THEORETICAL MODEL AND LOW-FREQUENCY VIBRATION ISOLATION CHARACTERISTICS OF STACKED MIURA-ORI VIBRATION ISOLATORS
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摘要: 低频振动衰减速度慢难以控制, 严重影响高端装备的精度、稳定性和可靠性. 线性隔振器因承载能力和隔振频带之间的固有矛盾无法有效隔离低频振动. 因此, 可通过引入非线性获得等效负刚度和非线性刚度构建准零刚度隔振器, 以实现低频隔振. 折纸结构具有双稳态、负泊松比和负刚度等独特的力学特性, 近年来得到了广泛关注. 受堆叠三浦折纸(stacked Miura-ori, SMO)启发, 提出了一种新型SMO隔振器, 由SMO结构和线性弹簧并联构成. 基于能量法推导了SMO结构的势能方程, 获得了竖直方向的力-位移关系, 揭示了SMO结构大行程负刚度机理. 研究了无应力条件下不同初始角和折痕刚度比对SMO隔振器的恢复力的影响规律, 阐明了SMO隔振器非线性刚度调节机制. 建立了SMO隔振器的理论模型, 基于谐波平衡法推导了位移传递率. 通过数值仿真研究了不同负载下的低频隔振性能. 最后研制了SMO隔振器的原理样机, 试验验证了SMO隔振器的恢复力和低频隔振性能. 结果表明, 折纸隔振器峰值频率可低至1.52 Hz, 最小峰值传递率为1.29. 本研究为低频折纸隔振结构的设计提供了一种新思路.Abstract: Low-frequency vibration is difficult to control due to the slow decay rate, which seriously influences the precision, stability and reliability of high-end equipment. The linear vibration isolator cannot effectively isolate the low-frequency vibration due to the inherent contradiction between the bearing capacity and the vibration isolation frequency band. Therefore, the quasi-zero-stiffness vibration isolator can be constructed by introducing nonlinearity to obtain equivalent negative stiffness and nonlinear stiffness to realize the low-frequency vibration isolation characteristics. Origami structures have unique mechanical properties such as bistability, negative Poisson's ratio, negative stiffness, etc., which have been concerned in recent years. Inspired by stacked Miura-ori (SMO) origami, a novel SMO vibration isolator is proposed, which consists of an SMO structure and a linear spring connected in parallel. The potential energy equation of the SMO structure is derived based on the energy method, and the force-displacement relationship is derived, which reveals the mechanism of the negative stiffness of the SMO structure with large stroke. The effects of different initial angles and crease stiffness ratios on the restoring force of the SMO vibration isolator under stress-free conditions are investigated, and the nonlinear stiffness adjustment mechanism of the SMO vibration isolator is revealed. The theoretical model of the SMO vibration isolator is established, and the displacement transfer rate is derived based on the harmonic balance method. The low-frequency vibration isolation performance under different loads was investigated by numerical simulation. Finally, a prototype of the SMO vibration isolator was developed, and the restoring force and low-frequency vibration isolation performance of SMO vibration isolator were verified experimentally. The results demonstrate that the peak frequency and transmissibility of the SMO vibration isolator are 1.52 Hz and 1.29. This study provides a new approach for the design of low frequency origami-inspired vibration isolators.
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引 言
振动响应严重影响高端装备的精度、可靠性、安全性和稳定性等性能指标. 通常采用减振超材料[1-3]、隔振[4-6]和吸振[7]等方式降低振动带来的危害. 一般而言, 由弹簧和黏性阻尼器构成的线性隔振器在大于系统固有频率$\sqrt 2 $倍频率区域开始隔振. 通过降低系统的刚度可拓宽隔振频率, 但会导致承载能力下降. 为了解决上述问题, 在隔振系统引入等效负刚度与非线性刚度从而实现准零刚度隔振, 可以解决重载支撑与低频隔振之间的矛盾.
根据等效负刚度实现机理, 准零刚度隔振器可分为三弹簧式、磁力调控、仿生结构、凸轮滚子式和屈曲梁式等. Carrella等[8]提出了一种“三弹簧”准零刚度结构, 由两个斜弹簧与一个竖直弹簧并联而成, 通过参数调节可在平衡位置区域实现准零刚度隔振. 严博等[9]设计了一种永磁式变磁刚度非线性隔振器, 基于磁非线性构建等效负刚度, 实现了大加速度激励下的高性能低频隔振. Hao等[10]提出一种正交六自由度非线性隔振器, 通过弹簧正刚度和磁负刚度结合的方式实现非线性刚度, 研究结果表明, 该结构能在短的响应时间内抑制多方向冲击. Yan等[11]提出一种仿趾状隔振器, 实现了大范围负载能力和刚度的调节. 并在此基础上, 采用磁斥力模拟猫爪脂肪垫, 与趾状结构并联, 设计了一种仿爪状隔振器, 实现了大位移区间的准零刚度[12]. Sun等[13]受鸟类颈部启发提出一种新型多层结构, 可在低频、大振幅和位移突变的外部激励下实现动态稳定效果. Yan等[14]将仿蟑螂隔振器与分支电路阻尼结合, 显著降低了峰值传递率. 更多仿生隔振可参考文献[15]. 周加喜等[16]利用竖直布置的凸轮和水平布置的弹性滚子, 设计了一种凸轮-滚子式隔振器, 在大幅值激励下具有良好的隔振性能. Dalela等[17]提出了一种由双稳态屈曲梁和曲梁并联而成的隔振超结构, 实现了大区间零刚度特性. Gatti[18]提出一种由成对斜弹簧组成的装置, 通过改变几何配置能够获得硬化、软化和大变形准零刚度等非线性力-位移特性. Liu等[19]提出一种双级变刚度摩擦阻尼器, 该结构具有变刚度和变阻尼特性, 通过理论和实验验证该结构可以显著抑制飞轮在轨和发射载荷下的动态响应. 而对于非线性引起的混沌和跳跃现象, Yan等[20]提出了混合时滞前馈和反馈的控制方法, 显著提升了非线性隔振器的隔振性能和稳定性, 更多关于非线性隔振及时滞控制研究可参考文献[21].
在复杂动力学环境下, 激励幅值不定、激励频率随机且覆盖范围宽, 导致非线性隔振器的低频隔振性能难以得到有效保证, 往往需要对结构尺寸进行优化设计或进行阻尼调控. 折纸起源于中国, 在日本得到长足发展, 其独特的力学性能为非线性隔振器设计提供新的思路. 方虹斌等[22]系统总结了折纸结构和折纸超材料在动力学方面的最新进展, 剖析了其独特的力学特性, 例如, 负泊松比[23-24]、负体积模量[25]和双稳态等[26-27], 为低频非线性隔振器的设计提供了参考. Zeng等[28]利用六面非欧几里得硬折纸耦合8根拉簧, 设计了一种管式折纸结构, 其隔振性能优于等效线性隔振系统. 周雅等[29]受Kresling折纸启发设计了一种折纸准零刚度隔振器, 由非线性正刚度Kresling结构和非线性负刚度Kresling结构并联而成, 实现了较大的准零刚度区间, 验证了低频隔振性能. 王超等[30]将Kresling双稳态构型与橡胶支座结合, 有效降低了橡胶支座的刚度, 拓宽了橡胶支座的隔振带宽. Ye等[31-33]用螺旋弹簧代替折痕, 设计了一种SMO (stacked Miura-ori)准零刚度隔振器, 通过理论和实验验证了该结构的低频隔振性能, 并进一步研究了结构自重对隔振性能的影响. Liu等[34]用带有压缩弹簧的弹性接头模拟折痕, 设计了一种Tachi-Miura弹性折纸结构, 研究表明设计的折纸结构在不同负载下均具备良好的低频隔振性能. Suh等[35]提出了一种多模块Yoshimura折纸管构成的隔振器, 通过模块参数调节, 实现复杂激励的自适应调节.
本文受SMO结构启发, 将SMO结构负刚度特性引入, 设计了一种由SMO结构和线性弹簧并联而成的隔振器. 研究了不同折痕刚度比和不同初始角度SMO隔振器的非线性力学特性. 通过数值仿真和试验验证了SMO隔振器的隔振性能. 在不改变基本几何尺寸的情况下, 通过配置不同折痕刚度和初始角度, 实现SMO隔振器的刚度调节以适应不同负载以及不同频带的外部激励.
1. SMO折纸构型及力学特性
1.1 SMO折纸构型
SMO折纸构型由两个标准的三浦折纸cell A和cell B组成, 如图1所示. 首先将两个三浦折纸单元分别按照折痕图折叠(虚线为谷折痕, 实线为山折痕), 然后将折叠的两个单元中间对应的边线连接形成SMO结构, 如图2所示. SMO在竖直方向运动有3种构型, 分别为外凸(${\theta _A} < 0$)、折平(${\theta _A} = 0$)和内凹(${\theta _A} > 0$).
标准的三浦折纸单元可以通过两条相邻折痕线${\alpha _i}$, ${b_i}$以及它们之间的夹角${\gamma _i}$($i = A,B$)定义. 为了保证SMO结构折纸平面刚性以及运动过程中的兼容性, 其几何参数应满足
$$\qquad\qquad {b_A} = {b_B} = b $$ (1) $$\qquad\qquad \frac{{\cos {\gamma _B}}}{{\cos {\gamma _A}}} = \frac{{{\alpha _A}}}{{{\alpha _B}}} $$ (2) $$\qquad\qquad \cos {\theta _B}\tan {\gamma _B} = \cos {\theta _A}\tan {\gamma _A} $$ (3) 式中${\theta _A}$和${\theta _B}$是两个三浦折纸单元相对于连接线所在水平面的折叠角, 并且由式(3)可知${\theta _A}$和${\theta _B}$具有非唯一对应关系, 即每个${\theta _B}$对应两个不同的${\theta _A}$.
SMO结构的折叠运动可以通过各相邻面之间的二面角描述, 各二面角与折叠角的关系满足
$$ \qquad {\rho _{i1}} = {\rho _{i3}} = {\text{π}} - 2{\theta _i} $$ (4) $$ \qquad {\rho _{i2}} = 2\arcsin \left( {\frac{{\cos {\theta _i}}}{{\sqrt {1 - {{\sin }^2}{\theta _i}{{\sin }^2}{\gamma _i}} }}} \right) $$ (5) $$\qquad {\rho _{i4}} = 2{\text{π}} - {\rho _{i2}} $$ (6) $$ \qquad{\rho _C} = {\theta _B} - {\theta _A} $$ (7) 式中, ${\rho _{i1}}$, ${\rho _{i2}}$, ${\rho _{i3}}$和${\rho _{i4}}$($i = A,B$)为二面角, ${\rho _C}$为连接线折痕对应的二面角. 需要注意的是, 当${\theta _A} < 0$时, ${\rho _{A2}} \in \left( {{\text{π}},{{2\text{π} }}} \right)$; 当${\theta _A} > 0$时, ${\rho _{A2}} \in \left( {0,{\text{π}}} \right)$. 竖直方向高度$H$可以用折叠角${\theta _A}$表示为
$$ H = {\alpha _B}\sin {\theta _B}\sin {\gamma _B} - {\alpha _A}\sin {\theta _A}\sin {\gamma _A} $$ (8) 1.2 力-位移关系
当SMO结构受到竖直向下的外力时, SMO结构通过折痕折叠和折纸平面弯曲而变形. 假设折纸面板是刚性的, 折痕产生的刚度用具有特定扭转刚度的弹性铰链代替. 如图3所示, 在SMO折纸构型二面角${\rho _{B2}}$和${\rho _{B4}}$对应的折痕位置引入扭转刚度${K_B}$, 二面角${\rho _C}$, ${\rho _{A2}}$和${\rho _{A4}}$对应的折痕引入扭转刚度${K_A}$, 结构的总势能$E$可表示为
$$\begin{split} &E = \frac{1}{2}\left[ {K_B}{\left( {\Delta {\rho _{B2}}} \right)^2} + {K_B}{\left( {\Delta {\rho _{B4}}} \right)^2} + 4{K_A}{\left( {\Delta {\rho _C}} \right)^2} + \right.\\ &\qquad \left.{K_A}{\left( {\Delta {\rho _{A2}}} \right)^2} + {K_A}{\left( {\Delta {\rho _{A4}}} \right)^2} \right]\end{split} $$ (9) 式中, $\Delta {\rho _{ij}} = {\rho _{ij}} - \rho _{ij}^0$($i = A,B$; $j = 2,4$), $\Delta {\rho _C} = {\rho _C} - \rho _C^0$, 其中$\rho _{ij}^0$和$\rho _C^0$为初始状态下${\rho _{ij}}$和${\rho _C}$的角度, $\theta _A^0$为初始状态时${\theta _A}$的角度. 根据式(8)可得SMO结构初始状态的高度${H_0}$为
$$ {H_0} = {\alpha _B}\sin \theta _B^0\sin {\gamma _B} - {\alpha _A}\sin \theta _A^0\sin {\gamma _A} $$ (10) 式中, $\theta _B^0$为初始状态时${\theta _B}$的角度.
竖直方向位移$y$随${\theta _A}$的关系为
$$ y = {H_0} - H $$ (11) 由式(9)可知结构的势能$E$主要受折痕刚度${K_A}$, ${K_B}$以及$\theta _A^0$的影响, 引入$\tilde E = {E}/{{{K_A}}}$和$\tilde y = {y}/{{{H_0}}}$对式(9)和式(11)进行归一化处理, 得到
$$\begin{split} &\tilde E = \frac{1}{2}\left[ k{\left( {\Delta {\rho _{B2}}} \right)^2} + k{\left( {\Delta {\rho _{B4}}} \right)^2} + 4{\left( {\Delta {\rho _C}} \right)^2} +\right.\\ &\qquad \left.{\left( {\Delta {\rho _{A2}}} \right)^2} + {\left( {\Delta {\rho _{A4}}} \right)^2} \right] \end{split}$$ (12) $$ \tilde y = 1 - \frac{H(\theta _A)}{{{H_0(\theta _A)}}} $$ (13) 式中, $k = \dfrac{{{K_B}}}{{{K_A}}}$为折痕刚度比.
结构的恢复力可以通过势能对位移求导得到, 将式(5) ~ 式(7)代入式(12), 并对式(13)求导, 得到SMO结构竖直方向的等效恢复力$\tilde F$和等效刚度$\tilde K$为
$$\qquad\qquad \tilde F = \frac{{{\text{d}}\tilde E}}{{{\text{d}}\tilde y}} = \frac{{{\text{d}}\tilde E}}{{{\text{d}}{\theta _A}}}\frac{{{\text{d}}{\theta _A}}}{{{\text{d}}\tilde y}} $$ (14) $$\qquad\qquad \tilde K = \frac{{{\text{d}}\tilde F}}{{{\text{d}}\tilde y}} $$ (15) 表1为SMO结构几何设计参数. 图4为当$\theta _A^0 = - {60^ \circ }$时不同折痕刚度比($k = 14,12,10$)对SMO结构力-位移曲线的影响. 可以看出, 当$k = 10$时$\tilde F$的值均为正值, 随着折痕刚度比$k$的增大, SMO结构逐渐从单稳态变为双稳态, 当$k = 14$时$\tilde F$的最小值为负值, 此时结构为双稳态构型. 并且折痕刚度比的增加也会拓宽负刚度区间, 负刚度更加显著, 如图5所示. 图6为当$k = 10$时, 不同初始角($\theta _A^0 = - {60^ \circ },- {55^ \circ }, - {50^ \circ }$)对SMO结构力-位移曲线的影响. 可以看出, $\tilde F$的值均为正, SMO始终为单稳态构型. 随着初始角$\theta _A^0$的减小, SMO结构负刚度更加显著, 负刚度区间增大, 如图7所示.
表 1 SMO结构几何参数Table 1. Geometric parameters of the SMOParameter Value ${\alpha _A}/{\text{mm}}$ 45 $b/{\text{mm}}$ 45 ${\gamma _A}/\left( ^\circ \right)$ 55 ${\gamma _B}/\left( ^\circ \right)$ 70 ![]()
图 4 不同折痕刚度比下SMO结构的力-位移曲线Figure 4. Force-displacement curves of SMO structure under different crease stiffness ratios![]()
图 5 不同折痕刚度比下SMO结构的刚度曲线Figure 5. Stiffness curves of SMO structure under different crease stiffness ratios![]()
图 6 不同初始角下SMO结构的力-位移曲线Figure 6. Force-displacement curves of SMO structure under different stress-free initial angles![]()
图 7 不同初始角下SMO结构刚度曲线Figure 7. Stiffness curves of the SMO structure with different stress-free initial angles2. SMO隔振器结构设计与理论建模
2.1 准零刚度特性分析
上一节研究发现, SMO结构存在一定的负刚度区间. 为此, 通过将SMO结构与刚度为${k_L}$线性弹簧并联, 使SMO结构压缩到负刚度段, 线性弹簧提供恒定正刚度时, 正负刚度叠加产生准零刚度(椭圆区域), 如图8所示. 基于此原理设计了一种新型SMO隔振器, 如图9所示. 该隔振器由负载平台、底板、SMO结构以及线性弹簧组成. 无负载时, 负载平台位于${H_0}$; 加载时, 负载平台压缩至静平衡位置${H_1}$.
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图 8 SMO隔振器的力-位移曲线示意图Figure 8. Schematic of force-displacement curves of SMO vibration isolatorSMO隔振器在竖直方向的归一化恢复力$\hat F$可表示为
$$ \hat F = \tilde F + {\tilde F_L} $$ (16) 式中, ${\tilde F_L} = \dfrac{{{k_L}y{H_0}}}{{{K_A}}}$.
所对应的归一化刚度$\hat K$为
$$ \hat K = \frac{{{\text{d}}\hat F}}{{{\text{d}}\tilde y}} $$ (17) 为了保证隔振器工作的稳定性, 在任意变形下应避免出现负刚度, 即满足
$$ \hat K \geqslant 0 $$ (18) 图10为不同折痕刚度比时SMO隔振器的力-位移曲线. 可以看出, 随着折痕刚度比$k$的减小, 准零刚度区间逐渐变宽(黑色箭头线段长度), 但会导致承载能力降低. 图11为不同初始角度下SMO隔振器的力-位移曲线, 随着无应力初始角$\theta _A^0$的增大, 准零刚度区间逐渐变宽(黑色箭头线段长度), 相应的承载能力降低.
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图 10 不同折痕刚度比下SMO隔振器力-位移曲线Figure 10. Force-displacement curves of SMO vibration isolator under different crease stiffness ratios![]()
图 11 不同初始角时SMO隔振器力-位移曲线Figure 11. Force-displacement curves of SMO vibration isolator under different stress-free initial angles2.2 SMO隔振器的隔振特性分析
根据上述分析, SMO隔振器具有明显的准零刚度区域, 为探究其隔振性能, 本节将对SMO隔振器的隔振性能进行分析.
图12为SMO隔振器的简化模型, 其中, ${k_S}$为SMO结构竖直方向的刚度, ${k_L}$为线性弹簧的刚度, $c$为SMO隔振器的等效黏性阻尼.
当隔振器受到${y_b} = {Y_b}\cos \left( {\omega t + \varphi } \right)$的基础加速度激励时, 负载平台的位移为${y_p}$, 其中${Y_b}$为激励幅值, $\omega $为激励频率. 系统的运动微分方程为
$$ m\ddot x + c\dot x + \sum\limits_{i = 0}^7 {{k_i}} {x^i} = - m{\ddot y_b} $$ (19) 其中, $x = {y_p} - {y_b}$为负载平台与底座之间的相对位移. 利用谐波平衡法求解系统动力学方程, 假设系统进入到稳态响应时的稳态解为$x = X\cos \left( {\omega t} \right)$, 将其代入式(19), 只考虑谐波激励下的主共振响应, 忽略高次谐波项, 可得
$$ \begin{split} & \frac{{35}}{{64}}{k_7}{X^7} + \frac{5}{8}{k_5}{X^5} + \frac{3}{4}{k_3}{X^3}+ \\ &\qquad \left( {{k_1} - m{\omega ^2}} \right)X = m{Y_b}{\omega ^2}\cos \varphi \end{split} $$ (20) $$ c\omega X = m{Y_b}{\omega ^2}\sin \varphi $$ (21) 进一步可以得到幅频关系和相频关系
$$\begin{split} &{\left[ \frac{{35}}{{64}}{k_7}{X^7} + \frac{5}{8}{k_5}{X^5} + \frac{3}{4}{k_3}{X^3} + \left( {{k_1} - m{\omega ^2}} \right)X \right]^2} +\\ &\qquad {\left( {c\omega X} \right)^2} = {\left( {m{Y_b}{\omega ^2}} \right)^2} \end{split}$$ (22) $$\begin{split} &\cos \varphi = \frac{1}{{m{Y_b}{\omega ^2}}}\left[ \frac{{35}}{{64}}{k_7}{X^7} + \frac{5}{8}{k_5}{X^5} +\right. \\ &\qquad \left.\frac{3}{4}{k_3}{X^3} + \left( {{k_1} - m{\omega ^2}} \right)X \right]\end{split} $$ (23) 因此, 系统的位移传递率可表示为
$$ T = \frac{{\left| {{y_b} + x} \right|}}{{\left| {{y_b}} \right|}} $$ (24) 图13为不同负载(m = 1000, 1040, 1080 g)时SMO的位移传递率, 结果表明, 不同负载下SMO隔振器的共振频率均小于2 Hz, 且当负载质量m = 1040 g时, 共振频率和传递率峰值最小, 此时共振频率为1.48 Hz, 传递率峰值为1.51.
3. 试验与讨论
3.1 SMO结构力学特性
为了验证SMO结构理论模型的正确性, 根据表1所示几何参数研制了原理样机, 如图14所示. 刚性折纸面板为0.3 mm厚不锈钢薄板, 通过激光切割的方式单独切割而成. 二面角${\rho _{A1}}$, ${\rho _{A3}}$, ${\rho _{B1}}$和${\rho _{B3}}$对应的折痕处用3D打印的铰链进行连接, 折痕处的扭转刚度可忽略不计. ${\rho _{B2}}$和${\rho _{B4}}$对应的折痕处粘贴0.2 mm厚的预弯${92^ \circ }$弹簧钢条(即$\theta _A^0 = - {50^ \circ }$), 提供较大的扭转刚度, 其余二面角对应的折痕使用带有背胶的双向拉伸聚丙烯(BOPP)薄膜连接, 由于预弯弹簧钢条和BOPP薄膜粘贴在折纸面板上, 预弯弹簧钢条和BOPP薄膜的角度随SMO结构折叠角同步变化, 因此预弯弹簧钢条和BOPP薄膜折痕处发生弹性变形从而提供扭矩, 且BOPP薄膜的扭转刚度远小于预弯弹簧钢条提供的扭转刚度, 折痕刚度的差异使SMO结构产生双稳态.
SMO结构的等效力, 可用7阶多项式拟合, 如下
$$ f = \sum\limits_{i = 0}^7 {{a_i}} {y^i} $$ (25) 其中, ${a_i}$为拟合系数, 如表2所示.
表 2 多项式系数Table 2. Polynomial coefficientsParameter Value ${a_0}$ 8.269 × 10−2 ${a_1}$ 760.9 ${a_2}$ −8.695 × 104 ${a_3}$ 5.118 × 106 ${a_4}$ −1.715 × 108 ${a_5}$ 3.213 × 109 ${a_6}$ −3.496 × 1010 ${a_7}$ 2.032 × 1011 使用压力计(HP-200, 最大负荷200 N, 精度$ \pm 0.5\% $)测试了SMO结构样机的性能, 如图15所示. SMO结构在压力计下的测量结果与多项式拟合结果吻合较好, 证明了SMO结构理论建模的有效性与正确性(图中所标3个位置分别对应外凸(${\theta _A} < 0$)、折平(${\theta _A} = 0$)和内凹(${\theta _A} > 0$)).
3.2 SMO隔振器隔振性能
为验证SMO隔振器隔振性能, 根据表3给出的参数制作了SMO隔振器样机, 如图16所示. SMO隔振器由负载平台、线性弹簧、轴承、导杆、SMO结构以及底板组成, 两轴承安装在负载平台下方并沿导杆在竖直方向运动. 搭建了试验平台进行试验验证, 如图17所示. 试验平台主要由计算机、振动控制器、功率放大器、激振器、加速度传感器和SMO隔振器组成. SMO隔振器的底板安装在激振器上, 激振器由控制系统和台面加速度传感器构成的闭环系统进行控制. 控制器输出控制信号, 经功率放大器放大, 驱动激振器. 两个加速度传感器分别固定在SMO隔振器的下底座和负载平台上, 分别测量激励和输出加速度, 从而获得幅频响应关系.
表 3 SMO隔振器的参数Table 3. Parameters of SMO vibration isolatorParameter Value ${k_L}/({{{\mathrm{N}}\cdot{\mathrm{m}}}}^{-1})$ 305 $c/\left({{{\mathrm{N\cdot s}}\cdot{\mathrm{m}}}}^{-1} \right)$ 10 ${Y_b}/{\text{mm}}$ 2 图18为不同负载下的传递率曲线. 当负载质量$m = 1040 \;{\mathrm{g}}$时, 其共振频率和峰值传递率分别为1.52 Hz和1.29, 均小于负载质量为$m = 1000 \;{\mathrm{g}}$(1.64 Hz和1.47)以及$m = 1080 \;{\mathrm{g}}$(1.81 Hz和1.58), 这与仿真结果基本一致. 此外, SMO隔振器对摩擦异常敏感, 低频时会产生摩擦锁定效应[36-37](椭圆区域), 影响隔振区的隔振效果. 因此, 需要对所提出的SMO隔振器进行进一步结构优化设计, 以适应不同的工况需求.
4. 结论
本文提出了一种SMO隔振器, 研究了折痕刚度比和初始角对SMO结构和SMO隔振器力学性能的影响规律. 建立了SMO隔振器的理论模型, 通过数值仿真和试验研究了不同负载下SMO隔振器的力学特性和隔振性能. 结果表明:
(1)较大的折痕刚度比和较小的无应力初始角能够使SMO结构负刚度效果更加显著, 因此需要刚度较大的弹簧并联, 以提升SMO隔振器的承载能力;
(2)随着折痕刚度比的减小, SMO隔振器准零刚度区间变宽、承载能力降低. 随着无应力初始角的增大, SMO隔振器的准零刚度区间变宽、承载能力降低. 因此在实际应用中应合理配置折痕刚度比和无应力初始角.
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图 4 不同折痕刚度比下SMO结构的力-位移曲线
Figure 4. Force-displacement curves of SMO structure under different crease stiffness ratios
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图 5 不同折痕刚度比下SMO结构的刚度曲线
Figure 5. Stiffness curves of SMO structure under different crease stiffness ratios
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图 6 不同初始角下SMO结构的力-位移曲线
Figure 6. Force-displacement curves of SMO structure under different stress-free initial angles
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图 7 不同初始角下SMO结构刚度曲线
Figure 7. Stiffness curves of the SMO structure with different stress-free initial angles
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图 8 SMO隔振器的力-位移曲线示意图
Figure 8. Schematic of force-displacement curves of SMO vibration isolator
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图 10 不同折痕刚度比下SMO隔振器力-位移曲线
Figure 10. Force-displacement curves of SMO vibration isolator under different crease stiffness ratios
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图 11 不同初始角时SMO隔振器力-位移曲线
Figure 11. Force-displacement curves of SMO vibration isolator under different stress-free initial angles
表 1 SMO结构几何参数
Table 1 Geometric parameters of the SMO
Parameter Value ${\alpha _A}/{\text{mm}}$ 45 $b/{\text{mm}}$ 45 ${\gamma _A}/\left( ^\circ \right)$ 55 ${\gamma _B}/\left( ^\circ \right)$ 70 
下载: 导出CSV
表 2 多项式系数
Table 2 Polynomial coefficients
Parameter Value ${a_0}$ 8.269 × 10−2 ${a_1}$ 760.9 ${a_2}$ −8.695 × 104 ${a_3}$ 5.118 × 106 ${a_4}$ −1.715 × 108 ${a_5}$ 3.213 × 109 ${a_6}$ −3.496 × 1010 ${a_7}$ 2.032 × 1011
下载: 导出CSV
表 3 SMO隔振器的参数
Table 3 Parameters of SMO vibration isolator
Parameter Value ${k_L}/({{{\mathrm{N}}\cdot{\mathrm{m}}}}^{-1})$ 305 $c/\left({{{\mathrm{N\cdot s}}\cdot{\mathrm{m}}}}^{-1} \right)$ 10 ${Y_b}/{\text{mm}}$ 2
下载: 导出CSV
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