Nonlinear Contact Stiffness Research Articles

An Energy Transfer-Based Bifurcation Detection Method for Nonlinear Rotating Systems: Enables Accurate Capture of Period-Doubling Bifurcation and Instability

Rotor systems are widely used in industrial power generation and propulsion. Once the nonlinear contact stiffness and oil film force are taken into account, the dynamics and stability of rotor systems become quite complex, often accompanied by super-harmonic and chaotic motions. Furthermore, conventional methods face limitations in real-time detection of the bifurcations and complex nonlinear motions. This research investigates the bifurcations and stability induced by nonlinear factors in a rotor-bearing system from an energy perspective. Dynamic equations of a rotor-bearing system considering cubic term stiffness are established, the steady responses are obtained by the fourth-order Runge–Kutta method. The relationship between the bifurcations and energy transfers is analyzed numerically, the proposed stability criterion is validated by comparing the Lyapunov exponents. The bistable phenomenon of period-3[Formula: see text] ([Formula: see text]) motion is discussed in terms of numerical results and experiments which corresponds to the asymmetric jumps of the generalized energy. It is found that bifurcations and unstable motions of the nonlinear system can be captured accurately by detecting the energy transfers, the proposed generalized energy curve exhibits more detailed information than the conventional speed-up curve. These findings provide a new perspective on studying bifurcations and stability of rotating systems, which can be further applied in the condition monitoring, stability prediction as well as the design of nonlinear energy sink, representing significant progress in converting theory into engineering applications for rotor systems.

Investigation on skidding behavior of a lubricated rolling bearing with fluid–solid-heat coupling effect

Skidding phenomenon is a critical cause of abnormal vibration, noise, and temperature rise of lubricated rolling bearings. It is affected by various factors such as the structure size, lubrication medium, thermal properties, and service conditions. Owing to the limited influence factors, the skidding mechanism of the lubricated rolling bearing under the fluid–solid-heat coupling effect is not well understood. Therefore, a fluid–solid-heat coupled dynamics model of the lubricated rolling bearing based on the dynamic and thermodynamic equations is developed in this study using the lumped parameter method and thermal network method. The time-varying geometrical parameter of the components, time-varying kinematic viscosity and density of the lubricant, and temperature raise and distribution of the bearing during the operation process are involved in this model, which can significantly affect the system parameters such as the equivalent nonlinear contact stiffness, friction coefficient, viscous drag, radial and circumferential clearances, and so on. To verify the accuracy of the proposed model, the cage slipping velocity and outer ring temperature rise are compared between simulation results and test data, respectively. The influence of the rotating speed of the inner ring and radial load on the temperature raise and skidding phenomenon is analyzed in detail. The results indicate that skidding behaviors and internal dynamic interactions are intensified under the thermal effect. Therefore, improving the heat dissipation efficiency of the bearing can alleviate skidding, especially under light load and high-speed conditions. It should be noted that the rotating speed of the ring and radial load are the primary factors affecting bearing temperature rise and skidding phenomenon.

Non-linear Dynamic Characteristics of High-Speed Wheel-Rail Interface Considering Surface Topography and Liquid Medium

In this work, the force-deflection characteristics, and the nonlinear vibration of the wheel-rail interaction with surface roughness and liquid medium are studied. The statistical micro-contact Greenwood and Williamson model is used to characterize the rough surface. The load sharing theory is used to determine the analytical elastic contact force-deflection and damping force-deflection relationships, which are further approximated by power law functions. The nonlinear contact stiffness and damping characteristics are illustrated for different rough surface topographies and running speeds. The natural frequency is solved by assuming small covariates and defining time variables of different scales. Effects of surface topography and liquid medium on the natural frequency of the wheel-rail interface system are studied. The first-order harmonic responses under harmonic excitation are further determined as well as the jump-up and jump-down responses for different rough surface topographies, excitation loads, liquid mediums and running speeds. It is shown that the resonance region and the peak value of the vibration amplitude increase as the surface roughness, running speed and excitation load increase; whilst the jump-up and jump-down frequencies decrease. In addition, the system can change from a monostable to a bistable structure when surface roughness and excitation force increase and large limit cycle oscillation can occur.

Analytical expressions of friction-induced self-excited vibration amplitudes of the marine rubber bearing-shaft system

By the analytical method, this paper investigates the mode-coupling friction-induced instability and the resulting stick-slip self-excited vibration of the marine stern water-lubricated rubber bearing-shaft system. The 2-DOF simplified theoretical model is reasonably developed to characterize stern water-lubricated rubber bearing-shaft system. The complex eigenvalue analysis is conducted to analyze the instability behaviors of the model. Results show the normal force and nonlinear stiffness often deteriorate the stability of the model. Subsequently, two types of analytical expressions for stick-slip vibration amplitude of the model are by the Krylov-Bogoliubbov-Mitropolsky method, which match well with the results from numerical integration. Then, parameter discussion about the influences of friction coefficient, damping, normal force and nonlinear contact stiffness on self-excited vibration amplitudes are performed. The results indicate that the bearing-shaft system can keep the steady-state static equilibrium or the oscillations with small amplitude if the proper parameter values of the variables are chosen, which will benefits controlling the friction-induced vibration of the water-lubricated rubber bearing-shaft system. The novelty of this work is to give analytical expressions for stick-slip self-excited vibration amplitude of the simplified model.

Nonlinear excitation and mesh characteristics model for spiral bevel gears

Accurate and rapid calculation of nonlinear excitation (NE) and loaded mesh characteristics (LMC) of gear pairs is a key to achieving rapid iterative optimization of gear system designs. Therefore, an analytical model (AM) was proposed for the accurate and numerically efficient calculation of NE and LMC for application to spiral bevel gears (SBG). First, a tooth contact analysis of an SBG was completed using Coons surface technology and mesh theory to determine the contact path and unloaded transmission error (UTE). Thus, an AM for the principal and relative normal curvatures of the SBG was derived using the Euler's formula. Based on the traditional calculation method for a Hertz contact ellipse, the relation between the relative curvatures was introduced to derive the formulas for the major and minor axes using the Muller's method, which could improve the programmability and avoid numerical instability of the program. Then, analytical models for the contact ellipse and pressure distribution on the contact tooth surfaces were derived using the Hertz model and geometric theory. Subsequently, based on a single tooth LMC, infinitesimal method, elasticity, and series stiffness model, a single mesh stiffness (SMS) model for SBGs was developed, which incorporates transverse tooth stiffness, transverse gear foundation stiffness, axial tooth stiffness, axial gear foundation stiffness, and Hertz nonlinear contact stiffness. Based on the SMS and UTE, compliance and load matrices were established, and the time-varying mesh stiffness, loaded transmission error, mesh force, and multi-tooth LMC models of the SBG were determined. Finally, the proposed method was validated against nonlinear FE simulations using examples. Compared to FE simulations, the proposed method requires significantly less computational effort and can be further extended to optimization or system analysis problems.

INTEGRATED CALCULATION AND EXPERIMENTAL TECHNOLOGY FOR ANALYSIS OF CONTACT INTERACTION WITH ACCOUNT OF ELASTIC INTERMEDIATE LAYER

The work delivers solution to a relevant scientific problem of solid mechanics. It consists in developing theoretical basis for computational and experimental research of contact interaction of complex-shaped bodies with account for micromechanical models of roughness and other intermediate or surface layers. New methods and models have been developed for the analysis of contact interaction of complex-shaped bodies with account for the micromechanical properties of surface and intermediate layers characterized by nonlinear local contact stiffness. The developed methods and models as well as the numerical analysis tools have been applied to a series of model and applied problems. The deformation behaviour of novel materials with network microstructures of one-dimensional elements has been determined. Macroscopical properties of these materials have been evaluated based on the special microscopic models and the homogenization methods. Regularities in contact distribution and its dependence on geometrical and physical factors have been determined for various complexshaped bodies. New design solutions for machine elements that improve their strength and durability have been justified. The developed methods and analysis tools have been introduced into the design of new engineering products. Their implementation resulted in improved technical characteristics of protection, structural and functional elements of transport vehicles of special purpose, gear transmissions, technical equipment, hydrovolumetric drives and so on. The conducted computational and experimental studies showed good agreement with the real response of network materials and the observed behavior of interacting elastic bodies with intermediate contact layer. Keywords: solid mechanics, complex-shaped body, contact interaction, contact pressure, intermediate layer

Nonlinear Contact Stiffness and Damping Identification of a Tenon Connection Structure Based on Its Dynamic Characteristics

A tenon connection structure is widely used for the blade-disk connection in turbomachinery, and its contact stiffness and contact damping have caused nonlinear dynamic characteristics. The nonlinear dynamic characteristics of a blade-disk structure are mainly affected by its contact stiffness and damping characteristics, which are difficult to accurately predict and affected by many factors such as the operating speed and the centrifugal force caused by the rotation speed. Based on a single blade and tenon groove structure and the use of the dynamic characteristics experiments and simulations, the influence law of the contact stiffness and damping of the tenon connection structure with the change of the blade root compression force on the dynamic characteristics of the structure is obtained. The average relative error of identifying the contact stiffness is 0.48%; thus, identification of the nonlinear contact stiffness and damping of multiple pairs of contact curved surfaces is realized.

Dynamic characteristic of thrust-vectoring nozzle adjusting mechanism considering clearance of motion joint

An essential device for managing the vectoring nozzle’s thrust direction is the thrust-vectoring nozzle adjusting mechanism. The attitude of an aircraft’s flight may be changed by adjusting the vectoring nozzle’s airflow, which affects the direction of thrust. The adjusting mechanism’s motion precision and operational stability are crucial in this procedure. In this paper, the influence of the clearance of the motion joint on the dynamic characteristics of the thrust-vectoring nozzle adjusting mechanism under different flight conditions are studied. First, a dynamic model of the thrust-vectoring nozzle adjusting mechanism that takes into account the clearance and friction properties of the motion joint is constructed by combining the Lagrange equation of the first sort with the mixed contact force model and the LuGre friction model. Second, the variation law of nonlinear contact stiffness is explored, and the precision of the dynamic model is confirmed by contrasting the results of the numerical simulation with those of the Simulink simulation. Finally, the simulation results reveal that while the motion joint’s clearance causes the adjusting blade’s velocity and acceleration to vibrate violently and momentarily during early operation, this clearance has minimal effect on the precision of blade motion throughout the stable phase.

Modeling of normal contact stiffness for surface with machining textures and analysis of its influencing factors

A new contact stiffness model for the rough surface with different machining textures is established. The studied surface consists of regular surface machining texture in the global scale and stochastic micro-asperities in the local scale. The proposed model improves upon previous work that considers only the effect of stochastic asperities and completely ignores that of the machining textures generated by different machining methods. The analytical expression to calculate the contact stiffness of the studied textured surface is derived and numerical results are obtained via the finite element method. A high-precision test bench to measure the interface contact stiffness of the workpiece with different surface textures is designed. The measured results and simulated results are directly compared to demonstrate the validity of the proposed method. The effects of different texture types and surface roughness values on the contact stiffness of the proposed textured surface are analyzed. The proposed method provides a research basis for the accurate calculation of nonlinear contact stiffness and dynamic characteristics of the surface with complex texture topography.

Influence of Tooth Surface Wear and Nonlinear Contact Stiffness on Dynamic Responses of Helical Gears

Background: Tooth surface wear is inevitable in helical geared transmission. Consequently, the worn profile deviates from the ideal involute one. As a result, the structural stiffness of worn tooth and contact stiffness of tooth-pair are both changed. Methods: This work presents an improved calculation method for structural stiffness of worn teeth by combining slicing and potential energy method, considering non-uniform distribution of wear amount along the tooth surface. Then, a nonlinear contact stiffness model is employed to investigate the influence of wear on contact stiffness. Meanwhile, taking wear as one kind of profile deviation, the analytical model of time-varying mesh stiffness (TVMS) of helical gear pair is derived. Furthermore, governing equations with 6 degree-of-freedom are established and influences of wear on dynamic responses are revealed. Results: Results indicate that structural stiffness of worn teeth decreases but contact stiffness does not always keep increasing or decreasing. The fluctuation of dynamic transmission error with the nonlinear contact model is not as significant as that from the constant contact stiffness model. Conclusion: The approach presented in this work is suitable for condition monitoring of helical gears in view of long-term service

THEORETICAL STUDY OF NONLINEAR CHATTER STABILITY ANALYSIS BASED ON SYNTHESIS OF LINEARIZATIONS IN OPERATING POINTS FOR DIFFERENT TURNING CONDITIONS

This paper deals with an approach to study nonlinear chatter analysis, which is based on the synthesis of linear theory for different turning conditions. Mainly, contact nonlinearities play a crucial role in the behavior of the machine tool and the stability of turning operation. However, the analysis of nonlinear systems is challenging because nonlinear system analyzes are often limited to a linearized solution, which may be insufficient to reveal the stability lobe diagram. The presented study proposes a theoretical approach which is based on a synthesis of linearization in the operating point. This approach provides several solutions of stability lobe diagrams which are strongly depending on the loading conditions of the nonlinear system and all solutions are integrated to the final stability lobe diagram. The presented method is applied to a simplified 2D structure between two supports with nonlinear contact stiffness. Resulted analysis of the lobe diagram are compared with results of the time-domine simulations and it shows a good match. This paper shows that the presented approach offers an effective solution for refining the stability estimate for nonlinear mechanical systems.

Design and Load Distribution Analysis of the Mismatched Cycloid-Pin Gear Pair in RV Speed Reducers

The current loaded tooth contact analysis of cycloid drives based on the assumption of theoretical positions of ring pins ignores the deviations caused by manufacturing errors and elastic deformations, which are not in agreement with reality. To fill this gap, an improved load distribution model of the mismatched cycloid-pin gear pair with ring pin position deviations is presented for a component-level analysis. Firstly, with the cycloid gear tooth profile geometry defined, the unloaded tooth contact analysis is applied as a pre-processor to determine the potential contact points, the gear backlash, and the rotation angle of the cycloid gear. Secondly, due to the statically indeterminate structure of the multi-tooth contact, a varying nonlinear contact stiffness is introduced to establish the relation between force and deformation. Then, the force and moment equilibrium equations with compatibility conditions are solved by using an iterative approach. With this, detailed parametric case studies are presented to verify the correctness of the proposed model by comparing it with those predicted by the current model and to demonstrate the influences of ring pin position deviations on the distributed load, contact stress, loaded transmission error, and instantaneous gear ratio of the mismatched cycloid-pin gear pair. This study provides a deeper investigation into the load distribution characteristics of the cycloid drive and therefore can be employed to assist in gear design.

An improved model for dynamic characteristics analysis of high-contact-ratio spur gears considering localised tooth spall defect

Tooth spall defects are one of the most common types of gear failures. The impact of the bottom features of the tooth spall on the dynamic properties of high-contact-ratio spur (HCRS) gears is investigated in this work. Considering the load sharing ratio, coupling structure effects, and nonlinear contact stiffness, a hybrid model was developed for evaluating the mesh stiffness of the HCRS gear, and a torsional dynamic model of the HCRS gear pair was built. The dynamic transmission errors and dynamic meshing forces are acquired by solving the above model; based on the dynamic responses of the healthy HCRS gear, the dynamic characteristics of two different spalled gear systems were compared. The results revealed that the proposed mesh stiffness model is effective within 5% calculation errors. The bottom shape of the tooth spall has important effects on the mesh stiffness but limited influence on the gear dynamic response. This study is expected to provide a theoretical foundation for future research on spall defect diagnosis in gear transmission devices.

A general contact stiffness model for elastic bodies and its application in time-varying mesh stiffness of gear drive

As a critical element of time-varying mesh stiffness (TVMS), contact stiffness of a gear drive has been defined based on simplified Hertzian contact stiffness or semi-empirical nonlinear Hertzian contact stiffness in previous works. This study proposes a general contact stiffness model for elastic bodies through piecewise linear interpolation of contact pressure. The TVMS of a spur gear drive is determined through potential energy method and proposed contact stiffness model verified by Hertzian contact theory and finite-element method. Then, the influence of applied load on contact stiffness is studied, and the differences among proposed contact stiffness, simplified Hertzian contact stiffness, and nonlinear Hertzian contact stiffness are analyzed. Results show that contact stiffness increases with the applied load, and the TVMS based on the proposed contact stiffness model is the smallest among the three contact stiffness models. Effects of tooth width and input torque on the TVMS are further discussed. The TVMS becomes bigger with increased tooth width and input torque, but the increase rate decreases as tooth width or input torque increases. These findings indicate that reasonable matching of design parameters is beneficial for increasing load capacity and optimizing the dynamic performance of gear systems.

Experimental study of nonlinear waves in phononic materials with rough contacts

Incorporating contacting interfaces in periodic media enables enriched wave dynamics from the combined effects of periodicity and nonlinearity. While Hertzian contacts have been extensively studied in the framework of engineered media, rough contacts have received relatively less attention. Recently, our numerical work showed that continuum phononic material with periodic rough contacts supports strongly nonlinear waves belonging to the family of solitary waves. In this talk, we experimentally study nonlinear waves in phononic material based on rough contacts, realized through an array of aluminum disks. The disks act as elastic layers and have roughness on either side, which is generated through surface treatment. First, we characterize a rough contact through optical imaging and ultrasonic measurements. We obtain the nonlinear contact stiffness–precompression relationship that informs the power exponent of the contact law. Then, we study nonlinear wave propagation through periodic rough contacts of the same roughness topography. We measure propagating waves through a laser vibrometer and investigate their frequency content, speed, and amplitude. This study develops a fundamental understanding of the role of rough contacts in the form of local nonlinearity on wave responses. Such understanding is useful in designing these phononic materials for wave propagation control.

Dynamic NVH Numerical Analysis of Power Steering in the Presence of Lubricant in the System.

The ongoing shift towards hybrid and electric vehicles has a strong impact on noise and vibration engineering. New, complex dynamic phenomena are brought to vehicle user attention due to the absence of internal combustion engines and the significant role in vehicle and drive feel perception. This paper presents an FEM (Finite Element Method) dynamic simulation model of an automotive Electric Power Steering assembly. Preliminary modal simulations and experiments as well as field data replication techniques were implemented to identify the phenomena and prepare and validate model components. A full dynamic model of an Electric Power Steering was presented, and fine-tuned including the presence of lubrication at the gear mesh interface. Experimental investigations were conducted alongside FEM simulations for various model setups. Linear and nonlinear contact stiffness models were implemented, as well as contact damping, and simulated at chosen assembly interfaces. The results indicated that in the case of NVH (Noise Vibration and Harshness) analysis of shock/impact originating problems, contact parameters used for static, quasi-static, and low velocity analyses were not applicable. Nonlinear and damped contact stiffness provided better results in such a case.

A Lagrangian approach for the electromechanical model of single-stage spur gear with tooth root cracks

This paper presents a unified electromechanical modeling framework of a single-stage spur gear with cracks at tooth root. The model has been developed following modified Lagrangian approach with the consideration of Rayleigh dissipative potential. The coupled electromechanical equations of motion represent a non-autonomous system. Improved analytical model based time-varying mesh stiffness (IAM-TVMS) at different crack depths is incorporated in the developed model to examine the impact of cracks on the dynamic behavior of the system. The IAM-TVMS model considers the effect of misalignment between base and root circle, accurate transition curve, nonlinear Hertzian contact stiffness, revised fillet foundation stiffness with the structural coupling effect when nearby tooth is loaded and improved crack model which considers the straight line for crack path and parabolic curve for limiting line. The current and vibration responses are taken after solving the coupled equations of motion through fourth order Runge–Kutta method. In this study, the impact of crack depths are effectively visualized using stator current as a primary source of signal. Even when the crack depth is significantly small, the residual signal clearly depicts the presence of crack in the cepstrum analysis. The cepstrum analysis of residual stator current signal is also compared and verified with the residual vibration signal for different crack depths. The developed analytical model is one of the first attempt of its kind in modeling approach; the model successfully shows the impacts of crack on the system with motor current signal.

Critical thresholds for mode-coupling instability in viscoelastic sliding contacts

Mode-coupling instabilities are known to trigger self-excited vibrations in sliding contacts. Here, the conditions for mode-coupling (or “flutter”) instability in the contact between a spherical oscillator and a moving viscoelastic substrate are studied. The work extends the classical 2-Degrees-Of-Freedom conveyor belt model and accounts for viscoelastic dissipation in the substrate, adhesive friction at the interface and nonlinear normal contact stiffness as derived from numerical simulations based on a boundary element method capable of accounting for linear viscoelastic effects. The linear stability boundaries are analytically estimated in the limits of very low and very high substrate velocity, while in the intermediate range of velocity the eigenvalue problem is solved numerically. It is shown how the system stability depends on externally imposed parameters, such as the substrate velocity and the normal load applied, and on contact parameters such as the interfacial shear strength tau _{0} and the viscoelastic friction coefficient. In particular, for a given substrate velocity, there exist a critical shear strength tau _{0,crit} and normal load F_{n,crit}, which trigger mode-coupling instability: for shear stresses larger than tau _{0,crit} or normal load smaller than F_{n,crit}, self-excited vibrations have to be expected.

Estimation of a Planetary Gear Mesh Stiffness: An Approach Based on Minimising Error Function

The mesh stiffness of gear teeth is one of the major sources of excitation in gear systems. Many analytical and finite element methods have been proposed in order to determine the mesh stiffness of gears especially parallel axis spur gears. Most of these methods are not trivial because they involve complicated analyses which incorporate parameters like gear tooth error, gear spalling sizes and shapes, nonlinear contact stiffness and sliding friction before mesh stiffness can be determined. In this work, a method is proposed to determine the sun-planet and ring-planet mesh stiffnesses of a planetary gear system. This approach involves fitting a relationship between the measured natural frequencies from an experimental modal test and natural frequencies predicted using an analytical model of a planetary gear. This method is relatively easier compared to the existing methods which involve complicated analyses. For this study, the average mesh stiffness estimated is 12.5 MN/m.

Estimation of a Planetary Gear Mesh Stiffness: An Approach Based on Minimising Error Function

The mesh stiffness of gear teeth is one of the major sources of excitation in gear systems. Many analytical and finite element methods have been proposed in order to determine the mesh stiffness of gears especially parallel axis spur gears. Most of these methods are not trivial because they involve complicated analyses which incorporate parameters like gear tooth error, gear spalling sizes and shapes, nonlinear contact stiffness and sliding friction before mesh stiffness can be determined. In this work, a method is proposed to determine the sun-planet and ring-planet mesh stiffnesses of a planetary gear system. This approach involves fitting a relationship between the measured natural frequencies from an experimental modal test and natural frequencies predicted using an analytical model of a planetary gear. This method is relatively easier compared to the existing methods which involve complicated analyses. For this study, the average mesh stiffness estimated is 12.5 MN/m.