Rotor response to unbalanced load and system performance considering variable bearing profile

Nomenclature

Symbol

description

a , b

profile parameters (m)

A, B

dimensionless profile parameters

c

clearance (m)

CS

dimensionless critical speed

e

eccentricity of the journal (m)

F _X , F _y

dimensionless forces

G ( z )

gap due to modification

h

film thickness (m)

H

dimensionless film thickness

K eq

dimensionless equivalent stiffness

L

width of the bearing (m)

P

dimensionless pressure

R

radius of the bearing (m)

U m

mean velocity (m/s)

W ¯

dimensionless load

z

coordinate along bearing width (m)

Z

dimensionless coordinate

∅

attitude angle (°)

δ h

horizontal misalignment (m)

Δ h

dimensionless horizontal misalignment

Δ h o

dimensionless horizontal misalignment at the bearing edge

δ v

vertical misalignment (m)

Δ v

dimensionless vertical misalignment

Δ v o

dimensionless vertical misalignment at the bearing edge

ε r

eccentricity ratio

η

lubrication viscosity (Pa s)

ρ

density of oil (kg/m³)

θ

angle of circumferential direction (°)

ω

angular velocity (rad/s)

Δ θ

step of θ (°)

Δ Z

step of Z

1 Introduction

Journal bearings are widely used to support high-speed rotating machinery such as compressors and turbines due to their superior durability and high levels of load-carrying capacity in addition to the relatively low manufacturing and maintenance cost. Therefore, they are very important machine elements for enhancing the performance of the rotating machinery [1]. The demand of the current industrial applications to generate high power levels results in a further increase in the speed of the types of machinery. On the other hand, high-speed levels may cause unstable oscillations due to self-excited vibrations. High oscillation amplitudes affect the stability of the system and may result in bearing damage. Therefore, determining the instability onset is an essential design step to ensure the stable operation of the system [2]. Unbalanced load represents the most important vibration source in the systems of rotor bearing. Unbalance is a result of uneven mass distribution about the axis of rotation of any rotating machine element and it is a common machinery fault [3]. The vibration associated with this form of excitation accompanies noise and may reduce the bearing life in addition to the resulting unsafe working conditions [4]. This topic has had the researcher’s consideration for decades due to the previously mentioned important aspects. One of the early studies in this direction was presented by Barrett et al. [5] in order to investigate the unbalanced load consequences on short-bearing performance. However, the short bearing solution resulted in considerable errors when the bearing ratio is more than 0.5 [6]. Another work was also performed by Brancati et al. [7] for the analysis of rotor bearing response to unbalanced load using short bearing theory. They used different unbalanced values of the rotor in their dimensionless form. Sghir and Chouchane [2] investigated the unbalanced effect on the rotor response supported by the short bearing. Adiletta et al. [8,9,10] studied the unbalance consequences on the response of the system as this problem inherently presents in any rotor. However, the investigations were focused on limited conditions of the system. The long bearings were also used by Chang-Jian and Chen [11] in investigating the effect of unbalanced rotating on the dynamic response of the rotor using a simplified model. Solution to the rotor response under the effect of mass imbalance and base excitations was presented by El-Saeidy and Sticher [12]. Eling et al. [13] performed an investigation to identify the rotor response to unbalanced load. Their results illustrated the interaction between oil film and the rotation of the journal that produced an unstable response featured by sub-synchronous motion. This is essentially the oil whirl and under such conditions, the whirling frequency is about half the journal speed.

The general performance of the rotor-bearing system is affected by many other causes such as shaft deviations. In industrial applications, the system is subjected to a level of deviation relative to the longitudinal axis of the bearing. Such deviation may result due to installation and manufacturing errors, large deformation, and bearing wear.

In the field of the shaft deviation simulation in journal bearings, a rich volume of literature has been published. Sun et al. [14,15] investigated the performances of journal bearing under different levels of misalignment caused by linear shaft deviation. They explained that the distribution shape of the pressure and film thickness was significantly changed due to misalignment. Jang and Khonsari [16] presented a model to investigate the effect of misalignment in journal bearings. Zheng et al. [17] proposed a model to examine the couple stress lubricants effects of journal bearings considering journal misalignment. Sun et al. [18,19,20] investigated the misalignment effect on the characteristics of thermohydrodynamic and turbulent lubrication of journal bearings with the consideration of surface roughness. Jang and Khonsari [21] proposed a model to study the consequences of wear on the misaligned bearings performance. Yang et al. [22] developed a numerical solution to investigate the misalignment effect resulting from a linear shaft deviation on the bearing wear levels as well as its consequences on the mixed lubrication performances.

Extensive works have been performed to explain the influences of the shaft deviations on the general system characteristics [14,23–27] in terms of pressure rise and the lubricant thickness reduction, which has consequences on the life of the bearing as a result of increasing the friction coefficient and wear rate. The negative effects of shaft deviation can be reduced by using variable profiles to increase the gap between the surfaces. This concept has been investigated by researchers using different methods. One of the first attempts was performed by Nacy [28] using a chamfered bearing to control the side leakage. Another method was used by Fillon and Bouyer [29] by using defects on the bearing to study their influences on reducing the deviation effects on the performance of the bearing. Strzelecki [30] used a variable profile over the whole bearing width using a hyperboloidal shape as an attempt to elevate the levels of the bearing-supported load under shaft deviation. Chasalevris and Dohnal [31] illustrated the possibility of enhancing the system stability by using variable geometry. Ren et al. [32] studied the influence of varying the bearing shape on the system performance. They concluded that using a quadratic shape enhanced the bearing characteristics. Jamali et al. [33,34] explained that the system characteristics can be improved by changing the bearing shape. Recently, the effects of shaft deviation have also been studied [35–37]. However, these studies have not considered the influence of using variable profiles on the bearing characteristics and rotor response under an unbalanced load for different bearing ratios.

In this work, the effect of an unbalanced load on the rotor dynamic response is studied using a numerical solution. Different bearing ratios are considered in the analysis. Furthermore, the effect of using the variable bearing profile on the performance of the system is considered under the presence of 3D shaft deviations. The influences of using this profile on the shaft center trajectory, friction coefficient, pressure, and film thickness levels are investigated. The finite difference method is used in the numerical solution. The dynamic response of the system is obtained by the fourth-order Runge-Kutta solution considering different conditions of operation. Therefore, the main contribution of this work is related to investigating the combined effect of unbalanced load, 3D misalignment, and variable bearing profile on the dynamic response of a rotor-bearing system in addition to their effects on friction coefficient, pressure, and film thickness levels.

2 Governing equations

Figure 1 illustrates a schematic drawing for the solution model. Figure 1a shows the ideal aligned case, Figure 1b illustrates the shaft deviation representation and Figure 1c illustrates the change in the shape of the bearing which is used in the current work to investigate its effect on the system performance and the response to the unbalanced excitation. The variables in these figures will be defined later with their related equations.

Figure 1

Solution model. (a) Ideal case, (b) shaft deviation mode, and (c) section explaining the change in the bearing profile. JCM: Center of the misaligned journal; JCA: Aligned journal center.

The governing equations of the hydrodynamic solution are given by [26,28]

where c is the clearance in the radial direction, h is the film thickness, p is the pressure, U m is the mean velocity, U m = U j + U b 2 , U j , U b are the shaft and bearing speed ( U b = 0 and U j = R ω ), η is the viscosity, ∅ is the attitude angle, ρ is the density, ε r is the eccentricity ratio, ε r = e / c , and e is the eccentricity distance.

The method of Reynolds boundary conditions is considered in the current work in solving the governing equations, which is characterized by [38],

where θ c is the cavitation position that is identified by an iterative method [38,39].

The dimensionless forms of Eqs. (1) and (2) are

where α = R 2 L 2 = 1 4 ( L / D ) 2 , P = p − p o 6 η ω c 2 R 2 , H = h c , x = R θ , and Z = z L

The dimensionless load W ¯ = w 6 η ω R L c R 2 and the attitude angle can be determined using the following equations [40]:

where

The shaft deviation representation shown in Figure 1b is used from a previous investigation [26]. The relations used to determine the deviation at any Z position are given by

where the dimensionless relations are given by Z = z / L and Δ = δ / c .

This 3D shaft deviation model provides a representation of the horizontal ( Δ h ( z ) ) shaft deviations and the vertical ( Δ v ( z ) ) shaft deviations. It is worth mentioning that, the shaft deviations result in variable attitude angle eccentricity along the bearing width (Z direction), which is not the case in the perfectly aligned bearing. Therefore, the following equations describe these variations in terms of the Z position [26]:

where ∅ m is the attitude angle at z = L / 2 and e m is the eccentricity at z = L / 2 .

The bearing profile variation explained by Figure 1c increases the space between the surfaces on both sides of the bearing as at these locations, the shaft deviation has its worst influence on the film thickness. Therefore, using such a bearing profile helps compensate for the decrease in the lubricant thickness resulting from shaft deviation.

The new profile is defined by two variables, which are the radial height ( a ) and the portion of the bearing width ( b ) where the change in the profile is performed at both bearing sides. Figure 1c illustrates the parameters a and b .

Using such a profile results in the following dimensionless gap in terms of the Z position,

where the dimensionless variables are given by, A = a / C   and B = b / L .

The effectiveness of this profile will be identified in the current work by using a wide range of the profile parameters A and B . Using these parameters in their dimensionless forms provides a more clear explanation of the required change in the bearing geometry in terms of the clearance ( A = a / C   ) and the bearing length ( B = b / L ).

The total resulting film thickness with the consideration of the shaft deviation and the change in the bearing shape can be calculated by coupling Eqs. (4), (9), and (11).

3 Dynamic response

The dynamic coefficients are determined using the linear stability solution. The nonlinear bearing forces are linearized around the equilibrium position. In such analysis, the Reynolds equation considering the time-depending term is given by

The corresponding equation for the lubricant film thickness is [41]

The time-dependent term ( ∂ h ∂ t ) is

Eqs. (12)–(14) result in,

where

The bearing forces in terms of the displacements and velocities in the horizontal and vertical directions are given by [41,42]

F y = F y ( x , y , x ̇ , y ̇ ) ,

The stiffness and dynamic coefficients are given by [43]

Lund and Thomsen [42] suggested the relations for these coefficients,

Eqs. (3), (18), and (19) yield

where

These equations need to be solved numerically to calculate the pressure derivatives that are used in the determination of the coefficients.

3.1 Unbalanced load

The dynamic response of the rotor bearing system illustrated in Figure 2 can be determined by solving the equations of motion after calculating the eight dynamic coefficients. These equations are given by [44]

(26) m x ̈ ′ = − F x − f u sin ω t ,

(27) m y ̈ ′ = − F y − f u cos ω t + W ,

where [44] F x ,   F y are the bearing forces, W is the load, ω is the speed of the shaft, f u is the unbalanced load, and x ′ , y ′ are the whirling axes.

Figure 2

Schematic drawing of the system [45 modified].

Eqs. (26) and (27) can be given by

(28) M X ̈ ′ = − F x − AM sin T ,

(29) M Y ̈ ′ = − F y − AM cos T + 1 ,

where x ′ = X ′ C , y ′ = Y ′ C , ω t = T , F X = F x W , F Y = F y W , M = m c ω 2 W , and AM = m u r ω 2 W .

The dynamic response of the system to the unbalanced load can be calculated by solving Eqs. (28) and (29). The first step in this direction is the determination of the rotor’s critical speed (CS). This can be achieved by ignoring the applied and unbalanced loads in Eqs. (28) and (29) and solving the resulting equations which are

(30) M X ̈ ′ + F x = 0 ,

(31) M Y ̈ ′ + F y = 0 .

The bearing forces ( F x and F y ) are determined in terms of the dynamic coefficients based on the linear stability analysis as follows [41]:

(32) F x = K x x X ′ + K x y Y ′ + C x x X ̇ ′ + C x y Y ̇ ′ ,

(33) F y = K y x X ′ + K y y Y ′ + C y x X ̇ ′ + C y y Y ̇ ′ .

The following equations are used to solve Eqs. (30) and (31) [41]:

(34) X ′ = a e i λ t , Y ′ = b e i λ t .

Substitutions of Eqs. (32)–(34) in Eqs. (30) and (31) yield,

(35) λ = ( k eq − K x x ) ( K eq − K y y ) − K x y K y x C x x C y y − C x y C y x ,

(36) K eq = K x x C y y + K y y C x x − K y x C x y − K x y C y x C x x + C y y .

The rotor CS can be determined now by the following equation:

(37) CS = k eq λ .

4 Numerical solution

The governing equations are discretized using the finite difference method. The main two equations are the Reynolds and film thickness equations. The shaft deviations and the change in the bearing profile, which affects the gap between the shaft and the bearing surfaces, are involved in these two equations as explained previously. The solution considered a successive over-relaxation method under the well-known Gauss-Seidel approach. At this step, the pressure field as well as the corresponding lubricant thickness are known. Therefore, the dynamic coefficients (stiffness and damping) can now be determined using numerical integration. After that, the following step is determining the equations of motion solution for obtaining the shaft trajectory under unbalanced load. This response is determined numerically based on the use of a fourth-order Runge-Kutta solution.

The discretization procedure gives the following equation for the pressure field at a given i, j node:

where Δ θ and Δ Z are the steps of discretization.

C 0 = H b 3 + H a 3 + α C 2 H c 3 + α C 2 H d 3 ,

C 1 = Δ θ 2 , C 2 = Δ θ 2 Δ Z 2 .

Detailed steps for the solution and discretization procedure are available in the study by Jamali and Al-Hamood [26].

The derivatives of pressure that are used to determine the eight coefficients are obtained by discretizing Eqs. (22)–(25). The general equation of the derivatives can be written as

where C 3 = − H b 3 − H a 3 − α C 2 H c 3 − α C 2 H d 3 and Q is the resulting right-hand side after the differentiation of the Reynolds equation.

After obtaining the equations discrete forms, the solution can be performed numerically. The convergence criterion of this numerical solution is given by

The determination of the pressure field is followed by calculating the hydrodynamic load using a numerical integration method. If the difference between the calculated load and the actual load are within the accuracy limits of ± 10 − 5 (actual load ± 10 − 5 × actual load), the results are accepted. If this accuracy is not obtained, ε r is changed and the solution is repeated where a new pressure field and a new gap between the surfaces are obtained, which also gives new values of the eight dynamic coefficients. This process will continue until both convergence criteria are achieved. Then, the Runge-Kutta method is adopted to find the rotor response to the unbalanced excitation. Figure 3 shows the main solution step.

Figure 3

Flowchart of the steps involved in the solution.

5 Results and discussion

The analysis is performed in the current work for a length-to-diameter ratio ( L / D ) between 1 and 2 in a step of 0.25. The resulting five ratios are used to investigate the effect of changing the bearing design on reducing the negative shaft deviation effects on the performance of the bearing. Using such a range provides a clear representation of the effectiveness of each parameter related to the bearing shape in a general concept where all the results are presented in their dimensionless forms. The numerical solution is examined at first to observe the independency of the outcome on the number of nodes in the radial and longitudinal directions used in the discretization scheme. A number of nodes of 32,761 is found sufficient to the solution of the current problem. However, a total number of 65,341 is used in the current analysis to obtain a higher level of accuracy in terms of discretization of the governing equations and their relation to the number of nodes. Furthermore, the outcome of the current solution method is verified with that given in the study by Chasalevris and Dohnal [31] as illustrated in Table 1. In this comparison, the Sommerfeld number is chosen to have relatively low and high values. The comparison shows very good agreement in the compared stiffness coefficients.

Before presenting the results, it is of great importance to distinguish between two concepts in the analysis of a range of L / D ratio. In the first concept, the whole range of L / D ratio is analyzed based on using the same load while in the second concept, the analysis is performed for the whole range using the same eccentricity ratio. There is a clear difference between these two methods of analysis. In the first method, the comparison between the results of different L / D ratios is more reasonable as the same power is used for the whole range while in the second method, the same minimum film thickness is used for the whole range. Using the same film thickness with different L / D ratios means that in other words applying different loads (i.e. different powers) for each L / D ratio, which makes the comparison less meaningful. Figure 4 illustrates the difference between the two concepts in terms of the key bearing performance characteristics, which are the maximum pressure (Po) and minimum film thickness (Ho) for the whole range of L / D ratio. Figure 4a illustrates how the difference is clear as using the same load decreases P _o with the increase in L / D ratio and the opposite results can be seen when the analysis is performed on the base of using the same eccentricity ratio. These results are understandable as using the same load over the higher supported area (larger L / D ratio) reduces the maximum pressure and vice versa. Figure 4b shows the corresponding results for H _o where using the same load for different L / D ratios increases H _o as L / D ratio is increased while H _o is constant for the whole L / D range. It is worth mentioning that the load used in producing these results is for ε r = 0.6 and L D = 2 , which explains why the results intersect at L / D = 2 . Therefore, the first concept (using the same load for the whole L / D range) is adopted in the current work.

Figure 4

Difference between analysis approaches for different bearing ratios on (a) max. pressure and (b) min. film thickness. Results are dimensionless.

The effects of an unbalanced load on the trajectory of the journal center are illustrated in Figures 5 and 6. Tieu and Qiu [44] used a dimensionless unbalanced load, which is scaled to the supported load AM = m u r ω 2 W of 0.2. This value corresponds to ε r = 0.6 and ω = 3,000 rpm ( ω = 314.159 rad / s ). It is clear from this relation that any change in the rotational speed produces different unbalanced excitation. Therefore, the following parameter, UB = r m u / W is used to represent the dynamic excitation as a function of speed where AM = UB ω 2 . Using AM = 0.2 and ω = 314.16 rad / s [44] gives a value of 2.026423 × 10 − 6 for UB . In the current investigation, three magnitudes of UB are used in the analysis. These values are 0.5 UB , UB , and 1.5 UB , which are considered to determine the influence of the unbalanced load on the performance of the rotor bearing system using different rotational speeds. In other words, the considered magnitudes of UB represent a value of AM = 0.1 , 0.2 , and 0.3, respectively, which correspond to a speed of 314.16 rad / s (3,000 rpm). The changes in rotational speed results in different values for AM .

Figure 5 illustrates the shaft center trajectories for the whole range of L / D ratio. In this figure, the considered speed is half the CS under the three values of U B . It can be seen that in all L / D ratios, the value of 0.5 UB (black line) produces low amplitude around the steady state position, which makes the journal relatively far from the bearing inner surface. The amplitude increases as in the case of 1.0 UB and 1.5 UB in the whole L / D range. It is of great interest to notice that the amplitude is higher for lower L/D ratios. The case of L / D = 1.0 shown in Figure 5a gives the highest and most dangerous amplitude particularly for the condition of 1.5 UB even at this relatively low rotational speed (half the CS). Comparing this case with the case of L / D = 2 shown in Figure 5e, the amplitude in the latter case is relatively low. Keep in mind that the two cases supported the same load as stated previously. This important outcome illustrates that using shorter bearing to avoid the misalignment effect may cause another problem related to the dynamic response to unbalanced excitations.

Figure 5

The trajectory of the shaft center under unbalanced excitation with three different amplitudes and an operating speed of half the system CS when the bearing ratio is (a) 1, (b) 1.25, (c) 1.5, (d) 1.75, and (e) 2.

Figure 6 shows the corresponding results for the whole L / D range. The operating speed in this figure is equal to the rotor CS at each L / D ratio. It can be seen that in all cases, the condition of 1.5 UB produces an extremely large journal amplitude and leads to contact with the bearing inner surfaces. Again here, the lower value of L / D produces a larger amplitude even at the condition of 1.0 UB . The contact occurs when L / D = 1.0 and 1.25 and the system continues to work at this condition when L / D = 1.5 ,1.75, and 2.0. These results emphasize the previous outcome where the lower L / D ratio has to be examined carefully in terms of their response to unbalanced excitation.

Figure 6

Trajectory of the shaft center under unbalanced excitation with three different amplitudes and an operating speed equal to the system CS when the bearing ratio is (a) 1, (b) 1.25, (c) 1.5, (d) 1.75, and (e) 2.

Figure 7 explains the influence of 3D shaft deviation parameters on P _o, H _o, and friction coefficient (f) when L / D = 1. The range of deviation is selected between 0 and 0.52. The value of zero deviation parameters represents the ideal case, which is difficult to achieve in the industrial use of journal bearing as explained previously. However, this perfect case is presented here to show how the shaft deviation affects the performance of the bearing. On the other hand, the maximum selected value of deviation, which is 0.52, is found to have severe effects on the bearing characteristics. The H _o particularly reduces significantly at this level, and going further by using deviation values higher than 0.52 is unrealistic as nearly metal-to-metal contact is almost taking place. It can be seen that in this figure the presence of shaft deviation even at a lower level affects the values of H _o. The effect of shaft deviation starts to have a clear negative effect on P _o and f at levels of deviation greater than 0.4.

Figure 7

Variation in P _o, f, and H _o with the values of deviation parameters when L / D = 1 .

It is worth mentioning that the value of P _o reduces at lower levels of deviation parameters which is attributed to the shape of pressure distribution at these relatively low levels of deviation which do not result in a pressure spike. The deviation parameters are scaled to the radial clearance as explained previously, therefore a 0.4 value means 0.4C in a dimensional form. To have a clear understanding of the influence of deviation value on system performance, suppose that c = 50 μm and a bearing width of L = 100 mm, which are typical values, the deviation parameter becomes 0.4 × 50 = 20  μm. The resulting deviation angle with respect to the longitudinal axis is 0.023 o . At this level, H _o drops from 0.266 to 0.105 (more than half the original value). The higher values of the angle of deviation result in significant increase in P _o and f as shown in Figure 7. This figure shows how the results change with the variation in the deviation parameters. Therefore, Figure 8 is used to illustrate the percentage change in the results in comparison with the corresponding ideal aligned case for the whole L/D ratios range.

Figure 8

Percentage change in bearing characteristics due to the increase in deviation levels. (a) max pressure, (b) min. film thickness, and (c) friction coefficient.

Figure 8a shows the percentage changes in P _o with the increase in the deviation parameters. Again here, the value of zero deviation represents the ideal aligned case. It can be seen that a reduction in P _o (negative change) is obtained for the whole L/D ratio range when the deviation parameters ≤ 0.4 except for the L / D ratio of 2. The change in P _o becomes positive (which represents a disadvantage) as the deviation parameters increased higher than 0.4 with a maximum change when a deviation level is 0.52. The percentage changes in P _o at this level are 65.9, 38.0, 25.3, 42.2, and 51.9% when L / D = 1, 1.25, 1.5, 1.75, and 2, respectively. The corresponding H _o results are illustrated in Figure 8b. The film thickness levels are very sensitive to the shaft deviation where even at lower values of deviations, H _o is significantly changed as explained previously. The maximum reductions are 88.6, 80.9, 74.9, 75.9, and 75.8% when L / D = 1, 1.25, 1.5, 1.75, and 2, respectively. Such reduction levels have severe negative consequences on the performance of the bearing as any change in the operating conditions may result in further reduction in these levels such as the presence of dynamic excitations due to position perturbation and vibrations. The percentage changes in the friction coefficient are shown in Figure 8c where the shaft deviations start to have considerable effect when their levels become higher than 0.4. The maximum changes at deviation levels of 0.52 are 9.25, 7.78, 7.94, 8.43, and 8.59% when L / D = 1, 1.25, 1.5, 1.75, and 2, respectively. The similar trends of the effects of shaft deviations on P _o, H _o, and f give a sense of generality of the deviation effect despite the change in L / D ratio for the same supported load.

It is clear from the previous results that the presence of shaft deviations (even at lower levels) has negative consequences on the bearing performance, particularly at the level of the film thickness. Keeping in mind that based on the previous studies as well as from the authors’ experience in this field, the shaft deviation is unavoidable. Therefore, it is of great importance to work on reducing the effect of shaft deviation on the bearing performance to ensure a safe operation and to increase the life of the bearing. Changing the design of the bearing is investigated here to evaluate its consequences in this direction. The severe level of deviation (0.52) presented in Figure 8, which has the most negative impact on the bearing characteristics, is used in this investigation. Figure 9 illustrates the influence of A on P _o, H _o, and f for the whole L / D range under a deviation level of 0.52. Six values of A are selected, which are 0.1, 0.2, 0.25, 0.3, 0.5, and 1, to produce the results presented in this figure in addition to the classical unchanged design of the bearing. Using the higher value of A causes an increase in P _o because of which the A range is limited to 1. Figure 9a shows the effect of A results on P _o where the red color represents the classical design without any change. It can be seen that when A ≤ 0.5 , P _o is reduced in the whole L / D ratio range while when A = 1 , P _o is increased above the value of the classical design. Therefore, the value of A has to be selected carefully. The best values of A are 0.2 when L / D ≤ 1.5 and 0.25 when L / D > 1.5 . However, the results of A = 0.2 and 0.25 are very close to each other. The reductions in P _o are 26.1, 16.4, 11.5, 19.6, and 19.9% when L / D = 1, 1.25, 1.5, 1.75, and 2, respectively. This represents an important general outcome for the whole L / D range. Figure 9b illustrates the H _o results. These results explain that the best A values are 0.25 when L / D ≤ 1.5 and 0.3 when L / D > 1.5 . The increases in H _o are 339.9, 224.7, 163.8, 173.7, and 160.3% when L / D = 1, 1.25, 1.5, 1.75, and 2, respectively. This is another important general result as the level of the lubricant thickness is enhanced significantly in the whole L/D range. Figure 9c presents the corresponding effects on the friction coefficient. This coefficient reduces due to the change in the bearing design for all A values in a similar trend for the other L/D ratios. The percentage reductions in f are 5.44, 4.42, 4.46, 4.78, and 4.72% when L / D = 1, 1.25, 1.5, 1.75, and 2, respectively.

Figure 9

Effect of design parameter A on the bearing characteristics at 0.52 level of misalignment and B = 0.25. (a) Max. pressure, (b) min. film thickness, and (c) friction coefficient.

The previous figure was obtained for a design variable B of 0.25 . Therefore, the influence of this parameter on the bearing characteristics is illustrated in Figure 10 when A = 0.25, which improves the bearing characteristics in the whole L/D ratio range. Also, six values of B are selected, which are 0.05, 0.1, 0.2, 0.25, 0.35, and 0.5 in addition to the classical unchanged design under a 3D shaft deviation level of 0.52. The maximum value of B, which is 0.5, corresponds to the change in the whole bearing length as the modification takes place from both bearing directions. The red color also represents the classical bearing design. It can be seen in Figure 10a–c that all values of B enhance P _o, H _o, and f, respectively. However, the optimum B value is 0.35 in terms of the improvement in H _o which is increased at this value by 450.5, 261.8, 189.7, 203.7, and 187.1% when L / D = 1, 1.25, 1.5, 1.75 and 2, respectively. This B value also reduces P _o by 26.9, 17.4, 11.9, 21.9, and 23.2% when L / D = 1, 1.25, 1.5, 1.75, and 2, respectively and reduces f by 7.02, 5.79, 5.85, 6.09, and 6.10% when L / D = 1, 1.25, 1.5, 1.75, and 2.0, respectively.

Figure 10

Influence of parameter B on the bearing characteristics at 0.52 level of misalignment and A = 0.25. (a) Max. pressure, (b) min. film thickness, and (c) friction coefficient.

The general outcome of Figures 9 and 10 is that using A = 0.25 and B = 0.35 enhances the bearing performance significantly for the whole considered L/D ratio values under severe deviation levels which may have a positive impact on the life of the bearing as well as the reliability of the bearing under such extreme conditions. The effect of adopting these design variables on the pressure and film thickness distribution is shown in Figure 11 when L / D = 2.0. This figure compares the outcome of the ideal case, the 3D shaft deviation case (deviation level of 0.52) and the 3D deviation case (deviation level of 0.52) under modified bearing design when A = 0.25 and B = 0.35 . It can be seen in these figures that how the modification increases the lubricant level on both sides of the bearing where the shaft deviation has its most negative influence. Furthermore, the modification reduced the pressure spikes associated with shaft deviations and also reduced the asymmetry in the pressure distribution. Similar behaviors also result when the other L / D ratios are used.

Figure 11

Comparison between aligned (left), misaligned with 0.52 deviation parameters (middle), and modified design with A = 0.25, B = 0.35 (right) when L / D = 2 ; upper figures: pressure distributions and lower figures: film thickness distributions.

Figure 12 shows the influence of varying the bearing shape on the shaft trajectory with respect to the equilibrium position when L / D = 2 . The rotational speed for the results presented in this figure is the same as the system CS. It is clear that changing the profile does not increase the amplitude of the trajectory which means in other words that the advantages of the modification in terms of f, P _o, and H _o do not come at the cost of affecting the rotor response. In fact, the modification even slightly reduces the amplitude of the trajectory. The reduction in the amplitude due to the modification of the profile is more clear in Figure 13 where the displacements in the horizontal and vertical directions are shown for the new bearing profile and the classical bearing design. Similar behaviors also result in the other bearing ratios. The important outcomes of this work require further investigation to use an optimization method for this L/D range to obtain the optimal design values under dynamic excitations.

Figure 12

Trajectories relative to the equilibrium position ( X ′ and Y ′ coordinates) at an operating speed equals to the CS when A = 0.25 and B = 0.35 .

Figure 13

Changes in X ' and Y ' with T for the classic and new profiles. (a) X ′ and (b) Y ′ .

6 Conclusion

The current investigation presents a solution to the hydrodynamic problem of journal bearing to determine the response of a rotor to unbalanced excitation under different amplitudes and operating speeds for a range of bearing ratios. The solution involves the incorporation of the shaft deviation in both directions and the effect of using the variable bearing profile on system performance. The problem is solved numerically and the fourth-Runge-Kutta method is considered to obtain the response of the system. Results show that for the same supported load, at the lower bearing ratio, the shaft suffers from a high amplitude of oscillation even at a relatively low rotational speed (half the CS). Using a variable profile for the bearing enhances the performance of the system for the whole range of the bearing ratio. The film thickness elevated significantly as a result of using such a profile and both the pressure levels and friction coefficient are also reduced in the whole bearing ratio range. Using design parameters of A = 0.25 and B = 0.35 enhances the bearing performance significantly for the whole considered L/D ratio values under a severe deviation level of 0.52, which may have a positive impact on the life of the bearing as well as the reliability of the bearing under such extreme conditions. Furthermore, the modification reduced the pressure spikes associated with shaft deviations and also reduced the asymmetry in the pressure distribution. Further optimization of the steady for the profile parameters is required for the selected range of the bearing ratio.