Force Balance Equation Research Articles

Geometrically nonlinear analysis of composite beams based on global–local superposition

A composite beam finite element is designed to capture through-thickness effects, specifically normal stress and strain and transverse shear, in the context of geometrically nonlinear analyses. The starting point for the formulation is a similar element already proposed for linear analyzes based on a global–local superposition approach, where local functions are defined in each layer of the laminate, and global functions are defined along the thickness. The consistency of the kinematic hypotheses is guaranteed by imposing the continuity equations of displacements through the thickness, the force balance equations along the thickness, directly or indirectly, by imposing the continuity of transverse stresses, and by applying the boundary conditions on the lower and upper surfaces of the elements. In the context of nonlinear analyzes, the imposition of continuity of displacements is straightforward. However, the continuity of the transverse stresses needs to be carefully imposed, as the relevant stresses are the second order Piola-Kirchhoff stresses and the strains are the Green-Lagrange strains, consistent with the total Lagrangian approach used. The constitutive equations are written in incremental form and a detailed analysis is conducted to ensure that the stresses and strains involved are physically consistent across the different reference frames employed. In order to assess the accuracy of the numerical model implemented, a unique semi-analytical technique is developed to obtain the response of asymmetrical laminated beams under compression.

Finite element analysis method for the sliding cable analysis: The sliding variable method

A sliding cable structure is typically characterized by uncertainty of sliding and complexity of friction at the cable-strut joints, which makes accurate and efficient analysis difficult. In this study, we have proposed the sliding variable method, to simulate the cable sliding process considering friction equivalently and analyze the internal force and deformation response at the equilibrium state quickly and accurately. The sliding variable method takes the sliding vector of the cable at cable-strut joints as the basic variable, extracts the slippage stiffness matrix and establishes the cable force balance equation to solve the sliding vector, and simulates the change of cable original length and cable force by applying equivalent temperature load. In this paper, first, the basic principles and calculation procedures of the sliding variable method without considering friction, are discussed in detail to simulate frictionless slip of continuous cable, using with roller-type cable-stayed joints. as an example. On this basis, a calculation model incorporating the effects of friction is analyzed, a method to judge the sliding situation and balance relation of cable-forces is revealed, and the calculation principle and steps of the method are further deduced. Meanwhile, choosing a multi-joint cable–pulleys model as an example, the accuracy and rationality of the sliding variable method with and without considering friction are verified respectively. This method requires a conventional finite element calculation and second calculation alternately. Finally, to improve the practicability and computational efficiency of the sliding variable method, a macro document was compiled using the secondary development function of ANSYS parametric programming language (APDL) according to the basic framework of the sliding variable method. The frictional sliding behaviors in two typical engineering cases, a cable girder and a prestressed fish-belly girder on a foundation pit, are analyzed by using the compiled macro document in finite element program. The results show that the compiled macro document can be both accurate and efficient in calculating and analyzing cable sliding problems and can be easily popularized and applied in engineering practices.

Neural force functional for non-equilibrium many-body colloidal systems

Abstract We combine power functional theory and machine learning to study non-equilibrium overdamped many-body systems of colloidal particles at the level of one-body fields. We first sample in steady state the one-body fields relevant for the dynamics from computer simulations of Brownian particles under the influence of randomly generated external fields. A neural network is then trained with this data to represent locally in space the formally exact functional mapping from the one-body density and velocity profiles to the one-body internal force field. The trained network is used to analyse the non-equilibrium superadiabatic force field and the transport coefficients such as shear and bulk viscosities. Due to the local learning approach, the network can be applied to systems much larger than the original simulation box in which the one-body fields are sampled. Complemented with the exact non-equilibrium one-body force balance equation and a continuity equation, the network yields viable predictions of the dynamics in time-dependent situations. Even though training is based on steady states only, the predicted dynamics is in good agreement with simulation results. A neural dynamical density functional theory can be straightforwardly implemented as a limiting case in which the internal force field is that of an equilibrium system. The framework is general and directly applicable to other many-body systems of interacting particles following Brownian dynamics.

Influence of axial clearance on load distribution in double supported tapered roller bearings of wind turbine main shaft

The operational safety and stability of wind turbines are highly determined by the characteristics of main bearings, which are influenced by the axial clearance. In this paper, for the arrangement type of double-supported tapered roller bearings on the wind turbine main shaft, the actual clearance distribution assumption is established, and a new method of calculating the internal load distribution taking into account the axial clearance is proposed. Firstly, the mathematical relationship between the clearance of tapered roller bearings and ring displacement, contact deformation, and load interval is described. Secondly, the load components of main bearings are calculated by force balance equation and load-displacement condition. Finally, a mathematical model of main bearings including axial clearance is established, and it is numerically solved to obtain the internal load distribution of the bearings. The calculation results of a large megawatt wind turbine example show that: the axial load of both main bearings is negatively correlated with the sum of axial clearances; the number of loaded rolling elements is negatively correlated with the sum of axial clearances; and there exists an optimal value of the sum of axial clearances for main bearings to reach the optimal load carrying state.

Formulating strain-based quadrilateral membrane finite elements with drilling rotations

Membrane finite elements with drilling degrees of freedom have sparked interest in many research works since they can be conveniently combined with plates to form shell elements. This study presents two non-conforming strain-based four-node quadrilateral membrane elements, SBQ13 and SBQ13E, for static analysis. SBQ13 partially satisfies the equilibrium equations, while SBQ13E completely fulfils the force balance equations. Both elements carry drilling rotations at each node. One difficulty when formulating quadrilateral elements is the singularity in the transformation matrix, which is addressed in this study by utilising the properties of singular matrices. Both quadrilateral elements were tested in several benchmark problems. It has been found that both elements passed the higher-order patch test but failed the constant stress patch test. Nevertheless, SBQ13 produced accurate responses in most numerical tests, but SBQ13E unreasonably overestimated the solution. Solving the eigenvalue problem revealed that the SBQ13E element has a near-zero energy deformation mode, which might explain the anomaly. Although fulfilling equilibrium does not always enhance solution accuracy, it is essential to overcome volumetric locking. Apart from the newly developed elements, this paper presents several new ideas that may apply to strain-based element formulations.

Predicting bond line thickness of polymeric thermal interface materials based on the rheological properties

Bond line thickness (BLT) is an essential parameter of thermal conductive composite gels as the thermal interface material (TIM). Extensive research has been performed on designing next-generation TIMs with high thermal conductivity; however, it remains elusive how to link laboratory measurements and the theoretically predicted BLT. Here, we propose a new model to estimate BLT based on a rheological property, in which the TIM is assumed to be a simple power law fluid. To avoid the unrealistic situation of the BLT tending to zero, we introduce a decaying exponential function to describe the influence from fillers during the lid attach process and rebuild the force balance equation. Compared with previously reported models, the theoretical prediction BLT based on the proposed model is in good agreement with the experimental data. Our model has a guiding significance in predicting the BLT, which may help to optimize the thermal performance of TIMs.

A gyrokinetic simulation model for 2D equilibrium potential in the scrape-off layer of a field-reversed configuration

The equilibrium potential structure in the scrape-off layer (SOL) of the field-reversed configuration (FRC) can be affected by the penetration of edge biasing applied at the divertor ends. The primary focus of the paper is to establish a formulation that accurately captures both parallel and radial variations of the two-dimensional (2D) potential in SOL. The formulation mainly describes a quasi-neutral plasma with a logical sheath boundary. A full-f gyrokinetic ion model and a massless electron model are implemented in the GTC-X code to solve for the self-consistent equilibrium potential, given fixed radial potential profiles at the boundaries. The first essential point of this 2D model lies in its ability to couple radial and parallel dynamics stemming from resistive currents and drag force on ions. The model successfully recovers the fluid force balance and continuity equations. These collisional effects on 2D potential mainly appear through the density profile changes, modifying the potential through electron pressure gradient. This means an accurate prescription of electron density and temperature profiles is important in predicting the potential structure in the FRC SOL. The Debye sheath potential and the potential profiles applied at the boundaries can be additional factors contributing to the 2D variations in SOL. This comprehensive full-f scheme holds promise for future investigations into turbulent transport in the presence of the self-consistent 2D potential together with the non-Maxwellian distributions and open boundary conditions in the FRC SOL.

Heavy-load Nonapod: A novel flexible redundant parallel kinematic machine for multi-DoF forming process

The high-performance multi-DoF forming process (MDFP) necessitates a 6-DoF forming machine tool with high normal and lateral stiffness to bear large normal and lateral forming force of millions of Newton (MN). However, the payload of parallel kinematic machine (PKM) is generally limited to thousands of Newton (kN), which restricts its application in MDFP. Therefore, this paper aims to develop a novel heavy load PKM with high stiffness for MDFP. To maximise the normal stiffness, a 6-PSS PKM with zero base angle and horizontal driver is proposed. Further, the inner force transfer model of 6-PSS PKM is established, indicating that the normal stiffness will be maximised when the link force approaches to be vertical. Consequently, a design criterion for maximising normal stiffness, i.e., the root mean square error (RMSE) for horizontal projection of all links should be minimised, is established. To maximise the lateral stiffness, general force balance equations of 6-PSS PKM are derived, indicating that lateral force can cause unintended negative force of links, significantly reducing the lateral stiffness. Thus, a novel auxiliary 3-SPS configuration is employed to provide additional force system to mitigate this negative force via hydraulic links. Correspondingly, a design criterion for maximising lateral stiffness, i.e., all link force should remain positive, is proposed. By combining aforementioned design criterion and kinetostatic models, a near-singular 6-PSS PKM with maximising normal stiffness is achieved, and dimension parameters of 3-SPS PKM with maximising lateral stiffness are optimised. On this basis, a novel flexible redundant 6-PSS/3-SPS PKM with both high normal and lateral stiffness is proposed, and a novel heavy load Nonapod with payload of 8 MN and payload-mass ratio of 40 is developed, showing good stiffness performance. The plastic deformation mechanisms of multi-DoF formed aviation bevel gear are revealed, and experimentally formed aviation bevel gear in the new Nonapod achieves good accuracy, microstructure and mechanical performance. This work provides a new methodology for synthesis of heavy load PKM with high normal and lateral stiffness, and has significant application prospect in PKM under heavy load working condition.

Effect of Indentation at Rolling Body Surfaces on the Film Formation in Elastohydrodynamic Lubrication Contact under Dynamic Loads

Vibration is one of the phenomena that can occur during the operation of roller bearing which can lead to dynamic loads being subjected to the contact and thus, affect the film formation in elastohydrodynamic (EHD) contacts. On the other hand, surface textures such as artificial indentations and grooves greatly affect the film formation as well as the corresponding pressure distribution in the contact area, since the depths of the textures are usually much greater than the film thickness in the contact. This paper investigates the influence of an artificial indentation located on the rolling element, which passes through a point contact under dynamic loads by means of numerical simulations. Solutions to the Reynolds equation are performed and calculations of the elastic deformation equation, force balance equation and lubricant properties equations to show the effects of both indentation passing through the contact and a sinusoidal dynamic load on the film thickness as well as pressure profile of EHD point contact. Moreover, the effect of frequency and amplitude excitation of the dynamic load on the film formation was investigated. The results revealed that the artificial indentation under sinusoidal dynamic load of EHD point contact induced a significant effect on the thickness of the film formation and pressure distribution.

Sensitivity analysis of parameters impacting the performance of an energy ship using airborne wind energy

This study assesses the impact of different model parameters on the power performance of energy ship systems using point-mass models, focusing on two operational modes. The first mode involves a kite operating in a pumping mode with adjustable tether lengths, aided by additional propellers for ship propulsion. It utilizes a quasi-steady kite model coupled with ship motion’s drag force and velocity triangles. The second mode features a kite pulling the ship, while underwater hydraulic turbines harness energy from ship velocity. Derived from a sail-based model, calculations deploy the force balance equation in line with ship motion, considering the drag forces of the ship and hydraulic turbines, alongside the propulsive force resulting from the kite’s interaction with the ship. The investigated parameters influencing the power performance include kite surface, ship length, wind speed, the angle between the ship’s motion direction and wind direction, and, for the pumping mode, ship speed. The power production of the propulsion mode is generally lower, with differences of at least 72%, depending on the combination of considered parameters.

Stage–discharge prediction in the multi-stage ice-covered compound channel

The multi-stage compound channel, which is a common pattern in natural alluvial rivers and the regulation projects of urban rivers, inevitably freezes in winter when it is situated in cold northern areas with high latitudes. Given that ascertaining the stage–discharge relationship for rivers is the foundation for the development of flood control schemes and water resources management, this study concentrates on proposing an analytical model for predicting the stage–discharge curves of multi-stage ice-covered compound channels. In deducing the analytical model, the cross section of the channel is first segmented into several homogeneous subregions that can be grouped into seven categories according to the geometric characteristics. Through analyzing the momentum transfer between adjacent subregions, the force balance equation for each subregion is then established to get the bulk mean velocity for the corresponding subregion, thereby obtaining the discharge by solving a tridiagonal matrix. Subsequently, measurements from two-stage and three-stage ice-covered compound channel experiments and three sets of experimental data from the literature are used to validate the performance of the proposed model. Good agreement between the predictions and the measured data suggests that the deduced model can accurately estimate the discharge for the multi-stage ice-covered compound channels when the flow depth is given. Finally, sensitivity analysis indicates that Manning's roughness coefficient of the channel bed has a more pronounced impact on the stage–discharge relationship than that of the ice cover. Moreover, when compared to the two-stage ice-covered compound channel, the multi-stage ice-covered compound channel offers greater potential for water resource utilization.

On the Effect of Self-Generated EM-Centrifugal Force within a Closed Circular-Loop-Electric Current and a New Type of Lamb-Shift

Force of electromagnetic interaction between a small element of current and the magnetic induction due to other element of current diameter-apart flowing within the same circular loop has been calculated in three different cases producing three different equations when applied to find out the expression for force-balance equations in these three different cases.

Theoretical study of electron mobility in AlGaN back-barrier Al2O3/InAlN/GaN heterostructures under optical phonon scattering

Considering the strong built-in electric field caused by polarization in wurtzite nitride heterostructures, the energy levels and wave functions of the two-dimensional electrons in AlyGa1−yN back-barrier Al2O3/InxAl1−xN/GaN heterostructures are calculated using the finite element difference method. The dispersion relations and electrostatic potentials of optical phonons are obtained by the transfer matrix method. The electron mobility under optical phonon scattering is studied based on the theory of Lei–Ting force balance equation. The influences of AlyGa1−yN back barrier, ternary mixed crystal effect, and size effect are also analyzed by comparing with the Al2O3/InxAl1−xN/GaN heterostructure without back barrier. It is found that the introduction of a back barrier can attenuate the effect of gate-insulating layer, but enhance the effect of barrier and channel layers on electronic states. Then, the interaction between electrons and optical phonons is weakened, so the electron mobility in AlyGa1−yN back-barrier Al2O3/InxAl1−xN/GaN heterostructure is higher than that in Al2O3/InxAl1−xN/GaN structure under the same conditions. It is also found that using a thinner AlyGa1−yN film with y = 0.25 as the back-barrier layer of Al2O3/InxAl1−xN/GaN heterostructure is more conducive to improving 2DEG mobility. These conclusions can provide references for the preparation of InxAl1−xN/GaN heterojunction devices.

Experimental and theoretical study of single bubble formation at the nozzle in a liquid oxygen crossflow

In this paper, the condensation bubbles generated by the mixing and condensation of gaseous and liquid oxygen crossflow experimentally and theoretically. The mixing and condensation experiment of the bubbles was carried out, employing gaseous and liquid oxygen as working fluids in vertical tubes. Three successive flow patterns were observed: initial gas plume, condensation oscillation, and separation condensation. The corresponding characteristics include gas mass aggregation, bubble surface smoothness and wrinkling alternately, and bubble disappearance after separation and condensation. Then, a Nusselt number (Nu) heat transfer correlation formula is fitted with a corresponding fitting error in the range of −31% ∼ 35%. In addition, bubble force balance and conservation equations were employed to establish a mixing condensation bubble model suitable for gaseous and liquid oxygen in vertical tubes, and the error between theoretical and experimental results was in the range of −25% ∼ 20%. Furthermore, the effects of working conditions and forces on bubble condensation length has been analyzed. The results showed that increasing the liquid oxygen flow rate, reducing the gas oxygen volume flow rate, or appropriately reducing the gas inlet diameter led to reducing the bubble condensation length. It was shown that the bubble drag and shear lift forces affect the condensation length mostly.

Investigation of thermal stress effects on subthreshold conduction in nanoscale p-FinFET from Multiphysics perspective

The rising temperature due to a self-heating or thermal environment not only degrades the subthreshold performance but also intensifies thermal stress, posing a severe challenge to device performance and reliability design. The thermal stress effects on the ON-state performance of the p-type fin field-effect transistor were previously studied. However, as far as we know, how thermal stress affects its subthreshold conduction remains unclear, which is studied in this manuscript. The impact of thermal stress due to the self-heating of adjacent devices on subthreshold conduction is investigated by solving the quantum transport, thermal conduction, and force balance equations for ballistic transport and dissipative transport with phonon scattering. Then, the thermal stress effects at different ambient temperatures are further discussed and analyzed. The simulation results show that the OFF-state leakage current can be reduced by thermal stress, even up to 9.28% for the (110)/[001] device operating at an ambient temperature of 550 K, and its reduction is the comprehensive result of the thermal stress effects on the band structure, potential profile, carrier distribution, and source-to-drain tunneling. In addition, the thermal stress has no significant effects on subthreshold swing although it can change the magnitude of the subthreshold current. Moreover, the effect of thermal stress on subthreshold conduction is highly dependent on the thermal environment of the device and the crystal orientation of the channel semiconductor material.

Extended gravitational vortex without dark matter

This paper analyzes, within the extended gravitoelectromagnetic (GEM) formulation, the equilibrium of a large scale gravitational system formed by rotating dust. The force balance equation gives the rotation velocity in terms of the GEM fields. Boundary conditions for the fields are introduced using Helmholtz’s decomposition and the virtual casing principle. Hydro-gravitomagnetic Cauchy invariance is implemented to relate the fluid and gravitomagnetic field vorticities. An energy conservation equation gives the rotation velocity in terms of the gravitational field and respective boundary values. A detailed solution is calculated for the case of rotating oblate spheroids. The equilibrium is in the form of a sheared rotational vortex, without introducing dark matter. The results are consistent with the Tully–Fisher relation and the Virginia Trimble correlations.

Modelling principal stress orientations in the Arabian Plate using plate velocities

Abstract We model maximum principal horizontal stress orientations in the Arabian Plate using a 3D finite element approach in conjunction with plate velocities. To capture the impact of geometry and tectonics, the model considers an accurate plate boundary shape and associated deformations. Three primary geological units represent plate architecture: sedimentary cover, crust and upper mantle. The mesh resolution varies to capture important geometrical features. Subsequently, we calculate the stress field using the force balance equation. Displacement boundary conditions are evaluated as accumulative deformation. NE–SW maximum principal horizontal stress (S Hmax ) azimuths dominate in northeastern Saudi Arabia and Kuwait, whilst NW–SE to NNW–SSE define the Dead Sea area. The Red Sea Basin and Saudi Arabia's interior is characterized by north–south S Hmax azimuths. Iraq's western area shows azimuths from NNW–SSE to NW–SE due to the collision at the Zagros Mountain Range, but changes to NE–SW in the east at the Zagros fold-and-thrust belt. An extensive literature review reveals publicly available S Hmax azimuth data which augment the sparse records compiled in the World Stress Map database. Our simulated S Hmax azimuths are consistent with these data. The results further corroborate ongoing tectonic processes, deepen our understanding of in situ stress variation drivers and inform current elastic deformation mechanisms in the Arabian Plate.

Simplified method for evaluating tunnel response induced by a new tunnel excavation underneath

AbstractTo estimate the tunnel response induced by a new tunnel excavation underneath, theoretical solutions are proposed in this study. The overlying tunnel is idealized as an infinite Timoshenko beam resting on the Kerr foundation model, then the vertical force balance equation is established. The unloading stress can be expressed as Fourier cosine series and a theoretical solution can be derived. The effectiveness of the proposed method is verified by a centrifuge test and an actual field measurement, which are extracted from previous investigations. Calculation results from the proposed method show good accord to centrifuge test results and field‐measured data. Meantime the predictions given by the proposed method are more accurate to estimate tunnel response compared with the previous method. Parametric analysis indicates that the volume loss ratio and the vertical clearance between two tunnels, are both significantly alleviate the tunnel response, but the effect of tunnel shear stiffness is slight. The theoretical solutions can be adopted to predict the overlying tunnel response induced by tunneling underneath in real projects.

The theory of kinetic effects on resistive wall mode stability in tokamaks

Tokamak fusion plasmas benefit from high pressures but are then susceptible to modes of instability. These magnetohydrodynamic (MHD) modes are macroscopic distortions of the plasma, but certain collective motions of individual particles can provide stabilizing effects opposing them. The presence of a resistive wall slows the mode growth, converting a kink to a resistive wall mode (RWM). A kinetic MHD model includes Maxwell's equations, ideal MHD constraints, and kinetic effects included through the pressure tensor, calculated with the perturbed drift-kinetic distribution function of the particles. The kinetic stabilizing effects on the RWM arise through resonances between the plasma rotation and particle drift motions: precession, bounce, and transit. A match between particle motions and the mode allows efficient transfer of energy that would otherwise drive the growth of the mode, thus damping the growth. The first approach to calculating RWM stability is to write a set of equations for the complex mode frequency in terms of known quantities and then to solve the system. The “energy principle” approach, which has the advantage of clarity in distinguishing the various stabilizing and destabilizing effects, is to change the force balance equation into an equation in terms of changes of kinetic and potential energies, and then to write a dispersion relation for the mode frequency in terms of those quantities. These methods have been used in various benchmarked codes to calculate kinetic effects on RWM stability. The theory has illuminated the important roles of plasma rotation, energetic particles, and collisions in RWM stability.

Stability of Individual Phases in the Elastic Matrix of a Composite

A methodology for analytical stability assessment of individual compressed flexible phases of discretely heterogeneous composite elements of engineering structures is proposed to ensure their reliable safe operation and minimise the consequences of emergencies. Based on the energy balance equation for the proportional forces of a normal resistance of the medium to the phase displacement, expressions are obtained for determining the critical force, taking into account the possible initial curvature (non-rectilinearity) of a single phase in the elastic medium of the matrix for three possible cases of stability loss. The approbation of the results of the study on the example of deformation features of reinforcing bars in a compressed reinforced concrete column is presented.