分段阻尼隔振系统的动力学特性分析

阮子悦，邢海军，卢学礼

引 en

分段阻尼隔振系统的动力学特性分析^*

阮子悦
机构：
石家庄铁道大学机械工程学院, 石家庄 050043
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，邢海军
机构：
石家庄铁道大学机械工程学院, 石家庄 050043
×
，卢学礼
机构：
石家庄铁道大学机械工程学院, 石家庄 050043
×

石家庄铁道大学机械工程学院, 石家庄 050043；

Dynamic Characteristic Analysis of a Piecewise Damping Vibration Isolation System^*

Ruan Ziyue
Affiliation：
School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043 , China
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，Xing Haijun
Affiliation：
School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043 , China
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，Lu Xueli
Affiliation：
School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043 , China
×

School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043 , China；

通讯作者:

邢海军,E-mail:xinghj@stdu.edu.cn

中图分类号:O322;TH113.1

文献标识码:A

文章编号:1672-6553-2023-21(5)-044-009

DOI:10.6052/1672-6553-2022-061

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出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 变阻尼装置及阻尼特性分析
1.1 变阻尼装置
1.2 分段阻尼特性
2 动力学方程及求解
3 幅频响应特性
3.1 幅频特性求解
3.2 幅频特性分析
4 积极隔振特性
4.1 力传递率求解
4.2 隔振特性分析
5 结论
参考文献

摘要

针对传统线性隔振器在降低共振峰值的同时会牺牲隔振性能的矛盾,设计了一种含分段阻尼的隔振器.首先,采用移动凸轮变阻尼装置,通过凸轮廓线的设计使系统的垂向阻尼系数的受振动位移大小控制并呈现分段线性特征,分析了该装置的阻尼特性.然后将分段阻尼装置应用于隔振器中,建立了含分段阻尼的积极隔振系统模型及其动力学方程,通过能量等效原理求出了分段阻尼系统的等效线性阻尼系数,求解了简谐力激励下系统响应的理论解,并用四阶龙格-库塔法数值仿真验证了理论解的正确性.最后研究了分段阻尼隔振系统的动态特性,分析了主要参数对幅频响应特性与力传递率特性的影响.结果表明,通过合理的参数选择,分段阻尼隔振器可兼顾无阻尼隔振器与线性阻尼隔振器的优点,既能有效降低系统的共振峰值,又能保证高频区域的优秀隔振性能,为新型非线性隔振器的设计提供了理论依据.

Abstract

In view of the contradiction that the traditional linear vibration isolator will sacrifice the vibration isolation performance while reducing the resonance peak, a vibration isolator with piecewise damping is designed. First of all, the variable damping device of moving cam is used. With the design of cam profile, the vertical damping coefficient of the system is controlled by the magnitude of vibration displacement and presents piecewise linear characteristics, and the damping characteristics of the device are analyzed. Then the piecewise damping device is applied to the vibration isolator, and the active vibration isolation system model with piecewise damping and its dynamic equation are established. The equivalent linear damping coefficient of the piecewise damping system is obtained through the energy equivalence principle, and the theoretical solution of the system response under simple harmonic force excitation is solved. The correctness of the theoretical solution is verified by the numerical simulation of the fourth-order Runge-Kutta method. Finally, the dynamic characteristics of the piecewise damping vibration isolation system are studied, and the effects of the main parameters on the amplitude frequency response characteristics and force transmissibility characteristics are analyzed. The results show that through reasonable parameter selection, the piecewise damping vibration isolator can take into account the advantages of undamped vibration isolator and linear damping vibration isolator, which can not only effectively reduce the resonance peak value of the system, but also ensure the excellent vibration isolation performance in the high frequency region. It provides a theoretical basis for the design of new nonlinear vibration isolators.

关键词

移动凸轮；分段阻尼；能量等效；积极隔振；幅频响应；力传递率

Keywords

translating cam ； piecewise damping ； energy equivalence ； active vibration isolation ； amplitude frequency response ； force transmissibility

引言
隔振是工程领域控制振动的一种重要方法，隔振器通过弹簧和阻尼器等弹性元件将振源与地基之间的近刚性连接改为弹性连接，以隔离或减少振动能量的传递^[1].传统线性振动理论认为，隔振的起始有效频率为系统固有频率的倍，为抑制低频共振，隔振器应具有一定的阻尼，但阻尼的存在会牺牲掉部分高频隔振效果^[2，3].因此阻尼恒定的线性隔振器难以满足日趋严苛的减振需求，近20年来，非线性隔振器因阻尼和刚度可调节的优越性而备受关注.
Ibrahim^[4]对非线性被动隔振器的研究成果及发展进行了综述.Guo等^[5]研究了基于非线性黏性阻尼的隔振系统，分析了力传递率与位移传递率特性，得出非线性阻尼隔振器能够在更宽频域内改善隔振性能.Peng等^[6-8]运用输出频响函数的概念，推导出了振动传递与非线性参数之间的显式解析关系; 进一步研究了含立方非线性阻尼的多自由度隔振系统; 运用谐波平衡法研究了立方非线性阻尼隔振系统的动态特性.陆泽琦^[9，10]在非线性隔振器的研究中既考虑刚度非线性又考虑阻尼非线性，设计并分析了新型双层非线性隔振器; 综述了非线性隔振理论和应用的十年进展.Sun等^[11]用平均法分析了几何非线性阻尼隔振器的力传递率特性.Xiao等^[12]分析了阻尼线性项对立方非线性阻尼的单层隔振效果的影响，得出不含线性阻尼的单纯立方非线性阻尼有更好的隔振特性.Laalej等^[13]试验验证了立方非线性阻尼隔振系统特性，可提升共振区隔振效果而不影响其他频域.赵祥等^[14]试验研究了一种新型变阻尼式调谐质量阻尼器（TMD）的力学性能.李茂^[15]研发设计了新型被动变阻尼耗能装置，分别计算了单阶梯变阻尼和多阶梯变阻尼装置的风振控制效果.付伟庆等^[16，17]设计制作了多阶梯被动变阻尼耗能装置，阻尼系数大小受速度控制.杜香刚^[18]提出了基于荷载特征的非线性隔振器，并对车辆-轨道-隧道耦合模型进行了仿真验证.蒲华燕等^[19]设计研究了具有频变阻尼特性的四参数隔振系统.刘国勇等^[20]利用两层变杆长X型机构，提出一种新型隔振器，除了能保证低刚度和高阻尼输出之外，还具有较好的稳定性与静承载能力.高雪等^[21]建立了双线性分段光滑隔振系统模型，研究了摒除不利于隔振的非线性动力学现象的设计方法.邵栋等^[22]从功率流角度研究非线性隔振，发现非线性刚度能拓宽隔振频带，非线性阻尼能保持较优高频隔振性能.刘兴天等^[23]在隔振器中引入库仑摩擦阻尼器，探究了其动态特性.张婉洁等^[24]研究了基于相对速度反馈的阻尼控制半主动隔振系统，讨论了主要参数对振动控制的影响.顾栋浩等^[25]通过圆环弯曲获得准零刚度，水平阻尼器引入非线性阻尼，研究了系统参数对传递率的影响.昌耀鹏等^[26]采用遗传算法优化双层主动隔振系统，提高了隔振性能.Zhou等^[27，28]提出了含凸轮-滚轮机构的准零刚度隔振器，通过实验测定了不同工况下的力传递率.Li等^[29]用凸轮-弹簧结构实现非线性刚度，在准零刚度隔振系统的动力学分析中考虑了摩擦.尹蒙蒙等^[30]提出了X-QZS隔振器，从动力学角度分析了其相对于一般准零刚度隔振器的优势.尹佑旺等^[31]基于柔性曲梁的力学特性设计了新型仿生柔性隔振结构，能够实现扭转方向的低频隔振，具有较大的轴向负载能力.综上所述，非线性隔振器可通过对系统非线性参数的设计与优化，得到优于线性系统的隔振效果.通过隔振器的结构设计实现非线性隔振，仍为当前值得研究的热点问题.
本文基于移动凸轮机构，提出一种分段阻尼隔振系统，其具有“大位移有阻尼，小位移无阻尼”的分段线性特征，以解决传统线性隔振中阻尼抑制共振与高频隔振效果的矛盾.理论求解与数值仿真结合，研究分段阻尼系统的幅频响应与隔振特性，为非线性阻尼隔振系统的设计及参数优化提供参考.
1 变阻尼装置及阻尼特性分析
1.1 变阻尼装置
基于移动凸轮机构，使得隔振器在工作中能够产生非线性阻尼力，其大小随系统在竖直方向上位移的变化而改变.图1（a）所示为移动凸轮变阻尼装置，图1（b）为变阻尼装置三维模型，通过对移动凸轮外缘轮廓线的设计，就能得到不同特性的阻尼形式，如分段阻尼、连续变阻尼等.
图1 移动凸轮变阻尼装置原理图
Fig.1 Schematic diagram of moving-cam variable damping mechanism
移动凸轮被动变阻尼装置主要由三部分结构组成:可在竖直方向上移动的凸轮、水平对称布置的线性阻尼器以及滚动轴承.凸轮块两侧轮廓对称，外缘导轨设计成T形槽的形式，将滚动轴承的内圈固定于阻尼器活塞杆外端，外圈嵌套于移动凸轮的T形导轨槽内.装置在工作时，移动凸轮在竖直方向发生位移，滚动轴承在T形导轨槽内滚动，带动活塞杆水平运动，就产生了水平方向的阻尼力.由于横向阻尼的布置无须预紧力，运动过程中的滚动摩擦力很小，可忽略系统的摩擦阻尼.
该装置完全由机械结构组成，结构简单、可经过设计凸轮廓线来控制阻尼器的非线性特征.如设凸轮廓线所对应的函数关系式为y=y（x），它可以是分段函数，又可以是连续函数，改变该函数关系式，就可得到各种不同的阻尼形式.本文主要针对分段阻尼形式展开研究.
1.2 分段阻尼特性
在有阻尼振动系统中，阻尼力实际上起到消耗系统能量的作用，故横向阻尼消耗的能量可等效为由垂向阻尼消耗，从而得到垂向等效阻尼系数.
如图1（c）所示，设垂向位移为x，水平位移为y，阻尼分段点为x=x₀，凸轮廓线倾角为α，横向总阻尼系数为c，由两个横向阻尼器产生的垂向等效阻尼系数为c′.根据阻尼耗能相等原则有

c^{'} \frac{d y}{d t} \frac{d y}{d t} = c \frac{d x}{d t} \frac{d x}{d t}

(1)

得到垂向等效阻尼系数的表达式

c^{'} = c {(\frac{d y}{d x})}^{2}

(2)

故分段阻尼的垂向等效阻尼系数为

c^{'} = \{\begin{matrix} 0 | x | ⩽ x_{0} \\ c \cdot {t a n}^{2} α | x | > x_{0} \end{matrix}

(3)

由式（3）可知，分段阻尼以x₀为分段点，具有小位移时无阻尼，大位移时有阻尼的特性.因此，当系统开始振动时，分段阻尼隔振器提供的阻尼力具有分段线性特征.
2 动力学方程及求解
将上节所述移动凸轮被动变阻尼装置应用于隔振器中，得到简谐力激励下分段阻尼隔振系统模型（积极隔振模型），如图2所示.本节以分段阻尼凸轮廓线倾角α=45°为例，建立起分段阻尼隔振系统的动力学方程并进行求解.
图2 分段阻尼隔振系统模型
Fig.2 Model of piecewise damping vibration isolation system
含有分段阻尼的单自由度隔振系统在简谐力激励下的动力学方程为

m \ddot{x} + c^{'} \dot{x} + k x = F_{0} s i n ω t

(4)

其中m为系统质量，k为系统刚度，x为系统位移， $\dot{x}$ 为位移对时间的一阶导数， $\ddot{x}$ 为位移对时间的二阶导数.注意到式（4）中分段阻尼c′具有分段线性特征，故难以直接求得方程的精确解析解.
当系统的阻尼不是很大时，响应近似为正弦曲线，可以利用能量等效原理对复杂阻尼进行线性化处理，得到等效黏性阻尼^[32].设c_eq为c′经能量等效后的阻尼系数.
引入以下参数:
$ω_{0} = \sqrt{\frac{k}{m}}, ξ_{e q} = \frac{c_{e q}}{2 m ω_{0}}, Δ_{s t} = \frac{F_{0}}{k}$
则式（4）可化为

\ddot{x} + 2 ξ_{e q} ω_{0} \dot{x} + ω_{0}^{2} x = ω_{0}^{2} Δ_{s t} s i n ω t

(5)

设系统的稳态解为

x = A s i n (ω t - φ)

(6)

其中参数A表示分段阻尼隔振系统的振幅.
等效阻尼在稳态一个周期内消耗的能量表示为

\int_{0}^{T} c_{eq} \dot{x} \dot{x} d t = c_{e q} π ω A^{2}

(7)

为了求解分段阻尼在稳态下的一个周期内的耗能，做出稳态时任意单个响应周期内的位移时程曲线，并用阴影区域标记阻尼耗能区间，系统位移与阻尼分段点关系如图3所示.可见，系统在单个稳态周期内滚动轴承四次经过分段点，分段阻尼耗能区间由对称的两部分组成，只有当系统位移绝对值超过分段点x₀时，阻尼器才开始耗能.
图3 单个响应周期内位移与阻尼分段点的关系
Fig.3 Relationship between displacement and damping segment point in a single response period
分段阻尼在稳态一个周期内消耗的能量表示为

2 \int_{\frac{φ 0}{ω}}^{\frac{π - φ 0}{ω}} c \dot{x} \dot{x} d t = A^{2} (π - 2 φ_{0} - s i n 2 φ_{0})

(8)

根据振动能量等效原理，当振动系统处于稳态时，在一个振动周期内，分段阻尼消耗的能量应当等同于等效黏性阻尼消耗的能量，故将式（7）与式（8）联立，解得

c_{e q} = c (1 - \frac{2 φ_{0}}{π} - \frac{s i n 2 φ_{0}}{π})

(9)

在式（9）中

φ_{0} = {s i n}^{- 1} \frac{x_{0}}{A}

(10)

由阻尼比与阻尼系数的关系，可以得到系统等效阻尼ξ_eq比的表达式为

ξ_{e q} = ξ [1 - \frac{2 {s i n}^{- 1} x_{0} / A}{π} - \frac{\sin (2 {s i n}^{- 1} x_{0} / A)}{π}]

(11)

式（11）中，参数ξ为横向阻尼比，与系统横向总阻尼系数c相对应.由以上推导可见，等效阻尼比ξ_eq的表达式与振幅有关，即与归一化频率λ相关，不同频率下的等效阻尼比不同.
这样经过能量等效，就将较为复杂的分段阻尼系统转化为了等效线性系统，可运用简谐激励下系统强迫振动的相关理论简化后续计算.设振幅放大因子为α，归一化频率λ=ω/ω₀，则等效线性系统的振幅放大因子表示为

α = \frac{A}{Δ_{s t}} = \frac{1}{\sqrt{{(1 - λ^{2})}^{2} + {(2 ξ_{e q} λ)}^{2}}}

(12)

将式（11）代入式（12），可得分段阻尼系统的振幅放大因子表达式，从而求得系统的稳态解为

\begin{matrix} x = \frac{Δ_{s t} s i n (ω t - φ)}{\sqrt{{(1 - λ^{2})}^{2} + ε^{2}}} \\ φ = {t a n}^{- 1} \frac{ε}{1 - λ^{2}} \end{matrix}\}

(13)

其中

ε = 2 ξ [1 - \frac{2 {s i n}^{- 1} x_{0} / A}{π} - \frac{\sin (2 {s i n}^{- 1} x_{0} / A)}{π}] λ

(14)

3 幅频响应特性
3.1 幅频特性求解
幅频响应特性即振动系统稳态响应振幅随频率比的变化关系，幅频响应曲线即α-λ曲线.
由式（11）、式（12）可见，等效线性系统的振幅放大因子α与横向阻尼归一化频率λ以及稳态响应振幅A有关，且式（12）等式两边都含有参数A.为了求解幅频响应特性，应先取一组n个稳态响应振幅:

(15)

对于每一个振幅A，都有一个振幅放大因子α，相位角φ₀与等效阻尼比ξ_eq与之对应:

(16)

(17)

(18)

反解式（12）得到归一化频率与振幅放大因子α和等效阻尼比ξ_eq的关系为

(19)

在式（19）中
$\begin{matrix} λ (i) = \sqrt{1 - 2 ξ_{e q} (i)^{2} \pm \sqrt{\frac{1}{α (i)^{2}} - 4 ξ_{e q} (i)^{2} + 4 ξ_{e q} (i)^{4}}} \\ (i = 1, 2, 3, \dots, n) \end{matrix}$
令归一化阻尼分段点 ${\bar{x}}_{0} = x_{0} / Δ_{s t}$ ，定义一组参数:取阻尼分段点 ${\bar{x}}_{0}$ =1，横向阻尼比ξ=0.2.根据式（19）与式（20）做出分段阻尼系统的幅频响应曲线，并应用四阶Runge-Kutta法对式（4）数值求解，理论求解结果与数值仿真结果对比如图4所示.
由图4可以发现，数值解曲线与理论解曲线之间具有良好的拟合度.在大部分频段内，两条曲线基本重合，共振区的误差在5%左右.这就说明，理论求解结果与数值仿真结果是互相印证的.
图4 幅频响应曲线理论解对比数值解
Fig.4 Comparison of theoretical solution and numerical solution of amplitude frequency response curve
3.2 幅频特性分析
为探究横向阻尼比、阻尼分段点参数对系统幅频特性的影响，分别做出归一化阻尼分段点 ${\bar{x}}_{0}$ =1时不同横向阻尼比下的幅频响应曲线，如图5所示; 以及横向阻尼比ξ=0.2时不同阻尼分段点下的幅频响应曲线，如图6所示.并将无阻尼系统和线性阻尼系统（ ${\bar{x}}_{0}$ =0，ξ=0.2）作为对照，与分段阻尼系统进行比较分析.
图5 不同横向阻尼比下的幅频响应曲线
Fig.5 Amplitude frequency response curves of different horizontal damping ratios
由图5可见，系统的共振峰值随横向阻尼比增大而衰减，而高频区振幅受横向阻尼比影响不明显; 由图6可见，系统的共振峰值随阻尼分段点的降低而衰减，而高频区振幅受分段点影响不明显.所以与无阻尼隔振器相比，分段阻尼隔振器可通过增大横向线性阻尼和适当降低分段点的方法来抑制共振，这与在线性阻尼隔振器中直接增大线性阻尼系数起到相似的作用.
图6 不同阻尼分段点下的幅频响应曲线
Fig.6 Amplitude frequency response curves of different damping piecewise points
4 积极隔振特性
4.1 力传递率求解
积极隔振是将隔振器置于振源与地基之间，以减少振源向地基传力，故通常用力传递率来评价隔振器性能的好坏，其定义为隔振后传递到基础力的幅值与激励力幅值的比值.等效线性系统的力传递率为

T_{a} = \frac{F_{a}}{F_{0}} = \sqrt{\frac{1 + {(2 ξ_{e q} λ)}^{2}}{{(1 - λ^{2})}^{2} + {(2 ξ_{e q} λ)}^{2}}}

(21)

同上节解法，取一组n个力传递率:

(22)

反解式（21）得到关于归一化频率的表达式

(23)

在式（23）中，有

\begin{matrix} λ (i) = \sqrt{1 - 2 ξ_{e q} (i)^{2} + \frac{2 ξ_{e q} (i)^{2}}{T_{a} (i)^{2}} \pm \frac{\sqrt{- 4 T_{a} (i)^{2} [T_{a} (i)^{2} - 1] + {[- 4 ξ_{e q} (i)^{2} - 2 T_{a} (i)^{2} + 4 ξ_{e q} (i)^{2} T_{a} (i)^{2}]}^{2}}}{2 T_{a} (i)^{2}}} \\ (i = 1,2, 3, \dots, n) \end{matrix}

(24)

4.2 隔振特性分析
根据振动理论，隔振系统的传递率特性往往受到阻尼大小的影响，本小节主要探究横向阻尼比参数对于分段阻尼隔振系统中力传递率的影响.
图7 不同横向阻尼比下的力传递率
Fig.7 Force transmissibility of different horizontal damping ratios
以频率比为界，低频区需要阻尼抑制共振，而在高频区阻尼越大隔振效果越差.故以阻尼分段点为佳，此时系统具有“低频有阻尼，高频无阻尼”特性.若分段点提升，会不利于低频共振抑制效果，若分段点降低，会不利于高频隔振性能.
当归一化阻尼分段点时，根据式（24）得到不同横向阻尼比下分段阻尼隔振系统的力传递率特性曲线，如图7所示.
由图7可知:系统的力传递率在频率比接近1时（共振区）达到峰值，此时横向阻尼比的增大有利于抑制振动中力的传递; 不同值下高频区力传递率趋于一致，这是由于分段阻尼在较高频段体现出无阻尼特性，不受横向阻尼比参数的影响.
为研究所述分段阻尼隔振系统相对传统隔振方法的性能提升，将合适参数的分段阻尼隔振系统（，）与无阻尼系统和线性阻尼系统（，）相对比，其力传递率曲线如图8所示.通过对比分析力传递率特性曲线，可以发现:分段阻尼隔振器的起始有效频率为系统固有频率的倍，这与线性系统一致; 与无阻尼隔振器相比，分段阻尼隔振器可有效抑制共振频率附近力的传递; 与线性阻尼隔振器相比，分段阻尼隔振器在具有相同共振抑制效果的前提下，高频区的隔振效果更优.
数据化的力传递率对比见表1.由表1可知，分段阻尼隔振器具有与无阻尼系统同样优秀的高频隔振效果，且频率越高，对比线性系统优势越大.
图8 分段系统、无阻尼系统与线性系统力传递率曲线对比
Fig.8 Comparison of force transmissibility curves among piecewise system, undamped system and linear system
表1 分段系统、无阻尼系统与线性阻尼系统力传递率对比
Table1 Comparison of force transmissibility among piecewise system, undamped system and linear system
5 结论
针对无阻尼与线性阻尼隔振器的不足，基于移动凸轮变阻尼装置设计了一种分段阻尼隔振器，通过阻尼耗能相等原则探究了移动凸轮变阻尼装置的阻尼特性.建立了分段阻尼积极隔振系统的动力学方程，求得等效线性阻尼系数，进而得到方程的理论解，并用四阶Runge-Kutta数值解法进行验证.分析了分段阻尼隔振系统的幅频响应与隔振特性，研究了关键参数对响应和力传递率的影响，并与线性系统及无阻尼系统进行对比.得出结论如下:
（1）移动凸轮变阻尼装置通过以凸轮-滚动轴承为核心的纯机械结构，使系统垂向阻尼大小受振动位移被动控制，无须外界能量输入，通过对凸轮廓线的设计可获得多种不同的非线性阻尼特性.
（2）分段阻尼隔振器的特性主要受横向阻尼比与阻尼分段点影响，增大横向阻尼、适当降低分段点可有效抑制共振且不影响高频隔振效果.
（3）通过合理的参数选择，分段阻尼隔振系统呈“低频区大位移有阻尼，高频区小位移无阻尼”特性.相比一般隔振方法，本系统既能有效降低低频区振幅和力的传递，又能保证抑制高频振动传递率的效果免受阻尼影响，使高频隔振性能（时）比线性系统提升不低于19%.
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- [27] ZHOU J X,WANG X L,XU D L,et al.Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam-roller-spring mechanisms [J].Journal of Sound and Vibration,2015,346:53-69.
- [28] 周加喜,王心龙,徐道临,等.含凸轮-滚轮机构的准零刚度系统隔振特性实验研究 [J].振动工程学报,2015,28(3):449-455.ZHOU J X,WANG X L,XU D L,et al.Experimental study on vibration isolation characteristics of the quasi-zero stiffness isolator with cam-roller mechanism [J].Journal of Vibration Engineering,2015,28(3):449-455.(in Chinese)
- [29] LI M,CHENG W,XIE R L.Design and experiments of a quasi-zero-stiffness isolator with a noncircular cam-based negative-stiffness mechanism [J].Journal of Vibration and Control,2020,26(21-22):1935-1947.
- [30] 尹蒙蒙,丁虎,陈立群.X型准零刚度隔振器动力学设计及分析 [J].动力学与控制学报,2021,19(5):46-51.YIN M M,DING H,CHEN L Q.Dynamic design and analysis of X-shaped quasi-zero stiffness isolator [J].Journal of Dynamics and Control,2021,19(5):46-51.(in Chinese)
- [31] 尹佑旺,郑佩.一种柔性隔振结构的动力学分析及设计方法 [J].动力学与控制学报,2021,19(6):16-24.YIN Y W,ZHENG P.Design and dynamic analysis of a flexible vibration isolation structure [J].Journal of Dynamics and Control,2021,19(6):16-24.(in Chinese)
- [32] DEN HARTOG J P.Mechanical vibrations [M].New York:McGraw-Hall Book Company,1947:414-418.

基本信息

中图分类号: O322;TH113.1
文献标识码: A
DOI: 10.6052/1672-6553-2022-061
文章编号: 1672-6553-2023-21(5)-044-009

基金信息

国家自然科学基金资助项目(U1934201)；
National Natural Science Foundation of China(U1934201)；

引用信息

稿件历史

收稿日期: 2022-10-02

参考文献

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[22] 邵栋,陆泽琦,陈立群.非线性刚度非线性阻尼隔振系统功率流研究 [J].振动工程学报,2017,30(5):764-773.SHAO D,LU Z Q,CHEN L Q.Power flow characteristics of a two-stage nonlinear vibration isolation system [J].Journal of Vibration Engineering,2017,30(5):764-773.(in Chinese)
[23] 刘兴天,陈树海,王嘉登,等.几何非线性摩擦阻尼隔振系统动力学行为研究 [J].力学学报,2019,51(2):371-379.LIU X T,CHEN S H,WANG J D,et al.Analysis of the dynamic behavior and performance of a vibration isolation system with geometric nonlinear friction damping [J].Chinese Journal of Theoretical and Applied Mechanics,2019,51(2):371-379.(in Chinese)
[24] 张婉洁,牛江川,申永军,等.一类阻尼控制半主动隔振系统的解析研究 [J].力学学报,2020,52(6):1743-1754.ZHANG W J,NIU J C,SHEN Y J,et al.Dynamical analysis on a kind of semi-active vibration isolation systems with damping control [J].Chinese Journal of Theoretical and Applied Mechanics,2020,52(6):1743-1754(in Chinese)
[25] 顾栋浩,陆泽琦,丁虎,等.圆环非线性隔振设计和动力学研究 [J].振动工程学报,2021,34(6):1223-1229.GU D H,LU Z Q,DING H,et al.Nonlinear vibration isolation design and dynamic study of circular ring [J].Journal of Vibration Engineering,2021,34(6):1223-1229.(in Chinese)
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[28] 周加喜,王心龙,徐道临,等.含凸轮-滚轮机构的准零刚度系统隔振特性实验研究 [J].振动工程学报,2015,28(3):449-455.ZHOU J X,WANG X L,XU D L,et al.Experimental study on vibration isolation characteristics of the quasi-zero stiffness isolator with cam-roller mechanism [J].Journal of Vibration Engineering,2015,28(3):449-455.(in Chinese)
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[30] 尹蒙蒙,丁虎,陈立群.X型准零刚度隔振器动力学设计及分析 [J].动力学与控制学报,2021,19(5):46-51.YIN M M,DING H,CHEN L Q.Dynamic design and analysis of X-shaped quasi-zero stiffness isolator [J].Journal of Dynamics and Control,2021,19(5):46-51.(in Chinese)
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分段阻尼隔振系统的动力学特性分析*

Dynamic Characteristic Analysis of a Piecewise Damping Vibration Isolation System*

摘要

Abstract

关键词

Keywords

1 变阻尼装置及阻尼特性分析

1.1 变阻尼装置

1.2 分段阻尼特性

(1)

(2)

(3)

2 动力学方程及求解

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

3 幅频响应特性

3.1 幅频特性求解

(15)

(16)

(17)

(18)

(19)

3.2 幅频特性分析

4 积极隔振特性

4.1 力传递率求解

(21)

(22)

(23)

(24)

4.2 隔振特性分析

5 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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分段阻尼隔振系统的动力学特性分析^*

Dynamic Characteristic Analysis of a Piecewise Damping Vibration Isolation System^*