Hertz Theory Research Articles

Stress State Analysis of Radial Stress Superposed Bending

Radial stress superposed bending is a sheet metal bending process, which superposes predetermined radial stresses. Stress superposition is mandatory to enable the reduction of the triaxiality in bending, resulting in delayed damage evolution and an improved product performance. The knowledge of the stress state is essential for damage-controlled bending as the triaxiality is the driving force for the void evolution. To control the stress state in radial stress superposed bending, an additional counter force responsible for the pressure in the outer fiber is applied. To predict the effect of the counter force on the radial stress and the triaxiality an analytical model is proposed. The prediction of the reaction forces in the system is required for the process design and for the calculation of the stress superposition. The stress state for plane strain bending with stress superposition is derived, and pressure calculations are made using the theory of Hertz. The model and the assumptions are verified in numerical and experimental studies for various counter pressures and bending ratios. Finally, a discussion of the load path depending on the transient counter pressure is carried out and experimental evidence for a inhibited damage evolution due to stress superposition is given.

Comparison of analytical and numerical solution of bearing contact analysis

In this paper, an effort has been put to analyze the rolling ball bearing using finite element analysis the contact pressure level, stress or displacement behavior of rolling ball bearing. The obtained results were then compared with the analytical results obtained through the methodology of Hertz theory, which means composition and solution of analytical formulas and equations based on specific ball bearing parameters in order achieve optimal results to demonstrate an almost identical similarity of results of numerical and analytical solution and correct setting up of the given simulation in computational software ANSYS Workbench.

Proposal of a novel approach for 3D tooth contact analysis and calculation of the static transmission error in loaded gears

The static transmission error in gears represents the main noise and vibration source of mechanical transmissions, both for self-excitation and for the excitation of powertrain components. An accurate determination of it is very complex, since it requires experimental tests or highly detailed models. In both cases the costs and the computational time are high, therefore the possible different types of geometries to analyse are significantly reduced. In the present work, a semi-analytical methodology is proposed, with the aim of evaluating the variation of the contact area in gears during their meshing and at different levels of applied torque. This model considers the tooth compliance (tooth shear and bending, foundation and rim, gear body) and the local contact effects. The present semi-analytical methodology allows a detailed analysis of the tooth compliance and the contact area on a high number of points along the meshing cycle (typically over 100 points). It also enables the definition of the correct contact area geometry between meshing teeth, depending on the possible geometry modification (micro-geometry and/or manufacturing errors) and on the input torque. In particular, the contact is simulated through a non-Hertzian model able to evaluate every contact shape, overcoming the limits of the Hertzian theory. The goal of the proposed methodology is to determine with high precision the static transmission error between gears in a limited time with respect to the classical finite element method. Furthermore, it is useful for a reliable prediction of the transmission dynamic behaviour, considering the load exchanged between the teeth in relation to spin speed and torque, in a computational time lower than the classical techniques.

A two-dimensional contact model between a multilayered solid and a rigid cylinder

Multilayer coatings are often reported in the papers on surface engineering and have been proved to be able to enhance the toughness, anti-wear and bonding strength, while an understanding of their mechanical behavior is essential to their optimal design and tribological application. In this paper, a two-dimensional contact model of a multilayered solid in line contact with a rigid cylinder is developed based on a semi-analytical method. The frequency response functions, which are essential to producing the influence coefficients of the displacements and stresses components in the contact model with a conversion method based on the fast Fourier transform, are obtained by solving a system of linear equations concerning the unknown constants in the elastic field general solution of layered materials with a numerical method. The present model is validated by comparisons of the contact pressure and von Mises stress between the solutions of the present model and those of the Hertz theory and the finite element contact model. The effect of the elasticity modulus changing mode as well as the layer number of the multilayered solid is further studied by utilizing the present model.

Dynamic analysis of rotating satellite de-spun using <?A3B2 ACK?>flexible brush

With the continuous exploration and expansion of human beings in space, the number of satellites in space is gradually increasing. In the process of catching the failed satellites, the rolling spin of the satellite has always been the main problem which is affecting the stability of the catching system between targets and servers. The derotation mechanism, a flexible deceleration brush, is proposed, which is placed at the end of a seven-degree-of-freedom manipulator to form a racemable space robot. The momentum of the rolling target is dissipated by the contact collision between the end flexible deceleration brush and the rolling target windsurfing board. The angular velocity deceleration of the freely rolling target is carried out. Based on the rope element of the large deformation cable beam element of the absolute nodal coordinate method (ANCF), the continuum model of the flexible deceleration brush is established. According to the Hertz theory, contact collision force between the flexible deceleration brush and the rolling target windsurfing board are calculated. For the non-cooperative target of free rolling in different speeds, the effects of the length, angle and other parameters of the flexible deceleration brush on the racemization effect are analyzed by dynamics simulation, and the optimal racemization parameters are obtained in the process of the dynamics simulation. Using these parameters to verify the simulation of the racemization, the results show that the strategy, which is using flexible deceleration brush to rack non-cooperative targets, can successfully eliminate the initial spin angular velocity and it is feasible and effective.

Theoretical and Finite Element Analysis of Residual Stress Field for Different Geometrical Features After Abrasive Waterjet Peening

Abrasive waterjet (AWJ) peening can be used for metal surface strengthening by introducing near-surface plastic strain and compressive residual stress. The present studies seldom focus on residual stress by AWJ peening of targets with different geometrical features. In fact, those targets usually exist on some machine parts including gear roots, shaft shoulders, and stress concentration areas. According to Hertz theory of contact and Miao's theoretical model for predicting residual stress of flat surface, this paper developed a theoretical model for investigating residual stress of targets with different geometrical features including concave arc surface, concave sphere surface, convex arc surface, and sphere surface. AWJ peening of targets with different geometrical features and different radii of Gaussian curved surface was simulated by abaqus. Theoretical results were consistent with numerical simulation results and published experimental results (H. Y. Miao, S. Larose, et al., 2010, “An analytical approach to relate shot peening parameters to Almen intensity,” Surf. Coat. Technol., 205, pp. 2055–2066; Cao et al., 1995, “Correlation of Almen arc height with residual stresses in shot peening process”, Mater. Sci. Technol. 11, pp. 967–973.), which will be helpful for predicting residual stress of gear roots, shaft shoulders, and stress concentration areas after AWJ peening. The research results showed that under the same peening parameters, σmax, σtop, dmax, and dbottom in concave surface (including concave arc surface and concave sphere surface) were the maximum; σmax, σtop, dmax, and dbottom in convex surface (including convex arc surface and sphere surface) were the minimum; for concave surface, σtop, σmax, dbottom, and dmax decreased, respectively, with target radius; for convex surface, σtop, σmax, dbottom, and dmax increased, respectively, with target radius.

Method for highly spatially resolved determination of residual stress by using nanoindentation

Measurement of highly spatially resolved residual stresses is a crucial task for tailoring the stress state during manufacturing to improve mechanical performance of metallic objects. Especially in sheet metal forming or blanking with sheet thicknesses down to 0.1 mm, a highly resolved pattern of the stress distribution over the sheet thickness is required for optimisation. For this purpose, a method which uses an extended Hertzian theory for calculating residual stresses from the results of nanoindentation is used to measure the stress profile in metallic cylinders. For verification of the indentation results, hole drilling is utilised to determine residual stresses in the near-surface area. This method provides similar assumptions about the stress state (biaxial). The measuring principle is also based on correlations between mechanical behaviour of the material and the residual stress distribution. The results yield a good accordance of the measured values despite the different scales of measurement. Therefore, a highly spatially resolved measurement of residual stresses with nanoindentation is possible and shows comparable values to the classic hole drilling technique.

Simulation and measurement study of metro wheel wear based on the Archard model

PurposeThe purpose of this paper is to study the wear evolution of metro wheels under the conditions of different track sequences, track composition and vehicle load and then to predict wheel wear and to guide its maintenance.MethodologyBy using the SIMPACK and MATLAB software, numerical simulation analysis of metro wheel wear is carried out based on Hertz theory, the FASTSIM algorithm and the Archard model. First of all, the vehicle dynamics model is established to calculate the motion relationship and external forces of wheel-rail in the SIMPACK software. Then, the normal force of wheel-rail is solved based on Hertz theory, and the tangential force of wheel-rail is calculated based on the FASTSIM algorithm through the MATLAB software. Next, in the MATLAB software, the wheel wear is calculated based on the Archard model, and a new wheel profile is obtained. Finally, the new wheel profile is re-input into the vehicle system dynamics model in the SIMPACK software to carry out cyclic calculation of wear.FindingsThe results show that the setting order of different curves has an obvious influence on wear when the proportion of the straight track and the curve is fixed. With the increase in running mileage, the severe wear zone is shifted from tread to flange root under the condition of the sequence-type track, but the wheel wear distribution is basically stable for the unit-type track, and their wear growth rates become closer. In the tracks with different straight-curved ratio, the more proportion the curved tracks occupy, the closer the severe wear zone is shifted to flange root. At the same time, an increase in weight of the vehicle load will aggravate the wheel wear, but it will not change the distribution of wheel wear. Compared with the measured data of one city B type metro in China, the numerical simulation results of wheel wear are nearly the same with the measured data.Practical implicationsThese results will be helpful for metro tracks planning and can predict the trend of wheel wear, which has significant importance for the vehicle to do the repair operation. At the same time, the security risks of the vehicle are decreased economically and effectively.Originality/valueAt present, many scholars have studied the influence of metro tracks on wheel wear, but mainly focused on a straight line or a certain radius curve and neglected the influence of track sequence and track composition. This study is the first to examine the influence of track sequence on metro wheel wear by comparing the sequence-type track and unit-type track. The results show that the track sequence has a great influence on the wear distribution. At the same time, the influence of track composition on wheel wear is studied by comparing different straight-curve ratio tracks; therefore, wheel wear can be predicted integrally under different track conditions.

COMBINED USE OF PARALLEL-PLATE COMPRESSION AND FINITE ELEMENT MODELING TO ANALYZE THE MECHANICAL PROPERTIES OF INTACT PORCINE LENS

The objective of this study is to explore the feasibility of a compression test for measuring mechanical properties of intact eye lenses using novel parallel plate compression equipment to compare the accuracy of implementing a classical Hertzian model and a newly proposed adjusted Hertzian model to calculate Young’s modulus from compression test results using finite element (FE) analysis. Parallel-plate compression tests were performed on porcine lenses. An axisymmetric FE model was developed to simulate the experimental process to evaluate the accuracy of using the classical Hertzian theory of contact mechanics as well as a newly proposed adjusted Hertzian theory model for calculating the equivalent Young’s modulus. By fitting the force-displacement relation obtained from FE simulations to both the classical and adjusted Hertzian theory model and comparing the calculated modulus to the input modulus of the FE model, the results demonstrated that the classical Hertzian theory model overestimated the Young’s modulus with a proportional error of over 10%. The adjusted Hertzian theory model produced results that are closer to original input values with error ratios all lower than 1.29%. Measurements of three porcine lenses from the parallel plate compression experiments were analyzed with resulting values of Young’s modulus of between 3.2[Formula: see text]kPa and 4.3[Formula: see text]kPa calculated. This study demonstrates that the adjusted Hertzian theory of contact mechanics can be applied in conjunction with the parallel-plate compression system to investigate the overall mechanical behavior of intact lenses.

Plastic deformation in zinc-blende AlN under nanoindentation: A molecular dynamics simulation

Using molecular dynamics (MD) simulations, we studied the mechanical response and nanostructure evolution of aluminum nitride with zinc-blende structure (zb-AlN). The defect types and structural evolutions as well as the formation and propagation of prismatic loops were discussed. The generalized stacking fault energy (GSFE) was also calculated to explain the glide of dislocations. It showed that perfect dislocations with Burgers vectors b = 1/2〈1 1 0〉 and Shockley partial dislocations with b = 1/6〈1 1 2〉 occur simultaneously for the indentation on (1 1 0) and (1 1 1) planes. The indentation force-depth (P-h) curve exhibits great compliance compared with that determined by the Hertz theory in the elastic phase. And the first three main drops on the P-h curve are related to dislocation nucleation and expansion. Moreover, a “lasso”-like formation mechanism of prismatic loops was found and the GSFE curves were also analyzed to assess the dislocation slips in zb-AlN. It was found that the dominant dislocation is of shuffle-set with Burgers vector b = 1/2 〈1 1 0〉, by comparing the GSFE along 111<1¯10> with that along 111<112¯>.

Analytical prediction of the geometry of contact ellipses and kinematics in a roller screw versus experimental results

This paper brings together the geometry of threaded contact areas and roller screw kinematics in order to shine some light on the local phenomena which dissipate power in the mechanism. The subject has received little attention in current literature, often because only the global kinematics is studied. Moreover, principal directions of curvature are usually assumed from the start. In this paper, differential geometry is used to calculate the result, which proves to be different from published research. Next, the classic Hertzian theory is adapted for slightly conforming contacts, which can be encountered in common roller screw designs. The generalized equations are used to deduce the shape, size and orientation of the contact ellipses for both the roller-screw and the roller-nut contacts. The two appear to be very different in terms of local kinematics. A new stationary model is introduced, which can calculate the sliding velocity field at any point within the contact area. The model has only one degree of freedom in the form of a slip ratio, which depends on lubricant properties and dynamics. An experimental setup was designed to measure this ratio and thus allow comparison with analytical models available in the literature.

Temperature rise of diamond-like carbon during sliding: Consideration of the real contact area

Abstract The real contact area based on modified Hertzian theory derived from sinusoidal asperity model is introduced and the measurement method for the real contact area is proposed. The real contact approach is compared with the Greenwood-Williamson theory showing good agreement between the two methods. The separation between the contact surfaces is estimated by taking into account the surface roughness. The temperature rise at the sliding interface between the DLC coating and E52100 steel is evaluated experimentally by the thermal indicator paint method, and the real contact area ratio is estimated from the temperature simulation data. The temperature rise is discussed from the entropy balance point of view induced from the real contact area and the frictional energy.

Kinematics and Force Analysis of Flexible Screw Mechanism for a Worm Robot

AbstractThis paper presents a new type of flexible screw mechanism (FSM), which is composed of a nut, flexible axle, and roller. It can be used in a worm robot to achieve flexible peristaltic motion, as well as curvilinear motion and deformation. This type of FSM uses a roller to decrease the friction. We investigated the transmission principle and the kinematic characteristics of this FSM, established the model of the velocity, acceleration of the roller, characterized the feed motion characteristics of the flexible shaft, and achieved an analytical solution of the flexible shaft's velocity. Furthermore, by considering the position of the pure rolling section of the roller, the spin slide model is proposed based on Hertz theory. To investigate the friction loss between the roller and the flexible axle, we established a friction work model of the entire FSM system. Finally, the motion characteristics of the FSM are evaluated through experiments.

An analytical model for accurate and numerically efficient tooth contact analysis under load, applied to face-milled spiral bevel gears

An analytical model that allows for a computationally efficient analysis of face-milled spiral bevel gears, is presented. The methodology builds on the consideration that the mating tooth flanks are designed to transmit motion in a nearly conjugate manner. A multibody approach to tooth contact analysis is proposed that assumes contact between rigid surfaces. By taking advantage of the action surfaces for each flank pair, contact is detected in a computationally efficient and accurate way. An analytical load distribution model is used to translate the detected penetration into resulting contact forces, under the assumption that the flank penetration matches the deformation of the teeth if they were flexible. To account for the global tooth deformation Tredgold’s approximation in combination with a set of expressions based on beam theory are utilized, while the local contact deformation is modeled based on Hertz theory. The methodology is validated against nonlinear finite element simulations. A comparison in terms of transmission error, contact patterns and contact pressure is provided. Contrary to FE simulations the proposed methodology requires significantly less computational effort, allowing further extension to optimization or system analysis problems.

Progression of plastic strain on heavy-haul railway rail under random pure rolling and its influence on crack initiation

The present work evaluates the plastic strain accumulation on railway rails for heavy-haul transportation and the effect of them on the life cycle and the stress-strain responses considering random loading passes of normal loads on the top of rail for wheel-rail pure rolling conditions. The normal pressure distribution on the contact region was estimated by the Hertz theory and applied on the surface of the rail in a tridimensional elastoplastic finite element model. Multiaxial Dang Van criterion was applied to estimate the life to crack nucleation. The results show that the level of plastic accumulation presents distinct magnitudes depending on the position analyzed, reaching similar plastic accumulation in analogous positions on the subsurface after several passes, for two different starting positions. That result is fundamental for life analyses, since indicates that, no matter the starting point, the computed life cycles always stabilizes after a few number of cycles. From that, the fatigue life to shelling can be estimated using high cycle models.

3D Finite Element Approach to Simulate the Contact Pressure between Two Deformable Cylinders. Application to a Spur Gear

Knowing the stresses and pressures in the contact between two deformable solids is fundamental in order to optimize the strength and the lifetime of mechanical components such as bearings or gears. These constraints can be determined by the calculation (finite element method or Hertz theory) or by experimental methods such as photoelasticity. The objective of this study is to model and compute the stress field and contact pressure using 3D finite element software. The validation of obtained results is done by comparison with the classical results of the non linear Hertz theory between two deformable cylinders. An application to spur gears with a circle involute profile is done and also validate with the same Hertz theory.

Responds of a helical triple-wire strand with interwire contact deformation and friction under axial and torsional loads

This paper aims to provide an analytical solution for describing the behaviors of a helical multi-wire strand under axial and torsional loads with the impacts of the local interwire contact. Both the interwire contact deformation and sliding friction are considered by using a theoretical approach, which divides the internal forces and moments of the multi-wire strand into two respective parts defined in two deformed configurations: one is related to the frictionless deformation state, and another is the friction deformation state of the multi-wire strand. The theoretical analysis has also conducted to understand the global responds of the strand, the local contact deformation and the friction contributions for a helical triple-wire strand. The overall axial forces, the twisting moments, and the stiffness components of the triple-wire strand with different helix angles under tensional and torsional loads are obtained The analytical results are in close agreement with the Finite Element (FE) predictions, validating the proposed analytical solution. The extended theoretical model and the proposed analytical solution well predict the responds of the helical triple-wire strand with a large range of helix angles (from 55° to 90°). The impacts of interwire contact deformation using Hertz and rigid contact theory, as well as with/without friction forces on the helical multi-wire strand behaviors have been discussed in details. The results show that the local interwire contact deformation and friction play a significant role on the strand responds as the helix angle decreasing.

Investigation on indentation rolling resistance of belt conveyor based on Hertz contact theory compared with one-dimensional Winkler foundation

Based on Hertz contact theory and one-dimensional Winkler foundation combination with viscoelastic theory, the author derived theoretical formulas of indentation rolling resistance, respectively. Using the laboratorial apparatus of indentation rolling resistance, the author mainly concentrates on the error analysis about two kinds of theoretical formula which bear on indentation rolling resistance compared with experimental result. The reason why author employs Hertz contact theory to discuss indentation rolling resistance is that indentation rolling resistance is a sort of contact resistance. As a result, Hertz contact theory is generally applicable to study it. On the other hand, because conveyor belt has viscoelastic property, it is appropriate to use viscoelastic theory by the aid of three-parameter Maxwell viscoelastic model combination with one-dimensional Winkler foundation. Ultimately, this article infers that theoretical formula based on the Hertz contact is brief and clear compared with one-dimensional Winkler foundation in principle. However, it is noticeable that when the belt is at high speed, the reliability of formula based on Hertz theory has decreased obviously. This conclusion can give a beneficial reference for the energy saving of belt conveyor.

Stress analysis and design of cylinder bearings supporting metal bridges

The present work presents a method for dimensioning solid cylinders supporting metal bridges at the side of the stationary hinge, which is known as cylinder bearing. The hinge constitutes of two plates with curved surfaces in contact with a cylinder between them. The imposed forces on the hinge cause contact pressures distributed on two antisymmetric circular arcs. If the arcs are too large, the Hertz theory can not be used for the calculation of stresses because it gives no sufficient results. For this particular problem an exact solution is given for normal and shearing stresses on the whole cylinder cross section. The octahedral shearing stress is used as a criterion to express the strength of the cylinder according to the maximum distortion energy theory, because it offers a more reliable tool to estimate the strength of the cylinder material. Moreover, a design equation is derived which may be useful to engineers, if they have to calculate the cylinder dimensions. The results show that the cylinder dimensions should be higher than estimated by other designers.

Worm gear efficiency model considering misalignment in electric power steering systems

Abstract. This study proposes a worm gear efficiency model considering misalignment in electric power steering systems. A worm gear is used in Column type Electric Power Steering (C-EPS) systems and an Anti-Rattle Spring (ARS) is employed in C-EPS systems in order to prevent rattling when the vehicle goes on a bumpy road. This ARS plays a role of preventing rattling by applying preload to one end of the worm shaft but it also generates undesirable friction by causing misalignment of the worm shaft. In order to propose the worm gear efficiency model considering misalignment, geometrical and tribological analyses were performed in this study. For geometrical analysis, normal load on gear teeth was calculated using output torque, pitch diameter of worm wheel, lead angle and normal pressure angle and this normal load was converted to normal pressure at the contact point. Contact points between the tooth flanks of the worm and worm wheel were obtained by mathematically analyzing the geometry, and Hertz's theory was employed in order to calculate contact area at the contact point. Finally, misalignment by an ARS was also considered into the geometry. Friction coefficients between the tooth flanks were also researched in this study. A pin-on-disk type tribometer was set up to measure friction coefficients and friction coefficients at all conditions were measured by the tribometer. In order to validate the worm gear efficiency model, a worm gear was prepared and the efficiency of the worm gear was predicted by the model. As the final procedure of the study, a worm gear efficiency measurement system was set and the efficiency of the worm gear was measured and the results were compared with the predicted results. The efficiency considering misalignment gives more accurate results than the efficiency without misalignment.