Quasi-static Solution Research Articles

Quasi-static problem solution using Tomlinson’s model

The article considers the interaction of nanoparticle chains. With the help of specialized software experimental data were obtained allowing to solve the quasistatic problem with certain input parameters in nanotribology. During the research, it was found that the change in the number of atoms in a chain directly affects the acceleration of force oscillations. Furthermore, it was shown that during the increase of time of force action by the moving the chain a, represented by a sinusoidal function, the amplitude of the displacement increases.

In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect.

A novel method to compensate the residual magnetic field for an atomic magnetometer consisting of two perpendicular beams of polarizations was demonstrated in this paper. The method can realize magnetic compensation in the case where the pumping rate of the probe beam cannot be ignored. In the experiment, the probe beam is always linearly polarized, whereas, the probe beam contains a residual circular component due to the imperfection of the polarizer, which leads to the pumping effect of the probe beam. A simulation of the probe beam's optical rotation and pumping rate was demonstrated. At the optimized points, the wavelength of the probe beam was optimized to achieve the largest optical rotation. Although, there is a small circular component in the linearly polarized probe beam, the pumping rate of the probe beam was non-negligible at the optimized wavelength which if ignored would lead to inaccuracies in the magnetic field compensation. Therefore, the dynamic equation of spin evolution was solved by considering the pumping effect of the probe beam. Based on the quasi-static solution, a novel magnetic compensation method was proposed, which contains two main steps: (1) the non-pumping compensation and (2) the sequence compensation with a very specific sequence. After these two main steps, a three-axis in situ magnetic compensation was achieved. The compensation method was suitable to design closed-loop spin-exchange relaxation-free magnetometer. By a combination of the magnetic compensation and the optimization, the magnetic field sensitivity was approximately 4 fT/Hz(1/2), which was mainly dominated by the noise of the magnetic shield.

Analysis of bearing stiffness variations, contact forces and vibrations in radially loaded double row rolling element bearings with raceway defects

A method is presented for calculating and analyzing the quasi-static load distribution and varying stiffness of a radially loaded double row bearing with a raceway defect of varying depth, length, and surface roughness. The method is applied to ball bearings on gearbox and fan test rigs seeded with line or extended outer raceway defects. When balls pass through the defect and lose all or part of their load carrying capacity, the load is redistributed between the loaded balls. This includes balls positioned outside the defect such that good raceway sections are subjected to increased loading when a defect is present. The defective bearing stiffness varies periodically at the ball spacing, and only differs from the good bearing case when balls are positioned in the defect. In this instance, the stiffness decreases in the loaded direction and increases in the unloaded direction. For an extended spall, which always has one or more balls positioned in the defect, this results in an average stiffness over the ball spacing period that is lower in the loaded direction in comparison to both the line spall and good bearing cases. The variation in bearing stiffness due to the defect produces parametric excitations of the bearing assembly. The qualitative character of the vibration response correlates to the character of the stiffness variations. Rapid stiffness changes at a defect exit produce impulses. Slower stiffness variations due to large wavelength waviness features in an extended spall produce low frequency excitation which results in defect components in the velocity spectra. The contact forces fluctuate around the quasi-static loads on the balls, with rapid stiffness changes producing high magnitude impulsive force fluctuations. Furthermore, it is shown that analyzing the properties of the dynamic model linearized at the quasi-static solutions provides greater insight into the time–frequency characteristics of the vibration response. This is demonstrated by relating the low frequency event that occurs when a ball enters a line spall to the dynamic properties of the bearing assembly.

Simulation of Store Separation using Low-cost CFD with Dynamic Meshing

The simulation of the store separation using the automatic coupling of dynamic equations with flow aerodynamics is addressed. The precision and cost (calculation time) were considered as comparators. The method used in the present research decreased the calculation cost while limiting the solution error within a specific and tolerable interval. The methods applied to model the aerodynamic forces are time-dependent dynamic meshes and quasi-static methods. In the time-dependent method, a dynamic unstructured tetrahedral mesh approach using combination of spring-based smoothing and local remeshing is employed in respect of bodies motion with an implicit, second-order upwind accurate 3d Euler solver. In this method, a 6dof dynamic code is coupled with the flow solver to update the store trajectory information. In the quasi-static method, a 3-D implicit, steady state Euler solver is automatically integrated with a grid generation software and a 6dof dynamic code. Although the time-dependent method is more precise and reliable, it is not proper and appropriate for the initial design of the separation system due to its high cost. The quasi-static solution is very fast, but unable to simulate realistically because of not satisfying the problem conditions due to solution divergence as the store speed increases. The method used in the present research decreased the calculation cost while limiting the solution error within a specific and tolerable interval. In this way, the time step can be enlarged, the solution can be carried out with a few calculation points, and the solution can have considerably more speed with a limited error magnitude. Simulation of the store separation using the automatic coupling of dynamic equations with flow aerodynamics with the new Low-Cost method is the innovative aspect of this paper. To validate the solution method, the transonic store separation was simulated that agreed well with the wind tunnel test outcomes.

3D numerical model for wave-induced seabed response around breakwater heads

This paper presents a three-dimensional (3D) integrated numerical model where the wave-induced pore pressures in a porous seabed around breakwater heads were investigated. Unlike previous research, the Navier-Stokes equation is solved with internal wave generation for the flow model, while Biot's dynamic seabed behaviour is considered in the seabed model. With the present model, a parametric study was conducted to examine the effects of wave and soil characteristics and breakwater configuration on the wave-induced pore pressure around breakwater heads. Based on numerical examples, it was found that the wave-induced pore pressures at breakwater heads are greater than that beneath a breakwater. The wave-induced seabed response around breakwater heads become more important with: (i) a longer wave period; (ii) a seabed with higher permeability and degree of saturation; and (iii) larger angle between the incident waves and breakwater. Furthermore, the relative difference of wave-induced pore pressure between fully-dynamic and quasi-static solutions are larger at breakwater heads than that beneath a breakwater.

Kinetic and kinematic differences between deadlifts and goodmornings

BackgroundIn order to improve training performance, as well as avoid overloading during prevention and rehabilitation exercises in patients, the aim of this study was to understand the biomechanical differences in the knee, hip and the back between the exercises “Goodmornings” (GMs) and “Deadlifts” (DLs).MethodsThe kinetics and kinematics of 13 subjects, performing GMs and DLs with an additional 25% (GMs), 25% and 50% (DLs) body weight (BW) on the barbell were analysed. Using the kinetic and kinematic data captured using a 3D motion analysis and force plates, an inverse approach with a quasi-static solution was used to calculate the sagittal moments and angles in the knee, hip and the trunk. The maximum moments and joint angles were statistically tested using ANOVA with a Bonferroni adjustment.ResultsThe observed maximal flexion angle of the knee was 5.3 ± 6.7° for GMs and 107.8 ± 22.4° and 103.4 ± 22.6° for DLs with 25% and 50% BW respectively. Of the hip, the maximal flexion angle was 25% smaller during GMs compared to DLs. No difference in kinematics of the trunk between the two exercises was observed. For DLs, the resulting sagittal moment in the knee was an external flexion moment, whereas during GMs an external extension moment was present. Importantly, no larger sagittal knee joint moments were observed when using a heavier weight on the barbell during DLs, but higher sagittal moments were found at the hip and L4/L5. Compared to GMs, DLs produced a lower sagittal moment at the hip using 25% BW while generating the same sagittal moment at L4/L5.ConclusionsThe two exercises exhibited different motion patterns for the lower extremities but not for the trunk. To strengthen the hip while including a large range of motion, DLs using 50% BW should be chosen. Due to their ability to avoid knee flexion or a knee flexion moment, GMs should be preferentially chosen over DLs as ACL rupture prevention exercises. Here, in order to shift the hamstring to quadriceps ratio towards the hamstrings, GMs should be favoured ahead of DLs using 50% BW before DLs using 25% BW.

A new variational principle for cohesive fracture and elastoplasticity

Variational methods for studying cohesive fracture and elastoplasticity have generally relied on minimizing an energy functional that is the sum of a stored elastic energy and a defect energy, corresponding to fracture or plasticity. The usual method for showing existence of minimizers is the Direct Method, whose success requires some properties of the defect energy that are not physically motivated, or in fact are contrary to physically desired properties. Here we introduce a new variational principle based on the idea of “necessity” of the defect, in the spirit of Garroni and Larsen (2009), reflecting the notion that these defects occur only if necessary in order for the elastic stress to be admissible, i.e., under the critical stress at which fracture or plasticity begins. The advantage is that the Direct Method only comes into play with a constraint on the defect set, which obviates some of the technical issues usually involved. The most significant advantage is that existence of global minimizers generally requires an infinite stress or strain threshold for plasticity or fracture, while our formulation is appropriate for finite thresholds. A further advantage is that the method produces local minimizers or locally stable states, rather than less physical global minimizers. General existence results will require new methods, but here we easily show existence in one dimension for both static and quasi-static solutions, even when global minima do not exist.

Explicit dynamic formulation to demonstrate compliance against quasi-static aircraft seat certification loads (CS25.561) – Part II: Influence of body blocks

Loading an aerospace and automotive seat statically through lap or body blocks is a complex and highly non-linear problem, as the key numerical challenge is to replicate the contact and slipping kinematics between seat, lap block and belt. In addition, severe element distortions and unexpected contact between parts can occur due to the large deformations involved, which result in implicit solvers struggling to find a converged solution. This paper focuses on the use of an explicit Finite Element Analysis (FEA) solver (LS-DYNA3D) for an aircraft seat subject to Certification Specifications CS25.561, although the ideas presented are equally applicable to automotive seat designers. Explicit codes are better able to overcome contact convergence issues and are often used with appropriate damping to achieve a quasi-static solution. This paper reviews the methodology presented in Part I, whereby issues relating to damping, mass and time scaling are outlined in order to overcome the high computational time step costs (Courant-Friedrichs-Lewy (CFL) condition), together with the procedural and error checks required to ensure a quasi-static response. This paper extends the methodology by considering load cases that use lap blocks, such as ‘forward 9g’ and ‘upward 3g’ certification requirements. Alternative modelling approaches to represent the loading mechanism and effect of lap block mass on solution accuracy are discussed. This paper concludes with a verification framework that outlines the quality checks on various model energies and their ratios, where the numerical results are validated against test in terms of displacements and seat kinematics. Thus, ‘Part I’ and ‘Part II’ cover all elements related with the application of an explicit dynamic integration scheme to demonstrate static seat compliance, and together, form a clear framework to assist a Computer Aided Engineering (CAE) analyst involved in applying an explicit integration scheme to solve non-linear quasi-static analyses.

Explicit dynamic formulation to demonstrate compliance against quasi-static aircraft seat certification loads (CS25.561) – Part I: influence of time and mass scaling

A finite element model of an aircraft seat subjected to static certification loads (Certification Specifications CS25.561) involves material, geometric and contact non-linearities. Implicit algorithms can model the physics of such problems appropriately but suffer from shortcomings such as significant finite element modelling efforts, high disk space and memory requirements and unconverged solutions. Explicit finite element schemes offer a more robust alternative for convergence for quasi-static loadcases but may come at an even higher computational cost as smaller solution time steps are required, in addition to unwanted inertial effects. A methodology to apply an explicit formulation for simulating static certification loading for an aircraft seat-structure is presented and validated in this article. The first part reviews the design novelties of the triple seat-structure considered, the safety regulations used in aircraft seat certification. The key theoretical aspects of an explicit solver are presented, together with the numerical challenges faced when applied to solving quasi-static problems. Time scaling, mass scaling and damping are common approaches to assist in artificially reducing the computational time but previous articles provide little insight into how to apply these techniques correctly and the level of checking that is required to ensure the quality of the results are unaffected by these modifications. The main focus of this article is to clearly define the procedure to establish appropriate factors for mass scaling, time scaling and damping. Quality checks, such as ratio of kinetic energy to internal energy and their time-histories have been investigated to ensure a quasi-static solution. finite element analysis results are validated against experimental testing for the 8.6 g downward loadcase. Parameters such as kinematic behaviour and deflections at key locations been used for comparison. An acceptable level of correlation between finite element analysis results and physical tests validates the proposed methodology, which will be extended in a future article (Part II) to consider additional contact complexities with the inclusion of body blocks.

An effective approach for transient thermal analysis in a functionally graded hollow cylinder

An effective approach is developed to analyze the transient thermal analysis in a functionally graded hollow cylinder. The heat conductivity, mass density and specific heat are assumed to vary along the radial direction with arbitrary grading pattern. The transient solution in the radial direction is obtained based on the laminate approximation theory. In solving technique, the transient solution is divided into two parts: One is quasi-static solution and the other is dynamic solution. The quasi-static solution is obtained by the state space method and the dynamic solution is obtained by the initial parameter method. In this analysis, all the manipulation is performed in time domain. In the illustrative examples, the effect of the inhomogeneity form on the transient thermal responses is investigated.

Charge Polarization and Current Distribution in a Conductive Particle in the Rayleigh Region

An investigation of the interaction of a conductive sphere with an electromagnetic wave with attention given to space-charge effects would appear timely, as there is much current interest in the electromagnetic properties of mesoscopic structures. In this study, the polarization of a small semiconductor sphere immersed in a dynamic electric field is explored analytically and numerically. In one approach, suitable for a sphere with low to moderate charge carrier concentration, the Poisson's equation is coupled with the transport equations of the carriers, leading to a quasi-static formulation under the weak-field approximation. Screening effects of the charges on the interior field are revealed, along with a current distribution that is essentially uniform over much of the volume of the sphere. Frequency dependence of the total induced dipole moment of the sphere displays strong dispersion and absorption near the bulk plasma frequency. Validity of the quasi-static assumption is assessed by comparison to results of calculations based on a full-wave formulation. As the nominal carrier concentration exceeds 1020 cm-3, the quasi-static solutions for interior field and current distribution begin to deviate from the full-wave solution and the latter must be employed to provide a realistic account of the charge-wave interaction within the sphere.

Dynamic explicit solution for higher-order crystal plasticity theories

Dynamic terms are included in the balance equations that govern a higher-order crystal plasticity formulation. The resulting equations are integrated in time by the central-difference explicit method. The methodology avoids solution methods based on the inversion of the associated stiffness matrices. This is a great advantage of the proposed methodology, considering that these matrices tend to be highly ill-conditioned. As the solution becomes dependent on the time-step size used in the integration, a stability study is presented. Only a rate independent constitutive law is taken into account. As examples, a crystal layer and a composite material under simple shear and the wedge indentation of a single crystal are considered. Comparisons with an implicit quasi-static solution method show the robustness of the proposed methodology, but reveal an elevated computational cost related to it.

Cosmological perturbations inF(R)gravity

The quasi-static solutions of the matter density perturbation in $F(R)$ gravity models have been investigated in numerous papers. However, the oscillating solutions in $F(R)$ gravity models have not been investigated enough so far. In this paper, the oscillating solutions are also examined by using appropriate approximations. And the behaviors of the matter density perturbation in F(R) gravity models with singular evolutions of the physical parameters are shortly investigated as applications of the approximated calculations.

A quasi-static model of wheel–tissue interaction for surgical robotics

Wheel-driven mobile in vivo robotic devices can provide an unconstrained platform for visualization and task performance. Careful understanding of the wheel–tissue interaction is necessary to predict in vivo performance of medical mobility systems. Here, an analytical study of the friction involving rolling contact of a surgical wheel, moving at constant velocities over soft tissue, is presented and verified. A quasi-static frictionless solution is first derived from existing theory, and newly developed theory considering frictional effects is later introduced. In this analysis, the effect of friction on wheel mobility over a viscoelastic substrate is analyzed with wheel velocity as the only changing variable. The analytical model is later verified by experiments and Finite Element Method (FEM) simulations. A simple application of this model to help design a surgical robot is also presented. Additional results indicate that the resistance force, which arises from the tissue viscosity, approaches zero for small and very large wheel velocities.

Extrusion of a plastic material from a circular sector with a small apex angle and a sink at the vertex

The plane inertialess extrusion of a perfect rigid-plastic material from a circular sector, the angle of which serves as a small parameter, is investigated using the asymptotic integration. The flow was initiated by the approach of the two sides of the sector and the existence in it of a sink of specified power. The principal velocity and stress approximations are obtained and the domains of applicability of the asymptotic expansions and the inertialess (quasistatic) solutions are found. Analogies are drawn with the solutions of the classical Prandtl problem and several of its extensions.

Failure initiation during impact of metal spheres onto ceramic targets

A combined analytical/numerical framework is developed for ascertaining the critical impact velocities required to initiate failure in flat, back-supported ceramic targets when impacted by spherical metal projectiles. Four inelastic modes are considered: yielding, microcracking followed by comminution, cone cracking, and radial cracking. The time histories of contact pressure and contact force for a wide range of projectile yield strength and impact velocity are obtained through finite element analyses (FEA). The results are used to assess and calibrate analytic models for peak values of contact stress and force. Additionally, temporal and spatial evolutions of subsurface stresses computed by FEA are analyzed. Within the velocity range of present interest (<1000 m/s), the maximum dynamic subsurface stresses due to impact can be well approximated by the quasi-static solutions for a uniform pressure load of appropriate magnitude. The maximum back face stress is also well approximated by the corresponding quasi-static solution. The computed stresses and forces are combined with established criteria for each of the four failure modes to ascertain the minimum projectile velocity necessary to activate each. In turn, the results are used to identify the sequence of failure modes that occur, specifically, upon impact of metal spheres onto a confined alumina target. Reasonably good quantitative correlations are obtained between predictions and corresponding experimental observations.

Low-velocity impact response of geometrically asymmetric slender sandwich beams with metal foam core

The low-velocity impact response of fully clamped slender metal foam core sandwich beams with geometrically asymmetric cross-section struck by a heavy mass at midspan is investigated. The exact and approximate yield criteria for metal sandwich cross-section are presented considering the effect of core strength. Using the yield criteria, the analytical solutions for large deflection of the fully clamped slender sandwich beams, including dynamic and quasi-static solutions, are derived respectively. Also, finite element (FE) calculations are carried out to validate the analytical model. Moreover, the effects of material elasticity and strain hardening of face sheets are considered in FE analysis. Comparisons of the FE results and analytical solutions reveal that the present analytical model enables the acceptable predictions of the low-velocity impact response of fully clamped asymmetric slender sandwich beams. Material elasticity and strain hardening of face sheets have minor effects on dynamic response.

Análisis asintótico y aproximación de Padé en problemas de propagacion de grietas con difusión controlada

En este trabajo, consideramos la fractura de difución controlada axisimétrica en un espacio infinito, y en el semiespacio. Un ejemplo importante del crecimiento de una fractura de difusión controlada es dado por el hidrógeno inducido en agrietamiento. En metales, el hidrógeno es típicamente disuelto en forma de protones. Cuando los protones alcanzan la superficie de la grieta, se recombinan con electrones y forman hidrógeno molecular en la cavidad de la grieta. Entonces, la fractura puede propagar aún en ausencia de cualquier carga externa, esto es, sólo bajo presión excesiva de gas hidrógeno acumulado dentro de la grieta.}&#x0D; Nuestros resultados muestran que en la aproximación asintótica a largo plazo (basada en la solución cuasiestática), la delaminación de difusión controlada propaga con velocidad constante. Nosotros determinamos una concentración crítica ́máxima que limita el uso de la solución cuasiestática. Una solución transitoria, que representa una aproximación asintótica de corto plazo, es usada cuando la concentración del gas excede la concentración crítica. Entonces apareamos estos dos casos usando el método de aproximaciones de Padé y presentamos soluciones en forma cerrada tanto para propagación de grietas de difusión controlada internas como cercanas a la superficie, en diferentes escalas de tiempo.

Instability of hyper-elastic balloon-shaped space membranes under pressure loads

This paper discusses the evaluation of quasi-static equilibrium solutions for inflatable space membrane structures. A Mooney–Rivlin hyper-elastic material model, with variable constitutive constants, is considered. A compressible weightless medium is used to introduce within the membrane a one-parametric over-pressure loading compared to an ambient pressure. Analytical instability results are shown for a spherical and derived for a cylindrical case. These are compared to numerical simulations based on a flat linearly interpolated triangular space membrane element. Path-following procedures are used to find generalized equilibrium paths, with different parameterizations. Numerical examples show that the methods developed can give information on the stability of the membranes, but that the medium and means for introducing the internal pressure are of importance for the interpretation of stability.

Solid–liquid phase change driven by internal heat generation

This article presents results of solid–liquid phase change, the Stefan Problem, where melting is driven internal heat generation, in a cylindrical geometry. The comparison between a quasi-static analytical solution for Stefan numbers less than one and numerical solutions shows good agreement. The computational results of phase change with internal heat generation show how convection cells form in the liquid region. A scale analysis of the same problem shows four distinct regions of the melting process.