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Article(id=1241838628465348616, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838624241681037, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.049, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1730304000000, receivedDateStr=2024-10-31, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774007001271, onlineDateStr=2026-03-20, pubDate=1740672000000, pubDateStr=2025-02-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774007001271, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774007001271, creator=13701087609, updateTime=1774007001271, updator=13701087609, issue=Issue{id=1241838624241681037, tenantId=1146029695717560320, journalId=1241755870837649424, year='2025', volume='46', issue='1', pageStart='1', pageEnd='148', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774007000265, creator=13701087609, updateTime=1774007104623, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241839062017966445, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838624241681037, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241839062017966446, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838624241681037, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=117, endPage=128, ext={EN=ArticleExt(id=1241838628809281545, articleId=1241838628465348616, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Multi-modal Self-Sustained Motion of Flat Paper Sheets in Hot Steam, columnId=1241831201674171363, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

Self-sustained motion has proven to be an effective approach for tackling complex problems and addressing various challenges across a variety of disciplines, such as bionics, soft robotics, and engineering, owing to its efficiency, resourcefulness, and flexibility. However, traditional single-mode self-sustained systems are often limited to specific tasks and lack adaptability to environmental changes. This study addresses these limitations by developing a multi-modal self-sustained system using circular silicone oil paper. It demonstrates that hot steam drives the silicone oil paper to achieve self-sustained motion, thereby constructing a self-sustained system. In this system, the circular silicone oil paper placed on a steam-supported surface continuously oscillates and tumbles under the influence of hot steam. The study analyzes the mechanisms behind these motions and establishes a geometric model for the self-sustained behavior of the circular silicone oil paper. Computational programming examines how the oscillation frequency and amplitude of the circular silicone oil paper relate to steam temperature and structural dimensions. Critical conditions for motion pattern transitions and phase diagrams are identified, with experimental studies validating theoretical predictions. The research findings reveal that by adjusting structural size and steam temperature, the circular silicone oil paper can freely switch among three modes: stationary, self-sustained oscillation, and self-sustained tumbling. The frequency and amplitude of self-sustained oscillation increase with higher steam temperatures, larger outer diameters, and an increased inner-to-outer diameter ratio. The multi-modal self-sustained system developed in this study can better adapt to diverse tasks and environments while reducing costs and energy consumption. Therefore, it holds significant potential for applications in fields such as autonomous robotics, medical devices, waste heat recovery, and thermo-mechanical conversion.

, correspAuthors=Jun Zhao, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Jun Zhao, Xiaodie Sun, Wei Zuo), CN=ArticleExt(id=1241838642306552131, articleId=1241838628465348616, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=蒸汽驱动下平面纸片的多模态自持续运动, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

自持续运动在仿生学、软机器人以及工程学等多个学科领域都取得了显著的进展，其高效性、机敏性和灵活性使其成为解决复杂问题的强大工具. 然而，针对不同的环境，单一模式的自持续运动通常只适用于特定类型的任务. 鉴于其运动通常缺乏对环境变化的适应能力，本文旨在发展多模式自持续系统. 文章发现了热蒸汽可以驱动硅油纸产生自持续运动，构建了圆环硅油纸片多模式自持续运动的系统，实现了两种自持续运动模式：摇摆和翻滚，并对两种运动模式的机理进行分析. 理论上建立了圆环硅油纸片自持续运动的几何模型，通过编程研究圆环硅油纸片的摇摆频率和振幅与热蒸汽温度、结构尺寸之间的关系. 给出了运动模式转换的临界条件和相图，通过实验研究验证了理论预测的合理性. 结果表明通过调整结构尺寸和蒸气温度，可以实现圆环硅油纸静止、自持续摇摆和自持续翻滚三种模式的自由转换. 增大蒸汽温度、圆环的外径和内外径比值可以增大系统自持续运动的振幅和频率. 文章构建的多模式自持续运动系统可以更好地适应多样化的任务和环境，同时降低成本和能源消耗. 因此，在自主机器人、医疗设备、余热回收、热-机械转化等领域具备一定的应用潜力.

, correspAuthors=赵俊, authorNote=null, correspAuthorsNote=
**E-mail：.
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Material properties and geometric parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
参变量表示符号单位数值
蒸汽温度 Tq℃20-50
蒸汽比热容 ρcJ/（kg·K）1
曲率影响系数 A/4.35
热传递系数 c、k/0.98、1
圆形半径 Rm0.01-0.018
圆环内外径比 R0/R1/0.2-0.8
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材料特性和几何参数

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参变量表示符号单位数值
蒸汽温度 Tq℃20-50
蒸汽比热容 ρcJ/（kg·K）1
曲率影响系数 A/4.35
热传递系数 c、k/0.98、1
圆形半径 Rm0.01-0.018
圆环内外径比 R0/R1/0.2-0.8
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蒸汽驱动下平面纸片的多模态自持续运动
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赵俊 ** , 孙晓蝶 , 左伟
固体力学学报 | 研究论文 2025,46(1): 117-128
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固体力学学报 | 研究论文 2025, 46(1): 117-128
蒸汽驱动下平面纸片的多模态自持续运动
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赵俊** , 孙晓蝶, 左伟
作者信息
  • 安徽建筑大学土木工程学院，合肥，230601

通讯作者:

Multi-modal Self-Sustained Motion of Flat Paper Sheets in Hot Steam
Jun Zhao** , Xiaodie Sun, Wei Zuo
Affiliations
  • College of Civil Engineering, Anhui Jianzhu University, Hefei, 230601
出版时间: 2025-02-28 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.049
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摘要
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自持续运动在仿生学、软机器人以及工程学等多个学科领域都取得了显著的进展，其高效性、机敏性和灵活性使其成为解决复杂问题的强大工具. 然而，针对不同的环境，单一模式的自持续运动通常只适用于特定类型的任务. 鉴于其运动通常缺乏对环境变化的适应能力，本文旨在发展多模式自持续系统. 文章发现了热蒸汽可以驱动硅油纸产生自持续运动，构建了圆环硅油纸片多模式自持续运动的系统，实现了两种自持续运动模式：摇摆和翻滚，并对两种运动模式的机理进行分析. 理论上建立了圆环硅油纸片自持续运动的几何模型，通过编程研究圆环硅油纸片的摇摆频率和振幅与热蒸汽温度、结构尺寸之间的关系. 给出了运动模式转换的临界条件和相图，通过实验研究验证了理论预测的合理性. 结果表明通过调整结构尺寸和蒸气温度，可以实现圆环硅油纸静止、自持续摇摆和自持续翻滚三种模式的自由转换. 增大蒸汽温度、圆环的外径和内外径比值可以增大系统自持续运动的振幅和频率. 文章构建的多模式自持续运动系统可以更好地适应多样化的任务和环境，同时降低成本和能源消耗. 因此，在自主机器人、医疗设备、余热回收、热-机械转化等领域具备一定的应用潜力.

蒸汽驱动  /  圆环纸片  /  自持续运动  /  多模态

Self-sustained motion has proven to be an effective approach for tackling complex problems and addressing various challenges across a variety of disciplines, such as bionics, soft robotics, and engineering, owing to its efficiency, resourcefulness, and flexibility. However, traditional single-mode self-sustained systems are often limited to specific tasks and lack adaptability to environmental changes. This study addresses these limitations by developing a multi-modal self-sustained system using circular silicone oil paper. It demonstrates that hot steam drives the silicone oil paper to achieve self-sustained motion, thereby constructing a self-sustained system. In this system, the circular silicone oil paper placed on a steam-supported surface continuously oscillates and tumbles under the influence of hot steam. The study analyzes the mechanisms behind these motions and establishes a geometric model for the self-sustained behavior of the circular silicone oil paper. Computational programming examines how the oscillation frequency and amplitude of the circular silicone oil paper relate to steam temperature and structural dimensions. Critical conditions for motion pattern transitions and phase diagrams are identified, with experimental studies validating theoretical predictions. The research findings reveal that by adjusting structural size and steam temperature, the circular silicone oil paper can freely switch among three modes: stationary, self-sustained oscillation, and self-sustained tumbling. The frequency and amplitude of self-sustained oscillation increase with higher steam temperatures, larger outer diameters, and an increased inner-to-outer diameter ratio. The multi-modal self-sustained system developed in this study can better adapt to diverse tasks and environments while reducing costs and energy consumption. Therefore, it holds significant potential for applications in fields such as autonomous robotics, medical devices, waste heat recovery, and thermo-mechanical conversion.

steam-driven  /  circular paper sheet  /  self-sustained motion  /  multi-modal
赵俊, 孙晓蝶, 左伟. 蒸汽驱动下平面纸片的多模态自持续运动. 固体力学学报, 2025 , 46 (1) : 117 -128 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.049
Jun Zhao, Xiaodie Sun, Wei Zuo. Multi-modal Self-Sustained Motion of Flat Paper Sheets in Hot Steam[J]. Chinese Journal of Solid Mechanics, 2025 , 46 (1) : 117 -128 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.049
正文
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0　引言
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长期以来，大自然一直被认为是启发人类开发具有适应性和智能的人工材料或设备的灵感源泉. 例如含羞草、太阳花和变色龙，引发了纳米医学策略[1]、柔性太阳能跟踪器[2,3]、变色软机器人[4,5]等的发明. 另一类具有自主、自我维持特征的刺激响应驱动在生物体中也无处不在，在各种工程和生命活动中发挥着重要作用[6-9]，例如，企鹅采取摇摇摆摆的步态，周期性地左右摇摆[10]. 据研究，这种摇摇摆摆的步态通过最大限度地交换重力势和动能，巧妙地减少了肌肉的工作量. 许多其他的例子包括鱼尾的摇摆、人类的心脏跳动和鸟的振翅[11-13]，这些都是有趣且值得人们学习研究的，同时对于设备创新来说也是具有挑战性的.
仿生学的进展催生了先进机器技术的崛起，通过模仿这些生物体，研究人员成功开发出具备柔性外骨骼和传感器的机器人. 其中，基于主动材料的机器人研究成为机器人技术的关键方向. 这些机器人不仅具备传统机械机器人的能力，还能够通过激活或控制其内部的主动材料来实现各种动作. 传统的控制方式包括远程操作、编程控制和反馈控制等，尽管它们十分强大，但在特定情况下存在一些挑战和缺陷. 例如，这些方式对环境的敏感性较高，难以应对环境的变化，可能不够通用或适应性差，需要在面对不同任务时进行重新训练或重新编程. 因此，基于主动材料的自控制方式具备轻量化设计、快速响应和高效率等特征，具有显著的优势. 其中，自持续控制方式尤为引人瞩目，能够减少对外部设备的依赖，实现长时间运行，同时具备出色的节能和适应性.
尽管这些具有自主性，由稳定刺激驱动的自持续运动结构或体系因其在机器人、自适应系统和能量收集装置中的潜在应用而备受关注[14-26]. 但主要集中在单一模式的自持续体系上，这在某种程度上带来了一定局限性，在众多的研究中，限制了机器人行为的精确控制和操纵，所以，构建多功能机器设备需要具有可控制的多模态自持续运动[27-29]. 然而在设计这些设备中，多模态自持续运动的预测建模发挥至关重要的作用，这通常是非常具有挑战性的，因为经常涉及高非线性和多场耦合[30-34].
基于以上原因，本文尝试采用生活中的剪纸材料来构建一种可实现多模态自持续运动的结构系统. 在日常生活中，常常会用硅油纸来隔离面食和蒸锅的隔层，防止面食粘连在隔层上，若先加热水，再放硅油纸，会发现硅油纸片会在隔层上来回卷曲运动. 通过实验发现，硅油纸具有热、汽敏感性特质，实际上普通的薄纸片也具有此性质，只是硅油纸片具有防水特性，运动的可持续性更强. 在本文研究中，采用了圆环硅油纸剪纸材料，它可以通过热蒸汽驱动执行自主自持续驱动，能够从外部环境中吸收热能，实现连续和不间断的热机械驱动. 这种材料在适当的热蒸汽场下展示出可编程的弯曲变形和仿生的自激运动. 此外，
基于特殊结构系统的设计，通过调整结构尺寸和蒸汽温度，能够控制系统的多模式运动转换，构建出多模式自持续体系，以满足复杂性和多样性挑战. 多模式自持续系统的自主性增强，能够节约资源和提高能源效率，为医疗、自主机器人、余热回收、热-机械转化和生物学等研究中产生广泛的应用潜力.
1　实验研究和机理分析
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实验前需要准备圆环硅油纸片，尺寸为外径R1=1.5 cm，内径R0=0.5*R1，厚度h=50微米，疏水透气膜；实验仪器为加热托盘、控温加热台. 实验过程是在加热托盘中加水，在托盘内水表面放上一层疏水透气膜，切割硅油纸片，将切割好的圆环硅油纸片放置于疏水透气膜平面，实验时室内温度保持15 ℃，使用控温加热台加热托盘中的水，热蒸汽会从疏水透气膜下输送至上表面，驱动圆环纸片运动. 通过调节水温，观察圆环硅油纸片的不同运动情况，并用摄像机记录下实验图片.
1.1　自持续摇摆运动
图1为圆环纸片自持续摇摆时程曲线图. 设定温度为35 ℃，在热蒸汽的驱动下圆环纸片发生左右摇摆，通过摄像机记录下圆环纸片的自持续摇摆过程. 图1给出了圆环纸片从中性位置到最右、最左，然后回到中性位置的完整的自持续摇摆过程，绘制出中性轴变化角度的时程曲线图.
图2为圆环纸片自持续摇摆机理图. 如图2所示，初始时刻剪纸未发生变形，在热蒸汽的作用下，剪纸底部膨胀变形，重心位置g上移，当重心位置达到此时底部接触点曲率半径对应的圆心位置ρ时，剪纸不再稳定，发生细微的扰动将向一侧倾倒，接触面发生变化，接触点随之发生变化，剪纸在蒸汽作用下重新变形，重心位置调整，产生“扶正”力矩，剪纸有着朝平衡位置转动的趋势，整个剪纸结构从热蒸汽中吸收能量用来抵抗系统的能量消耗，从而达到平衡，这样，就实现了摇摆运动的一个周期，自锁负反馈回路在不改变外部刺激的情况下维持这样的自持续摇摆运动.
1.2　自持续翻滚运动
图3为圆环纸片翻滚运动时程图. 设定温度为55 ℃，此时温度较高，在热蒸汽的驱动下圆环纸片发生翻滚，通过摄像机记录下圆环纸片的翻滚过程，并绘制出时间与夹角的关系图. 横坐标是时间，纵坐标是结构初始时刻上表面中心点法向y0轴与竖向坐标y轴之间的夹角，其夹角使用叠加法读取，时间与夹角变化反映了结构发生稳定的周期翻滚运动，通过实验得到可以通过控制温度使结构从自持续摇摆模式转变为翻滚模式.
图4为圆环纸片自持续翻滚机理图. 如图4所示，随着温度的增加，热蒸汽驱动圆环纸片摇摆的幅度进一步增加，重力产生的力矩将会逐渐增大，而此时输入能量足够，圆环将会实现跃迁而发生翻转，翻面之后原本受热膨胀面将处于上方，此时底面获得受热膨胀，使弯曲的圆环伸展铺平，将进入下一个循环变形过程，具体运动过程可见图3. 由此可以发现通过改变蒸汽温度不仅可以改变摇摆的频率与幅值，甚至可以调节运动状态，实现不同多模态运动.
2　模型与材料
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2.1　几何模型
如图5所示，建立了圆环硅油纸片模型，硅油纸片质量密度均匀，表面光滑质地轻薄且具有疏水性，圆环外边缘与内边缘均为标准的圆形，内径为R0，外径为R1，厚度为h，其中R1>R0≫h.
图5（a）是一个裁剪的圆环硅油纸模型，图5（b）是模型计算的简化图，5（c）显示了简化模型在蒸汽作用下的弯曲变形. 在剪纸底部加热产生蒸汽对其作用时，它会膨胀并诱导纸片发生变形. 为了简便计算将圆环沿直径方向简化为一维线条，在重心偏移过程中，结构始终保持着相似的宏观弯曲曲率轮廓，由初始的直线段逐渐变为弧线，弧线段上各处曲率随时间和环境温度差不断发生变化.
2.2　材料参数
通过实验研究硅油纸在蒸汽环境下的具体曲率变化情况，同时考虑时间和环境温度的影响，采用多元指数模型描述已知数据，曲率1/ρ可定义为：
其中ρ和A分别为曲率半径（单位：cm）和影响系数. T为环境温度差（单位：℃），T0为蒸汽响应纸片在长时间蒸汽作用下的极限温差系数. t为时间（单位：s），τ为蒸汽响应纸片与环境之间的热交换特征时间系数. 由此通过一系列实验数据，描述曲率变化规律，如图6所示.
图6（a）是时间对曲率半径影响变化实验图，图6（b）是温差对曲率半径影响变化实验图. 图6记录不同时刻和不同温度的纸片弯曲程度，计算出纸片与蒸汽平面接触点处的曲率半径，如图中黑点所示，再通过数据拟合出曲线图，获得时间和温度对材料曲率的影响规律，图6（a）和6（b）拟合曲线的确定系数分别为0.999和0.983. 通过设置不同环境温度差与时间计算出对应的实际曲率，确定出曲率影响系数，得出具体的表达式：
实验时采用纸片的Biot数非常小，假设热交换迅速且温度均匀. 纸片直接接触蒸汽时，每秒从蒸汽中吸收热量Tq. 同时，纸片与环境也进行热交换，假设热通量与纸片与环境之间的温度差T成线性关系. 当纸片处于蒸汽中时，温度差T可由下式计算得出[35]：
其中ρc为蒸汽比热熔，k是传热系数.
3　计算方法
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蒸汽驱动纸片运动过程中纸片各处的曲率随温度与时间不断变化，不同曲率又导致不同位置温度的变化，对此要获得解析解是非常具有挑战性的，本文通过Matlab软件编程进行数值求解.
图7给出了变形微段的圆心角、弧长和曲率半径之间的关系. 根据曲率在微分中的定义关系，对于变形微段的圆心角dθ，弧长ds以及曲率存在的关系为：
另外，由于圆环纸片不断发生变形，其各个位置与蒸汽方向的夹角不断发生变化，而不同夹角又会影响热传递，最终导致整体温度具有差异，因此（3）式可写为：
其中，c为热传递系数，α为纸片上点的切线与蒸汽垂直方向的夹角，并且假设蒸汽方向始终与接触点的切线方向垂直. 可以发现当夹角α>π/2或α<-π/2时上侧纸片将会被下侧纸片遮挡，处于蒸汽的背侧，此时Tq=0，此时温度关系则变为：
如图8所示，利用离散法求解，对均匀圆环纸片简化后的直线进行划分为n-1个微段，共n个离散点，则有ds=2R1/（n-1）.
当简化为n个离散点求重心时，由于密度分布不同，需要知道各个点的权重，这个密度比重关系可以被具体地表示为：
其中ϕi为圆环纸片简化后线段上第i个点的权重，R1为圆环外径，R0为圆环内径，di为线段上第i个点到线段中点的距离.
因此，该质量点体系的重心坐标为：
式中为重心的水平坐标，为重心的纵向坐标，xi和yi为第i个质量点的水平坐标和纵向坐标，和分别为质量相对于x轴和y轴的静矩，为总质量.
当重心的高度超过接触点的曲率半径ρ时，圆盘就会变得不稳定，其接触点的位置也会移到一个新的位置. 接触点转变的判定准则为：
设置时间步长为dt，总步数为j，则有dt=t/j. 根据控制方程（4）写出递推关系式为：
以直线段左端点为坐标原点，则有，，，其中θ为点的切线与横坐标轴的夹角，x，y分别为点对应的横纵坐标，它们满足的关系有：
和
当初始时刻即j=1时，纸片尚未发生变形，初始条件为，由此可以根据等式（10）依次计算出每个点的角度θ以及各自的横纵坐标值. 假设初始接触点m为线段中点即m=n/2+1，并且蒸汽方向始终与接触点处的切线方向垂直，则任一点处的切线与蒸汽垂直方向的夹角α可用下式计算：
再根据等式（5）、（6）得出温度关于角度α的递推关系式：
通过式（14）由每个点处的切线与蒸汽垂直方向的夹角，计算出每个点下一时刻的温度，代入等式（10）中计算出下一时刻各点的角度，再代入等式（11）和（12）分别计算横纵坐标值，由此循环计算出每一时刻各点的温度，角度，坐标及曲率，进而通过每个时刻n个点的坐标信息求出当前重心位置高于接触点m对应曲率半径的中心位置ρ时，触发圆环纸片的临界摇摆条件，如图9所示.
同样假设接触点需要满足一些条件. 首先，该接触点处的切线必须几乎垂直于重心与该点间的连线. 其次，重心到该点的路径需要是最优的，也就是满足两点间距离最短. 只有满足以上条件，接触点m才能成为下一时刻的接触点. 如果没有满足条件的点存在，则接触点将保持不变进入下一次迭代. 随着计算时间的推移，可以得到一系列接触点数值，这些点呈周期性往复变化，对应了实际的自持续摇摆运动状态.
当接触点摇摆至左右端点的位置时，可以认为纸片摇摆幅度过大，此时会从摇摆模式进入新的运动状态即翻滚模式. 通过上述模型和理论公式的推导，使用Matlab软件编程实现递推演化过程，并获得数值模拟计算结果.
4　数值结果与讨论
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圆环纸片模型数值计算的材料特性和几何参数如表1所示，总时间步长为20000，接下来研究圆环几何尺寸和蒸汽温度对其自持续运动的影响.
三种不同条件下圆环纸片的自持续摇摆状态如图10所示.
图10（a）、（d）对应的是无量纲圆环外径R1=0.9，蒸汽温度Tq=20时圆环纸片自持续摇摆状态的时程曲线和相图，图10（b）、（e）对应的是无量纲圆形环外径R1=1.2，蒸汽温度Tq=30时圆环纸片自持续摇摆状态的时程曲线和相图，图10（c）、（f）对应的是无量纲圆环外径R1=1.35，蒸汽温度Tq=40时圆环纸片自持续摇摆状态的时程曲线和相图，并且三种条件下固定了圆环内外径之比为R0/R1=0.5. 由此分别得出小外径低温度、中外径中温度和大外径高温度三种自持续摇摆. 图10（a）、（b）、（c）分别反映了三种条件下的左右摇摆幅度以及各自的频率，且都先是静止然后摇摆再逐渐趋于稳定的自持续摇摆. 图10（d）、（e）、（f）则是三者的相图，都各自呈现最终收敛于一个极限环上，也反映了系统在不同条件下都趋于稳定的自持续摇摆状态.
基于以上的数值分析，发现蒸汽温度与圆环外径的改变对结构的振动频率与幅度值有着较大的影响.
4.1　蒸汽温度的影响
图11反应了蒸汽温度对圆环硅油纸片自持续摇摆的影响，设定外半径R1=1.25，内外径之比R0/R1=0.5，T0=1/0.167，τ=1/0.485，ρc=1，A=4.35，c=0.98，k=1. 图11（a）是不同蒸汽温度条件下自持续摇摆的极限环，发现当蒸汽温度值设置较低时即Tq≤5.2，图中显示为一点为静止状态，因输入的能量不足，圆环轻微的变形不能达到很大的重心位置变化，整体仍处于稳定状态. 随着温度逐渐升高，输入的能量也不断增加，当Tq=20，35，50触发的自持续摇摆状态. 图11（b）为不同温度条件下圆环纸片自持续摇摆实验与数值模拟频率图，红线为数值模拟结果，蓝点为实验数据. 可以看出，随着温度的增加，摇摆的频率也在增加，实验数据点与数值模拟结果吻合性较强，可见，本文计算模型具有一定的准确性与可靠性.
4.2　圆环外径的影响
图12为圆环外径对圆环硅油纸片自持续摇摆的影响，设定蒸汽温度Tq=25，内外径之比R0/R1=0.5，T0=1/0.167，τ=1/0.485，ρc=1，A=4.35，c=0.98，k=1. 图12（a）绘制了不同圆环外径下对应自持续摇摆的极限环，其中将圆环外径值取的较小时即R≤0.72，在图12（a）中显示为一点，为静止状态，因圆环面积小，与蒸汽接触面积小，输入的能量也比较小，重心位置难以升高，故整体仍处于稳定状态. 随着半径逐渐增大，接触面积也随之增加，当R=1，1.4，1.8触发的自持续摇摆状态，它们的极限环如图12（a）所示. 图12（b）为不同圆环外径条件下自持续摇摆的频率和振幅，它们均随着圆环外径的增大而增大.
4.3　圆环内外径比的影响
图13为圆环内外径比值对圆环纸片自持续摇摆的影响，计算中保持圆环外径R1=1.25不变，通过控制内外径比值，确定不同的圆环内径大小. 图13（a）为蒸汽温度Tq=25条件下，内外径比值从0.2到0.8变化，得到相应的的频率与振幅；图13（b）为蒸汽温度Tq=35条件下，内外径比值从0.2到0.8变化，得到对应的频率与振幅. 可以发现通过增大内外径比值，即在圆环外径一定的情况下，增大圆环内径时，振动的频率和幅值逐渐增大.
5　模式转换
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通过前文研究发现可以通过增加蒸汽温度、圆环外径和内外径比值，来增大圆环纸片摇摆的频率与幅值，当幅值达到阈值就会发生跃迁，圆环纸片便会发生翻转，实现由自持续摇摆向翻滚模式的转变. 图14是圆环纸片模式转换相图. 圆环纸片固定内外径比值R0/R1=0.5，基于温度和尺寸两个影响因素，实验和数值模拟给出不同尺寸与温度条件下结构所处的运动模式，图中给出两个临界条件，其中红色曲线1是数值模拟给出的静止与自持续摇摆的分界线，蓝色曲线2是数值模拟给出的自持续摇摆与翻滚模式的分界线. 图中的三种背景则是大量实验获得的模式分布，可以看出数值模拟的临界条件曲线与实验临界条件吻合较好. 当条件控制在曲线1下方，即圆环外径较小或者蒸汽温度较低时，系统输入能量不足，被阻尼所消耗，最终均处于静止状态；当温度升高或者增加圆环外径，即条件控制在曲线1和2之间，输入能量刚好满足系统的能量消耗，实现了自持续摇摆的运动状态，并且该条件下自持续摇摆的频率与幅值随着温度和外径的增加而增加；当条件控制在曲线2上方时，即较高的蒸汽温度与较大的圆环外径，使得圆形剪纸摇摆频率和幅值足够大而实现自持续翻转运动. 可见，通过调整圆环几何尺寸与蒸汽温度，可以实现运动模态的转变.
6　结论
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硅油纸片剪纸材料在热蒸汽驱动下可以连续运动，这是由于其快速变化和不均匀的热膨胀所致，这种自持续运动是由其势能、弹性、热传导和热膨胀之间的相互作用决定的. 温度可以使材料产生热变形，改变其弯曲程度，进而改变重心位置，直至失去稳定而倾倒，提升温度或增大尺寸都将使结构重心位置远离结构中心点，会失去稳定导致自持续摇摆运动，进一步改变条件，重心位置将更加远离中心点，会产生更大的力矩，突破阈值而出现自持续翻滚模式. 综上所述，本研究的主要结论如下：
（1）发现用热蒸汽可以驱动硅油纸片产生自持续的运动，构建了圆环剪纸材料多模态自持续系统，实现了三种不同的自持续运动模式：静止，摇摆和翻滚. 通过实验发现结构尺寸和热蒸汽温度是影响运动模式的关键因素，调控这两种因素可以实现三种模式的相互转化，并给出了相互转化的临界条件和相图.
（2）建立了蒸汽驱动下圆环纸片自持续运动的几何模型，通过编程计算可以预测其摇摆频率与热蒸汽温度、结构半径尺寸之间的关系. 理论预测与实验结果吻合性强，为该自持续运动系统提供了可靠的理论解释.
（3）通过对硅油纸片及类似具有光、热敏感性的软薄材料进行结构设计，可以实现可控的多模态运动模式.
基金
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  • 国家自然科学基金面上项目(12172001)
  • 安徽省高校科研项目(2024AH050237)
文献
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文章信息
doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.049
  • 接收时间：2024-10-31
  • 首发时间：2026-03-20
  • 出版时间：2025-02-28
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出版历史
  • 收稿日期：2024-10-31
基金
国家自然科学基金面上项目(12172001)
安徽省高校科研项目(2024AH050237)
作者信息
    安徽建筑大学土木工程学院，合肥，230601

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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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