Second-order Analytical Solution Research Articles

Multi-scale analysis of harmonic resonance in cylindrical bubbles under acoustic excitation

In this paper, the dimensionless oscillation equation of a cylindrical bubble is analyzed using the multi-scale method, Lyapunov stability theory, and the Routh–Hurwitz stability criterion. The corresponding second-order analytical solution and stability criterion are obtained. By examining the cases of second-order super-harmonic resonance and 1/2-order sub-harmonic resonance, the harmonic resonance characteristics of cylindrical bubbles and the influencing factors are revealed. The conclusions are summarized as follows: (1) Super-harmonic resonance can exhibit up to three solutions, along with unstable phenomena such as jump and hysteresis. Sub-harmonic resonance, however, shows at most two solutions simultaneously, without jump or hysteresis phenomena. (2) As the acoustic excitation amplitude increases, both the response amplitude and the unstable zone significantly enlarge. An increase in nonlinear coefficients can reduce the response amplitude and increase instability. (3) When the acoustic excitation amplitude reaches a certain threshold, the oscillation mode of the bubble shifts from periodic to chaotic. Under the same initial conditions, the chaos threshold for sub-harmonic resonance is higher than that for super-harmonic resonance.

Double-pole dark-bright mixed solitons for a three-wave-resonant-interaction system

This Letter explores a three-wave-resonant-interaction system that is widely seen in fluid mechanics, plasma physics and nonlinear optics. With the aid of the iterated-form generalized Darboux transformation method, we obtain the second-order analytic solutions that illustrate the double-pole dark-bright mixed solitons. Under certain parameter conditions, the double-pole bright-dark-bright solitons are illustrated through the 3D, density and contour plots. Those plots demonstrate the stable attractive forces between the adjacent soliton components. Moreover, we conduct the asymptotic analysis on the double-pole bright-dark-bright solitons to investigate their physical properties, which are consistent with those in the plots. Our work might contribute to the studies of the dynamic properties of the curved dark-bright solitons and be extended to the double-pole dark-bright mixed solitons within the other integrable systems.

Investigation on the free vibration characteristics of vertical quasi-zero stiffness isolation system considering the disc spring's loading position

This study investigates a quasi-zero stiffness (QZS) isolation system designed for vertical vibration control in isolation structures. The QZS characteristics are achieved by connecting a negative stiffness disc spring in parallel with the linear positive stiffness spiral spring. Initially, the force-displacement relationship of the disc spring is established, taking into account the loading position, and validated through static experiments. Subsequently, the free vibration equations of the QZS system are derived. The Jacobi elliptic functions and the Newton-harmonic balance (NHB) method are then employed to study its free vibration characteristics without considering damping. The results demonstrate that the natural vibration period of the QZS system is related to the nondimensional amplitude and the height of the loaded inner cone. Furthermore, the second-order analytical approximate solution obtained via the NHB method exhibits remarkable accuracy. Finally, the critical damping ratio of the damped QZS isolation system is investigated using a numerical approach. The findings indicate that the critical damping ratio is only correlated with the nondimensional amplitude. This study addresses the gaps in existing research and provide recommendations for the parametric design of vertical QZS isolation system.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.

Analytical Solution and Quasi-Periodic Behavior of a Charged Dilaton Black Hole

With the vast breakthrough brought by the Event Horizon Telescope, the theoretical analysis of various black holes has become more critical than ever. In this paper, the second-order asymptotic analytical solution of the charged dilaton black hole flow in the spinodal region is constructed from the perspective of dynamics by using the two-timing scale method. Through a numerical comparison with the original charged dilaton black hole system, it is found that the constructed analytical solution is highly consistent with the numerical solution. In addition, several quasi-periodic motions of the charged dilaton black hole flow are numerically obtained under different groups of irrational frequency ratios, and the phase portraits of the black hole flow with sufficiently small thermal parameter perturbation display good stability. Finally, the final evolution state of black hole flow over time is studied according to the obtained analytical solution. The results show that the smaller the integral constant of the system, the greater the periodicity of the black hole flow.

Accurate analytical approximation to post-buckling of column with Ramberg−Osgood constitutive law

The post-buckling of a clamped column made of nonlinear elastic material and subject to axial compression is investigated in this paper. The Ramberg−Osgood type constitutive relation is adopted and it is expanded by using Taylor series. Based on Euler−Bernoulli beam theory, the exact governing equations are expressed in terms of rotation angle. Because the equations contain strongly nonlinear terms that are functions of the rotation angle, analytical solutions are virtually impossible. In this respect, this paper is focused on presenting an alternative method to construct concise yet accurate analytical approximate solutions for post-buckling of the Ramberg−Osgood column that is related to large rotation amplitude. The improved harmonic balance method is used to solve the nonlinear governing equations which are simplified via the Maclaurin series expansion and orthogonal Chebyshev polynomials. In addition, numerical solutions by applying the shooting method on the governing equations are obtained for comparison. The second-order analytical approximate solutions presented in this paper show excellent accuracy by comparing with numerical solutions. The analytical approximate method and numerical result presented in this paper can be applied as design guidelines for designing engineering structures that sustain large deformation, such as slender nonlinear compression of aluminum alloy columns, rods or braces.

Symmetry breaking in solitary solutions to the Hodgkin–Huxley model

This paper presents necessary and sufficient conditions for the existence of bright/dark solitary solutions in the Hodgkin–Huxley model. The second-order analytic solitary solutions are derived using the generalized differential operator technique. It is shown that the heteroclinic bifurcation in the Hodgkin–Huxley model yields a symmetry breaking effect. Trajectories of solitary solutions before the bifurcation lie on manifolds of one of the saddle points and the separatrix between periodic and non-periodic solutions. A new separatrix emerges after the heteroclinic bifurcation—but solitary solutions do not lie on this trajectory. This symmetry breaking effect is demonstrated using analytic and computational experiments.

Performance of Asymmetric Particulate Filter with Soot and Ash Deposits: Analytical Solution and Its Application

With ever-tightening emission regulations, particulate filters are critical for internal combustion engines to meet the stringent particulate matter emission standards. A fast way to predict the filter performance, instead of numerically solving the governing differential equations, is needed for filter design and selection, real-time control, malfunction detection, and deposit load sensing. Approximate analytical solutions for wall flow filters, considering asymmetric channels and arbitrary deposit amounts, are derived by a technique of successive approximation. The analytical predictions of filter pressure drop have been validated against both steady state and transient experimental measurements. Moreover, over a broad range of filter operating conditions, the accuracy of the second-order analytical solution is validated by comparisons with the numerical predictions. The derivation also provides analytical expressions for channel and wall velocity profiles along the filter length. This study reveals the...

Analysis of Duffing oscillator with time-delayed fractional-order PID controller

The free vibration of Duffing oscillator with time-delayed fractional-order Proportional-Integral-Derivative (FOPID) controller based on displacement feedback is studied. The second-order approximate analytical solution is obtained by KBM asymptotic method. The effects of the parameters in FOPID controller on the dynamical properties are characterized by some equivalent parameters. The correctness of the approximate analytical results is verified by the numerical results. The effects of the time-delayed FOPID controller with displacement feedback on control performances of Duffing oscillator are analyzed in detail by time response, and the stability conditions of zero solution and periodic motions are also presented. Finally, the control performances on Duffing oscillator with large damping are further analyzed. And the results show that one could take the advantage of time delay, when the parameters of time-delayed FOPID controller are chosen reasonably.

Approximate solution of a one-dimensional soil water infiltration equation based on the Brooks-Corey model

The solution of a one-dimensional unsaturated soil water infiltration equation can provide theoretical support when simulating soil water distribution. To solve the equation for one-dimensional vertical infiltration of soil moisture in unsaturated soil with initially uniform moisture content distribution, a new method is proposed based on the Principle of Least Action and the Variational Principle. We used second-, third- and fourth-order Taylor series expansions to obtain an approximate analytical solution for a one-dimensional, constant water head, vertical infiltration equation. The results were compared with a numerical solution obtained using the HYDRUS-1D software. Our results showed that the soil water content profile, accumulated infiltration and infiltration rate can be calculated by the new method, and the soil water content calculated by the second-, third- and fourth-order approximate analytical solutions was closer to the numerical solution in the higher water content areas. The relative error of the soil water content for clay loam, silt, loam and sandy loam ranged from 0.7187% to 3.0848%, 0.9667% to 2.2847% 0.6310% to 2.2943%, 1.7504% to 6.8823%, respectively. In four soils modeled numerically, the fourth-order results were the best, with the smallest relative errors and the best coefficients of determination. The longer the wetting front distance, the smaller the calculated errors. Thus, the fourth-order approximate analytical solution can be used to simulate the long-term vertical infiltration process. If the results do not require the highest error precision, the second-order approximate analytical solution was simple in form and also sufficient to be used to describe the water content of the vertical infiltration process. Our results show that the approximate analytical solution proposed provides an effective approach to predict soil water content in the vertical infiltration process.

Asymptotic solutions of coupled equations of supercritically axially moving beam

In supercritical regime, the coupled model equations for the axially moving beam with simple support boundary conditions are considered. The critical speed is determined by linear bifurcation analysis, which is in agreement with the results in the literature. For the corresponding static equilibrium state, the second-order asymptotic nontrivial solutions are obtained through the multiple scales method. Meantime, the numerical solutions are also obtained based on the finite difference method. Comparisons among the analytical solutions, numerical solutions and solutions of integro-partial-differential equation of transverse which is deduced from coupled model equations are made. We find that the second-order asymptotic analytical solutions can well capture the nontrivial equilibrium state regardless of the amplitude of transverse displacement. However, the integro-partial-differential equation is only valid for the weak small-amplitude vibration axially moving slender beams.

Improved Perturbative Solution of Yaroshevskii's Planetary Entry Equation

An improved approximate analytical solution is developed for Yaroshevskii's classical planetary entry equation for the ballistic entry of a spacecraft into planetary atmospheres at circular speed. Poincaré's method of small parameters is used to solve for the altitude and flight path angle as a function of the spacecraft's speed. From this solution, other important expressions are developed including deceleration, stagnation-point heat rate, and stagnation-point integrated heat load. The accuracy of the solution is assessed via numerical integration of the exact equations of motion. The solution is also compared to the classical solutions of Yaroshevskii and Allen and Eggers. The new second-order analytical solution is more accurate than Yaroshevskii's fifth-order solution for a range of shallow (−3 deg) to steep (up to −90 deg) entry flight path angles, thereby extending the range of applicability of the solution as compared to the classical Yaroshevskii solution, which is restricted to an entry flight path of approximately −40 deg.

Method of regularized sources for axisymmetric Stokes flow problems

Purpose– The purpose of this paper is to find solution of Stokes flow problems with Dirichlet and Neumann boundary conditions in axisymmetry using an efficient non-singular method of fundamental solutions that does not require an artificial boundary, i.e. source points of the fundamental solution coincide with the collocation points on the boundary. The fundamental solution of the Stokes pressure and velocity represents analytical solution of the flow due to a singular Dirac delta source in infinite space.Design/methodology/approach– Instead of the singular source, a non-singular source with a regularization parameter is employed. Regularized axisymmetric sources were derived from the regularized three-dimensional sources by integrating over the symmetry coordinate. The analytical expressions for related Stokes flow pressure and velocity around such regularized axisymmetric sources have been derived. The solution to the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary. The intensities of the sources are chosen in such a way that the solution complies with the boundary conditions.Findings– An axisymmetric driven cavity numerical example and the flow in a hollow tube and flow between two concentric tubes are chosen to assess the performance of the method. The results of the newly developed method of regularized sources in axisymmetry are compared with the results obtained by the fine-grid second-order classical finite difference method and analytical solution. The results converge with a finer discretization, however, as expected, they depend on the value of the regularization parameter. The method gives accurate results if the value of this parameter scales with the typical nodal distance on the boundary.Originality/value– Analytical expressions for the axisymmetric blobs are derived. The method of regularized sources is for the first time applied to axisymmetric Stokes flow problems.

Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method

Due to various sources of nonlinearities, micro/nano-electro-mechanical-system (MEMS/NEMS) resonators present highly nonlinear behaviors including softening- or hardening-type frequency responses, bistability, chaos, etc. The general Duffing equation with quadratic and cubic nonlinearities serves as a characterizing model for a wide class of MEMS/NEMS resonators as well as lots of other engineering and physical systems. In this paper, after brief reviewing of various sources of nonlinearities in micro/nano-resonators and discussing how they contribute to the Duffing-type nonlinearities, we propose a Homotopy Analysis Method (HAM) approach for derivation of analytical solutions for the frequency response of the resonators. Toward this aim, we first apply the HAM to the proposed Duffing equation, and through this procedure, we derive the first-order and second-order HAM-based analytical solutions for the frequency response of the resonator. As the main novelty, we show that the second-order solution benefits from a tunable parameter, known as the convergence-control parameter, which is a distinguishing aspect of the HAM and plays a key role in enhancing the accuracy of the obtained analytical expressions in strongly nonlinear problems. We use the obtained analytical solutions for the study of nonlinear dynamics in two types of electrostatically actuated MEMS resonators proposing hardening, softening or mixed behaviors near their primary resonance frequency. Numerical simulations are performed to validate the analytical results.

NONLINEAR BREAKER CHARACTERISTICS AND IMPULSE ON A SLOPING BOTTOM

This paper presents an experimental and theoretical investigation of nonlinear surface-wave propagation over a sloping bed. First, a second-order analytical solution for nonlinear surface-wave propagation over a sloping bed is derived using a perturbation method for the bottom slope α and the wave steepness e in an Eulerian system. Then, by transforming the flow field solution from an Eulerian system into a Lagrangian system, more accurate wave profiles are determined. New theoretical breaker characteristics and breaker impulses are derived using the kinematic stability parameter. Subsequently, a series of experiments to measure breaker characteristics and the breaker impulse are conducted in a wave tank. The theoretical solutions are compared with both the present experimental data and previously published experimental results. The results reveal that the analytical solution can reasonably describe the wave breaking phenomenon. In this paper, a new theoretical solution for the breaker characteristics and impulses is provided, which is proven to be a useful approach for follow-up studies to predict breaker characteristics and impulses.

Response of Fractional Oscillators With Viscoelastic Term Under Random Excitation

A system with fractional damping and a viscoelastic term subject to narrow-band noise is considered in this paper. Based on the revisit of the Lindstedt–Poincaré (LP) and multiple scales method, we present a new procedure to obtain the second-order approximate analytical solution, and then the frequency–amplitude response equations in the deterministic case and the first- and second-order steady-state moments in the stochastic case are derived theoretically. Numerical simulation is applied to verify the effectiveness of the proposed method, which shows good agreement with the analytical results. Specially, we find that the new method is valid for strongly nonlinear systems. In addition, the influences of fractional order and the viscoelastic parameter on the system are explored, and the results indicate that the steady-state amplitude will increase at a fixed point with the increase of fractional order or viscoelastic parameter. At last, stochastic jump is investigated via the received Fokker–Planck–Kolmogorov (FPK) equation to compute the stationary solution of probability density functions with its shape changing from one peak to two peaks with the increase of noise intensity, and the phenomena of stochastic jump is consistent with the solution of frequency–amplitude response equations.

An analytical solution for the interaction of two-dimensional currents and gravity short-crest waves

Short-crest wave and current coexist widely in the ocean environments. However, the interaction between them has been studied recently and the method of velocity potential function was generally applied. Unlike the previous study, this article considers that location variable‘x'and time variable‘t'are independent of each other and it does not take into account the capillary effect, thus leading to an addition of a first-order time item in the second-order velocity potential function. Based on the perturbation technique, a second-order analytical solution is derived. Comparisons between the result in this article, where variables‘x'and‘t'are dependent on each other and the capillary effect is considered, and Huang's solutions show the difference. The difference between wave profile and wave pressure on mudline will become apparent with the increase of wave height, indicating that the solutions obtained in this article will be much suitable for ocean conditions with larger wave height.

Analytical investigation of compressible oscillating flow in a porous media: A Second-order successive approximation technique

In this paper the field equations of compressible oscillating flow in the regenerator of Pulse Tube Refrigerators (PTRs) are solved. To obtain an analytical solution, successive approximation technique is applied to continuity, momentum, gas and matrix energy equations and ideal gas equation of state. In this respect, the flow field variables are expanded to second-order unsteady terms. To introduce the influence of consecutive terms in the expansions, a comparative study is conducted on first unsteady, second steady and second unsteady terms. The proposed model presents a powerful analytical tool, applicable to oscillating flow analysis in porous media and rapid optimization of the pulse tube cycle. Results proved that expanding the terms to second-order ones can enhance the solution precision considerably. Friction factor and Nusselt number for different meshes are proposed based on the second-order analytical solution.

Second-order wave diffraction by horizontal rectangular barriers

A complete second-order analytical solution for the nonlinear diffraction of waves by 2-D rectangular cylinders fixed at the free surface of finite-depth waters is presented. As a sequel to a previous linear treatment of the same problem, the present study revisits the nonlinear wave diffraction modelling in 2-D, in which a series of boundary value problems are built up by a classical perturbation expansion. Particular solution for the second-order potential is derived by means of a Fourier integral theorem, whilst the homogeneous second-order solution is obtained in a way similar to that of the linear problem. The present solution is validated by comparisons with experiments as well as with previous computational results of various researchers.

Improved Understanding of Vibrating-Wire Viscometer−Densimeters

The vibrating-wire technique has been widely applied in measurements of both viscosity and density. When tensioned by a suspended mass or sinker, the device may be used to measure these properties simultaneously. The sensitivity to density arises mainly from the effect of buoyancy on the tension and hence the resonance frequency of the vibrating wire. Earlier studies have used simplified working equations to relate the resonance frequency and the tension. In this work, we have employed the exact equations describing forced simple harmonic oscillations of a stiff wire under tension and also a second-order analytical solution of those equations. We show that the exact working equation describes the actual resonance frequency of tensioned tungsten wires with improved accuracy. This finding has been validated by means of experimental measurements of the resonance frequency at prescribed tensioning forces and at temperatures in the range (293 to 473) K. One consequence of this is that the true independently measured sinker volume may be used in preference to one obtained by calibration of the assembled sensor. The remaining parameters to be obtained are then the radius of the wire, Young’s modulus E for the wire, and an empirical parameter that accounts for the remaining departure from the theoretical expression. The values of wire radius obtained by calibration in a liquid of known viscosity have been compared with those from independent mechanical measurements having a relative uncertainty of 0.2 %. These comparisons have been made for both drawn wires and for a centerless ground rod with a diameter of 0.15 mm, and the differences are discussed in the light of scanning electron microscopy (SEM) studies of the circularity and uniformity of the wire. Finally, we provide experimental evidence that rough drawn wires may yield inaccurate viscosity measurements when applied to fluids having viscosities very different to the one used for calibration; conversely, a centerless-ground rod gave good results even when the viscosity to be measured was 60 times that of the calibration fluid.