RESEARCH ON TORSIONAL VIBRATION SUPPRESSION OF CHIRAL METAMATERIAL INERTER DYNAMIC VIBRATION ABSORBER
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摘要: 惯容结构常用于振动抑制中, 有着较为良好的减振效果. 而惯容与吸振器结合的新型吸振器具有轻量化的优点, 然而设计复杂的惯容结构限制了惯容吸振器在振动领域的广泛应用. 针对这一局限性, 设计一种具有简单高效惯容结构的手性超材料惯容吸振器(CIDVA). 首先引入了手性超材料的压缩−扭转耦合效应, 并利用该效应放大惯容盘的扭转行程, 形成惯容机制. 为了保证惯容机制的可行, 设计一种辅助机构来保证手性超材料的运动. 其次研究了CIDVA结构和工作原理并进行有限元仿真分析, 计算和验证其惯容放大常数. 并在此基础上建立了CIDVA−主系统的动力学方程, 对CIDVA−主系统在稳态和瞬态激励下的扭转振动抑制能力进行了研究, 并与锁定CIDVA进行了对比. 接着对惯容有效性进行了分析. 最后, 基于试验验证了CIDVA对主系统的扭转抑振能力. 结果表明, CIDVA能在瞬态和稳态激励下有效抑制主系统扭转振动, 且相较于传统DVA, 能节省自身10倍以上的转动惯量. 为DVA实现轻量化设计和高效的振动抑制提供了新思路和方法.Abstract: The inerter structure has proven to be highly effective in vibration suppression, exhibiting remarkable vibration reduction capabilities. The new type of dynamic vibration absorber combined with inerter and vibration absorber has the advantage of light weight. However, the complex design of traditional inerter structures hinders their widespread adoption in vibration control applications. In view of this limitation, a chiral metamaterial inerter dynamic vibration absorber (CIDVA) with simple and efficient inerter structure is designed in this paper. The CIDVA combines the advantages of both inerter and vibration absorber technologies, notably its lightweight design. Firstly, the compress-torsion coupling effect of chiral metamaterials is introduced, and the effect is used to amplify the torsion stroke of the inerter disk to form the inerter mechanism. In order to ensure the feasibility of the inerter mechanism, an auxiliary mechanism is designed to ensure the movement of chiral metamaterials. Secondly, the structure and working principle of CIDVA are thoroughly examined, and the finite element simulation analysis is carried out to accurately calculate and verify its inertance amplification constant. Building upon this foundation, the dynamic equation of the CIDVA-primary system is established, enabling a comprehensive study of the torsional vibration suppression abilities of the CIDVA-main system under steady-state and transient excitations. A comparison with the locked CIDVA configuration is also performed. Furthermore, the validity of the achieved inerter is meticulously analyzed. Finally, to validate the torsional vibration suppression capabilities of CIDVA on the main system, experimental verification is conducted. The simulation and experimental results demonstrate that CIDVA can effectively suppress the torsional vibration of the primary system under both transient and steady-state excitation, surpassing the performance of traditional DVA. Notably, CIDVA achieves significant weight savings, reducing its own moment of inertia by more than 10 times compared to traditional DVAs. and can save more than 10 times of its own moment of inertia compared with the traditional DVA. It provides new ideas and methods for DVA to achieve lightweight design and efficient vibration suppression. These findings contribute novel ideas and methodologies for achieving lightweight design and efficient vibration suppression in the field of DVAs.
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引 言
被动吸振是一种抑制结构振动的方法, 在可靠性和经济性等方面通常优于主动振动控制[1]. 其中, 动力吸振器(dynamic vibration absorber, DVA) 通常作为被动减振方法[2-3], 附加在主系统上以减轻力或运动的传递. 以往的研究表明, DVA以其高效、可靠和简便的特点被广泛应用于振动抑制方面[4-6]. 随着大型旋转设备在扭转振动抑制领域的发展, 动力吸振器的结构也愈加复杂, 因此追求其轻质化和简便化成为了学者研究的目标.
相较于传统DVA结构, 具有惯容结构的DVA(inerter dynamic vibration absorber, IDVA)采用一种机械机制来放大吸振器的“有效惯性”[7-9], 如飞轮齿轮惯性[10-11]、滚珠丝杠惯性[12-13]和液压惯性[14-15]等. 这种特点使得IDVA在保持结构稳定性和紧凑性的同时实现了轻量化设计. Hu等[16]结合不动点理论和代数解对IDVA进行了分析和优化, 有效提高结构体系的地震反应控制. Wang等[17-18]对4种基于惯容的新型负刚度DVA进行了研究和优化, 得到了闭环最优参数和改进的振动控制性能. Barredo等[19-21]对IDVA进行了大量理论和实验研究, 包括IDVA优化, 以及一种新型高性能非传统IDVA的设计与优化. Sui等[22]提出一种新的接地刚度惯容DVA, 确定了惯容的有效范围, 通过理论验证其具有良好的吸振能力. 日益复杂的惯容设计限制了IDVA结构在振动领域的广泛应用, 一种简单高效的惯容机构应用于DVA具有重要意义[23-24].
由于其独特的物理和机械特性, 超材料的出现为衰减振动提供了一种新的方法[25-26]. 在各种类型的超材料中, 自2017年Frenzel等[27]发现压缩−扭转耦合效应以来, 手性超材料受到了广泛关注[28-29]. 其中, 手性的概念已被引入到机械超材料的设计和振动抑制中[30]. Jamil等[31]提出了一种基于惯容的弹性超材料, 并证明了其有效性. Zhao等[32]开发了一种结合手性的晶格设计并应用于振动控制和隔离. 手性超材料已被成功地应用于具有更低和更宽带隙的惯容结构, 显示出其振动抑制的巨大潜力. 此外, Lin等[33]构建新型手性超材料并进行理论建模和有限元分析, 研究表明这种手性超材料拥有较好的压扭−耦合效应. 随着手性超材料的不断发展[26], 其在吸振领域的应用也日益广泛. 因此, 结合手性材料对惯容结构进行改进和简化是研究的热点问题[34-35].
目前, 通过附加动力吸振器的方法进行振动抑制已广泛用于消除机械系统的有害扭转振动中. 然而, 传统的吸振器质量较大, 这限制了它在主系统扭转振动抑制中的应用. 为了解决这一问题, 本文提出一种手性超材料惯容吸振器(CIDVA), 对其结构原理、抑振效果进行理论研究. 通过手性超材料的压扭耦合特性, 实现惯容机制的质量扩增, 减少实际应用的转动惯量, 抑制主系统的扭转振动, 并通过仿真和试验验证CIDVA的振动抑制能力. 为DVA实现轻量化设计和高效的振动抑制提供了新思路和方法.
1. 基于手性超材料的惯容机制
1.1 手性超材料的压−扭特性
本文所采用的手性超材料结构如图1(a) 所示, 包括2个振子盘和4段螺杆. 螺杆以振子盘的中心圆周分布, 并以手性的方式排列. 手性超材料的高度为h, 初始夹角为$\theta $ , 则有$h = l\sin \theta $.
手性超材料具有独特的压缩−扭转耦合效应, 其几何关系如图1(b) 所示.
设手性超材料上下振子盘分别为A1, B1, 斜杆初始夹角为θ, 假设某杆为PS, 振子盘B1固定不动, 对振子盘A1施加压力或者扭转力, 扭转角为θc, 扭转弧长PN2为
$$ P{N_2} = {r_1}{\theta _c} $$ (1) 手性超材料单元胞在Z方向产生轴向位移. 振子盘A1移动到A2, P点移动到N1点, 则螺杆旋转角度也为PS与N1S夹角. 螺杆长度l保持不变. 并且可以得到它们之间的关系
$$ P{N_1} = l{\theta _c} $$ (2) 设N1在振子盘A1的映射点为N2, 则手性超材料沿Z轴轴向位移N1N2表示为
$$ P{N_1} = \sqrt {{{\left( {P{N_2}} \right)}^2} + {{\left( {{N_1}{N_2}} \right)}^2}} $$ (3) 化简得${\theta _c}$与N1N2的关系为
$$ {N_1}{N_2} = \frac{{2{\text{π }}{r_1}n{\theta _c}}}{{{\text{tan}}\theta }} $$ (4) 1.2 惯性放大
作为一种两端元件, 惯容在各种机械系统中起着重要的作用. Smith等[10]提出的惯容根据齿轮与飞轮之间的质量关系决定运动放大系数. 而本文利用双手性超材料的压缩−扭转耦合效应来实现惯性放大. 为了实现这种惯性放大作用, 设计一种辅助机构限制手性超材料的局部运动. 如图2(a)所示, 移动约束限制双手性超材料沿z轴移动, 而旋转约束限制双手性超材料绕z轴旋转. 两者与手性超材料配合实现惯性放大, 其内环固定在支撑轴上. 两个移动约束分别与手性超材料CM1和CM2的上环和下环对齐. 为防止摩擦, 运动约束装置的支承梁不与振动环直接接触. 该机构允许在输入端和输出端产生绕z轴方向的旋转运动, 同时中介盘通过螺杆的空间变形实现沿z轴的轴向运动.
手性超材料加上辅助机构, 可以形成完整的惯容机制, 实现惯性放大. 运动原理如图2(b)所示, 其中CM1, CM2和中介盘分别用红、蓝、绿线表示. 带箭头的绿线表示中介盘的运动方向. CM1和CM2沿o1o2方向的轴向位移是共享的, 对应绿环的移动距离. 通过调整手性材料的参数, 可以获得适当的惯性放大系数.
根据1.1节推导的手性超材料CTC效应的几何关系, 可得吸振盘扭转角度θa和惯容盘θi分别与横向位移xa的关系为
$$\left.\begin{split} & {x_a} = \frac{{2{\text{π }}{r_1}{n_1}}}{{\tan {\theta _{c1}}}}{\theta _a} \\ & {x_a} = \frac{{2{\text{π }}{r_2}{n_2}}}{{\tan {\theta _{c2}}}}{\theta _i} \end{split}\right\} $$ (5) 式中, n1, r1和θc1分别为CM1的螺旋度、振子盘半径和初始夹角; n2, r2和θc2分别为CM2的螺旋度、振子盘半径和初始夹角.
简化得到θi和θa的关系
$$ {\theta _i} = \frac{{{r_1}{n_1}\tan {\theta _{c2}}}}{{{r_2}{n_2}\tan {\theta _{c1}}}}{\theta _a} $$ (6) 令$b = {r_1}{n_1}\tan {\theta _{c2}}/({r_2}{n_2}\tan {\theta _{c1}})$为惯性放大因子.
2. CIDVA结构与建模
2.1 CIDVA 结构
基于手性超材料惯容机理的CIDVA结构如图3所示. 它由DVA盘、惯容盘、双手性超材料和辅助机构组成. 双手性超材料用于连接DVA盘和惯容盘. 将旋转和移动约束附加到双手性材料上以提供方向限制. 该结构具有体积小、重量轻、没有铰链间隙、高运动精度等优点, 能够满足减振场景中对惯容的小尺寸、低重量和高可靠性需求.
2.2 有限元仿真
使用ANSYS有限元软件对CIDVA进行仿真分析, 验证CIDVA惯容机制的放大系数. 为了保证准确的比较和控制变量, 设置DVA盘和惯容盘的尺寸一致, 提供相等的转动惯量. 对初始角度递增的不同模型进行仿真分析, 取CM1初始夹角为10°, CM2的初始夹角由10°递增至70°, 其他参数为n1 = n2 = 0.25和r1 = r2 = 18 mm. 在理想情况和添加辅助机构两种情况下对CIDVA运动进行分析.
理想情况下, 在ANSYS中设置圆柱支撑约束, DVA盘和惯容仅允许切向运动, 而中介盘仅允许轴向运动. 仿真结果如图4(a)所示, CM2的扭转量随着初始角度的增大而增大, 而CM1的扭转量变化不大, 理论与仿真结果吻合较好. 由此可知, 通过调整CM2的初始角度, 将惯容的惯性放大系数调整到合适的范围是一种有效的方法.
取CM2初始角度为70°的模型, 添加不同厚度的旋转限制进行仿真分析, 结果如图4(b)所示. 理想情况下该模型的惯性放大系数为14.45倍, 而添加辅助机构后, 惯容盘的扭转减小. 当hz = 0.2 mm, 惯性放大系数降低至8倍, 当hz = 0.1 mm, 惯性放大系数降低至12.1倍. 由此可知厚度较小的旋转限制对惯容机制的影响较小. 但是, 当hz = 0.05 mm时, 旋转限制不能抑制CM1的扭转, 惯容机制失去放大效应.
2.3 主系统−CIDVA动力学建模
主系统−CIDVA结构如图5(a)所示, 由主系统和CIDVA两部分组成. 主系统包括底座、主系统盘和支撑轴, 其中主系统盘和支撑轴通过轴承连接. 主系统盘在扭矩作用下转动, 而支撑轴则固定不动.
图5(b) 为主系统−CIDVA扭转振动动力学模型, 根据牛顿第二定律, 该耦合系统的运动方程为
$$ \left. \begin{split} & {J_d}{{\ddot \theta }_d} + {c_d}{{\dot \theta }_d} + {k_d}{\theta _d} + {c_a}\left( {{{\dot \theta }_d} - {{\dot \theta }_a}} \right) = T \\ & {J_a}{{\ddot \theta }_a} - {c_a}\left( {{{\dot \theta }_d} - {{\dot \theta }_a}} \right) + {k_a}\left( {{\theta _d} - {\theta _a}} \right){\text{ + }}{T_i}\left( {{\theta _i}} \right) = 0 \\ & {J_i}{{\ddot \theta }_i} - {T_i}\left( {{\theta _i}} \right) = 0 \end{split} \right\} $$ (7) 式中, Jd 和kd分别为主系统盘的等效转动惯量和扭转刚度; Ja, ca和ka分别为DVA盘的等效转动惯量、扭转阻尼和扭转刚度. Ji为惯容盘的等效转动惯量; T = Tricos(ωt)为外激励扭矩, ω为转速. 主系统和CIDVA的扭转阻尼cd和ca分别表示为
$$ \left.\begin{split} & {c_d}{\text{ = }}2{\xi _d}\sqrt {{J_{{d}}}{k_{{d}}}} \\ & {c_a}{\text{ = }}2{\xi _a}\sqrt {{J_a}{k_a}} \end{split}\right\} $$ (8) 式中, 主系统的阻尼比$ {\xi _d} $ = 0.005, CIDVA的阻尼比${\xi _a}$ = 0.02.
将式(6)代入式(7), 则有
$$ \left. \begin{split} & {J_d}{{\ddot \theta }_d} + {c_d}{{\dot \theta }_d} + {k_d}{\theta _d} + {c_a}\left( {{{\dot \theta }_d} - {{\dot \theta }_a}} \right) = T \\ & \left( {{J_a} + {J_i}h} \right){{\ddot \theta }_a} - {c_a}\left( {{{\dot \theta }_d} - {{\dot \theta }_a}} \right) + {k_a}\left( {{\theta _d} - {\theta _a}} \right) = 0 \end{split} \right\} $$ (9) 为了评估CIDVA的抑振性能, 应建立附加锁定CIDVA的动力学方程, CIDVA的惯容盘仅有转动惯量对系统的动力学做贡献, 主系统−锁定CIDVA的动力学方程为
$$ \left( {{J_d}{\text{ + }}{J_a}{\text{ + }}{J_i}} \right){\ddot \theta _d} + {c_d}{\dot \theta _d} + {k_d}{\theta _d} = T $$ (10) 3. 仿真与讨论
3.1 参数设定
主系统盘的直径为120 mm, 厚度为40 mm. DVA盘和惯容盘的尺寸相同. 主系统盘的材料是钢, DVA采用3D打印技术制作, 材料为聚乳酸(polylactic acid, PLA). 另外, 为了保证柔性变形, 旋转限制采用黄铜切割加工而成. 基体材料参数见表1. 主系统−CIDVA仿真参数如下: Jd = 7.8 × 10−3 kg·m2; Ja = Ji = 8.0 × 10−5 kg·m2; kd = 70.8 N·m/rad ; ka = 9.1 N·m/rad ; 手性超材料对应参数为仿真分析b = 12.1时模型, 此时hz = 0.1 mm.
表 1 基体材料参数Table 1. Material parameterMaterial Elasticity modulus/Pa Density/(kg·m−3) Poisson ratio PLA 3.0 × 109 1250 0.3 iron 2.0 × 1011 7850 0.3 brass 1.0 × 1011 8550 0.28 3.2 CIDVA抑振性能分析
3.2.1 稳态响应
在稳态激励下, 分析CIDVA的扭转振动抑制特性. 主系统−CIDVA系统的初始条件设为
$$ {\theta _d}\left( 0 \right) = {\theta _a}\left( 0 \right) = {\dot \theta _d}\left( 0 \right) = {\dot \theta _a}\left( 0 \right) = {\ddot \theta _d}\left( 0 \right) = {\ddot \theta _a}\left( 0 \right) = 0 $$ (11) 采用以上这组参数对主系统−CIDVA进行稳态响应减振分析. 令周期激励Tri = 0.05 N·m. 主系统−CIDVA系统的稳态扭转振动响应如图6所示, 主系统一阶共振频率为15.1 Hz. 对纵坐标位移幅值进行归一化处理后, 附加锁定CIDVA时, θd的共振峰值为1°. 附加激活CIDVA后, θd的最大幅值为0.26°.
3.2.2 瞬态响应
本节研究了冲击激励下CIDVA的扭转振动抑制性能. 以初始角速度对系统进行初始激励, 将初始角位移减小到较低幅值(小于初始角位移的10%)所需的时间用于分析CIDVA的能量耗散速度. 如图7(a)所示, 主系统的初始角位移约为2.8°, 降至0.28°需要约3 s的时间. 与锁定CIDVA相比, 激活CIDVA提高了耗散速度. 主系统在1.5 s内衰减到0.28°左右, 耗散速度比锁定CIDVA快2倍. 相应的小波分析图如图7(b)所示.
3.3 惯容有效性
为了说明CIDVA的惯容机制的优越性, 将其与传统DVA进行了比较. 比较了两者在相同转动惯量下的抑振能力和相同抑振能力下所需的转动惯量. DVA扭转振动抑制百分比的目标函数定义为
$$ {A_d} = 100\% \times \frac{{\left( {{\theta _{d\_wo}} - {\theta _{d\_w}}} \right)}}{{{\theta _{d\_wo}}}} $$ (12) 式中, θd_wo为附加锁定DVA时的θd; θd_w为附加激活的DVA的θd.
假设传统DVA和CIDVA的转动惯量为9.7 × 105 kg·m2, 并且假设两者的刚度和阻尼参数相等. 而对位移幅值进行归一化处理后, 稳态响应对比曲线如图8(a)所示. 加入传统DVA后, 主系统的幅值基本没有减小. 这表明在该参数设置下, 没有惯容的辅助, 传统DVA对主系统没有抑制振动的作用. 而CIDVA展现出显著的扭转振动抑制能力. 其次, 比较了CIDVA与传统DVA在相同抑振效果下所需的转动惯量, 验证了惯容机制具有节省必要质量的能力. 对图8(b)和图8(c), 传统DVA的转动惯量需要比CIDVA大13.74倍, 才能达到相近的约60%的抑振能力. 传统DVA最大扭振抑制百分比为71.8%, 此时的转动惯量是CIDVA的12.47倍时, 但抑振能力仍略低于CIDVA.
综上所述, 在同等刚度、阻尼的参数条件下, 无惯容结构的传统DVA由于其转动惯量较小, 对主系统盘的减振效果较差. 而CIDVA的惯容机制有效地增强了其抑制振动的能力, 同时节省了10倍以上的转动惯量, 使其成为一种轻量化、简单的IDVA结构.
4. 试验验证
搭建主系统−CIDVA试验台, 如图9所示. 阶梯轴与底座连接, 底座固定在试验台上, 起支撑作用. CIDVA通过合适的扭转刚度弹簧和支座连接到主系统盘. DVA盘和惯容盘通过轴承与阶梯轴连接, 并随主系统盘的旋转而旋转. 激振器与主系统盘连接, 由信号发生器产生谐波信号经功率放大器传输. 而主系统盘上的另一端安装加速度传感器, 由LMS SCADAS系统采集振动响应信号并研究.
对主系统−CIDVA进行扫频. 扫描范围为1 ~ 80 Hz, 扫描时间为1.5 Hz/s. 未加CIDVA的时域曲线如图10(a)所示, 在20 ~ 30 s内, 红线处有较大的振动波动. 附加CIDVA后, 主系统的振动响应明显衰减, 如图10(b)中蓝色所示. 验证了CIDVA对主系统振动响应的抑制能力.
与图10(a)和图10(b)对应的colormap图如图10(c)和图10(d)所示. 未加CIDVA的主系统振动激励出现整数次谐波(1次谐波、2次谐波和3次谐波等), 加入CIDVA后, 谐波分量的强度被有效削弱, 15.1 Hz (固有频率)处的亮带消失. CIDVA大大降低了每一阶的幅值.
5. 结 论
本文提出了一种基于手性超材料惯容机制的CIDVA, 并将其应用于主系统的扭转振动抑制, 得出结论如下:
(1) CIDVA通过手性超材料的压扭耦合效应实现惯容结构, 进行质量扩增, 相较于传统复杂的惯容结构更为简便、静质量小且减振效果更优;
(2) CIDVA结构在瞬态激励和稳态激励下都展现出良好的振动抑制效果;
(3) 对比无惯容结构的传统DVA, 节省了相较自身10倍以上的转动惯量. 为DVA实现轻量化设计和高效的振动抑制提供了新思路和方法.
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表 1 基体材料参数
Table 1 Material parameter
Material Elasticity modulus/Pa Density/(kg·m−3) Poisson ratio PLA 3.0 × 109 1250 0.3 iron 2.0 × 1011 7850 0.3 brass 1.0 × 1011 8550 0.28 
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