H∞ DESIGN AND EXPERIMENTAL STUDY OF TWO TYPES OF DYNAMIC VIBRATION ABSORBERS WITH TUNABLE DAMPING
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摘要: 动力吸振器凭借其出色的吸振能力得到了学术界和工程界的广泛关注. 然而其对共振频率、阻尼系数等系统参数的高敏感性给系统设计及其应用带来了极大的挑战. 传统的油液式黏滞阻尼器难以实现阻尼的有效调节, 新兴磁流变阻尼器高昂的造价也限制了该技术的实际应用. 近年来可调电磁分支电路阻尼的发展为动力吸振器现场最优调谐提供了新的可行性. 文章基于直线式电磁分支电路阻尼器的结构特点, 采用永磁体同极对置的方法提升了阻尼器的机电耦合效率, 通过数值仿真得出了六段对置式电磁分支电路阻尼器的机电耦合系数及等效阻尼系数. 电磁分支电路阻尼器可通过调整外接分支电路的阻抗实现阻尼力的实时调整, 并将其应用于悬置式和接地式动力吸振器系统中, 实现了两类动力吸振系统基于H∞ 优化方法的最优化吸振效果, 有效抑制了主系统的振动响应. 两类动力吸振系统的试验测试结果与理论预测一致, 充分证明了可调电磁分支电路阻尼器的有效性, 为动力吸振器的进一步发展应用提供了设计准则和依据.Abstract: Dynamic vibration absorbers have attracted extensive attention from both academia and industry due to its excellent vibration absorption performance. However, the high sensitivity in resonance frequency and damping coefficient makes the system design and application a real challenge. The traditional oil-type viscous dampers are difficult to achieve effective damping adjustment, and the high cost of emerging magnetorheological dampers also limits the practical application of this technology. In recent years, the development of the adjustable electromagnetic shunt damping has provided new feasibility for on-site optimal tuning of dynamic vibration absorbers. In this paper, based on the structural characteristics of the translational electromagnetic shunt damper, the method of opposing permanent magnets is used to improve the electromechanical coupling efficiency of the damper. The electromechanical coupling coefficient and equivalent damping coefficient of the six-stage opposing electromagnetic shunt damper are obtained through numerical simulation. The electromagnetic shunt damper can achieve real-time adjustment of the damping force by adjusting the impedance of the external resistance. Then, the electromagnetic shunt damper is applied in the suspended dynamic vibration absorber system and ground-hooked dynamic vibration absorber system, achieving the optimal vibration absorption effect of the two types of dynamic vibration absorption systems based on the H∞ optimization method, and suppressing the vibration response of the main system effectively. The maximum displacement response of the main system is minimized with the tuned natural frequency ratio and the adjustable damping ratio. Moreover, the experimental results of both dynamic vibration absorber systems match well with the theoretical results, which provides design criterion and a basis for further application of dynamic vibration absorber systems.
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引 言
动力吸振器(dynamic vibration absorber, DVA), 又称调谐质量阻尼器(tuned mass damper, TMD), 凭借其优异的共振能量吸收性能, 被广泛应用于输液管道[1]、机床系统减振[2]、内燃机轴系扭转振动抑制[3]、汽车悬架隔振[4]、座椅悬置减振[5]、传动系统宽频减振[6]、铁路列车地板结构减振[7]和商用车方向盘怠速抖动振动控制[8]等车辆结构的被动式振动控制领域.
动力吸振器一般存在悬置式和接地式两种结构形式. Ormondroyd等[9]最早基于固定点理论推导出了悬置式动力吸振器的近似解析解. Ren[10]和Liu等[11]同样基于固定点理论对接地式动力吸振器的最优频率比和阻尼比进行了求解, 两者的结果虽然形式略有差异, 但实际数值接近.
Shen等[12]提出了一种新型的动力吸振器, 通过杠杆幅值放大原理连接主副系统, 实现了共振抑制和有效带宽的扩展. 陈依林等[13]利用不同类型水平弹簧的组合实现了动力吸振器刚度的非线性, 在保证线性动力吸振器优势的同时拓宽了吸振频带. 代晗等[14]提出了负刚度动力吸振器, 并对负刚度时滞反馈控制动力吸振器进行了等峰值优化设计, 实现了共振峰值的降低和作用频带的拓宽. 为提高负刚度吸振器在真实场景中的应用性, 刘海平等[15]对非接地的负刚度动力吸振器进行了动力学设计及优化, 实现了相对线性动力吸振器更优的振动控制效果.
为了得到接地式动力吸振器的全局最优参数, 邢子康等[16]采用盛金公式求解了接地式三要素动力吸振器固定点幅值的表达式, 并对比了接地式三要素动力吸振器与接地式动力吸振器的响应幅值, 得出了系统全局最优解. 隋鹏等[17]采用改进的固定点理论方法求解出接地刚度吸振器的固定点坐标, 并根据幅值特征进行全局优化, 拓宽了频带宽度并得出与接地阻尼吸振器一致的峰值响应结果.
动力吸振器在实现过程中需要精确的阻尼配比, 且结构阻尼及装配方式等容易造成系统阻尼的变化. 因此, 可调阻尼对动力吸振器的工程实现意义重大. 当前可调阻尼器的研究主要集中在孔道可调的油液式连续阻尼控制(CDC)系统[18-19]和改变磁场引起流体黏度变化的磁流变阻尼器[20-21]. 然而这两类阻尼器的泄露风险及磁流变阻尼器高昂的造价限制了其广泛使用. 电磁分支电路阻尼器[22-25]基于其优异调节性能, 在振动控制领域得到了学术界的高度关注. 然而其有限的调节范围限制了在大型装备的广泛应用, 当前对电磁分支电路阻尼器调节范围的提升主要集中在负阻抗电路、电流主动调节和本体机电耦合系数等方面.
负阻抗电路可以抵消线圈内阻以提升回路电流, 继而提升阻尼调节范围. 当前研究的负阻抗电路主要分为多个运算放大器的大功率负阻抗电路[26]和单个运算放大器的小功率负阻抗回路[27-28]两类. 以上两类负阻抗电路的运算放大器在工作时都需要额外的电源供给, 回路总阻抗趋近于0时电流过大, 会造成系统不稳定. 采用主动式的电磁作动器[29]可实现可调阻尼力的大幅提升, 振动抑制效果显著, 但需要外部的能源供应. 因此, 近年来也出现了一些针对本体机电耦合系数提升的相关研究, Sun等[30]研究了磁极对置式结构对线性圆柱式电磁分支电路阻尼器机电耦合效率的提升效果. 针对相同的永磁体体积及线圈长度, 进一步优化了对置式磁极数对机电耦合系数及阻尼调节范围的影响, 给出了最优化磁极对的一般性求解方法[31].
基于电磁分支电路阻尼优异的调节性能, 本工作基于作者开发的六段对置式电磁分支电路阻尼器, 将其应用于悬置式和接地式动力吸振系统. 通过频率比的匹配调节设计实现了系统响应曲线中固定点的调平, 通过电磁分支电路阻尼的调谐达到两类系统所需的最优阻尼比, 实现了两类动力吸振器作用下主系统的最佳抑振效果. 试验测试结果与理论分析结果一致, 验证了电磁分支电路阻尼器优异的调节性能.
1. 悬置式与接地式动力吸振器
如图1所示, 动力吸振器为附着于主系统质量块$ {M_1} $且相对质量和刚度较小的子振动系统. 如图1(a)所示的悬置式动力吸振器通过弹簧$ {K_2} $和阻尼器$ {c_e} $与主系统质量块相连, 与地面基座不直接接触. 在外界正弦波$ F = {F_0}\cos (\omega t) $的激励下, 主系统质量块$ {M_1} $的响应可表示为
$$ \begin{split} &{\left| {\frac{{{X_1}}}{{{X_{st}}}}} \right|_A} = \\ &{\left\{ {\frac{{{{\left( {{\gamma ^2} - {\lambda ^2}} \right)}^2} + {{\left( {2\xi \gamma \lambda } \right)}^2}}}{{{{\left[ {\left( {1 - {\lambda ^2}} \right)\left( {{\gamma ^2} - {\lambda ^2}} \right) - \mu {\gamma ^2}{\lambda ^2}} \right]}^2} + {{\left( {2\xi \gamma \lambda } \right)}^2}{{\left( {1 - \mu {\lambda ^2} - {\lambda ^2}} \right)}^2}}}} \right\}^{\tfrac{1}{2}}} \end{split}$$ (1) 式中, $ {X_{st}} = {{{F_0}} /{{K_1}}} $, 表示主系统质量作用下的静态变形量; $ \mu = {{{M_2}} \mathord{\left/ {\vphantom {{{M_2}} {{M_1}}}} \right. } {{M_1}}} $, 表示子系统与主系统的质量比; $ \gamma = {{{\omega _2}} \mathord{\left/ {\vphantom {{{\omega _2}} {{\omega _1}}}} \right. } {{\omega _1}}} = \sqrt {{{{K_2}} \mathord{\left/ {\vphantom {{{K_2}} {{M_2}}}} \right. } {{M_2}}}} /\sqrt {{{{K_1}} \mathord{\left/ {\vphantom {{{K_1}} {{M_1}}}} \right. } {{M_1}}}} $, 表示子系统与主系统的固有频率比; $ \xi = {{{c_e}} \mathord{\left/ {\vphantom {{{c_e}} {{c_c}}}} \right. } {{c_c}}} $, 表示子系统阻尼系数与其临界阻尼系数之比, 临界阻尼系数$ {c_c} $是指单自由度振动系统中, 随着阻尼系数增大, 系统不发生自由振动的阻尼系数临界值; $ \lambda = {\omega \mathord{\left/ {\vphantom {\omega {{\omega _1}}}} \right. } {{\omega _1}}} $, 表示激振频率与主系统固有频率之比.
H∞优化方法即最小化主系统的最大响应值, Ormondroyd等[9]基于固定点理论给出的最优频率比和阻尼比可表示为$ {c_e} $
$$ \left. \begin{aligned} & {{\gamma _{{\mathrm{opti}}}}{\text{ = }}{1 \mathord{\left/ {\vphantom {1 {\left( {1 + \mu } \right)}}} \right. } {\left( {1 + \mu } \right)}}} \\ & {{\xi _{{\mathrm{opti}}}} = {{\left\{ {{{3\mu } /\left[{8\left( {1 + \mu } \right)}\right]}} \right\}}^{\tfrac{1}{2}}}} \end{aligned} \right\} $$ (2) 图1(b)所示的接地式动力吸振器通过弹簧${K_2}$与主系统质量块相连, 通过阻尼器与地面基座相连. 主系统质量块$ {M_1} $的响应可表示为
$$ \begin{split} &{\left| {\frac{{{X_1}}}{{{X_{st}}}}} \right|_B} =\\ &{\left\{ {\frac{{{{\left( {{\gamma ^2} - {\lambda ^2}} \right)}^2} + {{\left( {2\xi \gamma \lambda } \right)}^2}}}{{{{\left[ {\left( {1 - {\lambda ^2}} \right)\left( {{\gamma ^2} - {\lambda ^2}} \right) - \mu {\gamma ^2}{\lambda ^2}} \right]}^2} + {{\left( {2\xi \gamma \lambda } \right)}^2}{{\left( {1 + \mu {\gamma ^2} - {\lambda ^2}} \right)}^2}}}} \right\}^{\tfrac{1}{2}}} \end{split}$$ (3) 同样基于固定点理论, Ren[10]给出了对应的最优频率比和阻尼比
$$ \left. \begin{aligned} & {{\gamma _{{\mathrm{opti}}}}{\text{ = }}{{\left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {1 - \mu } \right)}}} \right. } {\left( {1 - \mu } \right)}}} \right]}^{\tfrac{1}{2}}}} \\ & {{\xi _{{\mathrm{opti}}}} = {{\left\{ {{{3\mu } / \left[{8\left( {1 - 0.5\mu } \right)}\right]}} \right\}}^{\tfrac{1}{2}}}} \end{aligned} \right\} $$ (4) 当系统质量比$ \mu = 0.1 $时, 通过式(2)和式(4)确定系统的最优频率比, 实现固定点的调平, 然后根据式(1)和式(3)绘制出阻尼变化对主系统振动响应的影响规律曲线. 在如图2所示的频域响应图中, 不同颜色的曲线表示不同阻尼比$ \xi $作用下的主系统质量块$ {M_1} $的响应. 如图2(a)所示, 在悬置式动力吸振器作用下, 当阻尼比$ \xi \approx 0.24 $时, $ {{{X_1}} \mathord{\left/ {\vphantom {{{X_1}} {{X_{st}}}}} \right. } {{X_{st}}}} $的最大值达到最小化, 系统达到最优状态. 如图2(b)所示, 在接地式动力吸振器作用下, 当阻尼比$ \xi \approx 0.22 $, $ {{{X_1}} \mathord{\left/ {\vphantom {{{X_1}} {{X_{st}}}}} \right. } {{X_{st}}}} $的最大值达到最小化, 系统达到最优状态. 两类动力吸振器均可通过阻尼比的调节实现振动响应最大值的最小化, 即满足H∞ 准则的最优吸振效果, 悬置式动力吸振器对应的最优阻尼比略大于接地式动力吸振器.
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图 2 两类动力吸振器作用下阻尼比变化对主系统频域响应的影响曲线图Figure 2. The primary system response curves in frequency domain with the varied damping ratios of two types of DVAs相较于无动力吸振器的单自由度系统, 附加动力吸振器的系统变为两自由度的系统, 存在两个共振峰. 随着动力吸振器阻尼系数的增大, 第二个共振峰逐渐消失, 系统可等效退化为单自由度系统, 两类动力吸振器作用下的共振峰移动方向不同. 悬置式动力吸振器作用下的主系统响应峰值频率左移, 接地式动力吸振器作用下的主系统响应峰值频率右移. 其主要原因在于, 悬置式动力吸振器的引入相当于增加质量效应, 接地式动力吸振器的引入相当于增加刚度效应, 因此引发了对应的共振峰偏移现象.
2. 可调电磁分支电路阻尼器
本文采用了如图3所示的永磁体磁极对置式电磁分支电路阻尼器. 如图3(a)左侧所示的两段对置式电磁分支阻尼器采用圆环式永磁体, 螺杆穿过圆环内孔, 两端通过螺母将对置式永磁体压紧. 永磁体的同极对置点位于外部环绕线圈的垂向中心处. 两线圈以180°的相位差连接, 组合后的线圈两端外接可调电阻, 通过改变阻抗值实现对阻尼力的调节. 当永磁体磁柱与线圈结构发生相对运动时, 线圈切割磁感线产生感应电动势, 外接阻抗接入后回路产生感应电流, 继而生成与原磁场反向的磁场阻碍相对运动, 阻碍原运动的力可被视为等效的电磁分支电路阻尼力.
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图 3 永磁体磁极对置式电磁分支电路阻尼器Figure 3. The electromagnetic shunt damper with opposing permanent magnet pairs为获得更高的阻尼调节范围, 本文采用如图3(a)右侧所示的六段永磁体同极对置式电磁分支电路阻尼器, 永磁体的内径$ {d_{{\text{in}}}} $ = 4 mm, 外径$ {d_{{\text{out}}}} $ = 15 mm, 单节长度$ {L_0} $ = 10 mm, 采用12节永磁体拼合连接, 两端采用单节永磁体, 内部5个20 mm长的磁柱由两节永磁体顺置组合而成, 采用螺杆螺母将同极对置式永磁体压紧, 便得到六段对置式永磁体磁柱. 以磁柱底面圆心为原点, 径向位置$ r $为横坐标, 轴向高度$ L $为纵坐标, 建立直角坐标系. 将磁柱水平放置后, 磁柱侧面外围的径向磁感应强度$ {B_r} $随轴向高度$ L $和径向位置$ r $的变化曲线如图3(b)所示. 磁极对置点处的径向磁感应强度($ {B_r} \approx $2.1 T)相对于端部自由磁极的磁感应强度实现了双倍增长, 相应的机电耦合系数也得到了大幅提升.
然而, 随着径向位置$ r $的不断增大, 径向磁感应强度$ {B_r} $快速衰减. 如图3所示, 当$ r $ = 20 mm时, $ {B_r} $已趋近于零. 因此, 越小的空气间隙越有利于机电耦合系数的提高. 本文采用文献[31]中的六段磁极对置式电磁分支电路阻尼器, 其机电耦合系数可表示为
$$ {K_t}{{ = - }}\oint_{{\mathrm{loop}}} {{B_r}\left( {L,r} \right){\mathrm{d}}l} $$ (5) 其中$ {\mathrm{d}}l $代表单位长度的导线. 忽略系统极低的内部摩擦阻尼, 经过仿真与实验验证, 得出其机电耦合系数为6.21 $ {{\mathrm{V}}} \cdot {\text{s/m}} $. 电磁分支电路阻尼器的阻尼系数$ {c}_{e} $可表示为
$$ {c_e}{\text{ = }}\frac{{K_t^2}}{Z} $$ (6) 其中, $ {K_t} $为机电耦合系数, $ Z $为回路总阻抗. 经过仿真与实验对比验证, 相应的阻尼调节范围为0 ~ 14.6 $ {\text{N}} \cdot {\text{s/m}} $. 通过实时调整外界阻抗, 可以实现系统阻尼比的改变, 继而完成悬置式与接地式动力吸振系统的最优吸振参数调节.
3. 试验设计与结果分析
3.1 试验设计
图4为对置式电磁分支电路阻尼器应用于接地式与悬置式动力吸振系统的试验平台. 接地式动力吸振器采用磁吸座将电磁分支电路阻尼器的线圈接地固定, 动力吸振器平衡重块的位置由阻尼器的底部转移至顶部. 非接触式激振器提供试验台架的激励, 且不会对系统刚度产生额外影响. 基于微控制器的电磁继电器可根据需求快速有效地调整外接电路阻抗. 两个激光位移传感器固定在外部支架上, 非接触式测量主系统与动力吸振系统的位移. 力传感器作为主系统的构成部件, 安装在主系统平板中心位置, 上端与非接触式激振器的永磁体相连, 用于测量来自线圈的激振力.
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图 4 (a)接地式与(b)悬置式动力吸振器试验平台Figure 4. The experimental setup of the (a) ground-hooked and (b) suspended DVAs通过对激光位移传感器1测得的主系统位移$ {x_1} $, 力传感器测得的系统激振力$ F $和主系统的刚度$ {K_1} $进行频率响应函数(FRF)分析, 即可得到主系统单位激振响应$ {{{X_1}} \mathord{\left/ {\vphantom {{{X_1}} {{X_{st}}}}} \right. } {{X_{st}}}} $. 两类动力吸振系统的主要参数如表1所示, 基于给出的参量数值, 计算得出未添加动力吸振器时, 主系统的固有频率$ {f_{n1}} \approx $10.72 Hz; 悬置式动力吸振系统的质量比$ \mu = $
0.1034 , 最优频率比$ {\gamma _{{\mathrm{opti}}}} = $0.9063 , 调整配重后的频率比$ \gamma = $0.8911 ; 接地式动力吸振系统的质量比$ \mu = $0.0758 , 最优频率比$ {\gamma _{{\mathrm{opti}}}} = $1.0402 , 调整配重后的频率比$ \gamma = $1.0412 . 两类系统的调整后的频率比与理论计算值吻合较好. 试验测试采用0.5 $ {\text{Hz/s}} $的扫频速度在0.1 ~ 20 Hz范围内进行扫频激励.表 1 接地式与悬置式动力吸振系统参数Table 1. The parameters of the ground-hooked and suspended DVAsParameters Values stiffness of primary system K1/(N·mm−1) 20.961 mass of primary system M1/kg 4.6202 stiffness of DVA K2/(N·mm−1) 1.7213 mass of suspended DVA M2/kg 0.4778 mass of ground-hooked DVA M2'/kg 0.3500 3.2 动力吸振调谐
动力吸振调谐主要分为调整频率比和调整阻尼比两个步骤. 由于本系统采用的刚度固定的螺旋式弹簧, 选定弹簧刚度后, 可通过调整动力吸振系统重块质量实现固有频率比的调整. 在未调整至最优频率比的主系统频域响应图中, 两固定点P和Q的响应幅值不同. 调整至最优频率比后, 如图5所示, 固定点P和Q处主系统振动响应幅值相同. 进行阻尼比调整时, 通过基于微控制器的可调电阻改变电路阻抗, 从而实现电磁分支电路阻尼器阻尼系数的变化, 继而实现动力吸振系统阻尼比的改变. 如图5所示的两类系统在不同阻尼作用下的测试响应结果均通过固定点P和Q, 且响应结果随阻尼的变化趋势与图2中的理论计算结果一致.
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图 5 主系统振动响应随电磁分支电路阻尼器外接阻抗的变化曲线Figure 5. The primary system response curves in frequency domain with different external resistance of the electromagnetic shunt damper由图5结果可知, 通过固有频率的调整, 两种形式的动力吸振系统均可达到两个固定点响应幅值相等的状态. 通过进一步调整可调电阻器的阻抗值, 可知外接阻抗约为2 Ω时, 悬置式与接地式动力吸振系统均达到最优状态, 实现了最大响应幅值的最小化, 即达到了H∞准则的最优化吸振效果. 两类吸振系统的最优阻尼比对应的回路外接阻抗均在2 Ω左右, 但两者相应的阻尼系数并不完全相等. 由于电磁分支电路阻尼系数$ {c_e} $与回路阻抗$ Z $具有如式(6)的反比例关系, 当回路阻抗趋近于零时, 电磁分支电路阻尼系数$ {c_e} $与阻抗$ Z $为高敏感关系, 回路阻抗的轻微变化即可导致阻尼系数较大的改变; 且回路阻抗较低时, 回路电流较大, 电路发热会导致回路阻抗的变化, 增加了调节难度. 后续研究中可以通过提高阻尼器调节范围使目标阻尼值处于低敏感区, 降低最优化阻尼的调节难度.
3.3 最优响应结果
通过对以上两类动力吸振系统的参数调谐, 在回路外接阻抗约为2 Ω时, 均实现了最优阻尼系数的匹配, 主系统振动响应达到基于H∞准则的最优化效果, 对应的主系统时域响应曲线如图6所示. 在图6中, 悬置式和接地式两种动力吸振器的作用下, 主系统振动响应均出现两次峰值, 对应图5频域响应曲线的P和Q固定点. 然而两个峰值在数值上并不相等, 其原因在于如图6(b)所示的激振力的变化. 理想扫频激振力曲线是等幅值的频率递增曲线, 但由于共振峰值出现后激振力的减小, 导致第二峰值处激振力的下降及相应主系统响应幅值的下降. 该现象也揭示了采用力传感器以获取主系统单位激振响应$ {{{X_1}} \mathord{\left/ {\vphantom {{{X_1}} {{X_{st}}}}} \right. } {{X_{st}}}} $的必要性.
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图 6 最优配置的悬置式和接地式动力吸振系统作用下主系统时域响应Figure 6. The primary system response in time domain of the suspended and ground-hooked DVA systems with optimal parameters通过对比图6(a)中的主系统位移响应幅值可知, 接地式动力吸振器作用下的主系统最优响应幅值大于悬置式动力吸振器, 但该结果并不能表明接地式动力吸振器的吸振效果弱于悬置式动力吸振器; 通过对比图6(b)的两类激振力可知, 接地式动力吸振器作用下的主系统所受的激振力幅值也大于悬置式系统, 因此通过对测量所得的主系统激振力$ F $和主系统位移$ {x_1} $进行基于$ {H_1} $估计的有FRF分析, 可得主系统的单位激振响应$ {{{X_1}} \mathord{\left/ {\vphantom {{{X_1}} {{X_{st}}}}} \right. } {{X_{st}}}} $如图7(b)所示. 两种动力吸振器相应的理论最优响应曲线可通过式(1) ~ 式(4)绘制如图7(a)所示. 试验测试所得的结果与理论结果吻合较好, 验证了对置式电磁分支电路阻尼器调节性能, 实现了动力吸振器的实时调控.
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图 7 最优化的动力吸振器主系统频域响应曲线Figure 7. The primary system response curves in frequency domain of the DVA systems with optimal parameters图7(a)的理论结果中, 接地式动力吸振器作用下, 主系统的频域响应峰值右移, 且数值明显减小. 在图7(b)的试验测试结果中, 主系统的频域响应峰值也出现了右移现象, 但峰值的数值降低幅度不明显. 其主要原因在于试验过程中采用了调整质量比$ \mu $的方法进行最优参数匹配, 接地式动力吸振系统的质量比为$ \mu = $
0.0758 , 难以达到理论分析中$ \mu = $0.1的抑振效果. 试验条件满足时, 可通过调整动力吸振系统刚度的方式保证系统相同的质量比, 达到更加有效的抑振效果.试验结果中悬置式动力吸振器作用下的右侧峰和接地式动力吸振器作用下的左侧峰曲线出现波动. 其主要原因在于, 系统调整为最优阻尼比时, 电磁分支电路阻尼器的外接阻抗值较小, 进行阻抗调节时敏感度较高; 且相对较小的回路阻抗导致电流增大, 致使回路温度升高, 导致阻尼调控精度受限, 影响系统的实时阻尼比, 继而影响系统的响应曲线光滑度. 该问题可通过提升阻尼器的阻尼调节范围来解决, 将最优阻尼调节点移动至外接阻抗大阻值的低敏感区, 实现最优阻抗的稳定调节.
4. 结论
本文对悬置式动力吸振器与接地式动力吸振器进行了理论分析和试验研究. 悬置式动力吸振器的质量增强效应导致了响应频带的左移, 接地式动力吸振器的刚度增强效应导致了响应频带的右移. 两种动力吸振装置均可通过频率比及阻尼比的调谐实现最大响应幅值的最小化, 即达到了基于H∞ 准则的最优化效果. 频率比可通过改变系统有效质量便捷调整, 阻尼比则采用电磁分支电路阻尼器, 通过改变外接阻抗实现阻尼比的精确调整. 本文基于对置式永磁体的电磁分支电路阻尼可实现阻尼调节范围的大幅提升. 将其应用于悬置式和接地式动力吸振器后, 实现了两类动力吸振器基于H∞ 准则的最优化调谐. 试验测试结果与理论吻合较好, 充分验证了电磁分支电路阻尼器优异的调节性能, 并实现了动力吸振器的最优化调谐. 为动力吸振器的更广泛应用提供了新的解决思路, 并为基于电磁分支电路阻尼器的参数化实时调控奠定了基础.
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图 2 两类动力吸振器作用下阻尼比变化对主系统频域响应的影响曲线图
Figure 2. The primary system response curves in frequency domain with the varied damping ratios of two types of DVAs
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图 3 永磁体磁极对置式电磁分支电路阻尼器
Figure 3. The electromagnetic shunt damper with opposing permanent magnet pairs
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图 4 (a)接地式与(b)悬置式动力吸振器试验平台
Figure 4. The experimental setup of the (a) ground-hooked and (b) suspended DVAs
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图 5 主系统振动响应随电磁分支电路阻尼器外接阻抗的变化曲线
Figure 5. The primary system response curves in frequency domain with different external resistance of the electromagnetic shunt damper
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图 6 最优配置的悬置式和接地式动力吸振系统作用下主系统时域响应
Figure 6. The primary system response in time domain of the suspended and ground-hooked DVA systems with optimal parameters
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图 7 最优化的动力吸振器主系统频域响应曲线
Figure 7. The primary system response curves in frequency domain of the DVA systems with optimal parameters
表 1 接地式与悬置式动力吸振系统参数
Table 1 The parameters of the ground-hooked and suspended DVAs
Parameters Values stiffness of primary system K1/(N·mm−1) 20.961 mass of primary system M1/kg 4.6202 stiffness of DVA K2/(N·mm−1) 1.7213 mass of suspended DVA M2/kg 0.4778 mass of ground-hooked DVA M2'/kg 0.3500 
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