Effect of Thermal Response and Vibration on the Multicracked Rotor Immersed in a Fluid Medium Using Finite Element Method

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Introduction
In the last couple of years, the issue of health monitoring and open crack detection of structures has received significant attention. This could lead to catastrophic failure in mechanical systems, aerospace engineering, and civil structure and could be challenging to identify. It was found that cracks cause changes in the vibration response of the rotor shaft. It is essentially important for a correct prediction of the dynamic behavior of the rotor shaft system. Numerous techniques have been reported to determine the vibration signature of the rotor with the transverse crack in rotating machinery, but there is not a specific investigation of the multicracked rotor shaft in the different viscous fluid environments that have been discussed.
Curadelli et al. [1] examined the vibration aspects of spherical tanks under horizontal motion and determine the fundamental natural frequencies of the modes of this particular structure. Chouksey et al. [2] studied the model analysis of the rotor systems under the influence of the fluid-film forces and internal damping of rotor shaft material. Two nodes of finite Rayleigh beam element have been considered to determine the equation of motion. Singh and Tiwari [3] have proposed the transverse forced vibration analysis of a stationary cracked shaft in two orthogonal planes by using the finite element method (FEM). The two-stage method was used to identify the sizes of cracks, their location, and the number of cracks in the shaft. Han et al. [4] recommended the FEM for examining and modeling the slant crack geared rotor with a breathing crack. Darpe et al. [5] studied the analysis of bowed rotors with transverse cracks. The transient response of the rotor carrying transverse crack with and without gravity was obtained. Georgantzinos and Anifantis [6] projected the nonlinear FE analysis to analyze the quasistatic crack breathing mechanism in the spinning rotor shaft. Bachschmid and Tanzi [7] investigated the vibration signature of cracked round cross-section beams with axial, torsion, and shear loading. Sekhar [8] studied the dynamics analysis of the vibration signature of a multicracked rotor shaft system and also he obtained the slenderness ratio, stability of the rotor, and eigen frequencies with parameters of crack. Chasalevris and Papadopoulos [9] analyzed the vibration signature of doubled transverse crack circular cross-section beam considering the relative crack angle and crack depth as variable parameters. Nandi [10] proposed the standard technique of assumed solution for analyzing the stability of the rotor system. Mohamed et al. [11] investigated the dynamics behavior of multicracked shaft systems to identify the commencement and propagation of precracked on high-carbon steel rotor shaft systems. Baviskar and Tungikar [12] developed a finite element model of a rectangular beam and employed the inverse method for determining damage detection in moving parts. Hossain et al. [13] proposed the FEM by using ANSYS to identify the dynamic characteristics of the mini cantilever beam which is partially immersed in a different viscous medium. Polytech scanning vibrometer was used to extract the vibration response of the beam and resonant frequency. Georgantzinos and Anifantis [6] investigated to determine the dynamic response of a circular cross-section cracked cantilever beam which is subjected to twisting load using the nonlinear contact-FEM technique. Quasistatic approximation technique was used to evaluate the effect of damage in terms of the crack on the vibration signature of the beam. Kerboua et al. [14] identified the vibration signature of plates (rectangular cross-section) fixed with the fluid and velocity potential. Bernoulli's equation was employed to investigate fluid pressure. FE-based finite element approach and Sander's shell theory were proposed for developing the mathematical model of the plate. Darpe [15][16][17] reported a novel method to predict the fault in terms of crack size and location in the high-speed mild steel rotor shaft considering the breathing crack which is subjected to the transient torsional load. A non-linear breathing crack model was developed to compute the unsteadiness performance of the mild steel high-speed cracked rotor shaft. Developed the FE-based model of the cracked rotor and formulated the stiffness matrix. Silania et al. [18] analyzed the vibration characteristics of the cracked rotating shaft to consider breathing crack. Modified integration techniques were carried out to acquire the stiffness matrix for the cracked shaft element. FEM has been developed to compute the vibration signature of the cracked rotor shaft. Kulesza [19] developed the finite element modeling of the spinning shaft and employed the multisine method to recognize the crack depth and crack location in the rotor shaft. Sekhar and Prasad [20] proposed the finite element analysis (FEA) approach to evaluating the vibration characteristic of the cracked rotor-bearing system using stiffness and flexibility matrix for crack element associated with FEM analysis of the cracked rotor-bearing system. Arruda and Castro [21] investigated the dynamic characteristic of the linear structure using the time integration technique and proposed the FEM of hybrid-mixed stress for the identification of the vibration response of the structure. Singh and Tiwari [22] discussed the multicrack detection and localization algorithm for the recognition of crack location in fixed-fixed type shafts. Multicrack detection and localization algorithm results were validated with experimental analysis results. The algorithm is completely based on finding the discontinuity of the slope in the shaft. Rubio et al. [23] have proposed the technique as polynomial expressions of the flexibility for dynamic characteristics of the cracked shaft to subjected bending with an elliptical shape. Wang et al. [24] have developed the experimental setup and used the convolutional neural network for the identification of crack location via time domain response of vibration. Yuhong et al. [25] have suggested deep metric learning and the convolutional neural network method for the identification of crack locations in a hollow shaft rotor system. On the basis of transfer learning and the convolutional neural network method, damage features were automatically taken out for damage detection [26]. Babu Rao and Mallikarjuna [27] proposed the ANN technique for damage detection in rotor shaft systems using the discrete wavelet transform technique. The principal component analysis was employed to calculate the damage index for recognition of the position of the crack [28]. Xiang et al. [29] have proposed the orbit morphological characteristics method to obtain the position of the crack. Both signal-based and model-based models may be useful to enhance the analysis of performance. Ganguly and Roy [30] have proposed an analysis of the effect of breathing crack on the vibration characteristics of the rotor shaft. Analysis of an unbalanced transverse crack on the two-disc rotor shaft system has been carried out by Kushwaha and Patel [31]. A crack closure line position (CCLP) breathing model has been proposed for crack identification using the zero stress intensity factor model [32]. A sequence of discrete points divides the crack front into equal sections in this process, and the crack opening and closing locations are identified using the stress intensity factors. Wang et al. [33] have proposed analyses of the influence of unbalance on the cracked rotor shaft. Also, the authors have employed the finite element method to evaluate the vibration signature. CholUk et al. [34] have reported the investigation on the identification of damage in rotating rotor shaft bearing systems with transverse cracks using the finite element method. Liu et al. [35] have suggested the finite element method to analyze damaged rotor shafts for the control of crack propagation. Toke and Patil [36] have proposed the finite element method and experimental method technique to identify and control the open crack in the beam.
The above literature review indicates that many works have been carried out on the dynamic analysis of vibration response of cracked rotating structures like beams and rotor shafts in air medium. Very few investigators have done the dynamic analysis of structure-interaction with a fluid medium. However, it appears that there was not much investigation has been reported on the dynamic characteristic of the damaged (crack) rotor shaft rotating in the viscous fluid medium using the FEM method and subsequent experimental validation. In the present research, ANSYS is utilized to model the spinning multiple cracked cantilever rotor shaft system partially submerged in different viscous medium environments and measure the vibration response in both transverse directions of crack. The effect of diverse viscous fluid mediums on vibration response spinning cracked rotor system is calculated at the beginning. Further, FEM analysis results were compared and validated with the 2 International Journal of Energy Research experimental analysis results, and the result of FEM analysis is compared with the acquired experimental results.

Experimental Analysis Procedure
The responses of the frequency of the multicracked cantilever rotor were experimentally measured by the proposed experimental test rig shown in Figure 1. Firstly, sample of the minicantilever rotor was arranged from high-modulus stainless steel with the mentioned dimension in Table 1.
The physical parameter to vibration characteristic of cracked cantilever rotor is given in Table 2. The experimental test rig consists of the rotor shaft with power motor, extramass (i.e., disc), variac, microcontroller, 3 ultrasonic sensors, breadboard, USB connection serial port, and computer system (response display devices). The complete setup for experiments is the same, excluding the existence of a different crack depth and crack location of the shaft and viscous medium.
An experimental analysis has been performed on the rotor shaft with an attached disc at the mid-span under con-sideration of different fluid mediums (lubricant oil 4T, palm oil, and seawater), relative crack depth, and constant relative crack location for calculating the dynamic response of the    For the ball bearing, the following assumptions are made [37,38].
(1) The assembly of ball bearings is firm apart from the point of contact among the rolling bodies and raceways (2) The rolling surfaces of the bearing have completely smooth and free from flaws (3) Although the inner race of the ball bearing is supporting the heavy-weight flexible rotor shaft, the outer race of the ball bearing is firmly coupled to a firm shell The bearing effect is not considered here to rationalize the rotor system, whereas the rotor shaft speed is taken between 300 rpm and 700 rpm. The properties of the fluid were taken at the temperature of 25°C. While the rotor shaft spinning in the viscous medium increases, the temperature of the fluid due to sharing process. Because of the changing temperature, there were very small deviations observed in the viscosity of the fluid. The kinematic viscosity of the fluids has been taken at the 25°C temperature. While the rotor shaft revolves in the viscous fluid environment, there are minimal increases in the temperature of the liquid because of the shearing action. Because of the temperature difference, there are unusually small changes have occurred in fluid viscosity. This intention neglected the temperature effect on the kinematic viscosity of the fluid. In the current analysis, the major contributor, i.e., first mode has been considered. The power motor shaft is linked with the rotor shaft using a universal joint. The three ultrasonic sensors are organized with equal angles (i.e., 180°) around the revolving multicracked cantilever rotor shaft in the fluid medium. These three ultrasonic sensors are connected with an Arduino microcontroller with the help of a jumper wire and breadboard to attain the deflection data of the rotor shaft from starting to the end position. Every sensor has distinct deflection data of the rotor shaft motion. Three-time readings were taken at a similar speed to maintain accuracy. Every set has 1000 samples estimated by sensors in fifteen minutes. The sampling time was 1 sec/sample.

The Finite Element Analysis Procedure
FEM permits an independent platform for measuring the vibration response necessary for the proposed procedure. The FEA package ANSYS has been used to cultivate the FEM and simulation of cracked cantilever rotors partially immersed in different viscous mediums in the multiphysics platform. The dynamic behavior (natural frequencies and relative amplitude) at the parallel to the shaft axis and perpendicular to shaft axis directions has been calculated from block lanczos. The two-dimensional plain strains problem was considered in the case of the rotor shaft. As shown in Figure 2, only the longitudinal cross-section of the rotor shaft was modeled in ANSYS.
The cracked rotor and fluid are meshed and used in ANSYS built-in tetrahedral element 3D structural solid (SOLID187) and FLUID 30 element. The SOLID187 is a 10-node element that is used in the analysis of 3D models. The FLUID30 element was mainly well suited for modeling and analysis of fluid medium and fluid-structure interaction problems taking into account sound wave propagation and immersed structural dynamics. The fluid element and the rotor shaft at the interface shared the same node. The    Table 3 shows the properties of three different fluids. Equation (1) represents the mathematical model for the FEM which is used in this work. The fundamental mathematical modal equation is given by where ½K is the structural stiffness matrix, σ i is the real part of eigenvalues, fϕ i gis the eigenvector, λ i is the complex eigen valuesσ i ± jω i , ½C is the structural damping matrix, ω i is the imaginary part of eigen values (damped circular frequency), and j = ffiffiffiffiffi ffi −1 p ½M is the structural mass matrix. Several eigen solvers are available in model analysis using ANSYS. The "damp" eigen solver has been used for the viscous effect of the fluid medium, which computes the system damping matrix. The fluid-filled container is constrained from all three directions X, Y, and Z from the bottom side. Rotor shaft speed was fixed in the ranges of 400 to 800 rpm. The mesh size of the model has been considered based on the conversion test. The crack must be precisely captured; hence, a reasonable mesh density is necessary. The mesh convergence investigation was carried out for this purpose using various mesh sizes in the specimen's plane ranging from 0.15 mm to 0.3 mm for vibration response. Mesh convergence was attained with interface elements of size 0.25 mm.
A total of 4750 elements and 2512 nodes are generated in the FE model. Figures 2(a) and 2(b) show the mesh model of the cantilever mild steel cracked rotor shaft partly submerged in the fluid medium in a finite region.

Results and Discussion
The vibration response of the cracked cantilever mild steel rotor shaft in air and different fluid mediums was experimentally measured. Later on, for authentication, experimental analysis results were compared with the computed results of FEA. The effect of rotating speed on the amplitude of the rotor shaft without a crack in the form of dimensionless has been illustrated in Figures 6 and 7. It has been noticed that when increasing the length of the rotor shaft, decreases the amplitude of vibration. While the fluid viscosity increases, there is a shift in critical speed and a decrease in the amplitude of vibration.  The dimensional-less amplitude ratio (δ n * /ɛ * ) vs. frequency ratio (ω/ω 0 ) graphs are plotted in Figures 4 and 5 showing the comparison of experimental and FEM results. The three different types of viscous fluid (i.e., palm oil, water) and three different relative crack depths (β) with constant relative crack positions (α) are considered for the finite element result to validate the experimental result. The results acquired are in admirable understanding with each other. Table 4 shows the comparison between the acquired experiment and FEA results for the effect of altered fluid medium on the dynamic behavior of the cracked cantilever rotor shaft. Tables 5 and 6 show the comparison of results obtained from experimental and FEA for the influence of additional mass on the no-cracked rotor and cracked rotor shaft in the different fluid medium environments.
Tables 7 and 8 represent the comparison of experimental and finite element analysis results for the effect of the altered radius of the container which is filled by fluid medium on the dynamic behavior of the cantilever no-cracked and cracked rotor shafts, respectively. It has been noticed that if the radius of the container is increased, then the amplitude of the rotor shaft increases. Table 9 illustrates the comparison among the obtained finite element analysis result with the investigational result for the influence of changed crack size, depth, and positions on the dynamic behavior of the mild still cantilever-type rotor shaft in the direction parallel to the cracked axis and perpendicular to the crack axis. However, the results

Sl. no
Disc mass (M D ) in kg Experimental analysis (dimensionless amplitude ratio (δ * /ɛ * )) FEA (dimensionless amplitude ratio (δ * /ɛ * )) Error in s%       Sl. no Gap ratio (J) Experimental analysis (dimensionless amplitude ratio (δ * /ɛ * )) FEA (dimensionless amplitude ratio (δ * /ɛ * )) Error in % The current investigation can be utilized for the identification of the vibration response of the damaged rotor shaft in the fluid medium, for example, a long rotating shaft utilized for removing oil from the ocean bed by high-speed turbine rotors, drilling, etc.

Conclusions
In the present article, the dynamic behavior of the robust mild steel multicracked cantilever-type rotor shaft, in the air and submerged in the different viscous fluids, has been first evaluated by FEA using ANSYS. The following outcomes can be drawn from the research: (i) The FEM results to forecast the dynamic characteristics of the partly submerged multicracked cantilever rotor shaft in viscous fluid have been found to deviate to within 3%.
(ii) As the kinematic viscosity of the fluid rises, the amplitude decreases due to an increase in damping factor, and also the critical speed falls due to enhancing the virtual mass effect. The length of the shaft increases as well as decreases the amplitude of the shaft (iii) When the crack depth of a rotor shaft is increased, the stiffness of the rotor will be decreased. Simultaneously decreases the natural frequency and increases the amplitude of the shaft. The stiffness of the multicracked cantilever rotor shaft in the parallel direction of the crack is inferior to the stiffness in the perpendicular direction of the crack (iv) The amplitude of the 55-direction is smaller than the 44-direction of the crack of the rotor shaft. Also, when the gap ratio increases, the amplitude of the rotor increases due to the virtual mass effect and damping factor Results found from this parametric study can be used to identify the multiple transverse cracks in the rotor shaft immersed in different viscous mediums. The vibration response obtained from the FEA method can also be employed to design and develop fault identification and condition monitoring methods.

Data Availability
All the data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare no conflict of interest. International Journal of Energy Research