SINGLE-STAGE AND MULTI-STAGE CONSECUTIVE CONSTANT QUASI-ZERO STIFFNESS FOR VIBRATION ISOLATION AT LOW FREQUENCY
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摘要: 准零刚度隔振器具有高静态和低动态刚度特性, 在低频隔振领域受到广泛关注. 具有非线性刚度特性的准零刚度隔振器在承载质量变化时, 导致动态刚度增大从而降低低频隔振性能, 不适用于变承载质量隔振工况. 为解决此问题, 采用拉簧和斜杆提出一种具备恒值准零刚度特性的隔振器, 通过静态分析获得的刚度表达式推导出两个准零刚度参数条件, 可实现恒值准零刚度、恒值零刚度或恒力和非线性准零刚度特性; 进而基于恒值准零刚度提出了多级连续恒值准零刚度隔振方法. 采用谐波平衡法和增量谐波平衡法分别对运动方程求解并得到位移传递率. 制作了单级和多级恒值准零刚度实验样机, 开展两种机制的变承载质量隔振实验研究. 第一种为单级恒值准零刚度机制的隔振实验, 适用于小幅变承载质量, 如设计承载质量的10%以内; 第二种为多级连续恒值准零刚度机制的隔振实验, 适用于大幅变承载质量, 如设计承载质量的32%及以上. 两种隔振机制为准零刚度隔振器在变承载质量工况下的应用提供了可行方法.Abstract: Quasi-zero stiffness (QZS) isolators have the feature with high static and low dynamic stiffness and widely been focused in the field of the vibration isolation at low frequency. Isolators with nonlinear QZS can’t be used for the low frequency vibration isolation under variable mass loads because the increased dynamic stiffness caused by mismatched mass load raises the initial frequency of vibration isolation and increases the magnitude of transmissibility. In order to solve this problem and improve the performance of vibration isolation at low frequencies, an isolator has been designed by using tension springs and oblique bars. Two QZS conditions including parameters have been derived on the basis of zero values of the stiffness and its second-order derivative at the static equilibrium position. According to the parameter conditions, constant QZS, constant zero stiffness or constant force, and nonlinear QZS can be realized as any values desired. The vibration isolation method with multi-stage consecutive constant QZS is furtherly proposed based on the constant QZS. The displacement transmissibility has been derived and calculated by employing the harmonic balance method and the incremental harmonic balance method, and shows the same result with each other. Prototypes with single-stage and multi-stage constant QZS have been fabricated to experimentally study the two vibration isolation mechanisms of variable mass loads. The first mechanism is the single-stage constant QZS to successfully isolate vibrations with the small magnitude of variable mass loads, such as 10% variation of the designed mass load. The second mechanism is the multi-stage consecutive constant QZS, which can be applied to variable mass loads with the large magnitude, such as the mass load is 32% variation of the designed value or larger extent. The two mechanisms of the proposed QZS isolators provide practicable approaches for the vibration isolation at low frequencies under variable mass loads.
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引 言
线性隔振器的起始隔振频率是$ \sqrt{2} $倍共振频率, 若满足低频隔振, 需低动态刚度, 但无法承受重载荷. 准零刚度(quasi-zero stiffness, QZS)隔振器由正、负刚度并联构造, 负刚度可大幅减小正刚度, 获得较低的起始隔振频率和较宽的隔振频带, 同时保持较高的静态刚度和承载力, 这种低动态和高静态刚度特性解决了线性隔振器低频隔振与重载荷的矛盾, 获得了广泛关注[1-6]. 依据负刚度的构造结构, 现有文献提出了多种类型的QZS隔振器, 如凸轮滚子[6-9]、磁体[10-11]、屈曲梁[12-13]或板[14]、斜杆[15-17]、斜弹簧[18-19]、X形[20-22]、仿生[23-27]和折纸结构[28-29]等类型.
单一静态平衡点的QZS隔振器具有非线性刚度特性, 如斜弹簧类模型, 当隔振质量偏离静态平衡位置, 将引起动态刚度增大, 导致起始隔振频率增大、隔振频带变窄和传递率幅值增大的不利影响, 不适合变承载质量工况的低频隔振[30-34]. 另一方面, 在大幅激励工况, 非线性QZS将引起共振峰向右弯曲, 进而增大起始隔振频率和减小隔振频带, 同时增大传递率幅值, 降低非线性QZS隔振器的低频隔振性能, 但有利于提升振动能量俘获性能[35-38]. 这些问题对QZS隔振器在变承载质量工况下的应用带来挑战, 若解决这些问题可显著拓宽QZS隔振器的工程应用场景.
针对变承载质量低频隔振问题, 现有文献提出了一些新颖的隔振方法. Lu等[39]基于线圈和磁体提出了一种承载质量自适应的电磁QZS隔振器, 可通过调节电流并按线性特性控制质量载荷大小, 在设计载荷1.9 kg和激励幅值1.2 mm工况下, 动态测试的起始隔振频率为3.5 Hz. Zhang等[10]基于弯曲梁机械超材料设计出裁剪的阶梯状力位移曲线, 提出了多级QZS特性的隔振器, 在0.5 ~ 5 kg范围内测试了5种不同的隔振质量, 获得较好的低频隔振性能. Wang等[30]理论研究了基于惯容器的QZS隔振器, 结果表明变质量载荷降低低频隔振性能. Li等[40]研究多段抛物线凸轮滚子隔振器, 采用多段QZS特性解决离散变质量载荷的低频隔振问题, 实验结果表明起始隔振频率在2 ~ 14 Hz之间变化. Zhang等[41]采用空气弹簧设计承载质量自适应的QZS隔振器, 当承载质量为224 kg时, 在0.5 Hz到30 Hz频率范围内, 传递率小于1. Lan等[42]通过调节三弹簧QZS隔振器的正刚度弹簧高度, 研究变质量载荷低频隔振, 测试结果表明, 在激励幅值为2 mm和承载质量分别是2、4和6 kg工况下, 起始隔振频率可达2 Hz. Qi等[43]基于单对斜杆的QZS隔振器, 提出恒力调节变质量载荷的等效重力方法, 承载质量相对设计值2.25 kg偏离0.5 kg时, 起始隔振频率为2 Hz. Xiao等[44]提出软树脂3 D打印梁构造的QZS隔振器, 承载质量从25 g增加到65 g时, 共振频率从118.1 Hz降低到36.2 Hz. 樊东芝等[45]提出堆叠三浦折纸隔振器, 在承载质量变化80 g条件下隔振效果良好. 针对变质量载荷隔振问题, 现有文献取得了较好的隔振效果, 但在质量载荷连续变化工况, QZS隔振机制仍需进一步研究.
本文基于拉簧和斜杆构造的恒值QZS机制, 提出了变承载质量隔振的双重机制: 单级恒值QZS特性解决承载质量小幅变化的问题, 多级连续恒值QZS特性解决承载质量大幅变化的问题. 制作了实验模型, 静、动态测试结果验证了理论分析结果, 具有较好的一致性. 实验结果表明, 承载质量相比设计值增大32%的大幅变化工况, 起始隔振频率基本保持不变, 保持了较宽隔振频带和较低的传递率幅值. 本研究为变承载质量低频隔振问题提供了一种可行方法, 拓宽了QZS隔振器的应用范围.
1. 静态分析
1.1 单级恒值准零刚度隔振设计
恒值QZS具有高静态刚度、低动态刚度和恒值刚度的力学特性, 在特定工况下可保障较好的隔振性能, 如变承载质量下动态刚度不变、大幅激励下传递率频带不变. 恒值QZS的产生机理是恒值负刚度结构与恒值正刚度结构并联构造, 在位移范围内可实现恒值的QZS特性. 单级即具有一个静态平衡位置的恒值QZS特性. 单级恒值QZS隔振器如图1所示, 刚度和力曲线如图2(a)和图2(b)所示. 单对斜杆和刚度为$ {k}_{1} $的水平拉伸弹簧产生垂直方向的恒值或非线性负刚度, 抵消垂直方向上恒值正刚度$ {k}_{2} $, 可获得恒值或非线性QZS特性. 水平拉簧在初始状态下的预拉伸长度为$ \delta $. 在初始状态下, 斜杆在水平方向上的投影长度为$ a $. 从初始位置到静态平衡位置的距离为$ h $, 静平衡位置即斜杆处于水平的状态. O点从初始位置开始的位移为$ x $, 从静态平衡位置开始的位移为$ y $.
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图 1 基于拉簧构造的单级恒值QZS隔振器Figure 1. Single-stage constant QZS isolator composed of tension springs![]()
图 2 恒值QZS与非线性QZS (LP: linear positive; NLN: nonlinear negative; NL: nonlinear; CD: constant dynamic)Figure 2. Constant and nonlinear QZS (LP: linear positive; NLN: nonlinear negative; NL: nonlinear; CD: constant dynamic)恒值QZS在一定位移范围内存在承载力的变化且动态刚度不变, 本质上允许承载质量变化, 如图2(a)和图2(b)所示. 而非线性QZS的隔振质量偏离设计值时, 隔振质量远离静态平衡位置, 则动态刚度显著增大, 低频隔振效果变差, 因此本质上不允许承载质量变化, 如图2(c)和图2(d)所示. 一旦设定恒值QZS隔振器的参数, 则QZS区域对应的力变化范围或承载质量的允许变化量就确定了, 恒值QZS的力学特性不会再改变, 导致对变承载质量的适应能力有限, 只能适用于小范围的承载质量变化. 虽然这种隔振器具有恒值QZS特性, 在位移范围内刚度相同, 但依然是单静态平衡位置的隔振器, 因此称此结构为单级恒值QZS特性.
1.2 多级连续恒值准零刚度隔振方法
由于单级恒值QZS对应的力变化范围有限, 一般是设计承载质量的10%以下(承载质量的变化量与设计负载的比值); 当承载质量变化大时, 隔振质量将偏离静态平衡位置过大, 超出QZS范围, 导致隔振失效而无法满足应用需求, 因此基于单级恒值QZS隔振器, 提出了多级恒值QZS隔振方法. 具体设计如图3所示, 开关按钮与控制器的内置程序关联, 控制器连接驱动器控制步进电机转动; 电机的输出轴为丝杠, 丝杠的转动驱动丝母上下移动, 丝母与正刚度弹簧$ {k}_{2} $的顶端通过螺栓固定连接; 随着丝母的上下移动, 使正刚度弹簧也随着上下移动, 引起初始位置O点上下变化, 进而改变$ h $和承载质量的大小. 其中, 控制器中的程序与开关按钮一一对应, 每个开关按钮对应着相应的静态平衡位置或$ h $的大小; 当按下相应按钮时, 通过控制器中的程序驱动电机改变正刚度弹簧的位置, 使$ h $和承载质量达到需要的值. 每个按钮对应一个静态平衡位置, 整个隔振系统具有相临的多个静态平衡位置, 即具有多个单级恒值QZS, 其力位移呈现多级台阶式特性, 因此称为多级恒值QZS隔振器. 又因相临的每级恒值QZS的承载力(图4纵坐标)可不间断的连续连接在一起, 因此具有多级连续恒值QZS特性. 例如, 有$ {x}_{0} $、$ {x}_{1} $和$ {x}_{2} $, 甚至更多静态平衡位置, 为保持较低的起始隔振频率, 需要较低动态刚度值, 恒值QZS对应的力变化范围小, 导致每个静态平衡位置允许的隔振质量变化小, 如$ \pm 100 $ g, 但多个静态平衡位置所允许的承载质量变化范围将大幅增加, 并且承载质量可连续变化.
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图 3 基于单级恒值QZS隔振器提出的多级连续恒值QZS隔振方法Figure 3. Isolation method with multi-stage consecutive constant QZS based on the single-stage consecutive constant QZS isolator![]()
图 4 多级连续恒值QZS力学特性Figure 4. Mechanical feature of the multi-stage consecutive constant QZS1.3 刚度推导及QZS条件
单级和多级连续恒值QZS隔振器的关键是拉伸弹簧和斜杆结构, 这种结构的力、刚度和QZS条件推导如下. 力$ f $满足以下方程[18]
$$ \left(f + {f}_{2} + 2{f}_{1x}\right)\Delta x = 0 $$ (1) 其中, $ {f}_{2} $是正刚度$ {k}_{2} $的拉伸弹簧在垂直方向上的弹性恢复力, $ {f}_{1 x} $表示刚度$ {k}_{1} $的水平拉伸弹簧和斜杆在垂直方向上的力, 具体表达式为
$$ \left.\begin{aligned} &{f}_{2} = -{k}_{2}x\\ &{f}_{1x} = -{f}_{1}\frac{h-x}{\sqrt{{a}^{2} + {h}^{2}}}\end{aligned}\right\} $$ (2) 其中, $ {f}_{1} $表示水平拉簧沿斜杆轴向的力, 具体表达式为
$$ \left.\begin{aligned} &{f}_{1} = {f}_{h}\frac{\sqrt{{a}^{2} + {h}^{2}}}{\sqrt{{a}^{2} + {h}^{2}-{\left(h-x\right)}^{2}}}\\ &{f}_{h} = {k}_{1}\left\{2\left[\sqrt{{a}^{2} + {h}^{2}-{\left(h-x\right)}^{2}}-a\right] + \delta \right\}\end{aligned}\right\}$$ (3) 其中, $ {f}_{h} $表示水平拉簧作用在斜杆外侧端部向内的水平拉力. 结合式(1) ~ 式(3), 以坐标$ x $表示的力$ f $为
$$ f = {k}_{2}x + 2{k}_{1}\left(h-x\right)\left\{\frac{2\left[\sqrt{{a}^{2} + {h}^{2}-{\left(h-x\right)}^{2}}-a\right] + \delta }{\sqrt{{a}^{2} + {h}^{2}-{\left(h-x\right)}^{2}}}\right\} $$ (4) 为了表示力$ f $与静平衡位置之间的关系, 力$ f $也可以用坐标$ y $表示
$$ \begin{split} &f = {k}_{2}h +\\ &\quad \left\{{k}_{2}y-2{k}_{1}\left[2\left(\sqrt{{a}^{2} + {h}^{2}-{y}^{2}}-a\right) + \delta \right]\frac{y}{\sqrt{{a}^{2} + {h}^{2}-{y}^{2}}}\right\}\end{split} $$ (5) 式中等号右边的第一项为隔振器的承载力, 第二项表示动态力. 两个坐标之间的转换关系为$ y = x-h $. 式(4)除以$ {k}_{2}\sqrt{{a}^{2} + {h}^{2}} $, 获得无量纲形式为
$$ \hat{f} = \hat{x} + 2\alpha {p}_{1}\frac{2\sqrt{1-{{p}_{1}}^{2}}-2\hat{a} + \hat{\delta }}{\sqrt{1-{{p}_{1}}^{2}}} $$ (6) 其中, $ \hat{f} $表示无量纲力, $ \hat{x} $是从初始位置开始的无量纲位移, $ \alpha $表示水平拉簧刚度与垂直拉簧刚度的比值, $ \hat{a} $是在初始状态下斜杆投影到水平方向上的无量纲长度, $ \hat{\delta } $表示初始状态下水平拉簧的无量纲预拉伸长度. $ {p}_{1} = {q}_{1}-\hat{x} $, $ {q}_{1} = \sqrt{1-{\hat{a}}^{2}} $. 以无量纲坐标$ \hat{y} $表示的无量纲力为
$$ \hat{f} = \sqrt{1-{\hat{a}}^{2}} + \left(1-2\alpha \frac{2\sqrt{1-{\hat{y}}^{2}}-2\hat{a} + \hat{\delta }}{\sqrt{1-{\hat{y}}^{2}}}\right)\hat{y} $$ (7) 式(7)可以通过式(5)除以力$ {k}_{2}\sqrt{{a}^{2} + {h}^{2}} $获得, 也可以通过式(6)和坐标关系$ \hat{y} = \hat{x}-\sqrt{{a}^{2} + {h}^{2}} $获得. 式(6)和式(7)中的无量纲参数定义为
$$ \left.\begin{aligned} &\hat{f} = \frac{f}{{k}_{2}\sqrt{{a}^{2} + {h}^{2}}}\\ &\hat{x} = \frac{x}{\sqrt{{a}^{2} + {h}^{2}}}\\ &\hat{y} = \frac{y}{\sqrt{{a}^{2} + {h}^{2}}}\\ &\alpha = \frac{{k}_{1}}{{k}_{2}}\\ &\hat{a} = \frac{a}{\sqrt{{a}^{2} + {h}^{2}}}\\ &\hat{\delta } = \frac{\delta }{\sqrt{{a}^{2} + {h}^{2}}}\end{aligned}\right\} $$ (8) 无量纲参数$ \hat{a} $的取值范围为$ \left(0\sim 1\right) $, 无量纲参数$ \hat{\delta } $的取值决定于水平拉簧的预拉伸长度和自由长度. 静态平衡点的无量纲形式为
$$ {\hat{x}}_{e} = \frac{h}{\sqrt{{a}^{2} + {h}^{2}}} = \sqrt{1-{\hat{a}}^{2}} $$ (9) 无量纲动态刚度$ \hat{K} $可以通过式(6)对$ \hat{x} $求导获得, 具体形式如下
$$ \begin{split} &\hat{K} = 1-2\alpha \left[2 + \left(\hat{\delta }-2\hat{a}\right){\left(1-{{p}_{1}}^{2}\right)}^{-\tfrac{1}{2}}\right]-\\ &\qquad 2\alpha {{p}_{1}}^{2}\left(\hat{\delta }-2\hat{a}\right){\left(1-{{p}_{1}}^{2}\right)}^{-\tfrac{3}{2}}\end{split} $$ (10) 通过将式(7)对无量纲坐标$ \hat{y} $求导, 也可以获得无量纲动态刚度$ \hat{K} $
$$ \hat{K} = 1-2\alpha \left[2 + \left(\hat{\delta }-2\hat{a}\right){\left(1-{\hat{y}}^{2}\right)}^{-\tfrac{1}{2}} + {\hat{y}}^{2}\left(\hat{\delta }-2\hat{a}\right){\left(1-{\hat{y}}^{2}\right)}^{-\tfrac{3}{2}}\right] $$ (11) 隔振器处于静态平衡位置时, $ {p}_{1} $等于0. 令无量纲刚度$ \hat{K} $在静态平衡位置等于零, $ \hat{K} = 0 $, 推导出第一个QZS条件为
$$ \alpha = \frac{1}{2\left[2 + \left(\hat{\delta }-2\hat{a}\right)\right]} $$ (12) 为获得静态平衡位置附近宽范围的QZS特性, 令无量纲刚度$ \hat{K} $对$ \hat{x} $的二阶导数在静态平衡位置等于零, $ \dfrac{{{\mathrm{d}}}^{2}\hat{K}}{{\mathrm{d}}{\hat{x}}^{2}} = 0 $, 推导出第二个QZS条件为
$$ \begin{split} &6\alpha \left(\hat{\delta }-2\hat{a}\right)\left[-{\left(1-{{p}_{1}}^{2}\right)}^{-\tfrac{3}{2}}-6{{p}_{1}}^{2}{\left(1-{{p}_{1}}^{2}\right)}^{-\tfrac{5}{2}}-\right.\\ &\qquad \left.5{{p}_{1}}^{4}{\left(1-{{p}_{1}}^{2}\right)}^{-\tfrac{7}{2}}\right] = 0 \end{split}$$ (13) 其中
$$ \hat{\delta } = 2\hat{a} $$ (14) 将方程(14)代入式(12), 得到简化的$ \alpha $值
$$ \alpha = 0.25 $$ (15) 如果参数依据式(12)选取, 隔振器具有非线性刚度QZS特性; 当按式(14)选取参数时, 隔振器具有恒值动态刚度特性; 当基于式(14)和式(15)选取参数时, 隔振器具有恒值零刚度特性, 即恒力特性. 为拓宽QZS隔振器的应用范围, 在获得QZS特性的同时, 期望QZS能适应大幅激励、变承载质量工况, 即恒值QZS, 当按式(16)选取参数时可获得.
$$ \left.\begin{aligned} &\hat{\delta } = 2\hat{a}\\ &0 < \alpha < 0.25\\ &\alpha \to 0.25\end{aligned}\right\} $$ (16) 1.4 参数影响下的QZS特性
1.4.1 非线性QZS
如果参数符合公式(12), 隔振器具有非线性QZS特性, 如图5所示. 当参数$ \hat{a} $值给定时, 如图5(a)和图5(b)中$ \hat{a} = 0.9 $, 随着参数$ \hat{\delta } $值增大, 静态平衡点附近的QZS范围逐渐变宽, 可达到极小的动态刚度; 但静态平衡点处的承载能力不变. 当参数$ \hat{\delta } $给定时, 如$ \hat{\delta } = 1.6 $, 随着参数$ \hat{a} $的增大, 静态平衡点附近的QZS范围逐渐变窄, 静态平衡点处的承载能力逐渐降低, 如图5(c)和图5(d)所示. 参数$ \hat{\delta } $能调节静态平衡位置附近的QZS范围, 但对静态平衡位置处的承载力无任何影响; 参数$ \hat{a} $不仅能调节静态平衡位置附近的QZS范围, 还能改变静态平衡位置处的承载能力.
1.4.2 恒值动态刚度
如果参数满足方程(14), 隔振器可获得恒值动态刚度, 如图6所示. 当参数$ \alpha $值给定时, 如$ \alpha = 0.2 $, 虽然参数$ \hat{\delta } = 2\hat{a} $的值变化, 但恒值动态刚度不变, 如图6(a)所示; 对应的力位移曲线是平行的斜直线, 且随着$ \hat{\delta } = 2\hat{a} $值的增大, 静态平衡位置处的承载能力逐渐降低, 如图6(b)所示. 当参数$ \hat{\delta } = 2\hat{a} $值确定, 如$ \hat{\delta } = 2\hat{a} = 1.6 $, 随着参数$ \alpha $值的增大, 恒值动态刚度逐渐降低并趋近于零, 最后变成负值, 如图6(c)所示; 对应的力位移曲线具有不同的恒值斜率, 相交于相同的静平衡位置, 如图6(d). 参数$ \hat{\delta } = 2\hat{a} $不影响恒值动态刚度的值, 但决定着静平衡位置的承载能力; 参数$ \alpha $决定着恒值动态刚度, 但不影响静平衡位置的承载能力.
1.4.3 恒值零刚度或恒力特性
当隔振器参数满足式(14)和式(15)时, 存在一个独立变量$ \hat{\delta } = 2\hat{a} $, 可获得恒值零刚度和恒力特性, 如图7所示. 恒值零刚度是位移范围内的动态刚度始终为零. 随着$ \hat{\delta } = 2\hat{a} $值增大, 动态刚度恒定不变且为零, 但恒力的值逐渐降低.
1.4.4 恒值QZS
当隔振器的参数选取符合公式(16)时, 可以获得恒值QZS, 用于隔离低频振动. 恒值QZS是位移范围内的动态刚度相同且接近于零. 当参数$ \alpha $小于且接近于0.25, 如$ \alpha = 0.247 $, 恒值动态刚度的值可达到非常小, 接近于零, 并且参数$ \hat{\delta } = 2\hat{a} $的值对恒值动态刚度无影响, 如图8(a)所示. 对应的力位移曲线具有相同的斜率, 且随$ \hat{\delta } = 2\hat{a} $值的增大而降低, 如图8(b)所示. 这种力学特性提供了一种有效的低频隔振方法, 特别适合于变承载质量的隔振.
1.5 多级QZS特性
由力表达式(7)可知, 在静态平衡位置处的静态承载力为$ \sqrt{1-{\hat{a}}^{2}} $, 因此承载能力由参数$ \hat{a} $确定, 随着参数$ \hat{a} $的减小, 承载能力逐渐增大. 同时, $ \sqrt{1-{\hat{a}}^{2}} $也是$ h $的无量纲形式, 表示从初始位置到静态平衡位置的距离. 给定参数$ \hat{a} $, 对应的静态平衡位置是$ \sqrt{1-{\hat{a}}^{2}} $, 对应的静态承载力也是$ \sqrt{1-{\hat{a}}^{2}} $. 因此可以绘制多级QZS的刚度和力位移曲线.
多级非线性QZS如图9所示, 拓宽了QZS范围和承载能力, 但承载力不连续. $ \hat{a} = 0.95 $对应的静态平衡位置0.312, 对应的静态承载力0.312; $ \hat{a} = 0.9 $对应的静态平衡位置0.436, 对应的静态承载力0.436; $ \hat{a} = 0.85 $对应的静态平衡位置0.527, 对应的静态承载力0.527. 单个静态平衡位置对应的QZS范围窄, 3个静态平衡位置对应的QZS范围显著变宽, 但3个静态平衡位置的承载力是分级离散不连续的状态.
多级恒值QZS如图10所示, 每一级静态平衡位置对应的QZS范围宽且等值, 承载力是分级且连续状态. 每级静态平衡位置可适用于一定范围的承载质量变化, 且动态刚度不变. 相邻两级静态平衡位置对应的承载力是连续的, 即两个相邻的静态平衡位置具有共同的承载力. 如图10(b)所示, $ \hat{a} = 0.95 $和$ 0.9 $两条力位移曲线具有相同的隔振载荷重合区域. 因此, 具有多级连续的恒值QZS特性, 适合承载质量大幅变化工况. 恒值QZS与非线性QZS具有相同的静态平衡位置, 但承载能力却不同; 恒值QZS的承载能力是多级连续特性, 非线性QZS的承载能力是多级离散特性.
2. 动态分析
2.1 谐波平衡法推导位移传递率
QZS隔振器在初始位置放置合适的承载质量, 可达到静态平衡位置, 运动方程为[15]
$$ m\left(\ddot{z}-{\ddot{z}}_{e}\right) + c\dot{z} + f = mg $$ (17) 方程(17)中, 力$ f $来源于QZS隔振器的弹性恢复力, 包括恒力和动态力两项, 恒力与隔振质量抵消. 在谐波位移激励下, QZS隔振器的运动方程表示为
$$ m\ddot{z} + c\dot{z} + {k}_{L}z + {k}_{NL}{z}^{3} = m{\omega }^{2}{Z}_{e}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right) $$ (18) 其中, $ z = {z}_{e}-y $表示QZS隔振器的相对位移, $ {z}_{e} $是隔振器的位移激励, $ {Z}_{e} $是位移激励幅值, $ \omega $表示位移激励频率, $ c $是隔振器阻尼, $ {k}_{L}z $和$ {k}_{NL}{z}^{3} $表示QZS隔振器动态力泰勒展开的一次和三次项. 式(18)除以$ {k}_{2}\sqrt{{a}^{2} + {h}_{1}^{2}} $可得无量纲方程
$$ \acute{\acute{\bar{z}}} + 2\zeta \acute{\bar {z}} + {\mu }_{1}\bar {z} + {\mu }_{3}{\hat{Z}}_{e}^{2}{\bar {z}}^{3} = {\varOmega }^{2}\mathrm{c}\mathrm{o}\mathrm{s}\varOmega \tau $$ (19) 其中, $ \tau = {\omega }_{0}t $, $ {\omega }_{0} = \sqrt{\dfrac{{k}_{2}}{m}} $, $ \varOmega = \dfrac{\omega }{{\omega }_{0}} $, $ \dfrac{{\mathrm{d}}z}{{\mathrm{d}}t} = \dfrac{{\omega }_{n}{\mathrm{d}}z}{{\mathrm{d}}\tau } $, $ \dfrac{{{\mathrm{d}}}^{2}z}{{\mathrm{d}}{t}^{2}} = \dfrac{{\omega }_{n}^{2}{{\mathrm{d}}}^{2}z}{{\mathrm{d}}{\tau }^{2}} $, $ \zeta = \dfrac{c{\omega }_{0}}{2{k}_{2}} $, $ {\mu }_{1} = \dfrac{{k}_{L}}{{k}_{2}} $, $ {\mu }_{3} = \dfrac{{k}_{NL}}{{k}_{2}} $, $ {\hat{Z}}_{e} = \dfrac{{Z}_{e}}{\sqrt{{a}^{2} + {h}^{2}}} $, $ \hat{z} = \dfrac{z}{\sqrt{{a}^{2} + {h}^{2}}} $, $ \bar {z} = \dfrac{\hat{z}}{{\hat{Z}}_{e}} $, $ \acute{\acute{\bar {z}}} = \dfrac{{{\mathrm{d}}}^{2}\bar {z}}{{\mathrm{d}}{\tau }^{2}} $, $ \acute{\bar {z}} = \dfrac{{\mathrm{d}}\bar {z}}{{\mathrm{d}}\tau } $. $ {\omega }_{0} $是对应线性隔振器的共振频率, 所谓对应线性隔振器是指QZS隔振器去掉产生负刚度的水平拉伸弹簧. $ \varOmega $表示无量纲激励频率, $ m $是隔振质量, $ \zeta $表示阻尼比, $ {\mu }_{1} $和$ {\mu }_{3} $分别表示无量纲线性和非线性刚度系数. 假设方程(19)的响应为$ \bar {z} = \hat{Z}\mathrm{c}\mathrm{o}\mathrm{s}\left(\varOmega \tau + \varphi \right) $, 采用谐波平衡法求解可得
$$ \left.\begin{aligned} &\frac{3}{4}{Z}_{e}^{2}{\mu }_{3}{\hat{Z}}^{3} + \left({\mu }_{1}-{\varOmega }^{2}\right)\hat{Z} = {\varOmega }^{2}\mathrm{c}\mathrm{o}\mathrm{s}\phi \\ &-2\zeta \varOmega \hat{Z} = {\varOmega }^{2}\mathrm{s}\mathrm{i}\mathrm{n}\phi \end{aligned}\right\} $$ (20) 其中, $ \hat{Z} $表示相对位移传递率, $ \phi $是激励和响应之间的相位差. 将方程(20)等式两边平方, 然后再将两个等式两边相加, 可以得到关于$ \hat{Z} $的多项式方程
$$ {\left[\frac{3}{4}{Z}_{e}^{2}{\mu }_{3}{\hat{Z}}^{3} + \left({\mu }_{1}-{\varOmega }^{2}\right)\hat{Z}\right]}^{2} + {\left(2\zeta \varOmega \hat{Z}\right)}^{2}-{\varOmega }^{4} = 0 $$ (21) 幅频方程(21)中的非线性刚度$ {\mu }_{3} $越小, 位移传递率越低, 更适合于大幅激励下的低频隔振. 对于具有恒值QZS特性的隔振器, 频响方程(21)中的非线性刚度$ {\mu }_{3} $等于0.
谐波平衡法是计算位移传递率简单有效的方法, 但在一些复杂的动力学方程中, 如恒力、强非线性或曲线向右弯曲较大时, 一阶谐波平衡法在计算位移传递率时遇到问题, 如无法直接连续的计算出传递率曲线. 因此, 需要一种高效稳定的方法计算传递率.
2.2 增量谐波平衡法和连续算法求解位移传递率
为了在复杂工况下提高位移传递率的计算效率和可靠性, 引入一阶增量谐波平衡法和弧长连续算法. 方程(18)采用无量纲方式$ \tau = \omega t $, 其余无量纲参数与上述相同, 运动微分方程重写为[46]
$$ {\varOmega }^{2}\acute{\acute{\bar {z}}} + 2\varOmega \zeta \acute{\bar {z}} + \alpha \bar {z} + \gamma {{\hat{X}}_{e}}^{2}{\bar {z}}^{3} = {\varOmega }^{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\tau \right) $$ (22) 如果方程(22)采用一阶谐波平衡法求解, 设解$ \bar {z} = \hat{Z}\mathrm{c}\mathrm{o}\mathrm{s}\left(\tau + \phi \right) $, 得到的频响方程与式(21)相同. 因此, 方程(22)与方程(19)虽然具有不同的无量纲形式, 但具有相同的位移传递率. 因此, 无量纲形式不影响运动方程的动力学特性.
增量谐波平衡法求解运动方程(22)的过程如下. 方程中参数的增量为
$$ \left.\begin{aligned} &\bar {z} = {\bar {z}}_{0} + \Delta \bar {z}\\ &\varOmega = {\varOmega }_{0} + \Delta \varOmega \\ &{\hat{X}}_{e} = {\hat{X}}_{e0} + \Delta {\hat{X}}_{e}\end{aligned}\right\} $$ (23) 其中, $ {\bar {z}}_{0} $、$ {\varOmega }_{0} $和$ {\hat{X}}_{e0} $是运动方程中的一组已知解, 分别表示满足运动方程的响应、频率和激励幅值; $ \Delta \bar {z} $、$ \Delta \varOmega $和$ \Delta {\hat{X}}_{e} $分别表示响应、频率和激励幅值的增量. 将方程(23)代入式(22), 保留一阶增量得
$$ \left.\begin{aligned} &{{\varOmega }_{0}}^{2}\Delta \acute{\acute{\bar{z}}} + 2\zeta {\varOmega }_{0}\Delta \acute{\bar{z}} + \alpha \Delta \bar{z} + 3\gamma {{{\hat{X}}_{e}}_{0}}^{2}{{\bar{z}}_{0}}^{2}\Delta \bar{z} = R + \\ &\qquad \left[2{\varOmega }_{0}\mathrm{c}\mathrm{o}\mathrm{s}\left(\tau \right)-2{\varOmega }_{0}{\acute{\acute{\bar{z}}}}_{0}-2\zeta {\acute{\bar{z}}}_{0}\right]\Delta \varOmega -2\gamma {\hat{X}}_{e0}{{\bar{z}}_{0}}^{3}\Delta {\hat{X}}_{e}\\ & R=-\left[{{\varOmega }_{0}}^{2}{\acute{\acute{\bar{z}}}}_{0} + 2\zeta {\varOmega }_{0}{\acute{\bar{z}}}_{0} + \alpha {\bar{z}}_{0} + \gamma {{\hat{X}}_{e0}}^{2}{{\bar{z}}_{0}}^{3}-{{\varOmega }_{0}}^{2}{\mathrm{cos}}\left(\tau \right)\right]\end{aligned}\right\} $$ (24) 计算运动方程的幅频曲线时, 可以令方程(24)中的激励幅值增量$ \mathrm{\Delta }{\hat{X}}_{e} $等于0. 因此, 增量方程(24)改写成
$$ \boldsymbol{C}\Delta \boldsymbol{a} = \boldsymbol{R} + \boldsymbol{Q}\Delta \varOmega $$ (25) 其中, $ \boldsymbol{C} $是一个二维矩阵, $ \Delta \boldsymbol{a} $、$ \boldsymbol{R} $和$ \boldsymbol{Q} $分别是二维向量; $ \Delta \varOmega $表示频率增量, 是一个实数. 具体的表达式为
$$ \boldsymbol{C} = \left\{\begin{aligned} &{C}_{11} = {\int }_{0}^{2\text{π} }\left[\left(-{{\varOmega }_{0}}^{2}\sin\tau + 2\zeta {\varOmega }_{0}\cos\tau + \alpha \sin\tau + \right.\right.\\ &\qquad \left.\left.3\gamma {{\hat{X}}_{e0}}^{2}{\left({a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau + {b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \right)}^{2}\sin\tau \right)\right]\mathrm{s}\mathrm{i}\mathrm{n}\tau {\mathrm{d}}\tau \\ &{C}_{21} = {\int }_{0}^{2\text{π} }\left[\left(-{{\varOmega }_{0}}^{2}\sin\tau + 2\zeta {\varOmega }_{0}\cos\tau + \alpha \sin\tau +\right.\right. \\ &\qquad \left.\left.3\gamma {{\hat{X}}_{e0}}^{2}{\left({a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau + {b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \right)}^{2}\sin\tau \right)\right]\mathrm{c}\mathrm{o}\mathrm{s}\tau {\mathrm{d}}\tau \\ &{C}_{12} = {\int }_{0}^{2\text{π} }\left[\left(-{{\varOmega }_{0}}^{2}\cos\tau -2\zeta {\varOmega }_{0}\sin\tau + \alpha \cos\tau + \right.\right.\\ &\qquad \left.\left.3\gamma {{\hat{X}}_{e0}}^{2}{\left({a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau + {b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \right)}^{2}\cos\tau \right)\right]\mathrm{s}\mathrm{i}\mathrm{n}\tau {\mathrm{d}}\tau \\ &{C}_{22} = {\int }_{0}^{2\text{π} }\left[\left(-{{\varOmega }_{0}}^{2}\cos\tau -2\zeta {\varOmega }_{0}\sin\tau + \alpha \cos\tau + \right.\right.\\ &\qquad \left.\left.3\gamma {{\hat{X}}_{e0}}^{2}{\left({a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau + {b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \right)}^{2}\cos\tau \right)\right]\mathrm{c}\mathrm{o}\mathrm{s}\tau {\mathrm{d}}\tau \end{aligned}\right. $$ (26) $$ \boldsymbol{R} = \left\{\begin{aligned} &{R}_{11} = {\int }_{0}^{2\text{π} }-\left[{{\varOmega }_{0}}^{2}{\acute{\acute{\bar {z}}}}_{0} + 2\zeta {\varOmega }_{0}{\acute{\bar {z}}}_{0} + \alpha {\bar {z}}_{0} + \gamma {{\hat{X}}_{e0}}^{2}{{\bar {z}}_{0}}^{3}-\right.\\ &\qquad \left.{{\varOmega }_{0}}^{2}\cos\left(\tau \right)\right]\mathrm{s}\mathrm{i}\mathrm{n}\tau {\mathrm{d}}t\\ &{R}_{21} = {\int }_{0}^{2\text{π} }-\left[{{\varOmega }_{0}}^{2}{\acute{\acute{\bar {z}}}}_{0} + 2\zeta {\varOmega }_{0}{\acute{\bar {z}}}_{0} + \alpha {\bar {z}}_{0} +\gamma {{\hat{X}}_{e0}}^{2}{{\bar {z}}_{0}}^{3}-\right.\\ &\qquad \left.{{\varOmega }_{0}}^{2}\cos\left(\tau \right)\right]\mathrm{c}\mathrm{o}\mathrm{s}\tau {\mathrm{d}}t\end{aligned}\right. $$ (27) $$ \boldsymbol{Q} = \left\{\begin{aligned} &{Q}_{11} = {\int }_{0}^{2\text{π} }-\left[2{\varOmega }_{0}\left(-{a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau -{b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \right) + \right.\\ &\qquad \left.2\zeta \left({a}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau -{b}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau \right)-2{\mathrm{\varOmega }}_{0}cos\left(\tau \right)\right]\mathrm{s}\mathrm{i}\mathrm{n}\tau {\mathrm{d}}\tau \\ &{Q}_{21} = {\int }_{0}^{2\text{π} }-\left[2{\varOmega }_{0}\left(-{a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau -{b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \right) + \right.\\ &\qquad \left.2\zeta \left({a}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau -{b}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau \right)-2{\mathrm{\varOmega }}_{0}cos\left(\tau \right)\right]\mathrm{c}\mathrm{o}\mathrm{s}\tau {\mathrm{d}}\tau \end{aligned}\right. $$ (28) $$ \left.\begin{aligned} &{\bar {z}}_{0}\left(\tau \right) = {a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau + {b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \\ &\Delta \bar {z} = \Delta {a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\tau + \Delta {b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\tau \end{aligned}\right\} $$ (29) $$ \Delta \boldsymbol{a} = \left[\begin{array}{c}\Delta {a}_{1}\\ \Delta {b}_{1}\end{array}\right] $$ (30) 在一次频率增量和迭代周期内, 当频率增量确定后, 采用牛顿迭代求解$ \boldsymbol{C}\Delta \boldsymbol{a} = \boldsymbol{R} $, 当向量$ \boldsymbol{R} $的模小于给定的收敛精度时, 可获得频率增量对应的响应增量, 进而求出满足一定精度的运动方程解. 在强非线性和大激励幅值工况下, 运动方程的幅频响应中可能存在多尖点情况, 需要用到连续算法才可能顺利通过尖点[47]. 根据已知解预测下一个解的过程如图11所示, $ {\bar {z}}_{0} $、$ {\bar {z}}_{1} $、$ {\bar {z}}_{2} $和$ {\bar {z}}_{3} $是运动方程的已知解, $ \Delta s $是外推弧长, $ {\bar {z}}_{4} $是预测点, 是进行牛顿迭代的初始点. $ {\bar {z}}_{4} $的计算过程如下
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$$ \bar{z}_4=\sum_{i=0}^3\left(\prod_{\substack{j=0 \\ j \neq i}}^3 \frac{t_4-t_j}{t_i-t_j}\right) \bar{z}_i $$ (31) 其中
$$ \left.\begin{aligned} &{t}_{0} = 0\\ &{t}_{1} = {s}_{1}\\ &{t}_{2} = {t}_{1} + {s}_{2}\\ &{t}_{3} = {t}_{2} + {s}_{3}\\ &{t}_{4} = {t}_{3} + \Delta s\\ &{s}_{1} = \left|{\bar {z}}_{1}-{\bar {z}}_{0}\right| = \sqrt{\sum _{j = 1}^{3}{\left[{\bar {z}}_{1}\left(j\right)-{\bar {z}}_{0}\left(j\right)\right]}^{2}}\\ &{s}_{2} = \left|{\bar {z}}_{2}-{\bar {z}}_{1}\right| = \sqrt{\sum _{j = 1}^{3}{\left[{\bar {z}}_{2}\left(j\right)-{\bar {z}}_{1}\left(j\right)\right]}^{2}}\\ &{s}_{3} = \left|{\bar {z}}_{3}-{\bar {z}}_{2}\right| = \sqrt{\sum _{j = 1}^{3}{\left[{\bar {z}}_{3}\left(j\right)-{\bar {z}}_{2}\left(j\right)\right]}^{2}}\end{aligned}\right\}$$ (32) 弧长增量$ \Delta s $足够小才可成功计算到下一个预测点, 从而用于牛顿迭代的初始值, 计算出符合精度要求的解. 对弧长控制的一个方法为$ \Delta {s}_{j} = \dfrac{\Delta {s}_{j-1}{N}_{d}}{{I}_{j-1}} $, 其中$ \Delta {s}_{j-1} $是上一轮计算预测点的弧长增量, $ {I}_{j-1} $表示上一次计算的牛顿迭代次数. $ {N}_{d} $是本次牛顿迭代期望的次数, 略大于1将更好的收敛.
2.3 位移传递率
方程(21)通过一阶谐波平衡法求解方程(19)获得, 方程(25)通过增量谐波平衡法求解方程(22)得到, 两种方法得到的相对位移传递率相同, 代入下式可得到绝对位移传递率[48]
$$ {T}_{a} = \sqrt{1 + 2\hat{Z}\mathrm{c}\mathrm{o}\mathrm{s}\phi + {\hat{Z}}^{2}} $$ (33) 在不同的激励幅值下, 采用谐波平衡法和增量谐波平衡法获得的绝对位移传递率如图12所示. 两种方法在弱、强非线性时计算结果一致, 验证了增量谐波平衡法计算传递率的有效性.
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图 12 给定参数$ \alpha = 0.1 $、$ \gamma = 0.1 $和$ \zeta = 0.1 $时谐波平衡法(BM)和增量谐波平衡法(IHBM)计算绝对位移传递率Figure 12. The absolute displacement transmissibility calculated by harmonic balance method (BM) and incremental harmonic balance method (IHBM) when $ \alpha = 0.1 $, $ \gamma = 0.1 $ and $ \zeta = 0.1 $3. 单级恒值QZS隔振实验
3.1 静态实验
单级恒值QZS隔振器的实验模型如图13所示. 两个线性正刚度拉伸弹簧支撑承载质量, 每根拉伸弹簧的刚度为0.93 N/mm. 一对斜杆和水平拉伸弹簧产生负刚度, 用于抵消正刚度获得QZS. 每个水平拉伸弹簧的刚度是0.205 N/mm, 两个斜杆铰接中心点的距离为55 mm, 隔振质量为1.63 kg.
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图 13 单级恒值QZS隔振器实验 (1)垂向正刚度的拉伸弹簧; (2)产生负刚度的水平拉伸弹簧; (3)斜杆; (4)斜杆铰接杆; (5)水平直线轴承; (6)沿直线轴承滑动的光轴; (7)电动振动台; (8)采集隔振质量振动位移的激光传感器(Panasonic HL-G108); (9)采集振动台位移的激光传感器(Panasonic HL-G108); (10)振动台的功率放大器; (11)驱动振动台的信号发生器; (12) 信号采集器和向激光传感器供电的直流电源; (13)电脑Figure 13. Experiments of the single-stage constant QZS isolator (1) vertical tension springs with positive stiffness, (2) horizontal tension springs with negative stiffness, (3) oblique bars, (4) hinge connecting bar, (5) horizontal linear bearings, (6) hard shaft, (7) electric vibration table, (8) laser sensor with Panasonic HL-G108 to collect displacement of the loaded mass, (9) laser sensor with Panasonic HL-G108 to collect displacement of the vibration table, (10) controller of vibration table, (11) signal generator, (12) vibration acquisition instrument and DC power supply, (13) computer恒值QZS值为$ {K}_{{\mathrm{CQZS}}} = {k}_{2}\hat{K} $ N/mm, 设计的承载质量为$ {m}_{{\mathrm{design}}} $, 要求的起始隔振频率为$ {\omega }_{{\mathrm{design}}} $. 参数的设计准则为
$$ \frac{\sqrt{2}}{2\text{π} }\sqrt{\frac{{K}_{{\mathrm{CQZS}}}}{{m}_{{\mathrm{design}}}}}\leqslant {\omega }_{{\mathrm{design}}} $$ (34) 单级恒值QZS隔振器实验模型的额定承载质量为2.56 kg, 其他的3种承载质量工况相比于额定承载质量－100 g、 + 100 g和 + 150 g, 承载质量的变化范围为250 g, 是设计值的10%. 实验模型的动态刚度理论值为0.147 N/mm, 如图14所示, 测试力位移曲线与理论预测结果具有较好的一致性. 测试位移范围内的动态力变化为5 N, 对应着承载质量变化范围为0.5 kg, 这是理想状态. 然而, 实际应用中的承载质量变化范围要远小于这一理论值, 在实验中承载质量的变化范围为0.25 kg或2.5 N, 隔振质量分别为2.46、2.56、2.66和2.71 kg.
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图 14 单级恒值QZS隔振器的静态测试力位移曲线和理论预测值Figure 14. Force-displacement curves of the single-stage constant QZS isolator3.2 动态实验
实验装置固定在振动台上进行动态测试, 如图13所示. 信号发生器产生1 ~ 7 Hz的激励信号, 通过振动台功放驱动振动台产生5 mm的激励幅值. 由一个激光位移传感器采集振动台响应, 另一个激光位移传感器采集承载质量的响应. 在2 ~ 4 Hz采集的隔振质量响应与振动台输出位移如图15所示, 起始隔振频率明显小于2 Hz.
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图 15 承载质量(实线)和振动台(点线)的位移响应Figure 15. Displacements of the loaded mass (solid lines) and the vibration table (dotted lines)在不同承载质量工况下测试的位移传递率如图16所示. 单级恒值QZS隔振器的起始隔振频率小于2 Hz, 相应线性隔振器的共振频率为4.29 Hz和起始隔振频率为6 Hz. 两种隔振器的频带差大于4 Hz. 随着隔振质量的增加, 共振频带处的传递率大幅降低, 隔振频带内的传递率略微变化. 因此, 变承载质量的最小值应该设计成额定值, 以获得较低的传递率幅值. 线性隔振器的共振频率峰值可通过理论计算获得, 当阻尼比$ \zeta $ = 0.0275, 计算结果如图16所示.
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图 16 单级恒值QZS隔振器和相应线性隔振器在激励幅值5 mm和变承载质量工况的位移传递率Figure 16. Displacement transmissibility of the single-stage constant QZS isolator and its corresponding linear isolator under variable mass loads and excitation amplitude of 5 mm4. 多级恒值QZS隔振实验
4.1 静态实验
多级恒值QZS隔振实验模型及构造如图17所示. 电机通过螺栓固定在框架上, 电机的丝杠可转动, 丝杠与丝母构成旋转运动副, 丝母与悬挂垂直正刚度拉伸弹簧顶端的固定板通过螺栓紧固连接, 正刚度拉伸弹簧的底部挂在垂直滑轨的下端, 垂直滑轨的顶端安装承载质量, 垂直滑轨在滑块中上下移动, 垂直滑块、丝母与垂直拉簧的顶板通过螺栓固定连接在一起. 承载质量通过垂直滑轨将重力传递给垂直拉伸弹簧. 斜杆的外侧铰接水平滑块, 水平滑块在固定的水平滑轨上左右移动; 斜杆的内侧铰接在支座上, 支座与垂直滑轨通过螺栓固定连接且可上下滑动; 水平拉簧通过两端的挂钩挂在两侧斜杆外端的铰接轴上. 在通电状态下, 步进电机的丝杠处于锁紧状态, 保持力矩1.26 Nm, 丝杠直径12 mm, 相当于210 N的力锁住丝杠的转动, 对于设计的承载质量已足够牢固.
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图 17 多级连续恒值QZS隔振器静态实验 (1)滚珠丝杠步进电机, (2)垂向正刚度拉伸弹簧, (3)产生负刚度的水平方向拉伸弹簧, (4)产生负刚度的斜杆, (5)步进电机的丝杠, (6)步进电机的丝母, (7)万能试验机力传感器(2 kN量程)Figure 17. Static tests of multi-stage consecutive constant QZS isolator (1) stepping motor, (2) vertical tension springs with positive stiffness, (3) horizontal tension springs with negative stiffness, (4) oblique bars, (5) screw rod, (6) nut, (7) force sensor with the scope of 2 kN for the universal testing machine在实验模型中, 单个水平拉伸弹簧的刚度0.256 N/mm, 水平拉伸弹簧的预拉伸长度108 mm, 斜杆两端铰接中心点的距离55 mm, 从初始位置到静态平衡位置的距离10 mm, 单个垂直拉伸弹簧的刚度1.136 N/mm. 基于这些参数和式(4)计算的力位移理论预测值如图18(a)所示, 理论动态刚度值0.115 N/mm(线性正刚度值2.272 N/mm). 通过小量程万能试验机测试实验模型的力位移曲线如图18(a)所示, 三次测试的曲线一致度较好, 且与理论结果匹配较好. 同时, 实验结果具有较好的恒值动态刚度特性.
图18(a)是实验模型在设计承载质量工况下测试的力位移曲线, 与理论预测值具有较好的一致性. 在单一静态平衡位置下, 测试了－100 g、 + 0 g、 + 100 g 3种不同的承载质量工况, 此为单级恒值QZS隔振机制. 当承载质量大幅变化时, 需要改变静态平衡位置, 共设计了5个静态平衡位置, 相邻两个静态平衡位置之间的承载质量级差为200 g, 对应静态平衡位置的位移差为0.863 mm, 此为多级连续恒值QZS隔振机制, 如图18(b)所示. 4个级差800 g的承载质量变化, 达到设计承载质量的32%, 大幅提升了QZS隔振器对变承载质量工况的适应能力.
4.2 动态实验
将多级恒值QZS隔振实验模型固定到振动台上, 如图19所示. 按下开关中与目标隔振质量匹配的按键, 控制器触发并控制驱动器使电机按照控制器中的程序设定转动, 使正刚度顶端平台(丝母)上下运动控制$ h $的大小或承载质量大小, 即开关按键与目标隔振质量建立对应关系. 激励幅值为5 mm, 激励频率从1 ~ 9 Hz, 两个激光位移传感器分别采集隔振质量和振动台的位移信号. 当承载质量为设计质量、－100 g和+ 100 g时, 对应一个静态平衡位置, 此时为单级恒值QZS的变承载质量隔振机制, 适用于承载质量小幅变化的工况.
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图 19 连续多级恒值QZS隔振器动态实验 (1)电动振动台, (2) 多级连续恒值QZS隔振器, (3)激光传感器, (4)振动台控制柜, (5)步进电机控制器开关, (6)步进电机控制器, (7)步进电机驱动器和直流电源, (8)信号发生器和振动信号采集仪, (9)直流电源, (10)电脑及信号采集软件Figure 19. Dynamic tests of the multi-stage consecutive constant QZS isolator (1) electric vibration table, (2) multi-stage consecutive constant QZS isolator, (3) laser sensor, (4) controller of vibration table, (5) switches, (6) controller of stepping motor, (7) drivers and DC power supply, (8) signal generator and vibration acquisition instrument, (9) DC power supply for the laser sensor, (10) computer and signal acquisition software当承载质量大幅变化时, 需要用到多级连续恒值QZS的变承载质量隔振机制, 如每200 g的承载质量变化设置一个新的静态平衡位置. 按一下相应的开关按键, 按键触发控制器相应的程序, 控制驱动器使电机做出相应的转动, 最终达到新的静态平衡位置. 由于QZS具有恒值特性, 多个静态平衡位置的承载力是连续的连接在一起. 在单级恒值QZS和多级连续恒值QZS双重变承载质量隔振机制下, 相比设计的隔振质量2.5 kg, 承载质量变化范围从－100 ~ + 900 g, 达到40%的变化范围.
承载质量小幅变化时, 采用单级恒值QZS隔振机制, 实验位移传递率如图20所示. 0 g对应设计的承载质量, －100和 + 100 g表示在设计的承载质量基础上分别减去100 g和增加100 g. 3种承载质量工况具有相同的静态平衡位置. 实验结果表明, 承载质量变化范围相比设计值为8%, 传递率幅值及隔振频带基本不变, 相比线性隔振器的传递率, 保持宽频隔振和低传递率的隔振效果.
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图 20 单级恒值QZS的小幅变承载质量隔振传递率Figure 20. Displacement transmissibility of the single stage constant QZS isolator under small variable mass load承载质量大幅变化时, 采用多级连续恒值QZS隔振机制, 相邻静态平衡位置之间的承载力连续, 实验传递率如图21所示. 0 g对应设计的承载质量, + 200、 + 400、 + 600和 + 800 g表示在设计的承载质量基础上分别增加200 g、400 g、600 g和800 g. 实验结果表明, 承载质量相比设计值变化了32%, 但传递率基本保持不变, 相比线性隔振器的传递率, 依然保持着宽隔振频带和低传递率的优良隔振效果.
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图 21 多级连续恒值QZS的大幅变承载质量实验传递率Figure 21. Experimental displacement transmissibility of the multi-stage consecutive constant QZS isolator under large variable mass load图20线性隔振器的共振峰值可采用实验测试的阻尼计算. 当阻尼较小时, 阻尼比的计算方式为$ \zeta = \dfrac{\delta }{2\text{π} } $, 其中对数衰减率$ \delta = \dfrac{1}{j}\ln\frac{{A}_{i}}{{A}_{i + j}} $, $ {A}_{i} $和$ {A}_{i + j} $表示相隔$ j $个周期的两个峰值[49]. 将线性隔振器, 给定一个初始位移, 获得的衰减信号如图22所示, 取14个周期, 两个峰值分别为10.91和0.52, 计算得到阻尼$ \zeta $ = 0.034 6. 线性隔振器的共振频率为4.8 Hz, 此频率下传递率的最大值$ \dfrac{1}{2\zeta \sqrt{1-{\zeta }^{2}}} = 14.46 $.
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图 22 给定初始位移的线性隔振器时域衰减信号Figure 22. The attenuated signal of the corresponding linear isolator under given initial displacement5. 结论
提出了由拉簧和斜杆构造的QZS隔振器, 静态分析获得刚度与参数的表达式, 经刚度和刚度二阶导数在静态平衡位置分别等于零, 获得两个QZS条件. 当参数符合不同的QZS条件时, 隔振器具有非线性QZS、恒值动态刚度、恒值QZS和恒值零刚度特性.
基于恒值QZS特性, 提出了变承载质量的双重QZS低频隔振机制. 首先提出单级恒值QZS隔振器, 解决了承载质量小幅变化的低频隔振问题; 进一步提出多级连续恒值QZS隔振器, 解决了承载质量大幅变化的低频隔振问题.
设计、制作了单级恒值QZS隔振样机和多级连续恒值QZS隔振实验装置, 通过静、动态实验验证了理论分析结果. 两种变承载质量隔振机制均表现出了较好的隔振性能, 具有较低的传递率幅值和较宽的隔振频带. 单级恒值QZS隔振样机可适用于承载质量变化幅度在设计值10%以内的工况, 多级连续恒值QZS隔振样机可适用于承载质量变化幅度在设计值40%及更高的工况.
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图 1 基于拉簧构造的单级恒值QZS隔振器
Figure 1. Single-stage constant QZS isolator composed of tension springs
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图 2 恒值QZS与非线性QZS (LP: linear positive; NLN: nonlinear negative; NL: nonlinear; CD: constant dynamic)
Figure 2. Constant and nonlinear QZS (LP: linear positive; NLN: nonlinear negative; NL: nonlinear; CD: constant dynamic)
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图 3 基于单级恒值QZS隔振器提出的多级连续恒值QZS隔振方法
Figure 3. Isolation method with multi-stage consecutive constant QZS based on the single-stage consecutive constant QZS isolator
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图 4 多级连续恒值QZS力学特性
Figure 4. Mechanical feature of the multi-stage consecutive constant QZS
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图 12 给定参数$ \alpha = 0.1 $、$ \gamma = 0.1 $和$ \zeta = 0.1 $时谐波平衡法(BM)和增量谐波平衡法(IHBM)计算绝对位移传递率
Figure 12. The absolute displacement transmissibility calculated by harmonic balance method (BM) and incremental harmonic balance method (IHBM) when $ \alpha = 0.1 $, $ \gamma = 0.1 $ and $ \zeta = 0.1 $
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图 13 单级恒值QZS隔振器实验 (1)垂向正刚度的拉伸弹簧; (2)产生负刚度的水平拉伸弹簧; (3)斜杆; (4)斜杆铰接杆; (5)水平直线轴承; (6)沿直线轴承滑动的光轴; (7)电动振动台; (8)采集隔振质量振动位移的激光传感器(Panasonic HL-G108); (9)采集振动台位移的激光传感器(Panasonic HL-G108); (10)振动台的功率放大器; (11)驱动振动台的信号发生器; (12) 信号采集器和向激光传感器供电的直流电源; (13)电脑
Figure 13. Experiments of the single-stage constant QZS isolator (1) vertical tension springs with positive stiffness, (2) horizontal tension springs with negative stiffness, (3) oblique bars, (4) hinge connecting bar, (5) horizontal linear bearings, (6) hard shaft, (7) electric vibration table, (8) laser sensor with Panasonic HL-G108 to collect displacement of the loaded mass, (9) laser sensor with Panasonic HL-G108 to collect displacement of the vibration table, (10) controller of vibration table, (11) signal generator, (12) vibration acquisition instrument and DC power supply, (13) computer
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图 14 单级恒值QZS隔振器的静态测试力位移曲线和理论预测值
Figure 14. Force-displacement curves of the single-stage constant QZS isolator
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图 15 承载质量(实线)和振动台(点线)的位移响应
Figure 15. Displacements of the loaded mass (solid lines) and the vibration table (dotted lines)
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图 16 单级恒值QZS隔振器和相应线性隔振器在激励幅值5 mm和变承载质量工况的位移传递率
Figure 16. Displacement transmissibility of the single-stage constant QZS isolator and its corresponding linear isolator under variable mass loads and excitation amplitude of 5 mm
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图 17 多级连续恒值QZS隔振器静态实验 (1)滚珠丝杠步进电机, (2)垂向正刚度拉伸弹簧, (3)产生负刚度的水平方向拉伸弹簧, (4)产生负刚度的斜杆, (5)步进电机的丝杠, (6)步进电机的丝母, (7)万能试验机力传感器(2 kN量程)
Figure 17. Static tests of multi-stage consecutive constant QZS isolator (1) stepping motor, (2) vertical tension springs with positive stiffness, (3) horizontal tension springs with negative stiffness, (4) oblique bars, (5) screw rod, (6) nut, (7) force sensor with the scope of 2 kN for the universal testing machine
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图 19 连续多级恒值QZS隔振器动态实验 (1)电动振动台, (2) 多级连续恒值QZS隔振器, (3)激光传感器, (4)振动台控制柜, (5)步进电机控制器开关, (6)步进电机控制器, (7)步进电机驱动器和直流电源, (8)信号发生器和振动信号采集仪, (9)直流电源, (10)电脑及信号采集软件
Figure 19. Dynamic tests of the multi-stage consecutive constant QZS isolator (1) electric vibration table, (2) multi-stage consecutive constant QZS isolator, (3) laser sensor, (4) controller of vibration table, (5) switches, (6) controller of stepping motor, (7) drivers and DC power supply, (8) signal generator and vibration acquisition instrument, (9) DC power supply for the laser sensor, (10) computer and signal acquisition software
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图 20 单级恒值QZS的小幅变承载质量隔振传递率
Figure 20. Displacement transmissibility of the single stage constant QZS isolator under small variable mass load
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图 21 多级连续恒值QZS的大幅变承载质量实验传递率
Figure 21. Experimental displacement transmissibility of the multi-stage consecutive constant QZS isolator under large variable mass load
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