The investigation on the dynamic analysis of a large rotor system is very difficult due to high complexity. The complexity arises from the presence of multiple numbers of degrees-of- freedom (DOFs) in the system. It becomes a tedious task to measure all DOFs, especially the rotational ones, due to difficulties in accessing certain locations on the rotor and limitations in the availability of number of displacement sensors. Moreover, taking measurements at all DOFs would be more expensive and consume more time. Therefore, the researchers have proposed various approaches, which can reduce the large size of the complex rotor model involved in the mathematic formulation.
Usually, there are two different methods used for the reduction of degrees-of-freedom.
One of the method is static reduction method, whereas the other method is dynamic reduction method. Out of both the methods, the static reduction method is simpler and popular. In earlier times, the formal method has been developed by Guyan (1965). The degrees of freedom eliminated in the reduction process are called slave DOFs and those retained for the analysis are called master DOFs. The limitation of this method is that it is suitable for zero frequency or low range of frequency and the inertia terms are assumed to be negligible as compared to the stiffness term. Moreover, this method cannot produce the accurate eigen solutions.
Therefore, to overcome these concerns, Paz (1984) proposed a dynamic condensation method.
The proposed method provided exact eigenvalues and eigenvectors for all the modes considered in the reduced eigenproblem of a structural system. He concluded that the method is quite suitable from the fundamental mode to any desired number of higher modes.
O'Callahan (1989) improved the static reduction method with the help of improved reduction system method (IRS), which was approximated up to the first order of a binomial series expansion in the transformation of slave DOFs. Further, an improved dynamic condensation method based on iterative approach was proposed to calculate the accurate values of a large structural system eigen properties, i.e. natural frequencies and modes of vibration of systems (Singh and Suarez, 1992).
Friswell et al. (1995) extended an improved reduction system (IRS) method using two approaches, in order to obtain more accurate reduction in degrees-of-freedom. The first approach was using the equivalent transformation based on dynamic reduction instead of static reduction, whereas the second approach was based on introducing an iterative scheme in which the corrective term was generated iteratively. They also investigated the convergence of the natural frequencies of the reduced model to those of the full model. Later, they have developed a model reduction method for a damped and gyroscopic coupled systems. They also compared
its reduction results with reduction process for no damping in the systems and found that the reduction transformations generated from the undamped model contained more errors (Friswell et al., 2001). Xia and Lin (2004) proposed an iterative reduction order (IRO) method to improve the computational efficiency and estimated accurately the eigenvalues and eigenvectors of a large structural systems. They have also demonstrated the novel reduction technique practically in a plate structure with large DOFs and observed that the developed iterative based approach was very efficient. Jung et al. (2004) developed an iterative dynamic condensation scheme for the finite element model reduction in a structural system. They concluded that the proposed scheme was more advantageous than the other iterative schemes. The convergence was found to be faster especially at the condition when the eigen properties of reduced model was close to that of the full model.
A high-frequency and hybrid reduction schemes were proposed by Dharmaraju et al.
(2005) and Tiwari and Dharmaraju (2006) to reduce the number of measurement responses in their developed identification algorithm. They eliminated transverse rotational DOFs and identified beam crack flexibility coefficients and crack depth. They extended the hybrid reduction method for damping in the rotor system (Karthikeyan and Tiwari, 2010). Choi et al.
(2008) presented an iterated improved reduced system (IIRS) method combined with substructuring scheme for both undamped and damped structures. The method provided highly accurate eigenproperties without consuming expensive computational cost. Numerical simulations were performed to validate the proposed method and to evaluate the computational efficiency. Prasad and Tiwari (2018) proposed a gyroscopic dynamic reduction method to reduce rotational DOFs in a finite element modelled flexible rotor-AMB system. The transformation matrix in the condensation method also included the gyroscopic matrix along with the mass and stiffness matrices. The dynamic condensation scheme was also utilized by several authors (Lal and Tiwari, 2013; Singh and Tiwari, 2016; Singh and Tiwari, 2018; Kuppa
and Lal, 2019; Kuppa and Lal, 2020; Srinivas R et al., 2020) for utilizing the reduced response measurements in their identification algorithm to estimate system and multiple faults such as unbalance, crack and coupling misalignment parameters as well as bearing dynamic parameters in a rotor system.
Upon carrying out various literature survey in this section, it can be stated that research has been performed well in the condensation schemes, especially in dynamic reduction method due to its advantage over static reduction method. Moreover, still the researchers depend on the dynamic condensation method for eliminating the rotational DOFs in the analytical model.
This is done to compare the measured experimental data and numerically generated data with complete information during model updating.