1.5 Crack identification in rotor system
1.5.2 Condensation techniques
Different methodology involved in the identification of cracks are reported in the present section. The 2nd harmonics present in the frequency domain signal indicates the presence of crack in the rotor system. Also, on the basis of change in modal parameters and mode shapes the identification of cracks are carried out. Due to complexity in the identification of the cracks, the increase in the trend of 1st, 2nd and 3rd harmonics of the vibration response is considered. Inverse problem approach and 3D models are also utilized for identification of crack depth, location and stiffness.
A real system and its model varies according to DOFs considered in the analysis. But based on continuous system theory, physical system can have infinite DOFs. Hence, the model can be proposed to be large DOFs. However, a large-DOF model of the system need more capacity of storage data and computational power. Because of this a model reduction technique (or condensation technique) is useful to reduce DOFs necessary to develop the analysis. After reduction of DOFs the time and cost of computational reduces drastically. Therefore, the number of equations to be solved reduces as per the reduction of DOFs. Also in identification of crack parameters through model based methods due to limited measurement availability, the mathematical model has to be reduced accordingly. Also in practical case measuring rotational DOFs accurately is very difficult. For the reduction method, it is the most important to handle such situation. Thus in next section, the literature survey on the condensation techniques are reviewed.
DOFs, which are possible to measure experimentally. The condensation method was first proposed by Guyan (1965) and Irons (1965) in case of undamped system. Guyan (1965) reduction method is mostly preferred due to lower computational difficulties (Genta, 2007). The reduction method is applied in the original vectors and matrices q, f, M and K, which represent the displacement vector, force vector, mass matrix and stiffness matrix, respectively. These matrices and vectors are partitioned in the form of sub-vectors and sub-matrices corresponding to master DOFs (retained) and slave (eliminate or reduced) DOFs. On considering above aspect equations of motion becomes
mm ms m mm ms m m
sm ss s sm ss s
M M q K K q f
+ =
M M q K K q 0 (1.1)
where subscriptsm and s denotes the master and slave DOFs, respectively. On ignoring the inertia the following transformation can be obtained (Henshell and Ong, 1974)
m
m s m
-1
s ss sm
q I
= q T q
q -K K (1.2)
Further, the above Guyan reduction algorithm cannot estimate the correct eigen solutions because of ignoring the effect of mass and it is not rationally consistent when estimating the reduced matrices in the system (Jeong et al., 2012). The error obtained is typically small (Genta, 2007) at low frequencies only or at zero frequency. Therefore, the Guyan reduction scheme is also called the static reduction. When excitation frequency increases, in this situation, the inertia term which is neglected in the static reduction process plays a vital role in the process of reduction. Here, comes the dynamic reduction scheme for higher frequency system, where inertia terms influence in the reduction scheme. Henshell and Ong (1974) reported an iterative technique for the selection of master or slave DOFs in the static reduction on the basis of comparing the magnitude between
the elastic and inertia force terms, automatically. The procedure has the ability to remove one DOF in each iteration.
Leung (1978) illustrated an alternative dynamic reduction procedure for more accurate results but in contrary to this, the dynamic reduction developed by Paz (1984) was considered as better alternative to the static reduction method in which the reduction is exact for a particular frequency, which may be chosen arbitrarily. In the dynamic analysis of rotor, there is an advantage to choose the frequency of reduction independently, since the scheme may be utilized at the frequency of external forcing, which gives rise to a multiple/factor of the spin frequency of rotor. The reduction transformation at a frequency ω0, is expressed as (Singh and Tiwari, 2018)
m
1 m d m
2 2
s ss 0 ss sm 0 sm
q I
q T q
q K M K M (1.3)
O’Callahan (1989) and Gordis (1992) modified the static reduction technique by the use of the improved reduced systems (IRS) scheme in the transformation of slave DOFs, where approximation up to the first order of a binomial series expansion is done. Suarez (1992) presented an iterative procedure for the reduction of matrices based on the developed equation of motion of a system. Friswell et al. (1995, 1998) developed an iterative IRS (IIRS) procedure in order to obtain more accurate reduction. The IIRS method updates a transformation matrix between the master and slave DOFs in a repetitive way with the help of iteration procedures.
Also, Xia and Lin (2004) presented an iterative order reduction (IOR) technique to improve the computational efficiency in applying condensation in a system of equations. On the basis of the requirements in the numerical analysis and experimental model updating, condensation schemes were developed, for example a hybrid condensation (or high frequency) scheme for elimination
of transverse rotational degrees of freedom in identification of beam crack parameters (Dharmaraju et al., 2005).
The model reduction technique finds wide applications in the field of numerical modelling, and systems and control. With the advancement in the developments in the model reduction technique in one field does not influence the other field quickly, rather most of the developments are independent. Besselink et al. (2013) reviewed model reduction schemes in the above three fields and compared the popular methods of modelling, system and control fields against a common benchmark for the accuracy and the ease of implementation in detail. For the verification of the methodology experimentally, the numerical modelling aids maximum benefit to the practitioner.
The model improvement and updating can be done through experimental verification. The experiment works in the field of rotor cracks are less frequent than the faults, like misalignment and unbalance in the rotors. The most difficult work in the experimentation of the rotor cracks is the generation of crack in the laboratory. A review work on experimental evaluation on rotor cracks is discussed in the next section.