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Article(id=1243226198969532659, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, articleNumber=null, orderNo=null, doi=10.7511/jslx20240708001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1720368000000, receivedDateStr=2024-07-08, revisedDate=1729440000000, revisedDateStr=2024-10-21, acceptedDate=null, acceptedDateStr=null, onlineDate=1774337823860, onlineDateStr=2026-03-24, pubDate=1761580800000, pubDateStr=2025-10-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774337823860, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774337823860, creator=13701087609, updateTime=1774337823860, updator=13701087609, issue=Issue{id=1243226190786441246, tenantId=1146029695717560320, journalId=1242798230522609684, year='2025', volume='42', issue='5', pageStart='699', pageEnd='888', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1774337821909, creator=13701087609, updateTime=1774338282025, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243228120724128564, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243228120724128565, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=729, endPage=736, ext={EN=ArticleExt(id=1243226199586095352, articleId=1243226198969532659, tenantId=1146029695717560320, journalId=1242798230522609684, language=EN, title=Physics-informed neural networks algorithm for sloving multi-point friction-induced stick-slip vibration problems, columnId=1243226193193971746, journalTitle=Chinese Journal of Computational Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

Addressing the challenge of accurately solving unstable stick-slip vibration problems in non-smooth dynamics, this paper proposes a solution algorithm based on Physics-informed Neural Networks (PINN). Firstly, the classical stick-slip vibration problem is dynamically modeled using the linear complementarity theory under unilateral constraints. Then, the linear complementarity relationship is designed as a loss function to guide the training of the neural network, constructing a PINN algorithm for solving multi-point friction-induced stick-slip vibration problems. The accurate simulation of complex responses of multiple sliders'stick-slip vibrations in frictional systems is conducted. By comparing the numerical results with the Switching Model method that includes event detection and the traditional Time-Stepping method without event detection, the accuracy of the PINN algorithm is verified. The proposed PINN algorithm transforms the traditional optimization problem calculation into network training of the machine learning algorithm, making it suitable for stick-slip vibration analysis with multiple contact points. This method achieves accurate nonsmooth state transitions and provides a convenient and easy-to-use new approach for the accurate simulation of complex nonlinear vibration responses in multi-degree-of-freedom frictional systems.

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针对非光滑动力学中不稳定粘-滑振动问题难以准确求解的挑战，本文提出了一种基于物理信息神经网络PINN（Physics-informed Neural Networks）的求解算法。首先，基于单边约束的线性互补理论对经典滑块-传送带模型进行动力学建模，随后将线性互补关系设计为指导神经网络训练的损失函数，构建了基于PINN的多点摩擦诱发粘-滑振动问题求解算法，进行了摩擦系统中多个滑块粘-滑振动复杂响应的准确模拟。通过与存在事件检测的Switching Model方法和传统无事件检测的Time-Stepping方法的数值计算结果对比，完成了PINN算法准确性的验证。本文提出的PINN算法将传统优化问题计算转化为机器学习算法的网络训练，适用于具有多个接触点的粘-滑振动分析，可实现准确的非光滑状态转换，为多自由度摩擦系统中复杂非线性振动响应的准确模拟提供了一种便捷易用的新方法。

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李姿琳*（1988-），女，博士，副教授（）.

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李姿琳*（1988-），女，博士，副教授（）.

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tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198969532659, language=CN, label=图10, caption=5自由度算例的时间响应, figureFileSmall=HPaXVMGOUsdbQ35vFo7ZSw==, figureFileBig=kYg92NgdPIkJuiOGpr26GQ==, tableContent=null), ArticleFig(id=1243226241222950926, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198969532659, language=EN, label=Tab. 1, caption=

Comparison of results from different methods for two degree of freedom numerical examples

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算例方法粘-滑转换点位移均方根值误差
Case 1Switching Model m11.2543Comparison result
m22.3219
Conventional LCP m11.25980.4406%
m22.33250.4593%
PINN m11.25990.4482%
m22.33250.4602%
Case 2Switching Model m12.2606Comparison result
m24.6629
Conventional LCP m12.26890.3653%
m24.48040.3746%
PINN m12.26820.3334%
m24.67320.2196%
Case 3Switching Model m12.8238Comparison result
m26.7292
Conventional LCP m12.81740.2261%
m26.74250.1977%
PINN m12.81670.2526%
m26.73890.1439%
), ArticleFig(id=1243226241575272465, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198969532659, language=CN, label=表1, caption=

2自由度算例不同求解方法结果误差分析

, figureFileSmall=null, figureFileBig=null, tableContent=
算例方法粘-滑转换点位移均方根值误差
Case 1Switching Model m11.2543Comparison result
m22.3219
Conventional LCP m11.25980.4406%
m22.33250.4593%
PINN m11.25990.4482%
m22.33250.4602%
Case 2Switching Model m12.2606Comparison result
m24.6629
Conventional LCP m12.26890.3653%
m24.48040.3746%
PINN m12.26820.3334%
m24.67320.2196%
Case 3Switching Model m12.8238Comparison result
m26.7292
Conventional LCP m12.81740.2261%
m26.74250.1977%
PINN m12.81670.2526%
m26.73890.1439%
), ArticleFig(id=1243226242015674391, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198969532659, language=EN, label=Tab. 2, caption=

Comparison of results from different methods for five degree of freedom numerical examples

, figureFileSmall=null, figureFileBig=null, tableContent=
方法粘-滑转换点位移均方根值误差
Conventional LCP Δt=10－7 m12.7785Comparison result
m24.4424
m36.0170
m48.0101
m56.8254
Conventional LCP Δt=10－3 m12.77790.0226%
m24.44310.0168%
m36.01590.0185%
m48.01160.0183%
m56.82720.0260%
PINN Δt=10－3 m12.77560.1033%
m24.43910.0742%
m36.01160.0899%
m47.96470.5673%
m56.81400.1667%
), ArticleFig(id=1243226242464464926, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198969532659, language=CN, label=表2, caption=

5自由度算例不同求解方法结果误差分析

, figureFileSmall=null, figureFileBig=null, tableContent=
方法粘-滑转换点位移均方根值误差
Conventional LCP Δt=10－7 m12.7785Comparison result
m24.4424
m36.0170
m48.0101
m56.8254
Conventional LCP Δt=10－3 m12.77790.0226%
m24.44310.0168%
m36.01590.0185%
m48.01160.0183%
m56.82720.0260%
PINN Δt=10－3 m12.77560.1033%
m24.43910.0742%
m36.01160.0899%
m47.96470.5673%
m56.81400.1667%
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基于物理信息神经网络的多点摩擦诱发粘-滑振动问题算法
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张非凡 1 , 李姿琳 2 , 白金帅 3 , 王伟 2 , 卫洪涛 2 , 卫荣汉 1, 2
计算力学学报 | 研究论文 2025,42(5): 729-736
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计算力学学报 | 研究论文 2025, 42(5): 729-736
基于物理信息神经网络的多点摩擦诱发粘-滑振动问题算法
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张非凡1, 李姿琳2 , 白金帅3, 王伟2, 卫洪涛2, 卫荣汉1, 2
作者信息
  • 1.郑州大学　网络空间安全学院，郑州　450002
  • 2.郑州大学　力学与安全工程学院，郑州　450001
  • 3.清华大学　航天航空学院，北京　100084

    • 李姿琳*（1988-），女，博士，副教授（）.

Physics-informed neural networks algorithm for sloving multi-point friction-induced stick-slip vibration problems
Feifan ZHANG1, Zilin LI2 , Jinshuai BAI3, Wei WANG2, Hongtao WEI2, Ronghan WEI1, 2
Affiliations
  • 1.School of Cyber Science and Engineering, Zhengzhou University, Zhengzhou 450002, China
  • 2.School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou 450001, China
  • 3.School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
出版时间: 2025-10-28 doi: 10.7511/jslx20240708001
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摘要
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针对非光滑动力学中不稳定粘-滑振动问题难以准确求解的挑战，本文提出了一种基于物理信息神经网络PINN（Physics-informed Neural Networks）的求解算法。首先，基于单边约束的线性互补理论对经典滑块-传送带模型进行动力学建模，随后将线性互补关系设计为指导神经网络训练的损失函数，构建了基于PINN的多点摩擦诱发粘-滑振动问题求解算法，进行了摩擦系统中多个滑块粘-滑振动复杂响应的准确模拟。通过与存在事件检测的Switching Model方法和传统无事件检测的Time-Stepping方法的数值计算结果对比，完成了PINN算法准确性的验证。本文提出的PINN算法将传统优化问题计算转化为机器学习算法的网络训练，适用于具有多个接触点的粘-滑振动分析，可实现准确的非光滑状态转换，为多自由度摩擦系统中复杂非线性振动响应的准确模拟提供了一种便捷易用的新方法。

物理信息神经网络  /  摩擦自激振动  /  非光滑动力学  /  粘-滑振动  /  线性互补问题

Addressing the challenge of accurately solving unstable stick-slip vibration problems in non-smooth dynamics, this paper proposes a solution algorithm based on Physics-informed Neural Networks (PINN). Firstly, the classical stick-slip vibration problem is dynamically modeled using the linear complementarity theory under unilateral constraints. Then, the linear complementarity relationship is designed as a loss function to guide the training of the neural network, constructing a PINN algorithm for solving multi-point friction-induced stick-slip vibration problems. The accurate simulation of complex responses of multiple sliders'stick-slip vibrations in frictional systems is conducted. By comparing the numerical results with the Switching Model method that includes event detection and the traditional Time-Stepping method without event detection, the accuracy of the PINN algorithm is verified. The proposed PINN algorithm transforms the traditional optimization problem calculation into network training of the machine learning algorithm, making it suitable for stick-slip vibration analysis with multiple contact points. This method achieves accurate nonsmooth state transitions and provides a convenient and easy-to-use new approach for the accurate simulation of complex nonlinear vibration responses in multi-degree-of-freedom frictional systems.

physical-informed neural networks  /  friction-induced vibration  /  nonsmooth dynamics  /  stick-slip vibration  /  linear complementarity problem
张非凡, 李姿琳, 白金帅, 王伟, 卫洪涛, 卫荣汉. 基于物理信息神经网络的多点摩擦诱发粘-滑振动问题算法. 计算力学学报, 2025 , 42 (5) : 729 -736 . DOI: 10.7511/jslx20240708001
Feifan ZHANG, Zilin LI, Jinshuai BAI, Wei WANG, Hongtao WEI, Ronghan WEI. Physics-informed neural networks algorithm for sloving multi-point friction-induced stick-slip vibration problems[J]. Chinese Journal of Computational Mechanics, 2025 , 42 (5) : 729 -736 . DOI: 10.7511/jslx20240708001
正文
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1　引言
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在工业生产、航空航天、汽车制造等领域中，不当的摩擦管理往往导致效率降低和设备寿命减短。特别是摩擦引起的自激振动现象，又称作摩擦诱发振动FIV（Friction-induced vibration）[1-3]，在多个工程领域中认为是一个待以解决的难题，如发出尖叫声的制动器[1]、吱吱作响的关节[4]、切削工具的抖动[5]、机器人定位不准确[6]等。
近年来，粘-滑振动作为FIV的主要机理之一[7]获得广泛关注，相关研究取得了一定进展。Popp等[8]指出即使在单自由度滑动带模型中，干摩擦诱导的粘-滑振动也可能是周期性的、准周期的甚至混沌的。Leine等[9]提出了一种Switching model法，可以利用Matlab求解器准确计算单接触点粘-滑振动分析。Pascal[10]研究了两个滑块参与的两自由度集中质量-传送带模型的粘-滑运动，捕获了粘-滑转换，保证了运动方程在正确的运动状态使用。对于多点接触的摩擦问题处理是困难的[11-14]。关于多体系统的不连续性有着广泛的研究[15]，如机器人或转子系统[16]。Moreau[17]首次提出了差分测量和扫描过程理论作为非光滑动力学分析的基础。多体系统的单边接触问题可以转化为线性互补问题LCP（Linear complementary problem）。基于LCP的几种方法用于求解刚性或弹性单边接触问题[16,18,19]。Pfeiffer等[15]综述了多体系统非光滑动力学的数值方法。目前常用的两种方法是事件驱动法和时间步进法[12-14]。摩擦系统的动力学过程复杂且不断变化，对于摩擦诱发的自激振动，特别是粘-滑振动过程的深入理解和有效预测仍然存在显著的算法瓶颈，对于多点接触问题，基于事件驱动的方法过于冗余无法适用，基于时间步进的方法依赖于极小时间步长来保证算法的准确性。
神经网络作为一种先进的机器学习工具在处理复杂系统非线性动态问题时展现出了显著优势，随着物理信息神经网络PINN（Physics-Informed Neural Networks）的提出[20]，神经网络在物理问题处理上的应用达到了新的高度。PINN的核心优势是将物理定律直接融入网络的学习过程，使神经网络模型既符合数值规律又遵循系统中的物理约束。这种方法有效弥补了传统神经网络在模拟物理相关问题时的不足，为精确捕捉和预测复杂物理过程提供了一种新的策略。文献[21，22]综述了PINN在各个领域的发展。结果表明将神经网络与物理信息的结合不仅有效增强了神经网络在处理物理问题上的能力，而且开辟了广泛的应用前景。Tarkhov等[23]研究了使用PINN将微分方程边值问题的近似解适应建模物理实体数据变化的可能性。Zhou等[24]使用概率物理信息神经网络结合支撑疲劳机制的物理，对疲劳寿命分布参数化更好地理解了神经网络在疲劳寿命预测中的应用。
本文对基于物理信息神经网络的多点摩擦诱发粘-滑振动问题算法理论进行了介绍，主要由三部分组成，即经典滑块-传送带模型中存在的线性互补理论、物理信息神经网络的搭建、算法整体流程；并对算法进行了评估，在多种算例下进行了不同求解方法的求解测试；最后进行了总结。
2　基于物理信息神经网络的多点摩擦诱发粘-滑振动问题算法
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2.1　单边接触系统中的线性互补问题
从数学角度出发，单边接触可以用线性互补条件进行表述。以图1为例，将物体距离接触面的距离记为gN，接触力大小记为λN，则法向上单点刚性接触的互补关系可表示为
即gN≥0时，λN=0；gN=0时，λN≥0。
当考虑切向摩擦时，以λT表示摩擦力，γT表示相对速度。滑动摩擦力遵循库伦摩擦定律，可表示为λT=μλN。多点接触问题中的摩擦力和相对速度可以分解为以下两个互补关系：
式中
相对速度与摩擦力冲量互补关系为
式中，
对于单向接触多体系统的动力学建模，矩阵形式的运动方程一般写作
式中M为质量矩阵，h为系统中力的矢量集合，λN和γT分别为法向和切向上的力，q为全局系统中定义的位移量，法向和切向的约束函数表示为，。通过对运动状态进行时间间隔Δt=tE－tA离散，得到时域离散运动方程和状态矢量为
式中ΛN=λNΔt，ΛT=λTΔt。
利用gN和λN之间的互补关系，可以实现单边刚性接触，摩擦力与法向上的状态解无关，遵循式（3）给出的互补关系。可以得到对于单边接触的LCP公式矩阵形式为
式中
其中存在互补关系为
通过将单边接触描述为线性互补问题，多点接触问题可以表示为一组线性方程，避免了基于事件检测的求解法中不同运动状态组合引起的复杂逻辑判断，对多点接触问题具有出色的适应能力。
2.2　粘-滑振动问题的物理信息神经网络
考虑到粘-滑振动过程的复杂性且求解过程具有迭代性质，容易造成连续的误差累积，对振动过程的计算需要具有很高的精度。为应对这一问题，本研究引入了物理信息神经网络，该方法基于前馈神经网络，同时融入粘-滑振动过程中的物理约束，增强了求解的准确性和可靠性。PINN通过将LCP的理论与神经网络结合，把LCP中的物理约束作为网络参数优化的惩罚项，实现了在网络参数调整过程中引入明确的物理导向。
如图2所示，PINN的神经网络部分是一个全连接神经网络框架，在该框架中，各层内的神经元通过线性变换以及随后的非线性激活函数处理进行相互连接。在该网络架构中，除输出层外的所有层均采用双曲正切激活函数，双曲正切激活函数的表达形式为
第n层PINN可以表示为
式中wn和bn分别为当前隐藏层的权重矩阵和偏置向量，xn－1为上一层神经元的输出。
为了优化网络输出，使其更符合线性互补问题的约束条件，提升网络在求解时的收敛效率，对ReLU激活函数进行了改良，提出了一种新型激活函数。改进后的激活函数的表达形式为
式中c1和c2为校正因子。通过调整校正因子，可以在多个维度上对网络的输出进行约束，从而提升网络输出的适应能力。这种调节可实现对网络在面对特定物理约束条件时进行微调，加速网络收敛。
PINN的输出层可以表示为
式中wn和bn分别为输出层的权重矩阵和偏置向量，xn－1为上一层神经元的输出。
网络的输出需要满足LCP约束条件，对输出值x和y构造满足LCP理论描述的等式：
式中A，b根据系统状态由式（6）计算得到。
PINN的训练目标是寻找wi和bi，使得损失函数L小于给定值ε，损失函数可表示为
式中li和pi分别是向量l和p的元素，N为向量的元素数。
在PINN训练过程中，隐藏层数量及神经元个数的正确搭配十分重要。经过不断测试，选用3层隐藏层、每层5个神经元进行网络搭建。本研究借助TensorFlow中的自动微分技术，通过迭代优化参数使损失函数小于给定值ε。在优化过程中，选用基于梯度的L-BFGS优化器。与数据驱动方法不同，用于LCP求解的PINN输入数据是任意的。网络的训练目标是针对当前迭代过程中的LCP约束条件输出符合精度的x和y。训练完成后的PINN能够进行线性互补问题的求解。
2.3　基于物理信息神经网络的多点摩擦诱发粘-滑振动问题算法
粘-滑振动过程中的状态更新采用中心差分法进行，使用当前时刻系统状态进行前半段的状态更新，使用PINN输出的接触力参与下半段状态更新。用于系统状态更新的多体动力学公式可表示为
式中hM为物体在摩擦系统中所受外力，qM为位移，v为速度。
式中a为运动物体加速度，ΛT为物体所受接触力。
综上所述，基于PINN的多点摩擦诱发粘-滑振动问题算法的整体流程如图3所示，主要步骤如下，首先对研究的非光滑动力学模型过程进行分析，构建相对速度与接触力之间的线性互补关系；之后将线性互补关系融入到神经网络的搭建当中，选择适当的隐藏层层数、神经元个数、激活函数搭建网络，对PINN进行训练得到接触力；最后按照多体动力学公式更新系统状态。
3　算例
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滑块-传送带模型是用于测试摩擦定律或分析方法性能的经典动力学模型。摩擦接触并非单点接触，而是面与面的接触，接触面间存在局部运动不统一性。图4展示了一个两自由度的滑块-传送带模型。该模型由若干滑块、线性弹簧和阻尼器组成，这些滑块受到一个以恒定速度vo运动的传送带的驱动。传送带和滑块间存在摩擦，摩擦定律使用工业中普遍采用的Stribeck摩擦力模型。对于含有干摩擦的系统，由于静、动摩擦系统的不统一，在传送带开始运作后，系统各滑块可以处于滑动阶段或粘滞阶段，并在两种运动状态间反复交替进行。
3.1　2自由度算例模拟
为验证PINN求解粘-滑动振动问题的有效性，本节针对2自由度系统进行了仿真算例设计，采用Switching model法和传统LCP方法作为对比结果。其中Switching model法作为标准解，通过计算PINN结果及传统LCP结果与标准解的差距评估PINN算法的求解能力。在算例选择方面，使用Switching Model方法在其他系统参数保持不变的情况下，进行了压力工况从1 N到30 N变化的30组算例的计算，并统计了计算结果中存在的粘-滑状态转换点。图5展示这30组算例计算结果中粘-滑状态转换点的分布情况。其中横坐标表示压力工况，纵坐标为粘-滑转换发生时的位移，点越密集表示该工况下存在越多周期的不稳定粘-滑振动。如图5所示，红色标注的算例存在多个粘滑转换点，通过观察其庞加莱图可知，红色标注算例为混沌振动。蓝色标注的算例表示在该工况下，两个运动物体均表现为单周期粘-滑振动，形成一个单周期粘-滑极限环。绿色标注的算例显示出运动物体的振动一个为双周期粘-滑振动，形成两个粘-滑极限环，另一个为单周期粘-滑振动，形成一个粘-滑极限环。这三组算例的运动过程可以覆盖大部分情况，选定为用于比较算法性能的算例，最终算例选定情况为，质量系数m1=m2=1，刚性系数k1=1，k2=1，初始位移x=0.5，压力分别为11 N、15 N、24 N的三组算例，时间步长统一选定Δt=10－3。
图6~图8展示了滑块在不同工况下的运动状态，其中图6展现了物理信息神经网络在处理不规则粘-滑振动问题的计算结果。图7展示了算例2的时间响应和极限环图像。由图7（c，d）可以发现该算例计算结果达到稳定状态后，m1和m2均存在一次粘-滑转换，与图5展示的结果一致。图8展示了算例3的时间响应和极限环图像。由图8（c，d）可以发现该算例计算结果达到稳定状态后，m1存在两次次粘-滑转换，m2存在一次粘-滑转换，与图5展示的结果一致。以上对比表明，PINN在解决具有规律性的粘-滑振动问题时，其表现与传统数值方法匹配，能够准确模拟不同条件下的完整运动过程。
表1对比了PINN方法、传统方法以及Switching Model方法进行两自由度算例计算时粘-滑转换点发生处位移的准确性。通过对相同工况算例应用这三种方法，并对计算结果中粘-滑转换点发生处的位移进行统计，以Switching Model方法得到的结果作为标准结果（Ground truth results），计算了传统方法和PINN方法相较于标准结果的粘-滑转换点位移均方根RMS（Root mean square）的误差，评估了PINN的准确性。从表1结果可以看出，PINN方法对粘-滑转换点的判断具有较高准确性，在误差水平上与传统方法相同。
3.2　5自由度算例模拟
Switching Model方法由于算法结构的问题，当接触点过多时算法过于冗余，不适合对多点摩擦接触的动力学问题进行求解，相比之下，PINN和传统方法没有这一限制。如图9所示设计了一个5自由度摩擦系统的算例，旨在探究PINN在处理多点摩擦诱发粘-滑振动问题时的有效性。该5自由度模型在初始速度、质量系数、刚性系数、阻尼系数以及初始位移设置上与3.1节2自由度算例保持一致，施加的压力值统一设定为15 N。为确保对比结果的准确性，采用时间步长为Δt=10－7，通过传统方法得到的计算结果作为标准结果。
图10展示了5自由度算例计算结果，PINN与传统数值方法在模拟结果上的对比图9（a~e）分别对应5自由度滑动带模型中m1~5的粘-滑振动时间响应。结果显示，基于PINN的求解方法在处理5自由度系统的粘-滑振动问题时，其表现与传统方法相近，能够在5自由度系统中进行准确的数值模拟。
为了更直观地进行误差分析，采用3.1节同样的方式对计算结果中存在的粘-滑转换点进行了统计，并计算了这些转换点位移量的RMS。三种方法计算结果的对比分析列入表2，基于PINN的求解方法具有较好的表现。
4　结论
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本文提出了一种基于物理信息神经网络PINN（Physics-informed Neural Networks）的多点摩擦诱发粘-滑振动问题算法，实现了复杂非光滑动力学振动响应的高精度模拟。首先，针对经典滑块-传送带系统，基于单边接触问题的线性互补理论，提出了用于非光滑动力学响应模拟的神经网络框架。同时，通过对激活函数改造，提出一种新型激活函数约束网络输出，使网络训练更高效。最后，将PINN算法应用于2自由度和5自由度多滑块摩擦系统的动力学模拟，并将数值结果与传统时间-步进算法计算结果进行对比，验证了本文提出的PINN算法的可行性，具体结论如下。
（1）本文提出的算法能够对不同接触复杂程度的粘-滑振动过程进行准确模拟，PINN算法与传统的无事件检测Time-Stepping方法同样具有较高精度。
（2）基于PINN的求解方法通过将线性互补问题的传统优化问题计算转化为机器学习算法的网络训练，方法使用便捷易于应用。
（3）本文提出的PINN算法可通过修改激活函数和网络结构实现计算精度的提高。
基金
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  • 国家自然科学基金青年基金(12402242)
  • 河南省高等学校重点科研项目(24A130002)
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2025年第42卷第5期
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doi: 10.7511/jslx20240708001
  • 接收时间：2024-07-08
  • 首发时间：2026-03-24
  • 出版时间：2025-10-28
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  • 收稿日期：2024-07-08
  • 修回日期：2024-10-21
基金
国家自然科学基金青年基金(12402242)
河南省高等学校重点科研项目(24A130002)
作者信息
    1.郑州大学　网络空间安全学院，郑州　450002
    2.郑州大学　力学与安全工程学院，郑州　450001
    3.清华大学　航天航空学院，北京　100084
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