Approximate Analytical Solution Research Articles

Vibrational dynamics of a subjected system to external torque and excitation force

This work focuses on the dynamical response of a novel auto-parametric two-degrees-of-freedom (DOF) dynamical system when it is subjected to external torque and force. It consists of a forced oscillator with nonlinear stiffness that is connected to a linear spring. In accordance with the system’s DOF, an organizing system of equations of motion (EOM) is produced using Euler–Lagrange’s equations. To obtain the analytical approximate solutions (ASs) of the equations of this system, the multiple scales technique (MST) is employed. The provided solutions are being juxtaposed with the numerical solutions (NSs) for comparison to show how closely they agree, as well as the precision of the MST. In view of removing secular terms, the system’s solvability criteria are obtained. All arising resonance cases are classified, and two of them are then examined at once. The relevant system of modulation equations is solved numerically to illustrate the advantageous influence of diverse factors on the behavior of the modified phases and amplitudes. Based on the values of these parameters, the frequency response curves (FRCs) are drawn to explore different zones of stability and instability. A wide variety of values for the system’s parameters shows that its behavior is steady. The achieved outcomes are deemed novel, as the MST has been applied on a new dynamical model. The results have broad relevance and can be implemented in various engineering contexts, including the analysis of ship movements, in which nonlinear coupling between rolling and pitching motions can be found.

Multistep Reduced Differential Transform Method in Solving Nonlinear Schrodinger Equations

This paper obtains semi-analytical solutions for the nonlinear Schrodinger equations (NLSEs) using the multistep reduced differential transform method (MsRDTM). The implemented method yields an analytical approximate solution over a longer time frame, in which the method applied is treated as an algorithm in a sequence of small sub-division of intervals of identical length compared to the traditional reduced differential transform method (RDTM). Excluding the need of perturbation, linearization, or discretization, this method offers the benefit and reliability of the multistep algorithm. The outcomes show that the MsRDTM generated highly accurate solutions of NLSEs than the RDTM. In addition, the results show that the suggested method is straightforward to use, saves a significant amount of computing work when solving NLSEs, and has potential for broad application in other complex partial differential equations (PDEs) in the fields of engineering and science. The accuracy of the method is shown through the tables and graphical illustrations provided.

Amplitude deflection in a nonlinear MEMS resonator under parametric excitation

In this paper, the dynamic characteristics of a parametrically excited MEMS micro-resonator incorporating high-order nonlinearity and nonlinear damping are investigated experimentally and theoretically. An intriguing phenomenon of amplitude deflection is observed when the parametric excitation strength exceeds a certain value. We establish a dynamic model of the micro-resonator under parametric excitation and employ the averaging method to obtain an approximate analytical solution. Then the theoretical study is made to explain the amplitude deflection under large excitation, attributing it to the combined effects of high-order nonlinearity and nonlinear damping. These findings contribute to a more comprehensive understanding of the characteristics of parametrically excited systems.

Modified optimal auxiliary functions method for approximate-analytical solutions in fractional order nonlinear Foam Drainage equations

The study aims to obtain new approximate analytical solutions for the fractional order nonlinear Foam Drainage equation through a novel modification of the well-known Optimal Auxiliary Functions Method (OAFM). This method systematically constructs auxiliary functions, and by optimizing their parameters, it provides approximate solutions closely mirroring the original behavior of the problem. Numerical simulations, presented in tabular form, demonstrate the effectiveness of the proposed approach. Graphical representations of the solutions are utilized to further validate the applicability and accuracy of the suggested technique. Additionally, the results affirm that the modified OAFM stands out as one of the most effective tools for addressing nonlinear problems in the field of science and technology, thereby contributing valuable addition to the field.

Theoretical and experimental study of a stable state adjustable nonlinear energy sink

Due to high sensitivity to the changes in external excitation intensity, the implementation of nonlinear energy sinks (NESs) have been greatly limited in practical applications. This study presents a stable state adjustable NES (SA-NES) structure. By adjusting the structural geometric relationship, the SA-NES structure can easily switch its stable state among monostable, bistable, and tristable. The governing equation of the coupled system is obtained by Hamilton's principle. The approximate analytical solution is obtained by the harmonic balance method, and the slow invariant manifold is obtained complexification-averaging method, which are validated through numerical analysis. Through theoretical analysis and experimental validation, the dynamic response of the SA-NES and vibration reduction performance are studied under different levels of excitations. The obtained results show that the designed SA-NES can effectively suppress the vibration under different level excitation intensities. It is also shown that the SA-NES structure can be adjusted to the monostable state, and the vibration of the linear oscillator can be effectively suppressed by the strong modulation response under a certain excitation intensity. When the excitation intensity is weak or strong, the SA-NES can be adjusted to the bistable state and the tristable state, respectively, thus effectively suppressing the vibration through chaotic inter-well oscillation. This study reveals the vibration suppression mechanism of the SA-NES, and introduces a new NES configuration to suppress different levels of engineering excitations.

A viscoelastic nonlinear energy sink with an electromagnetic energy harvester: Narrow-band random response

Abstract Nonlinear energy sink is a passive energy absorption device that surpasses linear dampers, and has gained significant attention in various fields of vibration suppression. This is owing to its capacity to offer high vibration attenuation and robustness across a wide frequency spectrum. Energy harvester is a device employed to convert kinetic energy into usable electric energy. In this paper, we propose an electromagnetic energy harvester enhanced viscoelastic nonlinear energy sink (VNES) to achieve passive vibration suppression and energy harvesting simultaneously. A critical departure from prior studies is the investigation of the stochastic P-bifurcation of the electromechanically coupled VNES system under narrow-band random excitation. Initially, approximate analytical solutions are derived using a combination of multiple-scale method and a perturbation approach. The substantial agreement between theoretical analysis solutions and numerical solutions obtained from Monte Carlo simulation underscores the method’s high degree of validity. Furthermore, the effects of system parameters on system responses are carefully examined. Additionally, we demonstrate that stochastic P-bifurcation can be induced by system parameters, which is further verified by the steady-state density functions of displacement. Lastly, we analyze the impacts of various parameters on mean square current and mean output power, crucial for selecting suitable parameters to enhance energy harvesting performance.

Flexural pulse wave velocity in blood vessels.

Arteriosclerosis is a major risk factor for cardiovascular disease and results in arterial vessel stiffening. Velocity estimation of the pulse wave sent by the heart and propagating into the arteries is a widely accepted biomarker. This symmetrical pulse wave propagates at a speed which is related to the Young's modulus through the Moens Korteweg (MK) equation. Recently, an antisymmetric flexural wave has been observed in vivo. Unlike the symmetrical wave, it is highly dispersive. This property offers promising applications for monitoring arterial stiffness and early detection of atheromatous plaque. However, as far as it is known, no equivalent of the MK equation exists for flexural pulse waves. To bridge this gap, a beam based theory was developed, and approximate analytical solutions were reached. An experiment in soft polymer artery phantoms was built to observe the dispersion of flexural waves. A good agreement was found between the analytical expression derived from beam theory and experiments. Moreover, numerical simulations validated wave speed dependence on the elastic and geometric parameters at low frequencies. Clinical applications, such as arterial age estimation and arterial pressure measurement, are foreseen.

Intrinsically Localized Modes of bilinear FPU chains: Analytical Study

Present study concerns the analysis of stationary intrinsically localized modes emerging in bilinear and symmetric Fermi–Pasta–Ulam chains. Each element of the chain is coupled to its nearest neighbors through identical and symmetric (tension/compression) bilinear springs. The intrinsically localized modes are the time-periodic, localized vibration states which are manifested by the extreme energy localization on one and two bonds of the chain for bond - centered and site - centered symmetries, respectively. Approximate and exact analytical solutions of intrinsic localized modes are derived for the infinite as well as the finite chains, correspondingly. The derived analytical approximations allow to describe explicitly the amplitude wave profiles of these special localized states as well as to establish their zones of existence in the space of system parameters. In the second part of the study, we derive the exact analytical solutions for the same type of site- and bond- centered intrinsically localized modes supported by the finite bilinear Fermi–Pasta–Ulam chains and analyze their stability properties using Fillipov method.

Approximate analytical solutions to nonlinear oscillations via semi-analytical method

A modified algebraic method is proposed in this study as an effective semi-analytical method for obtaining analytical solutions for two types of nonlinear differential equations. The trial solution is assumed to be a polynomial, and the algebraic approach is used to find the unknown coefficients. To enhance the solution, first, we apply the Laplace transformation on the series solution and then create the Padé approximations of the resultant equation. Finally, for the nonlinear problems under discussion, the inverse Laplace transformation is used to obtain periodic solutions.

A NEW FRACTAL MODELING FOR THE NERVE IMPULSES BASED ON LOCAL FRACTIONAL DERIVATIVE

In this paper, a new fractal nerve impulses modeling is successfully described via the Yang’s local fractional derivative in a microgravity space, and its approximate analytical solution is obtained by a new Adomian decomposition method. The efficiency and accuracy analysis of the proposed method is elucidated according to the graphs. The result shows that our method is excellent and accurate in dealing with Local fractional differential equations.

A handy analytical approximate solution for the magnetohydrodynamic flow of blood in a porous channel

This work presents a new version of the Picard method, known as the boundary values problems Picard method (BVPP), to obtain an analytical approximate solution for a highly complex nonlinear differential equation that models the magnetohydrodynamic flow of blood through a porous channel. The proposed method is versatile and can produce compact and easily evaluated analytical expressions that accurately capture the scientific phenomena being studied, making it ideal for practical applications. BVPP transforms a differential equation into an integral equation and utilizes an iterative algorithm like that of the basic Picard method. However, unlike the basic method, BVPP allows for the selection of an appropriate initial function and involves several adjustable parameters that can be optimized to obtain a precise analytical approximate solution with minimal effort. Overall, BVPP represents a significant advancement in the analysis of complex nonlinear differential equations, particularly in the field of biomedical engineering.

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Nonlinear Reaction-Diffusion and Mediated Electron Transfer Process at Conducting Polymer Modified Ultramicroelectrodes

The primary goal of this article is to present novel analytical solutions for the coupled nonlinear equation found in the mediated electron transfer process at polymer-modified conducting ultramicroelectrodes. Taylor’s series method is utilized to obtain approximate analytical solutions for the reaction-diffusion equations, allowing for the determination of the substrate and mediator concentrations and the current response concerning the substrate concentration at the electrode’s surface. The impact of different factors on concentration and current is also explored. The derived analytical results are in strong agreement with numerical results and with other analytical outcomes from the literature.

Analyzing the dynamics of a charged rotating rigid body under constant torques

This study explores the dynamical rotary motion of a charged axisymmetric spinning rigid body (RB) under the effect of a gyrostatic moment (GM). The influence of transverse and invariable body fixed torques (IBFTs), and an electromagnetic force field, is also considered. Euler’s equations of motion (EOM) are utilized to derive the regulating system of motion for the problem in a suitable formulation. Due to the lack of torque exerted along the spin axis and the nearly symmetrical nature of the RB, the spin rate is nearly unchanged. Assuming slight angular deviations of the spin axis relative to a fixed direction in space, it is possible to derive approximate analytical solutions (AS) in closed form for the attitude, translational, and rotational movements. These concise solutions that are expressed in complex form are highly effective in analyzing the maneuvers performed by spinning RBs. The study focuses on deriving the AS for various variables including angular velocities, Euler’s angles, angular momentum, transverse displacements, transverse velocities, axial displacement, and axial velocity. The graphical simulation of the subsequently obtained solutions is presented to show their precision. Furthermore, the positive impacts that alterations in the body’s parameters have on the motion’s behavior are presented graphically. The corresponding phase plane curves, highlighting the influence of different values in relation to the electromagnetic force field, the GM, and the IBFTs are drawn to analyze the stability of the body’s motion. This study has a significant role in various scientific and engineering disciplines. Its importance lies in its ability to optimize mechanical systems, explain celestial motion, and enhance spacecraft performance.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

The present study focuses on a real finance nonlinear dynamic system (FNLDS), which has been shown to exhibit chaotic behavior. The solutions for such nonlinear dynamical systems (NLDSs) have typically been derived using numerical techniques. The objective of this study aims to; firstly, derive approximate analytical solutions for the complex FNLDS (CFNLDS) by constructing the Picard iterative scheme. The convergence of this scheme is proven, and the error analysis shows good tolerance, indicating the efficiency of the technique. Second, a novel criterion for synchronizing the real and imaginary parts of the system is presented, based on a necessary condition. Thirdly, a new method for constructing the extended center manifold is introduced. The 3D portrait reveals a feedback scroll pattern, while the 2D portrait, representing the mutual components, shows multiple pools. The synchronization of the real and imaginary parts of the system is demonstrated graphically. The FNLDS is tested for sensitivity dependence against tiny variations in the initial conditions, and it is found that the system components are moderately sensitive. Furthermore, the Hamiltonian and the extended center manifold establish a two-fold structure. It is observed that the effect of the α-β derivative leads to a delay in the behavior of the solutions.

Approximate analytical solutions of the 1:3 internal resonance response of piezoelectric energy harvester considering two types of coupled frequency components

The 1:3 internal resonance effect in the L-shaped piezoelectric vibration energy harvester leads to an expanded frequency band for energy harvesting and better matching of external excitation frequency in the environment. The electromechanical-coupled governing equations for this vibration energy harvester have been validated in our previous work (Nie et al., 2019). The effect of load resistance on the energy harvester’s natural frequency and damping ratio is addressed. The quadratic and cubic nonlinearities in this system leads to two types of coupled frequency components appear, they ultimately affect the approximate solution of the voltage response. The method of multiple scales is used to derive the two types of coupled frequency components of the system along with their contribution to the output responses. The revised approximate analytical solutions of the output responses are then presented. The system’s stability is assessed through the Jacobi matrix of modulation equations. The nonlinear softening phenomenon that favors the broadening of the bandwidth of the energy harvester is analyzed. The vibration patterns in regions with nonlinear softening and non-softening behavior are investigated through frequency spectra, phase trajectories, and Poincaré cross-sections. The findings reveal that the load resistance significantly affects the damping ratio of the harvester. Inclusion of quadratic and cubic nonlinearities results in three frequency components in the output responses spectrum, with an approximate ratio of 1:2:3. Two types of coupled frequency components appeared in the spectrum of the system’s output responses. The contribution of coupled frequency components to the output response proves crucial, and the revised approximate solutions are in good agreement with that of numerical solutions. The output responses are significantly affected by the initial conditions in the nonlinear softening region, the large and small initial conditions yield solutions in the upper and lower branches, respectively.

Estimation of the mass density of biological matter from refractive index measurements

The quantification of physical properties of biological matter gives rise to novel ways of understanding functional mechanisms. One of the basic biophysical properties is the mass density (MD). It affects the dynamics in sub-cellular compartments and plays a major role in defining the opto-acoustical properties of cells and tissues. As such, the MD can be connected to the refractive index (RI) via the well known Lorentz-Lorenz relation, which takes into account the polarizability of matter. However, computing the MD based on RI measurements poses a challenge, as it requires detailed knowledge of the biochemical composition of the sample. Here we propose a methodology on how to account for assumptions about the biochemical composition of the sample and respective RI measurements. To this aim, we employ the Biot mixing rule of RIs alongside the assumption of volume additivity to find an approximate relation of MD and RI. We use Monte-Carlo simulations and Gaussian propagation of uncertainty to obtain approximate analytical solutions for the respective uncertainties of MD and RI. We validate this approach by applying it to a set of well-characterized complex mixtures given by bovine milk and intralipid emulsion and employ it to estimate the MD of living zebrafish (Danio rerio) larvae trunk tissue. Our results illustrate the importance of implementing this methodology not only for MD estimations but for many other related biophysical problems, such as mechanical measurements using Brillouin microscopy and transient optical coherence elastography.

Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems

Abstract In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.

Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

Analytical, numerical and experimental observation of isolated branches of periodic orbits in 1DOF mechanical parametric oscillator

The aim of this study is to investigate the dynamic properties of an existing experimental stand of 1DOF mechanical parametric oscillator, with a focus on approximate analytical solutions of the observed isolated branches of periodic orbits. The experimental stand involves a cart moving along a rolling guide, with the stiffness consisting of two components: a time-varying linear element created by a rotating rod with a rectangular cross-section and a nonlinear hardening stiffness caused by magnetic springs. It was demonstrated that a rolling bearing’s nonlinear resistance to motion consists of viscous damping and a second component analytically compared to dry friction. The study utilises multiple scales and harmonic balancing methods to provide analytical solutions. It is then successfully validated using numerical simulations and experimental data. The study investigates how dry friction influences oscillator response and applies the modified Mathieu–Duffing equation to represent the system’s dynamics. Different branches of periodic orbits are researched to determine their function in energy harvesting and mechanical system improvement. This research demonstrates the distinctions across analytical, numerical, and experimental methodologies, providing a comprehensive understanding of investigating intricate nonlinear systems.