مدل رگرسیون خطی برای شناسایی عدم تعادل تغییرات در ماشین آلات دوار

This paper presents a numerical method for the full identification of multi-plane unbalance changes in a multi-bearing rotating machine. This new method is a further development of the methods for one and two-plane identification of unbalance changes. In modern rotating machinery, it is common practice to place permanent probes into the main supporting bearings as a means of “health monitoring” or “condition monitoring”. These probes pick up the real-time vibration signals from a machine during its operation. By reprocessing these monitored signals and comparing them against developed criteria, the location and magnitude of any unbalance change during the machine's operation is identified. This is achieved by using the algorithm that combines the processed signals with the use of a non-linear mathematical model for the rotating machine. Assumption is made that the steady state responses before and after the unbalance change takes are available for comparison, and that the mathematical model as well as the dynamic and static properties of the system under consideration are truly representative. Verification of the proposed algorithm has been conducted using computer simulations of a real machine.

The dynamic response of a multi-bearing rotor system depends not only on the dynamic properties of its subsystems but also on its con"guration and its residual unbalance. The subsystems include rotors, oil bearings and supporting structures.Studies of the dynamics of rotating machines and their subsystems have been reported using both linear and non-linear models by numerous researchers such as Krodkiewski and co-workers [1}3], Craggs [4, 5], Bishop and Gladwell [6],Goodman [8], Lund and Tonnesen [9] and Parszewski and Roszkoowski [10] among others.There have been a number of established procedures for balancing large rotating machines. Most of these procedures have been developed for balancing rotating machines prior to their installations. Many of them assume a linear model and require many test runs for the full identi"cation of residual unbalance in the speci"ed correction planes.In the procedure for the balancing of large turbogenerator units described by Cragg [4] the equivalent residual unbalance is found by multiplying the measured response vector by the in#uence coe$cient matrix, which is determined from the"nite element model of the turbogenerator considered. In the theory of modal balancing, the unbalance distribution is developed into a series of modal functions.The modal functions are either computed by means of the "nite element method or determined experimentally. The unknown modal coe$cients, which represent the participation of individual models, are determined mode after mode by using the data which are usually measured for speeds close to the critical ones. The in#uence coe$cient balancing methods allow for the computation of the correction weights from measurements of the system response taken with test masses attached to the rotor at various locations along its length. This method results in both the identi"cation of the linearized system considered and the identi"cation of the residual unbalance.During the operation of the system, the available data are usually limited to the supported cross-sections of the rotor only. Furthermore, in the case of large-amplitude vibration of a rotor supported upon oil bearings, the linearization of the system leads to poor assessment or the identi"ed parameters.In the case of a turbogenerator set, changes in the balancing conditions may occur during its operation*this may be due to the result of the machine losing one or more of its blades. Development of procedures for on-side identi"cation of unbalance changes has, therefore, drawn much attention. On-site identi"cation involves using real-time vibration signals measured from a machine during its operation. Since the monitored dynamic information is often limited, proposals have been made to combine the e!orts in measurements with the modelling and numerical analysis of the system to aid the balancing of a rotating machine such as a large turbine generator set.In many situations, large-amplitude vibrations may result from a dramatic change in balancing conditions such as the loss of blades during the operation of a turbogenerator. The presence of large-amplitude vibrations often implies a machine operating beyond the linearized equilibrium, particularly in the case of a rotating machine using oil bearings. The linearization of the system in these cases may lead to poor assessment of the identi"ed parameters.To handle large-amplitude vibration problems, studies have been reported to focus on the non-linear dynamic characteristics of rotating machines. Krodkiewskiet al. [1] presented a method that uses a non-linear mathematical model for the on-site identi"cation of unbalance change that may take place during the operation of a multi-bearing rotor system. The mathematical model includes the dynamic properties of the rotor and foundations as well as the non-linearity of the oil bearings.In brief the method proposes that the signals measured before and after the unbalance change takes place are used to compute the time history of the hydrodynamic forces generated by the supporting journal bearings. Both the measured signals and the bearing forces are processed using fast Fourier transform technique. The data obtained form this process are then combined with the physical properties of the system represented by mass and sti!ness matrices to form a system equation. By using error functions and developing certain criteria, the location and the magnitude of unbalance change is identi"ed.The attractiveness of the method is that it requires only the relative journal-to-bearing displacements or velocities as input parameters to identify the change of unbalance. In modern rotating machinery, it is a common practice to place permanent probes into the main supporting bearings as a means of &&health monitoring'' or &&condition monitoring''. The real-time vibration signals measured by these probes from a machine during its operation represent the journal-to-bearing displacements or velocities. Hence, the real-time information required by the method is readily available in some practical situations.There is a drawback in the above method, however. It assumes that the change in system responses is due to the change in unbalance at one plane of the rotor only(for example, a few blades are lost from one row of a turbogenerator set). Hence, it was proposed to identify one location each time where an unbalance change has taken place.It is often desirable to identify in a round of computation several locations along the length of the rotating component for blade losses. This paper presents a further development of the above method and provides the basis for the full identi"cation of unbalance changes during the operation of a rotating machine. It is worth noting again that the method is based on the monitored trajectories of the journals and the non-linear mathematical model of the system considered. The model includes the dynamic properties of the rotor and the non-linear dynamic characteristics of the oil bearings. In this paper, a method is presented for identi"cation of the plane of the rotor at which the change of unbalance has taken place. The new numerical method may be applied to the identi"cation of one or more sections of a turbine which has resulted in blade losses.

The linear regression method was developed for the identi"cation of unbalance changes taking place during the operation of multi-bearing rotor systems such as turbogenerator sets. The method incorporates a non-linear mathematical model for the rotor-bearing system and the subsystem modelling approaches. The subsystem modelling includes the "nite element analysis of rotating components and the "nite di!erence method for the generalized Reynolds equation for modelling the hydrodynamic forces. The numerical veri"cations showed that the method is useful for predicting the unbalance changes in most cases where the number of supported stations is greater than the number of unsupported stations (i.e., the balancing planes). The e!ect of the noise on the results of the identi"cation is omitted from the study reported in this paper. Therefore, further research and studies are needed to assess the sensitivity of the method to the di!erent noise levels. Experimental veri"cations of the method in the various areas will also provide additional assessment of the method in the various areas will also provide additional assessment of the method developed here.