Abstract:Multi-stable structures are deformable structures that can have large deformations under external excitation. Generally, multi-stable structures have at least two stable points and can jump from one to another. Because multi-stable structures have excellent nonlinear characteristics, they are widely used in many fields. In the field of energy harvesting, multi-stable structures are often obtained by means of cantilever beams. This is because the cantilever beam is simple to make, low in stiffness, and high in sensitivity, and can undergo large deformations under small excitation forces. Besides, by simply sticking magnets on its free end and its outside, various kinds of multi-stable characteristics can be constructed, such as bi-stable characteristics, tri-stable characteristics, quad-stable characteristics, etc. Furthermore, the cantilever beam and the magnet at its end can generally be simplified into an equivalent mass-spring-damping mechanical model, which is convenient for the analysis of system potential function and dynamics.In recent years, many vibration energy harvesters proposed by researchers have adopted the conventional multi-stable cantilever beams, which involve many bi-stable cantilever beams and tri-stable cantilever beams. However, if the cantilever beams need to introduce more stable points, the number of magnets required will also increase accordingly. As a result, the adjustable parameters are continuously increasing, which increases the complexity of structural optimization and the tediousness of dynamic analysis. In order to make up for the shortcomings of conventional multi-stable cantilever beams, in this paper we present a multi-stable cantilever beam with only two magnets, a ring magnet and a rectangular magnet. By changing the size of the rectangular magnet and the distance between the two magnets, this cantilever beam can have mono-stable, bi-stable, tri-stable or quad-stable characteristics. This multi-stable cantilever beam greatly simplifies the complexity of the system design, dynamic analysis, debugging and installation, and provides new ideas and technical methods for the design and application of the vibration energy harvester realized by the multi-stable cantilever beam.In this paper, firstly, the magnetizing current method is used to analyze the magnetic induction intensity of the ring magnet at any point in the three-dimensional coordinate system, and the simulation and experimental results prove its correctness. Secondly, two methods of calculating the position of the rectangular magnet at the free end of the cantilever beam are compared. Thirdly, the magnetic force between the ring magnet and the rectangular magnet is calculated and verified in experiment. Fourthly, the system potential functions under different structural parameters are analyzed and it is found that the change of the number of the stable points of the system is caused by the change of the magnetic force between the two magnets. Finally, the correctness of the number of stable points of the system under different parameters is verified in experiment and by dynamic simulations. Keywords:two magnets/ multi-stable cantilever beam/ magnetic force/ potential function
其中, ${{M}}$为磁化强度, 其值为常数, ${\hat { n}}$为磁铁表面的法向单位矢量. 对于一个如图2所示的圆形实心磁铁, 设其磁化强度为${{{M}}_{\rm{B}}}$、厚度为${t_{\rm{B}}}$、半径为${R_{\rm{1}}}$, 根据(9)式, 可以得到其表面磁化电流面密度: 图 2 空间坐标系及圆形实心磁铁的磁化电流示意图 Figure2. Schematic diagram of there-dimension coordinate system and magnetizing currents on the surface of the circular magnet.
设环形磁铁磁化强度也为${{{M}}_{\rm{B}}}$、厚度也为${t_{\rm{B}}}$、外环半径也为${R_{\rm{1}}}$、内环半径为${R_{\rm{2}}}$, 则其外环表面磁化电流在图2空间坐标系中任一点$P(x, y, z)$处的磁感应强度也如(13)式. 因为内环表面磁化电流与外环表面磁化电流方向相反, 所以内环表面磁化电流在图2空间坐标系中任一点$P(x, y, z)$处的磁感应强度为:
表2实验器材及其型号 Table2.Experimental equipments and models.
图 3 磁感应强度Bi, Bj随x的变化关系 (a) Bi随x的变化, y = 6.0 mm; (b) Bj随x的变化, y = 6.0 mm; (c) Bi随x的变化, y = 10.0 mm; (d) Bj随x的变化, y = 10.0 mm Figure3. The curves of Bi and Bj varying with x: (a) The curves of Bi varying with x, y = 6.0 mm; (b) the curves of Bj varying with x, y = 6.0 mm; (c) the curves of Bi varying with x, y = 10.0 mm; (d) the curves of Bj varying with x, y = 10.0 mm.
图 4 磁感应强度测量系统 (a)Bi测量; (b)Bj测量 Figure4. Magnetic induction intensity measurement system: (a) The measurement of Bi; (b) the measurement of Bj.
23.2.矩形磁铁磁化电流在空间中的位置 -->
3.2.矩形磁铁磁化电流在空间中的位置
悬臂梁在振动过程中会发生弯曲, 并使得梁自由端的磁铁发生偏转, 如图5所示, 因此需要准确计算磁铁位置才能确定矩形磁铁磁化电流在空间中的位置. 图 5 悬臂梁弯曲状态及其矩形磁铁的坐标位置 Figure5. The position of the rectangular magnet in coordinate system when the cantilever beam is bent.
为了比较两种经验计算方法对本文中所选取悬臂梁的适用性, 根据表1中悬臂梁的参数, 对(16a)式和(16b)式进行数值分析, 分别得到${y_{\rm{C}}}$随${x_{\rm{C}}}$的变化关系曲线, 如图6中的黑色实线和青色虚线. 选取表2中的激光位移传感器等仪器搭建图7所示的位移测量系统, 测量出${y_{\rm{C}}}$随${x_{\rm{C}}}$的变化关系曲线, 如图6中的蓝色圆圈点线. 比较图中实验测量结果与(16a)式和(16b)式数值分析结果不难发现, (16b)式得到的仿真曲线更贴近实验数据, 计算更准确, 所以本文选用(16b)式来确定${y_{\rm{C}}}$与${x_{\rm{C}}}$的关系. 图 6 梁自由端磁铁位置的两种计算方法 Figure6. Two kinds of calculation of the position of the magnet at the free end of the beam.
即矩形磁铁左右表面无磁化电流, 只有上、下、前、后表面存在磁化电流, 如图8所示. 图 8 矩形磁铁尺寸及磁化电流示意图 Figure8. Schematic diagram of the size of the rectangular magnet and the magnetizing currents on the surface of the rectangular magnet.
其中, ${S_1}$, ${S_2}$分别为矩形磁铁上、下表面的面积, ${S_1} = {S_2} = {t_{\rm{A}}}{w_{\rm{A}}}$, ${S_3}$, ${S_4}$分别为矩形磁铁前、后表面的面积, ${S_3} = {S_4} = {t_{\rm{A}}}{l_{\rm{A}}}$, 矩形磁铁尺寸如图8所示. 为了验证(20)式的合理性, 在图5的坐标系中, 任取d = 5.8 mm和d = 8.0 mm两个值, 根据表1中的参数, 对(20)式进行数值仿真, 得到${F_i}$和${F_j}$随${x_{\rm{C}}}$的变化关系曲线, 如图9中的蓝色实线. 选取表2中的推拉式测力计和激光位移传感器等仪器搭建图10所示的磁力测量系统, 在对应的数值模拟坐标空间值处测量相应的磁力, 得到磁力${F_i}$, ${F_j}$随${x_{\rm{C}}}$的变化关系曲线, 如图9中的红色星号点线. 由图9可以看出, 实验数据与仿真数据基本吻合, 验证了(20)式磁力分析模型的合理性. 需要说明的是, 由于实验中选用的测力计的最小输入值为0.05 N, 因此图9中小于0.05 N的磁力只能用零值点表示. 图 9Fi, Fj随xC的变化关系 (a) Fi随xC的变化关系, d = 5.8 mm; (b)Fj随xC的变化关系, d = 5.8 mm; (c) Fi随xC的变化关系, d = 8.0 mm; (d) Fj随xC的变化关系, d = 8.0 mm Figure9. The curves of Fi and Fj varying with xC: (a) The curves of Fi varying with xC, d = 5.8 mm; (b) the curves of Fj varying with xC, d = 5.8 mm; (c) the curves of Fi varying with xC, d = 8.0 mm; (d) the curves of Fj varying with xC, d = 8.0 mm.
图 10 磁力测量系统 (a) Fi 测量; (b) Fj测量 Figure10. Magnetic force measurement system: (a) The measurement of Fi; (b) the measurement of Fj.
根据表1, 设置矩形磁铁和环形磁铁的尺寸分别为10 mm × 10 mm × 3 mm和40 mm(φ1) × 20 mm(φ2) × 3 mm, 对(21)式进行数值分析, 得到图11不同磁铁间距d的系统势函数变化图像. 图 11 矩形磁铁(10 mm × 10 mm × 3 mm)与环形磁铁(40 mm (φ1) × 20 mm (φ2) × 3 mm)作用的系统势函数 (a)系统势函数三维图; (b)磁铁间距分别为d = 3 mm, d = 6 mm, d = 20 mm时系统势函数二维图 Figure11. The system potential function varying with d when the size of the rectangular magnet is 10 mm × 10 mm × 3 mm and the ring magnet is 40 mm (φ1) × 20 mm (φ2) × 3 mm: (a) Three dimensional diagram of system potential function; (b) two dimensional diagram of system potential function when d = 3 mm, d = 6 mm and d = 20 mm.
由图11可知, 随着磁铁间距的增大, 系统势函数从三稳状态逐渐变成单稳状态. 当磁铁间距d很小时, 如d = 3 mm, 系统中间势阱较深, 其两侧势阱相对较浅. 当d增大时, 如d = 6 mm, 系统的三个势阱均会变浅. 如果继续增大磁铁间距, 如d = 20 mm, 则系统的三个势阱几乎消失, 系统退化为一个单稳状态. 保持环形磁铁的尺寸不变, 当增大矩形磁铁尺寸时, 系统势函数随磁铁间距d的增大, 会出现由三稳状态或四稳状态先退化为双稳状态再退化为单稳状态的过程. 以20 mm × 20 mm × 3 mm和30 mm × 30 mm × 3 mm的矩形磁铁为例, 作出系统势函数随磁铁间距d的变化图像, 如图12所示. 可以看到, 当矩形磁铁的尺寸为20 mm × 20 mm × 3 mm时, 随着磁铁间距的增大, 系统势函数依次从三稳状态逐渐变成双稳和单稳状态; 当矩形磁铁的尺寸为30 mm × 30 mm × 3 mm, 随着磁铁间距的增大, 系统势函数依次从四稳状态逐渐变成双稳和单稳状态. 图 12 系统势函数随d的变化 (a)矩形磁铁尺寸为 20 mm × 20 mm × 3 mm; (b) 矩形磁铁尺寸为 30 mm × 30 mm × 3 mm Figure12. The system potential function varying with d: (a) The size of the rectangular magnet is 20 mm × 20 mm × 3 mm; (b) the size of the rectangular magnet is 30 mm × 30 mm × 3 mm.
24.2.系统势函数随磁铁尺寸的变化 -->
4.2.系统势函数随磁铁尺寸的变化
取磁铁间距d = 6 mm, 并保持环形磁铁尺寸不变, 令矩形磁铁的长度和宽度相等, 厚度保持3 mm不变, 以矩形磁铁长度lA为变量, 对(21)式进行数值分析, 得到如图13所示的系统势函数变化图像. 图 13 磁铁间距d = 6 mm, 不同矩形磁铁尺寸与环形磁铁(40 mm(φ1) × 20 mm(φ2) × 3 mm)作用的系统势函数 (a) 系统势函数三维图; (b) 矩形磁铁长度分别为lA = 3 mm, lA = 10 mm, lA = 20 mm, lA = 30 mm, lA = 45 mm时系统势函数二维图 Figure13. The system potential function varying with lA when d = 6 mm and the size of the ring magnet is 40 mm(φ1) × 20 mm(φ2) × 3 mm: (a) Three dimensional diagram of system potential function; (b) two dimensional diagram of system potential function when lA = 3 mm, lA = 10 mm, lA = 20 mm, lA = 30 mm and lA = 45 mm.
从图13可以看出, 随着矩形磁铁长度的增大, 系统势函数从单稳状态逐渐变成三稳状态和四稳状态. 当矩形磁铁长度很小时, 如lA = 3 mm, 系统几乎呈现单稳状态. 而当lA增大时, 如lA = 20 mm, 系统会呈现出三个势阱的三稳状态. 如果继续增大lA, 如lA = 30 mm, 则系统会由三稳状态转变为四稳状态. 24.3.讨 论 -->
4.3.讨 论
根据(21)式, 系统势函数与弹性恢复力、两磁铁间磁力做功有关, 若不改变悬臂梁本身的结构参数, 那么系统的稳态形式由两磁铁间磁力所决定. 现以三稳、四稳两种稳态形式为例, 解释系统的稳态形式发生变化的原因. 取4.2节中lA = 20 mm和lA = 30 mm两尺寸矩形磁铁, 根据(21)式分别作出两尺寸矩形磁铁的${W_2}$和${W_3}$随${x_{\rm{C}}}$的变化关系, 如图14(a)中的黑色和青色实线以及黑色和青色星号点线. 图 14 (a) lA = 20 mm和lA = 30 mm时, W2和W3随xC的变化关系;(b) lA = 20 mm和lA = 30 mm时, Fi随xC的变化 Figure14. (a) The curves of W2 and W3 varying with xC when lA = 20 mm and lA = 30 mm; (b) the curves of Fi varying with xC when lA = 20 mm and lA = 30 mm.
为验证上述势函数的分析, 分别从实验和动力学的角度考察系统的稳态特性. 选取表1参数对应的悬臂梁、环形磁铁和矩形磁铁, 构建双磁铁非线性悬臂梁结构, 如图15所示. 调节磁铁间距为d = 6 mm, 可以观察到系统的三稳特性, 这里只给出了3个稳态位置中的中稳态点和上稳态点, 根据对称性可知下稳态点. 图 15 三稳结构 (a)中稳态点; (b)上稳态点 Figure15. The structure concluding three stable points: (a) The middle state point; (b) the upper stable point.
将图15中10 mm × 10 mm × 3 mm尺寸的矩形磁铁替换为30 mm × 30 mm × 3 mm尺寸的矩形磁铁, 其他参数不变, 可以观察到系统的四稳特性, 这里只显示悬臂梁上半部分的两个稳态位置: 上1稳态点和上2稳态点, 如图16所示. 根据对称性可知悬臂梁下半部分的两个稳态位置. 图 16 四稳结构 (a)上1稳态点; (b)上2稳态点 Figure16. The structure concluding four stable points: (a) The upper stable point 1; (b) the upper state point 2.
与图13比较可知, 图15和图16的实验结果验证了势函数分析的有效性. 取(1)式中的PA为0—120 Hz的宽带随机激励, 其强度Q由激励的方差表示. 根据表1参数以及(2)式—(8)式, 为较好地展现系统的响应特性, 取机械阻尼比${\xi _{\rm{r}}} = 0.0732$、激励强度Q = 0.06, 可计算得到磁铁间距d = 6 mm时系统的三稳振动响应的时域波形和相位图, 如图17所示. 图 17 三稳振动响应 (a)时域图; (b)相位图 Figure17. The vibration response of the tri-stable cantilever beam: (a) The time domain chart; (b) the phase chart.
将图17中的矩形磁铁尺寸替换为30 mm × 30 mm × 3 mm, 调整激励强度为Q = 0.45, 可以得到系统的四稳振动响应的时域波形和相位图, 如图18所示. 图 18 四稳振动响应 (a)时域图; (b)相位图 Figure18. The vibration response of the quad-stable cantilever beam: (a) The time domain chart; (b) the phase chart.