Hertzian Contact Stiffness Research Articles

Evaluation of the time-varying mesh stiffness for gears with tooth spalls with curved-bottom features

Abstract Gear tooth spalling is one of the most common defects in gear transmission. The loss of surface materials due to tooth spall reduces the Time-Varying Mesh Stiffness (TVMS) of the gear pair, and thus modifies the vibration response of the gear transmission. The evaluation of the TVMS of the gear tooth pair under gear tooth spalling conditions plays an important role in gear dynamic simulation and the corresponding fault feature analysis. Common approaches assume that the tooth spall has a flat bottom with a constant tooth spall depth. This assumption implies an abrupt change in dent depth and cliff-like material loss. However, in practice, the flake of the surface material usually results in a gradually changing dent depth with a curve-shaped bottom. Thus, the modelling of tooth spalls with a flat bottom may result in mesh stiffness differences. To address this shortcoming, this paper proposes a curved-bottom shaped tooth spall to model the tooth spalling geometric features as observed in practice. The proposed method is constructed based on an ellipsoid geometry which is capable of varying radii in three dimensions to best fit the shape of the tooth spall. The foundation stiffness within the double tooth contact area in the proposed method is corrected and the non-linearity of Hertzian contact stiffness is considered. The effectiveness of the proposed method on modelling single and multiple tooth spalls with different shapes and severity conditions is then validated.

Implementation of Hertz Contact Theory and Validation of Applicability of Hertz Contact Stiffness Model for Helical Gear using Multi Body Dynamics

Implementation of Hertz Contact Theory and Validation of Applicability of Hertz Contact Stiffness Model for Helical Gear using Multi Body Dynamics

An improved time-varying mesh stiffness model for helical gear pairs considering axial mesh force component

Abstract An improved time-varying mesh stiffness (TVMS) model of a helical gear pair is proposed, in which the total mesh stiffness contains not only the common transverse tooth bending stiffness, transverse tooth shear stiffness, transverse tooth radial compressive stiffness, transverse gear foundation stiffness and Hertzian contact stiffness, but also the axial tooth bending stiffness, axial tooth torsional stiffness and axial gear foundation stiffness proposed in this paper. In addition, a rapid TVMS calculation method is proposed. Considering each stiffness component, the TVMS can be calculated by the integration along the tooth width direction. Then, three cases are applied to validate the developed model. The results demonstrate that the proposed analytical method is accurate, effective and efficient for helical gear pairs and the axial mesh stiffness should be taken into consideration in the TVMS of a helical gear pair. Finally, influences of the helix angle on TVMS are studied. The results show that the improved TVMS model is effective for any helix angle and the traditional TVMS model is only effective under a small helix angle.

Fault mode analysis and detection for gear tooth crack during its propagating process based on dynamic simulation method

Gearbox is one of the most important parts of rotating machinery, therefore, it is vital to carry out health monitoring for gearboxes. However, it is still an unsolved problem to disclose the impact of gear tooth crack fault on gear system vibration features during the crack propagating process, besides effective crack fault mode detection methods are lacked. In this study, an analytical model is proposed to calculate the time varying mesh stiffness of the meshing gear pair, and in this model the tooth bending stiffness, shear stiffness, axial compressive stiffness, Hertzian contact stiffness and fillet-foundation stiffness are taken into consideration. Afterwards, the vibration mechanism and effects of different levels of gear tooth crack on the gear system dynamics are investigated based on a 6 DOF dynamic model. Then, the crack fault vibration mode is studied, and a parametrical-Laplace wavelet method is presented to describe the crack fault mode. Furthermore, based on the maximum correlation coefficient (MCC) criterion, the optimized Laplace wavelet base is determined, which is then designed as a health indicator to detect the crack fault. The results show that the proposed method is effective in fault diagnosis of severe tooth crack as well as the early stage tooth crack.

Thermo-mechanical analysis of angular contact ball bearing

The thermal-mechanical character, which is difficult to ensure because of the lack of a corresponding theory and tool, has a significant effect on the dynamics of bearings. It even leads to a sudden failure of bearings in a working situation. In this research, a thermal deformation model was established, based on the analysis of temperature effect on the basic size of angular contact ball bearing. And the transmission from rolling size to bearing axial stiffness was explicit. On the basis of the variation of Hertz contact stiffness and the change of initial contact angle of angular contact ball bearing caused by temperature rise, a “Thermo- mechanical” model of bearing was proposed. According to this model, using the corresponding calculation procedure programmed by MATLAB, the effect of bearing temperature on the axial stiffness has been studied. And the correctness of this model was verified with experiments. Some design suggestions have been made for the decision of bearing preload: to prevent the bearing failure caused by overheating.

Dynamic Modeling of Rolling Ball Bearing with Localized Surface Defects Considering Three Dimensional Motions and Relative Slippage

Vibration responses and kinematic parameters when a ball rolls over and impacts with a localized surface defect can provide efficient inputs and theoretical foundations for bearing fatigue analysis and fault diagnosis. A dynamic model for rolling ball bearings with localized defects is proposed based on GUPTA model. In this model,every bearing component(i.e.,rolling ball,inner raceway and outer raceway) has 6 degrees of freedom. Relative slippage and lubricant traction are also considered. Moreover,defects are modeled completely with consideration of additional clearance due to material absence,changes of Hertzian contact stiffness and changes of contact force directions. The relationships between bearing accelerations and impact forces at balls and defects are analyzed. Furthermore,the effects of the shaft speed and the defect width on bearing vibrations are investigated. The simulation results show that,because the relative slippage and the impact force effect are all considered,the proposed model is capable of simulating the dynamic characteristics of rolling ball bearings with surface defects more reasonable and providing valuable conclusions for fatigue analysis and quantitive diagnosis of rolling ball bearings.

Study on stiffness of elastohydrodynamic line contact

The stiffness of oil film, contact body and the resultant stiffness of elastohydrodynamic line contacts with various radii of curvature, loads and entrainment velocities are studied based on the numerical analysis of oil film thickness and elastic deformation. It is found that the elastohydrodynamic contact stiffness is close to the Hertzian contact stiffness in the cases of high loads, low speeds and small radii of curvature. The critical conditions in which the elastohydrodynamic stiffness can be approximated by the Hertzian stiffness are investigated and verified by the dynamic simulation of the valve train in a diesel engine.

Design and control methodology of a 3-DOF flexure-based mechanism for micro/nano-positioning

A 3-DOF ( X – Y – θ Z ) planar flexure-based mechanism is designed and monolithically manufactured using Wire Electro-Discharge Machining (WEDM) technology. The compact flexure-based mechanism is directly driven by three piezoelectric actuators (PZTs) through decoupling mechanisms. The orthogonal configuration in the x and y directions can guarantee the decoupling translational motion in these axes. The rotational motion and translational displacement in the x direction can be decoupled by controlling the piezoelectric actuators in the x axis with the same displacement values in same and opposite motion directions, respectively. The static and dynamic models of the developed flexure-based mechanism have been developed based on the pseudo-rigid-body model methodology. The mechanical design optimization is conducted to improve the static and dynamic characteristics of the flexure-based mechanism. Finite Element Analyses (FEA) are also carried out to verify the established models and optimization results. A novel hybrid feedforward/feedback controller has been provided to eliminate/reduce the nonlinear hysteresis and external disturbance of the flexure-based mechanism. Experimental testing has been performed to examine the dynamic performance of the developed flexure-based mechanism. • The mechanical design methodology for a compact 3-DOF planar flexure-based mechanism has been developed. • Influences of the Hertzian contact stiffness on the dynamic performance of the developed mechanism have been explored. • A novel hybrid feedforward/feedback controller has been proposed to reduce/eliminate the hysteresis.

Nuclear magnetic resonance measurements of velocity distributions in an ultrasonically vibrated granular bed

We report the results of nuclear magnetic resonance imaging experiments on granular beds of mustard grains fluidized by vertical vibration at ultrasonic frequencies. The variation of both granular temperature and packing fraction with height was measured within the three-dimensional cell for a range of vibration frequencies, amplitudes and numbers of grains. Small increases in vibration frequency were found--contrary to the predictions of classical 'hard-sphere' expressions for the energy flux through a vibrating boundary--to result in dramatic reductions in granular temperature. Numerical simulations of the grain-wall interactions, using experimentally determined Hertzian contact stiffness coefficients, showed that energy flux drops significantly as the vibration period approaches the grain-wall contact time. The experiments thus demonstrate the need for new models for 'soft-sphere' boundary conditions at ultrasonic frequencies.

Vertical vibrations of composite bridge/track structure/high-speed train systems. Part 2: Physical and mathematical modelling

A theory of one-dimensional physical and mathematical modelling of the composite (steel-concrete) bridge/track structure/highspeed train system is developed including viscoelastic suspensions of rail-vehicles with two two-axle bogies each, non-linear Hertz contact stiffness and one-sided contact between wheel sets and rails, the viscoelastic and inertia features of the bridge, the viscoelastic track structure on and beyond the bridge, approach slabs, and random vertical track irregularities. Compared to the state-of-the-art, the physical model developed in the study accurately reproduces dynamic processes in the considered system. Division of the system into the natural subsystems, a method of formulation of the equations of motion partly in implicit form and the finite element method are applied. Vibrations in the vertical plane of symmetry are described by more than nine matrix equations of motion with constant coefficients. Couplings and non-linearity are hidden in the generalized load vectors. The equations of motion are integrated using the implicit Newmark average acceleration method with linear extrapolation of the interactions between the subsystems.

Vibrations of Bridge / Track Structure / High-speed Train System with Vertical Irregularities of the Railway Track

The study presents the vibration theory and random dynamic analysis of the series-of-types (designed by the author) of composite (steel-concrete) bridges loaded by a German ICE-3 high-speed train. The vibration theory includes: a continuously welded ballasted track structure, viscoelastic suspensions of rail-vehicles with two two-axle bogies each, non-linear Hertz contact stiffness and one-sided contact between the wheel sets and the rails, the viscoelastic and inertia features of the bridge, the viscoelastic track structure on and beyond the bridge, approach slabs. The basic random factor, i.e. vertical track irregularities of the track, is taken into consideration.

Transient response of structures with uncertain properties to nonlinear shock loading

A method is presented to predict the transient response of a structure at the driving point following an impact or a shock loading. The displacement and the contact force are calculated solving the discrete convolution between the impulse response and the contact force itself, expressed in terms of a nonlinear Hertzian contact stiffness. Application of random point process theory allows the calculation of the impulse response function from knowledge of the modal density and the geometric characteristics of the structure only. The theory is applied to a wide range of structures and results are experimentally verified for the case of a rigid object hitting a beam, a plate, a thin and a thick cylinder and for the impact between two cylinders. The modal density of the flexural modes for a thick slender cylinder is derived analytically. Good agreement is found between experimental, simulated and published results, showing the reliability of the method for a wide range of situations including impacts and pyroshock applications.

Effect of random track irregularities on selected composite bridge / track structure / high-speed train system vibrations

A theory of the quasi-exact physical and mathematical modelling of the composite (steel–concrete) bridge / ballasted track structure / high-speed train system (BTT) was developed, including viscoelastic suspensions of rail-vehicles on two-axle bogies, the non-linear Hertz contact stiffness and one-sided contact between the wheel sets and the rails, the viscoelastic and inertia features of the bridge, the viscoelastic track structure on and beyond the bridge, the approach slabs, and random track irregularities. Based on this theory, advanced computer algorithms for the BTT numerical modelling were written and a computer program to simulate the vertical vibrations of the BTT systems was developed. The bridge subject to the preliminary dynamic analysis and designed according to Polish standards has a 15.00 m span length. The bridge was loaded by the German ICE-3 high-speed train moving at the critical (180 and 270 km/h) and the maximum (300 km/h) operating speeds.

On the undamped free vibration of a mass interacting with a Hertzian contact stiffness

Three methods are used to determine the natural frequency of undamped free vibration of a mass interacting with a Hertzian contact stiffness. The exact value is determined using the first integral of motion. The harmonic balance method is used on a transformed equation for an approximate solution, and the multiple scales method is used on an approximate equation. The maximum initial displacement avoiding contact loss is also determined, and the corresponding exact natural frequency is also obtained analytically. The methods are evaluated by studying the free vibration of an elastic sphere on a flat rigid surface.

Study of milling stability with Hertz contact stiffness of ball bearings

This present work examines the stability and nonlinear responses of a spindle milling system supported by ball bearings. A shaft finite element based on Timoshenko beam theory is employed to model the spindle, and modal reduction method is therefore adopted for saving the numerical calculating time. The issues of evaluating the effects of the ball bearing Hertz contact stiffness are consequently addressed. It is found that suitable constant bearing stiffness can be adopted to replace the nonlinear nonsmooth Hertz stiffness in prediction of the critical cutting depth of the milling system in certain bearing configuration conditions. For the constant bearing stiffness can be obtained by experiment, this replacement will undoubtedly simplify the spindle-bearing milling system. But with the increase in the bearing clearance, the spindle milling system will present obvious nonlinear behaviors, and the nonlinear Hertz contact bearing stiffness will take over. Isolated islands of chatter vibration, which are induced by the nonlinear nonsmooth bearing Hertz stiffness, can be found exist in milling processes in large bearing clearance conditions.

Vibration analysis of stylus instrument for random surface measurement

This paper presents dynamic modeling and vibration analysis methodologies for investigating surface profile fidelity in stylus instruments, using a novel dynamic model that takes account of Hertzian contact stiffness between the stylus tip and surface. It briefly describes generation methods for modelling random surfaces and finite tips, which provide input for this nonlinear dynamic system. The influence of the damping ratio on the dynamic performance of the stylus instrument is examined and tends to confirm other recent work. The best combinations of signal fidelity and measurement speed occur with damping ratios considerably higher than conventionally used, mainly because the damping can suppress or delay tip flight. Residual vibrations associated with start-up transients are shown to have potentially serious effects on the signal fidelity. Hence, the responses of the stylus system to three typical velocity control signals are investigated. The normally used step control signal can generate relatively large residual vibration, while slope control and, even better, cycloidal control signal can suppress the residual vibration and guarantee the signal fidelity.

Dynamic analysis of gear pairs with a gross motion effect using the dynamic stiffness matrix method

A continuum approach using the dynamic stiffness matrix (DSM) method (DSMM) is presented for the dynamic analyses of spur gear pairs incorporating the gross motion effect. A continuum dynamic model of spur gears was formulated based on the dynamic stiffness using the non-uniform Timoshenko beam theory. A systematic DSM for a gear pair was assembled including the elemental DSMs of two mating gears and non-linear Hertzian contact stiffness. The DSMs were updated at each computing step to reflect the time-varying property due to the moving mesh points. A bisection procedure was used to calculate the natural frequencies as the systematic DSM of the gear pair became singular. The mass and stiffness matrices were calculated by taking a numerical derivative of the systematic DSM. Finally, the gear dynamic responses caused by the external torque and gross motion excitation resulted. The effect of gross motion is also discussed.

Braking Impact of Normal Dither Signals

Dither control is a method of introducing high-frequency control efforts into a system to suppress a lower-frequency disturbance. One application of dither control is the suppression of automotive brake squeal. Brake squeal is a problem that has plagued the automotive industry for years. Placing a piezoceramic stack actuator in the piston of a floating caliper brake creates an experimental normal dither system. Many theoretical models indicate a reduction in the braking torque due to the normal dither signal. Using a Hertzian contact stiffness model, the loss in friction is due to lowering the average normal force. There are also theories that the dither signal eliminates the “stick-slip” oscillation causing an effective decrease in the friction force. Yet another theory indicates that the effective contact area is reduced, lowering the mean coefficient of friction. A particular approach considering a single-degree-of-freedom friction oscillator predicts a maximum friction reduction of 10%, occurring at the primary resonance of the system. This paper will concentrate on validating this claim by experimentally determining braking torque reduction for a variety of dither control signals. Several dither control frequencies were chosen at system resonances, while others were chosen at frequencies most likely to provide control of the system. These frequencies were chosen based on previous squeal suppression research. The results indicate that dither control frequencies at system resonances have a greater impact on the braking system’s performance. In general, dither control reduces braking torque by no more than 2%.

Effect of elastic anisotropy on contact stiffness in resonance ultrasound microscopy

The classical Hertzian-contact theory for an isotropic material has been adopted to simplify quantitative evaluation of local elastic modulus by resonance-ultrasound microscopy (RUM). However, the validity of this simplified model must be confirmed because most materials show elastic anisotropy in small regions. This study investigates the influence of the elastic anisotropy of the tip and the specimen on the determination of the local elastic modulus in RUM by introducing the Hertzian-contact stiffness for orthorhombic materials. Numerical results reveal that specimen anisotropy significantly affects the contact stiffness and the quantitative evaluation of local elastic modulus even for specimens with weak anisotropy when we consider the anisotropy of the oscillator tip in RUM.

Wheel/Rail Non-linear Interactions With Coupling Between Vertical and Lateral Directions

Summary A theoretical model is developed to explore the high frequency wheel/rail interaction with coupling between the vertical and lateral directions. This coupling is introduced through the track dynamics due to the offset of the wheel/rail contact point from the rail centre line. Equivalent models of the railway track in the time domain are developed according to the rail vibration receptances in the frequency domain. The wheel is represented by a mass in each direction with no vertical-lateral coupling. The vertical wheel/rail interaction is generated through a non-linear Hertzian contact stiffness, allowing for the possibility of loss of contact between the wheel and rail. The lateral interaction is represented by a contact spring and a creep force damper in series and their values depend on the vertical contact force. The vibration source is the roughness on the wheel and rail contact surfaces which forms a relative displacement excitation in the vertical direction. Using the combined interaction model with this relative displacement excitation, the wheel/rail interactions with coupling between the vertical and lateral vibrations are simulated. It is found that the lateral interaction force caused by the offset is usually less than thirty percent of the vertical dynamic force. The lateral vibration of the rail is significantly reduced due to the presence of the lateral coupling, whereas the vertical interaction is almost unaffected by the lateral force.