), ArticleFig(id=1225751368191885695, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=CN, label=表2, caption=

Box‑Behnken试验序列

, figureFileSmall=null, figureFileBig=null, tableContent=
序号ABCDY1Y2
10.010.3560.9130.871
20.030.3560.8830.854
30.010.5560.9110.907
40.030.5560.8560.853
50.020.4450.8620.852
…………………
250.020.4560.9010.879
260.020.4560.9200.893
270.020.4560.9280.912
280.020.4560.8770.912
290.020.4560.9190.891
), ArticleFig(id=1225751368334492041, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=EN, label=Tab. 3, caption=

Regression equation model variance analysis （R2）（TRZZ，TRZX）

, figureFileSmall=null, figureFileBig=null, tableContent=
方差来源平方和(×10-5)自由度均方(×10-5)FP
TRZZTRZXTRZZTRZXTRZZTRZXTRZZTRZX
本文模型3900240014275.6172.34.96.0****
A862.5763.91862.5763.915.426.8****
B29.117.3129.117.30.50.6
C934.450.11934.450.116.61.8**
D60.50.9160.50.91.10.03
AB16.933.0116.933.00.31.2
AC501.00.0011501.00.0018.90.00005**
AD4.913.014.913.00.10.5
BC692.731.21692.731.212.31.1**
BD50.7527.4150.7527.40.918.5**
CD6.2242.616.2242.60.18.5*
A248.620.6148.620.60.90.7
B2164.4261.81164.4261.82.99.2*
C2493.9542.91493.9542.98.819.0***
D225.4153.1125.4153.10.55.4*
残差786.0399.41456.128.5
失拟项617.8317.41061.831.71.51.6
误差168.282.0442.020.5
总误差4600280028
), ArticleFig(id=1225751368443543953, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=CN, label=表3, caption=

回归方程模型方差分析（R2）（TRZZ、TRZX）

, figureFileSmall=null, figureFileBig=null, tableContent=
方差来源平方和(×10-5)自由度均方(×10-5)FP
TRZZTRZXTRZZTRZXTRZZTRZXTRZZTRZX
本文模型3900240014275.6172.34.96.0****
A862.5763.91862.5763.915.426.8****
B29.117.3129.117.30.50.6
C934.450.11934.450.116.61.8**
D60.50.9160.50.91.10.03
AB16.933.0116.933.00.31.2
AC501.00.0011501.00.0018.90.00005**
AD4.913.014.913.00.10.5
BC692.731.21692.731.212.31.1**
BD50.7527.4150.7527.40.918.5**
CD6.2242.616.2242.60.18.5*
A248.620.6148.620.60.90.7
B2164.4261.81164.4261.82.99.2*
C2493.9542.91493.9542.98.819.0***
D225.4153.1125.4153.10.55.4*
残差786.0399.41456.128.5
失拟项617.8317.41061.831.71.51.6
误差168.282.0442.020.5
总误差4600280028
), ArticleFig(id=1225751368531624345, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=EN, label=Tab. 4, caption=

Comparison of the predictive performance between the BP‑ANN model and the RSM‑BP‑ANN model

, figureFileSmall=null, figureFileBig=null, tableContent=
频响函数数据集BP‑ANNRSM‑BP‑ANN
R2RMSER2RMSE
TRZZ训练集0.9270.1470.9550.115
测试集0.9200.1410.9560.105
TRZX训练集0.9130.0500.9310.041
测试集0.9110.0550.9290.045
), ArticleFig(id=1225751368632287647, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1225751356099707721, language=CN, label=表4, caption=

BP‑ANN与RSM‑BP‑ANN模型预测性能比较

, figureFileSmall=null, figureFileBig=null, tableContent=
频响函数数据集BP‑ANNRSM‑BP‑ANN
R2RMSER2RMSE
TRZZ训练集0.9270.1470.9550.115
测试集0.9200.1410.9560.105
TRZX训练集0.9130.0500.9310.041
测试集0.9110.0550.9290.045
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利用响应面法优化人工神经网络的座椅频响函数预测模型与分析
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林森 1 , 张筱璐 1, 2
振动工程学报 | 2025,38(10): 2255-2263
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振动工程学报 | 2025, 38(10): 2255-2263
利用响应面法优化人工神经网络的座椅频响函数预测模型与分析
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林森1 , 张筱璐1, 2
作者信息
  • 1.北京工业大学机械与能源工程学院，北京　100124；
  • 2.北京工业大学汽车结构部件先进制造技术教育部工程研究中心，北京　100124

    • 林森（2000―），男，硕士研究生。E‑mail：

通讯作者:

张筱璐（1984―），男，博士，副教授。E‑mail：
Optimization and analysis of artificial neural network‑based model for the prediction of the seat transmissibility
Sen LIN1 , Xiaolu ZHANG1, 2
Affiliations
  • 1.College of Mechanical & Energy Engineering, Beijing University of Technology, Beijing 100124, China
  • 2.Engineering Research Center of Advanced Manufacturing Technology for Automotive Components, Ministry of Education, Beijing University of Technology, Beijing 100124, China
doi: 10.16385/j.cnki.issn.1004-4523.202311064
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摘要
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人工神经网络（ANN）已初步应用于研究人体视在质量的响应预测，但在评估人‑椅耦合系统的振动传递特性方面尚需进一步量化研究。本文以低频振动激励下的人‑椅系统振动试验为基础，寻求构建一种基于响应面法优化的反向传播人工神经网络模型（RSM‑BP‑ANN），以人员年龄、身高、坐高、膝盖高度、臀膝长度、体重、性别、BMI，以及坐垫处泡沫厚度、频率作为输入特征，探究优化后的人工神经网络模型对座椅频响函数的预测性能。基于超参数之间的交互影响建立模型超参数与预测性能指标的映射关系，优化并获取最佳超参数组合。结果显示，随着坐垫处泡沫厚度的增加，垂向同轴和水平前后向正交轴座椅频响函数的共振频率显著降低。BP‑ANN模型在建立人体体征参数、座椅结构特征与人‑椅系统振动传递特性的非线性关系方面表现出良好性能。与BP‑ANN模型相比，经过超参数优化的RSM‑BP‑ANN模型在预测垂向同轴和水平前后向正交轴座椅频响函数时的误差分别降低了25%与18%。因此，经过响应面法优化后的反向传播人工神经网络模型可以更准确地预测座椅频响函数，为快速有效地分析人‑椅系统振动传递特性提供了思路。

人‑椅系统  /  座椅频响函数  /  人工神经网络  /  响应面法

Artificial neural network modelling has been preliminarily employed to investigate effects on the biodynamic responses. In order to evaluate the vibration transmission characteristics of the seat‑occupant system, further quantitative research is needed. Drawing from a low frequency experimental investigation into whole body vibration, this study is aimed to develop an ANN model with the response surface method optimization. The age, stature, sitting height, knee height, buttock‑to‑knee, weight, gender, BMI, cushion thickness and frequency are used as network input to explore that these how to predict transmissibility from the seat base to the seat pan. Based on the interaction between hyperparameters, the mapping relationship between model hyperparameters and prediction performance indexes was established, and the optimal combination of hyperparameters was optimized and obtained. The results show that the resonance frequencies in the vertical inline and the fore‑and‑aft cross‑axis transmissibilities from seat base to seat pan decreased with increasing thickness of foam at the seat pan. BP‑ANN model has good performance in establishing the nonlinear relationship between the anthropometric, seat structure characteristics and vibration transmission characteristics of seat‑occupant system. Compared with BP‑ANN model, the error of RSM‑BP‑ANN model is reduced by 25% and 18% respectively in predicting vertical in‑line transmissibility and fore‑and‑aft cross‑axis transmissibility from seat base to seat pan. And this also provides an idea for adjusting the parameters of neural network models to improve the prediction accuracy of seat transmissibility.

seat‑occupant system  /  seat transmissibilities  /  aritificial neural network  /  response surface method
林森, 张筱璐. 利用响应面法优化人工神经网络的座椅频响函数预测模型与分析. 振动工程学报, 2025 , 38 (10) : 2255 -2263 . DOI: 10.16385/j.cnki.issn.1004-4523.202311064
Sen LIN, Xiaolu ZHANG. Optimization and analysis of artificial neural network‑based model for the prediction of the seat transmissibility[J]. Journal of Vibration Engineering, 2025 , 38 (10) : 2255 -2263 . DOI: 10.16385/j.cnki.issn.1004-4523.202311064
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交通运输环境中的各种振动会影响乘坐人员的驾乘舒适性，高强度、长时间的振动暴露甚至会引发健康问题[1]。研究和评估人‑椅系统的振动传递特性可为改善乘坐舒适性提供可靠依据，座椅频响函数（人‑椅界面处的加速度与地板上的加速度之间的传递函数）常用于量化人‑椅整体系统的振动传递特性[2]。因此，建立可表征座椅频响函数的模型并准确评估人‑椅系统振动传递特性，对于研究车辆乘坐舒适性具有重要意义。
为准确分析和预测坐姿人体的振动响应，动力学建模已被广泛用作试验的有效补充方法[3‑5]，主要包括集总参数模型和多体动力学模型。集总参数模型主要通过质量块、阻尼、弹簧等建立特定系统来评估生物动力学响应[6]；多体动力学模型则将人体系统简化为由质量、质心和转动惯量组成的刚体，并通过铰链和弹簧阻尼元件相互连接[7‑9]。在上述两种方法中，模型参数是基于测量数据或特定个体数据以及特定座椅条件获得的。因此，这些模型仅适用于特定的参数范围。然而，实测数据受个体差异性影响，具有一定的随机性，因此，传统动力学模型难以适用于多样化的人员个体体征以及座椅条件。
人工神经网络（ANN）已表现出在多输入特征影响下对复杂非线性系统建模的潜力[10‑11]，已被初步应用于预测部分条件下的坐姿人体生物动力学响应[12]。GOHARI等[13]依托ANN模型预测座椅底板到头部（seat‑to‑head transmissibility）的频响函数，该模型基于不同参试人员体征参数以及底板加速度值构建输入数据集，然而，直接以大量原始加速度作为输入特征会导致模型的输入维度较高，从而可能降低模型预测精度。在此基础上，ALSHABI等[14‑15]基于不同体征参数以及振动幅度作为输入特征改进ANN模型预测上述频响函数及视在质量（apparent mass），提高了模型的准确率。此外，坐姿人体的生物动力学反应也受座椅条件的影响[16]，因此，体征参数与座椅条件都可作为输入特征，以提高ANN模型预测视在质量的准确性。然而，由于振动响应的影响因素较为复杂，且ANN模型性能常受到各种超参数的影响，因此优化输入特征对模型预测性能的提高仍然有限。超参数的选择需要平衡模型的预测精度和计算效率，以确保模型能够准确地捕捉振动响应的特征并具有良好的泛化能力。上述研究均采用试错法确定ANN模型超参数，而试错法难以全面探索超参数空间的整体性能。此外，由于该方法集中于单个超参数的影响，忽视了不同超参数之间的交互效应，从而导致模型对于超参数组合的复杂性理解不足，无法充分挖掘其预测潜力。
现有ANN模型具备预测刚性座椅条件下的人体生物动力学响应的能力，但仅考虑人体对振动输入的响应特性，难以得到真实弹性座椅与坐姿人体的相互动态作用后的总体系统振动传递特性。座椅动力学特性被认为是座椅设计中的一个重要因素，其中开孔聚氨酯泡沫材料对座椅的振动传递特性影响尤为显著[2]。研究表明，开孔聚氨酯泡沫的动态特性取决于其结构、材料特性、预载荷、密度和厚度等因素，例如座椅泡沫刚度会随着泡沫厚度的减小而增加[17]。此外，支撑人体的弹性座椅会改变人体的生物动力学响应，当驾乘人员就坐于开孔聚氨酯泡沫座椅上时，弹性泡沫可有效减轻人体坐骨神经下方的压力并显著改变人‑椅接触面积、坐姿以及人体质量在座椅上的分布[17‑18]。因此，有必要探究和预测聚氨酯泡沫厚度对人‑椅系统振动特性的影响。
本文基于低频人体振动试验，建立反向传播人工神经网络模型，预测不同人体体征参数和座椅条件下的人‑椅系统频响函数及其正交轴效应，将响应面法引入ANN模型的超参数优化，系统研究神经网络超参数对于模型在座椅频响函数预测性能方面的影响，并建立模型超参数与性能指标之间的映射关联，为进一步探究人‑椅系统振动传递特性和改善驾乘舒适性提供参考。
1　人‑椅系统振动试验
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1.1　试验设置
振动环境下人‑椅系统动态响应的试验设置如图1所示，本试验采用本构关系相同的方块状开孔聚氨酯泡沫。靠背区域的泡沫厚度为80 mm，坐垫处的泡沫厚度分别为60、80、100 mm。在座椅底板上固定有三轴加速度计（PCB 356A16），采集振动平台的加速度信号。在座椅泡沫坐垫上布置有三轴SIT‑pad （PCB 356B41），以采集人‑椅交界面处的垂向及水平前后向加速度信号。激励信号采用Servo‑test六自由度液压振动平台模拟产生激励幅值为1 m/s²（r.m.s.）的限宽白噪声垂向激励信号（1~10 Hz）。在试验过程中，参试人员上半身直立坐于刚性座椅的海绵坐垫上，背部与靠背自然接触，如图1所示。
参试人员包括12名志愿者，年龄为24~56岁（中位数为29，标准差为10.63），身高范围160~177 cm（中位数为166，标准差为5.85），坐高81~98 cm（中位数为88，标准差为5.88），膝盖高度50~60 cm（中位数为53.5，标准差为3.29），臀膝长度49~62 cm（中位数为54.5，标准差为3.18），体重45~75 kg（中位数为61，标准差为9.16），BMI范围17.58~27.22 kg/m2（中位数为21.84，标准差为2.67）。
1.2　座椅频响函数
本文研究工作采用CSD互谱密度法计算座椅频响函数（seat transmissibility）[19]，本文共涉及两种座椅频响函数：座椅底板到坐垫处的垂向同轴频响函数TRZZ可表示为：
TRZZ=GazfaszGazf
式中，azf为座椅底板处垂向加速度输入信号；asz为坐垫处垂向加速度输出信号；Gazfasz为asz和azf的互功率谱密度；Gazf为azf的自功率谱密度。
座椅底板到坐垫处的水平前后向正交轴频响函数TRZX可表示为：
TRZX=GazfasxGazf
式中，asx为坐垫处水平前后向加速度输出信号。
图2为12名参试人员在三种坐垫处泡沫厚度条件下（60、80、100 mm）的座椅频响函数，由上至下三条曲线分别为最大值、中值及最小值。研究发现，在相同泡沫厚度下，不同参试人员在坐垫处垂向同轴频响函数的共振频率在3.5~4.2 Hz变化，坐垫处水平前后向正交轴频响函数的共振频率在3.4~4.6 Hz变化，且少数参试人员表现出二阶共振。不同参试人员的坐垫处垂向同轴以及水平前后向正交轴频响函数的幅值均不相同。随着泡沫厚度的增加，坐垫处垂向同轴频响函数和水平前后向正交轴频响函数的共振频率均显著降低，而共振频率处幅值显著增加。结果表明，相同座椅条件下，不同参试人员加载条件下的座椅频响函数具有明显区别，体征参数的个体差异性与泡沫厚度的变化对同轴及正交轴座椅频响函数均具有明显影响。
2　人工神经网络建模
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2.1　反向传播人工神经网络建模
反向传播人工神经网络（BP‑ANN）是一种多层前馈感知器，采用前向传播方式计算输出，并通过反向传播最小化预测输出与实际输出之间的误差，逐步提升模型精度[20]。BP‑ANN网络由多个神经元组成，分布在输入层、隐藏层和输出层，各神经元通过激活函数引入非线性因素，从而确保能够建立体征参数以及座椅条件与座椅频响函数之间复杂的非线性关联。本文建立的BP‑ANN模型结构的输入层包含10个输入特征：年龄、身高、坐高、膝盖高度、臀膝长度、体重、性别、BMI、坐垫处泡沫厚度、频率；输出层包含两个输出特征：TRZZ，TRZX，并选择两个隐藏层，如图3所示。
BP‑ANN模型的样本集合由上述振动试验获取，12名参试人员在三种不同厚度的坐垫工况下试验，共获得36组试验结果，每组数据以0.25 Hz的频率分辨率采集1~10 Hz范围内的加速度信号，最终生成1332个独立样本集。在本研究中，拟将111个样本用于最终的模型测试，以检验输入特征对BP‑ANN模型预测座椅频响函数的影响；其余样本集分别用于模型的训练（1066个）与验证（155个），所有输入和输出数据均进行归一化处理，并将每个BP‑ANN模型进行20次训练以减小随机误差。本文采用Levenberg‑Marquardt算法训练模型，并采用均方根误差RMSE和决定系数R²评估模型对座椅频响函数预测精度，如下式所示：
RMSE=(∑i=1P(ni-ti)2P)12
R2=∑i=1P(ni-ti)2∑i=1P(ti-tavg)2
式中，P表示训练的样本数；ni表示第i个样本的模型预测结果；ti表示第i个样本的试验测得结果；tavg表示所有样本的试验测试结果的平均值。
2.2　利用响应面法优化BP‑ANN模型超参数
在建立BP‑ANN模型过程中，确定每个隐藏层中神经元个数、学习率、动量因子等超参数通常需要在模型的预测能力与计算需求之间进行平衡。在模型用于预测座椅频响函数时，合适的神经元数量与学习率能够影响模型对频响函数的预测精度，而动量因子则可以调整模型在训练过程中的稳定性。因此，合理设置超参数可以使本模型更好地建立输入特征与座椅频响函数之间的关联，从而提高预测的准确性。
本文采用响应面法（RSM）进一步研究座椅频响函数预测模型的超参数对预测性能的影响，通过建立数学模型预测响应变量来实现优化和改进系统性能，如图4所示。通过单因素试验确定一组单因素下最优模型超参数集，并使用此超参数集建立BP‑ANN模型，同时缩小了每个超参数水平值的范围，并以此进行响应面法分析，使用Box‑Behnken试验设计，通过回归拟合建立模型超参数与座椅频响函数预测性能指标R²之间的映射关联，如下式所示，以获得各超参数单独及交互作用对模型性能指标值的影响。最终得到响应面法优化的超参数集，并以此建立经过响应面法优化后的神经网络模型（RSM‑BP‑ANN）。
γ=B0+∑ikBiXi+∑ikBiiX2+∑ijkBijXiXj
式中，γ表示响应函数；X表示不同试验因素；B0、Bi、Bii和Bij分别表示常数项、线性项、二次项和交互项系数。
其中，学习率取值范围设置为0.01~0.1，增量为0.01；动量因子取值范围设置为0.1~1.0，增量为0.1；第一层隐藏层节点数取值范围设置为3~12，增量为1；第二层隐藏层节点数取值范围设置为3~12，增量为1。
3　结果与讨论
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3.1　BP‑ANN模型的响应面法优化试验与分析
BP‑ANN模型超参数对座椅频响函数预测性能指标R2的影响规律如图5所示，模型在预测TRZZ与TRZX时，R2值随学习率、动量因子、第一层隐藏与第二层隐藏层节点数的增加而波动，并总体呈下降趋势。当学习率设定为0.05、动量因子为0.4、第一层隐藏层节点数为5、第二层隐藏层节点数为6时，BP‑ANN模型对于座椅频响函数的预测性能达到最佳水平。基于上述试验结果，采用Box‑Behnken试验设计方法开展响应曲面优化试验，试验设计因素及水平如表1所示，其中-1和1分别代表每个因素的较低和较高水平，以相对于单因素试验中确定的最佳水平的偏差来表示。试验序列数据如表2所示。将BP‑ANN模型对预测TRZZ的R2值Y1与预测TRZX的R2值Y2作为响应值，将学习率A、动量因子B、第一层隐藏节点数C和第二层隐藏层节点数D设定为试验设计因素，以此寻找模型中各超参数与其性能指标R2之间的关联，进而得到最优超参数集。
对表2中的Y1和Y2进行拟合，可建立BP‑ANN模型在预测座椅频响函数时预测性能指标与各超参数之间的二次多项式回归方程：
Y1=0.91-0.027A-4.927×10-3B+0.028C+7.102×10-3D+6.492×10-3AB+0.035AC-3.493×10-3AD-0.042BC+0.011BD-3.938×10-3CD-8.657×10-3A2-0.016B2-0.028C2+6.262×10-3D2
Y2=0.9-0.025A-3.797×10-3B+6.464×10-3C+8.617×10-4D-9.085×10-3AB-6×10-5AC+5.712×10-3AD+8.83×10-3BC-0.036BD+0.025CD-5.637×10-3A2-0.2B2-0.029C2-0.015D2
式（6）和（7）中，一次项反映单个因素对响应变量的影响，交互项表示两个因素的联合作用，而二次项则反映同一因素不同水平下的影响变化。该回归方程能够描述BP‑ANN模型性能指标R2与各超参数之间的非线性关系。
针对响应面优化试验中获得的回归方程进行方差分析，结果如表3所示，Y1与Y2回归方程的F检验值分别为4.9和6.0；P值均小于0.01，表明两个回归方程均具有显著性。失拟项的P值均大于0.05，说明回归方程拟合度较好，可信度较高。其中，BP‑ANN模型中各超参数对R2（TRZZ）的一阶影响的主次顺序是C > A > D > B，即第一层隐藏层节点数>学习率>第二层隐藏层节点数>动量因子。交互项AC、BC和二次项C2对预测结果（TRZZ）也具有显著影响。此外，超参数对R2（TRZX）的一阶影响的主次顺序为A > C > B > D，即学习率>第一层隐藏层节点数 > 动量因子>第二层隐藏层节点数。交互项BD、CD和二次项B2、C2与D2对预测结果（TRZX）也具有显著影响。
图6为学习率、动量因子、第一层中隐藏节点数、第二层中隐藏节点数对R2（TRZZ）相互作用的响应面和等高线。响应面图可视化了超参数对模型性能指标R2的影响程度，影响越显著，对应的响应面就越陡，斜率就越大。结合显著性影响由表3与图6可知，在预测TRZZ时，动量因子B与学习率A、学习率A与第一层隐藏层节点数C、动量因子B与第一层隐藏层节点数C、第一层隐藏层节点数C与第二层隐藏层节点数D的响应面形状呈现不规则陡状，斜率较大，且等高线呈椭圆形，表明上述超参数之间的相互作用对TRZZ的预测精度具有重要影响。相反，随着动量因子B与第二层隐藏层节点数D的增加，R2（TRZZ）呈现平缓波动的变化趋势，表明动量因子B与第二层隐藏层节点数D之间的交互作用对TRZZ预测精度可能缺乏显著影响。学习率A和第二层隐藏层节点数D之间的交互作用同样对模型在预测TRZZ时的预测精度影响较小。
模型在预测TRZX时，学习率A与动量因子B、学习率A与第一层隐藏层节点数C、学习率A与第二层隐藏层节点数D的响应面形状呈相似的变化趋势，如图7所示。R2（TRZX）随学习率A的增加而快速下降，这表明学习率A对模型预测TRZX精度具有显著影响。此外，第一层隐藏层节点数C与动量因子B、第二层隐藏层节点数D与动量因子B以及第一层隐藏层节点数C与第二层隐藏层节点数D之间的响应曲面均较为陡峭，斜率均较大，且等高线均呈椭圆状，表明上述超参数之间的交互作用同样对模型预测TRZX的精度产生重要影响。
上述分析表明，BP‑ANN模型在预测TRZZ与TRZX时，各超参数间存在交互作用，并对模型的预测精度均产生影响。其中，模型在预测TRZX时，超参数之间交互作用对预测精度的影响尤为显著。因此，响应面法的应用为全面考虑超参数之间的复杂关系提供了一种有效方法，增强了座椅频响函数预测模型的可靠性和可控性。最终，对响应曲面模型求解得到座椅频响函数预测模型的最优超参数组合：学习率为0.01，动量因子为0.37，第一个隐藏层节点数为5，第二个隐藏层节点数为7。
3.2　模型性能评价及特征重要性分析
3.2.1　BP‑ANN和RSM‑BP‑ANN模型预测性能分析
基于单因素试验获得的最优超参数集与经响应面法优化后的超参数集，分别用于构建BP‑ANN与RSM‑BP‑ANN模型，预测性能如表4所示。对于TRZZ，BP‑ANN模型的R2值在训练和测试阶段分别为0.927和0.920；对于TRZX的预测，R2值分别为0.913和0.911。然而已有研究表明，通过试错法确定BP‑ANN模型的最优超参数集后，BP‑ANN模型对于刚性座椅条件下的视在质量预测精度较高，R2值可达0.96[16]。这可能是由于人体在弹性座椅和刚性座椅上的振动响应存在差异，人体和座椅两个子系统的耦合作用增加了准确预测人‑椅整体系统振动特性的难度，这说明BP‑ANN模型需要进一步优化，以提高其适用性和鲁棒性。
与BP‑ANN模型相比，RSM‑BP‑ANN模型在TRZZ的预测中，RMSE在训练和测试阶段分别下降了21%和25%，同时R2均高于0.95；在TRZX的预测中，RMSE在训练和测试阶段均下降了18%，且R2均高于0.92。结果表明，相对于传统的试错法，经响应面法优化的BP‑ANN模型的预测性能更为出色。这可能是由于传统试错法在优化超参数时无法全面搜索超参数空间，亦未充分考虑超参数之间的复杂交互作用。试错法常依赖基于经验的有限尝试和观察，进而导致模型易陷入局部最优解，从而限制了其对座椅频响函数的准确预测能力。相比之下，响应面法采用数学模型来拟合模型超参数与座椅频响函数预测性能指标之间的关系，有助于确定全局性能最佳的超参数组合，从而提高预测性能。
3.2.2　RSM‑BP‑ANN模型预测与实测数据对比分析
进一步使用未经训练和验证的测试数据集对RSM‑BP‑ANN模型的预测精度和泛化能力进行评估，图8为RSM‑BP‑ANN模型预测数据和实测数据之间的对比分析。RSM‑BP‑ANN模型在不同坐垫厚度工况下预测的座椅频响函数与实测值拟合度较高，模型预测的TRZZ与TRZX的共振频率均在4~6 Hz的频率范围，这与已有试验研究结果一致[17]，表明该模型对于未知工况下的座椅频响函数数据具有较好的预测精度。
同时，坐垫处泡沫厚度差异也会对座椅频响函数的预测结果产生影响。当坐垫厚度增加时，RSM‑BP‑ANN模型预测的TRZZ与TRZX共振频率均降低。此外，模型预测的TRZZ的共振频率处幅值随坐垫厚度的增加而增加，而TRZX共振频率处幅值受坐垫厚度差异影响较小，这与文献[17]的实测数据一致，这一现象可能是由于泡沫增厚导致座椅刚度发生变化。RSM‑BP‑ANN模型能够准确捕捉座椅频响函数的共振频率以及共振频率随坐垫泡沫厚度的变化趋势，这为建立体征参数和座椅条件与座椅频响函数之间的关联提供了一种有效且直观的方法。
3.2.3　特征重要性分析
图9为进一步采用置换特征重要性分析方法评估RSM‑BP‑ANN模型中坐垫处泡沫厚度与体征参数对于整个频率范围内座椅频响函数预测结果的相对重要性，RSM‑BP‑ANN模型在TRZZ与TRZX的预测中，各体征参数与坐垫处泡沫厚度的特征重要性排序结果呈现出相似趋势，其中坐垫处泡沫厚度对座椅频响函数预测具有最高的重要性，体征参数中年龄的影响最为显著，其后依次是性别、BMI、坐高、身高、体重、膝盖高度、臀膝长度。
坐垫处泡沫厚度在所有特征中对座椅频响函数预测结果影响最为显著，这凸显了在座椅设计中选择适当的泡沫厚度的关键性。合理选择坐垫处泡沫厚度可以显著影响人‑椅系统的振动响应特性，从而改善乘坐的舒适性。此外，人体体征参数对座椅频响函数的预测也存在显著影响，因此，后续研究中可将体征参数作为输入特征，以降低由体征参数差异引起的模型预测误差。在本文涉及的体征参数中，对座椅频响函数预测影响较大的为年龄与性别。此外，尽管BMI与体重对坐姿人体垂向视在质量存在较大影响，但其对座椅频响函数的预测影响相对较小，这可能是因为BMI较高的参试人员与座椅接触面积更大，座椅泡沫的受压程度较小，参试人员BMI与体重的变化影响被座椅动态特性的变化影响抵消。
4　结论
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本文通过建立基于响应面法优化的人工神经网络模型，完成对不同人体体征参数和坐垫泡沫厚度下的同轴和正交轴座椅频响函数的准确预测。采用响应面法对模型超参数进行优化，并基于超参数之间的交互影响建立了模型超参数与预测性能指标的映射关系，最终获取最佳超参数组合。研究结果表明，RSM‑BP‑ANN模型能更准确地预测垂向同轴和水平前后向正交轴座椅频响函数。同时，基于RSM‑BP‑ANN模型进一步分析了坐垫处泡沫厚度变化对预测性能的影响，并确定了各体征参数以及坐垫处泡沫厚度对座椅频响函数的重要性贡献。
人体对坐垫处垂向振动和靠背处水平前后向振动均具有较高的敏感性，因此，可进一步探究机器学习建模方法在多轴振动环境下的预测性能和优化算法。同时，本文对机器学习算法的研究仅局限于人工神经网络，且可解释性分析为模型无关方法。后续可考虑引入和比较多种机器学习算法以及开展模型的可解释性分析，更全面地探索和总结不同机器学习建模方法在模拟人‑椅系统振动响应特性上的效能及其学习和决策过程。
基金
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  • 国家自然科学基金资助项目(51605010)
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doi: 10.16385/j.cnki.issn.1004-4523.202311064
  • 接收时间：2023-11-30
  • 首发时间：2026-02-04
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  • 收稿日期：2023-11-30
  • 修回日期：2024-01-27
基金
国家自然科学基金资助项目(51605010)
作者信息
    1.北京工业大学机械与能源工程学院，北京　100124；
    2.北京工业大学汽车结构部件先进制造技术教育部工程研究中心，北京　100124

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张筱璐（1984―），男，博士，副教授。E‑mail：
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
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