Calculation Of Contact Forces Research Articles

An FBG-Based Method for Load Measurement of a Cylindrical Rolling Element Bearing

Abstract Contact forces between raceways and rolling elements of a bearing stand as crucial operational aspects defining the bearing's performance. While monitoring bearing load through load cells or strain gauges on the shaft or housing is feasible, it may not precisely reflect the distributed load transmitted by the rollers directly to the raceways. This paper introduces a method to calculate these contact forces, relying on a linear assumption correlating the loads to the measured strains on the outer race of a bearing. In our research, we conducted both static and dynamic tests on cylindrical roller bearings through simulations and experiments. We designed an experimental test rig to conduct both static and dynamic experiments and utilized FBG optical fiber sensors for contact force measurement in bearings due to their multiple advantages, including high sensitivity, resistance to corrosion and magnetic interference, and ease of installation on the bearing outer race due to their shape and size. Our findings indicate a consistent linear relationship between contact forces and strains measured on the bearing's outer race. Furthermore, we calculated the linear coefficient K values from three test groups: static tests under simulation study, static tests under experimental study, and dynamic tests under experimental study. The K values obtained from static tests (simulation and experimental studies) and dynamic tests (experimental study) align consistently. Following this, we calculated contact forces in both static and dynamic experiments by multiplying the measured strains with K. Our calculations resulted in an error percentage of less than 2% for static tests and below 5% for dynamic tests, highlighting the accuracy of our approach in determining these contact forces.

A discrete element solution method embedded within a Neural Network

This paper introduces a novel methodology, the Neural Network framework for the Discrete Element Method (NN4DEM), as part of a broader initiative to harness specialised AI hardware and software environments, marking a transition from traditional computational physics programming approaches. NN4DEM enables GPU-parallelised computations by mapping particle data (coordinates and velocities) onto uniform grids as solution fields and computing contact forces by applying mathematical operations that can be found in convolutional neural networks (CNN). Essentially, this framework transforms a DEM problem into a series of layered “images” composed of pixels, using stencil operations to compute the DEM physics, which is inherently local. The method revolves around custom kernels, with operations prescribed by the laws of physics for contact detection and interaction. Therefore, unlike conventional AI methods, it eliminates the need for training data to determine network weights. NN4DEM utilises libraries such as PyTorch for relatively easier programmability and platform interoperability. This paper presents the theoretical foundations, implementation and validation of NN4DEM through hopper test benchmarks. An analysis of the results from random packing cases highlights the ability of NN4DEM to scale to 3D models with millions of particles. The paper concludes with potential research directions, including further integration with other physics-based models and applications across various multidisciplinary fields.

A contact force calculation approach for collision analysis with zero or non-zero initial relative velocity

A contact force calculation approach for collision analysis with zero or non-zero initial relative velocity

Study on calculation method for dynamic load distribution of thread pair of planetary roller screw mechanism

To reveal the dynamic load distribution characteristics of planetary roller screw mechanism under the high-speed operating conditions, based on the spiral surface equation and deformation coordination relationship, a static contact force calculation method of the roller-screw and roller-nut thread at the contact point is proposed, and a formula for calculating the contact angle after bearing is derived. On this basis, further considering the inertial force generated by using high-speed operation, a dynamic load distribution model for planetary roller screw mechanism thread was established. By comparing with the finite element contact model of planetary roller screw mechansim, the error is below 5%, which verifies the correctness of the model established. Finally, the influence of the parameters such as screw speed, axial load, pitch, and tooth flank angle on the distribution of dynamic load was analyzed. The results show that the inertia force causes the roller to have a tendency to move closer to the nut and away from the screw resulting in a decrease in dynamic contact force on the roller screw side and an increase in dynamic contact force on the roller nut side. The degree of uneven distribution of dynamic load is positively correlated with the screw speed and pitch, and negatively correlated with the tooth side angle. The size of axial load has a significant impact on the distribution of dynamic load.

Peridynamic contact models for fracture analysis based on the micro-beam bond

Contact problems are ubiquitous in engineering systems. Establishing contact models to predict the crack propagation and fracture is significant. Therefore, the present study proposes two peridynamic contact models for two-dimensional (2D) and three-dimensional (3D) contact problems based on the micro-beam bond and the Hertz theory, using the relative position of particles and the contact micromoduli to calculate the contact forces. Firstly, the strain is generated on the contact surface when two bodies come into contact. Simultaneously, the contact stiffness is obtained by the Hertz theory, and the contact forces are calculated according to the relationship between the strain of contact surface and the contact stiffness. Next, based on the peridynamic theory, the contact horizon and contact micromoduli are defined for the calculation of contact forces. Finally, the contact micromoduli for 2D and 3D contact problems are derived by equating the contact forces obtained from both theories. Additionally, this study defines the correction coefficients to reduce the calculation error of the proposed models and investigates the influence of different contact horizons on the calculation of contact forces. To validate the effectiveness of the proposed contact models, three numerical examples are compared with the results from the finite element method or experiments. It is demonstrated that the proposed contact models accurately calculate the contact forces and predict the crack propagation.

Research on the influence of flexible wheelset rotation effect on wheel rail contact force

PurposeUnder the high-speed operating conditions, the effects of wheelset elastic deformation on the wheel rail dynamic forces will become more notable compared to the low-speed condition. In order to meet different analysis requirements and selecting appropriate models to analyzing the wheel rail interaction, it is crucial to understand the influence of wheelset flexibility on the wheel-rail dynamics under different speeds and track excitations condition.Design/methodology/approachThe wheel rail contact points solving method and vehicle dynamics equations considering wheelset flexibility in the trajectory body coordinate system were investigated in this paper. As for the wheel-rail contact forces, which is a particular force element in vehicle multibody system, a method for calculating the Jacobian matrix of the wheel-rail contact force is proposed to better couple the wheel-rail contact force calculation with the vehicle dynamics response calculation. Based on the flexible wheelset modeling approach in this paper, two vehicle dynamic models considering the wheelset as both elastic and rigid bodies are established, two kinds of track excitations, namely normal measured track irregularities and short-wave irregularities are used, wheel-rail geometric contact characteristic and wheel-rail contact forces in both time and frequency domains are compared with the two models in order to study the influence of flexible wheelset rotation effect on wheel rail contact force.FindingsUnder normal track irregularity excitations, the amplitudes of vertical, longitudinal and lateral forces computed by the flexible wheelset model are smaller than those of the rigid wheelset model, and the virtual penetration and equivalent contact patch are also slightly smaller. For the flexible wheelset model, the wheel rail longitudinal and lateral creepages will also decrease. The higher the vehicle speed, the larger the differences in wheel-rail forces computed by the flexible and rigid wheelset model. Under track short-wave irregularity excitations, the vertical force amplitude computed by the flexible wheelset is also smaller than that of the rigid wheelset. However, unlike the excitation case of measured track irregularity, under short-wave excitations, for the speed within the range of 200 to 350 km/h, the difference in the amplitude of the vertical force between the flexible and rigid wheelset models gradually decreases as the speed increase. This is partly due to the contribution of wheelset’s elastic vibration under short-wave excitations. For low-frequency wheel-rail force analysis problems at speeds of 350 km/h and above, as well as high-frequency wheel-rail interaction analysis problems under various speed conditions, the flexible wheelset model will give results agrees better with the reality.Originality/valueThis study provides reference for the modeling method of the flexible wheelset and the coupling method of wheel-rail contact force to the vehicle multibody dynamics system. Furthermore, by comparative research, the influence of wheelset flexibility and rotation on wheel-rail dynamic behavior are obtained, which is useful to the application scope of rigid and flexible wheelset models.

A centroid measurement method of large components for active compliant assembly

In the active compliant assembly system of large components, the centroid parameter is an important factor affecting the dynamic model of the attitude adjustment mechanism. This paper proposes a novel automatic centroid measurement method. Firstly, the contact force calculation method between large components during active compliant assembly was described, and the impact of the centroid of large components on the accuracy of the contact force calculation was analyzed. Secondly, based on the principles of torque balance and rigid body rotation, a novel centroid calculation model considering lateral forces is established, and RANSAC is used to optimize the centroid solution. Then, numerical simulation methods were used to analyze the impact of random errors and weighing strategies on the centroid measurement accuracy, and suggestions were given to improve the centroid measurement accuracy. Finally, a compliant assembly system for large component was built in the laboratory, and centroid measurement and accuracy verification experiments were conducted. The experimental results show that the proposed method can significantly improve the accuracy of centroid measurement and can effectively reduce the contact force calculation error during the compliant assembly process of large components.

Meta-contact analysis based on contact theory in three-dimensional discontinuous deformation analysis

Accurate contact analysis between nonspherical blocks, such as polyhedral blocks, is crucial for modeling jointed rock masses. The Contact Theory provides a rigorous foundation for contact detection. However, designing a robust and efficient algorithm that conforms to the original Contact Theory remains challenging. This study introduces the meta-contact concept for polyhedral blocks, which corresponds to the contact cover on the entrance block of the Contact Theory. Using the meta-contact concept, contact type identification, determination of contact points and plans, and contact force calculation can be formulated while retaining the advantages of Contact Theory in dealing with irregular blocks with small edges and large penetration.

Variable Time-Step Physics Engine with Continuous Compliance Contact Model for Optimal Robotic Grinding Trajectory Planning.

In the transition from virtual environments to real-world applications, the role of physics engines is crucial for accurately emulating and representing systems. To address the prevalent issue of inaccurate simulations, this paper introduces a novel physics engine uniquely designed with a compliant contact model designed for robotic grinding. It features continuous and variable time-step simulations, emphasizing accurate contact force calculations during object collision. Firstly, the engine derives dynamic equations considering spring stiffness, damping coefficients, coefficients of restitution, and external forces. This facilitates the effective determination of dynamic parameters such as contact force, acceleration, velocity, and position throughout penetration processes continuously. Secondly, the approach utilizes effective inertia in developing the contact model, which is designed for multi-jointed robots through pose transformation. The proposed physics engine effectively captures energy conversion in scenarios with convex contact surface shapes through the application of spring dampers during collisions. Finally, the reliability of the contact solver in the simulation was verified through bouncing ball experiments and robotic grinding experiments under different coefficients of restitution. These experiments effectively recorded the continuous variations in parameters, such as contact force, verifying the integral stability of the system. In summary, this article advances physics engine technology beyond current geometrically constrained contact solutions, enhancing the accuracy of simulations and modeling in virtual environments. This is particularly significant in scenarios wherein there are constant changes in the outside world, such as robotic grinding tasks.

An Improved Contact Force Model of Polyhedral Elements for the Discrete Element Method

The discrete element method (DEM) serves as a robust tool for simulating the mechanical behavior of granular materials. The accuracy of the DEM simulation is determined by the calculation of contact forces between elements. Compared to spherical elements, the contact modes of polyhedral elements are more complex. In this work, the contact force between polyhedral elements under different contact modes was investigated by experiment. Based on the experimental results, the normal stiffness coefficient in the Cundall’s contact model was modified. The improved contact force model was then applied in the DEM simulation and validated by means of comparison with the results of packing experiments. The research results demonstrate that the improved contact force model can be effectively applied to the simulation of different contact modes between polyhedral elements. The results of the packing experiment highly coincide with the results of the DEM simulation, which confirms the accuracy and reliability of the improved contact force model.

Efficient implementation on accuracy improvement of the two-dimensional node-to-segment contact approach for explicit dynamic analysis

The penalty-method-based node-to-segment (NTS) approach is widely employed in the explicit dynamic analysis owing to its computational efficiency and implementation simplicity. However, the classical approach does not pass the contact patch test and results in severe inaccuracies. This study attempts the accuracy enhancement of an explicit dynamic contact analysis with minimum efficiency loss using the NTS algorithm with the modified area regularization technique (NTS-mAR). The computational procedure is compared to an allied modified penalty-method-based NTS approach, i.e., the virtual node-to-segment algorithm passing the patch test (VTS-PPT). Then, an extension to an explicit dynamic analysis framework is attempted, wherein the speed of the contact force calculation significantly influences the overall computational efficiency. The cost of the remaining computation was minimized by employing a lumped mass matrix and a one-point integration rule for the internal force. Elastoplasticity was considered to extend its application. The accuracy improvement compared to the classical one-pass NTS approach was similar for the modified approaches. The VTS-PPT approach requires more than twice the cost of contact force estimation compared with the classical one-pass NTS approach. In contrast, NTS-mAR approach induces a cost increase from 6 to 36% that of classical one-pass NTS approaches. For the given examples, the NTS-mAR approach is beneficial when an improvement in accuracy is desired with minimum efficiency loss.

Influence of two kinds of clearance joints on the dynamics of planar mechanical system based on a modified contact force model

This study takes the slider-crank mechanism with revolute joint and translational joint as the research object and studies the contact force model of the clearance joint and the influence of the hybrid clearance joints on the nonlinear dynamic behavior of the mechanism. A modified contact force model is established based on the simplified elastic oscillator model, which can be used as a normal force in clearance joint. In the new contact force model, the component n of the indentation depth can be arbitrarily selected and it can support the calculation of contact force for both fully elastic recovery, non-elastic recovery and fully inelastic recovery. Based on the LuGre friction model, the tangential friction model of the clearance joint is given. Thus, the normal force and tangential force during the dynamic contact of the clearance joint are formed. Combining Lagrange’s equations of the first kind with the modified normal force and tangential friction force, the dynamic equations of the multi-body system with clearance joints are established. The Baumgarte stabilization method is used to improve the numerical stability. The correctness of the dynamic prediction model in the mechanism with clearance joint is verified by experiment. The dynamic analysis of the slider-crank mechanism with mixed clearance joints shows that the revolute clearance joint has a greater influence on the mechanism than the translational clearance, and the revolute clearance joint plays a leading role in the dynamic response.

Helical buckling of coiled tubing with initial bending curvature in three-dimensional curved wellbores

In this paper, a new buckling model of coiled tubing (CT) in a three-dimensional curved wellbore is established, considering the influence of the curvature and toolface angle of the wellbore and the initial CT bending curvature. First, with the wellbore curvature and toolface angle, the CT buckling configuration is based on the lower or outer curved side of the wellbore. When the initial CT bending curvature is less than the critical bending curvature, the initial CT configuration in the wellbore undergoes sinusoidal buckling; otherwise, the CT undergoes helical buckling. Second, the buckling model is solved by the energy method. After solving this buckling model, the critical CT helical buckling load, the critical bending curvature, and the quantitative relationship between the axial load and pitch can be obtained. Third, the CT contact force calculation method under a helical buckling configuration in a three-dimensional curved wellbore is given by combining the buckling model and the beam-column model. The influence of the initial bending curvature is also considered. Finally, the calculated values of the buckling model and contact force calculation method were compared with existing experimental data and published models to validate the proposed model. Therefore, the methods presented in this paper can be used to not only predict helical CT buckling but also optimize downhole engineering design parameters considering the influence of the initial bending curvature in more realistic drilling wellbore conditions.

Dynamic Investigations on the Wear Behavior of a 3D Revolute Joint Considering Time-Varying Contact Stiffness: Simulation and Experiment

The existence of the relative radial and axial movements of a revolute joint’s journal and bearing is widely known. The three-dimensional (3D) revolute joint model considers relative radial and axial clearances; therefore, the freedoms of motion and contact scenarios are more realistic than those of the two-dimensional model. This paper proposes a wear model that integrates the modeling of a 3D revolute clearance joint and the contact force and wear depth calculations. Time-varying contact stiffness is first considered in the contact force model. Also, a cycle-update wear depth calculation strategy is presented. A digital image correlation (DIC) non-contact measurement and a cylindricity test are conducted. The measurement results are compared with the numerical simulation, and the proposed model’s correctness and the wear depth calculation strategy are verified. The results show that the wear amount distribution on the bearing’s inner surface is uneven in the axial and radial directions due to the journal’s stochastic oscillations. The maximum wear depth locates where at the bearing’s edges the motion direction of the follower shifts. These findings help to seek the revolute joints’ wear-prone parts and enhance their durability and reliability through improved design.

Deformation and Force Chain of Two-Dimensional Granular Systems under Continuous Loading.

A continuous loading experiment of a two-dimensional granular system was carried out and the experimental data were obtained by digital image correlation (DIC). The deformation field of the granular system and the changing laws of the deflection angle and coordination number of the granules on force chains with time were obtained. Based on the granule element method, the quantitative calculation of contact force was realized, and the internal force chains of the granular system were identified. The effects of contact force between granules and mechanical parameters on the evolution of force chains in a two-dimensional granular system under line loads were analyzed. The formation, evolution, and reconstruction of force chains in a granular system during loading, as well as the influence of the force chain network evolution on the macroscopic mechanical properties of granules were discussed. The experimental results indicated that the evolution of force chains was directly related to the number, geometric properties, and permutation distribution of granules in direct contact with the external load.

A DEM-based model for predicting worn surface morphology of ductile metal target by single-particle erosion via energy conversion and geometric reconstruction

The DEM (Discrete Element Method) can simulate target wear due to erosion by impacts of a number of particles. Predicting worn morphology of target by single-particle erosion is the basis of that by multiple-particle erosion. However, the current DEM prediction models for single-particle erosion are aimed at total volume loss rather than worn morphology of target. A DEM-based model for predicting crater-shaped morphology of target by single-particle erosion has been developed depending on energy conversion and geometric reconstruction. In our model, target is represented by a shell element in contact detection, while is taken as a statistically isotropic rough surface covered with deformable asperities in contact force calculation. Then, a modified contact stiffness formula is presented for addressing the elastic-plastic contact characteristic of the two contact objects during erosion. In prediction, volume losses of target due to deformation damage and cutting removal are separately measured based on normal and tangential collision energies using DEM, and the crater-shaped morphology is predicted via geometric reconstruction. Through numerical tests, our model has been verified to be an effective model for predicting the crater-shaped morphology of target by single-particle erosion in response to different impingement angles and at least two different target materials.

Increase in lateral contact force in the tibiotalar joint during walking in flatfoot patients with reduced stiffness of the spring ligament

Foot deformities in patients with flexible flatfeet, such as the flattened medial arch and hindfoot valgus, affect the force distribution around the tibiotalar joint during walking and increase the risk of secondary injuries. In this study, we developed a multi-segment foot model that could calculate the dynamics around the tibiotalar joint and investigated the difference in the kinetics between normal feet and feet with flatfoot. Ten participants with normal feet and ten with flexible flatfoot were enrolled in the study. The body kinematics, ground reaction force, and foot pressure of the participants were recorded during walking. A five-segment foot model was developed to calculate contact forces in the tibiotalar joint. A flatfoot model was developed by modifying the stiffness of the spring ligaments of a normal foot model. Ground reaction force was applied to the plantar surface of the foot models. The foot models were attached to a full-body musculoskeletal model to conduct inverse dynamic simulations of walking. Participants with flatfoot had significantly greater lateral contact force (1.19 BW vs. 0.80 BW) and more posteriorly located center of pressure (33.7 % vs. 46.6 %) in the tibiotalar joint than those with normal feet (p < 0.05). The average and peak posterior tibialis muscle forces were significantly larger in participants with flatfoot than in those with normal feet (3.06 BW vs. 2.22 BW; 4.52 BW vs. 3.33 BW). The altered mechanics may influence the risk of arthritis.

Subject-specific tribo-contact conditions in total knee replacements: a simulation framework across scales.

Fundamental knowledge about invivo kinematics and contact conditions at the articulating interfaces of total knee replacements are essential for predicting and optimizing their behavior and durability. However, the prevailing motions and contact stresses in total knee replacements cannot be precisely determined using conventional invivo measurement methods. Insilico modeling, in turn, allows for a prediction of the loads, velocities, deformations, stress, and lubrication conditions across the scales during gait. Within the scope of this paper, we therefore combine musculoskeletal modeling with tribo-contact modeling. In the first step, we compute contact forces and sliding velocities by means of inverse dynamics approach and force-dependent kinematic solver based upon experimental gait data, revealing contact forces during healthy/physiological gait of young subjects. In a second step, the derived data are employed as input data for an elastohydrodynamic model based upon the finite element method full-system approach taking into account elastic deformation, the synovial fluid's hydrodynamics as well as mixed lubrication to predict and discuss the subject-specific pressure and lubrication conditions.

Hybrid PD-DEM approach for modeling surface erosion by particles impact

Peridynamics (PD) theory is a promising technique for modeling solids with discontinuities. Short-range repulsive force models are commonly employed in PD impact event simulations. Despite their extensive usage, short-range force models do not take damping, friction, and tangential force components into account and hence are unable to effectively describe energy dissipation, leading to uncertainty in the calculation of contact forces. However, the accuracy of impact simulations using alternate contact models has not been extensively investigated in the context of PD impact simulations. The Discrete Element Method (DEM) has been proven to be the most reliable and effective approach to model collision processes between distinct solid objects. This work presents, a particle-based hybrid PD-DEM model to accurately predict the particle impact forces and the resulting damage to the target material. The present model brings together the unique capabilities of PD and DEM and has the potential to make use of the various DEM contact laws, which allow the development and adjustment of relevant contact forces in the normal and tangential directions. Furthermore, damping effects, friction, and intra-particle stiffness are incorporated into the simulations through DEM. The proposed method has been used for modeling material failure after being validated and verified for the contact parameters during the impact process. The predicted damage patterns and resulting material loss demonstrate good agreement with the experimental results reported in the literature.

A GPU based Hybrid Material point and Discrete element method (MPDEM) algorithm and validation

The interaction between granular material and rigid bodies or complex boundaries are common in geotechnical engineering and industry. A hybrid MPDEM algorithm is proposed to simulate these kinds of problems. It is combined with the ability for solving granular material in MPM and the advantages of contact detection in DEM, which can be applied to solve sliding and impacting behavior of rigid blocks. Furthermore, the coupling algorithm based on GPU can be adopted for large-scale simulation to improve the calculation efficiency which has powerful advantages in geologic hazards simulation with high efficiency. To validate the algorithm, the sliding friction and impact effects are conducted to check the contact detection and force calculation for interaction between granular and block or boundary. Besides, the simulation results of granular impacting on the blocks are in good agreement with the experiment, which represents the feasibility and ability of algorithm to solve the interaction with granular materials and rigid-blocks or boundary systems. Through efficiency analysis, it is concluded that the computational performance is higher with the increase in particle number, compared with the pure MPM contact method. Additionally, the Tesla V100 card is a better choice for large-scale (particle number more than 1 million) simulations.