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In this paper, we derive an analytical solution of the damped cubic–quintic Duffing oscillator which is based on a rational elliptic form used to obtain exact and approximate solutions of undamped oscillators. We examine different set of system parameter values to assess the accuracy of our derived solution. It is shown that theoretical predictions compares well with the numerical integration solutions obtained by a fourth order Runge–Kutta method. This demonstrates the applicability of rational elliptic forms to solve damped oscillators with higher nonlinear terms. [ABSTRACT FROM AUTHOR]

In this paper we propose a new approach to replace nonlinear ordinary differential equations by approximate cubic-quintic Duffing oscillators in which its coefficients depend on the initial amplitude of oscillation. It is shown that this procedure leads to angular frequency values with relative errors that are lower than those found by previously developed approximate solutions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive the exact solution of the cubic-quintic Duffing oscillator based on the use of Jacobi elliptic functions. We also showed that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind. [Copyright &y& Elsevier]

Abstract: In this paper, we derive a class of analytical solution of the damped Helmholtz–Duffing oscillator that is based on a recently developed exact solution for the undamped case. Our solution procedure indicates that this solution holds for specific system parametric choice values. [Copyright &y& Elsevier]

Abstract: New analytical approximate solutions based on Jacobi elliptic functions are formulated to compute the angular frequency value of a nonlinear singular oscillator. It is shown that when our derived analytical angular frequency solutions are compared to the exact angular frequency value, the third-order elliptic balance solution provides the best estimate to the exact value since the percentage error value is about 0.0059% which is 216 times lower than the best estimate percentage error value reported in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, the exact solution of the quadratic mixed-parity Helmholtz–Duffing oscillator is derived by using Jacobi elliptic functions. It is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind. At the end of the paper, we examine the stability of the system and determine the regions for which periodic and unbounded motions take place. [Copyright &y& Elsevier]

Abstract: A rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using Jacobi elliptic functions. To illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators, we first investigate the exact solution of the non-homogenous, undamped Duffing equation. Then, we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators. At the end of the paper, we compare the numerical integration values of the angular frequencies with approximate solution results, based on the proposed rational elliptic balance method. [ABSTRACT FROM AUTHOR]

We investigate the dynamical response of a beam-column system with hinged ends subjected to an axial pulsating force of elliptic type. It is shown that the resulting equation is of the formd2y1/dτ2+y1[d1+d2cn2(τ,k2)]=0,which is the well-known Lame´ equation . In this paper, we obtain the general exact solution of this equation that reveals stable behavior of the beam-column system if the assigned initial conditions are of the form y1(0)=y10 and y˙1(0)=0. It is also found that at a certain value of the modulus of the elliptic force, the lateral vibrational frequency is independent of the material properties of the beam-column system. [Copyright &y& Elsevier]

This article aims to investigate the solution of the fractional Troesch's equation that models the mechanism of current generation in porous electrodes by numerical integration of its equivalent power-form representation and by deriving its approximate Taylor series solution. Simulation results compare to experimental data demonstrate that the overpotential distribution and efficiency within a porous electrode, from the bottom to the top of the bed is predicted by using Troesch's fractal overpotential diffusion equation, when the fractal parameter values are α ≥ 1.2618 for a porous electrode that follows the Hausdorff dimension, or α ≥ 1.236 for El Naschie's fractal space-time. • An equation that models the mechanism of current generation in porous electrodes is investigated. • The equivalent representation power-form expression for Troesch's equation is derived • Two scale dimension transform is used to study fractal Troesch's equation. • The Taylor series method is used to solve the equivalent power-form representation of Troesch's equation. [ABSTRACT FROM AUTHOR]

This research describes the design and fabrication of an electromagnetic quadrupole installed inside a biaxial tensile test bench to induce magnetic field effects whose orientation can be adjusted to either 45° or 90° to develop magnetorheological elastomer (MRE) materials subjected to a biaxial stress condition. The objective of the quadrupole is to generate a homogeneous and oriented magnetic field in the central zone of the specimen. To achieve this, we used computational tools to simulate magnetic fields and applied the fundamentals of electrical machine design. The quadrupole can produce a uniform magnetic field ranging from 0 to 120 mT with a minimal gradient within the central area of the specimen. This feature allows us to comprehensively analyze the specimen under the influence of a uniform and controllable magnetic field. The quadrupole effectively aligns the microparticles within the polymeric matrix during the MRE curing process, resulting in a fractal structure with an average dimension of 1.78. While examining the MRE specimen's molecular structure, we observed fractal micro-particle clusters oriented at 45° and 90° from the positive x-axis. Because of the unexpected specimens' fractal molecular structures, we calculated their fractal dimension (FD-Avg), standard deviation (FD-Sd) and lacunarity (FD-Lac) to investigate a possible relationship between fractal dimensions and the specimens' physical properties. Consequently, we discovered that the molecular structure and physical properties, such as the shear modulus of the MRE specimens, are related to the resulting fractal dimensions using data collected from biaxial tests that consider material softening and its residual deformation effects. Our findings elucidate, for the first time, how the fractal dimension obtained from MRE specimens can be used to determine some of their physical properties. • Novel design of an electromagnetic quadrupole capable of generating a homogeneous magnetic field. • Quadrupole core assembly optimized for biaxial testing with partial winding. • Excellent agreement between predicted and measured magnetic flux density. • Successful magnetic alignment in elastomer curing. • An expression that relates the fractal dimension of the MRE with the material static shear modulus was proposed. [ABSTRACT FROM AUTHOR]

A magnetorheological elastomer based on silicone rubber with carbonyl iron micro-particles was developed. The influence of the different amount of iron particles was experimentally studied by means of XRD, SEM, FTIR, EDS, XPS, uniaxial tension and rheological and cyclic tests. Different contents of carbonyl iron particles (10–40 wt%) were used to obtain the ratio of magnetic particles/silicone rubber that could provide the best mechanical properties on the MRE material. It was found that the composite material can have an increase of about 95% in its tensile strength when adding 20% of carbonyl iron particles to the raw rubber material. SEM analysis indicates a good dispersion of the magnetic particles on the rubber matrix, and the FTIR and XPS techniques confirm, as expected, that there is no chemical interaction between the iron from the carbonyl iron particles and the silicone rubber matrix due to a proper coating of the particles with silicone oil used as coupling agent. The TGA results evidenced that the addition of coated carbonyl iron particles had an impact on the thermal stability of the MRE and on the formation of cross-linked structures. The viscoelastic behavior of the magnetorheological elastomer is described by running experimental test on a rheometer device. Furthermore, cyclic testing were performed on the material sample to characterize the Mullin's effect. [ABSTRACT FROM AUTHOR]