RESEARCH ON THE VIBRATION SUPPRESSION PERFORMANCE OF A SERIES TWO DEGREE OF FREEDOM NONLINEAR ENERGY SINK UNDER PULSE EXCITATION WITH DAMPING NONLINEARITY
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摘要: 非线性能量阱(nonlinear energy sink, NES)是一种优于线性吸振器的被动能量吸收装置, 其在各个领域的抑振中起着关键性的作用. 对具有线性阻尼的串联式两自由度非线性能量阱(2-DOF LNES)、具有非线性立方项阻尼的串联式两自由度非线性能量阱(2-DOF NNES)以及具有线性和非线性阻尼的组合式串联两自由度非线性能量阱(2-DOF CNES) 3种系统在脉冲激励下的振动抑制性能及能量传递形式进行了研究. 建立了3种系统的理论模型, 并利用复变量平均法对相应的系统近似解进行了推导, 得到了对应的慢变动力流方程. 使用4阶龙格库塔法对同种系统在不同初始能量下NES的减振效率及能量传递形式进行了纵向对比研究, 又对不同系统在同种初始能量下的减振性能和能量传递形式进行了横向对比研究. 结果表明, 在主结构耦合串联式两自由度NES模型中, NES振子对主结构的减振主要通过1:1:1瞬态共振俘获实现. 2-DOF LNES和2-DOF NNES分别在较小和较大的初始能量条件下对主结构具有较好的抑振能力. 2-DOF CNES则综合了2-DOF LNES及2-DOF NNES系统各自的优势, 不仅具有小的能量触发阈值, 并且在主结构受到较大的初始脉冲激励时也能保持较好的减振性能, 这种优势使其在应对不同的初始能量条件下时均具有较高的减振效率和鲁棒性.Abstract: Nonlinear energy sink (NES) is a passive energy absorption device that is superior to linear vibration absorbers and plays a crucial role in suppressing vibration in various fields. This paper presents a study on the series two degree of freedom nonlinear energy sink (2-DOF LNES) with linear damping. The vibration suppression performance and energy transfer form of the series two degree of freedom nonlinear energy sink (2-DOF NNES) with nonlinear cubic damping and the combination series two degree of freedom nonlinear energy sink (2-DOF CNES) with linear and nonlinear damping under impact load were studied. The theoretical models of the three systems were described, and the approximate solutions of the corresponding systems were derived using the complex variable averaging method. The corresponding slowly varying dynamic flow equation was obtained. The fourth order Runge-Kutta numerical method was used to longitudinally compare the vibration reduction efficiency and energy transfer form of NES for the same system at different initial energies. Additionally, a transverse comparative study was conducted on the vibration reduction performance and energy transfer form of different systems at the same initial energy. Results showed that the vibration suppression of the NES oscillator on the main structure is mainly achieved through 1:1:1 transient resonance capture. 2-DOF LNES has good vibration suppression ability on the main structure under small initial energy conditions. The 2-DOF LNES and 2-DOF NNES have better vibration suppression capabilities for the primary structure under smaller and larger initial energy conditions, respectively. 2-DOF CNES combines the advantages of the 2-DOF LNES and 2-DOF NNES systems, which not only has a small energy triggering threshold, but also maintains a higher vibration suppression performance when the primary structure is subjected to a larger initial pulse excitation. This advantage makes the system more efficient and robust when dealing with different initial energy conditions.
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引 言
非线性能量阱(nonlinear energy sink, NES)是由Vakakis[1]和Gendelman等[2]在2001年提出的一种新型非线性被动吸振器. NES通常由立方非线性刚度、阻尼和质量组成[3]. NES与一般线性动力吸振器的最大区别在于其刚度为非线性刚度, 因此NES没有固有频率. 这意味着NES与任何频率成分都可能发生共振[4], 且NES相较于一般线性动力吸振器具有更宽的振动抑制带宽及更优的能量传递形式(靶向能量传递)[5].
学者们已通过理论和试验证实了NES在结构振动抑制方面的良好性能. 范舒铜等[6]利用Runge-Kutta数值方法对简谐激励下黏弹性NES的减振性能及参数选取进行了一定研究. Geng等[7]利用数值及试验相结合的方法对具有分段线性刚度耦合非线性立方刚度的单自由度NES (S-DOF NES)在不同激励下的减振性能进行了研究. 由于传统S-DOF NES在许多方面具有明显劣势, 如需一定的能量触发阈值, 大激励下减振性能大幅下降等[8-9], 因此学者们在改进NES方面也做了大量研究, 李爽等[10]将预压式柔性铰链结构与弹簧并联, 构造出一种新型NES结构. 王菁菁等[11]将线性与非线性特点同时考虑至NES中, 提出了双参数NES模型. Zang等[12]将具有非对称弹性边界的杠杆式NES集成至黏弹性Euler-Bernoulli梁中, 从数值解和近似解析解两方面对系统的稳态响应进行了探究. Chen等[13]利用永磁体为NES产生了非接触且光滑的双稳态恢复力, 并将这种新型装置与传统单稳态NES系统进行了对比研究. 由于NES的高效减振性, 因此被广泛用于土木工程[14-15]、机械工程[16-17]、航空航天工程[18-19]和生命线工程[20-21]等领域的减振研究.
靶向能量传递是NES特有的能量传递形式, 它具有能量传递效率高和鲁棒性强等特点[22]. 在一定参数选取范围内, 靶向能量传递被激发, NES振子可以快速吸收并且耗散掉由主结构传来的能量[23]. 由于非线性耦合振子力学特征的复杂性, 因此能量在非线性振子之间相互传递的特性至今也未被解析完全. 大量学者利用近似解析解方法对非线性振子耦合的靶向能量传递现象进行了解析[24-25], 得出了引发靶向能量传递的两个条件: 一是主结构与非线性振子之间发生1:1共振俘获现象, 二是在非谐振条件下的非线性振子出现更高频的振动.
刚度和阻尼对NES的振动抑制能力有着十分重要的影响. 目前关于NES的非线性刚度对靶向能量传递和减振性能的影响已得到了大量的研究[26-27], 但在NES中同时考虑刚度和阻尼非线性对减振性能影响的研究较少. 张运法等[28-29]对具有非线性阻尼的单自由度NES在谐波激励下的减振性能及强调制响应进行了一系列的研究, 发现具有组合非线性阻尼的NES系统在简谐激励下具有更好的振动抑制能力, 由此可以看出NES阻尼对于NES减振作用的影响同样至关重要. 同时, NES在脉冲激励下的减振性能及其减振机理也受到部分学者关注. 李爽等[30]对主结构耦合S-DOF NES系统受脉冲激励下NES的减振性能及靶向能量传递机理进行了探究, 结果表明S-DOF NES系统抑振效果与外部冲击能量等级密切相关. 李继伟等[31]将冲击减震器与S-DOF NES相互耦合, 结果表明耦合了冲击减震器的S-DOF NES系统对外部初始能量的依赖性大幅降低. 张文勇等[32]基于靶向能量传递理论对具有线性阻尼的串联式两自由度非线性能量阱2-DOF LNES及含S-DOF NES的整星系统在脉冲激励下的减振性能进行了对比研究, 结果表明耦合串联2-DOF LNES的系统在脉冲激励下能进一步提升原NES 的减振性能. 然而, 具有非线性刚度和非线性阻尼特性的多自由度NES在脉冲激励下的减振性能和靶向能量传递机理的研究还有待探索.
本文采用数值解和近似解模型对2-DOF LNES、具有线性和非线性阻尼的组合式串联两自由度非线性能量阱(2-DOF NNES)以及具有非线性立方项阻尼的串联式两自由度非线性能量阱(2-DOF CNES) 3种系统受脉冲激励时的抑振能力和靶向能量传递机理进行研究, 为具有非线性阻尼特性的多自由度NES在减振中的应用提供理论参考.
1. 系统模型建立
本节对主结构耦合2-DOF LNES, 2-DOF NNES以及2-DOF CNES 3种系统的理论模型进行了描述. 如图1 ~ 图3所示, $ {m_1} $表示主结构的质量, $ {m_2} $表示一级NES质量, $ {m_3} $表示二级NES质量. 主结构中包含线性刚度$ {k_1} $及线性阻尼$ {c_1} $. 图1中, 2-DOF LNES系统模型中的一级NES阻尼$ {c_2} $及二级NES阻尼$ {c_3} $均为线性阻尼, 刚度分别为非线性刚度$ {k_2} $和$ {k_3} $. 图2中, 2-DOF NNES系统模型中一级阻尼$ {c_{n2}} $及二级阻尼$ {c_{n3}} $均为非线性立方项阻尼. 图3中, 2-DOF CNES系统模型中一级NES阻尼由线性阻尼$ {c_2} $以及非线性立方项阻尼$ {c_{n2}} $组成, 二级NES阻尼由线性阻尼$ {c_3} $以及非线性立方项阻尼$ {c_{n3}} $组成. $ {x_1} $, $ {x_2} $, $ {x_3} $分别表示主结构、一级NES和二级NES的位移, 外部脉冲激励$ P(t) $满足$ {\mathrm{Dirac}} $函数形式, 即$ P(t) = P\delta (t) $, 以集中的形式作用于主结构 $ {m_1} $上.
$$ \left. \begin{aligned} & {m_1}{{\ddot x}_1} + {k_1}{x_1} + {k_2}{({x_1} - {x_2})^3} + {c_1}{{\dot x}_1} + {c_2}({{\dot x}_1} - {{\dot x}_2}) = P\delta (t) \\ & {m_2}{{\ddot x}_2} + {k_2}{({x_2} - {x_1})^3} + {k_3}{({x_2} - {x_3})^3} + {c_2}({{\dot x}_2} - {{\dot x}_1})+ \\ & \qquad {c_3}({{\dot x}_2} - {{\dot x}_3}) = 0 \\ & {m_3}{{\ddot x}_3} + {k_3}{({x_3} - {x_2})^3} + {c_3}({{\dot x}_3} - {{\dot x}_2}) = 0 \end{aligned} \right\} $$ (1) $$ \left. \begin{aligned} & {m_1}{{\ddot x}_1} + {k_1}{x_1} + {k_2}{({x_1} - {x_2})^3} + {c_1}{{\dot x}_1} + {c_{n2}}{({{\dot x}_1} - {{\dot x}_2})^3} = P\delta (t) \\ & {m_2}{{\ddot x}_2} + {k_2}{({x_2} - {x_1})^3} + {k_3}{({x_2} - {x_3})^3} + {c_{n2}}{({{\dot x}_2} - {{\dot x}_1})^3}+ \\ &\qquad {c_{n3}}{({{\dot x}_2} - {{\dot x}_3})^3} = 0 \\ & {m_3}{{\ddot x}_3} + {k_3}{({x_3} - {x_2})^3} + {c_{n3}}{({{\dot x}_3} - {{\dot x}_2})^3} = 0 \end{aligned} \right\} $$ (2) $$ \left. \begin{aligned} & {m_1}{{\ddot x}_1} + {k_1}{x_1} + {k_2}{({x_1} - {x_2})^3} + {c_1}{{\dot x}_1} + {c_2}({{\dot x}_1} - {{\dot x}_2})+ \\ & \qquad {c_{n2}}{({{\dot x}_1} - {{\dot x}_2})^3} = P\delta (t) \\ & {m_2}{{\ddot x}_2} + {k_2}{({x_2} - {x_1})^3} + {k_3}{({x_2} - {x_3})^3} + {c_2}({{\dot x}_2} - {{\dot x}_1})+ \\ &\qquad {c_{n2}}{({{\dot x}_2} - {{\dot x}_1})^3} + {c_3}({{\dot x}_2} - {{\dot x}_3}) + {c_{n3}}{({{\dot x}_2} - {{\dot x}_3})^3} = 0 \\ & {m_3}{{\ddot x}_3} + {k_3}{({x_3} - {x_2})^3} + {c_3}({{\dot x}_3} - {{\dot x}_2}) + {c_{n3}}{({{\dot x}_3} - {{\dot x}_2})^3} = 0 \end{aligned} \right\}$$ (3) 根据文献[33-34]的方法引入以下变量, 对式(1) ~ 式(3)进行无量纲化处理, 可得
$$\left. \begin{aligned} &\bar{t} = \sqrt{\frac{{k}_{1}}{{m}_{1}}}t, {\bar{x}}_{i} = {x}_{i}, \frac{{m}_{2} + {m}_{3}}{{m}_{1}} = \varepsilon , \frac{{m}_{2}}{{m}_{1}} = \varepsilon \eta = {\varepsilon }_{2} \\ &\frac{{m}_{3}}{{m}_{1}} = \varepsilon (1-\eta ) = {\varepsilon }_{3}, \frac{{k}_{2}}{{k}_{1}} = \varepsilon {k}_{21} = {k}_{n2}, \frac{{k}_{3}}{{k}_{1}} = \varepsilon {k}_{31} = {k}_{n3} \\ &\frac{{{c_1}}}{{\sqrt {{k_1}{m_1}} }} = \varepsilon {\lambda _1} = {\zeta _1}, \frac{{{c_2}}}{{\sqrt {{k_1}{m_1}} }} = \varepsilon {\lambda _2} = {\zeta _2} \\ & \frac{{{c_3}}}{{\sqrt {{k_1}{m_1}} }} = \varepsilon {\lambda _3} = {\zeta _3} \\ & \frac{{{c_{n2}}}}{{{m_1}}}\sqrt {\frac{{{k_1}}}{{{m_1}}}} = \varepsilon {\lambda _{n2}} = {\zeta _{n2}}, \frac{{{c_{n3}}}}{{{m_1}}}\sqrt {\frac{{{k_1}}}{{{m_1}}}} = \varepsilon {\lambda _{n3}} = {\zeta _{n3}} \end{aligned}\right\} $$ (4) 将式(4)代入式(1) ~ 式(3)中, 为表述简洁易懂, 后续$ \bar t $, $ {\bar x_i} $仍用$ t $, $ {x_i} $进行表示, 可得无量纲化后的控制方程分别为
$$ \left. \begin{aligned} & {{\ddot x}_1} + {x_1} + {k_n}_2{({x_1} - {x_2})^3} + {\zeta _1}{{\dot x}_1} + {\zeta _2}({{\dot x}_1} - {{\dot x}_2}) = {P_1}\delta (t) \\ & {\varepsilon _2}{{\ddot x}_2} + {k_n}_2{({x_2} - {x_1})^3} + {k_{n3}}{({x_2} - {x_3})^3} + {\zeta _2}({{\dot x}_2} - {{\dot x}_1}) + \\ &\qquad {\zeta _3}({{\dot x}_2} - {{\dot x}_3}) = 0 \\ & {\varepsilon _3}{{\ddot x}_3} + {k_{n3}}{({x_3} - {x_2})^3} + {\zeta _3}({{\dot x}_3} - {{\dot x}_2}) = 0 \end{aligned} \right\} $$ (5) $$ \left. \begin{aligned} & {{\ddot x}_1} + {x_1} + {k_n}_2{({x_1} - {x_2})^3} + {\zeta _1}{{\dot x}_1} + {\zeta _2}{({{\dot x}_1} - {{\dot x}_2})^3} = {P_1}\delta (t) \\ & {\varepsilon _2}{{\ddot x}_2} + {k_n}_2{({x_2} - {x_1})^3} + {k_{n3}}{({x_2} - {x_3})^3} + {\zeta _{n2}}{({{\dot x}_2} - {{\dot x}_1})^3}+ \\ & \qquad {\zeta _{n3}}{({{\dot x}_2} - {{\dot x}_3})^3} = 0 \\ & {\varepsilon _3}{{\ddot x}_3} + {k_{n3}}{({x_3} - {x_2})^3} + {\zeta _{n3}}{({{\dot x}_3} - {{\dot x}_2})^3} = 0 \end{aligned} \right\}$$ (6) $$ \left. \begin{aligned} & {{\ddot x}_1} + {x_1} + {k_{n2}}{({x_1} - {x_2})^3} + {\zeta _1}{{\dot x}_1} + {\zeta _2}({{\dot x}_1} - {{\dot x}_2})+ \\ & \qquad {\zeta _{n2}}{({{\dot x}_1} - {{\dot x}_2})^3} = {P_1}\delta (t) \\ & {\varepsilon _2}{{\ddot x}_2} + {k_{n2}}{({x_2} - {x_1})^3} + {k_{n3}}{({x_2} - {x_3})^3} + {\zeta _2}({{\dot x}_2} - {{\dot x}_1})+ \\ &\qquad {\zeta _{n2}}{({{\dot x}_2} - {{\dot x}_1})^3} + {\zeta _3}({{\dot x}_2} - {{\dot x}_3}) + {\zeta _{n3}}{({{\dot x}_2} - {{\dot x}_3})^3} = 0 \\ & {\varepsilon _3}{{\ddot x}_3} + {k_{n3}}{({x_3} - {x_2})^3} + {\zeta _3}({{\dot x}_3} - {{\dot x}_2}) + {\zeta _{n3}}{({{\dot x}_3} - {{\dot x}_2})^3} = 0 \end{aligned} \right\} $$ (7) 为了便于分析, 求解过程中将脉冲激励$ P(t) $转化为系统初始条件, 有
$$\left. \begin{aligned} &{\dot{x}}_{1}(0)\text{ = }\eta \text{ }({\mathrm{m}}/{{\mathrm{s}}})\\ &{\dot{x}}_{2}(0)\text{ = }{\dot{x}}_{3}(0)\text{ = }0\text{ }({\mathrm{m}}/{{\mathrm{s}}})\\ &{x}_{1}(0)\text{ = }{x}_{2}(0)\text{ = }{x}_{3}(0) = 0\;({\mathrm{m}})\end{aligned}\right\} $$ (8) 图1 ~ 图3所示系统内部能量为$ {E_{{\mathrm{tol}}}} $
$$ \begin{aligned} & {E_{{\mathrm{tol}}}} = \frac{1}{2}{{\dot x}_1}^2 + \frac{1}{2}{\varepsilon _2}{{\dot x}_2}^2 + \frac{1}{2}{\varepsilon _3}{{\dot x}_3}^2 + \frac{1}{2}{x_1}^2 + \\ &\qquad \frac{1}{4}{k_{n2}}{({x_1} - {x_2})^4} + \frac{1}{4}{k_{n3}}{({x_2} - {x_3})^4}\end{aligned} $$ (9) 将初始条件$ {\dot x_1}(0){\text{ = }}\eta $代入$ {E_{{\mathrm{tol}}}} $中, 可得到系统初始能量值为$ {E_{{\mathrm{tol}}1}} $. $ {E_0} $, $ {E_{{\mathrm{NES}}1}} $和$ {E_{{\mathrm{NES}}2}} $分别表示主结构, 一级NES和二级NES的瞬态能量占系统初始总能量比值. $ {E_{{\mathrm{NES}}}} $表示为两自由度NES的瞬态能量占初始总能量比值, 其表达式分别为
$$ \left. \begin{aligned} & {E_0} = \left(\frac{1}{2}{{\dot x}_1}^2 + \frac{1}{2}{x_1}^2\right)\Bigg/{E_{{\mathrm{tol}}1}} \times 100\% \\ & {E_{{\mathrm{NES}}1}} = \left[\frac{1}{2}{\varepsilon _2}{{\dot x}_2}^2 + \frac{1}{4}{k_{n2}}{({x_1} - {x_2})^4}\right]\Bigg/{E_{{\mathrm{tol}}1}} \times 100\% \\ & {E_{{\mathrm{NES}}2}} = \left[\frac{1}{2}{\varepsilon _3}{{\dot x}_3}^2 + \frac{1}{4}{k_{n3}}{({x_2} - {x_3})^4}\right]\Bigg/{E_{{\mathrm{tol}}1}} \times 100\% \\ & {E_{{\mathrm{NES}}}} = {E_{{\mathrm{NES}}1}} + {E_{{\mathrm{NES}}2}} \end{aligned} \right\} $$ (10) $$ \left. \begin{aligned} & {E_{{\mathrm{LNES}}}}(t) =\\ & \frac{{{\zeta _2}{{\displaystyle\int_0^t {[{{\dot x}_1}(\tau ) - {{\dot x}_2}(\tau )]} }^2}{\mathrm{d}}\tau + {\zeta _3}{{\displaystyle\int_0^t {[{{\dot x}_2}(\tau ) - {{\dot x}_3}(\tau )]} }^2}{\mathrm{d}}\tau }}{{{\eta ^2}/2}} \times 100\% \\ & {{E_{{\mathrm{LNES}}}} = \mathop {\lim }\limits_{t \gg 1} {E_{{\mathrm{LNES}}}}(t)} \end{aligned} \right\}$$ (11) $$ \left. \begin{aligned} & {E_{{\mathrm{NNES}}}}(t) =\\ &\frac{{{\zeta _{n2}}{{\displaystyle\int_0^t {[{{\dot x}_1}(\tau ) - {{\dot x}_2}(\tau )]} }^4}{\mathrm{d}}\tau + {\zeta _{n3}}{{\displaystyle\int_0^t {[{{\dot x}_2}(\tau ) - {{\dot x}_3}(\tau )]} }^4}{\mathrm{d}}\tau }}{{{\eta ^2}/2}} \times 100\% \\[-6pt] & {{E_{{\mathrm{NNES}}}} = \mathop {\lim }\limits_{t \gg 1} {E_{{\mathrm{NNES}}}}(t)} \end{aligned} \right\} $$ (12) $$ \left. \begin{aligned} & {E_{{\mathrm{CNES}}}}(t) =\\ & \frac{{{\zeta _2}{{\displaystyle\int_0^t {[{{\dot x}_1}(\tau ) - {{\dot x}_2}(\tau )]} }^2}{\mathrm{d}}\tau + {\zeta _3}{{\displaystyle\int_0^t {[{{\dot x}_2}(\tau ) - {{\dot x}_3}(\tau )]} }^2}{\mathrm{d}}\tau }}{{{\eta ^2}/2}}+ \\[-4pt] & \frac{{{\zeta _{n2}}{{\displaystyle\int_0^t {[{{\dot x}_1}(\tau ) - {{\dot x}_2}(\tau )]} }^4}{\mathrm{d}}\tau + {\zeta _{n3}}{{\displaystyle\int_0^t {[{{\dot x}_2}(\tau ) - {{\dot x}_3}(\tau )]} }^4}{\mathrm{d}}\tau }}{{{\eta ^2}/2}} \times 100\% \\[-6pt] & {{E_{{\mathrm{CNES}}}} = \mathop {\lim }\limits_{t \gg 1} {E_{{\mathrm{CNES}}}}(t)} \end{aligned} \right\} $$ (13) 式(11) ~ 式(13)中$ {E_{ - {\mathrm{NES}}}}(t) $表示不同结构中任意t时刻NES阻尼所耗散的能量占初始能量的比例, $ {E_{ - {\mathrm{NES}}}} $表示不同系统趋于稳定后NES阻尼所耗散的能量占系统初始能量的比例. 选取各系统主结构充分达到稳定状态时所需时间作为积分上界$ t $(此处积分上界均选取$ t{\text{ = }}800\;{\mathrm{s}} $时进行数值仿真).
由于篇幅的限制, 这里以主结构耦合2-DOF CNES为例采用复变量平均法对该系统慢变动力流进行分析. 首先对式(7)作复变量代换
$$ \left. \begin{aligned} & {\psi _1} = {{\dot x}_1} + {\mathrm{j}}\omega {x_1}, {\psi _2} = {{\dot x}_2} + {\mathrm{j}}\omega {x_2}, {\psi _3} = {{\dot x}_3} + {\mathrm{j}}\omega {x_3} \\ & {{\dot x}_1}{\text{ = }}\frac{1}{2}({\psi _1} + {\psi _1}^*), {{\dot x}_2}{\text{ = }}\frac{1}{2}({\psi _2} + {\psi _2}^*), {{\dot x}_3}{\text{ = }}\frac{1}{2}({\psi _3} + {\psi _3}^*) \\ & {x_1} = \frac{{{\psi _1} - {\psi _1}^*}}{{2{\mathrm{j}}}}, {x_2}{\text{ = }}\frac{{{\psi _2} - {\psi _2}^*}}{{2{\mathrm{j}}}}, {x_3}{\text{ = }}\frac{{{\psi _3} - {\psi _3}^*}}{{2{\mathrm{j}}}} \\ & {{\ddot x}_1}{\text{ = }}{{\dot \psi }_1} - \frac{1}{2}{\mathrm{j}}({\psi _1} + {\psi _1}^*), {{\ddot x}_2}{\text{ = }}{{\dot \psi }_2} - \frac{1}{2}{\mathrm{j}}({\psi _2} + {\psi _2}^*) \\ & {{\ddot x}_3}{\text{ = }}{{\dot \psi }_3} - \frac{1}{2}{\mathrm{j}}({\psi _3} + {\psi _3}^*) \end{aligned} \right\} $$ (14) 式中$ {\mathrm{j}} $为虚数单位, 即$ {\mathrm{j}} = {( - 1)^{1/2}} $, 主要考虑主结构、一级NES振子和二级NES振子发生1:1:1内共振的情况, 并假定主结构、一级NES振子和二级NES振子响应频率为$ \omega = {\omega _0} = {\omega _1} = 1 $, 将式(14)代入式(7)中, 可得
$$ \left. \begin{split} & {{\dot \psi }_1} - \frac{1}{2}{\mathrm{j}}({\psi _1} + {\psi _1}^*) + \frac{{{\psi _1} - {\psi _1}^*}}{{2{\mathrm{j}}}} + {k_{n2}}\left(\frac{{{\psi _1} - {\psi _1}^*}}{{2{\mathrm{j}}}} - \frac{{{\psi _2} - {\psi _2}^*}}{{2{\mathrm{j}}}}\right)^3 + \\ &\qquad \frac{1}{2}{\zeta _1}({\psi _1} + {\psi _1}^*) + {\zeta _2}\left[\frac{1}{2}({\psi _1} + {\psi _1}^*)-\right. \left. \frac{1}{2}({\psi _2} + {\psi _2}^*)\right] + \\ &\qquad {\zeta _{n2}}{\left[\frac{1}{2}({\psi _1} + {\psi _1}^*) - \frac{1}{2}({\psi _2} + {\psi _2}^*)\right]^3} = 0 \\ & {\varepsilon _2}\left[{{\dot \psi }_2} - \frac{1}{2}({\psi _2} + {\psi _2}^*) {\mathrm{j}}\right] + {k_{n2}}{\left(\frac{{{\psi _2} - {\psi _2}^*}}{{2{\mathrm{j}}}} - \frac{{{\psi _1} - {\psi _1}^*}}{{2{\mathrm{j}}}}\right)^3}+\\ & \qquad {k_{n3}}{\left(\frac{{{\psi _2} - {\psi _2}^*}}{{2{\mathrm{j}}}} - \frac{{{\psi _3} - {\psi _3}^*}}{{2{\mathrm{j}}}}\right)^3} + {\zeta _2}\left[\frac{1}{2}({\psi _2} + {\psi _2}^*) -\right.\\ &\qquad \left.\frac{1}{2}({\psi _1} + {\psi _1}^*)\right] + {\zeta _3}\left[\frac{1}{2}({\psi _2} + {\psi _2}^*) - \frac{1}{2}({\psi _3} + {\psi _3}^*)\right]+\\ &\qquad {\zeta _{n2}}{\left[\frac{1}{2}({\psi _2} + {\psi _2}^*) - \frac{1}{2}({\psi _1} + {\psi _1}^*)\right]^3} +\\ &\qquad {\zeta _{n3}}\left[\frac{1}{2} ({\psi _2} + {\psi _2}^*) - \frac{1}{2} ({\psi _3} + {\psi _3}^*)\right]^3 = 0 \\ & {\varepsilon _3}\left[{{\dot \psi }_3} - \frac{1}{2} ({\psi _3} + {\psi _3}^*) {\mathrm{j}}\right] + {k_{n2}}{\left(\frac{{{\psi _3} - {\psi _3}^*}}{{2{\mathrm{j}}}} - \frac{{{\psi _2} - {\psi _2}^*}}{{2{\mathrm{j}}}}\right)^3} + \\ &\qquad {\zeta _3}\left[\frac{1}{2}({\psi _3} + {\psi _3}^*) - \frac{1}{2} ({\psi _2} + {\psi _2}^*)\right] + \\ &\qquad {\zeta _{n3}}\left[\frac{1}{2}({\psi _3} + {\psi _3}^*)- \right. \left. \frac{1}{2}({\psi _2} + {\psi _2}^*)\right]^3 = 0 \end{split} \right\} $$ (15) 式中“*”为共轭, 同时将系统响应分为快变部分$ {{{\mathrm{e}}^{{\mathrm{j}}t}}} $以及慢变调制部分$ {\varphi _i}\;(i = 1, 2, 3) $两部分, 引入新的复变量
$$ \left. \begin{aligned} & {\psi _1}(t) = {\varphi _1}{{\mathrm{e}}^{{\mathrm{j}}t}}, {\psi _2}(t) = {\varphi _2}{{\mathrm{e}}^{{\mathrm{j}}t}}, {\psi _3}(t) = {\varphi _3}{{\mathrm{e}}^{{\mathrm{j}}t}} \\ & {\psi _1}^*(t) = {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}, {\psi _2}^*(t) = {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}, {\psi _3}^*(t) = {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}} \\ & {{\dot \psi }_1}(t) = {{\dot \varphi }_1}{{\mathrm{e}}^{{\mathrm{j}}t}} + {\mathrm{j}}{\varphi _1}{{\mathrm{e}}^{{\mathrm{j}}t}}, {{\dot \psi }_2}(t) = {{\dot \varphi }_2}{{\mathrm{e}}^{{\mathrm{j}}t}} + {\mathrm{j}}{\varphi _2}{{\mathrm{e}}^{{\mathrm{j}}t}} \\ & {{\dot \psi }_3}(t) = {{\dot \varphi }_3}{{\mathrm{e}}^{{\mathrm{j}}t}} + {\mathrm{j}}{\varphi _3}{{\mathrm{e}}^{{\mathrm{j}}t}} \end{aligned} \right\}$$ (16) 将式(16)代入式(15)中, 可得
$$ \left. \begin{aligned} & {{\dot \varphi }_1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\mathrm{j}}{\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - \frac{1}{2}{\mathrm{j}}({\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}) + \frac{{{\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}} +\\ &\qquad {k_{n2}}{\left(\frac{{{\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}} - \frac{{{\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}}\right)^3} +\\ &\qquad \frac{1}{2}{\zeta _1}({\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} +{\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}})+ {\zeta _2}\left[\frac{1}{2} \left({\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right) -\right. \\ &\qquad \left.\frac{1}{2}\left({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)\right] + {\zeta _{n2}}\left[\frac{1}{2} \left({\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right) -\right. \\ &\qquad \left.\frac{1}{2} \left({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)\right]^3 = 0 \\ & {\varepsilon _2}\left[{{\dot \varphi }_2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\mathrm{j}}{\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - \frac{1}{2}{\mathrm{j}}\left({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)\right] +\\ &\qquad {k_{n2}}\left(\frac{{{\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}} - \frac{{{\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}}\right)^3 + \\ &\qquad {k_{n3}}{\left(\frac{{{\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}} - \frac{{{\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}}\right)^3} + \\ &\qquad {\zeta _2}\left[\frac{1}{2}\left({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right) - \frac{1}{2}\left({\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)\right] + \\ &\qquad {\zeta _3}\left[\frac{1}{2} ({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}) -\right. \left. \frac{1}{2}\left({\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)\right] + \\ &\qquad {\zeta _{n2}}\left[\frac{1}{2}({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}) - \frac{1}{2}\left({\varphi _1}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _1}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)\right]^3 + \\ &\qquad{\zeta _{n3}}\left[\frac{1}{2}\left({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)- \right. \left. \frac{1}{2}({\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}})\right]^3 = 0 \\ & {\varepsilon _3}\left[{{\dot \varphi }_3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\mathrm{j}}{\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - \frac{1}{2}{\mathrm{j}}\left({\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}\right)\right] + \\ &\qquad {k_{n3}}\left(\frac{{{\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}} - \frac{{{\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} - {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}}}{{2{\mathrm{j}}}}\right)^3 + \\ &\qquad {\zeta _3}\left[\frac{1}{2}({\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}) -\right. \left. \frac{1}{2}({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}})\right] + \\ &\qquad {\zeta _{n3}}{\left[\frac{1}{2}({\varphi _3}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _3}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}}) - \frac{1}{2}({\varphi _2}{{{\mathrm{e}}^{{\mathrm{j}}t}}} + {\varphi _2}^*{{\mathrm{e}}^{ - {\mathrm{j}}t}})\right]^3} = 0 \end{aligned} \right\} $$ (17) 对式(17)中快变部分进行平均化处理, 去除$ {{\mathrm{e}}^{2 {\mathrm{j}}t}} $和$ {{\mathrm{e}}^{3 {\mathrm{j}}t}} $等高次项的影响, 可得系统慢变动力流方程
$$ \left. \begin{aligned} & {{\dot \varphi }_1} + {\mathrm{j}}{\varphi _1} - \frac{1}{2}{\mathrm{j}}{\varphi _1} - \frac{3}{8}{k_{n2}}{\mathrm{j}}{\left| {{\varphi _1} - {\varphi _2}} \right|^2}({\varphi _1} - {\varphi _2}) + \\ & \qquad \frac{1}{2}{\zeta _1}{\varphi _1} + \frac{1}{2}{\zeta _2}({\varphi _1} - {\varphi _2}) + \frac{3}{8}{\zeta _{n2}}{\left| {{\varphi _1} - {\varphi _2}} \right|^2}({\varphi _1} - {\varphi _2}) = 0 \\ & {\varepsilon _2}\left({{\dot \varphi }_2} + \frac{1}{2}{\mathrm{j}}{\varphi _2}\right) - {k_{n2}} \frac{3}{8} {\mathrm{j}} {\left| {{\varphi _2} - {\varphi _1}} \right|^2}({\varphi _2} - {\varphi _1})- \\ &\qquad \frac{3}{8}{k_{n3}}{\mathrm{j}}{\left| {{\varphi _2} - {\varphi _3}} \right|^2}({\varphi _2} - {\varphi _3}) + \frac{1}{2}{\zeta _2}({\varphi _2} - {\varphi _1}) +\\ &\qquad \frac{1}{2}{\zeta _3}({\varphi _2} - {\varphi _3})+ \frac{3}{8}{\zeta _{n2}}{\left| {{\varphi _2} - {\varphi _1}} \right|^2}({\varphi _2} - {\varphi _1}) + \\ &\qquad \frac{3}{8}{\zeta _{n3}}{\left| {{\varphi _2} - {\varphi _3}} \right|^2}({\varphi _2} - {\varphi _3}) = 0 \\ & {\varepsilon _2}({{\dot \varphi }_3} + \frac{1}{2}{\mathrm{j}}{\varphi _3}) - \frac{3}{8}{k_{n2}}{\mathrm{j}}{\left| {{\varphi _3} - {\varphi _2}} \right|^2}({\varphi _3} - {\varphi _2}) + \frac{1}{2}{\zeta _3}({\varphi _3} - {\varphi _2})+ \\ &\qquad \frac{3}{8}{\zeta _{n3}}{\left| {{\varphi _3} - {\varphi _2}} \right|^2}({\varphi _3} - {\varphi _2}) = 0 \end{aligned} \right\} $$ (18) 式(18)中的$ {\varphi _i} $仍为复数形式, 用极坐标表示为$ {\varphi _i}(t) = {\rho _i}(t){{\mathrm{e}}^{{\mathrm{j}}{\theta _i}(t)}} $, 其中$ {\rho _1} $, $ {\rho _2} $和$ {\rho _3} $分别为主结构, 一级NES振子和二级NES振子的慢变振幅包络. $ {\theta _1} $, $ {\theta _2} $和$ {\theta _3} $分别为主结构, 一级NES振子和二级NES振子的相位. 定义相位差如下$ {\phi _1}(t) = {\theta _1}(t) - {\theta _2}(t) $, $ {\phi _2}(t) = $θ2(t) − θ3(t)
$$ \left. \begin{aligned} & {\varphi _1} = {\rho _1}{{\mathrm{e}}^{{\mathrm{j}}{\theta _1}}}, {\varphi _2} = {\rho _2}{{\mathrm{e}}^{{\mathrm{j}}{\theta _2}}}, {\varphi _3} = {\rho _3}{{\mathrm{e}}^{{\mathrm{j}}{\theta _3}}} \\ & {{\dot \varphi }_1} = {{\dot \rho }_1}{{\mathrm{e}}^{{\mathrm{j}}{\theta _1}}} + {\rho _1}{{\mathrm{e}}^{{\mathrm{j}}{\theta _1}}}{\mathrm{j}}{{\dot \theta }_1}, {{\dot \varphi }_2} = {{\dot \rho }_2}{{\mathrm{e}}^{{\mathrm{j}}{\theta _2}}} + {\rho _2}{{\mathrm{e}}^{{\mathrm{j}}{\theta _2}}}{\mathrm{j}}{{\dot \theta }_2} \\ & {{\dot \varphi }_3} = {{\dot \rho }_3}{{\mathrm{e}}^{{\mathrm{j}}{\theta _3}}} + {\rho _3}{{\mathrm{e}}^{{\mathrm{j}}{\theta _3}}}{\mathrm{j}}{{\dot \theta }_3} \end{aligned} \right\} $$ (19) 将式(19)代入式(18)可得
$$ \begin{aligned} & {{\dot \rho }_1} + {\rho _1}{\mathrm{j}}{{\dot \theta }_1} + {\mathrm{j}}{\rho _1} - \frac{1}{2}{\mathrm{j}}{\rho _1} - \frac{3}{8}{k_{n2}}{\mathrm{j}} ({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\varphi _1} + {\rho _2}^2)\cdot\\ &\qquad({\rho _1} - {\rho _2}\cos {\varphi _1} + {\rho _2}{\mathrm{j}}\sin {\varphi _1}) + {\zeta _1}\frac{1}{2} {\rho _1} + \\ &\qquad \frac{1}{2}{\zeta _2}({\rho _1} - {\rho _2}\cos {\varphi _1} + {\rho _2}{\mathrm{j}}\sin {\varphi _1}) + \frac{3}{8}{\zeta _{n2}}({\rho _1}^2 -\\ &\qquad 2{\rho _1}{\rho _2}\cos {\varphi _1} + {\rho _2}^2) ({\rho _1} - {\rho _2}\cos {\varphi _1} + {\rho _2}{\mathrm{j}}\sin {\varphi _1}) = 0 \\ & {\varepsilon _2}{\text{(}}{{\dot \rho }_2} + {\rho _2}{\mathrm{j}}{{\dot \theta }_2} + \frac{1}{2}{\mathrm{j}}{\rho _2}) - \frac{3}{8}{k_{n2}}{\mathrm{j}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos \varphi + {\rho _2}^2)\cdot\\ &\qquad ({\rho _2} - {\rho _1}\cos {\varphi _1} - {\rho _1}{\mathrm{j}}\sin {\varphi _1}) - \frac{3}{8}{k_{n3}}{\mathrm{j}}({\rho _2}^2 - \\ &\qquad 2{\rho _3}{\rho _2}\cos {\varphi _2} + {\rho _3}^2)({\rho _2} - {\rho _3}\cos {\varphi _2} + {\rho _3}{\mathrm{j}}\sin {\varphi _2}) + \\ &\qquad \frac{1}{2}{\zeta _2}({\rho _2} - {\rho _1}\cos {\varphi _1} - {\rho _1}{\mathrm{j}}\sin {\varphi _1}) + \frac{1}{2}{\zeta _3}({\rho _2} - {\rho _3}\cos {\varphi _2} + \\ &\qquad {\rho _3}{\mathrm{j}}\sin {\varphi _2}) + \frac{3}{8}{\zeta _{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\varphi _1} + {\rho _2}^2) \cdot\\ &\qquad({\rho _2} - {\rho _1}\cos {\varphi _1} - {\rho _1}{\mathrm{j}}\sin {\varphi _1}) + \frac{3}{8}{\zeta _{n3}}({\rho _2}^2 - \\ &\qquad 2{\rho _3}{\rho _2}\cos {\varphi _2} + {\rho _3}^2) ({\rho _2} - {\rho _3}\cos {\varphi _2} + {\rho _3}{\mathrm{j}}\sin {\varphi _2}) = 0 \\ & {\varepsilon _3}({{\dot \rho }_3} + {\rho _3}{\mathrm{j}}\mathop {{\theta _3}}\limits^. + \frac{1}{2}{\mathrm{j}} {\rho _3}) - \frac{3}{8}{k_{n3}}{\mathrm{j}}({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\varphi _2} + {\rho _3}^2)\cdot \end{aligned} $$ $$ \begin{aligned} &\qquad({\rho _3} - {\rho _2}\cos {\varphi _2} - {\rho _2}{\mathrm{j}}\sin {\varphi _2}) + \frac{1}{2}{\zeta _3}({\rho _3} - {\rho _2}\cos {\varphi _2} - \\ &\qquad {\rho _2}{\mathrm{j}}\sin {\varphi _2}) + \frac{3}{8}{\zeta _{n3}}({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\varphi _2} + {\rho _3}^2)({\rho _3} -\\ &\qquad {\rho _2}\cos {\varphi _2} - {\rho _2}{\mathrm{j}}\sin {\varphi _2}) = 0 \\[-12pt]\end{aligned} $$ (20) 式(20)分离实虚部、虚部作差并且化简可得
$$ \begin{aligned} & \dot{\rho}_1\text{ = }-\frac{3}{8}k_{n2}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)\rho_2\sin\phi_1- \\ & \qquad\frac{1}{2}\zeta_1\rho_1-\frac{1}{2}\zeta_2(\rho_1-\rho_2\cos\phi_1)-\frac{3}{8}\zeta_{n2}(\rho_1^2- \\ & \qquad2\rho_1\rho_2\cos\phi_1+\rho_2^2)(\rho_1-\rho_2\cos\phi_1) \\ & \dot{\rho}_2=k_{n2}\frac{3}{8}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)\rho_1\sin\phi_1/\varepsilon_2- \\ & \qquad\frac{3}{8}k_{n3}(\rho_2^2-2\rho_3\rho_2\cos\phi_2+\rho_3^2)\rho_3\sin\phi_2/\varepsilon_2- \\ & \qquad\frac{1}{2}\zeta_2(\rho_2-\rho_1\cos\phi_1)/\varepsilon_2-\frac{1}{2}\zeta_3(\rho_2-\rho_3\cos\phi_2)/\varepsilon_2- \\ & \qquad\frac{3}{8}\zeta_{n2}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)(\rho_2-\rho_1\cos\phi_1)/\varepsilon_2- \\ & \qquad\frac{3}{8}\zeta_{n3}(\rho_2^2-2\rho_3\rho_2\cos\phi_2+\rho_3^2)(\rho_2-\rho_3\cos\phi_2)/\varepsilon_2 \\ & \dot{\rho}_3=\frac{3}{8}k_{n3}(\rho_2^2-2\rho_2\rho_3\cos\phi_2+\rho_3^2)\rho_2\sin\phi_2/\varepsilon_3- \\ & \qquad\frac{1}{2}\zeta_3(\rho_3 - \rho_2\cos\phi_2)/\varepsilon_3 - \frac{3}{8}\zeta_{n3}(\rho_2^2 - 2\rho_2\rho_3\cos\phi_2 + \rho_3^2)\cdot \\ & \qquad(\rho_3-\rho_2\cos\phi_2)/\varepsilon_3 \\ & \dot{\phi}_1\text{ = }\frac{1}{2}+k_{n2}\frac{3}{8}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)(\rho_1-\rho_2\cos\phi_1)/ \\ & \qquad\rho_1 - \frac{1}{2}\zeta_2\rho_2\sin\phi_1/\rho_1 - \frac{3}{8}\zeta_{n2}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)\cdot \\ & \qquad\rho_2\sin\phi_1/\rho_1-\frac{3}{8}k_{n2}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)\cdot \\ & \qquad(\rho_2-\rho_1\cos\phi_1)/(\varepsilon_2\rho_2)-\frac{3}{8}k_{n3}(\rho_2^2-2\rho_3\rho_2\cos\phi_2+ \\ & \qquad\rho_3^2)(\rho_2-\rho_3\cos\phi_2)/(\varepsilon_2\rho_2)-\frac{1}{2}\zeta_2\rho_1\sin\phi_1/(\varepsilon_2\rho_2)+ \\ & \qquad\frac{1}{2}\zeta_3\rho_3\sin\phi_2/(\varepsilon_2\rho_2)-\frac{3}{8}\zeta_{n2}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)\cdot \\ & \qquad\rho_1\sin\phi_1/(\varepsilon_2\rho_2)+\frac{3}{8}\zeta_{n3}(\rho_2^2-2\rho_3\rho_2\cos\phi_2+\rho_3^2)\cdot \\ & \qquad\rho_3\sin\phi_2/(\varepsilon_2\rho_2) \\ & \dot{\phi}_2\text{ = }\frac{3}{8}k_{n2}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)(\rho_2-\rho_1\cos\phi_1)/ \\ & \qquad(\varepsilon_2\rho_2)+\frac{3}{8}k_{n3}(\rho_2^2-2\rho_3\rho_2\cos\phi_2+\rho_3^2)\cdot \\ & \qquad(\rho_2-\rho_3\cos\phi_2)/(\varepsilon_2\rho_2)+\frac{1}{2}\zeta_2\rho_1\sin\phi_1/(\varepsilon_2\rho_2)- \\ & \qquad\frac{1}{2}\zeta_3\rho_3\sin\phi_2/(\varepsilon_2\rho_2)+\frac{3}{8}\zeta_{n2}(\rho_1^2-2\rho_1\rho_2\cos\phi_1+\rho_2^2)\cdot \\ & \qquad\rho_1\sin\phi_1/(\varepsilon_2\rho_2)-\frac{3}{8}\zeta_{n3}(\rho_2^2-2\rho_3\rho_2\cos\phi_2+\rho_3^2)\cdot \\ & \qquad\rho_3\sin\phi_2/(\varepsilon_2\rho_2)-\frac{3}{8}k_{n3}(\rho_2^2-2\rho_2\rho_3\cos\phi_2+\rho_3^2)\cdot\end{aligned} $$ $$ \begin{split} & \qquad\rho_3\sin\phi_2/(\varepsilon_2\rho_2)-\frac{3}{8}k_{n3}(\rho_2^2-2\rho_2\rho_3\cos\phi_2+\rho_3^2)\cdot \\ & \qquad(\rho_3-\rho_2\cos\phi_2)/(\varepsilon_3\rho_3)-\frac{1}{2}\zeta_3\rho_2\sin\phi_2/(\varepsilon_3\rho_3)- \\ & \qquad\frac{3}{8}\zeta_{n3}(\rho_2^2-2\rho_2\rho_3\cos\phi_2+\rho_3^2)\rho_2\sin\phi_2/(\varepsilon_3\rho_3)\end{split} $$ (21) 式(21)为主结构耦合2-DOF CNES系统的慢变动力流方程. 对于主结构耦合2-DOF LNES系统及主结构耦合2-DOF NNES系统的慢变动力流方程由同样方法也可得出.
①主结构耦合2-DOF LNES系统的慢变动力流方程如下
$$ \left. \begin{aligned} & {{\dot \rho }_1}{\text{ = }} - \frac{3}{8}{k_{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2){\rho _2}\sin {\phi _1} - \\ &\qquad \frac{1}{2}{\zeta _1}{\rho _1}{\kern 1pt} - \frac{1}{2}{\zeta _2}({\rho _1} - {\rho _2}\cos {\phi _1}) \\ & {{\dot \rho }_2} = {k_{n2}}\frac{3}{8}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2){\rho _1}\sin {\phi _1}/{\varepsilon _2} -\\ &\qquad \frac{3}{8}{k_{n3}} ({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2){\rho _3}\sin {\phi _2}/{\varepsilon _2} - \\ &\qquad \frac{1}{2}{\zeta _2}({\rho _2} - {\rho _1}\cos {\phi _1})/{\varepsilon _2} - \frac{1}{2}{\zeta _3}({\rho _2} - {\rho _3}\cos {\phi _2})/{\varepsilon _2} \\ & {{\dot \rho }_3} = \frac{3}{8}{k_{n3}}({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\phi _2} + {\rho _3}^2){\rho _2}\sin {\phi _2}/{\varepsilon _3} - \\ &\qquad \frac{1}{2}{\zeta _3} ({\rho _3} - {\rho _2}\cos {\phi _2})/{\varepsilon _3} \\ & {{\dot \phi }_1}{\text{ = }}\frac{1}{2} + {k_{n2}}\frac{3}{8}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2) \cdot \\ &\qquad ({\rho _1} - {\rho _2}\cos {\phi _1})/{\rho _1} - \frac{1}{2}{\zeta _2}{\rho _2}\sin {\phi _1}/{\rho _1} - \\ &\qquad \frac{3}{8}{k_{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2) ({\rho _2} - {\rho _1}\cos {\phi _1})/ \\ &\qquad ({\varepsilon _2}{\rho _2}) - \frac{3}{8}{k_{n3}}({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2) \cdot \\ &\qquad ({\rho _2} - {\rho _3}\cos {\phi _2})/({\varepsilon _2}{\rho _2}) - \\ &\qquad \frac{1}{2}{\zeta _2}{\rho _1}\sin {\phi _1}/({\varepsilon _2}{\rho _2}) + \frac{1}{2}{\zeta _3}{\rho _3}\sin {\phi _2}/({\varepsilon _2}{\rho _2}) \\ & {{\dot \phi }_2}{\text{ = }}\frac{3}{8}{k_{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2)({\rho _2} - {\rho _1}\cos {\phi _1})/\\ &\qquad ({\varepsilon _2}{\rho _2}) + \frac{3}{8}{k_{n3}} ({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2)\cdot\\ &\qquad ({\rho _2} - {\rho _3}\cos {\phi _2})/({\varepsilon _2}{\rho _2}) + \frac{1}{2}{\zeta _2}{\rho _1}\sin {\phi _1}/({\varepsilon _2}{\rho _2})- \\ & \qquad \frac{1}{2}{\zeta _3}{\rho _3}\sin {\phi _2}/({\varepsilon _2}{\rho _2}) - {k_{n3}}\frac{3}{8}({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\phi _2} + {\rho _3}^2)\cdot \\ &\qquad ({\rho _3} - {\rho _2}\cos {\phi _2})/({\varepsilon _3}{\rho _3}) - \frac{1}{2}{\zeta _3}{\rho _2}\sin {\phi _2}/({\varepsilon _3}{\rho _3})- \\ & \qquad \frac{3}{8}{\zeta _{n3}}({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\phi _2} + {\rho _3}^2){\rho _2}\sin {\phi _2}/({\varepsilon _3}{\rho _3}) \end{aligned} \right\} $$ (22) ②主结构耦合2-DOF NNES系统的慢变动力流方程如下
$$ \left. \begin{aligned} & {{\dot \rho }_1}{\text{ = }} - \frac{3}{8}{k_{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2){\rho _2}\sin {\phi _1} - \frac{1}{2}{\zeta _1}{\rho _1} -\\ &\qquad \frac{3}{8}{\zeta _{n2}} ({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2)({\rho _1} - {\rho _2}\cos {\phi _1}) \\ & {{\dot \rho }_2} = {k_{n2}}\frac{3}{8}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2){\rho _1}\sin {\phi _1}/{\varepsilon _2} - \\ &\qquad \frac{3}{8}{k_{n3}} ({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2){\rho _3}\sin {\phi _2}/{\varepsilon _2} -\\ &\qquad \frac{1}{2}{\zeta _3}({\rho _2} - {\rho _3}\cos {\phi _2})/{\varepsilon _2}- \\ &\qquad \frac{3}{8}{\zeta _{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2)({\rho _2} - {\rho _1}\cos {\phi _1})/{\varepsilon _2} -\\ &\qquad \frac{3}{8}{\zeta _{n3}} ({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2)({\rho _2} - {\rho _3}\cos {\phi _2})/{\varepsilon _2} \\ & {{\dot \rho }_3} = \frac{3}{8}{k_{n3}}({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\phi _2} + {\rho _3}^2){\rho _2}\sin {\phi _2}/{\varepsilon _3} -\\ &\qquad \frac{3}{8}{\zeta _{n3}} ({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\phi _2} + {\rho _3}^2)({\rho _3} - {\rho _2}\cos {\phi _2})/{\varepsilon _3} \\ & {{\dot \phi }_1}{\text{ = }}\frac{1}{2} + {k_{n2}}\frac{3}{8}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2) \cdot \\ &\qquad ({\rho _1} - {\rho _2}\cos {\phi _1})/{\rho _1} - \frac{3}{8}{\zeta _{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2) \cdot \\ &\qquad {\rho _2}\sin {\phi _1}/{\rho _1} - \frac{3}{8}{k_{n2}} ({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2) \cdot \\ &\qquad ({\rho _2} - {\rho _1}\cos {\phi _1})/({\varepsilon _2}{\rho _2}) - \\ &\qquad \frac{3}{8}{k_{n3}}({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2)({\rho _2} - {\rho _3}\cos {\phi _2})/({\varepsilon _2}{\rho _2}) - \\ &\qquad \frac{3}{8}{\zeta _{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2){\rho _1}\sin {\phi _1}/({\varepsilon _2}{\rho _2})+ \\ & \qquad \frac{3}{8}{\zeta _{n3}}({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2){\rho _3}\sin {\phi _2}/({\varepsilon _2}{\rho _2}) \\ & {{\dot \phi }_2}{\text{ = }}\frac{3}{8}{k_{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2)({\rho _2} - {\rho _1}\cos {\phi _1})/\\ &\qquad ({\varepsilon _2}{\rho _2}) + \frac{3}{8}{k_{n3}} ({\rho _2}^2 - 2{\rho _3}{\rho _2}\cos {\phi _2} + {\rho _3}^2)\cdot \\ &\qquad ({\rho _2} - {\rho _3}\cos {\phi _2})/({\varepsilon _2}{\rho _2}) + \\ &\qquad \frac{3}{8}{\zeta _{n2}}({\rho _1}^2 - 2{\rho _1}{\rho _2}\cos {\phi _1} + {\rho _2}^2){\rho _1}\sin {\phi _1}/({\varepsilon _2}{\rho _2}) - \\ &\qquad \frac{3}{8}{\zeta _{n3}}({\rho _2}^2 - 2{\rho _3}{\rho _2} \cos {\phi _2} + {\rho _3}^2){\rho _3}\sin {\phi _2}/({\varepsilon _2}{\rho _2}) -\\ &\qquad \frac{3}{8} {k_{n3}}({\rho _2}^2 - 2{\rho _2}{\rho _3}\cos {\phi _2} + {\rho _3}^2)\cdot \\ &\qquad ({\rho _3} - {\rho _2}\cos {\phi _2})/({\varepsilon _3}{\rho _3}) - \frac{3}{8}{\zeta _{n3}}({\rho _2}^2 -\\ &\qquad 2{\rho _2}{\rho _3}\cos {\phi _2} + {\rho _3}^2) {\rho _2}\sin {\phi _2}/({\varepsilon _3}{\rho _3}) \end{aligned} \right\} $$ (23) 2. NES振动抑制效果及能量传递形式探究
根据文献[35]的参数选取方法给出本系统质量、刚度和线性阻尼参数为$ {m_1} $ = 1 kg, $ {m_2} $ = 0. 09 kg,$ {m_3} $ = 0. 01 kg, $ {k_1} $ = 1 N/m, $ {k_2} $ = 1 N/m3, $ {k_3} $ = 0. 008 N/m3, $ {c_1} $ = 0. 01 N·s/m, $ {c_2} $ = 0. 01 N·s/m, $ {c_3} $ = 0. 002 N·s/m.非线性阻尼参数选取为$ {c_{n2}} $ = 0. 5 N·s/m3, $ {c_{n3}} $ = 0.01 N·s/m3, 经过式(4)变换可得系统参数比值取值如表1和表2所示.
表 1 NES阻尼参数Table 1. Damping parameters of NESSystem $ {\zeta _2} $ $ {\zeta _3} $ $ {\zeta _n}_2 $ $ {\zeta _n}_3 $ 2-DOF LNES 0. 01 0. 002 — — 2-DOF NNES — — 0. 5 0. 01 2-DOF CNES 0. 01 0. 002 0. 5 0. 01 表 2 系统其他参数Table 2. Other parameters of systems$ {\varepsilon _2} $ $ {\varepsilon _3} $ $ {k_{n2}} $ $ {k_{n3}} $ $ {\zeta _1} $ 0. 09 0. 01 1 0. 008 0. 01 采用4阶龙格−库塔法对以上3种系统在不同初始速度下的NES能量耗散效率进行计算, 计算结果如图4所示. 由图4可知, 在小激励情况下时, 2-DOF LNES系统相较于2-DOF NNES系统具有较好的减振效率, 但随着系统初始能量的增大($ \eta {\text{ = 0}}{\text{.32}}\text{ m} /{\text{s}} $), 2-DOF LNES系统减振效率便有了明显下降趋势. 在主结构初始速度$ \eta {\text{ = 2}}\text{ m} /{\text{s}} $时, NES仅有57%的能量耗散效率. 2-DOF NNES系统则在初始能量较小时减振效率较差, 而在初始速度较大时, 该系统仍能保持较好的减振性能. 在主结构初始速度$ \eta {\text{ = 2}}\text{ m} /{\text{s}} $时, NES仍有79%的能量耗散效率. 值得注意的是, 2-DOF CNES系统结合了线性阻尼及立方阻尼的优势, 不仅降低了触发NES靶向能量传递所需的能量阈值, 且在较大的脉冲激励下仍能保持较为良好的减振性能.
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图 4 不同初始能量下系统能量耗散图Figure 4. Energy dissipation of the system with different initial energies选取图4中的3个代表性的激励速度来研究2-DOF NES振子对主结构的振动抑制性能. 其中, 利用小波变换对主结构与一级NES振子、一级NES与二级NES振子之间的相对位移进行时频分析. 图中$ D $表示位移响应, $ f $表示频率响应, $ E\% $表示主结构与各NES振子的瞬态能量占系统初始输入总能量比值. 当主结构与一级NES振子、一级NES振子与二级NES振子之间均发生1:1瞬态共振俘获现象时表明主结构、一级NES振子和二级NES振子之间发生1:1:1瞬态共振俘获现象.
当$ \eta {\text{ = 0}}{\text{. 03}}\text{ m} / {\text{s}} $时, 系统的动态响应如图5所示. 由图4可以得知, 2-DOF LNES, 2-DOF NNES和2-DOF CNES在主结构达到平衡状态时NES能量耗散总效率分别为49.5%, 4.1%和50.7%. 由图5(a)和图5(e)可知在初始激励较小时, 2-DOF LNES和2-DOF CNES系统中的能量由主结构缓慢流向NES中, 故具有一定减振能力. 2-DOF NNES系统中能量几乎未传入NES中, 基本不具备减振能力, 而在该工况下, 3种系统均未发生靶向能量传递现象, NES中瞬态能量占比最高只达到了系统初始总能量的2%.
由图5(b) ~ 图5(d)和图5(f) ~ 图5(h)的时程曲线和时频响应可知, 在2-DOF LNES及2-DOF CNES系统中仅少部分能量由主结构流入NES中, 一级NES振子与主结构、一级NES振子与二级NES振子均发生微弱的1:1共振俘获, 但后者1:1共振俘获时位移很小. 而 2-DOF NNES系统中, 仅主结构和一级NES振子之间发生了微弱的1:1共振俘获, 一、二级NES振子之间并未产生共振俘获现象.
当$ \eta {\text{ = 0}}{\text{.32}}\text{ m} / {\text{s}} $时, 系统的动态响应如图6所示. 由图4可知3种系统在主结构振幅达到稳定时, NES能量耗散效率均在92%左右, 无明显差异. 由图6(a)和图6(e)可知, 在该激励下, 3种系统均具有良好的减振效率, 它们的主结构均于25 s左右达到平衡状态. 在2-DOF LNES系统中, 能量于NES及主结构之间来回振荡传递, 但总体而言, 大部分能量都能够在短时间内传向NES中, 并被NES阻尼耗散掉. 在2-DOF NNES系统及2-DOF CNES系统中, 这种能量振荡传递的现象大幅减小, 能量传递具有一定的靶向性.
由图6(b)和图6(f)可知, 2-DOF LNES系统在能量传递过程中主结构与一、二级NES振子除了发生1:1:1瞬态共振俘获之外, 同时也伴随出现了大量较高频率, 这一现象在一级NES振子中尤为明显. 由图6(c)和图6(g)可知, 在2-DOF NNES系统中, 主结构与NES振子之间主要通过1:1:1共振俘获能量, 在NES振子中并未出现较高频率, 在主结构减振结束时, 一级NES振子中还存在着少量能量未被消耗, 这部分能量无法传递, 导致2-DOF NNES系统中一级NES振子持续处于低频的运动状态. 由图6(d)和图6(h)可知, 在2-DOF CNES系统中, NES振子中残留的能量由NES振子之间的线性阻尼快速消耗, 主结构迅速达到平衡状态.
当$ \eta\text{ = 1}\text{.4}\text{ m}/\text{s} $时, 系统的动态响应如图7所示. 由图4可知, 3种系统在主结构达到平衡状态时NES能量耗散总效率分别为65. 8%, 84. 2%, 84. 2%. 由图7(a)和图7(e)可知, 该激励下, 3种系统均有较好的减振效率, 且均出现了较为明显的靶向能量传递现象, 能量单向且高效地由主结构传递至NES中, 但2-DOF NNES系统及2-DOF CNES系统减振效率明显高于2-DOF LNES系统, 这是由于后者相较于前两者靶向能量传递效率更低所导致的.
由图7(b) ~ 图7(d)和图7(f) ~ 图7(h)可知, 与$ \eta\text{ = 0}\text{.32} $$ \text{ m} / {\text{s}} $相比, 2-DOF LNES系统能量由主结构传递至一级NES振子中时, 一级NES中产生了更高的频率分量, 但大部分能量还是通过1:1:1内共振进行耗散. 且在2-DOF NNES系统及2-DOF CNES系统中也出现了少量高频分量, 但相较于2-DOF LNES系统, 这些高频分量所占的比例很小.
孙敏等[36]研究发现串联2-DOF LNES系统中一、二级NES振子的质量比对主结构的减振效率会产生较大影响且一级NES振子的质量应数倍大于二级NES振子的质量. 为进一步探究系统NES质量参数对2-DOF LNES系统、2-DOF NNES系统和2-DOF CNES系统减振性能及能量传递形式的影响. 为此, 在保证系统NES总质量及其他参数不变的情况下将上述工况中的一、二级NES振子的质量比$ {\varepsilon _P} $由9. 0改为2. 3. 具体参数选取如表3所示.
表 3 系统对照参数Table 3. Comparison parameters of systems$ {\varepsilon _2} $ $ {\varepsilon _3} $ $ {k_{n2}} $ $ {k_{n3}} $ $ {\zeta _1} $ 0.07 0. 03 1 0.008 0. 01 对表1及表3中系统参数在初始激励$ \eta = 1{\text{. 4}}\text{ m} / {\text{s}} $情况下系统的响应进行数值分析, 结果如图8所示.
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图 8 $ \eta \text{ = 1}\text{.4}\text{ m}/ {\text{s}}, {\varepsilon }_{P}\text{ = 2}\text{.3} $时系统动态响应图Figure 8. System dynamic response diagram ($ \eta \text{ = 1}\text{.4}\text{ m}/ {\text{s}}, {\varepsilon }_{P}\text{ = 2}\text{.3} $)对比图7和图8可知, 当系统一、二级NES振子质量比为2.3时, 2-DOF LNES系统主结构的幅值响应发生了较明显的变化, 这也与参考文献[36]的结论一致. 此时2-DOF LNES系统主结构达到稳定状态的时间虽略有缩短, 但在主结构、一级和二级NES振子中均出现了更丰富的高频和低频分量, 且由于二级NES振子质量的增大, 一、二NES振子达到平衡所需时间均有一定延长. 2-DOF NNES系统及2-DOF CNES系统的振幅和能量传递形式受NES质量参数变化的影响并不明显. 系统减振过程中主结构和一、二级NES振子之间仍保持以1:1:1瞬态共振俘获的形式进行能量耗散. 但2-DOF NNES系统对NES振子中残留少量能量耗散慢的缺点依旧存在. 与2-DOF LNES和2-DOF NNES系统相比, 2-DOF CNES系统在NES振子质量参数发生较大变化时同样能保持其良好的减振性能及能量耗散形式.
3. 慢变动力流分析
本节将对上述3种系统进行慢变动力流分析, 即分析上述3种系统在不同激励下的慢变幅值和相位差, 进一步分析靶向能量传递过程的原因及机制. 对应慢变动力流方程初始参数选取如表4, 系统参数选取如表1和表2.
表 4 慢变动力流参数 (m)Table 4. The slow variable equations of the system parameters (m)$ {\rho _1} $(0) $ {\rho _2} $(0) $ {\rho _3} $(0) $ {\phi _1} $(0) $ {\phi _2} $(0) 0. 05 0. 000 1 0. 000 1 0 0 0. 8 0. 000 1 0. 000 1 0 0 系统慢变动力流响应幅值及相位差如图9和图10所示. 由图9可知, 在小初始能量条件下, 即ρ1(0) = 0.05 m时, 上述3种系统NES振子振幅均远小于主结构振幅, 且2-DOF LNES系统及2-DOF CNES系统在100 s时, 系统所剩能量已不足以使一级NES及二级NES产生振动, 在100 s之前时, 一级NES与二级NES均有小幅振动. 在图9(d)和图9(f)上显示为主结构与一级NES相位差在100 s之前持续增大, 100 s后各结构之间相位差趋于稳定的现象. 在图8(b)和图8(e)中, 2-DOF NNES系统中NES振子几乎未参与到主结构的减振中来, 主结构与一级NES振子之间相位差不断增大, 一级NES振子与二级NES振子之间相位差恒定为0的现象.
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图 9 主结构初始位移为ρ1(0) = 0. 05 m时系统的慢变幅值和相位随时间的变化Figure 9. Variation of slow-variation amplitude and phase of the system with time for an initial displacement of the main structure of ρ1(0) =0.05 m由图10可知, 在大初始能量条件下, 即$ {\rho _1} $(0) = 0. 8 m时, 上述3种系统中NES振子均参与到了主结构的减振当中, 在2-DOF NNES系统及2-DOF CNES系统中表现尤为明显. 由图10(d)可知, 2-DOF LNES系统在前80 s时, 主结构与一、二级NES振子之间发生1:1:1共振俘获现象, 在图中表现为锁相现象. 随着系统内能量的递减, 主结构与一级NES振子之间的1:1共振俘获现象被打破, 而一级NES振子与二级NES振子之间仍然进行1:1共振俘获现象, 至250 s时, 一级NES振子与二级NES振子之间的这种现象也逐渐消失. 由图10(e)和图10(f)可知, 在大初始能量条件下, 2-DOF NNES系统及2-DOF CNES系统在46 s前, 系统各结构之间出现了强烈的1:1:1共振俘获现象, 至46 s时, 系统中的大部分能量已被NES振子所耗散, 在46 s后由于系统所剩能量较小, 这部分能量对于2-DOF NNES系统很难耗散, 导致其一级NES振子持续振荡, 在图中反映为主结构与一级NES振子在46 s后相位差持续增大. 2-DOF CNES系统在46 s后, 这小部分能量仍可由线性阻尼所耗散, 从而使系统很快达到平衡状态.
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图 10 主结构初始位移为ρ1(0) = 0.8 m时系统的慢变幅值和相位随时间的变化Figure 10. Variation of slow-variation amplitude and phase of the system with time for an initial displacement of the main structure of ρ1(0) = 0.8 m为进一步探究NES质量参数对于系统内部能量传递形式的影响, 选取表1和表3中参数对各系统慢变动力流方程进行数值分析, 结果如图11所示.
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图 11 主结构初始位移为ρ1(0) = 0.8 m 且质量比为εp = 2.3时系统的慢变幅值和相位随时间的变化Figure 11. Variation of slow-variation amplitude and phase of the system with time for an initial displacement of the main structure of 0.8 m and a mass ratio of εp = 2.3对比图10和图11可知, 当NES振子的质量参数发生变化时, 2-DOF LNES系统的能量传递形式发生了较明显的变化. 图11所示2-DOF LNES系统中主结构与一、二级NES振子发生1:1:1瞬态共振俘获的时间大幅缩短. 由图10中的80 s缩短为25 s, 随后这种1:1:1瞬态共振俘获现象便被迅速破坏. 但2-DOF NNES系统及2-DOF CNES系统对主结构的减振形式仍以1:1:1瞬态共振俘获为主, 仅在1:1:1瞬态共振俘获被打破的时间节点上略有区别.
4. 结论
本文研究了2-DOF LNES系统, 2-DOF NNES系统及2-DOF CNES系统的振动抑制能力, 并对其靶向能量传递原理进行了探究. 首先对上述3种系统的数学模型进行了描述, 并利用复变量平均法推导了上述3种系统的慢变动力流方程, 对3种系统进行了时程、瞬态能量占比、小波变换及慢变动力流分析. 从同种系统在不同初始能量下NES的减振效率及能量传递形式进行了纵向对比研究, 又对不同系统在同种初始能量下的减振性能和能量传递区别进行了横向对比研究. 研究表明了2-DOF LNES系统在应对小初始能量时比2-DOF NNES系统更有优势, NES振子与主结构之间更易产生高频共振现象. 在大初始能量时, 2-DOF NNES系统的减振性能明显优于前者, 其随着初始输入能量的增大NES减振效率下降缓慢. 但对于残存在NES振子中的小部分能量耗散缓慢, 而2-DOF CNES系统则综合了前两者各自优势, 不仅具有小的能量触发阈值, 并且在主结构受大的初始脉冲激励时仍能保持较好的减振性能. 当系统NES振子质量参数在一定范围内发生变化时, 与2-DOF LNES和2-DOF NNES系统相比, 2-DOF CNES系统仍能保持其良好的减振性能及能量耗散形式.
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图 4 不同初始能量下系统能量耗散图
Figure 4. Energy dissipation of the system with different initial energies
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图 8 $ \eta \text{ = 1}\text{.4}\text{ m}/ {\text{s}}, {\varepsilon }_{P}\text{ = 2}\text{.3} $时系统动态响应图
Figure 8. System dynamic response diagram ($ \eta \text{ = 1}\text{.4}\text{ m}/ {\text{s}}, {\varepsilon }_{P}\text{ = 2}\text{.3} $)
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图 9 主结构初始位移为ρ1(0) = 0. 05 m时系统的慢变幅值和相位随时间的变化
Figure 9. Variation of slow-variation amplitude and phase of the system with time for an initial displacement of the main structure of ρ1(0) =0.05 m
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图 10 主结构初始位移为ρ1(0) = 0.8 m时系统的慢变幅值和相位随时间的变化
Figure 10. Variation of slow-variation amplitude and phase of the system with time for an initial displacement of the main structure of ρ1(0) = 0.8 m
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图 11 主结构初始位移为ρ1(0) = 0.8 m 且质量比为εp = 2.3时系统的慢变幅值和相位随时间的变化
Figure 11. Variation of slow-variation amplitude and phase of the system with time for an initial displacement of the main structure of 0.8 m and a mass ratio of εp = 2.3
表 1 NES阻尼参数
Table 1 Damping parameters of NES
System $ {\zeta _2} $ $ {\zeta _3} $ $ {\zeta _n}_2 $ $ {\zeta _n}_3 $ 2-DOF LNES 0. 01 0. 002 — — 2-DOF NNES — — 0. 5 0. 01 2-DOF CNES 0. 01 0. 002 0. 5 0. 01 
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表 2 系统其他参数
Table 2 Other parameters of systems
$ {\varepsilon _2} $ $ {\varepsilon _3} $ $ {k_{n2}} $ $ {k_{n3}} $ $ {\zeta _1} $ 0. 09 0. 01 1 0. 008 0. 01
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表 3 系统对照参数
Table 3 Comparison parameters of systems
$ {\varepsilon _2} $ $ {\varepsilon _3} $ $ {k_{n2}} $ $ {k_{n3}} $ $ {\zeta _1} $ 0.07 0. 03 1 0.008 0. 01
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表 4 慢变动力流参数 (m)
Table 4 The slow variable equations of the system parameters (m)
$ {\rho _1} $(0) $ {\rho _2} $(0) $ {\rho _3} $(0) $ {\phi _1} $(0) $ {\phi _2} $(0) 0. 05 0. 000 1 0. 000 1 0 0 0. 8 0. 000 1 0. 000 1 0 0
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