1. Introduction

2. Theory

2.1. Mathematical model description

In this research, a flexible rotor-tilting pad journal bearing (TPJB) dynamic behavior is simulated considering the bearing geometrical imperfections. Figure 1 illustrates the presented model for the flexible rotor and TPJBs. In this model, the five pad TPJBs are numerically simulated to obtain the equivalent damping and stiffness coefficients at the equilibrium position of the system. The dynamic coefficients are used to calculate the critical speeds of the flexible rotor and the system behavior at its equilibrium state. Furthermore, the TPJBs hydrodynamic and solid-to-solid contact forces were determined at different wear severities for an unbalanced rotor-TPJB system. The equations of motion of the rotor-TPJB system can be written as follows [36]:

$$\left[M_s\right]\left\{\ddot{u}\right\}+\omega \left[G\right]\left\{\dot{u}\right\}+\left[K_s\right]\left\{u\right\}=\left\{F_{unb}\right\}+\left\{F_g\right\}+\left\{F_{nl}\right\}$$

(1a)

$$J{\ddot{\delta }}_{p,l}=M_l,l=1,\dots ,N_{pads}$$

(1b)

$$J{\ddot{\delta }}_{p,r}=M_r,r=1,\dots ,N_{pads}$$

(1c)

Where $\left[M_s\right]$, $\left[K_s\right],$ and $\left[G\right]$ are the shaft mass, stiffness, and gyroscopic matrices, respectively. The force induced by the rotor mass unbalance, gravity force, and the nonlinear forces applied on the rotor are represented by $\left\{F_{unb}\right\}$, $\left\{F_g\right\},$ and $\left\{F_{nl}\right\},$ respectively. Moreover, $J$ and ${\delta }_p$ in Equation (1) stand for the p’th pad moment of inertia and tilt angle, respectively. Indices l and r are used for pad numbering in the left and the right bearings, respectively. The pad numbers are showed in Figure 1. In this study, hydrodynamic TPJBs with pad geometric imperfections are simulated, considering the effects of journal-bearing rub-impact on the system motion. The nonlinear force $\left\{F_{nl}\right\}$ on the rotor consists of the oil film hydrodynamic force and the rotor-bearing solid-to-solid contact force. The solid-to-solid contact force applies when the contact between the journal and the bearing occurs. Rotor-bearing contact may happen due to the high level of rotor vibration or the lack of sufficient oil film to separate the journal and bearing. Horizontal and vertical components of the nonlinear forces are calculated as follows [36]:

$$F_x=-\sum _1^{N_{pads}}\left[{\int }_{-\frac{L}{2}}^{\frac{L}{2}}{\int }_{{\theta }_{s,p}}^{{\theta }_{e,p}}\left(\left(P_{fld,p}+P_{c,p}\right)\cos \theta -sign\left(\left({\stackrel{⃑}{V}}_1-\left({\stackrel{⃑}{V}}_{2,p}.\stackrel{⃑}{e}\right)\stackrel{⃑}{e}\right).\stackrel{⃑}{e}\right)\left({\tau }_{fld,p}+{\tau }_{c,p}\right)sin\theta \right)R_{j}d\theta dz\right]$$

(2a)

$$F_y=-\sum _1^{N_{pads}}\left[{\int }_{-\frac{L}{2}}^{\frac{L}{2}}{\int }_{{\theta }_{s,p}}^{{\theta }_{e,p}}\left(\left(P_{fld,p}+P_{c,p}\right)\sin \theta -sign\left(\left({\stackrel{⃑}{V}}_1-\left({\stackrel{⃑}{V}}_{2,p}.\stackrel{⃑}{e}\right)\stackrel{⃑}{e}\right).\stackrel{⃑}{e}\right)\left({\tau }_{fld,p}+{\tau }_{c,p}\right)\cos \theta \right)R_{j}d\theta dz\right]$$

(2b)

where $F_x$ and $F_y$ constitute the nonlinear force components in x and y directions, respectively. $P_{fld,p}$ and $P_{c,p}$ are the oil film hydrodynamic pressure and the contact pressure between the journal and the bearing, respectively. Moreover, the shear stresses due to the oil film hydrodynamic pressure and the contact pressure are represented by ${\tau }_{fld,p}$ and ${\tau }_{c,p}$, respectively. Other parameters in Equation (2) including the pad starting angle, the pad ending angle, the pad circumferential angle, the journal surface speed, the p’th pad surface speed, and the pad axial length are depicted by ${\theta }_s$, ${\theta }_e$, $\theta$, ${\stackrel{⃑}{V}}_1$, ${\stackrel{⃑}{V}}_{2,k}$, and L, respectively. Furthermore, $\stackrel{⃑}{e}$ denotes the unit vector in the journal absolute velocity direction at the journal-bearing contact. The moment applied from the oil film hydrodynamic force and the journal-bearing solid-to-solid contact force on the pivot of the p’th pad can be determined by Equation (3) as follows [36]:

$$M_p=-{\int }_{-\frac{L}{2}}^{\frac{L}{2}}{\int }_{{\theta }_{s,p}}^{{\theta }_{e,p}}\left[-\left(R_j+t_p\right)\left(\theta -{\theta }_{p,k}\right)\left(P_{fld,p}+P_{c,p}\right)-sign\left(\left({\stackrel{⃑}{V}}_1-\left({\stackrel{⃑}{V}}_{2,p}.\stackrel{⃑}{e}\right)\stackrel{⃑}{e}\right).\stackrel{⃑}{e}\right)\left({\tau }_{fld,p}+{\tau }_{c,p}\right)t_p\right]\cos \left(\theta -{\theta }_{p,k}\right)R_{j}d\theta dz$$

(3)

where $M_p$ is the applied moment on the p’th pad pivot, ${\theta }_p$ denotes the p’th pad pivot angle, and $t_p$ shows the thickness of the p’th pad. Forces and moments should be calculated for journal position at each time step. The methods used for calculating the effect of the bearing geometric imperfections, oil film hydrodynamic pressure, and contact force are described in the following section.

Figure 1. Flexible rotor- worn tilting pad journal bearing model.

Figure 1. Flexible rotor- worn tilting pad journal bearing model.

2.2. Tilting pad bearing geometric imperfections

Journal-bearing wear occurs when the oil film formed on the loaded parts of the bearing is not sufficient to separate the journal and the bearing. This situation happens when the rotor speed is not high enough for sufficient oil film formation during the run-up and shot-down phases. Journal-bearing contact leads to the removal of the bearing material from the inner side of the loaded pads and subsequent pad geometrical imperfections. The worn pad material removal at low speeds almost occurs on the pad’s inner side and in front of the pad pivot [2]. In this study, it is assumed that TPJB geometric imperfections are induced by contact between the journal and the loaded pads during the run-up/shot-down phases. The effects of using worn pads with different wear severities on the system thermal and dynamic behavior are studied. The simulated TPJB has the load between pads (LBP) configuration. Since the rotor does not rotate in the middle of the bearing, wear severity is not equal on both loaded pads, as the pad closer to the journal center experiences more severe wear. It is assumed that the rotor rotates in the counter-clockwise direction, and there are 5 pads installed in the bearing. The pads numbering is depicted in Figure 1. In this case, the pads No. 4 and 5 experience journal-bearing wear and the journal equilibrium position will be closer to the pad No. 5. Therefore, the pad No. 5 experiences more severe wear. Moreover, the maximum wear depth on the 5^th pad is assumed to be twice that of the 4^th pad. A schematic representation of the pad No.5 with wear signature on its inner side is depicted in Figure 2. The wear pattern was chosen based on the presented mathematical model for worn hydrodynamic bearings in [3]. Furthermore, it is assumed that just the loaded pads experience wear. Therefore, the oil film thickness on the p’th pad of the TPJB, considering the worn pads geometrical imperfections, can be determined by:

$${\overline{h}}_p=\left\{\begin{array}{c}{\overline{h}}_{0,p}\left(\theta \right),p=\mathrm{1,2},3\\ {\overline{h}}_{0,p}\left(\theta \right)+d_{w,p}\left(\theta \right),p=\mathrm{4,5}\end{array}\right.$$

(4)

Where:

$${\overline{h}}_{0,p}\left(\theta \right)=\left(R_p-R_j\right)-X_{j}cos\theta -Y_{j}sin\theta -\left(R_p-R_b\right)\cos \left(\theta -{\theta }_p\right)-{\delta }_p\left(R_p+t_p\right)\sin (\theta -{\theta }_p)$$

(5)

In Equation (5), $R_p$ and $R_b$ represent the pad and the bearing radii, respectively. Furthermore, wear depth for a worn pad is denoted by $d_{w,p}$ which can be obtained by the formulation presented in [3]. Since the maximum wear depth occurs in front of the pivot [2], this formulation should be modified for TPJBs as follows:

$$d_{w,p}=\left\{\begin{array}{c}{d}_{max,p}-C_b\left(1-\cos \left(\theta -{\theta }_p\right)\right),{\alpha }_{s,p}<\theta <{\alpha }_{e,p}\\ 0,otherwise\end{array}\right.$$

(6)

where ${\alpha }_{s,p}$ and ${\alpha }_{e,p}$ are starting and ending angle of wear on the p’th pad with the following expressions:

$${\alpha }_{s,p}=\frac{\pi }{2}+{\cos }^{-1}\left(\frac{d_{max,p}}{C_b}-1\right)+{\theta }_p$$

(7a)

$${\alpha }_{e,p}=\frac{\pi }{2}-{\cos }^{-1}\left(\frac{d_{max,p}}{C_b}-1\right)+{\theta }_p$$

(7b)

Noteworthy, wear profile depends on diverse factors such as application, rotating direction, manufacturing errors, and the system properties. Equations (6) and (7) present the wear signature on pads as an example of common wear in bearing. These equations can be replaced by any other profile of wear in the same model. The other wear signatures can obtain using wear depth measurement on the worn bearings or using other mathematical models for wear. Some examples of other wear signatures are given in [2,23].

Figure 2. Pad number 5 schematic considering pad geometrical imperfections induced by wear.

Figure 2. Pad number 5 schematic considering pad geometrical imperfections induced by wear.

2.3. Tilting pad bearing thermo-hydrodynamic simulation

Hydrodynamic pressure distributions on each pad of a TPJB are obtained by the Reynolds equation. The mixed lubrication and the oil film turbulence may affect the behavior of the TPJBs experiencing heavy and impact loads [37]. Therefore, a modified version of the Reynolds equation considering the effects of surface roughness and the fluid turbulence on the oil film flow is provided in Equation (8) [38]:

$$\frac{\partial }{\partial \theta }\left({\varphi }_{\theta }^p\frac{{\overline{h}}_p^3}{k_{\theta }\mu }\frac{\partial P_{fld,p}}{\partial \theta }\right)+\frac{\partial }{\partial z}\left({\varphi }_z^p\frac{{\overline{h}}_p^3}{k_z\mu }\frac{\partial P_{fld,p}}{\partial z}\right)=\frac{V_1+V_2}{2}\frac{\partial \left(\rho {\overline{h}}_{T,p}\right)}{\partial \theta }+\frac{V_1-V_2}{2}\frac{\sigma \partial {\varphi }^s}{\partial \theta }+\frac{\partial \left(\rho {\overline{h}}_p\right)}{\partial t}$$

(8)

where ${\overline{h}}_p$ and ${\overline{h}}_{T,p}$ are the oil film thickness on the p’th pad and the expected value of the mean gap between the journal and the p’th pad, respectively. The standard deviation of the surface asperity height on the rough surface of the journal and the bearing is also denoted by σ. For large enough distances between the journal and the pad’s inner surface, the effect of surface roughness can be neglected, and the expected mean gap value ${\overline{h}}_{T,p}$ is equal to the oil film thickness ${\overline{h}}_p$. In the cases where the separation between the journal and the pad’s inner surface is less than 3σ, the surface roughness affects the mean gap value and the mean gap will be equal to the expected value of ${\overline{h}}_{T,p}$, considering the surface roughness effect. In this work, it is assumed that asperities have Gaussian distribution on both journal and pads. The formulation for calculating the expected value of ${\overline{h}}_{T,p}$, considering the effect of Gaussian distribution of the asperities, is given in [39]. Furthermore, the oil film viscosity and density are represented by $\mu$ and $\rho ,$ respectively. While ${\varphi }_{\theta }^p$, ${\varphi }_z^p,$ and ${\varphi }^s$ are the oil flow correction factors. The effects of surface roughness on oil film flow can be evaluated by these factors. The value of the oil flow correction factors depends on the surface roughness pattern and asperities heights. Patir and Cheng [39] presented a detailed description of the calculation procedure for oil flow correction factors. The effects of oil film turbulence can be obtained by $k_{\theta }$ and $k_z$ factors which can be calculated as follows [40]:

$$\left\{\begin{array}{c}\begin{array}{cc}\begin{array}{c}{k}_{\theta }=12\left(1+TBA.R_L^{ETA}\right)\\ k_z=12\left(1+TBB.R_L^{ETB}\right)\end{array}& TurbulentFlow\end{array}\\ \begin{array}{cc}{k}_{\theta }=k_z=12& LaminarFlow\end{array}\end{array}\right.$$

(9)

where TBA, TBB, ETA, and ETB are Ng-Pan [40] turbulent model constants as listed in Table 1. $R_L$ is the local Reynolds number. Boundary conditions for the oil film pressure distribution in a hydrodynamic TPJB can be written as:

$$P_{fld,p}\left({\theta }_s,z\right)=P_{fld,p}\left({\theta }_e,z\right)=P_{fld,p}\left(\theta ,\pm \frac{L}{2}\right)=P_{\mathrm{s}\mathrm{u}\mathrm{p}}$$

(10)

Reynolds equation given in Equation (8) can be solved using the finite element method based on the boundary conditions in Equation (10). Reynolds boundary condition is also applied on Equation (8) to consider cavitation effects in the model [2]. Pad surfaces are meshed via quadrilateral elements. The Galerkin method is used to obtain the hydrodynamic pressure on each node. A detailed description of the finite element solution of the Reynolds equation can be found in [41].

It should be noted that the oil film viscosity varies by film temperature as follows [42]:

$$\mu ={\mu }_0\exp \left(-\beta \left(T_p-T_0\right)\right)$$

(11)

where ${\mu }_0$ is oil film viscosity at the temperature of $T_0$, $\beta$ shows the thermo-viscosity index, and $T_p$ denotes the lubricant temperature in the p’th pad. The one-dimensional energy equation and the two-dimensional heat equation are iteratively solved to find the lubricant and the pad’s temperature distributions. Then, the calculated oil film temperature distribution is used to obtain the oil film viscosity. The lubricant temperature on the p’th pad, $T_p,$ can be determined based on the one-dimensional energy equation as follows [42]:

$$\rho {\overline{C}}_p\frac{\omega }{2}\frac{d{T}_p}{d\theta }=\frac{\mu {\left(R_j\omega \right)}^2}{{\left({\overline{h}}_p\right)}^2}-\frac{{H_{p1}\left(T_p-T_p^b\right)+H}_j\left(T_p-T_j\right)}{{\overline{h}}_p}$$

(12)

Where $\rho$ and ${\overline{C}}_p$ are the oil film density and specific heat capacity, respectively. Furthermore, $T_j$ and $T_b^p\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$represent the temperatures of the journal and the p’th pad which could be obtained using the calculated lubricant temperature as follows [42].

$$T_j=\frac{\sum _{p=1}^{N_{pads}}{\int }_{{\theta }_{s,p}}^{{\theta }_{e,p}}{T}_{p}d\theta }{N_{pads}{\mathsf{\Delta }}_p}$$

(13)

Table 1. Ng-Pan turbulence model constants [40].

Table 1. Ng-Pan turbulence model constants [40].

Model	TBA	ETA	TBB	ETB
Ng-Pan	0.00113	0.9	0.000358	0.96

The two-dimensional heat equation which gives the pads temperature distribution is given as follows:

$$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T_p^b}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{T}_p^b}{\partial {\theta }^2}=0;R_p\le r\le R_p+t_p,{\theta }_{s,p}\le \theta \le {\theta }_{e,p}$$

(14)

It is worth mentioning that the boundary conditions of Equation (14) include the conduction heat transfer in the pads, convection heat transfer between the oil film and the pad’s inner side, and convection heat transfer between the air, the pad’s edges and the pad’s backside [42].

In this paper, the finite difference method is used to solve the one-dimensional energy equation, while the finite element method is implemented to solve the two-dimensional heat conduction equation in pads (see Equations (12) and (14)). The following suggestion is used to calculate the lubricant temperature at the leading edge of each pad [42]:

$$T^p\left({\theta }_{s,p}\right)=T_{\mathrm{s}\mathrm{u}\mathrm{p}}+\frac{\min \left({\left\{{\overline{h}}_k\left({\theta }_{s,k}\right)\right\}}_{k=1,\dots ,N_{pads}}\right)}{{\overline{h}}_p\left({\theta }_{s,p}\right)}\mathsf{\Delta }T;p=1,\dots ,N_{pads}$$

(15)

where $\mathsf{\Delta }T$ is the oil film bulk temperature rise, and $T_{\mathrm{s}\mathrm{u}\mathrm{p}}$ shows the oil film supply temperature. The lubricant film bulk temperature rise can be determined by [42]:

$$\mathsf{\Delta }T=\frac{\sum _{p=1}^{N_{pads}}{R}_j\omega {\int }_{{\theta }_{s,p}}^{{\theta }_{e,p}}\mu \frac{R_j\omega }{{\overline{h}}_p}L{R}_{j}d\theta }{Q_{\mathrm{s}\mathrm{u}\mathrm{p}}\rho {\overline{C}}_p}$$

(16)

Equations (8), (12), and (14) should be solved iteratively to obtain the lubricant viscosity at the TPJB equilibrium position. The oil film viscosity can be finally calculated via Equation (11). A more detailed description of the iterative procedure for energy and heat conduction equation solution can be found in [42].

3. Results and discussion

Table 2. Rotor-tilting pad bearing properties.

Table 3. TPJB Thermal and Surface properties.

Figure 3. Oil film thickness on intact and worn TPJBs. a) Shaft rotational speed is equal to 3500rpm b) Shaft rotational speed is equal to the first critical speed of the system c) Shaft rotational speed is equal to 10200rpm. (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 3. Oil film thickness on intact and worn TPJBs. a) Shaft rotational speed is equal to 3500rpm b) Shaft rotational speed is equal to the first critical speed of the system c) Shaft rotational speed is equal to 10200rpm. (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 4. Hydrodynamic pressure distribution on intact and worn TPJBs. a) Shaft rotational speed is equal to 3500rpm b) Shaft rotational speed is equal to the first critical speed of the system c) Shaft rotational speed is equal to 10200rpm. (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 4. Hydrodynamic pressure distribution on intact and worn TPJBs. a) Shaft rotational speed is equal to 3500rpm b) Shaft rotational speed is equal to the first critical speed of the system c) Shaft rotational speed is equal to 10200rpm. (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 5. Pad temperature distribution at 3500rpm. a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Figure 5. Pad temperature distribution at 3500rpm. a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Figure 6. Pad temperature distribution at the first critical speed of the system. a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Figure 6. Pad temperature distribution at the first critical speed of the system. a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Figure 7. Pad temperature distribution at 10200rpm. a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Figure 7. Pad temperature distribution at 10200rpm. a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Figure 8. Pads inner surface temperature for intact and worn TPJBs a) Shaft rotational speed equals 3500rpm b) Shaft rotational speed equals to the first critical speed of the system c) Shaft rotational speed equals 10200rpm. (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 8. Pads inner surface temperature for intact and worn TPJBs a) Shaft rotational speed equals 3500rpm b) Shaft rotational speed equals to the first critical speed of the system c) Shaft rotational speed equals 10200rpm. (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 9. The effect of wear in loaded pads considering variable shaft speed a) Pad number 4 temperature in 71% of the pad’s arc length from its leading-edge b) Pad number 5 temperature in 71% of the pad’s arc length from its leading-edge. c) Journal eccentricity (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 9. The effect of wear in loaded pads considering variable shaft speed a) Pad number 4 temperature in 71% of the pad’s arc length from its leading-edge b) Pad number 5 temperature in 71% of the pad’s arc length from its leading-edge. c) Journal eccentricity (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 10. TPJB stiffness for intact and worn bearings cases at different speeds (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 10. TPJB stiffness for intact and worn bearings cases at different speeds (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 11. TPJB damping for intact and worn bearings cases at different speeds (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 11. TPJB damping for intact and worn bearings cases at different speeds (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 12. Rotor vibration amplitudes for intact and worn TPJBs considering $2\times {10}^{-6}\mathrm{k}\mathrm{g}.\mathrm{m}$ mass unbalance on disc a) Disc vibration amplitude b) Journal vibration amplitude (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 12. Rotor vibration amplitudes for intact and worn TPJBs considering $2\times {10}^{-6}\mathrm{k}\mathrm{g}.\mathrm{m}$ mass unbalance on disc a) Disc vibration amplitude b) Journal vibration amplitude (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Table 4. The effect of the bearing wear severity on the rotor-TPJB first critical speed and the variation of the rotor’s vibration amplitude.

Figure 13. Solid-to-solid contact pressure induced by journal-bearing contact at different speeds for pad No. 1 to pad No. 5 in the intact and worn bearing cases (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 13. Solid-to-solid contact pressure induced by journal-bearing contact at different speeds for pad No. 1 to pad No. 5 in the intact and worn bearing cases (Blue solid line: Intact bearing, Red dashed line: Bearing with maximum wear depth of $10\mathsf{\mu }\mathrm{m}$, Magenta dashed dot line: Bearing with maximum wear depth of $20\mathsf{\mu }\mathrm{m}$.).

Figure 14. The journal displacement bifurcation diagrams in different speeds for the intact and the worn bearing cases.

Figure 14. The journal displacement bifurcation diagrams in different speeds for the intact and the worn bearing cases.

Figure 15. Waterfall diagrams for disc velocity in a range of rotor speeds and different wear severities, a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Figure 15. Waterfall diagrams for disc velocity in a range of rotor speeds and different wear severities, a) Intact bearing b) Maximum wear depth: 10μm c) Maximum wear depth: 20μm.

Table 5. Peak amplitudes of the disc velocity signal spectra at 3500rpm in mm/s.

Table 6. Peak amplitudes of the disc velocity signal spectra at the first critical speed of the system in mm/s.

Table 7. Peak amplitudes of the disc velocity signal spectra at 10200rpm in mm/s.

Table 8. Effect of wear severity on disc vibration amplitude at 3500rpm and the first critical speed.

Table 9. Journal orbits in selected rotational speeds for intact and worn bearing cases.

Table 9. Journal orbits in selected rotational speeds for intact and worn bearing cases.

	3500rpm	5500rpm	10200rpm	11000rpm
Intact bearing
$d_{max,5}=10\mu m$
$d_{max,5}=20\mu m$

4. Conclusions

Nomenclature

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