SOMMERFELD EFFECT IN NON-IDEAL VIBRATION SYSTEMS DRIVEN BY MOTORS
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摘要: 真实转子系统的电机输出功率有限且无法提供理想恒定驱动, 随着电机输入电压升高, 在系统固有频率附近会出现长时间平均转速停滞和多稳态跳跃, 称为“索末菲(Sommerfeld)效应”. 这类非理想电机启动时会经过索末菲效应作用区, 而长期陷于这一区域时电机转速剧烈波动, 会严重影响电机寿命. 简化忽略转速波动而注重稳态平均转速, 无法观察系统内部受力变化, 因此借助数值分析研究电机系统与振动系统作用过程. 建立了非理想直流电机-质体-不平衡转子系统的机电耦合动力学模型, 通过Matlab/Simulink建模, 分析了振动系统各内力分量的变化规律, 考察了索末菲效应产生、演变和消失过程. 观察到非理想系统进入共振区后, 恢复力、惯性力和阻尼力将突然产生偶数倍频成份, 导致系统在平均转速与固有频率渐进一致的前提下, 产生快转、停转乃至反转剧烈交替切换的现象, 并在输入电压大于临界电压后结束跳跃. 研究描述了在非理想驱动下系统展现的非线性行为, 最后为振动系统动力设计提供指导并确保动力设备正常运行.Abstract: The motor output power of a real rotor system is limited and cannot provide ideal constant drive force. As the input voltage value of the motor increases, there will be long-term average rotor speed stagnation and the steady-state response of the system appears multiple values and switches by sudden jumps around the natural frequency of the system, which is called "Sommerfeld effect". This kind of non-ideal motor will pass through the Sommerfeld effect zone when starting, and the motor speed will fluctuate sharply when it is trapped in this Sommerfeld zone for a long time, which will seriously affect the performance life of the motor. The theoretical simplification ignores the speed fluctuation and focuses on the steady-state average speed, and cannot observe the internal energy change of the system. Therefore, the interaction process of the motor system and the vibration system is studied by means of numerical analysis method. The electromechanical coupling dynamic model of the non-ideal DC motor, mass and unbalanced rotor system is established. The change law of each internal force component of the vibration system is analyzed by Matlab/Simulink modeling, and the generation, evolution and disappearance process of Sommerfield effect are investigated. It is observed that when the non-ideal system enters the resonance region, the restoring force, inertia force and damping force will suddenly produce an even fold frequency component, resulting in the phenomenon of fast rotation, stop rotation and even reversal of the system under the premise that the rotor average speed is gradually consistent with the natural frequency of the system, and the jump ends when the input voltage value is greater than the critical voltage value. The study describes many nonlinear behaviors due to the non-ideal drive system, and finally provides some guidance for the dynamic design of the vibration system and ensures the normal operation of the power.
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Keywords:
- Sommerfeld effect /
- rotor system /
- vibrating system /
- nonlinear dynamics
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引 言
原动机是驱动机械设备的动力来源, 包括直流电机和交流电机等. 由于几乎所有原动机能够提供的功率是有限的, 因此也称为非理想原动机. 在非理想原动机驱动机械系统接近共振时, 机械系统激励振动, 当原动机输出功率不足时, 电机转子系统出现临界转速, 机械系统激振频率会在共振前长期停留并跳跃至共振后, 这类非线性跳跃现象称为Sommerfeld效应[1-3]. 通常, 这一类Sommerfeld效应由放置在弹性支撑上的不平衡电机(尤其是直流电机)的动力学描述[4-6].
对于一般旋转设备, 通常以高于其一个或多个临界速度的速度运行. 由此良好的隔振性能是必要的. 然而, 在转子系统中, 由于制造缺陷[7-8]、安装错误、转子叶片或轴承故障[9]、不对称负载[10-11]或自然磨损过程, 可能存在小的不平衡. 有时, 不平衡可能被有意添加以产生振动, 例如在振动筛[12]、混合器[13]和洗衣干燥机[14]中. 转子系统在其临界转速下具有较大的振动振幅, 并且在任何这些临界转速下持续运行都可能导致整个系统的故障. 在共振区域, 转子驱动装置(如发动机或电机)向旋转系统提供的大部分电力将用于增加结构振动, 而不是增加转子转速. 因此, 转子转速可能会在振动或涡动振幅过大的情况下发生共振.
考虑由于非理想振动系统引起的Sommerfeld效应时, 不同研究者从各种方面探究Sommerfeld效应的影响. Sinha等[15-17]研究了非理想驱动单自由度往复系统的动力学, 系统中出现了Sommerfeld效应, 同时为了平缓地通过系统的谐振, 给出了直流电机的功率要求. Bisoi等[18]研究了旋转对称平面动力系统中外部和内部阻尼以及陀螺力对Sommerfeld效应的影响. Bharti等[19]将直流电机视为由非理想原动机驱动的单盘刚性转子系统, 并在两个临界转速附近分析了Sommerfeld效应, 在他们的研究中, 当两个临界转速接近时, 系统表现出频率捕获和非线性跳跃现象.
在国内, 孔祥希等[20-22]针对电动机-转子系统研究了转子偏心质量与转子总质量之比等参数对Sommerfeld效应的影响和多个电机同步问题. 易园园等[23]建立了三相异步电机-多级齿轮系统的机电耦合动力学模型, 并对其冲击载荷下的动态特性进行了研究, 但并未对Sommerfeld效应进行研究. 张力豪等[24]对电机驱动转子系统进行升速过临界实验, 从其实验结果可以看出, 转子系统的振幅在临界转速之前平缓增大, 而在转速通过临界转速之后, 振幅并不是平缓降低, 而是突然急剧降低, 观察到了Sommerfeld效应, 但并未深入研究. 姜娇等[25-26]建立了梁-电动机耦合系统的模型, 使用平均摄动法, 在忽略转速波动情况下, 研究了感应电动机的不平衡质量和功率对Sommerfeld效应的影响规律, 同时研究了多机系统下出现的转速跳跃现象, 而且在梁结构共振频率前后系统的同步运动状态发生改变. Balthazar等[27-29]使用直接运动分离方法分析了非理想电源驱动摆系统, 获得了慢运动分量的方程, 分析了解的稳定性, 得出Sommerfeld效应规律, 并证明动力系统的特殊行为, 此时快变量演化过程被忽略, 而在Sommerfeld效应严重时快慢变量分离假设并不满足.
在实验测量方面, 大多研究者只在实验中观察到了大致现象, 无法准确测量旋转叶片转速, 目前通过数字图像相关法(DIC)进行测量是一种可行方案, Huang等[30-32]通过改进图像算法和优化散斑布置等方面在系统参数辨识和自同步方面取得了初步应用. 而理论方法分析Sommerfeld效应时, 由于系统高度非线性, 通过小参数法或直接运动分离法等方法解耦时, 往往忽略转速波动而只关注平均转速等慢变量, 无法观察到系统响应周期内演化过程和能量流动, 因此, 本文在此基础上充分考虑转子-电机系统与质体系统相互作用, 建立了非理想永磁直流电机驱动质体振动模型, 在转子质量与系统总质量之比不满足小参数假设下利用Matlab/Simulink得出精确完整响应过程, 通过分析质体受力各分量进一步详细分析Sommerfeld效应发生过程, 并描述由于非理想驱动系统展现的其他非线性行为, 最后为振动系统动力设计提供指导并确保动力设备正常运行.
1. 建立质体-电机动力学模型
本文研究整个系统模型如图1所示, 在图1中将一个大质量块$M$放置于$XY$水平面上, 并与一个刚度为$k$, 阻尼为$c$的弹簧相连. $M$上连接一个偏心转子, 质量为$m$, 连杆忽略质量且长度为$l$, 杆与$x$方向夹角为$\theta $, 转轴转动阻尼常数为${c_r}$, 转轴受到来自电机转矩${T_e}$作用. 而选择永磁直流电机输入电压${u_a}$, 内部电路等效电阻常数${R_a}$, 等效电感常数为${L_a}$, 等效感应电动势${E_a}$, ${i_a}$为电路电流. 采用质体$M$在其平衡位置附近的水平位移$x$和转子旋转的相位角$\theta $作为广义坐标.
整个质体-转子系统动能$E$和势能$U$为
$$\left.\begin{split} & E = \frac{1}{2}M{{\dot x}^2} + \frac{1}{2}m{\left( {\dot x - \dot \theta l\sin \theta } \right)^2} + \frac{1}{2}m{\left( {\dot \theta l\cos \theta } \right)^2} \\ & U = \frac{1}{2}k{x^2} \end{split}\right\} $$ (1) 其中$ ( \dot · ) = {\text{d}}( · )/{\text{d}}t,{\text{ }}( \ddot · ) = {{\text{d}}^2}( · )/{\text{d}}{t^2},{\text{ }}t $表示时间.
整个系统中非保守力所做虚功为
$$ {\text{d}}W = - \left( {c\dot x} \right){\text{d}}x + {T_e}{\text{d}}\theta - ({c_r}\dot \theta ){\text{d}}\theta = {Q_x}{\text{d}}x + {Q_\theta }{\text{d}}\theta $$ (2) 令$L = E - U$, 将式(1)和式(2)代入拉格朗日第二类方程
$$ \frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial L}}{{\partial {{\dot q}_j}}}} \right) - \frac{{\partial L}}{{\partial {q_j}}} = {Q_j}, \quad {q_j} = x,\theta $$ (3) 之后质体系统微分方程整理为
$$ \left. \begin{split} & (M + m)\ddot x(t) + c\dot x(t) + kx(t) = \\ &\qquad ml\ddot \theta (t)\sin \theta (t) + ml{{\dot \theta }^2}(t)\cos \theta (t) \\ & J\ddot \theta (t) + {c_r}\dot \theta (t) = {T_e}(t) + lm\ddot x(t)\sin \theta (t) \end{split} \right\} $$ (4) 其中$ J = m{l^2} $. 而对于永磁直流电机, 在忽略电机机轴的扭转时[33-34], 微分方程等效为
$$\left. \begin{split} & {k_m}{u_a} = {R_a}{\varGamma _e}(t) + {E_a}(t) + {L_a}{{\dot \varGamma }_e}(t) \\ & {i_a}(t) = {\varGamma _e}(t)/{k_m} \\ & {\varGamma _e}(t) = {c_r}\omega (t) + J\dot \omega (t) + {\varGamma _L}(t) \end{split} \right\} $$ (5) 其中, $ {E_a}(t) = {k_e}\omega (t) $; $\omega $为电机转速; ${k_m}$为电机转矩常数; ${k_e}$为电机转速常数; ${\varGamma _e}(t)和{\varGamma _L}(t)$分别为电机输出转矩和负载转矩.
将式(4)和式(5)最后一式对比, 可以得出质体和电机系统耦合后应满足下式
$$ \omega (t) = \dot \theta (t),{T_e}(t) = {\varGamma _e}(t),{\varGamma _L}(t) = - lm\ddot x(t)\sin \theta (t) $$ (6) 最后考虑机电耦合, 整理式(4)、式(5)和式(6)得到
$$\left.\begin{split} & (M + m)\ddot x(t) + c\dot x(t) + kx(t) = ml\ddot \theta (t)\sin \theta (t) + \\ &\qquad ml{{\dot \theta }^2}(t)\cos \theta (t) \\ & J\ddot \theta (t) + {c_r}\dot \theta (t) = {T_e}(t) + lm\ddot x(t)\sin \theta (t) \\ & {L_a}{{\dot T}_e}(t) + {R_a}{T_e}(t) = {k_m}\left( {{u_a} - {k_e}\dot \theta (t)} \right) \end{split} \right\} $$ (7) 式(7)的第1式描述了单自由度振动系统受来自转子的非理想激励而受迫振动, 其中质体无阻尼固有频率为${\omega _n} = \sqrt {k/(M + m)} $, 而式(7)的第2和第3式分别描述了转子系统受质体振动激励及电机作用下旋转和电机内部电压平衡关系, 3式并列表明此时系统是由质体、转子和电机3个系统相互耦合而成且高度非线性.
在理想情况下, 质体振动和转子旋转不相互耦合, 式(5)中负载转矩为零, 即${\varGamma _L}(t) = 0$. 然后可以单独求解电机微分方程组并将稳态转速代入式(4), 得出转子稳态转速${\omega _i}$和质体稳态位移响应$x(t)$为
$$ \left.\begin{split} & {\omega _i} = {k_m}{u_a}/({c_r}{R_a} + {k_m}{k_e}) \\ & x(t) = {a_x}\cos ({\omega _i}t - {\gamma _x}) \end{split}\right\} $$ (8) 式中, 频率比$\varpi = {\omega _i}/{\omega _n}$; 阻尼比$\zeta = c/[2\sqrt {k(M + m)} ]$; 幅值${a_x} = ml{\omega _i}^2/\left[k\sqrt {{{(1 - {\varpi ^2})}^2} + {{(2\zeta \varpi )}^2}} \right]$; 相位差${\gamma _x} = \arctan [2\zeta \varpi /(1 - {\varpi ^2})]$.
2. 建立机电耦合仿真模型并求解
根据建立的系统微分方程式(4)和式(5), 可以将其按状态方程分别改写为下式
$$ \left\{ \begin{gathered} {{\ddot {\boldsymbol{x}}}} \\ {{\dot {\boldsymbol{x}}}} \\ \end{gathered} \right\}{ = }\left[ {\begin{array}{*{20}{c}} {{ - }{{{\boldsymbol{M}}}^{{ - 1}}}{{\boldsymbol{C}}}}&{{ - }{{{\boldsymbol{M}}}^{{ - 1}}}{{\boldsymbol{K}}}} \\ {{\boldsymbol{I}}}&{{\boldsymbol{0}}} \end{array}} \right]\left\{ \begin{gathered} {{\dot {\boldsymbol{x}}}} \\ {{\boldsymbol{x}}} \\ \end{gathered} \right\}{ + }\left\{ {\begin{array}{*{20}{c}} {{{{\boldsymbol{M}}}^{{ - 1}}}{{\boldsymbol{F}}}} \\ {{\boldsymbol{0}}} \end{array}} \right\} $$ (9) $$ \left\{ {\begin{array}{*{20}{c}} {{{\dot i}_a}} \\ {\dot \omega } \end{array}} \right\} = {{\boldsymbol{A}}}\left\{ {\begin{array}{*{20}{c}} {{i_a}} \\ \omega \end{array}} \right\} + {{\boldsymbol{B}}}\left\{ {\begin{array}{*{20}{c}} {{u_a}} \\ {{\varGamma _L}} \end{array}} \right\},\left\{ \begin{gathered} {\varGamma _e} \\ \omega \\ \end{gathered} \right\} = \left[ {\begin{array}{*{20}{c}} {{k_m}}&0 \\ 0&1 \end{array}} \right]\left\{ \begin{gathered} {i_a} \\ \omega \\ \end{gathered} \right\} $$ (10) 其中${{\boldsymbol{x}}} = {\{ x,\theta \} ^{\text{T}}}$,${{\boldsymbol{F}}} = {\left\{ {0,{T_e}} \right\}^{\text{T}}}$
$$ {{\boldsymbol{M}}} = \left[ {\begin{array}{*{20}{c}} {M + m}&{ - ml\sin \theta } \\ { - ml\sin \theta }&J \end{array}} \right], {{\boldsymbol{A}}} = \left[ {\begin{array}{*{20}{c}} { - {L_a}^{ - 1}{R_a}}&{ - {L_a}^{ - 1}{k_e}} \\ {{J^{ - 1}}{k_m}}&{ - {J^{ - 1}}{c_r}} \end{array}} \right] $$ $$ {{\boldsymbol{C}}} = \left[ {\begin{array}{*{20}{c}} c&{ - ml\theta \cos \theta } \\ 0&{{c_r}} \end{array}} \right],{{\boldsymbol{K}}} = \left[ {\begin{array}{*{20}{c}} k&0 \\ 0&0 \end{array}} \right], {{\boldsymbol{B}}} = \left[ {\begin{array}{*{20}{c}} {{L_a}^{ - 1}}&0 \\ 0&{ - {J^{ - 1}}} \end{array}} \right] $$ 根据式(9)和式(10)在Matlab/Simulink中建立质体-电机耦合模型. 在电机模型中, 物理量${L_a},{R_a},{k_m}, {k_e},{c_r},J,{u_a}和{\varGamma _L}$作为输入参数, $\omega ,{i_a},{\varGamma _e},{\dot i_a}和\dot \omega $作为输出参数; 在质体振动模型中, 物理量$\theta ,\dot \theta ,\ddot \theta ,k,c,$$M,m,l和{c_r}$作为输入, $x,\dot x,\ddot x$作为输出. 对于任一确定的永磁直流电机, 参数${L_a},{R_a},{k_m}和{k_e}$均为常数, 最后考虑式(6)将两个模型连接得出完整质体-转子-电机耦合模型. 其中耦合后模型参数如表1.
表 1 系统参数Table 1. System parametersParameters Value Parameters Value $m/{\text{kg}}$ 20 $l/{\text{m}}$ 0.08 $M/{\text{kg}}$ 200 ${R_a}/{{\Omega }}$ 0.57 $c/({\text{N}} \cdot {\text{s}}\cdot {\mathrm{m}}^{-1})$ 235.57 ${k_m}/({\text{N}} \cdot {{{\mathrm{m}}\cdot{\mathrm{A}}^{-1}}})$ 0.57 ${c_r}/({\text{N}} \cdot {\text{m}} \cdot {{{\mathrm{s}}\cdot{\mathrm{rad}}^{-1}}})$ 0.03 ${L_a}/{\text{H}}$ 0.057 $k/({{{\mathrm{N}}\cdot{\mathrm{m}}^{-1}}})$ 630500 ${u_a}/{\text{V}}$ 0:0.1:100 ${k_e}/({\text{N}} \cdot {{{\mathrm{s}}\cdot{\mathrm{rad}}^{-1}}})$ 0.57 初始时刻振动质体满足$x(0) = 0{\text{ m}},\dot x(0) = 0{\text{ m/s}}$, 直流电机转子转角满足$\theta (0) = 0{\text{ rad}}$, 转子初始转速选择$\omega (0) = 0{\text{ rad/s}}$或者远离${\omega _n}$取$\omega (0) = 180{\text{ rad/s}}$, 仿真时间5 s, 采用Ode45求解器, 使用四阶龙格库塔法, 时间步长设置$\Delta t = {10^{ - 4}}{\text{ s}}$, 收敛相对误差为$5.0 \times {10^{ - 5}}$, 输入电压为逐点扫描, 在输入电压${u_a}$ = 50 V时, 偏心电机转子转速$\dot \theta (t)$和质体振动位移$x(t)$逐渐进入稳态过程如图2和图3所示. 而在转子质量$m$ = 1 kg时, 其他物理参数不变如表1所示, 仿真设置不变, 参数设置与已有文献[35]一致后, 得出转子系统稳态平均转速$\bar \omega $随输入电压变化结果如图4所示.
从图2和图3的仿真结果可以看出: ${u_a}$ = 50 V时, 在不同初始转速下转子转速$\dot \theta (t)$和质体振动$x(t)$在4 s左右进入稳态并存在两种不同稳态情况, 两种情况下, 转速波动和质体振动幅值都不同. 由此可以得出电机转速随时间明显波动, 系统稳态响应受转子电机系统初值影响. 其中平均转速计算方法为
$$ \bar \omega = \sum\limits_{i = {t_1}}^{i = {t_2}} {\dot \theta (t){\text{d}}t} \Biggr/({t_2} - {t_1}) $$ (11) 而${t_1}和{t_2}$分别是转速响应充分进入稳态时刻和稳态后经历多个波动周期循环对应时刻.
依据图4可以看出, 转子稳态平均转速在输入电压35 ~ 36 V时出现多值, 在36 V时突增, 35 V时突降, 文献中平均转速结果与数值仿真的结果基本吻合, 佐证了所建仿真模型的正确性.
在电机内部参数不变, 参考表1的情况下, 利用仿真分析振动系统参数和偏心转子质量对Sommerfeld效应的影响. 转子质量为1 kg并保持不变, 改变振动系统弹簧阻尼$c$, 弹簧刚度$k$; 在弹簧阻尼$c$和弹簧刚度$k$不变时改变转子质量$m$. 考察输入电压30 ~ 40 V结果, 得出转子平均转速$\bar \omega $随输入电压变化如图5(a) ~ 图5(c)所示.
从图5(a) ~ 图5(c)可以得出在其他参数不变, 单一变量下, 弹簧阻尼增加会抑制Sommerfeld效应, 弹簧刚度降低或转子质量减小同样可以抑制Sommerfeld效应. 因此在合适参数范围内, 对振动系统减振能有效抑制Sommerfeld效应.
在系统参数设置为表1所示, 转子质量为20 kg时, 转子旋转会剧烈影响振动系统, 小参数假设不在满足, 改变仿真输入电压${u_a}$, 仿真时间$t$为100 s并此后保持不变, 得出稳态后转子平均转速$\bar \omega $和理想系统结果对比, 如图6所示. 质体振动稳态位移响应幅值和理想系统结果对比如图7所示.
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图 6 转子平均转速$\bar \omega $($m$ = 20 kg)Figure 6. The average of the rotor speed$\bar \omega $($m$ = 20 kg)根据式(8)可知理想情况下, 系统不耦合时, 转子稳态转速随输入电压线性增加. 而从图6和图7可以得出, 在转子初始转速0 rad/s时, 转子稳态平均转速在电压大于32.1 V后大幅偏离理想转速, 并慢慢逼近质体系统无阻尼固有频率${\omega _n} = 53.53{\text{ rad/s}}$, 之后在大于82.4 V时突增到理想转速附近, 质体振动幅值则突降; 在转子初始转速为180 rad/s时, 随着输入电压减小, 转子平均转速在理想转速附近, 之后在50 V时突降到对应质体谐振频率附近, 质体振动幅值则突增. 因此对于转子系统, 转子稳态平均转速存在一个多值区间和无法稳定进入区间.
在图7中与理想线性系统相比, 质体响应表现出明显硬式非线性现象, 一方面转子旋转受质体振动影响, 转子平均转速始终低于理想转速, 长期停留在质体系统固有频率附近, 在输入电压增加过程中平均转速会出现跳跃现象, 并且受转子初始转速影响. 另一方面, 质体系统振动受转子旋转影响, 相比理想偏心激励质体振动, 质体系统脱离谐振所需电压会大大增加, 而振幅峰值低于理想系统. 在输入电压增加过程中, 质体长期处于谐振附近, 之后在转子平均转速跳跃同时振动幅值突降, 质体进入远谐振, 由此展现了剧烈的Sommerfeld效应.
3. Sommerfeld效应作用过程分析
在质体-转子系统耦合后, 转子转速受Sommerfeld效应的影响长时间停留在质体系统固有频率附近无法达到理想转速, 两个系统都无法理想运行, 而在Sommerfeld效应严重的情况下, 系统同时会出现一系列强非线性行为特征, 只考虑理论方法简化下的平均转速无法完整展现系统响应特征和描述Sommerfeld效应.
而在输入电压增加过程中, 转子稳态平均转速长期不变后突然跳跃, 同时在电机设备启动过程常常会经过系统谐振区, 对电机设备的危害性很大; 而输入电压降低过程一般只会发生在电机设备停机过程中, 设备会很快停止工作且很难停留在系统谐振区, 对电机设备正常运行影响很小.
因此下面利用仿真详细描述输入电压增加过程中转子稳态转速和质体系统受力情况.
3.1 非理想电机转速变化过程
在改变输入电压后, 系统参数如表1, 考虑转子初始转速为0 rad/s下, 仿真得出转子稳态转速的平均值、最小值和最大值情况如图8, 图8显示在不同输入电压阶段时, 转子稳态转速呈现不同的周期波动特征. 其中4个阶段考虑电压分别为32.1, 60, 80和90 V时, 系统稳态后最后1 s仿真时间内转子稳态转速以及与谐振频率${\omega _n}$对比具体体现在图9.
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图 9 转子稳态转速和${\omega _n}$对比(从上往下对应stage1 ~ stage4)Figure 9. Steady-state rotor speed and${\omega _n}$(stage1 ~ stage4)依据图8所示的转子稳态转速波动情况可把Sommerfeld效应萌芽、发展、剧烈和消失过程在输入电压上对应分为4段(stage1 ~ stage4): 输入电压为0 ~ 32.1 V为第1段, 此时转子稳态转速小幅波动并始终小于${\omega _n}$, 如图9中stage1阶段; 输入电压为32.1 ~ 65.3 V为第2段, 此时转子稳态转速在${\omega _n}$上下波动, 并且在平均转速上下呈等幅波动, 如图9中stage2阶段; 输入电压65.3 ~ 82.4 V为第3段, 此时转子稳态转速在${\omega _n}$上下波动, 但在平均转速上下呈不等幅波动, 转子转速在大于${\omega _n}$时出现双幅值, 因此虽然单位周期内转子平均转速几乎不动, 但是转子瞬时转速在快转、停转乃至反转之间剧烈交替出现, 如图9中stage3阶段. 最后, 输入电压82.5 ~ 100 V为第4段, 此时转子转速远大于${\omega _n}$, 转子转速在理想转速附近波动, 如图9中stage4阶段.
3.2 质体稳态位移响应频率和受力变化
在上述4个阶段, 不同对应输入电压区间下, 质体稳态响应同样呈现不同的频率特征. 在忽略高阶频率小量时, 稳态位移响应$x(t)$的频率随输入电压变化的结果如图10所示. 同样当输入电压分别为32.1, 60, 80和90 V, 初转速为0 rad/s时, 系统稳态后最后1 s仿真时间内质体稳态位移的自功率谱密度与对应响应结果, 分别如图11和图12所示.
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图 11 响应对应频谱(从上往下对应stage1 ~ stage4)Figure 11. Self-power spectral density of response (stage1 ~ stage4)![]()
图 12 稳态位移(从上往下对应stage1 ~ stage4)Figure 12. Steady-state displacement (stage1 ~ stage4)从图10和图11中得出, 在第1和第2阶段时质体稳态位移响应频率前3阶是奇数倍频, 主要包括基频$1{f_0}$, $3{f_0}$, $5{f_0}$. 而在第3阶段, 质体稳态位移响应频率包括$1{f_0}$, $3{f_0}$, $5{f_0}$和$2{f_0}$, $4{f_0}$, $6{f_0}$. 在第4阶段, 电压大于82.4 V后质体稳态位移响应频率恢复为3阶奇数倍频. 而从图11和图12中得出整个过程中稳态位移响应中基频分量$1{f_0}$占主导.
整个过程中, 从图6和图10对比得出质体稳态位移响应基频$1{f_0}$随电压变化和平均转速$\bar \omega $随电压变化结果一致, 因此转子平均转速对应质体位移响应基频, 转速波动对应质体位移响应多频. 而在转速剧烈波动的第3阶段, 质体稳态位移突然出现偶数倍频分量并在大于临界电压82.4 V后消失.
在图10中第1阶段出现输入电压6.6 V时${f_0} = {\omega _n}/5$, 在输入电压11.0 V时${f_0} = {\omega _n}/3$. 在第2和第3阶段时${f_0} \approx {\omega _n}$, 在第4阶段${f_0} \gg {\omega _n}$. 与理想系统结果对比可以得出在非理想系统内, 质体系统稳态响应和来自转子的偏心外激励力都会出现奇数倍频或偶数倍频, 而当外激励力的基频接近系统固有频率或者奇数倍频接近系统固有频率都引起质体系统谐振.
下面分析质体位移频率和转子转速之间的关系. 依据前面稳态转子平均转速和质体稳态响应的数值结果对式(7)微分方程组中转子稳态转速和角位移给出一个数值解, 即
$$\left. \begin{split} & \dot \theta (t) = \bar \omega + {A_n}\cos (n\bar \omega t + {\varphi _n}) \\ & \theta (t) = \bar \omega t + {B_n}\sin (n\bar \omega t + {\varphi _n}) + q\end{split}\right\} $$ (12) 其中${A_n},{B_n}$为幅值常数, ${\varphi _n}$为相位差常数, $n,q$为常数, 当认为转速波动远小于平均转速时$\bar \omega \gg {A_n}$.
在满足$\bar \omega \gg {A_n}$时可以忽略$\ddot \theta $并将式(12)代入质体振动微分方程式(4)的第1式, 可以得出质体稳态位移响应为
$$ x(t) = A\left( {\bar \omega } \right)\cos \left[ {\theta (t) - {\varphi _x}\left( {\bar \omega } \right)} \right]$$ (13) 式中, $A(\bar \omega )$和${\varphi _x}(\bar \omega )$是转子稳态平均转速为自变量且不随时间变化的幅值常数和相位差常数.
对式(13)右边的时变项进行频率分析, 即在忽略常数项情况下分析$\cos [\bar \omega t + {B_n}\sin (n\bar \omega t)]$如何展开成多个单频谐波叠加. $n = 2$时, 这一时变项为
$$\begin{split} & \cos [\bar \omega t + {B_2}\sin (2\bar \omega t)] = \cos (\bar \omega t)\cos [{B_2}\sin (2\bar \omega t)] - \\ & \qquad \sin (\bar \omega t)\sin [{B_2}\sin (2\bar \omega t)] \end{split} $$ (14) 使用傅里叶级数展开得出
$$ \left.\begin{split} & \cos [{B_2}\sin (2\bar \omega t)] = \frac{{{b_0}}}{2} + \sum\limits_{i = 1}^\infty {{b_i}} \cos (4i\bar \omega t) \\ & \sin [{B_2}\sin (2\bar \omega t)] = \sum\limits_{i = 1}^\infty {{c_i}} \sin [\left( {4i - 2} \right)\bar \omega t] \end{split}\right\} $$ (15) 其中各常数为
$$\begin{split} & {b_0} = \frac{2}{T}\int\limits_{ - T/2}^{T/2} {\cos [{B_2}\sin (2\bar \omega t)]} {\text{d}}t \\ & {b_i} = \frac{2}{T}\int\limits_{ - T/2}^{T/2} {\cos [{B_2}\sin (2\bar \omega t)]} \cos (4i\bar \omega t){\text{d}}t \\ & {c_i} = \frac{2}{T}\int\limits_{ - T/2}^{T/2} {\sin [{B_2}\sin (2\bar \omega t)]} \sin (\left[ {4i - 2} )\bar \omega t\right]{\text{d}}t \\ & T = \frac{\text{π} }{{\bar \omega }},\quad i = 1,2,3 \cdots \end{split} $$ 将式(15)代入式(14)并整理得出
$$ \begin{split} & \cos (\bar \omega t)\cos [{B_2}\sin (2\bar \omega t)] - \sin (\bar \omega t)\sin [{B_2}\sin (2\bar \omega t)] = \\ &\qquad \frac{{{b_0} - {c_1}}}{2}\cos (\bar \omega t) + \frac{{{b_1} + {c_1}}}{2}\cos (3\bar \omega t) + \\ &\qquad \frac{{{b_1} - {c_2}}}{2}\cos (5\bar \omega t) + \cdots + \frac{{{b_{i - 1}} - {c_i}}}{2}\cos [\left( {4i - 3} \right)\bar \omega t] + \\ &\qquad \frac{{{b_i} + {c_i}}}{2}\cos [\left( {4i - 1} \right)\bar \omega t] + \frac{{{b_i}}}{2}\cos [\left( {4i + 1} \right)\bar \omega t] \end{split} $$ (16) 从式(16)可以得出这一时变项是由平均转速基频项和其奇数倍频项叠加而成. 因此在Sommerfeld效应的第1, 2, 4阶段, 转子转速除了平均转速项还存在以平均转速2倍为频率的波动项, 而正是波动项引起的质体位移响应多倍频的情况.
而在Sommerfeld效应的第3阶段, 转速不光存在以平均转速2倍为频率的波动项, 还有平均转速为频率的另一波动项, 即此时$n = 1$, 同理可以证明, 此时$\cos [\bar \omega t + {B_1}\sin (\bar \omega t)]$可以展开为平均转速为基频项和其偶数倍频项叠加.
最后分析质体稳态后振动系统内部受力变化, 对于质体系统, 根据达朗伯原理, 受力合力为零. 其中恢复力${F_k}(t)$和偏心激励力$F(t)$分别为
$$ {F_k}(t) = kx(t),F(t) = - ml\left[ {\ddot \theta (t)\sin \theta (t) + {{\dot \theta }^2}(t)\cos \theta (t)} \right] $$ (17) 从频率结果可以得出, 质体稳态位移响应$x(t)$是由一系列倍频谐波叠加而成, 因此使用傅里叶级数拟合$x(t)$, 进而由式(17)得出${F_k}(t)和F(t)$, 由于输入电压0 ~ 5 V时系统响应与理想系统结果一致, 因此这里考虑在输入电压为5 ~ 100 V时, ${F_k}(t)$和$F(t)$各频率谐波分量的幅值情况如图13和图14所示.
从图13和图14中质体稳态时所受恢复力${F_k}(t)$和激励力$F(t)$的各频率分量情况可以看出, 在对应4个电压段内, ${F_k}(t)$的5倍频分量$5{f_0}$和3倍频分量$3{f_0}$由于系统谐振分别在电压6.6 V和11 V时出现局部峰值, 而整个过程中质体稳态位移都是基频分量$1{f_0}$占主导. 对于$F(t)$来说, 同样在电压6.6 V和11 V由于谐振幅值出现峰值, 在第3阶段质体稳态出现偶数分量并在65.4 V时2倍频分量$2{f_0}$超过基频分量$1{f_0}$, 而在第4阶段偶数倍频分量消失, 基频分量$1{f_0}$占主导.
因此在Sommerfeld效应发生的过程中, 即输入电压0 ~ 82.4 V增加过程中, 由于系统耦合, 偏心激励力输入$F(t)$呈多频, 并且对应平均转速的基频分量随电压缓慢增加, 高频分量快速增加, 高频输入对低频质体振动相当于参数激励, 且频率比从1:1附近逐渐变为2:1附近, 最后在输入电压大于临界电压82.4 V后质体系统脱离谐振.
4. 结 论
将直流电机动力学模型引入转子模型, 在充分考虑电机与转子系统相互耦合作用下, 利用仿真得出了系统响应较精确数值解, 在此基础上通过理论和仿真实验分析得出以下结论.
(1)理想驱动情况下, 电机转速由输入电压线性控制, 稳态后质体受力幅值和相位关系固定, 非理想系统耦合后转子转速受质体谐振频率影响出现平均转速锁定、非线性跳跃和多值现象, 同时选择合适振动系统参数可以抑制这一现象.
(2)由于系统耦合, 相比理想系统, 电机转速呈周期波动, 依据转速波动特征将Sommerfeld效应划分为萌发、发展、剧烈和消失4个阶段. 此外质体系统稳态响应呈多频, 在Sommerfeld效应剧烈作用时突然出现偶数倍频分量, 在Sommerfeld效应突然消失时偶数倍频分量也会突然消失.
(3)在质体系统进入共振区并且受力出现偶数倍频成分后, 质体对应所受外激励力逐渐以2倍频分量为主, 而位移响应以基频为主, 此时高频输入对低频质体振子相当于参数激励, 且频率比接近为2:1, 因此转子转速快转、停转乃至反转并在周期内剧烈交替切换, 最终在大于临界电压后伴随电机转速跳跃脱离谐振.
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图 6 转子平均转速$\bar \omega $($m$ = 20 kg)
Figure 6. The average of the rotor speed$\bar \omega $($m$ = 20 kg)
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图 9 转子稳态转速和${\omega _n}$对比(从上往下对应stage1 ~ stage4)
Figure 9. Steady-state rotor speed and${\omega _n}$(stage1 ~ stage4)
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图 11 响应对应频谱(从上往下对应stage1 ~ stage4)
Figure 11. Self-power spectral density of response (stage1 ~ stage4)
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图 12 稳态位移(从上往下对应stage1 ~ stage4)
Figure 12. Steady-state displacement (stage1 ~ stage4)
表 1 系统参数
Table 1 System parameters
Parameters Value Parameters Value $m/{\text{kg}}$ 20 $l/{\text{m}}$ 0.08 $M/{\text{kg}}$ 200 ${R_a}/{{\Omega }}$ 0.57 $c/({\text{N}} \cdot {\text{s}}\cdot {\mathrm{m}}^{-1})$ 235.57 ${k_m}/({\text{N}} \cdot {{{\mathrm{m}}\cdot{\mathrm{A}}^{-1}}})$ 0.57 ${c_r}/({\text{N}} \cdot {\text{m}} \cdot {{{\mathrm{s}}\cdot{\mathrm{rad}}^{-1}}})$ 0.03 ${L_a}/{\text{H}}$ 0.057 $k/({{{\mathrm{N}}\cdot{\mathrm{m}}^{-1}}})$ 630500 ${u_a}/{\text{V}}$ 0:0.1:100 ${k_e}/({\text{N}} \cdot {{{\mathrm{s}}\cdot{\mathrm{rad}}^{-1}}})$ 0.57 
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