Hertzian Impact Theory Research Articles

Online mass-flow-rate measurement of high-intensity gas-conveyed particle flow for dry mineral processing

Measuring the mass flow rate of high-intensity gas-conveyed particle flow (GCPF) in real time is of great importance for enhancing efficiency and productivity in dry mineral processing circuits. It also poses challenges beyond the capability of most existing technologies. In this article, we introduce a novel, cost-effective, online mass flow rate measurement system utilizing a force transducer. A theoretical model of the relationship between the average impact force and the mass flow rate was developed using the Hertz theory of impact. A prototype sensing system was then designed and constructed, followed by a series of experiments to validate the theoretical model. The experimental results corroborated the proposed model and showed the accuracy of the measurement was within 2% for mass flow rates above 1.741 g/s. This demonstrated the system’s capability to measure the mass flow rate of GCPF in real-time using measurements of the average impact force.

Design of robust particle dampers using inner structures and coated container walls

Classical particle dampers suffer from their non-robust damping behavior, i.e. they can only be efficiently applied to a specific frequency range and amplitude range. The reason for that is that particle motion, also called motion mode, and damper efficiency show a strong correlation. By changing particle or container properties the motion modes are shifted to other excitation conditions but their efficient range is not much affected. To increase the damping performance and robustness of particle dampers, two approaches are presented here by introducing new motion modes. Therefore, the particle dampers are analyzed experimentally using a shaker setup and numerically using the discrete element method. The first design approach uses inner structures inside the particle damper, manufactured by a 3D printer. The inner structures consist of different numbers of beams, placed perpendicular to the container moving direction. They lead to a much more robust damper as the transition between the motion modes gets smoother. For the second approach, the container walls are equipped with different soft polymers. In this way a new motion mode at low excitation intensities is observed, leading to a high efficiency possibly on a large excitation intensity range. For an easy calculation of the necessary wall’s Young’s modulus an analytical formula based on Hertz impact theory is derived.

“Touch‐aware” contact model for peridynamics modeling of granular systems

AbstractThis article presents an innovative “touch‐aware” frictional contact model for peridynamics to efficiently simulate the contact behavior between irregularly shaped particles. Conventional short‐range force contact model formulates the contact force field in such a way that leaves a large gap between particles, leading to a softer stiffness of contact and inaccurate simulation of densely packed granular materials. The developed “touch‐aware” contact model initiates the contact force calculation when contacting bodies are in touch, which leaves no gap between contacting bodies and provides a stiffer contact in an accurate and efficient way. It simulates frictional contact behaviors and damping between contacting bodies with irregular shapes. Different examples are given to verify the touch‐aware contact model with theoretical solutions such as Hertz theory of contact, Hertz theory of impact, Coulomb's law and damping formulation. The model is also compared with conventional peridynamics contact model and the discrete element method. Finally, the touch‐aware contact model is used to simulate condensation of an aggregate of irregularly shaped particles. These examples demonstrate excellent capacity of the “touch‐aware” model in simulating packed granular systems.

Modelling the multi-physics of wind-blown sand impacts on high-speed train

The wind-blown sand effect on the high-speed train is investigated. Unsteady RANS equation and the SST k–ω turbulent model coupled with the discrete phase model (DPM) are utilized to simulate the two-phase of air-sand. Sand impact force is calculated based on the Hertzian impact theory. The different cases, including various wind velocity, train speed, sand particle diameter, were simulated. The train's flow field characteristics and the sand impact force were analyzed. The results show that the sand environment makes the pressure increase under different wind velocity and train speed situations. Sand impact force increases with the increasing train speed and sand particle diameter under the same particle mass flow rate. The train aerodynamic force connected with sand impact force when the train running in the wind-sand environment were compared with the aerodynamic force when the train running in the pure wind environment. The results show that the head car longitudinal force increase with wind speed increasing. When the crosswind speed is larger than 35m/s, the effect of the wind- sand environment on the train increases obviously. The longitudinal force of head car increases 23% and lateral force of tail increases 12% comparing to the pure wind environment. The sand concentration in air is the most important factor which influences the sand impact force on the train.

An impact model of a ball bouncing on a flexible beam

In this work we investigate a model for the description of the impact of a ball with a flexible beam. The coupling of the kinematic parameters of the ball (the velocity components) with the vibration modes of the beam are taken into account by using the Hertz theory of impact. We solve the model equations numerically in a few physically relevant cases and illustrate the variation of the modal coefficients of the beam and of the velocity components of the ball during the impact. The restitution coefficient, the size of the deformation region and the impact duration are also reported as functions of the impact velocity and of the impact location.

Influence of coating thickness on the impact damage mode in Fe-based amorphous coatings

The impact behavior of Fe-based amorphous coatings with various thicknesses fabricated by high-velocity-air-fuel thermal spraying was systematically investigated via 3D X-ray tomography. The results showed that the impact damage occurred principally by cracking in the coating, plastic deformation in the substrate, and delamination at the coating/substrate interface. It was found that the size of the interfacial delamination was a nonlinear function of the normalized coating thickness tc/a (i.e. the ratio of coating thickness (tc) to the contact radius (a)). Three thickness regions of distinctive damage modes were identified: thin coating region (tc/a ≤ 0.33), intermediate-thickness region (0.33 < tc/a ≤ 0.54), and thick coating region (tc/a > 0.54). For tc/a ≤ 0.33, no delamination appeared in the coating/substrate interface. With an increase from tc/a = 0.33 to tc/a = 0.54, the extent of interfacial delamination increased quickly and reached a maximum. Finally, when tc/a > 0.54, the extent of delamination decreased when tc/a increased. In addition, it was indicated that the residual stress was found to decrease with the coating thickness, reflecting that the residual stress was excluded as a predominant factor of impact damage. Furthermore, a good agreement with the Hertz impact theory and finite element modeling was identified in the case of interfacial damage: significant delamination was observed in the 600 μm coating when the maximum shear stress occurred close to the coating/substrate interface. This finding can be used to assist the design of coated components in load-bearing applications.

On-line size measurement of pneumatically conveyed particles through acoustic emission sensing

Acoustic emission (AE) methods have been proposed for on-line size measurement of pneumatically conveyed particles in recent years. However, there is limited research on the fundamental mechanism of the AE-based particle sizing technique. In order to achieve more accurate measurement of particle size, the impact between particles and a waveguide should be described in a more realistic way. In this paper, an improved model based on the Stronge impact theory is presented to establish the relationship between the resulting AE signal and the particle size being measured. The improved model is validated with experiments on a single-particle test rig. A total five sets of glass beads with a mean diameter of 0.4, 0.6, 0.8, 1.0 and 1.2 mm, respectively, are used as the test particles with an impact velocity ranging from 22 m/s to 37 m/s. It is proven that the Stronge impact theory is more accurate to describe the collision process than the Hertzian impact theory and is thus more suitable for the particle size inversion, which is validated by comparing the inversion results using these two impact theories. Meanwhile, a good agreement is observed between the measured and reference particle sizes under different experimental conditions. The mean relative error between the measured and reference diameters is mostly within 12%.

FEA-based metal sphere signal map for mass estimation of simulated loose part in reactor coolant system

For safety management of nuclear power plants, accurate impact mass estimation of loose parts is very important. A center frequency method based on Hertz's impact theory, a frequency ratio method, and so forth are studied for mass estimation, but it is known that these approaches can hardly provide accurate information on impact response for identifying the impact source. In this paper, a finite element analysis (FEA) model to simulate the propagation behavior of the bending wave, generated by a metal ball impact, was validated by performing a series of impact tests and the corresponding finite element analyses for a flat plate as well as a curved plate (a half scale model of steam generator vessel). Various impact parameters such as amplitude, center frequency, group velocity and attenuation ratio of bending wave acceleration signal were investigated to verify the usefulness of the FEA model. Also, an FEA-based metal sphere signal map was developed, and then blind tests were performed to verify the map. This study provides an accurate simulation method for predicting the metal impact behavior and for building a metal sphere signal map, which can be used to estimate the mass of loose-parts on site in nuclear power plants.

Plate bending wave propagation behavior under metal sphere impact loading

A loose part can cause component damages and material wear on nuclear power plants; thus, a mass estimation of the loose part is crucial to safety management. A bending wave propagation of a structure under the loose part impact loading is precisely simulated to accurately estimate the mass of a loose part. Lamb’s general solution for an arbitrary impact force function and Hertz impact theory have been used to identify the characteristics of the bending wave that is impacted by a metallic loose part in reactor pressure boundary components. However, these approaches cannot provide accurate information on the acceleration response that is required to identify the impact source. In this study, the bending wave propagation behavior of plate structures under a simulated loose part (Metal sphere) impact loading was modeled using a Finite element analysis (FEA) technique. The characteristics (e.g., Maximum acceleration amplitude, primary frequency, and bending wave velocity) of the impact response signal from a metal sphere were analyzed with the FEA results and were verified with experimental results. In addition, the correlation between plate thickness and characteristic length was presented. Results from this study can be utilized to estimate the location and mass of a loose part for condition monitoring and diagnostics in nuclear power plants.

Bending Wave Propagation Analysis Induced by Metal Ball Impacts on Reactor Pressure Boundary Structure

Recently, a model-based prognostic approach is becoming one of noticeable techniques to predict the fault behaviors of mechanical components in nuclear power plant. Lamb’s general solution for an arbitrary impact force function and Hertz impact theory have been used to identify the bending wave characteristics impacted by a metallic loose part in reactor pressure boundary components. However. these approaches can hardly provide an information on accurate acceleration response for identifying the impact source. In this study, the impact response characteristics such as maximum acceleration amplitude and primary frequency of the impact response signal by a simulated loose part (metal ball) are analyzed using finite element analysis (FEA) and the FEA-based model is verified by experimental results. It is expected that the developed FEA-based model can be utilized in model-based prognostics for localization and estimation of mass of loose part in nuclear power plants.

Markov chain-based mass estimation method for loose part monitoring system and its performance

Abstract A loose part monitoring system is used to identify unexpected loose parts in a nuclear reactor vessel or steam generator. It is still necessary for the mass estimation of loose parts, one function of a loose part monitoring system, to develop a new method due to the high estimation error of conventional methods such as Hertz's impact theory and the frequency ratio method. The purpose of this study is to propose a mass estimation method using a Markov decision process and compare its performance with a method using an artificial neural network model proposed in a previous study. First, how to extract feature vectors using discrete cosine transform was explained. Second, Markov chains were designed with codebooks obtained from the feature vector. A 1/8-scaled mockup of the reactor vessel for OPR1000 was employed, and all used signals were obtained by impacting its surface with several solid spherical masses. Next, the performance of mass estimation by the proposed Markov model was compared with that of the artificial neural network model. Finally, it was investigated that the proposed Markov model had matching error below 20% in mass estimation. That was a similar performance to the method using an artificial neural network model and considerably improved in comparison with the conventional methods.

Simulation of ball motion and energy transfer in a planetary ball mill

A kinetic model is proposed for simulating the trajectory of a single milling ball in a planetary ball mill, and a model is also proposed for simulating the local energy transfer during the ball milling process under no-slip conditions. Based on the kinematics of ball motion, the collision frequency and power are described, and the normal impact forces and effective power are derived from analyses of collision geometry. The Hertzian impact theory is applied to formulate these models after having established some relationships among the geometric, dynamic, and thermophysical parameters. Simulation is carried out based on two models, and the effects of the rotation velocity of the planetary disk Ω and the vial-to-disk speed ratio ω/Ω on other kinetic parameters is investigated. As a result, the optimal ratio ω/Ω to obtain high impact energy in the standard operating condition at Ω = 800 rpm is estimated, and is equal to 1.15.

Applicability of the macro-scale elastic contact theories for the prediction of nano-scaled particle collision with a rigid flat surface under non-adhesive and weakly-adhesive conditions

Applicability of the macro-scale elastic contact theories for the prediction of the collision dynamics involving nano-sized particles is examined. Essential parameters controlling the nano-scale collision are found by continuum-based Hertz (Hertz, 1896) and the JKR (Johnson et al., 1971) theories. Collision parameters of Lennard-Jones particles comprised of 2899 and 17,789 molecules onto a rigid flat surface are numerically obtained by a molecular dynamics simulation (MDS) method. MDS results validate the theories in terms of elastic limit velocity, the maximum compression force and the maximum radius of contact area. The elastic limit velocity of the nano-sized particles is accurately predicted by imposing the macro-scale elastic limit criterion. For the maximum compression force acting on the colliding nano-particle, Hertz impact theory shows good agreement with MDS result in both elastic and inelastic collision regimes. The theories incorporated with the correction factor for conforming contact accurately predicted the contact radius. Both theories are valid within the limit of the elastic collision.

Confirmation of Hertz Theory during Solid Particle Flow Measurement

The article is focused on solid particle flow measurement by acoustic emission method. The solid particle impacts a waveguide and generates an impulse acoustic emission. It is possible to obtain information about parameters of impacting particles (e.g. specific mass, velocity, dimensions). The main focus of the article is a confirmation of model based on Hertz theory of impact. From performed measurements was created a model that was subsequently this model was compared with theoretical model. The models are similar, but further measurements will be necessary.

Dependence of electric energy output from a lead zirconate titanate ceramic piezoelectric element on impact conditions

In this work, dependence of electric energy output of impacted piezoelectric element on impact conditions is experimentally investigated. Investigated impact parameters include impact position, radius of impact sphere, orientation of the hitting object, and material and configuration of the buffer layer. It is found that the electric energy output of impacted piezoelectric plate may be increased by impacting the buffer layer at its center, decreasing the contact area between the hitting object and buffer layer, reducing internal energy absorption in the mechanical buffer layer and avoiding wave reflection in the buffer layer. The experimental phenomena are theoretically explained and analyzed, based on the piezoelectric constitutive equation and Hertz's impact theory.

Measurement of Impact Acceleration of Apple Fruits and Its Analysis by Hertz's Impact Theory

Measurement of Impact Acceleration of Apple Fruits and Its Analysis by Hertz's Impact Theory

Fracture of diamond coatings by high velocity sand erosion

This paper describes a study of the behaviour of diamond coatings when subjected to solid particle erosion from sand particles. The coatings were deposited by chemical vapour deposition (CVD) onto tungsten substrates and tested using a high velocity air–sand erosion test facility. The erosion tests were conducted using particle impact velocities of between 33 and 268 m/s. Examination of the eroded test specimens showed that the principal damage features were circumferential cracks and pin-holes. Comparison with Hertz impact theory revealed that the measured circumferential crack diameters were more than double the predicted Hertzian contact diameter. Moreover, a trend of increasing circumferential crack diameter with coating thickness, which is not predicted by Hertz, was found. Instead, the crack diameters showed good agreement with those predicted by the theory of stress wave reinforcement, which is more commonly associated with liquid impact damage of brittle materials. During impact, the bulk compression and shear waves are reflected at the rear surface of debonded regions of the coating to return to the front surface and reinforce the Rayleigh surface wave, which generates a tensile stress. Where this stress exceeds the local tensile strength of the coating, a ring of cracks surrounding the area of impact is created. The results from the present study therefore suggest that stress wave reflection is responsible for the formation of the cracks at locally debonded regions of the coating. This hypothesis was supported by images acquired using scanning acoustic microscopy, which showed that circumferential cracks and pin-holes were only found on areas of the coating that had become delaminated by multiple particle impacts during the erosion tests.

Asymptotic modeling of the impact of a spherical indenter on an elastic half-space

The impact of a rigid sphere on a homogeneous, isotropic elastic half-space in the absence of friction and adhesion is considered. The influence of the superseismic stage immediately following the moment of first contact upon the impact process is investigated in the frame of the Hertzian impact theory. The first order asymptotic approximation for the contact force in a three-dimensional dynamic contact problem with the slowly moving contact zone boundary is obtained and the corresponding asymptotic model of impact is developed. The motion of the indenter as it indents and rebounds from the elastic medium is analytically described. Explicit formulas are derived for the peak indentation depth, contact time, and rebound velocity as functions of the initial impact velocity, indenter mass, and characteristics of the elastic half-space.

Elastic contact mechanics-based contact and flash temperature analysis of impact-induced head disk interface damage

Shock-induced impulsive slider-disk contact during operation poses a major challenge for modern hard disk drives (HDD). The high contact pressure and surface temperature that are usually associated with such impact could lead to the loss of data or even catastrophic failure of the head-disk interface, rendering the HDD inoperable. In this work, an elastic contact mechanics-based analysis was performed to investigate the impact between a slider corner and a disk surface. First, the analysis uses a homogeneous, smooth contact based on Hertzian impact theory, and trends of critical contact parameters were discussed for various combinations of impact velocities and slider corner radii. From these results, disk plastic failure was identified based on the mean contact pressure. Then, a similar approach was adopted to extend the impact analysis to consider the effects of mechanical properties and thickness of various layers, adhesion, and roughness on the severity of slider-disk impact. Lastly, flash temperatures during the impact were calculated for various conditions, and it was found that thermal erasure could occur under current head-disk impact conditions.

Modelling of comminution processes in Spex Mixer/Mill

It is well known that mathematical models which simulate comminution processes represent a useful tool in several fields of academic and industrial research, with particular emphasis on nano-material and pharmaceutical production. In the present work a mathematical model which is able to quantitatively describe comminution processes in a ball milling system (i.e., Spex-Mixer/Mill) has been developed. The proposed approach takes into account three different contributions: dynamics of the vial, dynamics of spheres motion and simulation of the comminution process. The vial dynamics has been modelled by taking advantage of an appropriate roto-translation matrix. Model results have been successfully compared with literature experimental data. The spheres motion within the Spex Mixer/Mill has been simulated using a 3D dynamic model based on classical mechanics as well as the so-called discrete element method, which is widely adopted to quantitatively describe multi-body collision behaviour. In particular, existing models of impact with dissipation as well as the classical Hertz impact theory have been taken into account. This part of the global model allows one to obtain, for different operating conditions, the impact specific energy and impact velocity as a function of time. The latter ones represent input parameters for the simulation of comminution processes that is performed through suitable population balances, where different breakage functions as well as appropriate breakage probabilities have been considered. Model results are reported in terms of granulometric distribution of powders within the mixer-mill as a function of time, minimal grain size obtainable and time needed to complete the comminution process for various operating conditions (i.e., mill frequency and charge ratio).