Second-order Approximate Solution Research Articles

A simplified model and its asymptotic solution for the crashworthiness design of graded cellular material

Cellular materials have strong energy absorption capacity and have been extensively used in crashworthiness design of structures. The mechanical responses of cellular materials are related to their relative density. Introducing a density gradient to cellular materials may further improve their crashworthiness. A backward design method based on a nonlinear plastic shock model was developed to meet the crashworthiness requirement of protecting a mass when impinging a graded cellular rod. For the case of constant impact force, an asymptotic solution to the density distribution of graded cellular rod is obtained in this paper. The density distributions between the explicit asymptotic solution and the exact solution numerically obtained with a fourth-order Runge-Kutta scheme have been compared. The first-order approximate solution is a linear approximation, and the second-order approximate prediction is basically consistent with the numerical solution. For the impact force history, the first-order approximate prediction has a large deviation, and the second-order approximate prediction has a slight deviation. For practical application the second-order approximate solution seems to be accurate enough. Cell-based finite element models are employed to verify the asymptotic solution. The finite element models have been constructed and computed via ABAQUS/Explicit code. Samples of graded cellular rods designed from the asymptotic solution are tested numerically. The predictions using the second-order approximate solution of density distribution is very close to the finite element results. The design using the first-order approximate solution is not accurate enough, but in fact it is not so bad. Compared the uniform density distribution, the second-order approximate prediction is good for the crashworthiness optimization. The design of density distribution for a specific impact scenario is found to be inappropriate for much high velocity impact. In summary, a case study shows that the second-order approximate solution is very close to the exact solution and it is good enough for the design. The backward design method and the asymptotic solution of density distribution are helpful to guide the crashworthiness design of graded cellular materials.

Periodic Solution and Stability Behavior for Nonlinear Oscillator Having a Cubic Nonlinearity Time-Delayed

The current paper investigates the dynamics of the dissipative system with a cubic nonlinear time-delayed of the type of the damping Duffing equation. A coupling between the method of the multiple scales and the homotopy perturbation has been utilized to study the complicated dynamic problem. Through this approach, a cubic nonlinear amplitude equation resulted in at the first-order of perturbation; meanwhile, a quintic equation appears at the second-order of perturbation. These equations are combined into one nonlinear quintic Landau equation. The polar form solution is used, and linearized stability configuration is applied to the nonlinear amplitude equation. Also, a second-order approximate solution is achieved. The numerical illustrations showed that the damping, delay coefficient, and time delay play dual roles in the stability behavior. In addition, the nonlinear coefficient plays a destabilizing influence.

Multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations

• The second-order two-scale (SOTS) approximate solutions are constructed based on multiscale asymptotic analysis. • The error estimates for the SOTS solutions are obtained both in the pointwise sense and integral sense. • A multiscale numerical algorithm based on finite element method (FEM) is brought in details. • Our SOTS method can save computer resources. Only SOTS solutions can capture the microscale oscillation information. • The asymptotic homogenization theory in cartesian coordinate system is extended to the curvilinear coordinate system. This study develops a novel multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Firstly, the multiscale asymptotic analysis for these multiscale problems is given successfully, and the formal second-order two-scale approximate solutions for these multiscale problems are constructed based on the above-mentioned analysis. Then, the error estimates for the second-order two-scale (SOTS) solutions are obtained. Furthermore, the corresponding SOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and effectiveness of our multiscale computational method. Moreover, our multiscale computational method can accurately capture the local thermoelastic responses in composite block structure, plate, cylindrical and doubly-curved shallow shells.

Active magnetic bearing-based tuned controller to suppress lateral vibrations of a nonlinear Jeffcott rotor system

Active magnetic bearing system is an up-to-date technology that supports rotors without physical contacts and facilitates the vibration control in rotating machinery. Within this research, a tuned positive position feedback controller is proposed to control the lateral vibrations in a Jeffcott rotor system having cubic and quadratic nonlinearities. The controller is integrated into the system via four electromagnetic poles that act as actuators. The nonlinearity due to the electromagnetic coupling is included in the system model. A second-order approximate solution to the system governing equations is sought by utilizing asymptotic analyses. Bifurcation behaviours are investigated for both the system and controller parameters. The influence of the air-gap size, bias current, disc eccentricity, feedback gain, and control gain on the vibration amplitudes has been explored. The analytical results approved that the proposed controller can reduce the vibration amplitudes close to zero at any spinning speed even at large disc eccentricity. Then, numerical confirmations for the acquired analytical results have been performed that illustrated an excellent agreement with the analytical ones. Finally, a comparison with already published articles is included.

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.

On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force

This paper deals with the vibration analysis of a horizontally supported Jeffcott rotor system. Both nonlinear restoring force and the rotor weight are considered in the system modeling. The model shows a small difference between the natural frequencies of the vertical and horizontal mode. The multiple scales perturbation technique is utilized to obtain a second-order approximate solution at the simultaneous resonance case. The bifurcation analyses are conducted. The stability of the obtained solution is investigated by applying Lyapunov first method. The influences of all the parameters on the system behavior are explored. The Effect of both the negative and positive values of the nonlinear stiffness coefficient is studied. At the large rotor eccentricity, the analysis revealed the following: (1) the existence of three different stable solutions in an interval of the rotational speed. (2) The disk exposed to two consecutive jumps if its speed crossed the resonant speed. (3) For a soft spring, localized and nonlocalized oscillation in both the horizontal and vertical mode occurs. (4) For a hard spring, nonlocalized oscillation occurs in the two directions in addition to the localized motion in the vertical direction only (5) The system is very sensitive to initial conditions. Then, numerical simulations are performed to confirm the accuracy of the approximate results. It is found that the predictions from the analytical solutions are in a good agreement with the numerical simulations. Finally, a comparison with previously published work is included.

Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system

This paper investigates the vibration control of a horizontally suspended Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The system frequency (angular speed) -response curve is plotted at different values of the rotor eccentricity. The analysis illustrated that the system has a high oscillation amplitude and exhibits some nonlinear behaviors before control. A Proportional-Derivative (PD)-controller is integrated into the system via two pairs of electromagnetic magnetic poles. The nonlinearity due to the electromagnetic coupling is considered in the system model. A second-order approximate solution is obtained by utilizing multiple scales perturbation method. The bifurcation analyses of the controlled system are conducted. The results showed the high efficiency of the controller to mitigate the nonlinear vibrations of the considered system. Numerical simulations are carried out to validate the accuracy of the analytical results. The numerical results confirmed the excellent agreement with the analytical solutions. Then, the optimal working conditions of the system are concluded. Finally, a comparative study with previously published work is reported.

Primary resonance of coupled cantilevers subjected to magnetic interaction

Based on a distributed-parameter model, the forced vibration of a cantilever pair excited by a sinusoidal base movement is analyzed. Two cantilevers are coupled at their free ends by a linear spring. A nonlinear concentrated magnetic force acts on the tip of one cantilever, serving at the nonlinear boundary condition of the continuous model. The magnetic force is modeled as a fractional function, strongly dependent on the distance between two magnets. Via the method of multiple scales, the primary resonance is analyzed for all modes. A second-order approximate solution and its stability condition are analytically captured. It is revealed that the frequency–response curves are sensitive to the distance between the two magnets. The curve may exhibit the hardening-type, softening-type or linear behavior due to the existence of the quadratic nonlinearity. The outcomes are supported by the numerical simulations very well.

A simple harmonic balance method for solving strongly nonlinear oscillators

In this paper, a simple harmonic balance method (HBM) is proposed to obtain higher-order approximate periodic solutions of strongly nonlinear oscillator systems having a rational and an irrational force. With the proposed procedure, the approximate frequencies and the corresponding periodic solutions can be easily determined. It gives high accuracy for both small and large amplitudes of oscillations and better result than those obtained by other existing results. The main advantage of the present method is that its simplicity and the second-order approximate solutions almost coincide with the corresponding numerical solutions (considered to be exact). The method is illustrated by examples. The present method is very effective and convenient method for solving strongly nonlinear oscillator systems arising in nonlinear science and engineering.

A method to stochastic dynamical systems with strong nonlinearity and fractional damping

In this paper, a new technique is proposed to deal with strongly nonlinear stochastic systems with fractional derivative damping and random harmonic excitation. Combining the advantages of Linstedt–Poincare (L–P) method and multiple scales method, introducing a different frequency expansion form and a time transformation, a series of perturbation equations is obtained according to the powers of parameter. Then, we eliminate the secular producing terms to solve the perturbation equations to derive the second-order approximate solution. Furthermore, the steady-state frequency–amplitude function in deterministic case is analyzed, and the first-order and second-order steady-state moments of the amplitude are also discussed in the presence of random harmonic excitation. In order to explore the effectiveness of the proposed approximate method, two classical examples were proposed to verify the theoretical results through numerical simulations. Especially, the method can be used to investigate some types of extremely strong odd nonlinear terms via the discussions of each example.

A theoretical and experimental investigation on large amplitude free vibration behavior of a pretensioned beam with clamped–clamped ends using modified homotopy perturbation method

In this study, large amplitude free vibration behavior of a pretensioned Euler–Bernoulli beam is investigated both theoretically and experimentally. The Hamilton's principle is used to derive the beam governing equation of motion. By implementing the Galerkin's method and assuming the clamped–clamped boundary condition, the partial differential equation is converted to an ordinary nonlinear differential equation. Because of the large coefficient of the nonlinear term, the new homotopy perturbation method proposed by He, is modified to solve the governing nonlinear equation. Comparing the first- and the second-order approximate solutions of the modified homotopy perturbation method (MHPM) and those available in the literature demonstrates that the second-order MHPM leads to a more accurate solution which is valid for a wide range of the vibration amplitudes. The results have been validated by the experimental tests and the MHPM method. Also, the results show that an increase in the vibration amplitude and/or the pretension load increases the fundamental resonance frequency ratio. Besides, it would decrease with increasing the beam slenderness ratio.

Analysis of a nonlinear magnetic levitation system vibrations controlled by a time-delayed proportional-derivative controller

A time-delayed proportional-derivative controller is proposed in this paper to reduce the horizontal vibration of a magnetically levitated system having quadratic and cubic nonlinearities to primary and parametric excitations. Applying multiple scales perturbation technique, a second-order approximate solution is sought to analyze the nonlinear behavior of the system. The effects of the time delay are studied to indicate the stable range of time delays for the best performance. The results are compared with the numerical simulations, and this shows a good verification for the approximate solution and for the control algorithm used in this paper. A comparison with previously published work is included at the end.

Effect of soil–structure interaction on the nonlinear response of an inextensional beam on elastic foundation

The nonlinear response of a beam on elastic foundation subjected to the harmonic excitation is investigated, and the effect of soil–structure interaction on the primary resonance of the beam is analyzed. Considering the inextensional condition, the nonlinear equation of motion of a beam on elastic foundation is proposed via the Hamilton principle. Then, the method of multiple scales is used to obtain the frequency–response equation and second-order approximate solution of the dynamic response of the beam on elastic foundation. Finally, numerical results are presented to investigate the effects of the second-order moment, foundation models, Winkler parameter and excitation amplitude on the primary resonance of the beam on elastic foundation.

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.

The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials

In this paper, a kind of second-order two-scale (SOTS) computation is developed for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials, where the convection part is composed of long thin parallel pipes with periodic distribution, the conduction part occupied by solid materials and the radiation part is on the pipe's walls and the surfaces of cavities. First of all, by asymptotic expansion of the temperature field, the homogenization problem, first-order correctors and second-order correctors are obtained successively. Then, the error estimation of the second-order two-scale approximate solution is derived on some regularity hypothesis. Finally, the corresponding finite element algorithms are proposed and some numerical results are presented. The numerical tests indicate that the developed method can be successfully used for solving the integrated heat transfer problem, which can reduce the computational efforts greatly.

Responses of Duffing oscillator with fractional damping and random phase

This paper aims to investigate dynamic responses of stochastic Duffing oscillator with fractional-order damping term, where random excitation is modeled as a harmonic function with random phase. Combining with Lindstedt–Poincare (L–P) method and the multiple-scale approach, we propose a new technique to theoretically derive the second-order approximate solution of the stochastic fractional Duffing oscillator. Later, the frequency–amplitude response equation in deterministic case and the first- and second-order steady-state moments for the steady state in stochastic case are presented analytically. We also carry out numerical simulations to verify the effectiveness of the proposed method with good agreement. Stochastic jump and bifurcation can be found in the figures of random responses, and then we apply Monte Carlo simulations directly to obtain the probability density functions and time response diagrams to find the stochastic jump and bifurcation. The results intuitively show that the intensity of the noise can lead to stochastic jump and bifurcation.

Comparison of two iteration procedures for a class of nonlinear jerk equations

A modified Michens iteration procedure and a direct Michens iteration procedure are applied to determine approximate periods and analytical approximate periodic solutions of a class of nonlinear jerk equations. When we use the modified Michens iteration procedure to deal with the nonlinear jerk equations, we need to solve the nonlinear algebra equation for determining the approximate angular frequency. The higher the number of iterations, the more difficult it is to deal with the nonlinear algebra equation. This is because the kth-order approximate solution obtained by the modified Michens iteration procedure is a function of the angular frequency. However, since the kth-order approximate solution obtained by the direct Michens iteration procedure is independent on the kth-order approximate angular frequency, the form of nonlinear algebra equation that determines the kth-order approximate angular frequency of nonlinear jerk equation is similar. The second-order approximate period obtained by the direct Michens iteration procedure provides very accurate results for the large initial velocity amplitudes. But the second-order approximate period obtained by the modified Michens iteration procedure is invalid for all amplitudes B of the initial velocity. A comparison of the first- and second-order approximate periodic solutions obtained by the direct Michens iteration procedure with the numerically exact solutions shows that the second-order approximate periodic solution is much more accurate than the first one. Thus, the direct Michens iteration procedure is very effective for the class of nonlinear jerk equations.

Bifurcation in Perturbation Analysis:Calvo Pricing Examples

In recent macro models with staggered price and wage settings, the presence of variables such as relative price and wage dispersion is prevalent, which leads to the source of bifurcations. In this paper, we illustrate how to detect the existence of a bifurcation in stylized macroeconomic models with Calvo (J Monet Econ 12(3):383---398, 1983) pricing. Following the general approach of Judd (Numerical methods in economics, 1998), we employ l'Hospital's rule to characterize the first-order dynamics of relative price distortion in terms of its higher-order derivatives. We also show that, as in the usual practice in the literature, the bifurcation can be eliminated through renormalization of model variables. Furthermore, we demonstrate that the second-order approximate solutions under this renormalization and under bifurcations can differ significantly.

Multi-scale method for the quasi-periodic structures of composite materials

In this paper, we shall present a new second-order and two-scale approximate solution of boundary value problems for the linear elasticity systems with quasi-periodic structures by multi-scale analysis. The computation of strain and stress is presented here. This idea can apply to the damage computation of composite materials. Because the computation of the local stress by this method having more precision, it can provide some inspiration for construct optimal design. Finally numerical results show that the method presented in this paper is effective and reliable.