Hertzian Contact Force Research Articles

Calibration of parameters for DEM simulations of solar particle receivers by bulk experiments and surrogate functions

Bauxite particles known as proppants from the fracking industry are an emerging heat transfer medium for solar thermal power plants. The discrete element method (DEM) has just recently been applied to these systems and the information on parameters for contact force models is scarce; calibrated parameters cannot be found. In this work a novel, three-stage calibration approach based on multiple bulk experiments and surrogate functions is presented. The experiments comprise the angle of repose, the residence time on a horizontal conveyor and the impact on an inclined plate. Particle-particle and particle-wall parameters for the Hertz contact force model and the modified linear spring dash-pot rolling friction model are determined for five different proppants and two wall materials. The assumptions of the calibration procedure and the calibration results themselves are checked and discussed on the basis of a sensitivity analysis. It is shown that almost all parameters can be constricted to unique values by the approach and that most parameters only gradually differ between particle types. Additionally, the influence of coarse graining on the parameters is examined. The behavior of the restitution coefficients is explained in detail by kinetic gas theory. It is found that only the particle-wall restitution coefficient is invariant to coarse graining, but not the particle-particle restitution coefficient, which must decrease with coarse-graining factor.

Research on vibration performance of the nonlinear combined support-flexible rotor system

A finite element model of the cantilever combined support rotor included two discs established in this paper to predict and explain the nonlinear vibration phenomenon. The model considered local nonlinear support force generated by the combined support and the coupling effects of the rotor with the combined support. The nonlinear factors contained the clearance, the Hertz contact force and the vibration of the rolling bearing and the nonlinear force of the squeeze film damper. The combined support included bearing, squeeze film damper and squirrel cage elastic support. Finally, the high-dimensional partial differential equations with local nonlinearity were derived and solved by Newmark- $$\upbeta $$ combined with Runge–Kutta method. Based on the above theoretical results, the bifurcation phenomenon of vibration response at different positions was analysed and the dynamic response of the rotor system at different positions due to the rotational speed was compared.

Vibration Modelling and Verification for Rotor-Support-Casing Coupling System with Misalignment Fault

Misalignment is one of the common malfunctions that occur in rotating machines. Effects of misalignment on the casing vibration response of a rotor-support-casing (RSC) coupling system is investigated in detail. The model of an RSC coupling system is established using the lumped mass method. The coupling effects between the rotor, support and casing are fully considered. A misalignment model is proposed, and equivalent misalignment force is applied on corresponding lumped mass points. Nonlinear factors of bearing, such as the clearance of bearing, oil film force, nonlinear Hertzian contact force, and the varying compliance vibration are developed. The influences of oil film thickness and bearing size are considered in the nonlinear oil film force model. By using a numerical method, the governing equations of the system are solved to obtain the steady-state vibration. The simulation results from a coupling model are compared with the experimental results and the effectiveness of the new model is verified. It has been found that spectrums and orbit plots are effectively used to reveal the unique nature of misalignment faults, leading to reliable misalignment diagnostic information.

On the predictive modeling of nonlinear frequency behaviors of an archetypal rub-impact rotor

Due to the existence of Duffing-type nonlinearity, Hertzian contact and bearing clearance, the dynamic characteristics of rub-impact rotor system are seriously affected. The paper presents a new kinematic equation in plural form for the typical rotor system by employing the Lagrange equation, which consists of a disk fixed on a shaft of Duffing-type nonlinearity, a stator of Hertz contact force and Coulomb friction. Furthermore, based on the complex harmonic balance method, the analytical frequencies of this nonlinear rotor system under unperturbed and perturbed rotating cases are comprehensive developed, respectively. Numerical simulations of the Runge–Kutta method were conducted to verify the analytical results. The comparison showed that out of the resonant range, the theoretical and numerical results were in good agreement.

Elucidation of the role of cohesion in the macroscopic behaviour of coarse particulate systems using DEM

Ignoring the effect of cohesion between particles in Discrete Element Method (DEM) may incur in errors in the predictions of bulk properties. In this research work, we investigated the role of the cohesion force on macroscopic and microscopic characteristics of monodisperse and bidisperse iron ore granule systems, specifically on the angle of repose and coordination number of the assemblies. In order to simulate the cohesion between particles, a combination of the Hertz contact force and the pull-off force as predicted by the JKR model was employed. The monodisperse systems were made of particles whose sizes ranged between 3 and 10 mm. In order to analyse the effect of bidispersity on the characteristics of the particle assemblies, combinations of 10 mm particles with smaller particles (from 3 to 7 mm) were considered. The results showed that similarly to the rolling friction force, the cohesion force caused a considerable increase in the angle of repose, for the whole particle size range here studied. More importantly by increasing particle size, the rate at which the angle of repose increased as a result of the cohesion force, decreased significantly. This influence became more outstanding by increasing the value of the rolling friction coefficient, signifying the interplay between rolling friction force and cohesion force. In contrast to the case of non-cohesive particles, the effect of particle size on the angle of repose was independent of the rolling friction force in the presence of cohesion force and led to a decrease in the angle of repose for the entire range of rolling friction coefficient values. An empirical analysis of the obtained angles of repose confirmed the strong dependency of the angles on the Bond number and the rolling friction coefficient for cohesive particles. In addition, the coordination number of all studied systems was influenced by both parameters, as the value of coordination number increased by increasing cohesion force but decreased by increasing rolling friction force.

Nonlinear normal modes and primary resonance for permanent magnet synchronous motors with a nonlinear restoring force and an unbalanced magnetic pull

The unbalanced magnetic pull (UMP) in permanent magnet synchronous motors (PMSMs) leads to the generation of vibrations and noise because of mechanical and magnetic coupling effects. This paper investigates the nonlinear oscillations of a PMSM based on a Jeffcott rotor-bearing system, and the effects of the UMP, nonlinear restoring forces due to the Hertz contact force and bearing clearance, rotor weight and rotor mass eccentricity are considered. A second-order approximate solution in the primary resonance case is obtained by applying the multiple-scale perturbation method. The Routh–Hurwitz criterion is utilized to investigate the stability of the obtained solution. The linear natural frequencies of the horizontal and vertical modes show little difference based on the constructed model. The localized oscillations and nonlocalized oscillations of the rotor-bearing system are analyzed through its frequency response curves by introducing different values of system parameters. The response curves exhibit hard spring characteristics with unstable regions and jump phenomena. Detailed numerical research including Poincare map and bifurcation diagram reveals the effect of excitation amplitude on the system. Additionally, this work provides some theoretical and practical guidance for the design of PMSMs with strong dynamic behaviors.

Nonlinear characteristics of rubbing rotor–bearing system of locomotive excited by rotating speed of the traction motor

The paper aims to study the nonlinear characteristics excited by rotating speed of the traction motor. Considering the rub-impact force, Hertzian contact force and unbalanced rotor, a typical of dynamic model of rotor-bearing system for locomotive is established, and Runge-Kutta method is adopted to solve the nonlinear response. Based on bifurcation diagrams, axis trajectory curves, phase plane portraits, Poincaré maps and amplitude spectrums, nonlinear characteristics of the system are analysed when rotating speed of the traction motor is changed. The simulation reveals complex dynamic behaviour of the system including periodic, multi-periodic and chaotic motion. The results will help us understand and control the nonlinear behaviour of the proposed traction system for railway locomotive in the future research.

An innovative dynamic model for vibration analysis of a flexible roller bearing

Due to the centrifugal forces and external loads, the components of a cylindrical roller bearing (CRB) will produce elastic deformations, which will significantly influence the bearing vibrations. In this work, an innovative analytical dynamic model is proposed to provide an in-depth vibration analysis of a CRB with the flexible rollers, races, shaft, and housing. In this model, the structure deformations of the roller and races caused by the centrifugal forces and external loads are formulated, as well as the shaft and housing deformation. Both the nonlinear Hertzian contact force, lubricating oil film, and roller mass are also modelled. The above parameters cannot be completely described by the previous rigid dynamic models in the reported literature. Influences of the roller deformations, race deformations, external load, and rotational speed on the vibrations of the flexible and rigid CRBs are analyzed. The results indicate that the elastic deformations of the rollers and races have a significant influence on the vibrations of the CRB. Comparisons of the listed results show that this work can provide a more accurate and reasonable dynamic modelling method for a flexible CRB system.

Local defect modelling and nonlinear dynamic analysis for the inter-shaft bearing in a dual-rotor system

• A force model for the inter-shaft bearing with a local defect on surface of the outer or inner race. • The system may resonant in some non-critical-speed region due to the local defect. • Parameters affecting the resonant frequencies and vibration amplitudes of the abnormal resonances. • A comprehensive parametric analysis is carried out to show the nonlinear resonant response characteristics. This paper presents a force model for the inter-shaft bearing with a local defect on the surface of the outer race or the inner race, and the nonlinear dynamic characteristics of a dual-rotor system affected by the local defect are investigated. A simplified dual-rotor system is presented with the consideration of the inter-shaft bearing's nonlinearities such as the Hertzian contact force and the radial clearance. The local defect is considered as a regular dent with a constant depth, thus the radial clearance will increase when rolling elements go through the range of the local defect. The motion equations of the system with eight degrees of freedom are formulated by using the Lagrange's equation. The nonlinear vibration responses of the dual-rotor system affected by the local defect are obtained using numerical method. The results show that there exist four abnormal resonances on the amplitude frequency curves of the system due to the effect of the local defect, apart from the couple of primary resonances excited by the unbalance excitations of the two rotors. With the aid of the characteristic defect frequency analysis, it is revealed that one pair of the abnormal resonances are excited by the characteristic defect frequency, and the other pair of the abnormal resonances are excited by the combination frequency. Furthermore, a comprehensive parametric analysis is carried out to give an insight into the nonlinear resonant response characteristics affected by parameters such as the depth and the span of the defect, the rotation speed ratio, the unbalances of two rotors, the stiffness and the damping of the linear elastic spring, and the radial clearance, the stiffness and the roller number of the inter-shaft bearing. The results show that the vibration amplitudes for the abnormal resonances are mainly determined by the depth and the span of the defect, while the resonant frequencies for the abnormal resonances are mainly influenced by the rotation speed ratio and the roller number of the inter-shaft bearing. However, the rotors’ unbalances mainly affect the corresponding primary resonance rather than the abnormal resonances. The obtained results will contribute to a better understanding of the nonlinear resonant response characteristics of dual-rotor systems with a local defect on the inter-shaft bearing, which are helpful for the fault diagnostics of the inter-shaft bearing in a dual-rotor system.

Frictional viscoelastic based model for spherical particles collision

In this paper, collision of two balls in three dimensions with regard to energy loss and friction is studied. This investigation intends to propose the procedure which can determine the condition of pure sliding, sliding with rolling or sticking, and rolling or sticking at the beginning of contact. Furthermore, the other aim of this research is to suggest the closed-form relation for post-collision angular and linear velocities in which the possibility of three regimes of impact, pure sliding, sliding with rolling, and pure rolling, are considered. In this investigation, viscous Hertz contact force describes the normal interaction force. Moreover, stick regime is not taken into account and during sliding, Coulomb friction is considered as a tangential force, which is endorsed by collision experimental data in case the collision regime is not stick regime. Considering aforementioned issues along with employing momentum conservation, the condition for the possibility of transformation of sliding to sticking or rolling during collision, and furthermore, the analytical relation for post-collision velocities are achieved. Eventually, in order to demonstrate the procedure of solving two balls collision, numerical examples are presented.

Non-linear dynamic response analysis of cylindrical roller bearings due to rotational speed

The dynamic behaviour of cylindrical roller bearings is presented, in both balanced and unbalanced conditions as a function of speed. The stiffness and damping non-linearities at the contact points (due to Hertzian contact force between rollers and races), radial internal clearance and unbalanced rotor force make the bearing system non-linear. Presently, the differential equations representing the dynamics of the cylindrical roller bearings have been obtained using Lagrange’s equation and solved numerically using modified Newmark-β method. The results of the analyses of various motion behaviours are presented as time–displacement responses, orbit plots, phase portraits, Poincaré maps and Fast Fourier Transform plots. The obtained responses revealed the sensitive behaviour of the system from periodic to quasi-periodic and chaotic with speed variations for both balanced and unbalanced rotor conditions. Also, intermittent chaotic behaviour has been observed. A pattern of the interaction between rotational and variable compliance vibration is observed with speed variations. The frequency pattern analysis (with different techniques used like phase/orbit plots and Poincaré plots) for healthy cylindrical bearing and different rotor conditions under different applied non-linearity consideration is a new attempt to analyse dynamic behaviour of the bearing. This analysis is helpful for online monitoring of fault-free cylindrical roller bearings and for studying the impact of speed on system’s dynamical behaviour.

Bifurcations of periodic motion of a horizontally supported nonlinear Jeffcott rotor system having transversely cracked shaft

This article investigates the periodic motion bifurcations of a horizontally supported nonlinear Jeffcott rotor system having transversely cracked shaft. The nonlinear spring characteristics due to Hertz contact force and bearing clearance, disc weight, disc eccentricity, breathing of the shaft crack, and angle between the crack and imbalance directions are included in the system model. A mathematical model governing the cracked system lateral vibrations is derived and then analyzed utilizing asymptotic analysis in the primary resonance case. Effects of disc eccentricity, creak depth, and angle between the crack and imbalance directions on the system response curves are studied. The analysis revealed that at a small crack depth, the system executes both forward and backward whirling motions at a specific range of the disc spinning speed, while the backward whirling orbits disappear as the crack depth increases. In addition, at zero disc eccentricity, the cracked system does not oscillate unless the system linear stiffness coefficient is reduced by about 11% as a result of shaft crack. Moreover, there is a spinning speed range of the rotating shaft at which two stable periodic solution attractors appear beside the trivial solution one when the linear stiffness coefficient of the system is reduced to 20% or more. The obtained analytical results are confirmed numerically that showed a very good agreement with the numerical ones. Finally, the acquired results are compared with the work published in the literature.

Small nanoparticles, surface geometry and contact forces.

In this molecular dynamics study, we examine the local surface geometric effects of the normal impact force between two approximately spherical nanoparticles that collide in a vacuum. Three types of surface geometries-(i) crystal facets, (ii) sharp edges, and (iii) amorphous surfaces of small nanoparticles with radii R<10 nm-are considered. The impact forces are compared with their macroscopic counterparts described by nonlinear contact forces based on Hertz contact mechanics. In our simulations, edge and amorphous surface contacts with weak surface energy reveal that the average impact forces are in excellent agreement with the Hertz contact force. On the other hand, facet collisions show a linearly increasing force with increasing compression. Our results suggest that the nearly spherical nanoparticles are likely to enable some nonlinear dynamic phenomena, such as breathers and solitary waves observed in granular materials, both originating from the nonlinear contact force.

Role of the Adhesion Force during Copper Chemical Mechanical Planarization

The adhesion force between slurry particles and the wafer surface has a significant effect on the material removal and ultra-smooth surface formation during the chemical mechanical polishing/planarization (CMP) process. Based on zeta potentials measurements, we calculate the value of adhesion force, which cannot be ignored compared with the Hertz contact force for small sized abrasive particles at low polishing pressures. More over, the impacts of slurry (particle size and pH value) and wafer surface roughness on the adhesion force are evaluated. Theoretical analyses manifest that the adhesion force mainly originates from the van der Waals force and electrical double-layer force, which increases with the increase of the particle size and slurry pH, while decreases with the increase of the wafer surface roughness. The relationships between the adhesion force and scratch depth indicate that scratching would be avoided while enhancing the adhesion force during CMP. These findings would provide some inspirations to understand the CMP technology in depth, and they are also beneficial to engineering super-smooth and super-lubricant surfaces with nano-scaled roughness.

ON SURFACE FRACTURE OF RAIL HEADS

Purpose. The formation of crack-like defects in rails of railway tracks is a serious problem for engineering practice because of the danger of creating emergency situations. The purpose of this work is to establish theoretically the characteristic angle of propagation of surface cracks in the rail heads of railway rails, which is basic in the formation of typical surface contact fatigue damages, such as pitting, «checks» and «squat». It is also necessary to find the conditions for determining this angle. Methodology. The investigations were carried out on the basis of the method of singular integral equations. The rail damaged by the surface crack was modeled with a half-plane with an edge cut, and the action of the wheel on the rail by unidirectional repeated translational movement along the edge of the half-plane of the Hertzian contact forces with the tangential component. The problem of determining the stress intensity factors in the vicinity of the crack tip in the rail head was reduced to a system of two real singular integral equations which were solved numerically by the Gauss-Chebyshev mechanical quadrature method. The complexity of the problem consists in the fact that the boundaries of the contact areas and the opening of the crack faces are unknown beforehand and they change when the model contact forces move. These boundaries were determined simultaneously with solving the integral equations of the problem from additional conditions by the iteration method. Findings. The presence of the characteristic angle of propagation of mode II surface cracks in the rail head has been established theoretically and the conditions for its determination have been put down. The results obtained are in good agreement with engineering and experimental data. Originality. For the first time, the values of the characteristic angle were theoretically determined, under which at the initial stage, the surface contact fatigue cracks propagate in the head of the railway rail under the action of the wheels. Conditions for determining this angle have been also put down. Practical value. The received data are of great importance for engineering practice, since they reveal the nature of surface contact fatigue defects under various operating conditions and allow to predict their contact strength and durability.

Transformation of sliding motion to rolling during spheres collision

In this research, we have investigated the three-dimensional elastic collision of two balls, based on friction in the tangential plane. Our aim is to offer analytical closed form relations for post collision parameters such as linear and angular velocities, collision time and tangential and normal impulse in three dimensions. To simplify the problem, stick regime is not considered. In other words, balls have a low tangential coefficient of restitution. Sliding, sliding then rolling, and rolling at the beginning of contact are three cases that can occur during impact which have been considered in our research. The normal interaction force is described by the Hertz contact force and dimensionless analysis is used for investigating normal interaction force; furthermore, Coulomb friction is considered during sliding. Experimental data for collisions show when sliding exists through the impact, tangential impulses can be taken as frictional impulses using the Coulomb law if the dynamic regime is not stick regime. To identify transformation of sliding motion to rolling or sticking during the impact process, linear and trigonometric functions are considered as an approximation for the normal interaction force. Afterwards, we have obtained the condition for the possibility of this transformation; moreover, we can estimate the duration of sliding and rolling or sticking. We have obtained an analytical solution for maximum force and deformation, collision time, impulses and post-collision linear and angular velocities in three dimensions.

Characteristics analysis of aero-engine whole vibration response with rolling bearing radial clearance

For the influence of rolling bearing radial clearance on the whole vibration in the aero-engine whole system, a real engine rotorbearing- casing whole model is established. The rotor and casing systems are modeled by means of FEM; the support systems are modeled by lumped-mass model; rolling bearing radial clearance and strong-nonlinearity of Hertz contact force at four different supports are considered. The coupled system response is obtained by the numerical integral method. The characteristics of the whole vibration response are analyzed. For rolling bearing at a typical support, the rotor, outer ring of rolling bearing and casing response characteristics at different rotating speeds are analyzed. The changing law of contact forces for each ball and the global contact forces at different speeds are analyzed. The influence of the radical clearance on the contact forces on the whole vibration is analyzed. The results show that the contact forces will be larger and the acceleration amplitude jumps obviously when the radial clearance is increased, and due to the variable stiffness of the rolling bearing, the natural frequency will appear when the stiffness changes fiercely, that is frequency-locked phenomenon. When the radical clearance is larger and the rotating speed is between two critical speeds, the rotor squeezes the outer ring now and then. Reducing the radical clearance can reduce the whole vibration and increase the rotor’s stability.

Thermoplastic polyurethane/glass fabric composite laminates: Low velocity impact behavior under extreme temperature conditions

This paper reports the low velocity impact behavior of thermoplastic laminated structures based on thermoplastic polyurethane reinforced with woven glass fiber highlighting effects related to sample thickness and test temperature with the aim to verify their potential applications. Plaques obtained by film stacking and compression molding technologies are tested using an instrumented falling dart machine at room temperature, −25°C and −50°C. Impact results are reported in terms of typical load-deflection and energy-time curves, but also discussed with reference to the well known ductility index (DI) parameter. As expected, the cooling is reflected in an increased stiffness of tested specimens and their higher propensity to the damage. Increases of plate thickness and reductions of the test temperature lead to enhanced friction phenomena at the material-dart contact, responsible of significant deviations from well established semi-empirical models about the Hertz contact force and the penetration energy.Finally, no delamination phenomena seem to be verified for all investigated samples as supported by visual inspections of front and rear impacted areas.

Analysis of the Viscoelastic Sphere Impact against a Viscoelastic Uflyand-Mindlin Plate considering the Extension of Its Middle Surface

In the present paper, the problem on impact of a viscoelastic sphere against a viscoelastic plate is considered with due account for the extension of plate’s middle surface and local bearing of sphere and plate’s materials via the Hertz theory. The standard linear solid models with conventional derivatives and with fractional-order derivatives are used as viscoelastic models, respectively, outside and within the contact domain. As a result of impact, transient waves (surfaces of strong discontinuity) are generated in the plate, behind the wave fronts of which up to the boundaries of the contact domain the solution is constructed in terms of one-term ray expansions due to short-time duration of the impact process. The motion of the contact zone occurs under the action of extension forces acting in the plate’s middle surface, transverse force, and the Hertzian contact force. The suggested approach allows one to find the time-dependence of the impactor’s indentation into the target and the Hertzian contact force.

Low-velocity impact response of a pre-stressed isotropic Uflyand-Mindlin plate

The low-velocity impact response of a precompressed circular isotropic elastic plate is investigated in the case when the dynamic behavior of the plate is described by equations taking the rotary inertia and transverse shear deformations into account. Contact interaction between the rigid impactor and the target is modeled by a generalized Hertz contact force, since it is assumed that the viscoelastic features of the plate represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives due to the fact that during the impact process decrosslinking occurs within the domain of the contact of the plate with the sphere, resulting in more free displacements of molecules with respect to each other, and finally in the decrease of the plate material viscosity in the contact zone.