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Article(id=1243301634370028023, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243301630683234768, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2025.01.014, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1720886400000, receivedDateStr=2024-07-14, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774355809060, onlineDateStr=2026-03-24, pubDate=1737302400000, pubDateStr=2025-01-20, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774355809060, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774355809060, creator=13701087609, updateTime=1774355809060, updator=13701087609, issue=Issue{id=1243301630683234768, tenantId=1146029695717560320, journalId=1240685776644648972, year='2025', volume='29', issue='1', pageStart='1', pageEnd='169', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=0, createTime=1774355808181, creator=13701087609, updateTime=1774355986739, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243302379672678863, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243301630683234768, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243302379672678864, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243301630683234768, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=145, endPage=155, ext={EN=ArticleExt(id=1243301634961424897, articleId=1243301634370028023, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=Dynamic characteristics analysis of vibration isolation system with nonlinear energy sink on flexible foundation, columnId=1242129247389684400, journalTitle=Journal of Ship Mechanics, columnName=Hydro/Structural/Acoustics, runingTitle=null, highlight=null, articleAbstract=

For low-frequency line spectrum control in naval and other engineering fields, a nonlinear energy sink vibration isolation system dynamics model considering a flexible foundation was established, and the nonlinear dynamics of the system was analyzed using the harmonic balance method and Runge-Kutta method. The influence of the foundational parameters on the amplitude-frequency characteristics of the system was explored, and the nonlinear dynamical behavior of the system at the resonant frequency was analyzed. The influence law of mass ratio, cubic stiffness, and damping on the damping effect of nonlinear energy sink was explored with vibration power flow as the evaluation parameter. For the computational model analyzed, research shows: when the foundation stiffness is weak, the nonlinear energy sink has a large effect on the amplitude and frequency response characteristics of the equipment; for specific parameters, the state of motion of the base and the equipment at the first order resonant frequency may be different; the nonlinear energy sink has a superior damping effect, reducing the peak power flow to the base by 13.95 dB in this example.

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针对舰船等工程领域低频线谱控制难题，建立考虑柔性基础的非线性能量阱隔振系统动力学模型，利用谐波平衡法和Runge-Kutta法分析系统的非线性动力学特性。探讨基础参数对系统幅频特性的影响，并分析系统在共振频率处的非线性动力学行为。以振动功率流为评价参数，探讨非线性能量阱质量比、立方刚度、阻尼对减振效果的影响规律。针对本文计算模型的研究表明：当基础刚度较弱时，非线性能量阱对设备的幅频响应特性影响较大；对于特定的参数，基础和设备在一阶共振频率处的运动状态可能是不同的；非线性能量阱减振效果良好，在本算例中传递到基础的功率流峰值降低了13.95 dB。

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通讯作者，E-mail：
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陈森（1998-），男，硕士研究生

朱翔（1980-），男，博士，教授，通讯作者，E-mail：。

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陈森（1998-），男，硕士研究生

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陈森（1998-），男，硕士研究生

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朱翔（1980-），男，博士，教授，通讯作者，E-mail：。

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柔性基础非线性能量阱隔振系统动力学特性分析
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陈森 1a, 1b , 朱翔 1a, 1b, 2 , 李天匀 1a, 1b, 2 , 陈树鑫 1a, 1b , 戴维 1a, 1b, 2
船舶力学 | 流体与结构声学 2025,29(1): 145-155
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船舶力学 | 流体与结构声学 2025, 29(1): 145-155
柔性基础非线性能量阱隔振系统动力学特性分析
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陈森1a, 1b, 朱翔1a, 1b, 2 , 李天匀1a, 1b, 2, 陈树鑫1a, 1b, 戴维1a, 1b, 2
作者信息
  • 1a.华中科技大学　船舶与海洋工程学院，武汉　430074
  • 1b.华中科技大学　船舶和海洋水动力湖北省重点实验室，武汉　430074
  • 2.上海交通大学　高新船舶与深海开发装备协同创新中心，上海　200240

    • 陈森（1998-），男，硕士研究生

    朱翔（1980-），男，博士，教授，通讯作者，E-mail：。

通讯作者:

通讯作者，E-mail：
Dynamic characteristics analysis of vibration isolation system with nonlinear energy sink on flexible foundation
Sen CHEN1a, 1b, Xiang ZHU1a, 1b, 2 , Tian-yun LI1a, 1b, 2, Shu-xin CHEN1a, 1b, Wei DAI1a, 1b, 2
Affiliations
  • 1a.School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
  • 1b.Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics (HUST), Huazhong University of Science and Technology, Wuhan 430074, China
  • 2.Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai Jiao Tong University, Shanghai 200240, China
出版时间: 2025-01-20 doi: 10.3969/j.issn.1007-7294.2025.01.014
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摘要
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针对舰船等工程领域低频线谱控制难题，建立考虑柔性基础的非线性能量阱隔振系统动力学模型，利用谐波平衡法和Runge-Kutta法分析系统的非线性动力学特性。探讨基础参数对系统幅频特性的影响，并分析系统在共振频率处的非线性动力学行为。以振动功率流为评价参数，探讨非线性能量阱质量比、立方刚度、阻尼对减振效果的影响规律。针对本文计算模型的研究表明：当基础刚度较弱时，非线性能量阱对设备的幅频响应特性影响较大；对于特定的参数，基础和设备在一阶共振频率处的运动状态可能是不同的；非线性能量阱减振效果良好，在本算例中传递到基础的功率流峰值降低了13.95 dB。

柔性基础  /  非线性能量阱  /  谐波平衡法  /  振动功率流

For low-frequency line spectrum control in naval and other engineering fields, a nonlinear energy sink vibration isolation system dynamics model considering a flexible foundation was established, and the nonlinear dynamics of the system was analyzed using the harmonic balance method and Runge-Kutta method. The influence of the foundational parameters on the amplitude-frequency characteristics of the system was explored, and the nonlinear dynamical behavior of the system at the resonant frequency was analyzed. The influence law of mass ratio, cubic stiffness, and damping on the damping effect of nonlinear energy sink was explored with vibration power flow as the evaluation parameter. For the computational model analyzed, research shows: when the foundation stiffness is weak, the nonlinear energy sink has a large effect on the amplitude and frequency response characteristics of the equipment; for specific parameters, the state of motion of the base and the equipment at the first order resonant frequency may be different; the nonlinear energy sink has a superior damping effect, reducing the peak power flow to the base by 13.95 dB in this example.

flexible foundation  /  nonlinear energy sink  /  harmonic balance method  /  power flow analysis
陈森, 朱翔, 李天匀, 陈树鑫, 戴维. 柔性基础非线性能量阱隔振系统动力学特性分析. 船舶力学, 2025 , 29 (1) : 145 -155 . DOI: 10.3969/j.issn.1007-7294.2025.01.014
Sen CHEN, Xiang ZHU, Tian-yun LI, Shu-xin CHEN, Wei DAI. Dynamic characteristics analysis of vibration isolation system with nonlinear energy sink on flexible foundation[J]. Journal of Ship Mechanics, 2025 , 29 (1) : 145 -155 . DOI: 10.3969/j.issn.1007-7294.2025.01.014
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0　引言
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工程中的大多数振动是有害的，如潜艇低速航行时，由机械设备周期性运转造成的机械噪声将严重影响潜艇的声隐身性能，因此各种振动的控制方法与技术[1-3]得到了广泛的关注与研究。其中被动控制技术具有结构简单、可靠性高、实现成本低等优点，被广泛应用于工程中，包括调谐质量阻尼器（TMD）、非线性能量阱（nonlinear energy sink，NES）等。与TMD相比，由于NES的刚度为强非线性，其自振频率并不恒定，因此NES可以与主系统组合后形成一系列非线性共振条件，从而具有宽频减振的优点[4]。目前，NES技术正逐渐被应用到各个工程领域的减振中[5]。
许多学者针对NES的消极隔振、积极隔振系统的振动开展了大量研究工作。Gendelman等[6]指出在不同参数下，系统存在着混沌响应、强（弱）准周期响应以及周期响应；Zang等[7]通过谐波平衡法和Runge-Kutta法研究了NES系统的复杂动力学行为；AL-Shudeifat[8]提出了一种高效的双稳态NES。刘中坡等[9]对附加轨道型非线性能量阱的高层结构开展了振动台实验，结果表明这种NES能同时抑制结构多个模态的振动；Zhang等[10]利用惯容器代替了传统NES中的附加质量，数值结果表明：当附加质量很小时，仍然有优异的减振效果；Geng等[11]利用间隙振子提出了一种限幅型非线性能量阱装置，通过理论和实验研究表明，NES的振幅在合理的范围内，仍然对主系统有良好的减振效果，促进了NES在工程中的应用；范舒铜等[12]提出了一种黏弹性非线性能量阱，其减振性能优于传统的NES。这些工作为动力吸振器的设计提供了新思路。楼京俊等[13]设计了一种可垂向承载的柔性铰链型NES装置，试验结果表明：与线性吸振器相比，该装置具有较好的宽频减振效果；李爽等[14]讨论了激励力幅值对具有NES隔振系统全局分岔特性的影响；郑中强等[15]将NES应用在受随机波浪载荷作用下的固定式导管架海洋平台，与调谐质量阻尼器（TMD）相比，NES具有更强的鲁棒性。在公开发表的文献中，非线性能量阱技术在舰船领域开展的相关研究较少。
目前，在计算NES这一类非线性系统的稳态响应时，通常采用的是复平均法[16-17]、谐波平衡法[18-19]这两类近似解析研究方法和Runge-Kutta数值积分方法。李响等[20]通过对比复平均法、谐波平衡法这两种近似解析方法发现，对于强非线性系统，谐波平衡法更准确。
对于积极隔振系统，许多学者是基于刚性基础进行研究分析的，且多采用力传递率评价减振效果。对于舰船、航空航天等领域的积极隔振系统，基础的柔性往往不可忽略。此外，对于耦合了NES的积极隔振系统，以基础的振动能量来评估减振效果也越来越受到重视，这也是需要考虑基础的原因所在。对于舰船上的隔振系统，在理论分析中，一般将基础考虑成等效质量[21]、梁基础[22]、板基础[23]。在原隔振系统引入NES可能会导致复杂的非线性动力学行为[5]，为此，本文参考文献[21]将柔性基础简化为弹簧-质量构成的离散系统。
而在以往的研究中，利用振动功率流来评价NES减振效果的工作开展得相对较少。Yang[24]、Shi[25]、Zhu[26]等在利用振动功率流评估非线性减/隔振分析方面做了大量工作，研究了光滑非线性系统、非光滑非线性系统、复合材料结构等系统的功率流特性。这些工作具有比较重要的理论意义和工程应用价值。
综上所述，对于耦合NES的积极隔振系统，许多学者是基于刚性基础进行研究分析的，且多以力传递率评价减振效果，而实际的船舶隔振系统往往为柔性基础。因此，本文结合工程实际，建立柔性基础的非线性能量阱隔振系统动力学模型，利用谐波平衡法和Runge-Kutta法分析系统的动力学特性，并通过振动功率流分析非线性能量阱的减振效果。
1　理论模型
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本文分析的柔性基础非线性能量阱隔振系统如图1所示。在图1中，将柔性基础等效为弹簧-阻尼-质量构成的离散系统，其中基础的等效质量为M，等效刚度为k0，等效阻尼为c0；机械设备m1通过线性刚度k1、线性阻尼c1与柔性基础相连；NES由附加质量m2、立方刚度kN、线性阻尼cN构成，并与机械设备m1耦合连接，并记质量比μ=m2/m1。简谐激励力Fcos（ωt）集中作用于机械设备m1上。
根据牛顿第二定律，系统动力学方程为
式中，x0、x1、x2分别为基础、设备、NES的位移。
1.1　谐波平衡法
根据谐波平衡法（HBM），系统的近似解可假设为如下形式：
根据式（2）可以得到系统的速度与加速度。式中下标i=0、1、2，j为谐波阶数，谐波系数向量Ri与S（τ）分别为
将上述近似解代入方程（1），根据Galerkin法有
令谐波系数向量Ri为常数，对各阶谐波进行平衡，可得到一个非线性代数方程组。利用Newton-Raphson迭代法与伪弧长延伸法求解各阶谐波系数，进而得到系统的幅频响应特性曲线。保持谐波系数向量Ri为时间的变量，得到一个常微分方程组，根据Floquet稳定性理论可以判断系统解的稳定性。
1.2　数值方法
针对以上系统，可采用四阶Runge-Kutta法（R-K）对上述非线性方程组进行数值求解，并以此验证谐波平衡法分析的近似结果。利用MATLAB中的ode45函数实现Runge-Kutta法，求解时取计算时长为5000个周期，周期T=2π/ω，时间步长取为T/100，并取最后200个周期的稳态响应数据进行计算分析，即取最后200个周期的均方根值作为系统的响应幅值。
2　数值验证与分析
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2.1　方法验证
由于NES的刚度为立方非线性，故可忽略偶次和高阶谐波的影响，在谐波平衡法中近似解只需假设1阶和3阶谐波[11]。本文采用谐波系数的均方根值来描述系统的幅频响应特性曲线，即
式中，xir为系统位移的均方根值。
柔性基础非线性能量阱隔振系统的各参数取值如下：M=100 kg，k0=1×107 N/m，c0=50 N·s/m，m1=30 kg，k1=6×104 N/m，c1=50 N·s/m，m2=3 kg，kN=5×107 N/m3，cN=80 N·s/m，激励力幅值F=50 N。
基础位移、设备位移的近似解析解与数值解对比如图2所示。从图2可以看出，近似解析方法（HBM）与数值方法（R-K）吻合得很好，故解析解与数值解均是准确可信的。
2.2　讨论分析
首先，初步讨论NES的非线性刚度kN、阻尼cN对设备位移响应的影响，如图3所示。从图3（a）可以发现，随着非线性刚度kN的增大，设备位移响应呈现了不同的响应特性，当kN较大时（图中对应kN=5×108 N/m3），设备的位移幅频响应存在明显的硬化特性。并可由此推断，存在一个最优值kN使得设备位移响应的峰值最小；在图3（b）中，对比不同阻尼cN对设备位移的影响可以发现，随着阻尼cN的减小，设备位移响应的峰值呈现下降趋势。
当kN=5×108 N/m3、cN=50 N·s/m时，分别对设备位移稳态响应的稳定性进行判断，如图4所示。在图4（a）中，设备位移幅频响应的硬化区域有两段解的曲线是不稳定的；在图4（b）中，设备位移幅频响应在一阶共振频率附近是不稳定的。
3　柔性基础对含NES系统的影响
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3.1　柔性基础对系统幅频响应的影响
下面进一步研究基础刚度、质量对含NES的设备幅频响应峰值的影响。刚性安装的设备幅频响应峰值记为A1r，对应的频率为f1r；弹性安装的设备幅频响应峰值记为A1，对应的频率为f1。定义弹性安装设备与刚性安装设备的参数之比RA、Rf分别为
当激励力分别为F=10 N、50 N、100 N时，基础刚度、质量对带有NES的设备幅频响应峰值影响的二维等高线图如图5所示。
由图5中可知，设备幅频响应的共振频率与共振峰值随着基础刚度变化呈现出“梯田”式的变化趋势，说明基础刚度对设备幅频响应共振峰的影响占主导地位。当基础刚度接近设备隔振器刚度时，基础质量才会对设备幅频响应产生影响。随着基础刚度的减小，设备的共振频率向低频移动，而共振峰值在不断地增大。当激励力较小（F=10 N）或较大（F=100 N）时，随着基础刚度的减小，几乎同时影响设备的共振频率与共振峰值；当F=50 N时，基础刚度首先影响的是共振频率，进一步减小刚度，才会影响共振峰值。
综上所述，参考不同激励力下的等高线图，根据图5（e）中的等高线，本文给出了一个基础刚度参考值，取lg（k0）=6.25 N/m，即基础刚度k0与设备刚度k1之比k0/k1>30时，可以不考虑基础对含有NES的隔振系统的影响。
对于舰船上的隔振系统，一般来说，基础的刚度还是比较大的，因为较小的刚度会使基础的响应快速增大。在下面的计算分析中，基础参数保持原始参数不变。
3.2　系统的非线性动力学行为分析
下面采用非线性动力学时间序列分析方法，利用时间历程图、频谱图、相图和庞加莱映射图研究NES阻尼参数对系统运动状态的影响。在一阶共振频率ω=44.92 rad/s时，采用四阶Runge-Kutta法求解系统的稳态响应，系统的初始值均取为0。
图6分别对应阻尼cN=30 N·s/m、50 N·s/m、80 N·s/m时，基础和设备的系统动力学特性。从图6（a）、（d）中可见，当cN较小时，基础与设备均处于混沌运动状态；从图6（b）、（e）中可见，随着cN增大，设备处于准周期运动状态，但基础却处于混沌运动状态；从图6（c）、（f）中可见，当cN较大时，设备处于周期运动状态，但基础却处于弱准周期运动状态。综上所述，尽管此时的基础参数对设备的幅频特性影响很小，但是基础与设备二者在一阶共振频率处的运动状态可能是不同的。
4　振动功率流评估减振效果
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振动功率流可以综合性地预测和评估系统的动态特性，被广泛应用于系统隔振效果的理论分析。系统在一个周期内的时间平均输入功率流，以及传递到基础的时间平均功率流分别为
未控系统在基频处的时间平均输入功率流峰值记为，传递到基础的时间平均功率流峰值记为；带有非线性能量阱的系统在基频处的时间平均输入功率流峰值记为，传递到基础的时间平均功率流峰值记为。上述物理量均转换成分贝的形式，参考功率的基准值为10-12 W。下面分别定义两个功率流评价指标Rin、Rt，来评估非线性能量阱的吸振效果，即
从图4（b）可知，弱阻尼系数下的系统是不稳定的，在实际的工程应用中应该避免。下面讨论NES非线性刚度、阻尼对系统振动功率流的影响，其中阻尼系数cN=[70，120] N·s/m。对于不同的NES附加质量m2=1.5 kg（μ=5%）、3 kg（μ=10%），当非线性刚度和阻尼同时变化时，振动功率流评价指标Rin、Rt变化的二维等高线图如图7所示。
在图7（a）、（b）中，当NES的非线性刚度和阻尼同时变化时，Rin、Rt有着比较复杂的变化趋势，但二者的等高线图变化趋势基本一致。当刚度较小时，阻尼越小减振效果越好；但当刚度较大时，阻尼越大减振效果越好。在减振性能最优区域形成了明显的山峰，故非线性刚度存在着一个最优解，对应的Rin和Rt的最大值分别为3.56 dB、7.15 dB。在图7（c）、（d）中，Rin和Rt的变化趋势与图7（a）、（b）基本一致，也有着明显的山峰，但对应最优的非线性刚度增大。由于NES质量增大，减振效果更加显著，其对应的Rin和Rt的最大值分别为7.47 dB、13.95 dB。
从图7亦可知，当NES的非线性刚度和阻尼同时变化时，由于未控系统的振动功率流峰值始终不变，则可以认为：系统的时间平均输入功率流，以及传递到基础的时间平均功率流有着相似的变化趋势。
下面讨论非线性能量阱的质量和非线性刚度这两个关键设计参数对振动功率流评价指标Rin、Rt的影响，其变化曲线如图8所示。当NES的质量较小时，如2 kg以下，功率流评价指标Rin、Rt对非线性刚度kN并不敏感；随着附加质量的增大，减振效果对非线性刚度kN比较敏感，此时适度的kN有利于改善减振效果，但太大的kN又会抑制减振效果；当附加质量较大，达到3 kg以上时，增大kN可以显著改善减振效果，且质量m大于3 kg，即质量比大于10%时，Rin、Rt逐渐收敛，但kN太大的情况下随着质量增大，对应的Rin、Rt会缓慢减小。
5　结论
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本文建立了柔性基础的非线性能量阱隔振系统的动力学模型，利用谐波平衡法和Runge-Kutta法分析了系统的动力学特性，并通过振动功率流分析了非线性能量阱的减振效果。针对本文分析的计算模型，得到了以下结论：
（1）在基础参数中，刚度对主系统的影响最大，并给出了一个参考值，当基础刚度大于30倍的主系统刚度时，可以不考虑基础刚度对主系统振动特性的影响。
（2）引入了NES后，基础和设备在一阶共振频率处的运动状态可能是不同的，如当设备处于准周期运动时，基础却处于混沌运动。
（3）非线性能量阱减振效果优越，其非线性刚度存在着一个最优解。在本算例中，附加质量存在着一个上限值，上限质量比约为设备的10%，此时传递到基础的功率流峰值降低了13.95 dB；系统的时间平均输入功率流与传递到基础的时间平均功率流变化趋势相似。
基金
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  • 国家自然科学基金资助项目(51879113; 51839005; 12202152)
文献
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文章信息
doi: 10.3969/j.issn.1007-7294.2025.01.014
  • 接收时间：2024-07-14
  • 首发时间：2026-03-24
  • 出版时间：2025-01-20
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  • 收稿日期：2024-07-14
基金
国家自然科学基金资助项目(51879113; 51839005; 12202152)
作者信息
    1a.华中科技大学　船舶与海洋工程学院，武汉　430074
    1b.华中科技大学　船舶和海洋水动力湖北省重点实验室，武汉　430074
    2.上海交通大学　高新船舶与深海开发装备协同创新中心，上海　200240

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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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