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1、英 文 翻 译系 别专 业班 级学生姓名学 号指导教师 Control of suspension bridge flutter instability using pole-placement techniqueAbstrat The closed-loop state feedback control scheme by pole-placement technique, which is widely used in control literature, is applied to control the flutter instability of suspension bridge
2、s. When the mean wind speed U at the bridge site increases beyond the critical flutter wind speed, the real part of the dominant pole of the system is forced to a desired negative value by properly designing a state feed back gain matrix to control the flutter instability. The control force, which i
3、s expressed as a product of gain matrix and state vector in modal coordinates, is applied in the form of an active torsional moment in the middle of the bridge span. The values of the state variables are estimated by designing a full order observer system. The application of the control scheme for i
4、ncreasing the critical wind speed for flutter of suspension bridges is demonstrated by considering the Vincent Thomas Bridge as the numerical example. The efficiency of the method for controlling the bridge deck flutter is investigated under a set of parametric variations. The results of the numeric
5、al study show that the control scheme using pole-placement technique effectively brings down the divergent oscillation of the bridge at wind speeds greater than the critical wind speed for flutter, to almost zero value within few seconds. 2004 Elsevier Ltd. All rights reserved.1. Introduction Long-s
6、pan suspension bridges, due to their flexibility and lightness, are much prone to the flutter instability. Flutter is a wind-induced instability in the bridge deck at a critical wind velocity leading to an exponentially growing response. The evaluation of flutter condition of suspension bridges is o
7、ne of the most important phases in the design of these bridges. In recent years, many researchers have focused their attention on increasing the critical flutter wind speed of the cable-supported bridges using different types of control devices. Wilde et al. proposed a passive aerodynamic control of
8、 flutter by adding two additional surfaces to generate stabilizing forces and by putting an additional pendulum to control the torsional motion. Other studies also have been carried out to control the critical flutter wind speed of long-span bridges using eccentric mass on the bridges. In 2002, the
9、authors proposed a passive control of critical flutter wind speed of suspension bridges using a combined vertical and torsional tuned mass damper (TMD)system. The proposed TMD system has two degrees of freedom, which are tuned close to the frequencies corresponding to vertical and torsional symmetri
10、c modes of the bridge, which get coupled during flutter. Even with these advances, challenges still exist in increasing the critical flutter wind speed by applying reasonable external devices such as active control force or passive control properties, and trying to find practical methods to control
11、the flutter condition of these bridges. In particular, application of pole-placement technique for control of bridge vibration is much less compared to that of optimal control theory . Meirovitch and Ghosh used modal control for suppressing the suspension bridge flutter, but they essentially used op
12、timal control theory in modal space. Since control of bridge flutter is associated with the problem of making the system stable from an unstable state, the pole-placement technique should find as good application for this problem. In this paper, the closed-loop state feedback control method by pole-
13、placement technique, which has been used for other control problems, is applied to stabilize the flutter instability condition of suspension bridges. For this purpose, the equation of motion of the system is obtained by multi-mode finite element modeling (beam element) of the bridge deck using consi
14、stent mass matrix. The consistent mass matrix and structural stiffness matrix are evaluated using energy approach, which duly considers the effects of suspended cables. The final controlled equation of motion of the system is obtained in states space in terms of the generalized modal coordinate vect
15、or. The control force w is considered proportional to the values of the state vector, which are estimated by designing a full-order observer system. The control scheme is applied to suppress the flutter instability of Vincent Thomas Bridge and its effectiveness for flutter control of suspension brid
16、ge is investigated for different mean wind speeds through a numerical study. 2. Assumptions The following assumptions are made in the analysis: (1) All stresses in the bridge elements obey the Hookes law, and therefore no material nonlinearity is considered. (2) The control force and the output of t
17、he system are considered as scalar quantities. (3 )The initial dead load is carried by cables without causing any stress in the suspended structure. (4) Small defection theory is applied to obtain the dynamic equation of motion of the suspension bridge.3. Equation of motion of the bridge The equatio
18、n of motion of the system is obtained by multi-mode finite element method in time domain using the energy approach and applying the Hamiltons principle. For this purpose, the entire bridge is discretized into two-dimensional beam elements, each consisting of two nodes at its ends. At each node four
19、degrees of freedom, namely vertical displacement;bending rotat ; torsiona rotation; and warping displacement ; as shown in Fig.1, are considered. The governing equation of motion for flutter can be written as (3.1) In one of the experimental studies on model bridge cross-section Singh et al. showed
20、that there is a strong possibility of the dependence of lift on lateral (sway) motion. As a result, the lateral motion may significantly affect flutter wind speed if verticallateral or verticallateraltorsional mode of vibration predominantly governs the critical flutter wind speed. Since literature
21、on the effect of lateral motion in modifying the critical wind speed for flutter of long span bridges in quantifiable terms is scanty, the present study considers only the verticaltorsional flutter condition for the control problem. If lateral motion is also included in the flutter problem, the pole
22、 placement technique can be easily extended for controlling such flutter condition.The forces obtained by Eqs. (3a) and (3b) are considered to be constant along the element. In order to evaluate the aeroelastic force vector F in Eq. (3.1), the distributed aeroelastic forces are lumped at the element
23、 nodes as shown in Figs. 2(c) and (d). The mass and stiffness matrices in Eq. (1) are considered for double symmetric bridge deck and may be expressed as (3.2) (3.3) whereand are the mass and stiffness matrices, respectively in bending vibration, and andare those of the torsional vibration. Note tha
24、t there is no modal coupling between the vertical and torsional modes of vibration in linear analysis for doubly symmetric bridge decks . Using the total potential and kinetic energies of the bridge and applying the Hamiltons principle, the structural mass and stiffness matrices for vertical and tor
25、sional motions of the system can be evaluated The structural damping matrix C is assumed to be proportional to mass and stiffness matrices. Using Eqs. (3a) and (3b) and lumping of the aeroelastic forces at the element nodes the (4n1) Aeroelastic force vector F can be expressed in the form (3.4)where
26、 x is defined in Eq. , and matrices and are given in the Appendix A. Substituting Eq. (3.4) into Eq. (3.1), the final equation of motion can be expressed as (3.5)It is convenient to solve problem of flutter control in modal space as carried out by Meirovitch and Ghosh . By considering the displaceme
27、nt vector x in terms of modal matrix and generalized modal coordinate vector as （3.6）and using a standard modal transformation, the ith modal equation can be obtained as (3.7) m is the number of modes considered, is the ith modal mass, and and are the elements of the matrices D and E defined as and
28、(3.8) Eq. (3.7) can be written in a matrix form as （3.9）in which I is the identity matrix of order m, P and Q are the square matrices of size mm for which the elements can be defined as （3.10） （3.11） Finally, by choosing the modal coordinates as the state variables, the state equation can be written
29、 in the standard state-space form as follows （3.12）in which the (2m1) state vector z is defined as （3.13）The state matrix A, and the input matrix B are, respectively, given by （3.14）where 0 and I are the zero and identity matrices of order mm; respectively; and the other matrices previously are defi
30、ned.4. Design of control system by pole placement A ording to the second method of Liapunov stability analysis, for a system represented by Eq. (3.12), the eigenvalues of the system need to be investigated. For this purpose, the eigenvalues of the following equation is obtained: （3.15）If all eigenva
31、lues of matrix A have negative real part, the system is asymptotically stable. Further, a necessary and sufficient condition for all eigenvalues to have negative real part is that they have determinant with positive coefficient of the leading terms of their characteristic polynomial. The eignvalues
32、of the matrix A are called the poles of the original system. The stability of a linear closed-loop system can be determined from the location of the closed-loop poles in the complex s plane, in which s are the poles of the system. If any of these poles lie on the right half of the s plane, then the
33、system is unstable. Therefore, closed-loop poles on the right half of s plane are not permissible in the usual linear control system. If all closed-loop poles lie to the left half of s plane, then the system is stable. Whether a linear system is stable or unstable is a property of the system itself
34、and does not depend on the input of the system. Thus, the problem of absolute stability can be solved readily by choosing no closed-loop poles on the right half of plane, including the io-axis. Mathematically, closed-loop poles on the io-axis will yield oscillations, the amplitude of which is neithe
35、r decaying nor growing with time. If dominant complex-conjugate closed-loop poles lie close to the io-axis, the transient response exhibits excessive oscillations or it may be very slow. Therefore, to guarantee fast, yet well-damped, transient response characteristics, it is necessary that the close
36、d- loop poles of the system lie in a particular region far away from the io-axis. For a second-order closed-loop control system, it is shown that the poles can be written as （3.16）where is the damping ratio, is the undamped natural frequency (rad/s), and is the damped natural frequency. Without any
37、loss of generality, for a multi-dof second-order closed-loop control system also, the poles can be written in the above form in terms of the modal damping ratio and natural frequency. 使用极点配置技术控制悬索桥颤振失稳摘要闭环状态反馈的极点配置技术，可以广泛应用于控制体制控制文学，适用于控制悬索桥颤振失稳。当在桥址的平均u速度增加到超过颤振临界风速时，通过适当设计一个状态反馈增益矩阵去控制颤振失稳会迫使占系统支配
38、地位的主极点实部达到一个理想的负值状态。控制力，它表示为一个增益矩阵和状态矢量坐标模态的产品，是适用于一个桥跨中间的积极扭矩的形式。状态变量的价值是通过设计一个完整的订单观察员制度来评估的。那个提高悬索桥颤振的临界风速管制计划的申请文被森托马斯大桥的数值例子给论证了。控制桥面颤振方法效率是根据设定的一系列参数变化进行研究的。该数值研究结果表明利用极点配置技术控制体制有效地造成桥在几乎零值几秒钟之内风速比颤振临界风速较大时的发散震荡 。 2004年Elsevier有限公司。保留所有权利。1 导言大跨度悬索桥由于其柔韧性和重量小非常容易遭受颤振失稳。颤振是当一个关键的风速指数增长时在桥面引起的失稳
39、 。悬索桥颤振条件评价是这些桥梁的设计过程中最重要的阶段之一。 近年来，许多研究人员把他们的注意力集中在提高当使用不同类型的控制装置时索支承桥梁的颤振临界风速。王尔德等提出了通过添加两个额外的表面产生稳定的力量气动控制被动的颤振，并通过安置一个额外的钟摆去控制扭动。其他研究也已开展了以控制使用偏心质量的大跨度桥梁的颤振临界风速。2002年，作者提出了一个使用联合纵向和扭转调谐质量阻尼器系统（TMD）被动控制悬索桥颤振风速。提议的调谐质量阻尼器系统（TMD）有两个自由度， 它调整的频率与桥梁的纵向和扭转对称模式接近，颤振期间得到耦合。 即使有这些进步，挑战仍然存在于运用合理的外部设备诸如主动控制
40、力量或被动控制性能来提高颤振临界风速，并试图找到切实可行的方法来控制这些桥梁的颤振条件。特别是，对桥梁振动控制的极点配置技术的应用与最优控制理论相比要少得多。迈罗维奇和戈什采用模态控制来抑制悬索桥颤振，但他们在模态空间主要利用最优控制理论。由于桥梁颤振控制与使系统从一个不稳定到稳定状态的问题有关，极点配置技术应该找到这个问题的良好应用。在这个文件中，闭环状态反馈的极点配置技术，已用于其他控制问题，采用控制方法，适用于稳定悬索桥颤振失稳状态。因此，该系统的运动方程，是由桥面使用一致质量矩阵的多模有限元模型（梁单元）得来的。一致质量矩阵和结构刚度矩阵用使用能源的方式进行评估，这充分的考虑了悬索的影
41、响。最后的控制系统的运动方程，由状态空间在广义模态坐标向量获得的。控制力w被认为是与状态向量的值成正比的，这是通过设计一个全维观测系统进行估计。控制体制是用于抑制文森托马斯大桥颤振失稳及它中止桥梁的颤振控制的效力是通过数值研究不同的平均风速调查的。2. 假设 以下假设作出的分析：（1）桥梁中的所有的应力元素都服从胡克定律，因此材料都被认为是线性特性。（2）控制力和系统的输出被视为标量数量。（3）初始的恒载由钢索承载，而不会对悬挂结构造成任何压力。（4）小叛逃理论应用于取得悬索桥的动力学方程议案。3. 桥梁的运动方程 该系统的运动方程是多模态有限元法在时域里利用能源的方法和应用Hamilton原
42、理得到的。因此，整个大桥被离散成二维梁单元，在它的每个端部都有两个节点组成，在每个节点4个自由度，叫做垂直位移弯曲旋转扭转 和翘曲位移。如图1所示，占支配地位的颤振运动方程可以写为 （3.1）辛格等人的桥梁断面模型的实验研究之一表明，横向位移的发生很有可能依赖于升降机，因此，如果以垂直横向或纵向横向扭振型为主的模式控制颤振临界风速，横向运动可能会严重影响颤振风速。由于对大跨度桥梁颤振的量化计算在修改临界风速对横向运动的影响的文献是不足的，目前的研究考虑的只有垂直扭转颤振条件的控制问题，如果横向移动也包括在颤振问题中，那么极点配置技术就可以很容易扩展为控制此类颤振得条件。 由均衡器获得的力量(3
43、a)和(3b) 被认为是沿构件不变。为了评估气动弹性力向量（F在方程3.1中）， 分布式气动弹性力集中在如图2(c)和(d)所示的构件节点处。 方程1中的质量和刚度矩阵被认为是双层对称桥面并可以表示为： （3.2） （3.3） 其中和在弯曲振动中分别是质量和刚度矩阵，并且和都是扭振的。请注意，在对双对称桥面的线性分析的纵向和扭转模式之间没有模态耦合，利用桥的总潜力和动能以及应用汉密尔顿原则，结构质量和刚度矩阵对于该系统的纵向和扭转运动是可以被评价出来的。结构阻尼矩阵被假定为与质量和刚度矩阵成比例的。 使用均衡器(3a)和(3b)和把构件节点的气动弹性力结成块 (4n1) 气弹力向量F可用以下形
44、式表达： (3.4)其中（x）是定义均衡器Eq. 和矩阵和 是载列于附录A，把方程(3.4)代入方程(3.1), 最后的运动方程可以表示为 (3.5) 这是方便的解决颤振控制模态空间问题 由迈罗维奇和戈什实行的。通过考虑在模态矩阵的条件下的位移向量（x）和广义模态坐标向量关系可表示如下 （3.6） 并且使用标准模式的转变，第i个模态方程可为（3.7） m是模式考虑的数量, 是第i个模态质量，而且和都是矩阵 D和E 的元素可以被定义为 and (3.8) 方程(3.7)用矩阵形式表达如下 （3.9）这I是m阶矩阵，P和Q是mm的方形矩阵 ，这些元素可以被定义为 （3.10） （3.11）最后，选
45、择模态坐标作为状态变量，状态方程可以在标准状态空间里表达形式如下 （3.12）其中(2m1)状态向量 z被定义如下 （3.13）状态矩阵A与输入矩阵B依次表示如下 (3.14) 其中0和I是mm的零矩阵和恒等矩阵；依次地，其他矩阵可以在前面被定义。4.用极点配置技术控制系统的设计 作为对Liapunov稳定性分析的第二个方法，以方程式（3.12）为代表的系统，该系统的特征值需要研究。为此，下列公式得到的特征值： （3.15）如果所有的矩阵特征值A有负实数部分，该系统是渐近稳定的。此外，一个让所有特征值有负实数部分的必要和充分条件是他们有决定其特征多项式的主导条件的的正系数。矩阵A的特征值被称为原系统的极点。一个线性闭环系统的稳定性 可以从复杂的S平面闭环极点的位置来确定，其中S是系统的极点。如果这些极点取于S平面的右半部分，那么系统是不稳定的。因此，S平面右半部分的闭环极点是不允许在平常的线性控制系统。如果所有的闭环极点都取
