Equivalent Force Research Articles

Legolas 2.0: Improvements and extensions to an MHD spectroscopic framework

We report on recent extensions and improvements to the Legolas code, which is an open-source, finite element-based numerical framework to solve the linearised (magneto)hydrodynamic equations for a three-dimensional force- and thermally balanced state with a nontrivial one-dimensional variation. The standard Fourier modes imposed give rise to a complex, generalised non-Hermitian eigenvalue problem which is solved to quantify all linear wave modes of the given system in either Cartesian or cylindrical geometries. The framework now supports subsystems of the eight linearised MHD equations, allowing for pure hydrodynamic setups, only one-dimensional density/temperature/velocity variations, or the option to treat specific closure relations. We discuss optimisations to the internal datastructure and eigenvalue solvers, showing a considerable performance increase in both execution time and memory usage. Additionally the code now has the capability to fully visualise eigenfunctions associated with given wave modes in multiple dimensions, which we apply to standard Kelvin-Helmholtz and Rayleigh-Taylor instabilities in hydrodynamics, thereby providing convincing links between linear stability analysis and the onset of non-linear phenomena.

Seabed seismic analysis using exact responses of solid-liquid free field for 2D oblique incident waves by finite/infinite element method

The seabed seismic analysis of a viscoelastic half-space with overlying liquid under two-dimensional (2D) oblique incident P- or SV-waves is performed by the finite/infinite element method (FIEM). Firstly, the exact solutions of the free-field solid-liquid responses in frequency domain are derived for the waves with arbitrary incident angles, including the critical angles of reflection for SV-waves and of transmission for P- and SV-waves. Next, the equivalent seismic forces to be imposed on the near-field boundaries are computed for the solid and liquid assuming that the earthquake data are available on the solid-liquid interface. Then, the equations of motion for the seabed modeled by the finite/infinite elements are presented with the solid-liquid interface implemented and the key parameters selected. The reliability of the solutions has been verified by comparison with Wolf's in frequency domain, and by inversely computing the input ground motions in time domain. New findings from this paper include: (1) The suppressive frequencies exist in the presence of overlying liquid, for which the vertical solid displacements are reduced for P- and SV-waves both for under-critical cases. (2) Both the horizontal and vertical displacements of the solid surface are reduced by overlying liquid for SV-waves with over-critical incident angles.

Bifurcations from a normally degenerate cycle in forced planar differential equations

Bifurcations from a normally degenerate cycle in forced planar differential equations

Design method and experimental investigation of compression members in a space structure strengthened by folded sleeves

The compression member strengthened by the sleeve can effectively enhance the bearing capacity. In this paper, the calculation equation for the contact force between the core and the outer sleeve is deduced, and a bolt design suggestion of the folded sleeve is provided. The test results show that the peak load of the folded sleeve member is increased by over 55.3%. When the number of bolted regions of the folded sleeve increases, the peak load only slightly increases. There are two failure modes of folded sleeve members, including the global buckling and the end bending of the unrestrained segment of the specimen. The number of bolted regions of the folded sleeve has minimal influence on the failure mode. The performance of the folded sleeve is better than that of the assembled sleeve and similar to that of the seamless sleeve. The Abaqus numerical model of the specimen is established and can effectively simulate the mechanical properties of folded sleeve member. The comparison between the theoretical and numerical simulation results of the contact force shows that the theoretical calculation equation of contact force has high safety. Based on the experimental and numerical simulation results, the safety factor, λ, of the bolt spacing of the folded sleeve member is defined.

Study on aerosol thermophoretic force and thermophoresis deposition model in reactor building

During nuclear reactor severe accident, the input of external passive containment cooling system will result in significant aerosol thermophoresis deposition in reactor building, which plays important factor affecting the safety assessment of nuclear reactor. Ratio of aerosol particles radius and gas molecule mean free path (Kn number) in reactor building is not greater than 2, corresponding to the near-continuous medium zone and transition zone. Firstly the typical thermophoretic force equations for these zones are evaluated. Combined with key influencing factors analysis, the correction factor connected to Kn number is introduced to address the deviation in the prediction for the transition zone of the current equations. The modified thermophoretic force equation obviously improves the underestimation of the original equation and is then used to obtain the thermophoresis deposition coefficient. According to experimental validation, the obtained coefficient is more accurate than the model that is widely employed in analysis codes in predicting the thermophoresis deposition of aerosol particles. Finally, fluid continuity, momentum, energy and particle continuity equations are established with deduced thermophoresis deposition coefficient and solved numerically. The thermophoresis deposition model considering natural convection (CT model) is established by theoretical derivation and fitting with the numerical solutions. By introducing the effects of natural convection on temperature distribution and aerosol concentration distribution the CT model could accurately describe the aerosol thermophoresis deposition behavior under natural convection conditions for Kn number is not greater than 2, which can be applied to aerosol thermophoresis deposition prediction in reactor building.

The Dynamics of a Turning Ship: Mathematical Analysis and Simulation Based on Free Body Diagrams and the Proposal of a Pleometric Index

This study attempts to shed new light on the dynamics of a turning ship using the principles of free body diagrams (FBDs). Unexpectedly, the literature gap is defined by incomplete and flawed FBDs. The method behind this new approach involves the FBD of a turning ship, with all the essential forces included, namely propulsive force, sideward thruster force (producing the initial turning moment), drag force, lift force, centrifugal force, inertial force, and hydrodynamic force couple. From these forces, the force and moment equations are derived. The accelerations are calculated from the force and moment equilibria to simulate the dynamics from input parameters such as mass m, length L, draught D, and fluid density ρ. The turning dynamics are explained in terms of velocities, accelerations, forces, and moments, based on two conditions: flat and steep angles of attack (AoA) and long and short turning radii R. A critical result is the proportionality of lift and centrifugal forces, leading to the proposal of a pleometric index (m·L–2·D–1·ρ–1), which is nonlinearly proportional to the product of AoA and R/L, characterising the dynamics of a turning ship. The FBD approach of this study also identified missing databases required for accurate simulation of turning dynamics, such as drag and lift coefficients of different hull geometries.

Investigating horizontal-axis wind turbine aerodynamics using cascade flows

The simplest aerodynamic model of horizontal-axis wind turbines is the blade element momentum theory, which assumes that the blades behave as airfoils, but a correct two-dimensional representation is an infinite cascade of lifting bodies. This study analyzes the conventional and impulse forms of the forces on cascades of airfoils at spacings and pitch angles typical of wind turbine applications. OpenFOAM software was used to simulate steady, incompressible flow at a Reynolds number of 6×106 through cascades of NACA 0012 airfoils. The force equations agree well (less than 1% error) with the forces determined directly from OpenFOAM for four spacing ratios. We concentrate on the “wake vorticity” term, which is ignored in blade element momentum analysis. At a pitch angle of 90°, this term balances the viscous drag when the angle of attack is zero. At zero pitch, which models the outer region of a wind turbine blade at a high tip speed ratio, the term can account for 27% of the axial thrust when the angle of attack is about 4°. The normal force equation, like the angular momentum equation for wind turbines, has no viscous term, which forces the body drag to contribute to the circulation in the wake. It is shown that the airfoil assumption is conservative in that cascade elements have higher lift-to-drag ratios than airfoils at the same angle of attack. An associated result is that separation occurs at higher angles of attack on a cascade element compared to an airfoil.

Seismic damage characteristics of high arch dams under oblique incidence of SV waves

This study investigates the seismic damage characteristics of high arch dams considering the oblique incidence of seismic waves. Based on the dynamic implicit finite element method, a viscous-spring artificial boundary is applied to consider radiation damping of the infinite foundation, the seismic wave input is converted into an equivalent nodal force of the artificial boundary, and the equivalent nodal loads of the wave input under oblique incidence of the SV wave is deduced. Taking JP Ⅰ arch dam as an example, combined with the concrete damaged plasticity (CDP) model, the nonlinear dynamic response of the dam under 25° oblique incidence of the SV wave is conducted considering the geometrical nonlinearity of contraction joints and the dynamic mechanical properties of concrete materials. Based on the incremental dynamic analysis (IDA) method, the relative displacement and transverse joint opening are selected as the damage measures for fragility analysis of the arch dam. According to the damage distribution from the analysis of a large number of seismic time-history responses, the damage levels of the arch dam are classified. Three limit states of slight damage, moderate damage and severe damage are proposed for the arch dam. The average responses of the 10 earthquakes are considered as the limit values to quantitatively describe the damage of the arch dam. By fitting the results of incremental dynamic analysis, the seismic probabilistic demand model and seismic fragility curves for the two damage measures to predict the probability of the arch dam reaching damage levels under different seismic actions have been established. The fragility curve shows that the probability of severe damage is almost zero under the maximum design earthquake (PGA = 0.1971 g) and less than 5% under the maximum credible earthquake (PGA = 0.2299 g). It is concluded that JP Ⅰ arch dam has a high degree of safety redundancy and fully satisfies the “two levels” design requirements.

Nonlinear nonlocal damped free and forced vibrations of piezoelectric SWCNTs under longitudinal magnetic field due to surface effects using a two steps perturbation method

In the present work, damped free and forced vibrations of single-walled piezoelectric carbon nanotubes under longitudinal magnetic field due to surface effects surrounded on a non-linear viscoelastic medium using the nonlocal Euler-Bernoulli beam theory and multiple time scales method are investigated. Lorentz force equation is used to obtain the vertical force due to the applied voltage to the system. The surface effects as well as a combinational non-linear viscoelastic foundation are considered, and finally, the dynamic equilibrium equations are used, and non-linear equations of motion are extracted. In the following, the Galerkin and multiple time scales methods are used, and finally, analytical solutions are extracted as the non-linear free and forced vibrational responses of the system. The relevant coefficients of the extracted analytical solutions are discovered for two both simple support (S-S) and clamped (C-C) boundary conditions. In the following, , and the effects of the different parameters such as non-local parameter as well as electric-magnetic fields, effect of hardness-linear damping parameters of nonlinear considered viscoelastic foundation, applied magnetic field, base modes for different forms considering surface effects, and etc. will be studied. As one the results of this study, the presence of a non-local parameter has increased the curvature deviation to the right and the stiffening effect. In other words, the non-local parameter is a factor to increase the nonlinear effect of the system. Also, it is predictable that as the load affect position moves away from the center of the single-walled piezoelectric carbon nanotube toward the supports, the amplitude of the dynamic response decreases significantly, and this relative reduction is greater for the C-C boundary condition than for the S-S boundary condition. It is also important to note that the location of the load has no effect on the rate of deviation of the curve peak, and the degree of nonlinearity of the vibrational response of the system.

Acoustic radiation force for analyzing the mechanical stress in ultrasound neuromodulation

Objective. Although recent studies have shown that mechanical stress plays an important role in ultrasound neuromodulation, the magnitude and distribution of the mechanical stress generated in tissues by focused ultrasound transducers have not been adequately examined. Various acoustic radiation force (ARF) equations used in previous studies have been evaluated based on the tissue displacement results and are suitable for estimating the displacement. However, it is unclear whether mechanical stress can be accurately determined. This study evaluates the mechanical stress predicted by various AFR equations and suggests the optimal equation for estimating the mechanical stress in the brain tissue. Approach. In this paper, brain tissue responses are compared through numerical finite element simulations by applying the three most used ARF equations—Reynolds stress force ((RSF)), momentum flux density tensor force, and attenuation force. Three ARF fields obtained from the same pressure field were applied to the linear elastic model to calculate the displacement, mechanical stress, and mean pressure generated inside the tissue. Both the simple pressure field using a single transducer and the complex standing wave pressure field using two transducers were simulated. Main results. For the case using a single transducer, all three ARFs showed similar displacement. However, when comparing the mechanical stress results, only the results using the RSF showed a strong stress tensor at the focal point. For the case of using two transducers, the displacement and stress tensor field of the pattern related to the standing wave were calculated only from the results using the RSF. Significance. The model using RSF equation allows accurate analysis on stress tensor inside the tissue for ultrasound neuromodulation.

Alternative optical phase of the conformable ferromagnetic and antiferromagnetic media

In this manuscript, we describe several types of alternative conformable magnetic curves. We consider the fractional differential geometric tools to investigate how one can deal with both geometric and physical phenomena. In particular, we focus on the α-relatively parallel formulae and alternative scalar conformable quantities. Thus, we aim to determine particular characterizations of the charged particle when its motion is governed by the Lorentz force equations and conformable fractional derivative. Moreover, we study the inextensibility and non-stretching motions of the alternative conformable curves. By doing this, we attempt to obtain both their general and particular evolution equations including ferromagnetic and antiferromagnetic evolution equations. As a result, we find an alternative phase space of the fractionally conformable evoluted ferromagnetic and antiferromagnetic media by computing the anholonomy density and total phase of the system.

Structural Design and Sealing Performance Analysis of a Nanofluidic Self-Heating Unsealing Rubber Cylinder

As a crucial component for temporary blocking of layer segments in the segmental fracturing process, bridge plugs are difficult to unseal by conventional methods and may cause major downhole accidents if not handled properly. In this paper, a nanofluidic self-heating unsealing rubber cylinder is designed, which is equipped with a nanofluidic self-heating unsealing sandwich inside the conventional rubber cylinder, consisting of a nanofluidic system and an annular flexible heater. When unsealing, the nanofluidic self-heating unsealing sandwich is heated by the annular flexible heater, and the nanofluidic system can help the bridge plug rubber cylinder shrink in volume and unseal smoothly by the characteristics of heat shrinkage and cold expansion. The nanofluidic system, consisting of porous carbon with an exceptionally large specific surface area and glycerol, serves as a prime example for filling the sandwich layer, and the design parameters calculation was carried out. The sealing performance of the designed nanofluidic self-heating unsealing rubber cylinder was analyzed based on the Mooney–Rivlin principal structure by finite element modeling. The results show that the maximum contact stress between the nanofluidic self-heating unsealing rubber cylinder and the casing wall increases by 9.73%, the compression distance reduces by 24.47%, and the maximum equivalent force decreases by 12.17% on average compared with a conventional rubber cylinder under the same seating load. The designed nanofluidic self-heating unsealing rubber cylinder can satisfy the requirements of pressure-bearing capacity and sealing performance and performs better than a conventional rubber cylinder.

Numerical investigation on a floating self-reacting resonant wave energy converter

A self-reacting resonant C-shaped wave energy converter (SR-C-WEC) consisting of a pair of resonant sub-C-WECs connected by a rigid frame is proposed, which uses the two sub-C-WECs to provide mutual reaction force to solve the efficiency problem caused by floating WECs being hard to provide enough reaction force on WEC and uses the controllable resonant WEC to solve the efficiency problem caused by the temporal variability of waves. Resonant control and equivalent wave force control models are proposed to maximize the absorbed power. The effect of spacing between two sub-C-WECs, natural period, PTO damping and mooring stiffness to absorbed power is investigated. It is found that SR-C-WEC can achieve resonance at each wave condition and the power generation is maximum when the spacing is equal to half wavelength, which is comparable to the power generation when the frame is fixed. In a real sea condition, when the spacing is adjusted or not, the difference in average annual power is only 8.3%. Therefore, constant spacing can be used to simplify the structure and facilitate the application of SR-C-WEC. It is predicted that SR-C-WEC can reach 3638 effective power generation hours per year in real seas off China.

A coupled SBFETI-BDEs method for solving 2D static and dynamic contact problem with friction

In this study, we propose a coupled approach called the SBFETI-BDEs method for solving the 2D elastic frictional contact problem. The contact system is decomposed into multiple subdomains, similar to the finite-element tearing and interconnecting (FETI) algorithm. Each subdomain can be regarded as a scaled boundary finite element (SBFE), where only the boundaries are discretized. To avoid computing the global contact flexibility matrix using the force method, we consider the generalized inverse of the stiffness matrix for each subdomain as a subcontact-flexibility matrix. We uniformly formulate the contact equations, displacement continuity equations, and the equilibrium equations of force for the floating subdomains, as a B-differentiable equations (BDEs) system, which we solve using the B-differentiable damped Newton-Raphson method that possesses a global convergence property. The proposed method offers increased computational efficiency without compromising accuracy when compared to the original BDEs algorithm without domain decomposition. We present the dynamic scheme of the SBFETI-BDEs method within the framework of the HHT time-integration scheme, where the equilibrium equations of force for the floating subdomains disappear due to their nonsingular equivalent stiffness matrices. Numerical examples are provided to demonstrate the convergence, accuracy, efficiency, and capability in solving engineering problems of the proposed method.

Analytical simulation of vibration demeanor of robotic arm immersed in incompressible dynamic fluid

The use of underwater mechanical arms is extensively considered to be the most suitable tool for performing mechanical operations underwater. These mechanical arms can transport samples and patterns in water environments. A simulation of the fluid flow provoked vibrations in the arm of the robot is presented in this paper. Under the influence of the force generated through the surrounding fluid flow and considering the flexibility of the robotic arm, the governing forced vibration equations are derivative of the Rayleigh beam hypothesis. With the displacement couple pattern, the force from the surrounding fluid is tested for the arm. For analyzing the equations of movement, MATLAB software has been utilized to examine the influence of variables like the mechanical and geometric features and the liquid flow rate on the forced vibration the responding robot arm. Numerical simulations indicate that fluid inertial forces increase with increasing fluid velocity. Therefore, at higher levels of the fluid, the result of extra mass-produced by the fluid causes the range of oscillations to increase suddenly, which decreases the performance and accuracy of the arm. By identifying the lock-in area, the system parameters should be selected in such a way as to prevent the arm from entering the lock-in area.

Loading–shape interaction analysis of membrane structures subjected to deformation dependent hydrostatic loadings

Due to their flexibility, membrane structures loaded and/or supported by gas and/or fluid undergo large deformations when subjected to external loads, for example when a relatively flat membrane deflects under accumulative ponding. The most noteworthy feature during the deformation process is the loading–shape interaction since changes in the form of the structure induce changes in the loading profiles, and vice versa. Description of the hydrostatic loading is complex since the discontinuity of structural domain and fluid domain occurs with the deformation of membrane. In this paper, an interactive model dealing with accurate hydrostatic load description and nonlinear finite element analysis is employed to analysis the structural response. The hydrostatic loading is described in vectors by fluid level and loading direction in current mesh of the membrane. In case to conserve a specific volume, the free surface height is calculated with a Newton algorithm. Equivalent nodal forces evaluation on different elements are deduced especially when they are partially or fully loaded by judging the spatial relation between the element and the fluid domain, and integrating with shape functions and area coordinates in a local coordinate system. In case of multiple fluid domains, the hydrostatic load description procedure is proposed to be conducted for each fluid domain with care in identifying respective loading direction and then sum them up to get the total hydrostatic loading forces. Two numerical examples, (a) single fluid domain case: a flat membrane subjected to accumulative water loading, and (b) multiple fluid domains case: an air-inflated dam partially filled with water and loaded by lateral pressures of water, are presented to demonstrate correctness, performance and effectiveness of the proposed method in large displacement analysis of flexible membrane structures subjected to deformation dependent hydrostatic loading.

Effects of the current-wave interaction on a cylinder platform

ABSTRACT This article investigates the motion responses of floating production storage and offloading systems (FPSO) under wave-current interactions. Experiments were conducted on a cylindrical FPSO in a wave-current combined wave basin. Two methods were employed to compare the effects of currents: applying an equivalent force to a specific location in the platform and generating real currents in the basin. The primary objective of this study is to explore the sensitivity characteristics of the platform's dynamic response in the presence of wave-current coupling. The findings reveal significant phase shifts in the low-frequency motions of the sway and pitch responses due to the interaction between waves and currents. Moreover, the six degree-of-freedom motions of the platform experience amplified overall energies in both low-frequency and wave-frequency motions. A notable observation is that wave-current interaction leads to a considerable increase in surge motion amplitude, even when the reduced velocities are smaller than the lock-in region.

Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

Seismic analysis of underground tunnels subjected to 3D oblique incident P and SV waves by 2.5D approach

This paper presents, for the first time, the consideration of three-dimensional (3D) oblique incident P and SV waves in calculating the 3D seismic response of a lined tunnel embedded in a half-space by the 2.5D finite/infinite element method (FIEM). Firstly, the applicability of the 2.5D FIEM for 3D seismic analysis is summarized. With the exact solutions obtained for the free field in the Appendix, the equivalent seismic forces are rationally computed for the near-field boundary, considering the horizontal and vertical excitations of the Chi-Chi Earthquake. By performing seismic analysis of the half space embedded with a tunnel using the 2.5D FIEM, the time-domain responses of the tunnel are obtained. The accuracy of the present solutions is verified against those of de Barros and Luco. Conclusions drawn from the parametric study include: (1) Stress concentration for the principal stress under oblique incident seismic waves occurs at the polar angles of 0° (vault), 90°, 180° (inverted arch), and 270° of the lining wall. (2) The vault and inverted arch are the weakest parts of the tunnel during earthquakes. (3) The accelerations of the tunnel during earthquakes can be regarded as of the rigid body type. (4) The responses of the tunnel lining caused by SV waves of an earthquake are much more critical than those by P waves. (5) For arbitrary seismic waves, the maximum longitudinal acceleration azmax is of the same order of magnitude as the maximum horizontal acceleration axmax.

Developing a Biomechanical Testing Setup of the Pelvis-Part I: Computational Design of Experiments.

Biomechanics of the human pelvis and the associated implants are still a medical and engineering debated topic. Today, no biomechanical testing setup is dedicated to pelvis testing and associated reconstructive implants with accepted clinical relevance. This paper uses the computational experiment design procedure to numerically design a biomechanical test stand that emulates the pelvis physiological gait loading. The numerically designed test stand reduces the 57 muscles and joints' contact forces iteratively to only four force actuators. Two hip joints' contact forces and two equivalent muscle forces with a maximum magnitude of 2.3 kN are applied in a bilateral reciprocating action. The stress distribution of the numerical model of the developed test stand is very similar to that of the numerical model of the pelvis with all 57 muscles and joint forces. For instance, at the right arcuate line, the state of stress is identical. However, at the location of superior rami, there is a deviation ranging from 2% to 20% between the two models. The boundary conditions and the nature of loading adopted in this study are more realistic regarding the clinical relevance than state-of-the-art. The numerically developed biomechanical testing setup of the pelvis in this numerical study (Part I) was found to be valid for the experimental testing of the pelvis. The construct of the testing setup and the experimental testing of an intact pelvis under gait loading are discussed in detail in Part II: Experimental Testing.