Uncoupled Equations Research Articles

Unsymmetricity effects on seismic performance of multi-story buildings

The unsymmetrical configurations in buildings lead to non-uniform distributions in their strength, mass, and stiffness, and they are consequently prone to damage during seismic hazards. In this study, the seismic performance of multi-story buildings with 5, 8, 10, and 12 stories of square, ‘L’, ‘T’, and ‘U’-shaped buildings have been investigated. The research deals with the variation of the natural time periods and how it affects the seismic performance of unsymmetrical multi-story buildings. The coupled and uncoupled equations of motion, based on the symmetricity of the buildings about both axes, were solved to obtain natural time periods that influence the spectral acceleration of the ground accelerations. Six important ground accelerations were considered. Nonlinear static analysis, such as pushover analysis, was also carried out on all the buildings. Comparisons were made on the seismic behavior of both the symmetrical and unsymmetrical structures. The results revealed that the spectral acceleration influences dynamic responses, such as base shear, base moment, base torsion, roof displacement, roof rotation, and story drifts of the buildings. Moreover, it was found that even though the ‘L’-shaped buildings are unsymmetrical about both axes, they are less vulnerable than the ‘T’ and ‘U’-shaped buildings, which are unsymmetrical about one axis.

Extended Modified Bridge System (EMBS) method for decoupling seismic vehicle‐bridge interaction

AbstractSeismic vehicle‐bridge interaction (SVBI) is the study of vehicle‐bridge interaction (VBI) in the presence of earthquake excitation. SVBI is an interdisciplinary problem of increasing importance to the design and safety of railways. This study deploys a consistent methodology to decouple the vehicle‐bridge system and solve independently the bridge and vehicle subsystems, bypassing multiple challenges the seismic response analysis of a coupled vehicle‐bridge system entails. The proposed approach builds upon the previously established Extended Modified Bridge System (EMBS) method for decoupling vehicle‐bridge systems (in the absence of earthquake excitation). Its premise is to first characterize and then assess the relative importance of the VBI effect on the bridge and vehicle responses and replicate it by modifying the pertinent uncoupled equations of motion (EOMs). The formulation deployed accommodates multi‐degree of freedom models for both the vehicle and bridge and can thus tackle complex systems. The analysis examines the ability of the proposed decoupling approach to predict the response of a realistic system vehicle‐bridge system under a suit of historical earthquake records. The decoupled results are in excellent agreement with the coupled solutions for all earthquake records and scenarios (i.e., earthquake excitation solely in the transverse direction of the bridge, as well as in both the transverse and vertical directions simultaneously).

Experimental nonlinear model of a set of connecting elements in view of nonlinear modal coupling

The development process of mechanical systems involves the evaluation of its modes of vibrations in the frequency range of interest. In general, a linear modal analysis is sufficient to determine whether the system can operate in dynamic conditions. However, in some cases the assembly is composed of many subsystems connected through nonlinear connections which make the response depend on the amplitude and frequency of the excitation. In those cases, Linear Normal Modes (LNMs) are not sufficient to fully describe the dynamics of the system and Nonlinear Normal Modes (NNMs) must be used. Using substructuring techniques it is possible to treat nonlinear joints as independent subsystems. However, a reliable nonlinear model is needed to use this approach. The experimental characterization is the only way to correctly estimate the nonlinear behavior of the connection. Thus, the aim of this work is to develop and validate the experimental characterization to build an experimental nonlinear modal model of a strongly nonlinear element that can be used to connect different linear subsystems and can be regarded as a localized source of nonlinearity. The NNMs identification is performed using the single-point single-harmonic phase resonance method, and the experimental nonlinear modal model of the NLCE is obtained by retaining only the first harmonic term of each NNM in order to get nearly uncoupled modal equations.

Geometric approach to solving inverse kinematics of six DOF robot with spherical joints

Inverse kinematics is a fundamental concept in robotics that plays a crucial role in a robot’s ability to perform tasks. In this contribution, we propose a novel geometric approach based on vector calculus to solve the inverse kinematics problem. The primary advantage of this approach originates from the solutions, which exhibit a linear form and uncoupled equations. To validate the effectiveness and correctness of our proposed method, we constructed a six-degrees-of-freedom robot. This robot is controlled by an Arduino Mega 2650 on which we have implemented the inverse kinematics algorithm. The validation process involved considering various desired trajectories of the end-effector, which were simulated in Matlab and then performed by the physical robot. Importantly, our findings confirm that the end-effector successfully tracks the predefined trajectories. Furthermore, we conducted a comparative analysis between Paul’s method and the results obtained from joint angles using our proposed approach. Interestingly, our study reveals a significant similarity between the two sets of results, reaffirming the accuracy and validity of the approach presented in this study.

On the geometric mechanics of assignment flows for metric data labeling

Metric data labeling refers to the task of assigning one of multiple predefined labels to every given datapoint based on the metric distance between label and data. This assignment of labels typically takes place in a spatial or spatio-temporal context. Assignment flows are a class of dynamical models for metric data labeling that evolve on a basic statistical manifold, the so called assignment manifold, governed by a system of coupled replicator equations. In this paper we generalize the result of a recent paper for uncoupled replicator equations and adopting the viewpoint of geometric mechanics, relate assignment flows to critical points of an action functional via the associated Euler–Lagrange equation. We also show that not every assignment flow is a critical point and characterize precisely the class of coupled replicator equations fulfilling this relation, a condition that has been missing in recent related work. Finally, some consequences of this connection to Lagrangian mechanics are investigated including the fact that assignment flows are, up to initial conditions of measure zero, reparametrized geodesics of the so-called Jacobi metric.

Fast method for the determination of radiation-relevant eigenmodes of underwater structures when using FEM shell elements (“modal reduction”)

A self-developed code based on the BEM for the determination of the backscattered sound pressure level in the far field was extended so that additional FEM shell elements and thus elastic material properties can be considered. For this purpose, a separate FEM equation system is built up and directly integrated into the system of the BEM via corresponding transformation matrices. A common variant for solving the FEM-specific parts is to use a solver based on eigenvalue calculations, which provides the corresponding eigenfrequencies and eigenvectors of the uncoupled FEM equation system for a given upper cutoff frequency. It can be observed that only a part of the determined modes has a considerable sound radiation efficiency and is therefore relevant for the result. An algorithm determines these modes by a fast post-processing routine considering a given percentage, reduces the size of the eigenvalue equation system accordingly, and thus shortens the solution time. Furthermore, it was investigated whether the condition of the entire system of equations is improved by removing the irrelevant modes when using iterative solution methods and whether the number of iterations can thus be reduced. The paper describes the basics of FEM coupling with BEM, presents the reduction algorithm used and shows the results obtained for corresponding test structures depending on the reduction percentage used.

Uncoupling Techniques for Multispecies Diffusion–Reaction Model

We consider the multispecies model described by a coupled system of diffusion–reaction equations, where the coupling and nonlinearity are given in the reaction part. We construct a semi-discrete form using a finite volume approximation by space. The fully implicit scheme is used for approximation by time, which leads to solving the coupled nonlinear system of equations at each time step. This paper presents two uncoupling techniques based on the explicit–implicit scheme and the operator-splitting method. In the explicit–implicit scheme, we take the concentration of one species in coupling term from the previous time layer to obtain a linear uncoupled system of equations. The second approach is based on the operator-splitting technique, where we first solve uncoupled equations with the diffusion operator and then solve the equations with the local reaction operator. The stability estimates are derived for both proposed uncoupling schemes. We present a numerical investigation for the uncoupling techniques with varying time step sizes and different scales of the diffusion coefficient.

Intensity of Dynamic Stresses of Longitudinal Shear in a Periodically Layered Composite with Penny-Shaped Cracks

We consider the problem of harmonic torsional loading of an infinite elastic composite formed by alternating plane layers made of two different materials in the presence of a penny-shaped crack in one of the components of periodic structure. By using the integral representations for displacements and stresses in the frequency domain and satisfying the conditions of periodicity and perfect contact on the interfaces, we deduce a system of uncoupled boundary integral equations for the functions of tangential dynamic crack-opening displacement in a two-layered representative element of the composite. We perform the numerical analysis of the mode-III dynamic stress intensity factors in the vicinity of the analyzed crack depending on the wave number, the thicknesses of constituent layers, and the properties of their materials.

Three-Dimensional Stress Fields in Thick Orthotropic Plates with Sharply Curved Notches under In-Plane and Out-of-Plane Shear.

In this paper, an analytical solution for the stress fields in the close neighbourhoods of radiused notches in thick orthotropic plates under shear loading and twisting is provided. In the first step, the equations of the three-dimensional theory of elasticity are successfully reduced to two uncoupled equations in two-dimensional space. Later, the 3D stress field solution for orthotropic plates with radiused notches is presented and its degree of accuracy is discussed by comparing theoretical results and numerical data from 3D FE analyses. The solution proposed can be satisfactorily used to characterise the stress field in plates made with polymeric composite materials, such as fibre-reinforced polymers and natural composites.

Dynamic analysis of axially loaded cantilever shear‐beam under large deflections with small rotations

AbstractThe modal analysis and seismic response of a vibrating cantilever, with or without tip mass and rotary inertia, are investigated in this study using a shear deformable beam model and including the effect of vertical load. Based on existing approaches, an original method is proposed that does not use fourth‐order uncoupled equations to determine modal deflection and rotation. In fact, the approach presented herein transforms the second‐order coupled system into a first‐order system which can be solved more easily using matrix algebra and Laplace transform. Furthermore, the proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self‐adjoint, and the solution in the case of forced response using modal superposition. In addition, even if the solution presented herein is applicable only to the cantilever with a tip mass and rotary inertia, the scope is general, and the approach can be applied to shear deformable beams with other boundary conditions. Finally, the seismic response by modal superposition is shown, and some examples are proposed and discussed for the case of uniform or continuously varying cross‐sectional properties.

Reliability of the bi-stable piezoelectric energy harvester under random excitation

The bi-stable piezoelectric energy harvester can produce higher voltage and efficiency than traditional energy harvester. In this manuscript, the reliability of the bi-stable piezoelectric energy harvester excited by Gaussian white noise is investigated. The equivalent uncoupled equation of system is deduced on the basis of the generalized harmonic transformation for the bi-stable piezoelectric energy harvester original coupling system. The averaged stochastic differential equation for system energy of the equivalent uncoupled system is established using the stochastic averaging method. The conditional reliability function and the mean first-passage time are derived through the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged stochastic differential equation. The influences of noise intensity, electromechanical coupling coefficient, and time constant ratio on the conditional reliability function and mean first-passage time are discussed. The analytical results are highly consistent with the numerical results that are obtained by the Monte Carlo numerical simulation.

Modeling of a Porous Piezoelectric Nano Energy Harvester Based on Timoshenko-Beam Theory

Collecting electric power from the vibration of piezoelectric beams is practical and has been widely used for decades. This paper investigates a porous piezoelectric cantilever Timoshenko-beam energy harvester at nanoscale. The aim is to propose an optimized Timoshenko beam in order to gain more power with the same working conditions. For this purpose, the width of the beam is assumed to be exponentially changing and pores are also introduced into the beam to relieve the inherent brittleness of piezoelectric materials. Based on Timoshenko beam theory and porous piezoelectric elasticity, the uncoupled motion equations of the porous piezoelectric nanoTimoshenko beam are deduced. The analytical solutions to the output voltage and power density are obtained. According to the numerical calculation, it can be concluded that the electrical output can be improved by increasing the section parameter. Moreover, the surface effects and pore properties on the energy harvesting characteristics are discussed. In addition, when there is a concentrated mass at the free end of the beam, the displacements and output power are studied.

Application of Carrillo's method in unsaturated soil consolidation

The feasibility of Carrillo's method is confirmed for consolidation problems of unsaturated soils under instantaneous loading in this study. In the plane strain and axial symmetry consolidation models, the coupled governing equations expressed by the excess pore-air and pore-water pressures are transformed into the equivalent uncoupled equations expressed by two functions of excess pore pressure using a decoupled method. Then, two rigorous theorems are used to prove that Carrillo's method is suitable for unsaturated soil consolidation under the plane strain and axial symmetry conditions. Finally, the mathematical formulas by Carrillo's method for unsaturated soil consolidation under plane strain and axisymmetric conditions are presented, and are verified against those of the existing analytical solutions. It turns out that the combinations of existing solutions to the conditions of a single direction flow by the proposed formulas can directly furnish the solutions to unsaturated soil consolidation with both horizontal and vertical flows.

Surface waves on a coated incompressible elastic half-space

Abstract Recently, Khajiyeva et al. (2018) studied the dynamics of surface wave propagation in a pre-stressed incompressible half-space when accompanied by both vertical and tangential loads. In this study, however, we extend the work of Khajiyeva et al. to include the presence of a coating layer above the half-space, upon which the loads are acted. Moreover, we prescribe sufficient perfect continuity conditions between the two layers, resulting in an inhomogeneous composite structure. Furthermore, related effective boundary conditions within the long-wave assumption are acquired for the model through the application of the asymptotic approximation method. Finally, approximate uncoupled pseudo-differential equations are derived on the surface, thereby admitting all the results of Khajiyeva et al. as limiting cases of concern.

Analysis of Free Vibrations in Homogeneous Isotropic Rigidly Fixed Thermoelastic Circumferential Spherical Curved Plates

The 1st and 2nd class vibrations of rigidly fixed thermoelastic spherical curved plates have been presented in this article. After mathematical modeling, the obtained equations have been solved by applying normal mode analysis. The uncoupled equation has been considered as 1st class vibrations named toroidal vibrations, remains independent of temperature variations. The coupled system of equations which is considered as 2nd class vibrations named spheroidal vibrations. The series solution of matrix Fröbenius method has been applied to coupled system of ordinary differential equations to obtain potentials and temperature. Fixed point iteration numerical technique has been applied to analytical results with the assistance of MATLAB software tools. Computational techniques have been applied to analytical results and numerically generated data has been represented graphically for frequencies, dissipation factor, displacements and temperature. HIGHLIGHTS The 1st and 2nd class vibrations of rigidly fixed thermoelastic spherical curved plates have been presented in this article The uncoupled equation has been considered as 1st class vibrations named toroidal vibrations, remains independent of temperature variations The coupled system of equations which is considered as 2nd class vibrations named spheroidal vibrations The series solution of matrix Fröbenius method has been applied to couple system of ordinary differential equations to obtain potentials and temperature Computational techniques have been applied to analytical results and numerically generated data has been represented graphically for frequencies, dissipation factor, displacements and temperature GRAPHICAL ABSTRACT

Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction.

Coupled Boussinesq equations are used to describe long weakly nonlinear longitudinal strain waves in a bi-layer with soft bonding between the layers (e.g., a soft adhesive). From a mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations for the right- and left-propagating waves in each layer. However, the models impose a "zero-mass constraint"; i.e., the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow time variables. By construction, the Ostrovsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean, while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behavior of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalized) conservation laws of the coupled Boussinesq equations for strains are derived and used to control the accuracy of the numerical simulations.

Theoretical manual of ‘YTU DEEP’ SHIP motion program

The theoretical background for an in-house seakeeping code ‘YTU DEEP’ that calculates heave, roll and pitch motions of displacement ships is established in this paper. The code consists of two main parts: frequency and time-domain calculations. Two-dimensional added mass and damping coefficients are calculated by using Ursell's multipole expansion theory. Lewis conformal mapping technique is used together with Tasai's method to compute the sectional hydrodynamic coefficients. The strip theory of Salvesen et al. is used to obtain the global hydrodynamic coefficients. The damping values for the roll motion are calculated by using Ikeda's method. Excitation terms for pitch and heave motions are computed by using head seas approximation. The excitation term for roll motion is predicted by using low steepness waves approximation. For frequency domain code validation, Response Amplitude Operators for the AME CRC hulls motions are plotted at different forward speeds and in head waves. The results are compared with the experimental results and a good agreement is observed. For time-domain calculations, Cummins' method is utilized which used the sectional hydrodynamic coefficients in the frequency domain. The coupled pitch & heave and uncoupled roll motion equations are solved via the 4th order Runge Kutta Method. The solution of Cummins equations is tested for a container and passenger ship form simulation in bow quartering waves. A user-friendly interface is created for performing both frequency and time-domain simulations. Pre-processing, solution, and post-processing are applied with a MATLAB code. The proposed seakeeping code aims to create a quick and efficient ship motion program.

Analytical Solution for Thermoelastic Stress Wave Propagation in an Orthotropic Hollow Cylinder

The problem of thermoelastic stress wave propagation in an orthotropic hollow cylinder is investigated using analytical methods. The fully coupled classical theory of thermoelasticity is used to extract the equations for an orthotropic cylinder. To solve the boundary value problem, heat conduction equation and equation of motion are divided into two different sets of equations, the first set consists of uncoupled equations with considering boundary conditions and the second set comprises coupled ones with initial conditions. Finite Hankel transform (Fourier-Bessel expansion) is utilized to solve the problem with respect to radial variable. Two different cases, pure mechanical load and pure thermal load, were studied numerically to show the effect of considering the thermomechanical coupling term in the heat conduction equation. To show the effect of considering the coupling term in the heat conduction equation, the temperature history is plotted for the pure mechanical load case, where the temperature rises without applying any thermal load. By applying boundary conditions on the inner surface of the cylinder, initiation of the stress waves from the inner surface of the cylinder, propagation through the thickness in the radial direction and reflection from the outer surface were observed in the plotted figures.

Stabilization by memory effects: Kirchhoff plate versus Euler–Bernoulli plate

In this paper we study the asymptotic behavior of solutions of a dissipative coupled system where we have interactions between a Kirchhoff plate and an Euler–Bernoulli plate. The dissipative mechanism is given by memory terms that act either collaboratively (in both equations) or unilaterally (in only one equation). We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We found explicit decay rates that depend on the fractional exponents of the memory in each of the following cases: when the memory only acts in the Kirchhoff equation, or only in the Euler–Bernoulli equation, or in both. We also show that all decay rates found are the best. The results obtained are surprising for the following facts: in the collaborative case, the best decay rates of the system are given by the worst decay rates of the uncoupled equations, and in the unilateral case, we conclude that the memory effects in the Euler–Bernoulli equation dissipate the system more slowly than memory effects in the Kirchhoff equation.

Local Thermal Nonequilibrium (LTNE) Modeling of a Partially Porous Channel With Spatial Variation in Biot Number

Abstract The local thermal nonequilibrium (LTNE) model has been used widely for analyzing heat transfer during internal flow through porous media, including when a channel is only partially filled with a porous medium. In such problems, the Biot number describes the rate of convective heat transfer between solid and fluid phases. While uniform Biot number models are commonly available, recent advances in functionally graded materials necessitate the analysis of spatially varying Biot number in such geometries. This paper presents LTNE-based heat transfer analysis for fully developed flow in a channel partially filled with porous medium and with spatially varying Biot number to describe solid-fluid interactions in the porous medium. Fully uncoupled ordinary differential equations for solid and fluid temperature distributions are derived under three different boundary condition models. Solid and fluid temperature fields are presented for a variety of Biot number distributions, including quadratically and periodically varying functions. An explanation of the nature of temperature distribution predictions for such problems is provided. For special cases, the results presented here are shown to reduce to past work on constant Biot number. This work improves the theoretical understanding of porous media heat transfer and facilitates the use of such theoretical models for functionally graded materials.