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294 Vibration Control, system This approach has been employed for parameter and signal estimation in nonlinear. and linear vibrating mechanical systems where numerical simulations and experimental. results show that the algebraic identification provides high robustness against parameter. uncertainty frequency variations small measurement errors and noise Beltr n et al 2005. Beltr n et al 2006 In addition algebraic identification is combined with integral. reconstruction of time derivatives of the output GPI Control using a simplified. mathematical model of the system where some nonlinear effects stiffness and friction. were neglected in spite of that the experimental results show that the estimated values. represent good approximations of the real parameters and high performance of the. proposed active vibration control scheme which means that the algebraic identification and. GPI control methodologies could be used for some industrial applications when at least a. simplified mathematical model of the system is available Beltr n et al 2005. Some numerical simulations and experiments are included to show the unbalance. compensation properties and robustness when the rotor is started and operated over the. first critical speed,2 System description,2 1 Mathematical model. The Jeffcott rotor system consists of a planar and rigid disk of mass m mounted on a flexible. shaft of negligible mass and stiffness k at the mid span between two symmetric bearing. supports see figure 1 a when a b Due to rotor unbalance the mass center is not located. at the geometric center of the disk S but at the point G center of mass of the unbalanced. disk the distance u between these two points is known as disk eccentricity or static. unbalance Vance 1988 Dimarogonas 1996 An end view of the whirling rotor is also. shown in figure 1 b with coordinates that describe its motion. In our analysis the rotor bearing system is modeled as the assembly of a rigid disk flexible. shaft and two ball bearings This system differs from the classical Jeffcott rotor because the. effective shaft length can be increased or decreased from its nominal value In fact this. adjustment is obtained by enabling longitudinal motion of one of the bearing supports right. bearing in figure 1 a to different controlled positions into a small interval by using some. servomechanism which provides the appropriate longitudinal force With this simple. approach one can modify the shaft stiffness moreover one can actually control the rotor. natural frequency during run up or coast down to evade critical speeds or at least reduce. rotor vibration amplitudes Our methodology combines some ideas on variable rotor. stiffness Sandler 1999 and rotor acceleration scheduling Millsaps 1998 but completing. the analysis and control for the Jeffcott like rotor system. The rotor bearing system is not symmetric when the position of the right bearing changes. from its nominal value i e a b l 2 for the Jeffcott rotor l 0. For simplicity the following assumptions are considered flexible shaft with attached disk. gravity loads neglected insignificant when compared with the actual dynamic loads. equivalent mass for the base bearing mb linear viscous damping cb between the bearing base. and the linear sliding force actuator to control the shaft stiffness F angular speed. controlled by means of an electrical motor with servodrive and local. Proportional Integral PI controller to track the desired speed scheduling in presence of. small dynamical disturbances The mathematical model of the four degree of freedom. Jeffcott like rotor is obtained using Newton equations as follows. www intechopen com, Active Vibration Control of Rotor Bearing Systems 295. Fig 1 Rotor bearing system a Schematic diagram of a rotor bearing system with one. movable right bearing and b end view of the whirling rotor. mx cx kx m u sin u 2 cos 1,my cy ky m u 2 sin u cos 2. J z mu2 c p m xu,mb b c b b F 4, where k and c are the stiffness and viscous damping of the shaft Jz is the polar moment of.

inertia of the disk and t is the applied torque control input for rotor speed regulation In. is the rotor angular velocity The coordinate b denotes the position of the movable. addition x and y denote the orthogonal coordinates that describe the disk position and. right bearing which is controlled by means of the control force F t servomechanism. In our analysis the stiffness coefficient for the rotor bearing system is given by Rao 2004. 3EIl a 2 ab b 2,be controlled I 64, where l a b is the total length of the rotor between both bearings with b the coordinate to. D4 is the moment of inertia of a shaft of diameter D and E is the Young s. modulus of elasticity E 2 11 1011 N m2 for AISI 4140 steel The natural frequency of the. rotor system is then obtained as follows Rao 2004, In such a way that controlling b by means of the control force F one is able to manipulate n. to evade appropriately the critical speeds during rotor operation. The proposed control objective is to reduce as much as possible the rotor vibration. amplitude denoted in adimensional units by, for run up coast down or steady state operation of the rotor system even in presence of. small exogenous or endogenous disturbances Note however that this control problem is. www intechopen com,296 Vibration Control, quite difficult because of the 8th order nonlinear model many couplings terms. underactuation and uncontrollability properties from the two control inputs F. 3 Active vibration control,3 1 Speed control with trajectory planning.

In order to control the speed of the Jeffcott like rotor system consider equation 3 under. the temporary assumption that the eccentricity u is perfectly known and that c 0 to. reference trajectories of speed t and acceleration scheduling t for the rotor. simplify the analysis Then the following local PI controller is designed to track desired. J z v1 c kux sin kuy cos,v1 t t 0 t dt,tracking error e1 t. The use of this controller yields the following closed loop dynamics for the trajectory. e1 1 e 1 0 e1 0 9, Therefore selecting the design parameters 1 0 so that the associated characteristic. polynomial for equation 9,p s s 2 1s 0, is a Hurwitz polynomial one guarantees that the error dynamics is asymptotically stable. The prescribed speed and acceleration scheduling for the planned speed trajectory is given. t t ti t f f ti t t f, where i and f are the initial and final speeds at the times ti and tf respectively passing. through the first critical frequency and t ti tf is a B zier polynomials with t ti tf 0 and. t ti tf 1 described by,t t t t t t,t ti t f t ti 3 6.

t f ti t f ti t f ti t f ti, with 1 252 2 1050 3 1800 4 1575 5 700 6 126 in order to guarantee a. sufficiently smooth transfer between the initial and final speeds. The fundamental problem with the proposed feedback control in equation 8 is that the. eccentricity u is not known except for the fact that it is constant The Algebraic identification. methodology is proposed to on line estimate the eccentricity u which is based on the. algebraic approach to parameter identification in linear systems Fliess Sira 2003. www intechopen com, Active Vibration Control of Rotor Bearing Systems 297. 3 2 Algebraic identification of eccentricity, Consider equation 3 with perfect knowledge of the moment of inertia Jz and the shaft. stiffness k and that the position coordinates of the disk x y and the control input are. available for the identification process of the eccentricity u. Multiplying equation 3 by t and integrating by parts the resulting expression once with. respect to time t one gets,0 t dt J z 0 t c dt J z 0. t y cos x sin dt,t 1 t ku t, Solving for the eccentricity u in equation 12 leads to the following on line algebraic.

identifier for the eccentricity,J zt J z t tc dt,k t y cos x sin dt. where is a positive and sufficiently small value, Therefore when the denominator of the identifier of equation 13 is different to 0 at least. for a small time interval 0 with 0 one can find from equation 13 a closed form. expression to on line identify the eccentricity, 3 3 An adaptive like controller with algebraic identification. The PI controller given by equation 8 can be combined with the on line identification of the. eccentricity in equation 13 resulting the following certainty equivalence PI control law. J z v1 c kux sin kuy,v1 t 1 t 0 t dt,J zt J z t tc dt. k t y cos x sin dt, Note that in accordance with the algebraic identification approach providing fast.

identification for the eccentricity the proposed controller 14 resembles an adaptive control. scheme From a theoretical point of view the algebraic identification is instantaneous Fliess. Sira 2003 In practice however there are modeling errors and other factors that inhibit. the algebraic computation Fortunately the identification algorithms and closed loop system. are robust against such difficulties Beltr n et al 2005. 3 4 Simulation results, Some numerical simulations were performed using the parameters listed in table 1. Figure 2 shows the identification process of eccentricity and the dynamic behavior of the. adaptive like PI controller given by equation 14 which starts using the nominal value. www intechopen com,298 Vibration Control, u 0 m One can see that the identification process is almost instantaneous The control. objective is to take from the rest position of the rotor to the operating speed f 300 rad s. mr 0 9 kg D 0 01 m a 0 3 m,mb 0 4 kg rdisk 0 04 m c 1 5 10 3 Nms rad. cb 10 Ns m u 100 m b 0 3 0 05 m,Table 1 Rotor system parameters. Eccentricity,0 0 5 1 1 5 2 2 5 3 3 5 4 4 5 5,Rotor Speed.

0 10 20 30 40 50,0 10 20 30 40 50, Fig 2 Close loop system response using the PI controller a identification of eccentricity. b rotor speed and c control input, The desired speed profile runs up the rotor in a very slow and smooth trajectory while. passing through the first critical speed This control scheme is appropriate to guarantee. stability and tracking The resulting rotor vibration amplitude system response when F 0. is shown in figure 3 for three different and constant positions of the right bearing i e. b 0 25 m 0 30 m 0 35 m using the PI controller, The purpose of these simulations is to illustrate how the position of the bearing truly affects. the rotor vibration amplitudes for the desired speed profile The nominal length of the shaft. is l 0 60 m A smaller length l 0 55 m leads to a higher natural frequency and a bigger. length l 0 65 m leads to a smaller natural frequency see figure 3 Hence to get a minimal. unbalance response the rotor length should start at l 0 55 m and then abruptly change to. l 0 65 m This change of the bearing position must occur exactly when the response for. l 0 55 crosses the response for l 0 65 in order to evade the resonance condition because. the rotor speed is different from the natural frequency of the rotor bearing system. www intechopen com, Active Vibration Control of Rotor Bearing Systems 299. Unbalance Response m m,0 50 100 150 200 250 300,Rotor Speed rad s.

Fig 3 Unbalance response R for different and constant positions of the movable bearing. 3 5 Simulation results, It is evident from equations 5 and 6 that controlling the position of the movable right. bearing b applying the control force F and according to a pre specified speed profile t. the modification of the rotor amplitude response to the unbalance is possible As a matter of. fact this methodology is equivalent to a dynamic stiffness control for the Jeffcott like rotor. system enabling smooth changes on coordinate b, To design a controller for position reference tracking consider equation 4 Then one can. propose the following Generalized Proportional Integral GPI controller for asymptotic and. robust tracking to the desired position trajectory b t for the bearing position and velocity. which employs only position measurements of the bearing For more details on GPI control. see Fliess et al 2002,F mb v2 cb b,v2 b t 2 b b t 1 b b t 0 b b t dt. where b is an integral reconstructor of the bearing velocity which is given by. b b b 0 F d, The use of the GPI controller given yields the following closed loop dynamics for the. trajectory tracking error e2 b b t,2 2 e 2 1 e2 0 e2 0.

www intechopen com,300 Vibration Control, Therefore selecting the design parameters 0 1 2 such that the associated characteristic. polynomial for equation 17 be Hurwitz one guarantees that the error dynamics be globally. asymptotically stable The desired trajectory planning b t for the bearing position and. velocity is also based on B zier polynomials similar to equation 10. 3 6 Results and discussion, The proposed methodology for the active vibration control of the transient run up or coast. down of the rotor bearing system consists of the following steps. 1 Define the trajectory planning for the speed trajectory profile t to be asymptotically. tracked by the use of the adaptive like PI controller with the algebraic identifier of the. eccentricity i e limt t t, 2 Establish an appropriate smooth switching on the position of the movable bearing b t. to be asymptotically tracked by the application of the GPI controller i e. limt b t b t The switching time has to be at the crossing point leading to minimal. unbalance response in figure 3, Figure 4 shows the unbalance response of the rotor bearing system when rotor speed PI. 12 Active Vibration Control of Rotor Bearing Systems Andr s Blanco 1 Gerardo Silva 2 Francisco Beltr n 3 and Gerardo Vela 1 1Centro Nacional de Investigaci n y Desarrollo Tecnol gico 2Centro de Investigaci n y de Estudios Avanzados del IPN 3Universidad Polit cnica de la Zona Metropolitana de Guadalajara M xico 1 Introduction Rotating machinery is commonly used in many mechanical

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