主动控制压电旋转悬臂梁的参数振动稳定性分析1)

图1主动控制压电旋转悬臂梁

Fig.1Piezoelectric rotating cantilever beam with active control

根据一阶 Galerkin 离散法,压电悬臂梁的横向位移 $w(x,t)$ 可近似表示为

(1)$w(x,t) = W(x)q(t) $

其中,$W(x)$ 为旋转悬臂梁的第一阶振型,$q(t)$ 表示其位移响应.

基于 Yoo 等^[29]的研究成果,$W(x)$ 可用悬臂梁的振型近似表示^[30-31]

(2)$W(x) = \cosh s_{1} x - \cos s_{1} x + v_{1} (\sinh s_{1}x - \sin s_{1} x) $

式中,$s_{1} $ 与 $v_{1}$

表示一阶振型系数,其中 $s_{1}$ 的值应满足 $s_1 l_{\rm s}=1.875\,1$ ($l_{\rm s}$ 为悬臂梁的长度),系数 $v_{1}$ 则具有如下形式

(3)$v_{1} = - \dfrac{\sinh s_{1} l_{\rm s} - \sin s_{1} l_{\rm s} }{\cosh s_{1} l_{\rm s} + \cos s_{1} l_{\rm s} } $

设本文所研究的压电梁为细长结构,忽略剪切变形的影响.根据 Euler-Bernoulli 梁理论的假设,基础梁的正应力、正应变和横向位移之间的关系可表示为

(4)$\sigma _{\rm s} = E_{\rm s} \varepsilon _{\rm s} \qquad \ \ \ $

(5)$\varepsilon _{\rm s} = - z\dfrac{\partial ^2w(x,t)}{\partial x^2} $

其中,$\sigma _{\rm s} $,$E_{\rm s} $ 和 $\varepsilon _{\rm s}$ 分别为基础梁的正应力,弹性模量和正应变.

同理,悬臂梁的压电层位移场也有相同的关系

(6)$\sigma _{\rm p} = E_{\rm p} \varepsilon _{\rm p} \qquad \ \ \ $

(7)$\varepsilon _{\rm p} = - z\dfrac{\partial ^2w(x,t)}{\partial x^2} $

其中,$\sigma _{\rm p} $,$E_{\rm p} $ 和 $\varepsilon _{\rm p}$ 分别代表着压电层的正应力、弹性模量和正应变.

引入压电层以后,压电悬臂梁的中性层不再与几何形心重合,中性轴的位置可以利用材料力学中性轴的概念予以确定.梁在弯曲过程中任意横截面上内和外力之和等零,故有

(8)$\int_{ - h_{\rm s} + Z_{\rm n} }^{Z_{\rm n} } {b\sigma _{\rm s} dz} + \int_{Z_{\rm n}}^{Z_{\rm n} + h_{\rm p} } {b\sigma _{\rm p} dz = 0} $

将式 (4) $\sim $式 (7) 代入式 (8) 中,可得

(9)$Z_{\rm n} = \dfrac{E_{\rm s} h_{\rm s} ^2 - E_{\rm p} h_{\rm p} ^2}{2\left( {E_{\rm s} h_{\rm s} + E_{\rm p} h_{\rm p} } \right)} $

当压电层的极化方向沿 $z$ 轴的正方向,其电场强度 $E_{3}(t)$ 可以表示为

(10)$E_3 (t) = \left\{ \!\!\begin{array}{ll} { - v} /{h_{\rm p} } \,, & Z_{\rm n} \leqslant z \leqslant Z_{\rm n} + h_{\rm p} \ \\0 \,, & Z_{\rm n} - h_{\rm s} \leqslant z < Z_{\rm n}\end{array} \right. $

式中,$v$ 是电压源产生的控制电压.工程中,在压电材料基础上施加适当的控制电压能形成主动阻尼,从而提高系统的稳定性.为了实现这一目的,本文采用速度负反馈控制方法,即在悬臂梁某一点 $l_{\rm c}$ 处安装速度传感器,所采集速度信号传送到控制器,控制器通过速度负反馈方案计算控制电压,并作用压电材料而产生控制力.为满足控制要求,控制电压为

(11)$v(t) = K\dfrac{ dw(l_{\rm c} ,t)}{ dt} $

式中,$K$ 为反馈增益系数,$t$ 是时间变量.由于压电效应的存在,压电层的本构关系^[13]具有如下形式

(12)$\sigma _{\rm p} = E_{\rm p} \varepsilon _{\rm p} - E_{\rm p} d_{31} E_3$

(13)$D_3 = E_{\rm p} d_{31} \varepsilon _{\rm p} + \xi ^{\rm e}E_3$

其中,$d_{31}$,$D_3 $ 以及 $\xi^{\rm e}$ 分别表示压电材料的压电应变常数,电位移以及介电常数.

当悬臂梁绕定轴转动时,旋转所产生的离心轴力 $F_t(x,t)$ 可以表示为

(14)$F_t (x,t) = \int_x^{l_{\rm s} } {m\varOmega ^2(t)} (R + \varsigma )\varsigma =\\\ \qquad \dfrac{1}{2}m [(R + l_{\rm s} )^2 - (R + x)^2]\varOmega ^2\left( t \right)=\\\ \qquad \overline F (x)\varOmega ^2\left( t \right) $

式中,$m = \rho _{\rm s} bh_{\rm s} + \rho _{\rm p} bh_{\rm p}$ 代表单位长度梁的质量,其中 $\rho_{\rm s}$, $\rho_{\rm p}$ 分别表示为基础梁与压电层的密度. $\overline F (x) = m[(R + l_{\rm s} )^2 - (R + x)^2] / 2$,代表轴力的分布情况,$\varOmega (t)$ 为悬臂梁定轴转动的角速度,$R$ 为指轮毂的半径.

压电旋转梁的动能和势能分别为

(15)$ T = \int_{V_{\rm s} } {\dfrac{1}{2}\rho _{\rm s} \left( {\dfrac{\partial w(x,t)}{\partial t}} \right)^2 dV} + \int_{V_{\rm p} } {\dfrac{1}{2}\rho _{\rm p} \left( {\dfrac{\partial w(x,t)}{\partial t}} \right)^2 dV} =\qquad \int_0^{l_{\rm s} } {\dfrac{1}{2}\rho _{\rm s} bh_{\rm s} \left( {\dfrac{\partial w(x,t)}{\partial t}} \right)^2 dx} + \qquad\int_0^{l_{\rm s} } {\dfrac{1}{2}\rho _{\rm p} bh_{\rm p} \left( {\dfrac{\partial w(x,t)}{\partial t}} \right)^2} dx = \qquad \int_0^{l_{\rm s} } {\dfrac{1}{2}m\left( {\dfrac{\partial w(x,t)}{\partial t}} \right)^2 dx}$

(16)$U = \int_{V_{\rm s} } {\dfrac{1}{2}\sigma _{\rm s} \varepsilon _{\rm s} dV} +\int_{V_{\rm p} } {\dfrac{1}{2}\sigma _{\rm p} \varepsilon _{\rm p} dV}+ \qquad \int_0^{l_{\rm s} } {\dfrac{1}{2}} F_t (x,t)\left( {\dfrac{\partial w(x,t)}{\partial x}} \right)^2 dx$

其中,$V_{\rm s}$ 和 $V_{\rm p}$ 分别为基础梁的体积和压电层的体积. 将式 (4) $\sim$ 式 (6),式 (10) 和式 (12) 代入式 (16),可将系统势能进一步化为

(17)$ U = \int_0^{l_{\rm s} } {\int_{ - h_{\rm s} + Z_{\rm n} }^{Z_{\rm n} } {\dfrac{1}{2}} } bE_{\rm s} z^2\left( {\dfrac{\partial ^2w(x,t)}{\partial x^2}} \right)^2 dz dx \,+\\\ \qquad \int_0^{l_{\rm s} } {\int_{Z_{\rm n} }^{Z_{\rm n} + h_{\rm p} } {\dfrac{1}{2}} } bE_{\rm p} z^2\left( {\dfrac{\partial ^2w(x,t)}{\partial x^2}} \right)^2 dz dx \,-\\\ \qquad \int_0^{l_{\rm s} } {\int_{Z_{\rm n} }^{Z_{\rm n} + h_{\rm p} } {\dfrac{1}{2}} } bE_{\rm p} d_{31} \Big ( - \dfrac{v}{h_{\rm p} }\Big )\Big ( - z\dfrac{\partial ^2w(x,t)}{\partial x^2}\Big )dz dx \,+\\\ \qquad \int_0^{l_{\rm s} } {\dfrac{1}{2}} F_t (x,t)\left( {\dfrac{\partial w(x,t)}{\partial x}} \right)^2 dx =\\\ \qquad \int_0^{l_{\rm s} } {(EI)^ * } \left( {\dfrac{\partial ^2w(x,t)}{\partial x^2}} \right)^2 dx - \int_0^{l_{\rm s} } \varXi v\dfrac{\partial ^2w(x,t)}{\partial x^2} dx \,+\\\ \qquad \int_0^{l_{\rm s} } {\dfrac{1}{2}} F_t (x,t)\left( {\dfrac{\partial w(x,t)}{\partial x}} \right)^2 dx $

其中

$$ (EI)^ * = \dfrac{1}{6}bE_{\rm s} (3Z_{\rm n} ^2h_{\rm s} - 3Z_{\rm n} h_{\rm s} ^2 + h_{\rm s} ^3) \, +\\\ \qquad \dfrac{1}{6}bE_{\rm p} (3Z_{\rm n} ^2h_{\rm P} + 3Z_{\rm n} h_{\rm P} ^2 + h_{\rm P} ^3) \\\ \varXi = \dfrac{1}{2}E_{\rm P} d_{31} b(2Z_{\rm n} + h_{\rm p} ) $$

系统的电能可以表示为

(18)$W_{\rm e} = \int_{V_{\rm p} } {\dfrac{1}{2}E_3 D_3 dV} $

将式 (10) 和式 (13) 代入式 (18) 中,可得

(19)$W_{\rm e} = \int_0^{l_{\rm s} } {\int_{Z_{\rm n} }^{Z_{\rm n} + h_{\rm p} } {\dfrac{1}{2}E_{\rm p} bd_{31} \dfrac{v}{h_{\rm p} }z\dfrac{\partial ^2w(x,t)}{\partial x^2}dz dx} } \,+\\\ \qquad \int_0^{l_{\rm s} } {\int_{Z_{\rm n} }^{Z_{\rm n} + h_{\rm p} } {\dfrac{1}{2}b\xi ^{\rm e}\left( {\dfrac{v}{h_{\rm p} }} \right)^2 dz dx} } =\\\ \qquad \int_0^{l_{\rm s} } {\varXi v\dfrac{\partial ^2w(x,t)}{\partial x^2}} dx + \int_0^{l_{\rm s} } {\dfrac{1}{2}} b\xi ^{\rm e}\dfrac{v^2}{h_{\rm p} } dx $

外力所作功为

(20)$W = \int_0^{l_{\rm s} } {\left( {f(x,t) - c_{\rm s} \dfrac{\partial w(x,t)}{\partial t}} \right)w(x,t)} dx $

式中,$f(x,t) = \overline {f_1 } (x)\overline {f_2 } (t)$

代表分布载荷,$c_{\rm s}$ 为结构的阻尼系数.

广义 Hamilton 变分原理可以写为

(21)$\int_{t_1 }^{t_2 } {[\delta (T - U + W_{\rm e} + W)]} dt = 0 $

将式 (1)、式 (11)、式 (15)、式 (17)、式 (19) 以及式 (20) 代入式 (21),

可以得到主动控制压电旋转悬臂梁的一阶近似控制方程

(22)$a_1\dfrac{d^2q(t)}{dt^2} + a_2 \dfrac{dq(t)}{dt} + (a_3 + a_4 \varOmega ^2(t))q(t) = F\overline {f_2 } (t) $

各项系数分别是

$$ a_1 = m\int_0^{l_{\rm s} } {W^2(x) dx} + \dfrac{bK^2\xi ^{\rm e}l_{\rm s} }{h_{\rm p} }W^2(l_{\rm c})\\\ a_2 = c_{\rm s} \int_0^{l_{\rm s}} {W^2(x)dx} - 2\varXi KW(l_{\rm c} )\int_0^{l_{\rm s} } {W''(x) dx} \\\ a_3 = 2(EI)^ * \int_0^{l_{\rm s}} {W''^2} (x) dx \\\ a_4 = \int_0^{l_{\rm s}} {\overline F } (x)W'^2(x) dx \\\ F = \int_0^{l_{\rm s}} {\overline {f_1 } } (x)W(x) dx $$

2 稳定性分析

由式 (22) 不难发现,当轮毂角速度含有脉动分量时,旋转悬臂梁将会发生参数振动.为了分析参数振动的稳定性,采用多尺度法进行摄动求解.不妨假设旋转的角速度满足下列表达式

(23)$\varOmega (t) = \varOmega _0 + \varOmega _{\rm d} \cos \omega _{\rm r} t $

式中,$\varOmega _0 $ 为轮毂角速度平均值,$\varOmega _{\rm d}$ 和 $\omega _{\rm r} $ 分别为轮毂角速度脉动幅值和脉动频率.将式 (23) 代入式 (22) 中, 可以得到

(24)$a_1 \dfrac{d^2q(t)}{dt^2} + a_2 \dfrac{dq(t)}{dt} + \left[ {a_3 + a_4 \left( {\varOmega _0 ^2} \right.} \right. +\\\ \qquad \left. {\left. { 2\varOmega _0 \varOmega _{\rm d} \cos \omega _{\rm r} t + \varOmega _{\rm d}^2\cos ^2\omega _{\rm r} t} \right)} \right]q(t) = 0 $

考虑到脉动角速度一般很小,此处不妨假设

(25)$\varOmega _{\rm d} \ll \varOmega _0 $

忽略高阶小量,式 (24) 可简化为

(26)$\dfrac{d^2q(t)}{dt^2} + 2\zeta _{\rm r} \omega \dfrac{dq(t)}{dt} + \omega ^2(1 + 2\mu _{\rm r} \cos \omega _{\rm r} t)q(t) = 0 $

式中

$$\omega ^2 = \dfrac{a_3 + a_4 \varOmega _0 ^2}{a_1 }\,, \ \ \zeta _{\rm r} = \dfrac{a_2 }{2\omega a_1 }\,, \ \ \mu _{\rm r} = \dfrac{a_4 \varOmega _0 \varOmega _{\rm d} }{a_3 + a_4 \varOmega _0 ^2}$$

引入下列无量纲量

$$\bar {q} = \dfrac{q}{l_{\rm s} }\,, \ \ \bar {\omega } = \dfrac{\omega }{\omega _0 }\,, \ \ \bar {t} = \omega _0 t\,, \ \ \bar {\omega }_{\rm r} = \dfrac{\omega _{\rm r} }{\omega _0 }$$

式中出现的 $\omega _0 = \sqrt {a_3 / a_1 }$,为压电悬臂梁的第一阶固有频率. 将上述无量纲量代入式 (26) 中,可以得到

(27)$\dfrac{d^2\bar {q}}{d\bar {t}^2} + 2\zeta _{\rm r} \bar {\omega }\dfrac{d\bar {q}}{d\bar {t}} + \bar {\omega }^2(1 + 2\mu _{\rm r} \cos \bar {\omega }_{\rm r} \bar {t})\bar {q} = 0 $

为书写方便,无量纲量上部的`---'符号在后文中将被忽略. 基于多尺度法的原理, 引入如下系数变换

$$\zeta _{\rm r} \to \varepsilon \zeta\,, \ \ \mu _{\rm r} \to \varepsilon \mu \,, \ \ T_n = \varepsilon ^nt \ \ (n = 0,1,2 \cdots )$$

并代入式 (27) 中,可得

(28)$\dfrac{d^2q}{dt^2} + 2\varepsilon \zeta \omega \dfrac{dq}{dt} + \omega ^2(1 + 2\varepsilon \mu \cos \omega _{\rm r} t)q = 0 $

上式的解应具有如下形式

(29)$q(t,\varepsilon ) = q_0 (T_0 ,T_1 , \cdots ) + \varepsilon q_1 (T_0 ,T_1 , \cdots ) + \cdots $

当时间尺度进行上述变换后,对时间求导表达为

(30)$\left.\begin{array}{l} {\dfrac{d}{dt} = \dfrac{\partial }{\partial T_0 } + \varepsilon \dfrac{\partial }{\partial T_1 } + \cdots = D_0 + \varepsilon D_1 + \cdots } \\\ {\dfrac{d^2}{dt^2} = \dfrac{\partial ^2}{\partial T_0 ^2} + 2\varepsilon \dfrac{\partial ^2}{\partial T_0 \partial T_1 } + \cdots = D_0 ^2 + 2\varepsilon D_0 D_1 + \cdots } \end{array}\!\! \right \} $

这里,主要关注 $\omega = \dfrac{\omega _{\rm r}}{2}$ 亚谐波参数共振,为此设

(31)$\omega = \dfrac{\omega _{\rm r} }{2} + \varepsilon \omega _1 + \varepsilon ^2\omega _2 + \cdots $

将式 (29) $\sim$式 (31) 代入式 (28) 中,并且令方程两端 $\varepsilon$ 的同次幂系数相等,可得

(32)$D_0 ^2q_0 + \dfrac{\omega _{\rm r} ^2}{4}q_0 = 0$

(33)$ D_0 ^2q_1 + \dfrac{\omega _{\rm r} ^2}{4}q_1 = - 2D_0 D_1 q_0 - \xi \omega _{\rm r} D_0 q_0 - \qquad \dfrac{\omega _{\rm r} ^2}{2}\mu q_0 \cos \omega _{\rm r} t - \omega _{\rm r} \omega _1 q_0$

根据微分方程理论,式 (32) 复数形式的解可设为

(34)$q_0 = a(T_1 )\exp ({\rm i}\dfrac{\omega _{\rm r} }{2}T_0 ) + \bar {a}(T_1 )\exp ( - {\rm i}\dfrac{\omega _{\rm r} }{2}T_0 ) $

式中,$a$ 与 $\bar{a}$ 为一对共轭复数,将式 (34) 代入式 (33), 可以得到

(35)$D_0 ^2q_1 + \dfrac{\omega _{\rm r} ^2}{4}q_1 = \Big ( - {\rm i}\omega _{\rm r} \dfrac{ da}{dT_1 } - {\rm i}\zeta \dfrac{\omega _{\rm r} ^2}{2}a - \mu \dfrac{\omega _{\rm r} ^2}{4}\bar {a} - \omega_{\rm r} \omega _1 a \Big ) \cdot \\\ \qquad \exp \Big({\rm i}\dfrac{\omega _{\rm r} }{2}T_0 \Big ) - \mu \dfrac{\omega _{\rm r} ^2}{4}a\exp \Big({\rm i}\dfrac{3}{2}\omega _{\rm r} T_0 \Big ) + cc $

式中,$cc$ 表示等式右端各项的共轭复数部分.为消除持久项,下列条件应当成立

(36)${\rm i}\omega _{\rm r} \dfrac{da}{ dT_1 } + {\rm i}\zeta \dfrac{\omega _{\rm r} ^2}{2}a+\mu \dfrac{\omega _{\rm r} ^2}{4}\bar {a} + \omega _{\rm r} \omega _1 a = 0 $

此处不妨假设上式的解具有如下形式

(37)$a = \dfrac{A}{2}\exp ({\rm i}\varphi ) $

式中,幅值$A$与辐角 $\varphi $ 均为实数,且与变量 $T_1 $ 相关.将式 (37) 代入式 (36),并分离实部与虚部,可得

(38)$\dfrac{ dA}{dT_1 } = - \dfrac{1}{2}\zeta \omega _{\rm r} A + \dfrac{1}{4}A\mu \omega _{\rm r} \sin 2\varphi$

(39)$\dfrac{d\varphi }{dT_1 } = \omega _1 + \dfrac{1}{4}\mu \omega _{\rm r} \cos 2\varphi$

不妨令 $\dfrac{dA}{dT_1 } = \dfrac{d\varphi}{dT_1 } = 0$,可得到上式的定常解为

(40)$\sin 2\varphi = \dfrac{2\zeta }{\mu } = \dfrac{2\zeta _{\rm r} }{\mu _{\rm r}}$

(41)$\omega _{1} = \pm \dfrac{1}{4}\mu \omega _{\rm r} \sqrt {{1} - \dfrac{4\zeta _{\rm r}^2}{\mu _{\rm r} ^2}}$

将式 (41) 代入式 (31), 可得系统的稳定边界摄动解

(42)$\delta = 1\pm \dfrac{1}{2}\sqrt {\mu _{\rm r} ^2 - 4\zeta _{\rm r} ^2} $

其中

$$\delta = \dfrac{2\omega }{\omega _{\rm r} }$$

满足式 (42) 的 $\mu _{\rm r} $ 和 $\zeta _{\rm r}$ 分别为保证系统处于临界稳定状态的最大无量纲参数和最小阻尼比.这个最小阻尼比在本文中被定义为临界阻尼比.根据李雅普诺夫稳定性理论,可判定压电旋转悬臂梁系统的不稳定区间为

(43)$1 - \dfrac{1}{2}\sqrt {\mu _{\rm r} ^2 - 4\zeta _{\rm r} ^2} < \delta < 1 + \dfrac{1}{2}\sqrt {\mu _{\rm r} ^2 - 4\zeta _{\rm r} ^2} $

观察上式可以发现,无量纲参数$\mu _{\rm r} $的增大和阻尼比$\zeta_{\rm r} $的减小会扩大参数共振的不稳定区间. 也就是说,可以利用增大主被动阻尼和减小轮毂角速度脉动幅值的方式来提高系统的稳定性. 由于无量纲参数 $\mu _{\rm r}$ 和阻尼比 $\zeta _{\rm r}$ 对稳定性的直接影响,在后文中,这两个参数将作为重要的特征参数来评价系统的稳定性能.其中 $\mu _{\rm r} $ 所对应的临界值越大越好,$\zeta _{\rm r}$ 所对应的临界阻尼比越小越好. 系统处于临界稳定状态时,$\mu _{\rm r}$ 所对应的临界值越大意味着系统能承受的轮毂角速度脉动幅值越大,$\zeta_{\rm r}$ 所对应的临界阻尼比越小意味着维持系统稳定所需要的阻尼越小.此外,当频率比 $\delta =1$ 时,系统稳定的条件将简化为

(44)$\mu _{\rm r} < 2\zeta _{\rm r} $

为了验证式 (42) 所表示的稳定边界是正确的,本文采用文献^[32,33,34]中所提到的直接分析方法,求出压电旋转悬臂梁系统的稳定边界,所得到结果与解析方法相比较,从而说明多尺度方法所得到系统摄动解的有效性.

基于 Brown 等^[35]的研究,式 (28) 的解应具有如下形式

(45)$q = \sum_{k = 1,3, \cdots } {a_k \exp \Big( {\rm i}\dfrac{k\omega _{\rm r} }{2}} t\Big) + \bar {a}_k \exp \Big( -{\rm i}\dfrac{k\omega _{\rm r} }{2}t\Big) $

取一阶近似,因此,将上式的一阶表达式代入式 (28),可得

(46)$\left( { - \dfrac{\omega _{\rm r}^2}{4}a + {\rm i}\zeta _{\rm r} \omega \omega _{\rm r} a + \omega ^2a + \omega ^2\mu _{\rm r} \bar {a}} \right) \exp\Big( {\rm i}\dfrac{\omega _{\rm r} }{2}t\Big )\, +\\\ \qquad \omega ^2\mu _{\rm r} \bar {a} \exp \Big( {\rm i}\dfrac{3\omega _{\rm r}}{2}t\Big ) + cc = 0 $

为了分析简便, 仅研究第一项,可得

(47)$- \dfrac{\omega _{\rm r} ^2}{4}a + {\rm i}\zeta _{\rm r} \omega \omega _{\rm r} a + \omega ^2a +\omega ^2\mu _{\rm r} \bar {a} = 0 $

将式 (37) 代入式 (47),得压电旋转悬臂梁系统的稳定性边界为

(48)$\delta ^2 \Bigg (1\pm \mu _{\rm r} \sqrt {1 - \dfrac{4\zeta _{\rm r} ^2}{\delta ^2\mu _{\rm r} ^2}}\,\Bigg )= 1 $

考虑到发生 1/2 亚谐波参数共振时 $\delta ^2 \approx 1$,上式可以进一步简化为

(49)$\delta = \dfrac{1}{\sqrt {1\pm \sqrt {\mu _{\rm r} ^2 - 4\zeta _{\rm r} ^2} } } $

显然,当 $\sqrt {\mu _{\rm r} ^2 - 4\zeta _{\rm r}^2} \ll 1$ (发生参数共振) 时,式 (49)与式 (42) 所得到的压电旋转悬臂梁的参数振动稳定性边界是近似一致的.这可以说明本文所得到摄动解是有效的.

3 算例分析

为分析轮毂半径,轮毂角速度平均值和脉动幅值,梁长以及反馈增益系数对主动控制压电旋转悬臂梁的参数振动稳定性的影响,

系统的结构参数取如表 1中的数据进行算例分析.

表1旋转悬臂梁的结构参数^[21]

Table 1 Structural parameters of rotating cantilever beam ^[21]

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通过上节分析可知,增大主动阻尼可以提高系统的稳定性,这意味着速度传感器安装在能使压电层提供最大阻尼的位置有利于提高旋转悬臂梁的稳定性.为确定这一位置,图 2分析了不同安装位置对系统 1/2 亚谐波参数共振稳定性的影响.结果表明,传感器安装位置越靠近自由端,系统稳定性越好.

图2

图2传感器安装位置对压电旋转悬臂梁稳定性的影响($R =0.05$ m, $l_{\rm s} =1$m, $\varOmega _{0}=50$ rad/s, $K =50$)

Fig.2The effect of installation site of sensor on the stability of piezoelectric rotating cantilever beam ($R =0.05$ m, $l_{\rm s} =1$ m, $\varOmega _{0}=50$ rad/s, $K =50$)

给出了发生 1/2 亚谐波参数共振时梁长对临界阻尼比 $\zeta_{\rm r} $ 的影响. 在亚谐波共振点 ($\delta=1$) 之前临界阻尼比会随着频率比的增大而增大,但频率比超过共振点后,临界阻尼比反而会随着频率比的提高而降低,这意味着频率比越靠近共振点,系统越不稳定,系统维持稳定状态所需要的临界阻尼比就越大.此外,随着梁长的增大,临界阻尼比的曲线也会随之上升,这意味着梁长的增大会减弱参数振动的稳定性.

图3

图3梁长对发生 1/2 亚谐波参数共振时临界阻尼比的影响($R =0.05$ m, $\varOmega _{0}=50$ rad/s, $\varOmega_{\rm d}=5$ rad/s)

Fig.3The effect of beam length on the critical damping ratio of 1/2 sub-harmonic parametric resonance ($R =0.05$ m, $\varOmega _{0}=50$ rad/s, $\varOmega_{\rm d}=5$ rad/s)

描绘了不同的轮毂半径下旋转悬臂梁的临界阻尼比随频率比的变化规律,随着轮毂半径的增大,系统稳定区域的边界曲线将升高,参数共振的不稳定区域会随之扩大.这说明降低轮毂半径可以提高系统的稳定性.

图4

图4轮毂半径对发生 1/2 亚谐波参数共振时临界阻尼比的影响($l _{\rm s} =1$ m, $\varOmega _{0}=50$ rad/s, $\varOmega_{\rm d}=5$ rad/s)

Fig.4The effect of hub radius on critical damping ratio of 1/2 sub-harmonic parametric resonance ($l _{\rm s} =1$ m, $\varOmega _{0}=50$ rad/s, $\varOmega_{\rm d}=5$ rad/s)

通过观察图5可以发现,随着轮毂角速度平均值的增大,临界阻尼比的曲线会逐渐升高,但曲线增长的比率会降低,当超过某一临界值后,临界阻尼比和轮毂角速度平均值之间具有相反的关系.为解释这一现象,无量纲参数 $\mu_{\rm r}$ 与轮毂角速度平均值的关系在图 6中进行分析.结果表明,当轮毂角速度的平均值超过 100 rad/s,无量纲参数 $\mu _{\rm r} $ 随着轮毂角速度平均值的增大而减小.这表明轮毂角速度平均值对系统稳定性的影响并不是单调的,在临界值之前,轮毂角速度平均值的增大会降低系统稳定性,但在临界值之后,反而会提高系统稳定性.图 7曲线表明,增大轮毂角速度脉动幅值可明显地增大临界阻尼比.这表明脉动幅值的增大会降低系统的稳定性,并且将其与图 3$\sim$图 5相比较,可发现轮毂角速度脉动幅值和梁长变化对系统在发生参数共振时稳定性的影响较为明显.

图5

图5轮毂角速度平均值对发生 1/2 亚谐波参数共振时临界阻尼比的影响($R =0.05$ m, $l _{\rm s} =1$ m, $\varOmega_{\rm d}=5$ rad/s)

Fig.5The effect of average value of hub angular velocity on the critical damping ratio of 1/2 sub-harmonic parametric resonance ($R =0.05$ m, $l _{\rm s} =1$ m, $\varOmega_{\rm d}=5$ rad/s)

图6

图6轮毂角速度平均值对无量纲参数$\mu _{\rm r} $的影响 ($R=0.05$ m,$l _{\rm s}=1$ m)

Fig.6The effect of average value of hub angular velocity on the dimensionless parameter $\mu _{\rm r}$ ($R=0.05$ m,$l _{\rm s}=1$ m)

图 7

图 7轮毂角速度脉动幅值对发生 1/2 亚谐波参数共振时临界阻尼比影响 ($R =0.05$ m, $l _{\rm s} =1$ m, $\varOmega_{0}=50$ rad/s)

Fig.7The effect of pulsating amplitude of hub angular velocity on critical damping ratio of 1/2 sub-harmonic parametric resonance ($R =0.05$ m, $l _{\rm s} =1$ m, $\varOmega_{0}=50$ rad/s)

通过观察图8不难发现,随着反馈增益系数的增大,无量纲参数 $\mu_{\rm r} $ 的曲线也会随之上升,不稳定区域会随之减小.这表明反馈增益系数的增大会明显的提高系统的稳定性,即使在较大的变动转速作用下,悬臂梁系统也会趋向稳定的.总之,反馈增益系数越大,系统能承受的旋转角速度能力就越强.比较只含被动阻尼 ($K =0$) 和只含主动阻尼 ($K=50$) 的曲线,可以发现主动控制远大于被动阻尼对系统稳定性的影响.这说明与悬臂梁的结构阻尼相比,压电层提供的主动阻尼可大幅度地提高系统振动的稳定性.另外,在靠近亚谐波参数共振点附近数值分析方法与多尺度法所得稳定边界基本上是重合的,这证明了本文应用摄动方法分析压电旋转悬臂梁参数共振稳定性是有效的.

图8

图8反馈增益系数对压电旋转悬臂梁稳定性的影响 ($R =0.05$ m, $l _{\rm s}=1$ m, $l _{\rm c}=1$ m, $\varOmega_{0}=50$ rad/s)

Fig.8The effect of feedback gain coefficient on the stability of piezoelectric rotating cantilever beam ($R =0.05$ m, \\ $l _{\rm s}=1$ m, $l _{\rm c}=1$ m, $\varOmega_{0}=50$ rad/s)

为进一步分析压电主动控制与系统稳定性的关系,从阻尼比的角度,在图 9中给出了不同轮毂角速度平均值下反馈增益系数对稳定性的影响.结果显示反馈增益系数的增加,可以提高系统阻尼比,但增长比率逐渐降低,而轮毂角速度平均值的增大则会降低阻尼比.这说明主动控制对稳定性的影响随着轮毂角速度平均值的增大而逐渐减弱.

图9

图9不同轮毂角速度平均值下阻尼比随反馈增益系数 $K$ 的变化关系 ($R =0.05$ m, $l _{\rm s}=1$ m, $l _{\rm c}=1$ m)

Fig.9The variation of damping ratio versus the feedback gain coefficient $K$ for different average values of hub angular velocity ($R =0.05$ m, $l _{\rm s}=1$ m, $l _{\rm c}=1$ m)

4 小结

本文建立了主动控制压电旋转悬臂梁的一阶近似动力学方程,运用多尺度法找到了系统在发生第一阶模态参数共振时稳定性边界摄动解,通过分析得出来如下结论：

(1)当系统的第一阶固有频率接近 1/2 轮毂角速度脉动频率时,系统会发生参数共振而可能失稳.

(2) 无量纲参数$\mu _{\rm r} $减小以及无量纲阻尼比$\zeta _{\rm r}$增大均可以提高系统参数振动稳定性.

(3)发生参数共振时,梁长、轮毂半径和脉动幅值的增大会降低系统稳定性,而反馈增益系数的增大可以提高系统稳定性,但提高的抑制效果会逐渐减弱.

(4) 轮毂角速度平均值对系统稳定性的影响并不是单调的,在临界值之前,轮毂角速度平均值的增大会降低系统稳定性,但在临界值之后,反而会提高系统稳定性. 另外,轮毂角速度平均值的增大会削弱主动压电旋转悬臂梁的振动抑制效果.

本研究所得结论对于旋转机构的设计具有重要的理论指导意义.

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A study on the buckling and dynamic stability of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces is performed in this research. The static and dynamic governing equations of the nanobeam are established with Galerkin method and under Euler-Bernoulli hypothesis. The buckling, post-buckling and nonlinear dynamic stability character of the nanobeam is presented. The quasi-elastic method, Leibnitz's rule, Runge-Kutta method and the incremental harmonic balanced method are employed for obtaining the buckling voltage, post-buckling characteristics and the boundaries of the principal instability region of the dynamic system. Effects of the electrostatic load, van der Waals force, creep quantity, inner damping, geometric nonlinearity and other factors on the post-buckling and the principal region of instability are investigated. (C) 2013 Elsevier B.V.

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Chen , XC , Lu

, Li

Free vibration, buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium

Applied Mathematical Modelling, 2019 , 67 : 430 - 448