# Identification of Close Modes on Frequency in Rotating Systems
## 1 Introduction  
Modal balancing is a technique used in the field of turbomachinery balancing which allows balancing a specific mode without affecting the other modes present [1]. From a practical point of view, this is very convenient because in theory, a rotor could be balanced in a single run, as presented in [2]. However, although the theory has been in place for decades, its systematic application in the field is not so simple. Some current limitations are related mainly to the need to know the modal shapes of the rotor in resonance, instrumentation problems, non-linearity of the system, errors during the modal parameter extraction process, the presence of close modes in frequency, and the non-consideration of the main stiffness directions [3]. In the first three problems mentioned, there have been significant advances in using the finite element tool and the continuous improvement and development of vibration sensors and data acquisition and processing boards. Likewise, there are modal parameter extraction methodologies, such as those presented by [PERSON] in [4], designed for structures and which can be adapted for rotating systems through circle adjustment, such as the AMODAL software owned by the National Institute for Electricity and Clean Energy. However, these methodologies are focused on extracting the modal parameters in one mode for separate frequencies. Different investigations into structures to identify close modes and their extraction of some modal parameters have been carried out: [PERSON] et al., in [5], present a method in frequency which uses the stepped reassignment of the Wavelet transform to allow identification of the natural frequency and damping ratio of a three-degree-of-freedom structure, using a 5% difference between natural frequencies to define close modes in frequency. Likewise, in the frequency domain, a different approach is presented by [PERSON] in [6] by means of dummy modifications in Bode diagramsin amplitude using a fictitious mass at specific measurement points and comparing the measured diagram with those obtained using the proposed graphical technique. [PERSON] et al., in [7], used the method in time by using the Ibrahim Time Domain (ITD) technique to study close modes on a plate; by applying the finite element technique, the dynamic response of the structure is obtained to identify the natural frequency of the first five vibration modes with close frequencies. In [8], [PERSON] and [PERSON] present a method in time to identify close modes in structures using the Vector Backward Auto-Regressive (VBAR) technique, determining the modal parameters of natural frequency and mode damping. In [9], [PERSON] and [PERSON] present the measurement of the degree of correlation between modal shapes in structures to detect close modes. Along the same line of research, [PERSON] and [PERSON], in [10], present an investigation to identify close modes in structures through high sensitivity to small changes in the system, considering two modal forms through the theory of the local correspondence principle. In [11], [PERSON] and [PERSON] combine Wavelet Packet Decomposition algorithms and Complex Envelope Displacement Analysis for the identification of close modes in structures, applying their methodology in a two-degree-of-freedom model. On the other hand, [PERSON] and [PERSON], in [12], present the mathematical bases to identify close modes in structures using Bayesian operational model to determine the covariance matrix and propose the uncertainty identification term. The authors propose beginning development of what they call the uncertainty laws for the identification of the problem posed. [PERSON] et al., in [13], present a method that allows estimating the modal parameters of two close modes through the use of FRFs in a quasi-axisymmetric structure. The results presented were validated through an experimental study and finite element simulation. One use of the simulation is to determine the location of close frequencies. Recently, [PERSON] et al.