The three-disk, four-support bearing-rotor system shown in Fig. 1 represents a flexible camshaft-type multi-support journal-bearing-rotor system subjected to asymmetric loading. Shaft flexibility transmits the end load among the supports, producing nonuniform journal eccentricity and inclination and consequently altering the local oil-film thickness, pressure distribution, and dynamic coefficients. An equivalent end load is introduced to retain the essential mechanism of support-reaction redistribution while avoiding unnecessary geometric complexity. The structural and hydrodynamic fields are coupled through the journal attitudes and bearing reaction forces at all four supports. For a prescribed end load and rotational speed, deformation of the common shaft simultaneously changes the journal-center displacement and inclination at each bearing, while the resulting oil-film reaction forces are fed back into the global structural equilibrium. Consequently, the four bearing operating states are determined as mutually coupled equilibrium quantities rather than as independently prescribed misalignment conditions. The converged support-specific states are subsequently used to determine the oil-film dynamic coefficients, thereby defining the deformation-dependent support conditions used in the global rotordynamic model.

The rotor shaft is formulated using a small-strain, small-rotation Timoshenko beam model whose structural stiffness matrix is constructed in the reference configuration. Thus, second-order axial elongation caused by transverse bending and the associated configuration-dependent geometric stiffness are not included in the baseline structural model. This assumption applies only to the structural kinematics: journal eccentricity and centerline inclination are retained explicitly in the lubrication geometry, and the oil-film pressure, reaction forces, and dynamic coefficients are updated with the converged FSI state. The subsequent modal analysis is therefore a small-perturbation analysis about each state-dependent hydrodynamic equilibrium. An a posteriori kinematic check at the four bearing sections and a deliberately conservative stress-stiffening sensitivity estimate confirm that geometric nonlinearity produces only small corrections within the investigated operating range; a detailed assessment is provided in the Supplement.

The lubricant is modeled as incompressible, isoviscous, and laminar, and the analysis is restricted to converged hydrodynamic equilibria and small perturbations about those equilibria (Zhang et al., 2019). Inlet-pressure effects are neglected because the supply pressure is small relative to the hydrodynamic film pressure in the investigated range (Feng et al., 2019). Thermal transport, asperity contact, and mass-conserving cavitation are not solved explicitly; a half-Sommerfeld treatment is used for the divergent region. The first-order velocity-dependent squeeze contribution is retained in the linearized damping coefficients, whereas finite-amplitude transient squeeze, moving cavitation/reformation boundaries, and mixed-lubrication contact are outside the present formulation. Supplementary sensitivity checks show that the laminar assumption is well satisfied and that reasonable viscosity variations do not change the support-redistribution ordering, although the exact pressure peaks, minimum film thickness, and extreme J4 coefficient amplitudes remain model-dependent in the limiting thin-film state.

A finite-element formulation is employed to analyze the multi-support rotor–journal bearing system shown in Fig. 2. The shaft is modeled globally using Timoshenko beam theory so that both bending deformation and shear deformation, as well as their influence on the attitude of each supported journal section, can be represented. The global axial coordinate is denoted by z, the transverse x axis is taken as positive to the right, and the vertical y axis is taken as positive downward. In each transverse vibration plane, u (z) and ϕ (z) denote the lateral displacement and cross-sectional rotation, respectively. The fundamental relations of the Timoshenko beam are given in Eq. (1). 1M=EIdϕdzQ=kGAdudz-ϕ Here, Q is the shear force, k is the shear correction factor, G is the shear modulus, A is the cross-sectional area, and M is the bending moment. The shaft is divided axially into two-node beam elements. Each node contains lateral displacement and sectional-rotation degrees of freedom, with two degrees of freedom in each of the x and y planes. The element degree-of-freedom vector is denoted by qe. Based on the principle of virtual work, the element stiffness matrix Ke and consistent mass matrix Ke are derived as shown in Eq. (2): 2Ks=∑e=1neLeTKeLe,M=∑e=1neLeTMeLeq=xNϕx,NyNϕy,NT. Here, Le is the assembly matrix from element to global degrees of freedom. The structural static stiffness operator Ks is then obtained, and the static deflection and sectional attitude are solved under the prescribed external loads and bearing oil-film reaction forces. For an axisymmetric shaft made of homogeneous material, the structural equations have the same form in the two transverse planes. The external loads acting on the shaft include self-weight, the end load F, and concentrated forces induced by the disks. The oil-film reaction force at the ith bearing is denoted by Foil,i. The equilibrium point is first obtained by enforcing global balance among structural internal forces, external loads, and oil-film reactions, as expressed by Eq. (3): 3Ksq=fg+fext+∑i=14BiTFoil,i. In Eq. (3), Ks is the static stiffness operator obtained from the Timoshenko beam discretization, Bi is the mapping matrix from the support force to the structural degrees of freedom, and q is the structural degree-of-freedom vector. By extracting the actual centerline slope at the bearing section, the slope components Sx,i,Sy,i can be obtained as Eq. (4): 4Sx,i=ϕx,i+Qx,ikGASy,i=ϕy,i+Qy,ikGA. Thus, for the i-th bearing section z=zi, the eccentric displacement of the journal center relative to the bearing center is expressed as xi,yi. By normalizing it with respect to the radial clearance C, the dimensionless displacement components and eccentricity ratio are obtained as ζi=xiC,ηi =yiC,ϵi =ζi2+ηi2. The parameters required for the subsequent lubrication characteristic calculation are then extracted using Eq. (5). 5φi=atan2ηi,ξiαi=atan2Sy,i,Sx,iγi=arctanSx,i2+Sy,i2, where φi is the attitude angle of the journal, describing the direction of the journal-center eccentricity vector in the x–y plane; αi is the inclination azimuth angle, representing the in-plane direction of the journal axis inclination vector; and γi is the inclination angle, used to characterize the degree of inclination of the journal axis relative to the bearing axis. The magnitude of the corresponding slope is tanγi, and under the small-angle assumption, tan γi≈γi. To simultaneously capture the coupling effect between journal eccentricity and inclination, a flexible-deformation index Mr is introduced to quantify the coupled eccentricity–inclination state, as defined in Eq. (6). 6Mr=ϵ2+λLb2Ctanγi2 In Eq. (6), ϵ is the eccentricity ratio and represents the translational displacement of the journal center, whereas qr=Lb2C tanγi is the dimensionless inclination term, describing the additional edge displacement caused by journal tilting relative to the radial clearance. The weighting coefficient λ balances the relative contributions of the translational and inclination components. It should be emphasized that Mr is used only as a post-processing index for interpreting the journal attitude and does not enter the oil-film coefficient interpolation or the rotor dynamic equations.

To assess the robustness of the weighting coefficient, a global RMS balancing analysis was performed over the operating database, giving a statistical reference value of λRMS=2.3545; a 1000-sample operating-condition cluster bootstrap yielded a 95 % confidence interval of 2.3486–2.3603. Changing λ from 1.5 to this reference value changes the numerical magnitude of Mr but does not alter the support ordering or the characteristic trend: J1 decreases, J2 remains comparatively stable, J3 is transitional, and the loaded-end support, J4, strengthens markedly with load. The value λ=1.5 is therefore retained to preserve a compact O(1) interpretation scale, while the RMS-derived value is used only as an independent reference for sensitivity. The calibration and fixed-state sensitivity results are provided in the Supplement.

When rotor flexible deformation is considered, as shown in Fig. 3, the oil-film thickness at any point θ,z within the bearing clearance can be expressed as the superposition of an eccentricity term and an inclination term. Let the bearing length be Lb. The dimensionless oil-film thickness of the bearing is defined as Hi=hiC, as given in Eq. (7). 7Hi(θ,z*)=1-ϵicosθ-φi-kiZ*cosθ-αi

Here, ϵi, φi, αi, and γi are obtained from the structural solution; z* is the dimensionless axial coordinate; and ki is the dimensionless inclination coefficient, which characterizes the contribution of journal inclination to the axial variation in oil-film thickness. The detailed expression is given in Eq. (8). 8z*=z-12LbLb∈-12,12ki=LbCtanγi Under the assumptions of incompressibility, constant viscosity, and laminar flow, the Reynolds equation used in this study is written as Eq. (9): 9∂∂θh3∂p∂θ+R2∂∂zh3∂p∂z=6μR2Ω∂h∂θ+12μR2∂h∂t, where R is the journal radius, μ is the lubricant viscosity, and Ω is the journal angular speed. 10p(θ,0)=p(θ,Lb)=0∂p∂θθ=0=∂p∂θθ=2π=0 The half-Sommerfeld condition is implemented by truncating negative pressure in the divergent region. This treatment does not enforce mass conservation in the cavitated zone. During the static FSI equilibrium calculation, ∂h/∂t=0. During dynamic-coefficient extraction, however, positive and negative journal-velocity perturbations activate the film-thickness time-derivative term, thereby retaining the first-order squeeze contribution in the linearized damping coefficients. Nonlinear finite-amplitude squeeze, moving cavitation boundaries, and film reformation are not considered. The oil-film reaction force is then obtained by integrating the calculated pressure distribution over the bearing inner surface, as expressed in Eq. (11). 11Foil=FxFy=-∫0Lb∫02πp(θ,z)cosθsinθRdθdz The negative sign indicates that the pressure acting normal to the bearing surface is opposite to the reaction force acting on the journal. Numerical integration is performed using the composite Simpson rule, and (Fx, Fy) from each bearing is supplied to the equilibrium iteration.

For the four-support system, the journal operating states cannot be determined independently because all bearing reaction forces act on the same flexible shaft. A change in the reaction force at one support modifies the global shaft configuration and thereby alters the eccentricity and inclination conditions experienced by the remaining supports. The equilibrium problem is therefore formulated at the system level: the shaft deformation determines the journal-center displacement and inclination at all four bearings, while the corresponding oil-film reactions are simultaneously returned to the structural model. Iteration is continued until the structural configuration and the complete set of hydrodynamic support reactions become mutually consistent. This global closure enables the spatial redistribution of journal attitudes and hydrodynamic support conditions to emerge from the coupled solution rather than being imposed through prescribed local misalignment parameters. The rotor system contains four bearings, and the unknown vector of bearing operating points is defined in Eq. (12). 12χ=ξη=ξ1⋯ξnb,η1⋯ηnbT, where ξi and ηi are the dimensionless components of the journal-center displacement at the ith bearing section. For a prescribed rotational speed and external load, the bearing forces are calculated from the instantaneous journal attitudes and returned to the global structural equilibrium. The operating-state vector χ in Eq. (12) is therefore updated together with the shaft deformation until the structural and hydrodynamic fields become mutually consistent. The numerical iteration is monitored using the force-equilibrium residual in Eq. (13) at the free structural degrees of freedom. 13Rf(k)=Ksq(k)-fext-∑i=14BiTFoil,i(k), where the external-load vector contains gravity, disk loads, and the prescribed end load. A damped Newton-type correction is used to update the coupled equilibrium state, with a relaxation factor of 0.4. The normalized equilibrium residual is defined in Eq. (14). 14rrel(k)=Rf(k)maxfext,f,10-6, where the subscript f denotes the free structural degrees of freedom. The production calculations use the dimensionless normalized tolerance rrel<10-5. The implementation additionally monitors an absolute free-DOF residual threshold of 10⁻¹⁰ N and an auxiliary equilibrium-closure criterion |g|<10-6, with a maximum of 200 iterations. Representative convergence data show that the normalized equilibrium and closure criteria are satisfied within 45 iterations for the examined states; detailed residual statistics and the distinction between the diagnostic and internal stopping criteria are provided in the Supplement.

To reduce the computational cost of the subsequent dynamic analysis, an offline database is constructed over F∈0,30 N and Ω∈1000,8000 rpm. For each valid converged hydrodynamic operating point, the equilibrium states of the four bearings are obtained, and the corresponding oil-film stiffness and damping coefficients are extracted by differential perturbation. The index Mr is recorded only to characterize the eccentricity–inclination state and is not used for interpolation. During the state-space analysis, the required bearing coefficient matrices are obtained by bilinear interpolation and assembled separately at J1–J4 into the global finite-element model.

The converged support-specific dynamic coefficients are subsequently incorporated into the finite-element rotordynamic model shown in Fig. 4. In particular, an equivalent end load is applied at the rightmost node of the model shown in Fig. 4 to represent asymmetric external loading in the camshaft-type system. Let q (t) be the generalized coordinate vector of the system, including lateral displacement and rotational degrees of freedom at each node. Each shaft node in the present model contains four degrees of freedom: two lateral translations and two cross-sectional rotations.

After assembling the shaft, disks, and four bearing supports, the global dynamic equation of the three-disk four-support rotor system is obtained as Eq. (15): 15Mq¨+Cs+Coil+ΩGq˙+Ks+Koilq=f(t), where M is the global mass matrix, Ks is the structural stiffness matrix, G is the gyroscopic matrix, Cs is the structural damping matrix, f (t) is the unbalance excitation force vector, and Koil and Coil are assembled from the oil-film dynamic coefficients obtained from Eq. (16). 16Kxx,i=-Fx,ixi*+Δx-Fx,ixi*+Δx2ΔxCxx,i=-Fx,ix˙i*+Δx˙-Fx,ix˙i*+Δx˙2Δx˙Kyy,i=-Fy,iyi*+Δy-Fy,iyi*+Δy2ΔyCyy,i=-Fy,iy˙i*+Δy˙-Fy,iy˙i*+Δy˙2Δy˙Kxy,i=-Fx,iyi*+Δy-Fx,iyi*+Δyy2ΔyCxy,i=-Fx,iy˙i*+Δy˙-Fx,iy˙i*+Δy˙2Δy˙Kyx,i=-Fy,ixi*+Δx-Fy,ixi*+Δx2ΔxCyx,i=-Fy,ix˙i*+Δx˙-Fy,ix˙i*+Δx˙2Δx˙ The obtained dynamic characteristic coefficients are assembled into the global matrices, as shown in Eq. (17): 17Koil=Kxx,iKxy,iKyx,iKyy,i,Coil=Cxx,iCxy,iCyx,iCyy,i. The baseline dimensionless displacement and velocity perturbation sizes are both 10⁻⁴ and are applied using centered differences. Varying either perturbation from 0.25 to 4 times the baseline changes the stiffness and damping matrices by at most 0.122 % and 0.072 %, respectively, while all coefficient signs are preserved. Reynolds-mesh and perturbation-size sensitivity results are provided in the Supplement.

Because closely spaced complex modes may coexist near a synchronous intersection, modal identity is not determined from frequency proximity alone. For each prescribed end load, the support-specific oil-film stiffness and damping matrices are interpolated from the two-dimensional load-speed database at every trial rotational speed, and the synchronous critical speed of a tracked branch is obtained from the intersection between the branch frequency and the 1X excitation line. Modal continuity between adjacent loading states is evaluated using the mass-weighted complex modal assurance criterion (cMAC): 18cMACϕa,ϕb=ϕaHMϕb2ϕaHMϕaϕaHMϕb. Starting from the identified low-load synchronous modes, each branch is continued incrementally in F by selecting the candidate with the largest cMAC relative to the preceding state, preventing neighboring synchronous branches from being interpreted as a single branch. To interpret branch-dependent sensitivity, the mass-normalized modal vectors are further used to evaluate generalized bearing-stiffness participation and bearing-location displacement participation. The complete non-symmetric bearing stiffness and damping matrices remain unchanged in the eigenvalue calculation; the participation measures are diagnostic projections only. Their definitions and the finite boundary-substitution decomposition are provided in the Supplement.

The FSI program is validated by comparing the maximum pressure with the values calculated by Sun and Gui (2004) and Lv et al. (2018). The bearing parameters used for comparison are as follows: journal diameter 0.06 m, length-to-diameter ratio 1.1, clearance 30 µm, lubricant viscosity 9×10×10-3 Pa s, and rotational speed 3000 rpm.

The results are listed in Table 1. For inclination angles of 0, 0.004, and 0.007°, the present pressure maxima agree well with Sun and Gui (2004) and Lv et al. (2018), confirming the main pressure-field trend of the misaligned finite-length bearing. At 0.010°, the present value is 371.3 MPa compared with 409.8–414.8 MPa in the references, giving a deviation of approximately 9 %–11 %. This limiting state is characterized by severe local film thinning, for which the pressure peak becomes increasingly sensitive to grid resolution, viscosity, cavitation treatment, and the possible onset of mixed lubrication. Because the present model uses an isothermal constant-viscosity formulation with a non-mass-conserving half-Sommerfeld treatment, the exact extreme pressure peak and associated coefficient amplification should therefore be regarded as model-dependent predictions. The model is used here primarily to resolve the support-redistribution mechanism in the converged hydrodynamic regime; detailed lubrication-regime and equivalent-viscosity sensitivity checks are reported in the Supplement.

The numerical procedure is summarized in Fig. 5. For each sampled end load and speed condition, the shaft deformation and oil-film pressure field are iterated to a mutually consistent FSI state, after which the support-specific dynamic coefficients are extracted and stored. During online dynamic analysis, the required matrices are obtained by bilinear interpolation and assembled at J1–J4. Of the 211 001 sampled load-speed states, 200 304 return valid direct FSI solutions (94.93 %). Only these directly converged states are used in the physical statistics and independent validation. Missing coefficient entries are filled solely to maintain database lookup continuity and are not interpreted as directly converged operating states.

Figure 6 compares the present results with Lund and Orcutt's (1967) results for the non-dimensional stiffness and damping coefficients of a journal bearing with a length-to-diameter ratio of 1. The comparison shows good agreement, indicating that the method used in this study is suitable for the subsequent dynamic analysis.

To validate the flexible rotor model and the modal solution procedure, the mode shapes obtained using the MATLAB model were compared with those predicted by an ANSYS three-dimensional finite-element model under the operating condition of 3000 rpm and F=0 N. As shown in Fig. 7, the normalized bending mode shapes near 53 and 89 Hz obtained using the two models exhibit good overall agreement. The nodal positions near the supports and the deformation peaks at the disk locations are consistently reproduced, with only minor local differences in amplitude. The agreement supports the structural discretization and modal-solution implementation of the MATLAB model. Together with the lubrication and dynamic-coefficient validations above, it provides an independent structural reference for the subsequent modal-branch analysis.

Furthermore, an a posteriori validity check is conducted for the small-deformation assumption of the structure. Among the four bearing sections over all directly converged database states, the maximum recovered centerline inclination is 0.08869°, which corresponds to a second-order geometric elongation term of 0.5 tan² γ = 1.20 × 10⁻⁶. For further verification, to deliberately amplify the possible stress-stiffening effect, a conservative sensitivity case is constructed by assuming that the maximum local second-order elongation acts uniformly along the full shaft and is fully restrained axially. The relative variations in critical speed for the four synchronous branches, i.e., FW, BW, H1, and H2, are 0.1021 %, 0.1000 %, 0.0777 %, and 0.0742%, respectively, with all associated cMAC values above 0.999998. These results demonstrate that within the scope of the present study, geometric stress stiffening only imposes limited corrections on the synchronous critical speeds and does not induce substantial reconstruction of the reported modes.

Figure 8 presents the variation in the flexible-deformation index Mr at the four supports under combined load and speed conditions. At F = 0, the nearly coincident values of J1/J4 and J2/J3 indicate an approximately symmetric initial journal-attitude state. End loading rapidly breaks this symmetry: within 0–25 N, Mr at J4 increases by 266.5 %–402.2 %, whereas J1 and J3 decrease, and J2 changes only slightly. At 25 N, the Mr ratios J4/J1 and J4/J3 reach 4.19–6.02 and 1.76–2.33, respectively, showing that shaft flexibility concentrates the eccentricity–inclination coupling at the loaded-end support while redistributing the deformation response of the inner supports. Although increasing speed generally suppresses Mr, this effect weakens at J4 under high loads, indicating that loaded-end attitude amplification becomes dominant.

Figure 9 shows the evolution of journal attitude, oil-film thickness, and pressure distribution with increasing Mr. At Mr=0.3, the nearly parallel journal produces an approximately axially uniform film and an eccentricity-dominated pressure field. As Mr increases to 0.6 and 0.9, journal inclination introduces pronounced axial asymmetry: Hmin decreases and shifts toward the bearing end, while the pressure field becomes increasingly localized and Pmax increases markedly. These results indicate that Mr effectively characterizes the transition from a primarily eccentric state to an eccentricity–inclination coupled lubrication state under shaft flexibility.

Figures 10 and 11 show that the oil-film dynamic coefficients evolve non-uniformly with Mr at 3000 rpm. The loaded-end support J4 exhibits the strongest amplification: its non-dimensional stiffness trace Tr (K) and damping trace Tr (C) increase by 4298.0 % and 293.6 %, respectively. The cross-coupled terms also become strongly nonlinear, with Kxy reversing sign near Mr=0.96, indicating enhanced pressure-field asymmetry at high inclination levels. In contrast, J3 undergoes marked weakening, with Tr (K) and Tr (C) decreasing by 62.4 % and 36.8 %, while the variations at J1 and J2 remain limited. These results indicate that shaft-flexibility-induced journal inclination locally compresses the converging oil-film wedge at J4 and amplifies its incremental hydrodynamic reaction, whereas the inner supports respond mainly through reaction redistribution. Flexible deformation, therefore, produces a strongly nonuniform redistribution of oil-film stiffness and damping rather than uniform strengthening of all bearings.

Figure 12 presents the evolution of the direct-stiffness trace Tr (Ki) =Kxx,i+Kyy,i and the corresponding relative support-stiffness ratios at 2884 rpm. Tr (Ki) is used only as a scalar descriptor of the direct stiffness level; the complete non-symmetric 2×2 bearing matrices remain in the eigenvalue analysis. With increasing end load, Tr (K4) increases from 2.66×106 to 1.25×108 N m⁻¹, approximately 47.0 times its initial value. By contrast, Tr (K2) increases by only 15.9 %, while Tr (K1) and Tr (K3) decrease by 16.1 % and 63.8 %. The ratios Tr (K4) /Tr (K2) and Tr(K4) /Tr (K3) increase from 0.137 to 5.56 and 17.82, respectively. Thus, the multi-support boundary evolves from a comparatively shared distribution toward a strongly nonuniform, loaded-end-dominated support-stiffness distribution.

Because Mr is a derived descriptor of the eccentricity–inclination state rather than an interpolation variable of the dynamic-coefficient database, modal identities were tracked continuously with the physical end load F. The continuation analysis identifies two neighboring low-order synchronous branches near 2.89 krpm. The FW branch varies from 2883.6771 rpm at the low-flexible-deformation state to 2883.6665 rpm at the high-flexible-deformation state, whereas the BW branch changes from 2894.9847 to 2894.9960 rpm, corresponding to relative variations of only -0.00037 % and +0.00039 %, respectively. Their minimum adjacent-state cMAC values are 0.999966 and 0.999979, while the maximum inter-branch cMAC is only 0.0885, confirming that they are two distinct neighboring branches. As shown in Fig. 13, the higher-order synchronous branches H1 and H2 exhibit larger critical-speed variations of +0.01402 % and +0.14772 %, respectively, and an additional high-order synchronous intersection appears at 5843.2 rpm in the high-flexible-deformation state. The synchronous critical-speed response is therefore strongly branch-dependent.

To determine whether this branch dependence originates from global mode-shape reconstruction, Fig. 14 compares the spatial modes of the FW, BW, H1, and H2 branches under increasing deformation. The global modal patterns remain highly consistent, whereas the critical-speed sensitivities differ by more than 2 orders of magnitude. In particular, the FW and BW variations remain below 0.001 %, while H2 reaches 0.14772 %. The higher sensitivity of H2 therefore does not arise from a change in modal identity or a pronounced reconstruction of the global mode shape but from stronger coupling of that branch to the support locations.

Figure 15 further clarifies the support-participation mechanism. From FW and BW to H1 and H2, both the generalized bearing-stiffness participation and the bearing-location displacement participation increase in the same order as the critical-speed sensitivity. For H2 in the low-deformation state, these measures reach 1.449 % and 0.0568 %, approximately 23.6 and 22.1 times the corresponding FW values, while its critical-speed variation reaches 0.14771 %. This consistent ordering demonstrates that the dynamic consequence of support redistribution is governed by the generalized modal coupling at the bearing locations rather than by the local stiffness amplification alone.