本科毕业（设计）论文开题报告学生姓名赵孝威学号07B********指导教师吴元生系别机电工程系专业机械设计制造及其自动化交稿日期教务处制一、开题报告二、阅读文献目录三、文献综述注意：学生阅读文献后，必须写出3000字左右的综述，作为开题内容之一。
（可增页）电动绞车简介：电动绞车广泛用于工作繁重和所需牵引力较大的场所。
单卷筒电动绞车的电动机经减速机带动卷筒，电动机与减速器输入轴之间装有制动器。
为适应提升、牵引和回转等作业的需要，还有双卷筒和多卷筒装置的绞车。
一般额定载荷低于10T的绞车可以设计成电动绞车结构组成：卷筒，减速机，制动器，电气系统。
1、卷筒设计的基本原则在右面的图表中，弹簧卷筒“水平放线”和“垂直放线”的两种应用方式都有表示。
横向电缆安装：- 可以向1或2个方向出线。
- 电缆在连续平面上拖拽或用间隔小于1m的支撑将电缆撑离地面。
-安装高度从电缆拖拽平面到卷筒中心不超过1m。
- 不考虑电缆转向- 运行速度为：10米/分钟--60米/分钟-最大加速度可达0.3 m/s²垂直电缆安装：- 卷筒安装在顶端- 电缆出现方向垂直向下-运行速度为：10米/分钟--40米/分钟-最大加速度可达0.3 m/s²-自由下垂的电缆长度L0不考虑。
三、卷筒和电缆的选择如果应用在如下极端特殊的环境，强烈建议跟我们联系：-运行速度小于10 m/min- 强烈震动- 应用在海上，盐或者腐蚀性空气中-温度低于-15 °C - 急弯路径- 需要强制导向（见下栏）如果安装了强制导向器必须保证导向器与卷筒之间的距离为卷筒宽度的6倍。
弹簧卷筒必须安装在电缆可以自由放出和卷起，并且没有障碍物的位置上。
强制导向和尺寸太小的导向轮在任何情况下都不允许的。
四、文献翻译图1.三相电动机的图解模型用q–d框架参考转换建立了一个三相步进电机的数学模型。
下面给出了三相绕组电压方程v a = Ria+ L*dia/dt − M*dib/dt − M*dic/dt + dλpma/dt ,v b = Rib+ L*dib/dt − M*dia/dt − M*dic/dt + dλpmb/dt ,v c = Ric+ L*dic/dt − M*dia/dt − M*dib/dt + dλpmc/dt , (1)其中R和L分别是相绕组的电阻和感应线圈，并且M是相绕组之间的互感线圈。
λpma , λpmband λpmc是应归于永磁体的相的磁通，且可以假定为转子位置的正弦函数如下λpma = λ1sin(Nθ),λpmb = λ1sin(Nθ− 2/3),λpmc = λ1sin(Nθ - 2/3), (2)其中N是转子齿数。
本文中强调的非线性由上述方程所代表，即磁通是转子位置的非线性函数。
使用Q ,d转换，将参考框架由固定相轴变换成随转子移动的轴（参见图2）。
矩阵从a,b,c 框架转换成q,d框架变换被给出了[8](3)例如，给出了q,d参考里的电压(4)在a,b,c参考中，只有两个变量是独立的（ia + ib+ ic= 0），因此，上面提到的由三个变量转化为两个变量是允许的。
在电压方程（1）中应用上述转换，在q,d框架中获得转换后的电压方程为v q = Riq+ L1*diq/dt + NL1idω + Nλ1ω,v d = Rid+ L1*did/dt − NL1iqω, (5)注意：请将外文文献原文复印件附在后面。
Oscillation, Instability and Control of Stepper MotorsLIYU CAO and HOWARD M. SCHWARTZDepartment of Systems and Computer Engineering, Carleton University, 1125 Colonel By Drive,Ottawa, ON K1S 5B6, Canada(Received: 18 February 1998; accepted: 1 December 1998)Abstract. A novel approach to analyzing instability in permanent-magnet stepper motors is presented. It is shown that there are two kinds of unstable phenomena in this kind ofmotor: mid-frequency oscillation and high-frequency instability. Nonlinear bifurcation theory is used to illustrate the relationship between localinstability and midfrequencyoscillatory motion. A novel analysis is presented to analyze the loss of synchronism phenomenon, which is identified as high-frequency instability. The concepts of separatrices and attractors in phase-space are used to derive a quantity to evaluate the high-frequency instability. By using this quantity one can easily estimate the stability for high supply frequencies. Furthermore, a stabilization method is presented. A generalized approach to analyze the stabilization problem based on feedback theory is given. It is shown that the mid-frequency stabilityand the high-frequency stability can be improved by state feedback. Keywords: Stepper motors, instability, nonlinearity, state feedback.1. IntroductionStepper motors are electromagnetic incremental-motion devices which convert digital pulse inputs to analog angle outputs. Their inherent stepping ability allows for accurate position control without feedback. That is, they can track any step position in open-loop mode, consequently no feedback is needed to implement position control. Stepper motors deliver higher peak torque per unit weight than DC motors; in addition, they are brushless machines and therefore require less maintenance. All of these properties have made stepper motors a very attractive selection in many position and speed control systems, such as in computer hard disk drivers and printers, XY-tables, robot manipulators, etc.Although stepper motors have many salient properties, they suffer from an oscillation or unstable phenomenon. This phenomenon severely restricts their open-loop dynamic performance and applicable area where high speed operation is needed. The oscillation usually occurs at stepping rates lower than 1000 pulse/s, and has been recognized as a mid-frequency instability or local instability [1], or a dynamic instability [2]. In addition, there is another kind of unstable phenomenon in stepper motors, that is, the motors usually lose synchronism at higher stepping rates, even though load torque is less than their pull-out torque. This phenomenon is identified as high-frequency instability in this paper, becauseit appears at much higher frequencies than the frequencies at which themid-frequency oscillation occurs. The high-frequency instability has not been recognized as widely as mid-frequency instability, and there is not yet a method to evaluate it.Mid-frequency oscillation has been recognized widely for a very long time, however, a complete understanding of it has not been well established. This can be attributed to the nonlinearity that dominates the oscillation phenomenon and is quite difficult to deal with.384 L. Cao and H. M. SchwartzMost researchers have analyzed it based on a linearized model [1]. Although in many cases, this kind of treatments is valid or useful, a treatment based on nonlinear theory is needed in order to give a better description on this complex phenomenon. For example, based on a linearized model one can only see that the motors turn to be locally unstable at some supplyfrequencies, which does not give much insight into the observed oscillatory phenomenon. In fact, the oscillation cannot be assessed unless one uses nonlinear theory.Therefore, it is significant to use developed mathematical theory on nonlinear dynamics to handle the oscillation or instability. It is worth noting that Taft and Gauthier [3], and Taft and Harned [4] used mathematical concepts such as limit cycles and separatrices in the analysis of oscillatory and unstable phenomena, and obtained some very instructive insights into the socalled loss of synchronous phenomenon. Nevertheless, there is still a lack of a comprehensive mathematical analysis in this kind of studies. In this paper a novel mathematical analysis is developed to analyze the oscillations and instability in stepper motors.The first part of this paper discusses the stability analysis of stepper motors. It is shown that the mid-frequency oscillation can be characterized as a bifurcation phenomenon (Hopf bifurcation) of nonlinear systems. One of contributions of this paper is to relate the midfrequency oscillation to Hopf bifurcation, thereby, the existence of the oscillation is provedtheoretically by Hopf theory. High-frequency instability is also discussed in detail, and a novel quantity is introduced to evaluate high-frequency stability. This quantity is very easyto calculate, and can be used as a criteria to predict the onset of thehigh-frequency instability. Experimental results on a real motor show the efficiency of this analytical tool.The second part of this paper discusses stabilizing control of stepper motors through feedback. Several authors have shown that by modulating the supply frequency [5], the midfrequencyinstability can be improved. In particular, Pickup and Russell [6, 7] have presented a detailed analysis on the frequency modulation method. In their analysis, Jacobi series was used to solve a ordinary differential equation, and a set of nonlinear algebraic equations had to be solved numerically. In addition, their analysis is undertaken for a two-phase motor, and therefore, their conclusions cannot applied directly to our situation, where a three-phase motor will be considered. Here, we give a more elegant analysis for stabilizing stepper motors, where no complex mathematical manipulation is needed. In this analysis, a d–q model of stepper motors is used. Because two-phase motors and three-phase motors have the same q–d model and therefore, the analysis is valid for both two-phase and three-phase motors. Up to date, it is only recognized that the modulation method is needed to suppress the midfrequency oscillation. In this paper, it is shown that this method is not only valid to improve mid-frequency stability, but also effective to improve high-frequency stability.2. Dynamic Model of Stepper MotorsThe stepper motor considered in this paper consists of a salient stator with two-phase or threephase windings, and a permanent-magnet rotor. A simplified schematic of a three-phase motor with one pole-pair is shown in Figure 1. The stepper motor is usually fed by a voltage-source inverter, which is controlled by a sequence of pulses and produces square-wave voltages. Thismotor operates essentially on the same principle as that of synchronous motors.One of major operating manner for stepper motors is that supplying voltage is kept constant and frequencyof pulses is changed at a very wide range. Under this operating condition, oscillation and instability problems usually arise.Figure 1. Schematic model of a three-phase stepper motor.A mathematical model for a three-phase stepper motor is established using q–d framereference transformation. The voltage equations for three-phase windings are given byv a = Ria+ L*dia/dt − M*dib/dt − M*dic/dt + dλpma/dt ,v b = Rib+ L*dib/dt − M*dia/dt − M*dic/dt + dλpmb/dt ,v c = Ric+ L*dic/dt − M*dia/dt − M*dib/dt + dλpmc/dt ,where R and L are the resistance and inductance of the phase windings, and M is the mutual inductance between the phase windings. _pm a, _pm b and _pm c are the flux-linkages of thephases due to the permanent magnet, and can be assumed to be sinusoid functions of rotor position _ as followλpma = λ1sin(Nθ),λpmb = λ1sin(Nθ− 2/3),λpmc = λ1sin(Nθ - 2/3),where N is number of rotor teeth. The nonlinearity emphasized in this paper is represented by the above equations, that is, the flux-linkages are nonlinear functions of the rotor position.By using the q; d transformation, the frame of reference is changed from the fixed phase axes to the axes moving with the rotor (refer to Figure 2). Transformation matrix from the a; b; c frame to the q; d frame is given by [8]For example, voltages in the q; d reference are given byIn the a; b; c reference, only two variables are independent (ia C ib C ic D 0); therefore, the above transformation from three variables to two variables is allowable. Applying the abovetransformation to the voltage equations (1), the transferred voltage equation in the q; d frame can be obtained asv q = Riq+ L1*diq/dt + NL1idω + Nλ1ω,v d =Rid+ L1*did/dt − NL1iqω, (5)Figure 2. a, b, c and d, q reference frame.where L1 D L C M, and ! is the speed of the rotor.It can be shown that the motor’s torque has the following form [2]T = 3/2Nλ1i qThe equation of motion of the rotor is written asJ*dω/dt = 3/2*Nλ1iq− Bfω– Tl ,where Bf is the coefficient of viscous friction, and Tl represents load torque, which is assumed to be a constant in this paper.In order to constitute the complete state equation of the motor, we need another state variable that represents the position of the rotor. For this purpose the so called load angle _ [8] is usually used, which satisfies the following equationDδ/dt = ω−ω,where !0 is steady-state speed of the motor. Equations (5), (7), and (8) constitute the statespace model of the motor, for which the input variables are the voltages vq and vd. As mentioned before, stepper motors are fed by an inverter, whose output voltages are not sinusoidal but instead are square waves. However, because the non-sinusoidal voltages do not change the oscillation feature and instability very much if compared to the sinusoidal case (as will be shown in Section 3, the oscillation is due to the nonlinearity of the motor), for the purposes of this paper we can assume the supply voltages are sinusoidal. Under this assumption, we can get vq and vd as followsv q = Vmcos(Nδ) ,v d = Vmsin(Nδ) ,where Vm is the maximum of the sine wave. With the above equation, we have changed the input voltages from a function of time to a function of state, and in this way we can represent the dynamics of the motor by a autonomous system, as shown below. This will simplify the mathematical analysis.From Equations (5), (7), and (8), the state-space model of the motor can be written in a matrix form as followsẊ = F(X,u) = AX + Fn(X) + Bu , (10) where X D T iq id ! _U T , u D T!1 Tl U T is defined as the input, and !1 D N!0 is the supply frequency. The input matrix B is defined byThe matrix A is the linear part of F._/, and is given byFn.X/ represents the nonlinear part of F._/, and is given byThe input term u is independent of time, and therefore Equation (10) is autonomous.There are three parameters in F.X;u/, they are the supply frequency !1, the supply voltage magnitude Vm and the load torque Tl . These parameters govern the behaviour of the stepper motor. In practice, stepper motors are usually driven in such a way that the supply frequency !1 is changed by the command pulse to control the motor’s speed, while the supply voltage is kept constant. Therefore, we shall investigate the effect of parameter !1.3. Bifurcation and Mid-Frequency OscillationBy setting ! D !0, the equilibria of Equation (10) are given asand ' is its phase angle defined byL1/R) . (16)φ = arctan(ω1Equations (12) and (13) indicate that multiple equilibria exist, which means that these equilibria can never be globally stable. One can see that there are two groups of equilibria as shown in Equations (12) and (13). The first group representedby Equation (12) corresponds to the real operating conditions of the motor. The second group represented by Equation (13) is always unstable and does not relate to the real operating conditions. In the following, we will concentrate on the equilibria represented by Equation (12).。
