1.3 Literature Review on Faults in Rotating Machinery
1.3.1 Unbalance in Rotor
parameters can be further used for the quantification and prognosis. The accuracy of results in mathematical model based techniques is highly sensitive to the model accuracy.
There are mainly two methods for balancing of the flexible rotor i.e., modal balancing method and influence coefficient method. The first balancing method was developed by (Bishop and Gladwell, 1959; Gnielka, 1983), that requires the accurate knowledge of the modal parameters of the machine; however, the second method (developed by Drechslen (1980)) utilizes the vibration amplitude and phase measurement for the calculation of balance correction masses. At first, the unbalance response is determined at the rotor measuring planes at a given speed without any correction masses. Afterwards, a trial mass is given in one of the balancing planes and the rotor response is obtained for all of the measuring planes. This process is repeated for all of the balancing planes and from this data the influence coefficient matrix is obtained. The required correction masses can be calculated by multiplying the inverse of the influence coefficient matrix with the original unbalance response vector. The obtained trial masses will be helpful in the rotor balancing. Hence, the second method requires less a priori knowledge of the system model parameters.
A modal balancing method was proposed by Morton (1985) to balance a flexible shaft without the use of trial weights and a knowledge of the support bearings characteristics. The shaft gyroscopic effect and rotating damping of shaft were ignored for modelling of the rotor system. He concluded from numerical results that developed balancing technique can be utilized together with any other technique and is quite suitable for balancing of flexible shafts in the multiple range of critical speeds. Later, Krodkiewski et al. (1994) presented a method for identification of the plane of the rotor with the changes in residual unbalance responses.
They developed non-linear mathematical model of the rotor-bearing system and further numerically illustrated in a system consisting of flexible rotor (modelled using Timoshenko beam finite elements with four degrees-of-freedom at each node), four identical three-sleeve journal bearings and a rigid concrete foundation. The method was also tested with the addition of white noise in the signals and found to be accurate in identification.
A theoretical model was proposed by Lees and Friswell (1997) to estimate the unbalance distributions of flexible shafts and eccentricities of rigid disks supported on flexible bearings.
This method was based on finite element method (FEM) for analysing the steady-state responses of rotor-bearing system. Each bearing was modelled with eight number of linearized coefficients, i.e. four stiffness and four damping parameters. The identification technique used simulated measured response data (measured at one free end in the shaft) in frequency domain and least-squares fitting approach for estimation of the unbalance fault parameters. Further, Shih and Lee (1997) used measured pedestal vibration to determine the imbalance condition of a rotating machinery. For this purpose, they modelled rotor, bearings and the supporting structurein a consistent manner. The method was found to be insensitive to measurement noise in determining the imbalance fault parameters although the stiffness and mass terms showed moderate sensitivity to uncertainties.
Edwards et al. (2000) validated experimentally the proposed algorithm of Lees and Friswell (1997) to investigate the state of unbalance in rotating machinery by utilizing single run-down machine data. Along with the excitation due to unbalance fault, the excitation coming from bow in shaft was taken into consideration for the result analysis. The parameters related to elastic support structure (i.e., flexible foundation) of machine was also identified along with the unbalance parameters. The values of measured and estimated fault parameters were almost matching which signifies the accuracy of the proposed method. One of the most important observations was made from this experimental exploration that this single-shot balancing technique could reduce approximately 92% of the vibration after balancing the flexible rotor.
Zhou and Shi (2001) reviewed on dynamic analysis, different techniques for active balancing and active vibration control of the rotating system. To present his review in a completeness manner, they also presented the mathematical model for both the simple rigid rotor and complex flexible rotor model, separately. The complicated rotor system was
discretized into various elements, such as flexible shaft model, rigid disc model, linearized bearing model, coupling model, etc. Further, the system equations were obtained by assembling the equations of motion for each of those components. They concluded that the active balancing method can suppress the induced vibrational response of the unbalanced system. Moreover, the active balancing of rotor can enhance quality of the product in industries and increase machine fatigue life and reduce overall cost of the system.
A new method was proposed by De Queiroz (2009) for identification of the unknown unbalance parameters of a simple Jeffcott-like rotor by exploiting a dynamic robust control mechanism. He used unbalance disturbance forces by active feedback control mechanism to identify the unbalance-related parameters. He also demonstrated the effectiveness of the proposed identification strategy using numerical simulation.
Sudhakar and Sekhar (2011) developed three different approaches, i.e. equivalent loads minimization method, equivalent loads minimization method with modified theoretical fault model and vibration minimization method for the identification of unbalance fault in a rotor system. These approaches are the types of model based fault identification technique. In modified theoretical fault model, the effect of modal expansion gets nullified or reduced in the difference operation of least-squares minimization, which results in reduction of errors in the fault parameters identified. In vibration minimization method, the difference between measured and calculated vibrations is minimized using least-squares algorithm by varying fault parameters and this method does not depend on the number of measured degree of freedom.
They compared all three methods and found that error gets reduced in modified theoretical fault model as compared to equivalent loads minimization method without modification. They also found that equivalent loads minimization method with modified theoretical fault model and vibration minimization methods are equally effective in identifying unbalance fault with
reduced error even in case of measured degree of freedom as low as two. Unbalance fault was identified using proposed methods by measuring transverse vibrations at only one location.
An identification scheme was presented by Menshikov (2013) for identifying the characteristics of the unbalance fault in a rotor supported on two flexible supports. They developed the mathematical model of the system and used vibrational response of the rotor in the two transverse directions as the main information in the developed identification scheme.
Afterwards, they used inverse problem to estimate the unbalance and bearing support parameters. Pennacchi et al. (2013) proposed an estimation methodology for identification of the unbalance in a large rotor system. All components of the system, such as flexible rotor, bearings, flexible foundation, were accurately modelled and analysed before using in the identification method. The frequency domain data of displacement responses were utilized in the identification algorithm to identify the fault location and its severity along the shaft line.
They also performed an experimental work on a large size steam turbine (about 1.3 GW) and validated the numerical simulated results.
Chatzisavvas and Dohnal (2015) explored the concept of equivalent load approach as discussed in (Markert et al., 2001) and used for identification of single and double unbalance in a simple rotor-bearing system. They also applied the assumption of a sparse equivalent force vector to improve the unbalance identification without a priori information of the number of faults. The identification of fault parameters was done using both the time domain and frequency domain displacement signals. The proposed method provided effective and satisfying results even in the ill-conditioned problem arising due to an inadequate number of measuring locations and machine running at a constant speed.
For the identification and optimisation of unbalance parameters in a rotating machinery, the two different approaches were proposed by Yao et al. (2018). The first approach was based
on modal expansion method along with the concept of optimization algorithms, while the second approach was related to the combined use of modal expansion method, the inverse problem and optimization procedure. To overcome the practical issue of measuring responses at every locations of the shaft, the modal expansion technique was utilized. The first approach was illustrated in a single disc rotor-bearing system for the identification of unbalance fault, whereas the second approach was applied for the unbalance identification in a rotor system consisting of two discs. Both the numerical simulations and experimental works were conducted based on the two approaches of identification method. The method was capable of identifying the unbalance fault, not only their locations but also their magnitudes and severities.
Recently, a joint-input state and Kalman filter based input estimation techniques were utilized to identify unbalance force in a rigid rotor-conventional bearings system. The proposed techniques were a model-based method, which required a mathematical model of the rotor system along with response measurements. Accelerations were measured at bearing pedestals and used for unbalance parameters estimation. Bearing stiffnesses were also estimated using a frequency domain parameter estimation technique with measured unbalance responses.
Sensitivity analysis of the proposed method was also performed by changing the values of these estimated stiffnesses. The results for distinct spin speeds of shaft and measurement noise signal levels were found stably and validated experimentally in a single disc rotor-bearing system (Shrivastava and Mohanty, 2019; Shrivastava and Mohanty, 2020).
The literature presented in this section discussed about the effect of severe unbalance fault in a rotor system. The research performed in the balancing techniques including modal balancing method and influence coefficient method has been well described. For identification of rotor unbalance fault, the model based methods along with the measured vibrational response have gained more importance in the rotor dynamic field as these techniques can identify both location and severity of the fault.