This paper presents a novel control strategy for hybrid rotor–bearing systems integrating hydrodynamic journal bearings with active magnetic bearings (AMBs) to address the persistent challenge of nonlinear vibrations in high-speed rotating machinery. The study introduces the application of a state-dependent parameter proportional-integral-plus (SDP-PIP) controller designed within a non-minimal state-space framework, offering a significant advancement over conventional control approaches. A four-degree-of-freedom model incorporating short-bearing approximation for hydrodynamic forces and nonlinear electromagnetic force characterization is developed to capture the complex system dynamics. The controller performance is evaluated through numerical simulations over a range of rotational speeds from 130 to 500 rad/s, together with sensitivity analyses under parameter variations and comparisons with a conventional PID controller. The results show that the proposed controller effectively suppresses nonlinear vibrations and stabilizes oil-whirl and oil-whip instabilities over the investigated operating conditions. In comparison with the PID controller, the SDP-PIP controller provides improved vibration attenuation and maintains stable journal motion with lower oscillation amplitudes, particularly near unstable operating regimes. These findings demonstrate the potential of the SDP-PIP control strategy for enhancing the dynamic performance and operational stability of hybrid journal bearing systems.

High-pressure pump rotors are critical components in a wide range of industrial and mechanical systems, including turbomachinery, hydraulic systems, and aviation engines. This study presents a dynamic model for a single-stage high-speed high-pressure centrifugal pump supported by hydrodynamic journal and thrust bearings. The model is based on a rigid rotor framework and incorporates the coupling effects between the journal and thrust bearings. The influence of key bearing parameters is systematically investigated, with particular attention given to the impact of introducing a central groove in the left bearing. Furthermore, the analysis explores the effects of integrating a third central journal into the system which acts also as rear wear ring. The study also examines the role of bearing boundary conditions, which can be regulated through auxiliary lines, and evaluates the influence of unbalance forces on system performance. The results demonstrate that boundary conditions significantly affect the load-carrying capacity and stability of the bearings, potentially leading to bearing failure if not properly managed. Additionally, while a coupling effect between the journal and thrust bearings is observed, its influence is found to be relatively minor in the context of the rigid rotor model employed in this analysis. The findings of this study provide valuable insights for the design and optimization of high-speed centrifugal pumps, offering a deeper understanding of the dynamic interactions and operational constraints inherent in such systems.

This article presents a comprehensive investigation into the coupled behavior of hydrodynamic journal and thrust bearings, employing a 5-degree-of-freedom model and focusing on the impact of considering the second-order analysis on the evaluation system's stability. By employing the finite difference method (FDM) and the infinitesimal perturbation technique, the bearing coefficients derived from the governing Reynolds equation are obtained. The influence of Sommerfeld and misalignment angle on the first- and second-order dynamic coefficients of journal bearings is investigated. Furthermore, a dynamic analysis of a rotor-bearing system is conducted, considering the coupling between journal and thrust bearings, to investigate the influence of the second-order dynamic coefficients on system dynamics and stability. Additionally, the investigation focuses on the stability of Hopf bifurcation and the continuation of the system's steady state solution. The analysis findings emphasize the significant influence of the interaction between journal and thrust bearings on the stability of the rotor-bearing system. Key results reveal that Sommerfeld number and misalignment angle exhibit nonlinear influences on dynamic coefficients. The study also demonstrates that the second-order dynamic analysis is critical for accurately predicting Hopf bifurcation stability, where high stiffness combined with large Sommerfeld numbers yields supercritical bifurcation, while very low Sommerfeld numbers give rise to subcritical bifurcation. These insights advance the understanding of coupling-induced nonlinear dynamics in rotor-bearing systems.

Abstract Monitoring the changes in mortality patterns and levels requires studying mortality models and measures of lifespan length and variation. Measures of lifespan length and variation can be obtained from life tables or from mortality modes. In this paper, measures of lifespan length and variation are presented for mixture model introduced by Zanotto, L., V. Canudas-Romo, and S. Mazzuco. 2021. “A Mixture-Function Mortality Model: Illustration of the Evolution of Premature Mortality.” European Journal of Population 37 (1): 1–27. This was applied to the case of Egypt for males and females in the period 1950–2021. The mixture model was able to fit Egypt’s mortality pattern well for both males and females. The calculated measures of lifespan length and variation show that there is an increasing longevity and a declining variation of age at death distribution across time for both males and females.

Studying rotor-bearing systems involving fluid film bearings is essential for designing and assessing the dynamic responses and performance of rotating machinery. They are involved in many applications such as pumps, turbines, and engines. Water-lubricated bearings are often used in many applications where the use of oil-based lubricants is not desirable, such as in environmentally sensitive areas such as water desalination. In this study, dynamic analysis is performed to identify the stability regions that prevent the application of water-lubricated journal bearings. This is achieved by solving the system equations of motion and then using an infinitesimal perturbation method to evaluate the second-order bearing coefficients of a journal bearing. In this paper, a steel shaft supported by two symmetrical journal bearings was used to investigate the system stability analysis. A test rig is designed and manufactured to examine the rotor’s dynamic behavior and verify the theoretical outcomes of the FE model, utilizing the bearing coefficients based on second-order analysis. Furthermore, this study compares the two fluids, both theoretically and experimentally, investigating their impact on the rotor-bearing system at different rotational speeds. The theoretical findings indicate that the threshold speed for journal bearings is significantly higher when using water as the lubricant fluid film compared to using oil as the lubricant fluid. Additionally, because of the low viscosity of water, water-lubricated bearings are susceptible to significant wear and noise in operating conditions. Our experiments show that an oil lubricant provides less response than a water lubricant for unbalanced rotors within the tested speed range.

This research introduces a non-linear dynamical model containing Jeffcott rotor with flexible properties and an unbalanced rotating disc. The rotor is supported on two short journal bearings. The disk is surrounded with an elastic snubber ring with significant radial clearance between the disc and the snubber ring. This work considers the dynamic interaction between the rotor disc and the much stiffer snubber ring (stator) using a four degree-of-freedom model to account for inherent nonlinearities. The bearing forces are evaluated by solving Reynolds equation considering short bearing approximation. The system equations of motion are solved numerically by direct integration using the state-space technique. The nonlinear dynamic response results due to the changes in the frequency ratio arising from rotor disk–snubber ring (stator) interactions are evaluated. The present work results show that the system’s response is highly dependent on the frequency ratio, which is defined as η = Ω/ω (forced frequency/natural frequency. Rich dynamics are recorded ranges from periodic, quasi periodic and chaotic.

Abstract BACKGROUND: Analyzing the vibrations for rotor-bearing systems is a critical issue in the field of rotor dynamics. It is crucial to identify rotary machine vibrations, behaviors, and stability conditions. The main causes of vibration in rotating machinery are unbalanced masses, misalignment, mechanical looseness, shaft cracks, and other defects. METHODS: This paper investigates the experimental verification of a theoretical model, using a steel shaft with a disc set at the midpoint, supported by two symmetric fluid film bearings. The study examines the effect of unbalance on the dynamic behavior of the rotor and its vibration characteristics. The experimental investigation involved setting up a test rig, installing the journal bearing and rotor, and measuring relevant parameters. The theoretical analysis employed the solution of the Reynolds equation to determine the bearing coefficients, which were then modeled as a function of the Sommerfeld number using a polynomial fit. A finite element model with a consistent matrix formulation was used to simulate the shaft, including the external load and four degrees of freedom per node.. RESULTS: The theoretical model was validated against experimental results in both the time and frequency domains, considering the effect of unbalanced masses. The results are presented using orbit plots, system responses, and FFT spectra. The vibration analysis results show the whirl phenomena before it becomes uncontrollable and leads to self-excited vibration. CONCLUSIONS: The theoretical model based on nonlinear analysis was in agreement with the experimental analysis for all rotational speed ranges. The resonant speeds of 5107 rpm and 5850 rpm were observed in both the theoretical and experimental studies, respectively. However, a noticeable discrepancy was observed when the speed exceeded the threshold speed in both the first and second-order theoretical analyses.

The rotor-bearing system is a crucial component of rotating machinery, such as turbines, pumps, compressors, and turbogenerators, which are widely used in various advanced engineering fields. This work presented a methodology for studying the behavior of an elastic Jeffcott rotor supported by two similar fluid film journal bearings. A finite element model using consistent matrix formulation was employed to simulate the shaft, including the external load, with four degrees of freedom per node. The small perturbation method was used to evaluate the second-order bearing coefficients of a journal bearing of finite length. These coefficients were further integrated with the finite element model to evaluate the dynamic response of the flexible rotors. Moreover, the system equations of motion were presented in dimensional form. The results of the second-order bearing coefficients analysis agreed with nonlinear analysis when the speed was less than the threshold speed, while there was a pronounced difference in second-order analysis when the speed was above the threshold speed. The behavior of the rotor-bearing system was studied using dynamic response and orbit diagrams, revealing that changes in rotational speed significantly affected the rotor's stability.

The hydraulic turbocharger plays a vital role in harnessing the energy stored in brine within reverse osmosis desalination plants. To optimize the efficiency and durability of this equipment, it is crucial to develop accurate dynamic models of the turbocharger rotor. An improved understanding of rotor dynamics enables the integration of innovative technologies such as Hydraulic Energy Management Integration, effectively enhancing efficiencies in systems characterized by small capacities and high rotational speeds. This study presents a dynamic modeling methodology for the hydraulic turbocharger. The analysis involves approximating the turbocharger rotor with an equivalent finite element shaft line model. Verification of the model’s natural frequency is conducted using three-dimensional finite element analysis, employing the ANSYS modal analysis module. Computational fluid dynamics is employed to evaluate the fluid forces, while the Reynolds equation is utilized to assess the journal bearing forces. The resulting model is employed to investigate the nonlinear dynamics of the rotor, examining the impact of various system parameters, including rotational speed, unbalance forces, and shaft geometrical parameters. The results highlight the significance of balancing the turbine and pump disks for optimal performance. Furthermore, the research demonstrates that increasing the shaft length reduces the rotor’s threshold speed, while increasing the shaft diameter initially raises the threshold speed until it reaches a critical value. Beyond this critical value, further increases in shaft diameter lead to a decrease in the threshold speed.

Rotor bearing systems are a crucial component in many engineering applications, with bearings are typically classified as either rolling contact or sliding contact. In this context, we focus specifically on sliding contact bearings, and more specifically on fluid film journal bearings. The fluid film between the rotor and stator in such bearings can be modeled using Reynolds equation, with the solution of this equation being important for evaluating the bearing’s nonlinear forces. In order to solve the dynamics of a rotor bearing system, Reynolds equation must be solved at each time step. However, this process can be time-consuming, leading many researchers to use approximate formulae to solve the equation more efficiently. In the present study, a continuation analysis was introduced to examine the dynamics of a rigid rotor bearing system without relying on any such approximations. Subsequently, a comparison was made between the findings of this analysis and those of two previously published models, one based on four variable polynomial regression [1] and the other based on short bearing approximation [2]. The results suggest that all three methods are effective in assessing the threshold speed. However, it was observed that the short bearing model exhibits reduced accuracy beyond the threshold speed.

In this study, the linear free vibration of intact and cracked functionally graded material plates is investigated numerically and experimentally. The experimental work is limited to the isotropic materials. The numerical work is based on finite element, where a code is developed to obtain the natural frequencies of intact plates based on the first-order shear deformation theory (FSDT) using MATLAB software. Also, a model of through-cracked FGM plate is developed using ANSYS Workbench with the help of APDL coding. The material properties of the plates under study are graded in one, two, and three directions. The novelty of this study emerges through its examination of the synergistic impacts resulting from variations in FGM material properties, crack length, crack orientation, and crack location. These effects are comprehensively discussed in the results section. The result of the present model shows that the use of three-directional FGM reduces the natural frequency compared with the other cases of two-directional and unidirectional FGM. Also, the results show that the effect of FGM gradient on the frequency of intact and cracked plate is high when the gradient index n < 3 . The present paper results are useful for the design of FGM plates especially when cracks exist.

PurposeThrust bearings play a critical role in high-speed rotating machinery, such as turbines, pumps, compressors, and turbogenerators, by transferring axial loads between the collar and the bearing pads. The lubricating fluid prevents contact, friction, and wear between solid parts and acts as a cooling medium. The purpose of this study is to evaluate the first- and second-order bearing coefficients of an inclined pad hydrodynamic thrust bearing, which have not been previously investigated, to improve the accuracy of modeling thrust bearings.MethodsIn the analysis of thrust bearing systems, the lubricating fluid film is modeled as a massless spring-damper system. The finite perturbation method for the governing Reynolds equation is used to calculate the dynamic coefficients of the thrust bearing. This is done using the finite difference method (FDM) in polar coordinates.ResultsThis study investigates the influence of misalignment, rotating speed, and mesh size on the bearing coefficients of the thrust bearing. The results show that misalignment angle and film thickness have a clear effect on the dynamic coefficients, while changing the rotational speed has no effect on the damping coefficients. The study also investigates the axial force and moments and dynamic coefficients of an inclined pad thrust bearing.ConclusionThis study concludes that evaluating the first- and second-order bearing coefficients of the thrust bearing using the finite perturbation method can improve the accuracy of modeling thrust bearings. The study also highlights the significant influence of misalignment and film thickness on the dynamic coefficients of the thrust bearing. The results presented in this study provide valuable data that can be used as input for rotor dynamics analyses in high-speed rotating machinery.

Modeling the nonlinear dynamics of rotors supported by finite length journal bearings is of great importance in various engineering applications. In this study, four-dimensional polynomial functions are evaluated to represent the nonlinear hydrodynamic force based on a previously evaluated database. These functions are then used to model the dynamics of flexible rotor/bearing systems. The quasi statics and dynamics of rotor-bearing systems are investigated, and the results are compared with the numerical solution obtained by solving the Reynolds equation at each timestep. The findings indicate that the current analysis yields favorable agreement with the direct solution of Reynolds equation in both perturbation analysis from the equilibrium position and dynamic analysis. Moreover, the analysis reveals that the computational time required to solve the dynamics of rotor-bearing systems is significantly lower than that of solving Reynolds equation at each timestep to acquire the bearing forces.

Investigating dynamics and stability of rotors supported on journal bearings is a crucial step in the design of an efficient and reliable rotating machine. In the current work, a model for flexible rotor supported on two symmetric journal bearings is investigated. The nonlinear bearing forces are evaluated by either using direct solution of Reynolds equation or analyzing Reynolds equation to obtain linear and nonlinear bearing stiffness and damping coefficients using time dependent second order perturbation method. These coefficients are obtained for different operating conditions and bearing parameters such as length to diameter ratio, groove angle or applied groove pressure. The present results are validated with the previous literature and a perturbation analysis is used to investigate the validity range of the bearing linear and nonlinear coefficients. A novel technique based on polynomial fitting is used to present the bearing coefficient as a function of the bearing parameters. This enables the investigation of the dynamics of flexible rotor model using numerical continuation technique. Also, the effect of the bearing design parameters such as groove angle, length to diameter ratio and static pressure on the system stability is investigated.

In this work, a novel method to evaluate the nonlinear bearing forces is introduced. The method depends on using polynomial surface fitting to evaluate the bearing forces as a function of the journal center position. Then, these forces are used to investigate the stability and bifurcations of an elastic rotor-bearing model. The Hopf bifurcation analysis and limit cycle continuation are investigated using the proposed analysis for obtaining the bearing forces for un-grooved bearing with L/D = 0.5. The stability results are compared with short bearing approximation at several shaft flexibilities. In addition, the present obtained threshold speed is compared with that computed using bearing coefficients method. The present method time response results are compared with the numerical solution of Reynolds equation and a good agreement is recorded.

Hydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

The successful implementation of the PID+ control persuades the authors to investigate the same controller using the state-dependent parameter transfer function (SDP-TF) model so as to improve the performance of PID+ control when used on nonlinear systems. For this reason, this chapter introduces the SDP-PID+ control, for which the PID+ control is used on the four-degrees-of-freedom manipulator arm when modeled using SDP model structure, where the parameters of the TF change as a function of the state variables. Here, the additional input and proportional compensators that exist in the PID+ approach fight the effect of the discrete-time SDP-TF associated with samples time delay greater than unity and order greater than two. These additional compensators enable the use of the full-state feedback to develop the time-variant state variable feedback (SDP-SVF) control action for the SDP-PID+ controller. In this work, two tuning techniques are introduced for the SDP-PID+ controller; they are the LQ cost function through using the SDP-NMSS of the SDP-TF model and the pole placement. Both approaches provide an appropriate performance in addition to reject output and input disturbance with retrieving the zero steady-state error in a suitable time.

This paper develops a novel approach for discrete time proportional integral derivative (PID) control, namely PID+, with application to manipulator arm with four-degrees of freedom. The approach utilizes extra proportional and input compensators to counteract the influence of the discrete time transfer functions higher than second order and having samples time delay more than unity, respectively. These plus compensators allow the exploiting of the full state feedback signals to construct the state variable feedback control law for the proposed PID+ controller. The work introduces two tuning approaches for the compensators of the novel PID+ controller; they are the Linear Quadratic (LQ) cost function by exploiting the non-minimal state space, namely the novel NMSS-PID+ form, and the pole placement. Simulation results verify the applicability of the proposed PID+ controller. Both tuning approaches, PID+/LQ and PID+/pole placement, show satisfactory steady state response with good control action, along with deadbeat response when applying the pole placement approach, in simulation. Finally, the paper shows successful practical implementation for the PID+ controller, when applied to the four degrees of freedom manipulator arm, for which all the control design criteria (tracking with acceptable response time, and input/output disturbance rejection) are met.