Dynamic modelling and predictive health monitoring for vibration control and resonance of rotating machinery

The aim of this study is to investigate the critical speed analysis and response of a rotating machinery. The search for increasingly high performances in the field of the vibration phenomena which is subject rotor are increasingly important and can lead to system instability. The use of the finite element method makes to establish dynamic equations of the movement. Numerical calculations of the model developed, can extract the natural frequencies and modal deformed of the rotor, and this reduce is nonlinear. The Campbell diagram plot used to determine the critical speeds. Experimentally the study of the rotor in transient system allowed the determination of the spectral responses due to the unbalances and various excitations.


INTRODUCTION
A detailed understanding of vibration problems associated with rotating systems is currently a major issue in the industrial field.To optimize the dynamic behavior of rotors and dimension to the best of such systems, Stodola (1972) worked out an iterative method to calculate the fundamental frequency of a vibrating system based on a standard form of eigen mode; it has been noted that the gyroscopic effect is a factor impacting the critical speed of a rotor.Si-Chaib et al. (2008) presented models based on the finite elements of flexible rotors to calculate the critical speeds and the eigen modes.In these models, the gyroscopic effects and the axial loadings are not taken into account.In the last few years, the most employed model has been developed based on the finite element method (Nelson, 1980;Tran, 1981); which allowed the determination with good precision the eigen frequencies and the damping ratios, as well as the response to the various excitations.Moreover, this approach is modular because each element of the rotor is defined separately.Elements can thus be added or withdrawn according to the studied phenomena.The finite element method was thus used to study the embarked phenomena of damping in dynamics of the rotors (Duchemin, 2003).It was also applied to the study of the rotors whose shaft turns at variable speed (Al Majid, 2003).Many results concerning the dynamics of the rotors, whose support is fixed, have been reported for the models of Rayleigh-Ritz and the finite elements (Lalanne, 1998).
The experimental studies concerning the dynamics of the rotors are concentrated on inaccessible points by the theoretical way; these are the characteristics of bearing, dampings, actual stiffnesses, coefficients of sealing of the labyrinths and finally the flexibility of the foundations.Vance et al. (1987) compared the numerical and experimental results of the "free-free" modes and evaluate the precision of the model according to the coupling of the rotor with the discs.The Rayleigh-Ritz method is used to set up a model making it possible to treat simple cases and to highlight basic phenomena.A finite element model is developed to treat real systems.The equations of motion of the rotor are obtained by application of Lagrange equations.
The vibration indicators are used for fault detection and can track its evolution to manage the predictive maintenance A fault diagnosis approach has been proposed by Wei et al., (2015) for rotating machines based on a new method for extracting and evaluating the statistical function; health condition is classified as the threshold corresponding is exceeded.
In this work, the dynamic behaviour of a rotor is discussed.The numerical study determines the deformed modes, evaluate Campbell's diagram and the response to the unbalance of the rotor in the vicinity of critical speeds.
The result for the estimation of the term vibration source on a particular line is shown in this paper, and the terms source estimated by both experimental and numerical methods are in excellent agreement with the theory.

Modelling of the rotors
The finite element method is very much used for the calculation of the complex structures is also efficient in dynamics of the rotors (Lalanne, 1998).
The elements of a rotor are: Disc, shaft and bearing.The kinetic energy T, the strain energy U and virtual work δW of external forces are calculated for all system elements to obtain the general equations of motion of a rotor.
It is necessary to define the finite elements making it possible modelling the rotors: discs, shaft, bearing and to represent external forces in particular those due to the unbalances.

The disc
The disk is assumed to be rigid.Only its kinetic energy is considered.The coordinate system x, y and z is connected to the coordinate system X, Y and Z through the angles  ,  and  .

Chellil et al. 203
The angular velocity vector reflecting the position of the disc is written as: x, y and z respectively.The general expression of the kinetic energy of the disc d T is then written: Or this expression can be simplified when the disc is symmetrical dx I = dz I .And when the angles  and  are small and the angular velocity is constant, the Equation (3) becomes:

The shaft
The shaft is assimilated to a beam of circular section and characterized by its kinetic and potential energies.
The expression of the kinetic energy is:  is the density, and S is the shaft section, I is the moment of inertia transverse (diametric).
The first integral of Equation 5 corresponds to the expression of the kinetic energy of a beam in bending, the second to the inertia effect due to the rotation and the last integral represents the gyroscopic effect.
The strain energy depends only on the stress and therefore the strain of the shaft relative to the support.In this calculation, one neglects the shear effects.E is the Young modulus of the material,  and  represent respectively the strain and the stress, u* and w* are the displacements of the geometric center along the axes x and z (in the moving coordinate system).
The strain in bending of a point of the shaft with coordinate x and z in the reference frame R is With the linear strain is given by: The non linear strain is given by: The general expression for the strain energy of the rotor in bending is then: Where is the volume of the shaft and  is the stress in bending.
The relationship between the stress and the strain is Because of the symmetry of the shaft relative to the axes x and y, are obtained: The third term of the integral (9) represents the effect of an axial force and is not considered in this study.Using the Equation 6gives: By symmetry, the third term of Equation 12 is zero and, introducing the inertia of section: To avoid periodic terms, explicit function of time, it is necessary given considering the properties of bearings, to express the strain energy depending on u and w components of the displacement in the initial frame.
The passage of u*, w* at u, w is: Replacing u * and w * by their values (15): For a symmetrical shaft, the expression of the strain energy becomes:

Bearings
A bearing has the characteristics of stiffness and damping in the two planes.The forces applied by the bearings are due to displacement of the shaft relative to the support.
The virtual work P W  of external forces acting on the shaft of the first bearing is written as: Similarly to the second bearing

Mass unbalance
The initial unbalance is generally distributed so as continuous on the rotor.The expression of the kinetic energy T b of the unbalance is: is constant and will not intervene in the equations.The mass of the unbalance is negligible compared to the rotor mass; the expression of the kinetic energy can be approximated by: The terms of the kinetic energy, strain energy and virtual work being established.The expressions of displacements in the X and Z directions are respectively set in the (separation of variables method): Where 1 q and 2 q are generalized independent coordinates.
The second order derivative of u and w displacements is necessary to express the elastic energy of the shaft: The functions ) ( y g and ) ( y h represents the first derivative and the second derivative respectively.
Given that the angular displacements,  and  are small, they are approached by: Chellil et al. 205 The displacement function f is chosen to represent exactly the form of the first mode of a constant section beam in bending on two simply supported situated at its ends.
Substituting in the expressions of ( 25) where (4), we find: The kinetic energy of the disk D T can be written as follows: The expression for the kinetic energy of the shaft Ts is: The strain energy of the shaft is: The kinetic energy of the ensemble disk -rotor is given by: The kinetic energy of the unbalance is: are the respective kinetic energies of the disc, shaft, of the unbalance.The application of Lagrange equations: Which in general form are written: Solving this system of equations provides the expressions of frequencies and deflections of the tree line in each of its points.

Modal analysis and numerical simulation
It is about the calculation of the dynamic behaviour of a rotor step by step in time.The objective is to present a model of calculation using a simplified approach.The construction of the grid is made with ANSYS software starting from the characteristics of the various elements of the rotor.Modelling by finite elements requires the supply of data relating to the geometry (coordinated nodes), boundary conditions, and description of the elements (disc, bearing, and additional elements) of mechanical characteristics of materials and bearing, function of the rotation speed and on the information relative to the excitations.
The finite element Model of the rotor is carried out with the code ANSYS.The shaft is discredited by the beam element to 4 degrees of freedom by node: two displacements U and W and two rotations according to Y and Z.The disc is supposed to be rigid.This model makes it possible to carry out temporal simulations of the dynamic behaviour of the rotor.
The disc is assumed to be perfectly rigid and it's modeling using a pipe element (Pipe16).To model the shaft using the beam element (Beam189).For modeling the bearings are used spring-damper element (Combin14).
The finite element model of the rotor consists of a shaft, disc and bearings.The length and outer diameter of the shaft is

   
The calculation is made from the discretization of the shaft into several elements, once the data is entered the geometric model is established and it resulted in the rotor mesh (Figure 1).All numerical data on the mesh of the rotor is summarized in Table 1.The model comprises 51 nodes, or 204° of freedom and ale modal base consists of five modes.

Boundary conditions
The characterization of the rotor can be made by its decomposition into flexible finite element and therefore study node by node boundary conditions.
During the movement, the center line of the shaft does not remain confused with the original right are (Ux, Uy, Uz) displacements of the shaft, Uy and Uz are variables while Ux is considered as constant since only the shaft deflection movements are studied.
So for the nodes of the two bearings we annul all degrees of freedom, for the nodes of the rotor we annul the degrees of freedom of translation and rotation along the x axis (Figure 2), there are only four degrees of freedom per node, two rotations according y and z, and two translations according y and z.

Evolution of the stress
A force is applied on the disc, and thereafter determines the equivalent Von Mises stress in static analysis (Figure 3).The concentration of stress is observed at the connection between the shaft and the disc.The maximal stress is 201 Mpa; this is a sensitive area for the appearance of defects.

Modal analysis
The modal base consists of five modes.The objective of this study is to determine the eigen frequencies and eigen modes and the stresses of bending vibration of the rotor.The eigen frequencies are represented on the Table 2.The modal base contains five modes.The allure of the three modes of the rotor and their orbits of mode shapes at any rotational speed is shown in Figures 4 to 6.These modes are characterized by local bending.

Campbell diagram
The critical speeds are given by the intersection points of the excitation sources (Harmonics 1, 4, 5 ...) with the natural modes to direct precession and reverse precession.We get Campbell diagram (Lalanne, 1998;Genta, 1992;Muszynska, 1996;Mogenier, 2012;Saad and Seamus, 2011).The diagram of Campbell (Figure 7) shows that the first critical speeds (for the harmonic in the order 1, 4 and 5).For the harmonic of order 1, which corresponds to the first mode of the shaft at the speed of 2500 rev/min (40 Hz)?For the harmonic of order 4, the first critical speed is 600 rev/min (40 Hz); the second critical speed is 4000 rev/min (270Hz) and finally 4500 rev/min (310 Hz).

Spectrum of frequency response
The results in the Fourier spectrum Figure 8  Figure 8 represents the frequency response curve for speeds of rotation from 0 to 500 RPM .The frequency at which the peak response occurs 18 and 37 Hz is.The first frequency is explained by an initial shock due to the applied force.The second frequency corresponds to the frequency of the critical speed of the associated linear system, it amounts to the initial unbalance.(Kumar et al., 2013;Kumar and Kumar, 2014;Jalan and Mohanty, 2009;Yang andCourt, 2009, Chellil et al., 2015).This study will primarily investigate techniques based on the time domain, frequency domain and time-frequency domain [Ding et al., 2016] detailed spectral analysis is presented by Yamamoto et al. (2016), the results clearly show the feasibility of this experimental intelligent diagnostic approach for measuring vibration and unbalance fault detection in rotating machinery.

Vibration
A case of industrial survey was treated for validate the approach and to show that in some cases exceeding the thresholds imposed by the standards represent a defect synonymous.We will make the establishment of the rotor vibration analysis following the method of OFF LINE system.The measurements are performed at regular time intervals (periodically) by portable systems increasingly informatics.The aim of this work is to do a vibration analysis (monitoring and diagnostic) to study vibration phenomena emerged the rotor.
The combustion air ventilators have the task of feeding the steam generator combustion area required in power plants.They aspire outside air and do to the burners through air preheater.In this case the ventilator is composed of a shaft, impeller and two bearings (Figure 9).The high quality steel shaft, the impeller is high quality steel; it is mounted on a cylindrical seat on the shaft by means of an interference fit oil seal.The shaft is mounted in two bearings at bearings type FAG SN 328 and FAG 22328 ES.

Equipments used for test
The measurements were realized using the following equipment.

Vibration sensor
This is an AS-065 piezoelectric accelerometer connected to the VIBROTEST 60 analyzer manifold.This sensor is used to measure vibration acceleration (Figure 10b).

Vibrotest 60 analyzer
The VIBROTEST 60 (Figure 10c) is a practical acquisition apparatus for making: global vibration measurements, process parameter, time signals and spectral signals.
The implementation of the accelerometer on the machines is very important.Each measurement companion must be performed at specific points and always the same.We try to close as possible points of measurements of bearings; it allowed us to get the most loyal mechanical defects images.The first spectral line in the first spectrum (Figure 4a) has an amplitude of 0.47 mm/s, this value exceeds the warning threshold, the frequency of this line is 9.88 Hz (95 rev / min), the speed is the rotational speed of the rotor.

RESULTS AND DISCUSSION
The 2nd spectral line in the same spectrum (Figure 4a) has amplitude of 0.9 mm/s and a frequency 153.76 Hz.This value is below the alert threshold and not a danger.
The first spectral line in the first spectrum (Figure 4b) has an amplitude of 0.235 mm/s when it exceeds the threshold.The frequency of this line is 50 Hz (477.70 rev / min), the speed is the speed of rotation of the rotor.
The 2nd spectral line in the same spectrum (Figure 4b) has an amplitude of 0.14 mm/s and a frequency 153.76 Hz.This value is below the alert value and is not dangerous.
The 1st two spectral lines on the two spectra (Figure 4a  and b) are on the same frequency of rotation of the shaft and we have high vibration speed on the three directions of measurements, so this symptom tells us about the existence of a defect unbalance.
The two spectral lines that follow have frequencies equal to twice the rotation frequency of the shaft, the

Conclusion
The model of finite elements of the rotor was designed starting from a well defined geometry.An effective interpretation of the frequencies and modal deformations of the model were simulated.The distribution of the stress along the rotor was identified.In the present case, the maximum stress is Mpa were concentrated at the connection between the shaft and the disc.The differences on the level of the maximum amplitude of the vibrations can be easily reduced significantly with the geometrical model.Nevertheless, as the critical speed depends on the considered shape, one can determine the intensity of the amplitude on the response curve to unbalance.The layout of the response to an unbalance presents brutal increases in the amplitude of vibration of the rotor.In this article, an detection system based on industrial signals treated in the frequency domain is proposed.The results were presented to identify and detect unbalanced faults in bearing.The analysis of measurement results shows the presence of an unbalance fault on the rotating machinery.This defect is the cause of a long shutdown and poor conditions present during shutdown, it is recommended to remedy a bearing balancing the rotating machinery.


are the components of the angular velocity vector according to x, y and z .considering u and w coordinates of the center O of the disc along OX, OZ; the following coordinated OY remaining constant.The disk mass is I are the moments of inertia along the axis constant, has no influence on the equations of motion and represents the kinetic energy of the rotating disc at the rotational speed Ω, if all other displacement are zero.The last term .15 m respectively.The inner diameter and outer diameter of the disc is r 1 = 0.15 m and r 2 = 0.625 m respectively.The material is a special steel assumed homogeneous and isotropic, with density the damping for the first bearing is:

Figure 3 .
Figure 3. Evolution of the Von Mises equivalent stress.

Figure 7 .
Figure 7.The diagram of Campbell.
are obtained from the numerical simulation.The rotor is subject to an unbalance of 0001 kg and an asynchronous force with the intensity is 1 N placed at node number 19 for L = 1.4 m, for a rotational speed equal to 500 rpm.After running, the responses of each node is obtained, the results of the responses at the node Number 19 are presented as graphs showing the spectral response.

Figure
Figure 11a and b represent the measurements of vibrations in the horizontal direction (Bearing 1) and the vertical direction (Bearing 2) of the pump.The first spectral line in the first spectrum (Figure4a) has an amplitude of 0.47 mm/s, this value exceeds the warning threshold, the frequency of this line is 9.88 Hz (95 rev / min), the speed is the rotational speed of the rotor.The 2nd spectral line in the same spectrum (Figure4a) has amplitude of 0.9 mm/s and a frequency 153.76 Hz.This value is below the alert threshold and not a danger.The first spectral line in the first spectrum (Figure4b) has an amplitude of 0.235 mm/s when it exceeds the threshold.The frequency of this line is 50 Hz (477.70 rev / min), the speed is the speed of rotation of the rotor.The 2nd spectral line in the same spectrum (Figure4b) has an amplitude of 0.14 mm/s and a frequency 153.76 Hz.This value is below the alert value and is not dangerous.The 1st two spectral lines on the two spectra (Figure4a and b) are on the same frequency of rotation of the shaft and we have high vibration speed on the three directions of measurements, so this symptom tells us about the existence of a defect unbalance.The two spectral lines that follow have frequencies equal to twice the rotation frequency of the shaft, the

Table 1 .
the number of element, nodes and degrees of freedom of the rotor.
Figure 2. Boundary conditions of the rotor.

Table 2 .
Eigen frequencies of the rotor.