OPEN ACCESS
The dynamic behavior of the rotors is influenced by the bearings mounted on them. The aim of this research is to study the effect of the stiffness, shaft diameters and dimensions of lubricated journal bearings on the stability of the system using DOE methodology. We first calculated the values of the real frequency and imaginary frequency of the rotor by the Matlab program, then we prepared a matrix containing forty eight tests according to the Placket Berman plan representing the number of experiments and nine columns representing the factors: the rotor diameters and the dimensions of the hydrodynamic bearing and the system stiffness, to find out the most factors affecting the frequency in The real part or the fictional part, and any of them, affects system stability. The results showed that the diameter D1 has the more significant positive effect on the real part frequency, compared to outer journal diameter Db that has a negative effect on the imaginary part frequency and vice versa, as affirmed by the plot of principal effects. By with this criterion one can estimate the frequency as well as the mode for which the system will become unstable. On the other hand, a finite element code has been written in MATLAB to know the eigenvalues, and critical velocity that correspond to the speed at which the unbalanced excitation coincides with the eigenvalue. The results of the tests showed that the finite element method (FEM) was very effective studying system stability.
PlakettBurman, hydrodynamic bearings, stability, stiffness, gyroscopic forces, critical rotational speeds
The development of efficient and accurate numerical procedures to analyze the dynamic phenomena involved on rotorbearing systems have been motivated mainly by the analysis, design and commissioning of highspeed rotating machinery [1, 2]. In the modeling of rotating machines, some of the main features of the system should be considered, such as gyroscopic effects, rotatory inertia and the bearing contribution. When establishing the stiffness and damping coefficients associated with hydrodynamic rotating bearings in the rotor model, it may be demonstrated that they play an important role in the rotor response.
Statistical experimental design, also known as design of experiments (DOE), is the methodology of how to conduct and plan experiments in order to extract the maximum amount of information with the lowest number of analyses [3]. A désignes experiment is a tool or set of tools used for gathering test data. Typical characteristics of an experimental design are planned testing, data analysis approach, simultaneous factor variability and scientific approach [4]. Various industrial applications relate to rotating machines, such as gas turbines, turbojets, turbochargers, and others. The rotor is the main element which generates vibrations in all rotating machines, particularly during their operations, these vibrations are the results of an unbalance fault [5].
A rotor can be defined as a combination of shaft and multiple discs suspended on rigid or flexible bearings that allow it to rotate freely about a fixed axis. The discs are often considered to be rigid and the shaft, flexible. The calculation and subsequent control of unbalance response amplitude and stability limit speed have always been of utmost importance in all areas related to turbo machinery. This is especially true in the case of highspeed machines where uncontrolled vibration and instability of rotating machinery can lead to catastrophic results. Another crucial demand in modern designs is the need for a higher powertoweight ratio. Delivering better performance with lighter machines has added advantages of easier maintenance and transport as well. Considering all such requirements, rotors are designed to be lighter, more flexible, and easily operable at higher speeds.
Earliest among the studies on optimization techniques applied in the field of rotor dynamics has been minimizing a singleobjective function of the response due to system unbalance. Pilkey et al. [6] employed a more efficient linear programming model as opposed to trialanderror methods that were prevalent before. Bhat et al. [7] investigated the influence of journal bearing parameters like bearing diameter, clearance, and oil viscosity in optimum rotor design by minimizing unbalance response amplitude in the operating speed range by using the method of feasible directions. Stocki et al. [8] minimized the vibration amplitude of a compressor shaft while subject to multiple practical constraints on displacements that caused rubbing effects. Helfrich et al. [9] used optimization techniques to maximize the first critical speed of a multidisc rotor system with bearing and disc dimensions as the design variables. Yucel et al. [10] optimized unbalance response amplitude of a rotorbearing system experimentally using the Taguchi method to determine the design variables having the greatest influence on the objective function. While the system used in Pilkey et al. [6] and Bhat et al. [7] do not account for mass of the shaft or gyroscopic effects, all the studies discussed above consider optimization of a singleobjective function.
Therefore, it’s essential to reduce vibrations to ensure safe and stable operation of considered machine. This can be realized by proper investigation of the system dynamics. The modal analysis is carried out to get an idea of the dynamic behavior of the system.
A very basic model of a rotor was provided by Jeffcott [11]. First, he considered three hypotheses which are: (i) the rotor carries a point mass, (ii) axially symmetrical rotor, and (iii) no damping is related to the rotor. Then, the model was extended to take damping into account. Irretier et al. [12] constructed a mathematical formulation for the modal analysis of the entire shaftrotor system at first as a linear time independent system (LTI) and later as a Linear Time Varying system (LTV). Rotor rotation generates additional forces such as gyroscopic, tangential, and rotating damping forces [13]. Because of the effects of these forces, the structure of system matrices becomes asymmetric and depends strongly on the speed [14].
The stability analysis of symmetrical rotor bearing systems was studied by Laszlo [15] using finite element method, he taken in consideration the internal damping. His findings showed that the whirling motion of the rotor system becomes unstable beyond the critical speed of instability; he found that the rotor stability is enhanced by increasing of bearings damping.
Fegade et al. [16] and Patel et al. [17] in their works, thy studied the harmonic analysis of the rotor to identify the frequency using the variation of the diameter by design optimization (DOE) and by parametric design using ANSYS software. In addition, another study was carried out to develop an alternative procedure called harmonic analysis to identify the frequency of a system through critical speed, amplitude, and phase angle curves using ANSYS.
A study was performed using the PlakettBurman statistical method on the experimental designs in order to define the influence of the stiffness coefficients on the rotating machines dynamics in particular on the diameters which produce these high frequencies [18]. The factors interactions can affect by increasing or decreasing the principal effects as affirmed by the interaction and surface graphs. Their results show that the inclusion of the stiffness coefficients on the dynamic analysis of rotating machines supported on hydrodynamic bearings play a significant role on the estimation of the unbalance response of rotors.
Naouri Abdallah et al. [19]. In their study, they considered that the dynamic behavior of fluid film bearings is one of the main factors, which affects the rotating machine performances. In this study, a rigid rotor supported by two identical hydrodynamic bearings is taken into consideration. The principal goal of this work is to predict the effect of the damping film of the hydrodynamic bearings on the rotating machines stability.
This paper Concerns the optimization and the modeling of the stability and reliability of operation of a rotor system using the methodology of design of experiments (DOE).
First, we calculated the values of real frequency and imaginary frequency of the rotor by the Matlab program then set up a matrix containing forty eight tests according to the Placket Berman plan which represents the number of experiments and nine columns representing the factors, which are the following the diameters of the rotor, and the dimensions of the hydrodynamic bearing, and the rigidity of the system, in order to know the most influencing factors on the frequency in the real part or the imaginary part, and which of them have an impact on the destabilization of the system. It was found by the results that the factors having a positive effect on the increase in frequency in the real part have a negative effect on the frequency in the imaginary part. Finite element code is written in MATLAB to determine the eigenvalues and eigen vectors. Eigenvalues are calculated in imaginary parts, that show the system natural frequencies, which are used to plot the Campbell diagram. The stability diagram is plotted using the maximum real fraction of all eigenvalues with rotational velocity. Eigenvalues are also used to study the effect of a typical damping factor on rotational speed. These diagrams show that the stability of the system can be studied and the rotational speed stability limit can be determined.
Figure 1 illustrates the cross section of a "plain" journal bearing [20]. The bearing main dimensions are: $D_{B}=2 R_{B}$  inner bearing diameter, $D=2 R$ outer journal diameter, $L$  bearing length, $C=R_{B}R$  bearing clearance, $e=\overline{o_{B} o_{J}}$ eccentricity, $\phi$  attitude angle and $h$  fluid film thickness.
Figure 1. Geometry of a plain cylindrical bearing
It is necessary to present the following dimensionless engineering parameters: $L / D$  lengthtodiameter ratio; $C / R$  clearance ratio; $\varepsilon=e C$ eccentricity ratio; and $\bar{h}==h / C$ normalized film thickness.
The journal rotates anticlockwise with a constant angular speed $\Omega=2 \pi N(\mathrm{rad} / \mathrm{sec})$, where $N$ represents the journal spin speed in rps.
3.1 Effects of the bearings
The bearings act as external forces acting on the rotor. They are characterized by their stiffness and damping.
$F_{u}=K_{x x} uk_{x z} wC_{x x} \dot{u}C_{x z} \dot{w}$
$F_{w}=K_{z z} wk_{z x} uC_{z z} \dot{w}C_{z x} \dot{u}$ (1)
In matrix form:
$[F]=[K] \delta[C] \ddot{\delta}$ (2)
With stiffness With bearing damping
$[K]=\left[\begin{array}{cccc}k_{x x} & 0 & k_{x z} & 0 \\ 0 & 0 & 0 & 0 \\ k_{z x} & 0 & k_{z z} & 0 \\ 0 & 0 & 0 & 0\end{array}\right]\left[C_{d}\right]=\Omega\left[\begin{array}{cccc}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{D y} \\ 0 & 0 & I_{D y} & 0\end{array}\right]$ (3)
3.2 Equations of the rotor movement
The application of the Lagrange equations on the different energies gives:
For the shaft:
$\frac{d}{d t}\left(\frac{\partial T}{\partial \dot{\delta}}\right)\frac{\partial T}{\partial \delta}=\left(M+M_{S}\right) \ddot{\delta}+C \dot{\delta}$ (4)
With mass and gyroscopic effect matrices. The mass matrix is symmetrical. The matrix C is antisymmetric.
$\frac{\partial U}{\partial \delta}=\left(k_{c}+k_{F}\right) \delta$ (5)
K_{c} takes into account the effect of shear, K_{F} is due to axial forces.
For the disc:
$\frac{d}{d t}\left(\frac{\partial t}{\partial \dot{\delta}}\right)\frac{\partial T}{\partial \delta}=M_{d} \ddot{\delta}+C_{d} \dot{\delta}$ (6)
$\left[M_{d}\right]=\left[\begin{array}{cccc}k_{x x} & 0 & 0 & 0 \\ 0 & M_{D} & 0 & 0 \\ k_{z x} & 0 & I_{D x} & 0 \\ 0 & 0 & 0 & I_{D z}\end{array}\right][C]=\left[\begin{array}{cccc}c_{x x} & 0 & c_{x z} & 0 \\ 0 & 0 & 0 & 0 \\ c_{z x} & 0 & c_{z z} & 0 \\ 0 & 0 & 0 & 0\end{array}\right]$ (7)
From equations of the shaft, the disk and the bearings the equation of the rotor motion is written in the form:
$M \ddot{\delta}+C(\Omega) \delta+\dot{K} \delta=0$ (8)
The mass matrix contains the mass of the rotor and disks. The stiffness matrix contains the stiffness of the shaft and bearings. C contains the gyroscopic effect of the shaft and the disks and the matrix damping of the bearings.
The assembly of the displacement vectors of all the rotor nodes into finite elements gives the global displacement vector {X} and the global matrices. In this case the equation of rotor motion becomes:
$[M]\{\ddot{X}\}+\left(\Omega\left[C_{g}\right]+\left[C_{p}\right]\right)\{\dot{X}\}+\left(\left[K_{r}\right]+\left[K_{p}\right]\right)\{X\}=0$ (9)
The gyroscopic effect of the shaft and disks and the matrix damping of the bearings. F can represent unbalance or any other external forces.
3.3 Some important phenomena in rotor dynamics
We will now consider some important phenomena of rotor dynamics, and see in particular the concepts of critical velocities and instability related to rotational damping and the role that differences in rotor dynamics play.
The critical velocity corresponds with the speed at which the unbalanced excitation coincides with one of the natural frequencies of the system. In machines consisting of organs with significant moments of polar inertia, we observe a strong dependence on eigen patterns versus rotational velocity due to gyroscopic effects. Thus, we observe the duality of the eigen patterns of the system (in the case of the axial symmetry system) due to the gyroscopic forces as follows:
 (FW) as the rotor rotates in the same direction as it rotates. Then, under gyroscopic effects, the associated resonant frequency increases and this is called "direct precession".
 (BW), where the rotor rotates in the opposite direction to its preliminary motion, resulting in a softening effect and thus a decrease in the critical velocity and this is called the "retrograde precession".
In the study of stability of rotating machines where damping corresponds to one of the determinants of sizing systems, damping can be responsible for unstable phenomena at high speed, which may lead to rupture of rotor components. In order to ensure system integrity, the damping ratios on both fixed and moving parts must be estimated.
Stability analysis in the study of the vibrational and dynamic behavior of a flexible rotor is essential because it is considered a dynamic system governed by systems of differential equation. The definition of stability covers Leibunov's definition of equilibrium stability analysis and the Poincaré definition of the concept of orbital stability [21].
We can know the limits of dynamic system instability and in particular in rotor dynamics through the following techniques:
The use of this criterion is interesting for systems with a few degrees of freedom, since the analytical expressions of the characteristic polynomial associated with the perturbed motion can be inferred. However, it becomes complicated for systems that include a large number of degrees of freedom. Additionally, this criterion does not provide an instability frequency.
These two criteria studying the stability of a dynamic system restrict particular cases, or when they are described by linear models, for example R. Sino [23]. In the subject of his thesis, he uses these two methods to study and analyze the stability of a rotor due to rotating damping. A more general method is based on Floquet's theory.
The main objective of this work is to model and optimize the operation of the rotor system using DOE methodology and to know the gyroscopic effect on the subjective values of the rotorbearing systems through the Campbell scheme, and to calculate the imbalance responses mainly during the passage of critical velocity. In the use of finite element method. We designed a mathematical model under the rotor kit name in the matlab program that contains the engineering data for the rotor group element (tree data, disk data, and engineering data for the rotor group component.
In addition to the stiffness and damping matrices in the form of a group of nodes and elements to calculate the values of hardness and the imaginary and real frequency in the presence of speed 010000 rpm. This calculation mode should be able to give the geometry of the rotor in finite elements. The search for eigenvalues is an essential process in the study of rotor dynamics.
Figure 2. Schematic model of the used model of rotor kit
The used model is a rotor kit with a length of 0.42m as shown in Figure 2. A mass of 1.415 kg is mounted on the shaft which is supported by two bearings respectively 0.09 (m) and 0.42 (m) apart from the left end. four stations are considered during harmonic analysis as shown in Figure 1, where station numbers denote different nodes in the model (1) Disc, (2) First bearing node, (3) Disk, (4) Second bearing. For the distributed rotor and the concentrated disc (1), the material density is 7850 kg / m^{3} and the modulus of elasticity is 2.06E11 N / m^{2}. with a mass of 0.81 kg, disk (4) with a mass of 0.6050 kg polar inertia (5.783e4, 0 ,4.320e4) kg.m^{2} and diametral inertia (3.357e4,0, 2.433e4) kg.m^{2}.
Calculation of real and imaginary system frequency values
A mathematical model was designed under the name of rotor kit using Matlab which includes the data of each element of the rotor kit (shaft, disc, and bearing), and also the stiffness and damping matrices in the form of a set of nodes and elements were used to determine the values of the stiffness and the real and imaginary frequency at a speed interval ranging from 0 to 10,000 rpm (Table 1).
Table 1. Geometric data of rotorbearing element
Element Node No 
Node Location (cm) 
Bearing and Disk 
Inner Diameter (cm) 
Outer Diameter (cm) 
1 
0 
disc 
0 
0.009525 
2 
0.09 
Bearing 
0 
0.009525 
3 
0.22 
disc 
0 
0.009525 
4 
0.42 
Bearing 
0 
0.009525 
5.1 Response
The response chosen is the real frequency and the imaginary frequency of a rotor system  bearings, calculated by matlab software.
5.2 Determination of factors and field of study
The factors examined in this study are:
 The diameters of the rotor kit d1, d2, d3.d4.
 The dimensions of the hydrodynamic bearings
Lb = length of bearings [m].
Db = outer journal diameter [m].
Cb = bearing cleareance [m].
 The shaft is mounted on two fluid film bearings where the stiffness (Kyy, Kzz) was determined using Matlab software.
 Kyz= Kyz = 0.
 The components of damping are taken as: Czz = Cyy = 61, 4000 (Ns / m). The imbalance response for a counted disc center eccentricity of 0.635 (cm) at station (2) was estimated for a speed interval of 0 to 10000 rpm.
5.3 Choice of the experimental design
First, a screening plan is used. This is a first degree plan which allows you to sort the factors by highlighting the most influential. We chose the PlackettBurman plan because it is generally the most used in similar studies due to its economy in terms of number of tests.
The nine factors studied and their field of study were grouped together in Table 2.
Table 2. Caption
Levels 
Units 
Symbol 
Factors 

+1 
1 



m)) 
d1=d2=d3=d4 
The diameters 
0.050 0.10 0.00009 
0.045 0.09 0.00005 
)m) (m) m)) 
Lb Db Cb 
The dimensions of the hydrodynamic bearings 
2,2e5 
0 
Ns/ m 
Kyy= Kzz 
Stiffness 
6.1 Carrying out the tests
The tests are conducted according to the PlackettBurman plan for the 9 factors. The experiments took place according to the matrix of this plan. It represents the fixing of factors at different levels, as shown in Table 3.
Table 3. PlackettBurman plan based on the experimental matrix

d1 
d2 
d3 
d4 
Lb 
Cb 
Db 
Kyy 
Kzz 
FRQ img 
FRQ real 
1 
0.009475 
0.009475 
0.009575 
0.009475 
0.045 
0.1 
0.00009 
2.2E+05 
0.0E+00 
16.463 
310.192 
2 
0.009575 
0.009475 
0.009575 
0.009575 
0.05 
0.1 
0.00009 
0.0E+00 
0.0E+00 
44.0043 
311.976 
3 
0.009575 
0.009575 
0.009575 
0.009575 
0.045 
0.1 
0.00009 
2.2E+05 
2.2E+05 
145.212 
14.0174 
4 
0.009575 
0.009475 
0.009575 
0.009475 
0.045 
0.09 
0.00009 
2.2E+05 
0.0E+00 
16.463 
310.263 
5 
0.009575 
0.009475 
0.009475 
0.009575 
0.05 
0.1 
0.00009 
0.0E+00 
2.2E+05 
16.617 
279.268 
6 
0.009475 
0.009475 
0.009475 
0.009575 
0.05 
0.09 
0.00009 
2.2E+05 
0.0E+00 
16.616 
323.984 
7 
0.009575 
0.009475 
0.009475 
0.009475 
0.045 
0.1 
0.00005 
0.0E+00 
0.0E+00 
44.5628 
325.573 
8 
0.009475 
0.009575 
0.009475 
0.009475 
0.045 
0.09 
0.00009 
2.2E+05 
0.0E+00 
16.395 
324.338 
9 
0.009575 
0.009575 
0.009575 
0.009475 
0.045 
0.09 
0.00005 
2.2E+05 
0.0E+00 
16.234 
310.591 
10 
0.009475 
0.009475 
0.009575 
0.009575 
0.05 
0.1 
0.00005 
2.2E+05 
2.2E+05 
142.206 
13.7369 
11 
0.009475 
0.009475 
0.009475 
0.009475 
0.05 
0.09 
0.00005 
0.0E+00 
0.0E+00 
44.5475 
325.565 
12 
0.009475 
0.009575 
0.009475 
0.009575 
0.045 
0.09 
0.00005 
2.2E+05 
2.2E+05 
142.17 
13.5302 
13 
0.009475 
0.009575 
0.009575 
0.009475 
0.05 
0.1 
0.00005 
0.0E+00 
2.2E+05 
16.236 
310.516 
14 
0.009475 
0.009575 
0.009475 
0.009575 
0.045 
0.09 
0.00009 
2.2E+05 
2.2E+05 
142.17 
13.5302 
15 
0.009475 
0.009475 
0.009575 
0.009575 
0.05 
0.09 
0.00009 
0.0E+00 
2.2E+05 
16.463 
310.192 
16 
0.009575 
0.009475 
0.009475 
0.009475 
0.045 
0.1 
0.00009 
0.0E+00 
2.2E+05 
16.617 
324.052 
17 
0.009575 
0.009575 
0.009575 
0.009575 
0.045 
0.09 
0.00005 
0.0E+00 
0.0E+00 
16.234 
310.591 
18 
0.009575 
0.009575 
0.009475 
0.009575 
0.05 
0.09 
0.00005 
2.2E+05 
0.0E+00 
16.396 
324.409 
19 
0.009575 
0.009575 
0.009575 
0.009575 
0.05 
0.09 
0.00005 
0.0E+00 
0.0E+00 
44.4545 
312.228 
20 
0.009475 
0.009475 
0.009575 
0.009575 
0.045 
0.1 
0.00005 
2.2E+05 
0.0E+00 
16.463 
310.192 
21 
0.009475 
0.009575 
0.009475 
0.009475 
0.05 
0.1 
0.00009 
0.0E+00 
2.2E+05 
16.395 
324.338 
22 
0.009475 
0.009475 
0.009475 
0.009475 
0.05 
0.1 
0.00005 
2.2E+05 
0.0E+00 
16.616 
323.984 
23 
0.009575 
0.009475 
0.009475 
0.009575 
0.05 
0.1 
0.00005 
2.2E+05 
0.0E+00 
16.617 
324.052 
24 
0.009475 
0.009475 
0.009575 
0.009475 
0.045 
0.09 
0.00005 
2.2E+05 
2.2E+05 
142.206 
13.7369 
25 
0.009475 
0.009575 
0.009475 
0.009475 
0.045 
0.1 
0.00009 
0.0E+00 
2.2E+05 
16.395 
324.338 
26 
0.009575 
0.009475 
0.009475 
0.009575 
0.045 
0.09 
0.00009 
2.2E+05 
2.2E+05 
144.574 
13.9962 
27 
0.009475 
0.009475 
0.009475 
0.009475 
0.045 
0.09 
0.00005 
0.0E+00 
0.0E+00 
44.5475 
325.565 
28 
0.009575 
0.009475 
0.009575 
0.009475 
0.05 
0.09 
0.00005 
2.2E+05 
2.2E+05 
145.563 
14.1185 
29 
0.009575 
0.009575 
0.009575 
0.009475 
0.05 
0.09 
0.00009 
0.0E+00 
0.0E+00 
44.4545 
262.112 
30 
0.009475 
0.009475 
0.009575 
0.009575 
0.045 
0.1 
0.00009 
0.0E+00 
0.0E+00 
43.9888 
311.966 
31 
0.009475 
0.009475 
0.009475 
0.009575 
0.045 
0.09 
0.00005 
0.0E+00 
2.2E+05 
16.616 
323.984 
32 
0.009475 
0.009475 
0.009475 
0.009575 
0.05 
0.09 
0.00009 
0.0E+00 
2.2E+05 
16.616 
323.984 
33 
0.009575 
0.009575 
0.009475 
0.009575 
0.045 
0.1 
0.00005 
0.0E+00 
0.0E+00 
44.9909 
325.855 
34 
0.009475 
0.009575 
0.009475 
0.009475 
0.05 
0.1 
0.00009 
2.2E+05 
0.0E+00 
16.395 
324.338 
35 
0.009575 
0.009575 
0.009475 
0.009475 
0.05 
0.09 
0.00005 
2.2E+05 
2.2E+05 
144.904 
13.8962 
36 
0.009575 
0.009475 
0.009575 
0.009475 
0.05 
0.09 
0.00005 
0.0E+00 
2.2E+05 
16.463 
310.263 
37 
0.009475 
0.009575 
0.009575 
0.009475 
0.045 
0.1 
0.00005 
0.0E+00 
2.2E+05 
16.395 
324.338 
38 
0.009575 
0.009575 
0.009475 
0.009575 
0.05 
0.1 
0.00009 
2.2E+05 
0.0E+00 
16.396 
324.409 
39 
0.009475 
0.009575 
0.009575 
0.009575 
0.045 
0.1 
0.00005 
2.2E+05 
0.0E+00 
16.236 
310.516 
40 
0.009575 
0.009575 
0.009475 
0.009575 
0.045 
0.1 
0.00005 
0.0E+00 
2.2E+05 
16.396 
324.409 
41 
0.009575 
0.009475 
0.009475 
0.009475 
0.05 
0.1 
0.00005 
2.2E+05 
2.2E+05 
144.574 
13.9962 
42 
0.009475 
0.009575 
0.009575 
0.009475 
0.05 
0.09 
0.00009 
0.0E+00 
0.0E+00 
44.4388 
312.217 
43 
0.009475 
0.009575 
0.009575 
0.009575 
0.05 
0.09 
0.00009 
2.2E+05 
2.2E+05 
142.457 
13.6378 
44 
0.009575 
0.009475 
0.009575 
0.009475 
0.045 
0.1 
0.00009 
2.2E+05 
2.2E+05 
145.563 
14.1185 
45 
0.009475 
0.009575 
0.009575 
0.009575 
0.05 
0.1 
0.00005 
0.0E+00 
2.2E+05 
44.4388 
312.217 
46 
0.009575 
0.009575 
0.009575 
0.009475 
0.05 
0.1 
0.00009 
2.2E+05 
2.2E+05 
145.212 
14.0174 
47 
0.009575 
0.009475 
0.009575 
0.009575 
0.045 
0.09 
0.00009 
0.0E+00 
0.0E+00 
44.0043 
311.976 
48 
0.009575 
0.009575 
0.009475 
0.009475 
0.045 
0.09 
0.00009 
0.0E+00 
0.0E+00 
44.9909 
325.855 
The processing of the experimental data was carried out by linear regression multiple using the MINITAB17 software.
7.1 Graphic representation of effects
This diagram (Figure 3) makes it possible to extract the most important parameters. Among all the factors studied and at the chosen confidence level (α=0.05), the strong factors (kzz) and (kyy) appear to be very influential factors, kzz, kyy having the most significant positive effect in the imaginary part frequency and affects negatively the frequency in real part.
A. Pareto chart:
Figure 3. Pareto plot of normalized effects
Figure 4 reveals that kzz and kyy has the greatest significant positive effect on the rightmost imaginary frequency of the response line. However, the figure reveals a significant reduction effect of kzz and kyy on the real frequency of its effect is positioned to the left of the answer line.
B. Pareto plot of normalized
Figure 4. Pareto plot of normalized effects
C. Main effects diagram
The main effects diagram tells us about the simultaneous influence of all factors on the frequency. We can from this diagram (Figure 5) conclude that the stiffness kzz and kyy are the most influential factors positively on the imaginary part frequency, and negatively influential on the real part frequency.
Figure 5. Diagram of the main effects on frequency imaginary real
7.2 Determination of significant effects and coefficients of the model
The effects values and the coefficients of regression of the model are given as bellow in Tables 4 and 5.
Table 4. Factorial Regression: FRQ img versus d1; d2; d3; d4; Lb; Cb; Db; Kyy; Kzz
Source 
DF 
Adj SS 
Adj MS 
FValue 
PValue 
Model 
9 
71923 
7991.5 
5.07 
0.000 
Linear 
9 
71923 
7991.5 
5.07 
0.000 
D1 
1 
3984 
3984.4 
2.53 
0.120 
D2 
1 
0 
0.2 
0.00 
0.991 
D3 
1 
1986 
1985.5 
1.26 
0.269 
D4 
1 
2 
2.2 
0.00 
0.970 
Lb 
1 
240 
239.7 
0.15 
0.699 
Cb 
1 
3577 
3577.1 
2.27 
0.140 
Db 
1 
0 
0.2 
0.00 
0.992 
kyy 
1 
29663 
29662.7 
18.80 
0.000 
kzz 
1 
36092 
36092.3 
22.88 
0.000 
error 
38 
59946 
1577.5 


Total 
47 
131869 



Model Summary
S Rsq Rsq (adj) Rsq (pred)
39.7180 54.54% 43.77% 27.61%
Table 5. Factorial Regression: FRQ real versus d1; d2; d3; d4; Lb; Cb; Db; Kyy; Kzz
Source 
DF 
Adj SS 
Adj MS 
FValue 
PValue 
Model 
9 
595380 
66153 
10.83 
0.000 
Linear 
9 
595380 
66153 
10.83 
0.000 
D1 
1 
21791 
21791 
3.57 
0.067 
D2 
1 
0 
0 
0.00 
0.999 
D3 
1 
14753 
14753 
2.42 
0.128 
D4 
1 
12 
12 
0.00 
0.965 
Lb 
1 
1032 
1032 
0.17 
0.683 
Cb 
1 
19436 
19436 
3.18 
0.082 
Db 
1 
187 
187 
0.03 
0.862 
kyy 
1 
264891 
264891 
43.38 
0.000 
kzz 
1 
296031 
296031 
48.48 
0.000 
error 
38 
232052 
6107 


Total 
47 
827431 



Summary of model
S Rsq Rsq (adj) Rsq (pred)
78.1449 71.96% 65.31% 55.35%
Table 6. The used Runs in DOE

d1 
d2 
d3 
d4 
Lb 
Cb 
Db 
Kyy 
Kzz 
FRQ img 
FRQ real 
1 
0.009475 
0.009475 
0.009475 
0.009575 
0.045 
0.09 
0.00009 
0.0E+00 
2.2E+05 
16.616 
323.984 
2 
0.009475 
0.009575 
0.009575 
0.009475 
0.05 
0.09 
0.00009 
2.2E+05 
0.0E+00 
44.439 
312.217 
3 
0.009475 
0.009475 
0.009575 
0.009575 
0.045 
0.09 
0.00005 
2.2E+05 
2.2E+05 
142.46 
13.6378 
4 
0.009575 
0.009475 
0.009575 
0.009475 
0.045 
0.1 
0.00009 
2.2E+05 
2.2E+05 
144.9 
13.8962 
5 
0.009475 
0.009475 
0.009575 
0.009575 
0.05 
0.1 
0.00005 
0.0E+00 
2.2E+05 
44.439 
312.217 
6 
0.009575 
0.009475 
0.009475 
0.009475 
0.05 
0.1 
0.00009 
0.0E+00 
2.2E+05 
16.463 
310.263 
7 
0.009575 
0.009575 
0.009575 
0.009475 
0.05 
0.1 
0.00005 
0.0E+00 
0.0E+00 
44.455 
262.112 
8 
0.009475 
0.009575 
0.009475 
0.009475 
0.05 
0.1 
0.00005 
2.2E+05 
0.0E+00 
16.395 
324.338 
9 
0.009575 
0.009575 
0.009475 
0.009475 
0.05 
0.09 
0.00005 
2.2E+05 
2.2E+05 
145.56 
14.1185 
10 
0.009475 
0.009475 
0.009475 
0.009475 
0.05 
0.1 
0.00005 
0.0E+00 
2.2E+05 
16.395 
324.338 
11 
0.009575 
0.009475 
0.009475 
0.009575 
0.045 
0.09 
0.00005 
2.2E+05 
0.0E+00 
144.57 
13.9962 
12 
0.009475 
0.009575 
0.009475 
0.009475 
0.05 
0.09 
0.00005 
0.0E+00 
0.0E+00 
44.548 
325.565 
13 
0.009475 
0.009475 
0.009575 
0.009475 
0.045 
0.1 
0.00005 
2.2E+05 
2.2E+05 
144.57 
13.9962 
14 
0.009475 
0.009575 
0.009475 
0.009575 
0.05 
0.09 
0.00005 
0.0E+00 
2.2E+05 
44.991 
325.855 
15 
0.009575 
0.009575 
0.009475 
0.009575 
0.045 
0.09 
0.00009 
0.0E+00 
0.0E+00 
16.616 
323.984 
16 
0.009475 
0.009575 
0.009575 
0.009575 
0.05 
0.09 
0.00009 
2.2E+05 
2.2E+05 
44.439 
312.217 
17 
0.009575 
0.009575 
0.009575 
0.009575 
0.045 
0.1 
0.00005 
2.2E+05 
2.2E+05 
142.46 
13.6378 
18 
0.009575 
0.009475 
0.009575 
0.009575 
0.045 
0.09 
0.00005 
0.0E+00 
2.2E+05 
145.56 
14.1185 
19 
0.009575 
0.009475 
0.009575 
0.009575 
0.05 
0.1 
0.00009 
0.0E+00 
2.2E+05 
44.439 
312.217 
20 
0.009575 
0.009575 
0.009475 
0.009575 
0.045 
0.1 
0.00009 
2.2E+05 
0.0E+00 
16.463 
310.263 
21 
0.009475 
0.009475 
0.009575 
0.009575 
0.045 
0.1 
0.00009 
2.2E+05 
0.0E+00 
16.395 
324.338 
22 
0.009575 
0.009575 
0.009475 
0.009575 
0.045 
0.1 
0.00009 
0.0E+00 
2.2E+05 
16.396 
324.409 
23 
0.009575 
0.009575 
0.009475 
0.009575 
0.05 
0.09 
0.00009 
2.2E+05 
0.0E+00 
16.236 
310.516 
24 
0.009475 
0.009475 
0.009575 
0.009575 
0.05 
0.09 
0.00009 
2.2E+05 
0.0E+00 
16.396 
324.409 
25 
0.009475 
0.009575 
0.009475 
0.009575 
0.05 
0.09 
0.00005 
0.0E+00 
2.2E+05 
16.616 
323.984 
26 
0.009475 
0.009575 
0.009475 
0.009475 
0.05 
0.1 
0.00005 
0.0E+00 
2.2E+05 
16.395 
324.338 
27 
0.009575 
0.009475 
0.009575 
0.009475 
0.05 
0.09 
0.00005 
2.2E+05 
0.0E+00 
145.56 
14.1185 
28 
0.009575 
0.009575 
0.009475 
0.009475 
0.045 
0.1 
0.00005 
2.2E+05 
2.2E+05 
144.57 
13.9962 
29 
0.009575 
0.009575 
0.009575 
0.009475 
0.045 
0.09 
0.00009 
2.2E+05 
2.2E+05 
145.21 
14.0174 
30 
0.009575 
0.009575 
0.009475 
0.009575 
0.045 
0.09 
0.00009 
0.0E+00 
0.0E+00 
44.991 
325.855 
31 
0.009575 
0.009575 
0.009475 
0.009475 
0.05 
0.1 
0.00009 
2.2E+05 
0.0E+00 
16.395 
324.338 
32 
0.009575 
0.009575 
0.009575 
0.009475 
0.045 
0.1 
0.00009 
0.0E+00 
0.0E+00 
44.455 
262.112 
33 
0.009475 
0.009575 
0.009575 
0.009575 
0.045 
0.1 
0.00005 
0.0E+00 
0.0E+00 
43.989 
311.966 
34 
0.009575 
0.009475 
0.009475 
0.009475 
0.05 
0.1 
0.00009 
0.0E+00 
2.2E+05 
16.396 
324.409 
35 
0.009475 
0.009575 
0.009575 
0.009475 
0.05 
0.1 
0.00005 
0.0E+00 
2.2E+05 
16.236 
310.516 
36 
0.009575 
0.009475 
0.009575 
0.009475 
0.05 
0.09 
0.00005 
2.2E+05 
0.0E+00 
145.56 
14.1185 
37 
0.009475 
0.009575 
0.009475 
0.009575 
0.045 
0.1 
0.00005 
0.0E+00 
2.2E+05 
43.989 
311.966 
38 
0.009475 
0.009475 
0.009575 
0.009575 
0.045 
0.09 
0.00009 
0.0E+00 
0.0E+00 
16.395 
324.338 
39 
0.009475 
0.009475 
0.009475 
0.009575 
0.045 
0.1 
0.00005 
0.0E+00 
0.0E+00 
16.616 
323.984 
40 
0.009575 
0.009575 
0.009575 
0.009575 
0.045 
0.1 
0.00009 
0.0E+00 
0.0E+00 
16.396 
324.409 
41 
0.009475 
0.009475 
0.009575 
0.009475 
0.045 
0.1 
0.00005 
2.2E+05 
2.2E+05 
142.21 
13.7369 
42 
0.009475 
0.009575 
0.009575 
0.009475 
0.05 
0.09 
0.00009 
0.0E+00 
2.2E+05 
16.395 
324.338 
43 
0.009575 
0.009475 
0.009475 
0.009575 
0.045 
0.09 
0.00005 
2.2E+05 
0.0E+00 
144.57 
13.9962 
44 
0.009475 
0.009475 
0.009475 
0.009475 
0.045 
0.09 
0.00009 
0.0E+00 
0.0E+00 
44.548 
325.565 
45 
0.009575 
0.009475 
0.009575 
0.009475 
0.05 
0.09 
0.00009 
2.2E+05 
2.2E+05 
144.9 
13.8962 
46 
0.009475 
0.009475 
0.009475 
0.009475 
0.05 
0.09 
0.00009 
2.2E+05 
0.0E+00 
16.617 
324.052 
47 
0.009575 
0.009475 
0.009475 
0.009575 
0.05 
0.1 
0.00009 
2.2E+05 
0.0E+00 
16.616 
323.984 
48 
0.009475 
0.009475 
0.009575 
0.009475 
0.045 
0.09 
0.00005 
2.2E+05 
0.0E+00 
142.21 
13.7369 
7.3 Mathematical model equation
To build the model equation representing the relationship between the frequency (imgreal) and the 9 factors studied, we use the regression coefficients shown in Tables 4 and 5. This model has been simplified, and the ranking of factors is done according to the diagram of Pareto (Figures 3 and 4).
FRQ img = 2705 + 182863 d1 – 1239 d2 + 128641 d3  4289 d4 – 897 Lb 1733 Cb + 2952 Db + 0.000226 Kyy + 0.000252 Kzz
FRQ real = 7438 – 427644 d1 – 204 d2 – 350633 d3 – 9860 d4 + 1862 Lb+ 4039 Cb 98695 Db 0.000675 Kyy 0.000721 Kzz
• Optimization method
To enhance the seven remaining response factors, the values are set to P < 0.05, the red reference line was changed to zero by moving manually the columns of the base design matrix, maintaining the corresponding frequency values for each row, and keeping the matrix balanced until building a final matrix as shown in Table 6 [18].
The goal is therefore to find the optimal polynomial equation. From the previous statistical analysis, eliminating the quadratic terms yields a new wellfitting model. The results are presented in Tables 7 and 8.
Table 7. Factorial Regression: FRQ img versus d1; d2; d3; d4; Lb; Cb; Db; Kyy; Kzz
Source 
DF 
Adj SS 
Adj MS 
FValue 
PValue 
Model 
9 
142685 
15853.9 
97.80 
0.000 
Linear 
9 
142685 
15853.9 
97.80 
0.000 
D1 
1 
22047 
22044.9 
135.99 
0.000 
D2 
1 
3084 
3083.6 
19.02 
0.000 
D3 
1 
7235 
7235.1 
44.63 
0.000 
D4 
1 
6476 
6476.2 
39.95 
0.000 
Lb 
1 
17393 
17393.5 
107.30 
0.000 
Cb 
1 
15817 
15816.9 
97.57 
0.000 
Db 
1 
32677 
32677.4 
201.58 
0.000 
kyy 
1 
15627 
15627.0 
96.40 
0.000 
kzz 
1 
10429 
10429.3 
64.34 
0.000 
error 
38 
6160 
162.1 


Total 
47 
148845 



S Rsq Rsq (adj) Rsq (pred)
12.7320 95.86% 94.88% 93.37%
FRQ img = 1947 + 461686 d1  166044 d2 + 256755 d3  255436 d4 8302 Lb 3822 Cb 1379219 Db+ 0.000180 Kyy + 0.000141 Kzz
Obs FRQ img Fit Resid Std Resid
35 16.24 41.07 24.84 2.13 R
Table 8. Factorial Regression: FRQ real versus d1; d2; d3; d4; Lb; Cb; Db; Kyy; Kzz
Source 
DF 
Adj SS 
Adj MS 
FValue 
PValue 
Model 
9 
943547 
104839 
409.14 
0.000 
Linear 
9 
943547 
104839 
409.14 
0.000 
D1 
1 
158263 
158263 
617.64 
0.000 
D2 
1 
27375 
27375 
106.84 
0.000 
D3 
1 
38406 
38406 
149.88 
0.000 
D4 
1 
46773 
46773 
182.54 
0.000 
Lb 
1 
112402 
112402 
438.66 
0.000 
Cb 
1 
75656 
75656 
295.25 
0.000 
Db 
1 
193402 
193402 
754.77 
0.000 
kyy 
1 
123233 
123233 
480.93 
0.000 
kzz 
1 
67846 
67846 
264.78 
0.000 
error 
38 
9737 
256 


Total 
47 
104 



S Rsq Rsq (adj) Rsq (pred)
16.0075 98.98% 98.74% 98.37%
FRQ real = 4452  1237034 d1 + 494735 d2  591557 d3 + 686467 d4 + 21104 Lb + 8358 Cb + 3355373 Db  0.000506 Kyy  0.000360 Kzz
Obs FRQ real Fit Resid Std Resid
45 13.90 54.53 40.63 2.81 R
The employed model incorporates both principal effects and twoway interaction. We employed the values of (P) to estimate the coefficients and effects. To find the main effects using α = 0.05, the principal effects of diameter values of D1 to kzz and their interactions which are statistically important; where their (P) values are lesser than 0.05.
The imaginary part
Diameter d1 and Stiffness kzz, kyy and their related interactions are all important α = 0.05 (see Figure 6).
Figure 6. Representation of the standardized effects as an imaginary part
The hydrodynamic bearings dimensions of Db, Lb, Cb and their linked interactions are all important (α = 0.05) (see Figure 7).
Figure 7. Representation of the standardized effects as a real part
Figure 8. Pareto chart of the standardized effects imaginary part
We note that it is the factor that positively affects the frequency in the imaginary part has negatively affected the frequency in the real part.
The effects absolute values are displays by Minitab on the Pareto chart (see Figures 8, 9). All effects behind the reference line are significant at the level of 0.05, in the imaginary part. The diameter D_{1} and stiffness kyy are all important (α = 0.05), in the real part we find the opposite.Db and Lb all important (α = 0.05).
Figure 9. Pareto chart of the standardized effects real part
Figure 10. Main effects plot for frequency imaginary part
Figure 11. Main effects plot for frequency real part
Then, the principal effect plots are sketched in MINITAB 17 as illustrated in Figures 10 and 11. The different diameters effects, the stiffness, and the hydrodynamic bearing dimensions on the excitation frequency show:
The diameter d1 and the stiffness kzz, kyy have significant effect where they augment the excitation frequency. The plot also reports that:
• The diameter d1 has more influence on the frequency compared to the stiffness kzz, kyy.
• The other hydrodynamic bearing diameters and dimensions of (Lb, Db, and Cb( don’t have an important effect on the excitation frequency.
The hydrodynamic bearing dimensions of (Db, Lb, and Cb ( and diameters of D4 and D2 have significant effect where they augment the excitation frequency. The plot also reports that:
• The hydrodynamic bearing dimensions of (Db, Lb, and Cb( have significant effect on the frequency compared to the diameters D4 and D2.
• The other diameters and the stiffness don’t have a significant effect on the excitation frequency.
An improvement chart gives the effect of each factor (columns) on responses (rows) (Figure 12). The vertical red lines on the graph represent the current operator settings. The numbers displayed above the column indicate the current factor level settings (red color). The blue horizontal lines and numbers represent responses to the current factor level.
Minitab calculates in the imaginary part the diametre d1 and kzz are manimized when all factors are at their highest settings (d1=0,0096, kzz=195555,556).
In real part the hydrodynamic bearing dimensions of (Db, Lb, and Cb (and diameters of D4 and D2, are manimized when all factors are at their Its lowest settings (Db=0.0001, Lb=0.0450, Cb= 0.090, D4= 0.0095, D2 =0.0095).
Figure 12. Optimum solution for nine factors
You like to see how the responses change when the stiffness kzz =166666.667. In interactive mode, you can move the factor level (red) line for stiffness kzz, or enter 166666.667 in the stiffness kzz.The graphic shows that the expected response in the imaginary part (86.0828) and the expected response in the real part (191.2315).
Pareto charts Figure 13 are a type of bar chart in which the horizontal axis represents attributes of interest, rather than a continuous scale. Typically, these features are "disadvantages". When arranging bars from largest to smallest, a Pareto chart can help you identify errors that are made up of a small number of vital elements and which are of little significance. The Cumulative Percentage line helps you determine the added contribution of each of the categories. Pareto Charts can also help focus improvement efforts on areas where the greatest gains can be made.
a The imaginary part:
The vital few shortness in the imaginary part is represented by the following values of the imaginary frequency (Figure 13).
FRQ img =142.210, FRimg =142.60, FRQ img =144.570
FRQ img =144.900, FRimg =145.210, FRimg =145.560
Figure 13. Pareto charts " imaginary part"
bThe real part:
The vital few shortness in the real part is represented by the following values of the real frequency (Figure 14).
FRQ real =310.263, FR real =310.516. FR real =323.984.
FRQ real =324.052, FR real =324.409, FR real =325.855.
Figure 14. Pareto charts
After determining d1 as the main diameter affecting excitation frequency imaginary part, and the Db outer journal diameter as the main diameter affecting excitation frequency in real part, optimization charts are drawn based on values of frequencies obtained from matlab software. Figure 15 shows the model for rotor kit with various sections, disc and bearings. Figure 16 illustrates the Campbell diagram of the rotorshaft system, where the shafts internal material damping is taken into consideration. The graph is plotted using the whirl frequencies (found from imaginary part of the eigenvalues), which are two positions, the first position in reverse rotation "BW", where the rotor rotates in the opposite direction. The second position is the rotation "FW", where the rotor rotates in the direction of rotation. The critical speed corresponding to the first position and the critical speed corresponding to the second position appear, the values of the first critical speeds identified.
Figure 15. Rotor kit with various sections
Figure 16. Campbell diagram
Critical speeds – Total number of modes studied 5.
Mode (Hz) (rpm)
3 2.6233e002 1.5740e+000
4 2.3259e001 1.3955e+001
5 4.5778e+001 2.7467e+003
Figure 17 presents the evolution of the damping constant as a function of the rotational speed. Through the diagram, we notice that the values of the damping factor are negative and thus indicate that the rotor is stable.
Figure 17. Stability diagram
Onset of instability speeds
Mode (Hz) (rpm)
1 2.8279e+001 1.6967e+003
2 9.8980e+001 5.9388e+003
3 0.0000e+000 0.0000e+000
4 0.0000e+000 0.0000e+000
5 0.0000e+000 0.0000e+000
Table 9 gives the result of the shapes of the modes and the shape precession and the rotational speed for the modes correspondent to critical speeds. The modes 3, 4, 5, are direct precession (the rotor rotates in the direction of rotation).
Figure 18 gives relative deviation as a function of tree length and confirms the results obtained in Table 9.
Figure 18. Forms of modes and precession of forms of modes at 0 rpm
Table 9. Forms of modes and precession of forms at 0 rpm
Modes 
Precession 
Spin speed rpm 
3 
direct 
0 rpm 
4 
direct 
FB=56.2856 rpm 
5 
direct 
FB=2741.7408 rpm 
The PlackettBurman method in DOE and finite elements were employed to optimize the rotor and investigate the effects of the hydrodynamic bearings dimensions and the stiffness on the rotating machines dynamics and also to know the diameters which produce considerable effects on the excitation frequency as well as the reactions which can increase or decrease the principal effects and the gyroscopic effect on the eigenvalues of rotorbearing systems. Our results show that:
The diameter D1 has the more significant positive effect on the real part frequency which can be seen in the right of the response line compared to outer journal diameter Db that has a negative effect on the imaginary part frequency which is shown in the left of the response line and vice versa, as affirmed by the plot of principal effects.
By the consideration of the internal material damping of the rotor. During the forward whirl, damping decreases, as the spin speed augments and in backward whirl, damping augments, as the spin speed augments. So, the system stability is determined. the stability of the system indicates the positive value of damping factor.
For the system stability, it’s required to be operated at a speed lesser than a critical speed. From the maximum real part vs the spin speed plot, the stability can be determined. It’s concluded that the system is unstable for the positive value of the maximum real part, and it’s stable for the negative value of the real part. As conclusion, the modal analysis is a main tool to get an important idea about the system dynamic behavior.
Lb 
length of bearings [m]. 
Db 
outer journal diameter [m]. 
Cb 
bearing clearance [m]. 
Kyy, Kzz 
stiffness coefficients 
Cyy, Czz 
damping coefficients a gyroscopic effect 
k1, k2 
stiffness 
ω 
The speed of rotation of the shaft (rd / s) 
Ω 
angular velocity (shaft) (rd / s) 
[K b] [C b] 
dimensionless stiffness coefficients. dimensionless damping coefficients. 
DF 
Degrees of freedom from each source 
SS 
Sum of squares. 
MS 
Mean squares. 
F 
Calculate by dividing the factor MS by error. 
P 
Use to determine whether a factor is signif. 
Secoff 
Standard error of the coefficient. 
S 
Estimated standard deviation of the error. 
Seq SS 
Sequential sum of squares. 
Adj SS 
Adjusted sum of squares. 
F 
The degrees of freedom for the test 
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