Numerical Fourth-order Runge-Kutta Research Articles

Nonlinear effect of Low-velocity impact on tapered laminated composite structures using spline finite strip method

In present paper, the behavior of tapered composite plate under low-velocity impact was studied. The tapered plates were used in many structures according to their stiffness to weight ratio, while, the behavior of such plates did not evaluate under impact load in the literatures. Geometrically nonlinear Von-Karman strain was taken into account with respect to large deformations. The spline finite strip method (SFSM) was used to predict the impact behavior of composite laminates. The numerical fourth-order Runge-Kutta integration technique was used to solve the governing differential equations. The displacement field was defined according to the refined plate theory. The four most common internal taper configurations were considered. Local indentation was modeled with the help of contact Hertz’s laws. The governing differential equation was written based on the principal of virtual displacement and Hamilton’s principle for dynamic systems. The obtained results on eccentric impact showed that contact force and indentation are increased when indenter contact near supports.

Measuring the Pollutants in a System of Three Interconnecting Lakes by the Semianalytical Method

Pollution has become an intense danger to our environment. The lake pollution model is formulated into the three-dimensional system of differential equations with three instances of input. In the present study, the new iterative method (NIM) was applied to the lake pollution model with three cases called impulse input, step input, and sinusoidal input for a longer time span. The main feature of the NIM is that the procedure is very simple, and it does not need to calculate any special type of polynomial or multipliers such as Adomian polynomials and Lagrange’s multipliers. Comparisons with the Adomian decomposition method (ADM) and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the NIM is a powerful alternative for differential equations providing more realistic series solutions that converge very rapidly in real physical problems.

The Multistage Homotopy Perturbation method for solving Hyperchaotic Chen system

Finding accurate and efficient methods for solving nonlinear hyper chaotic problems has long been an active research undertaking. Like the well-known Adomian decomposition method (ADM) and based on observations, it is hopeless to find HPM solutions of hyper chaotic systems valid globally in time. To overcome this shortcoming, we employ the multistage homotopy-perturbation method (MHPM) to the nonlinear hyperchaotic Chen system. Based on the cases investigated, MHPM is more stable for a longer time span than the standard HPM. Comparisons with the standard HPM and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the MHPM is a powerful alternative for nonlinear hyper chaotic system. The new algorithm and the new technique for choosing the initial approximations were shown to yield rapidly convergent series solutions.

Forced vibration analysis of a system equipped with the nonlinear displacement-dependent (NDD) damper

In this paper, forced vibration analysis of a mass-spring system equipped with a nonlinear displacement-dependent (NDD) damper is elaborated. To this end, the nonlinear governing differential equation of the system is derived for two types of soft- and hard- periodic excitations. In order to obtain the displacement of the excited system, the approximate analytical solution of the governing equation is developed using the multiple scales method (MSM). The proposed analytical formulations are performed for several cases of hard- and soft-excitation and are also verified by the numerical fourth-order Runge-Kutta method. Moreover, the performance of the NDD damper is analyzed and compared with the traditional linear damper used in the both hard- and soft-excited vibration analyses. For a same external periodic force, a comparison has also been carried out between the responses of the hard- and soft-excitations. It is found that the organizing the external force based on its amplitude into two types as soft- and hard- excitation, leads to a better estimated response in the forced vibration analysis. Moreover, the NDD damper has a superior performance in reducing the vibration amplitude thorough tending to the steady-state solution compared to the traditional linear damper.

Nonlinear Ion Harmonics in the Paul Trap with Added Octopole Field: Theoretical Characterization and New Insight into Nonlinear Resonance Effect.

The nonlinear harmonics within the ion motion are the fingerprint of the nonlinear fields. They are exclusively introduced by these nonlinear fields and are responsible to some specific nonlinear effects such as nonlinear resonance effect. In this article, the ion motion in the quadrupole field with a weak superimposed octopole component, described by the nonlinear Mathieu equation (NME), was studied by using the analytical harmonic balance (HB) method. Good accuracy of the HB method, which was comparable with that of the numerical fourth-order Runge-Kutta (4th RK), was achieved in the entire first stability region, except for the points at the stability boundary (i.e., β = 1) and at the nonlinear resonance condition (i.e., β = 0.5). Using the HB method, the nonlinear 3β harmonic series introduced by the octopole component and the resultant nonlinear resonance effect were characterized. At nonlinear resonance, obvious resonant peaks were observed in the nonlinear 3β series of ion motion, but were not found in the natural harmonics. In addition, both resonant excitation and absorption peaks could be observed, simultaneously. These are two unique features of the nonlinear resonance, distinguishing it from the normal resonance. Finally, an approximation equation was given to describe the corresponding working parameter, q nr , at nonlinear resonance. This equation can help avoid the sensitivity degradation due to the operation of ion traps at the nonlinear resonance condition.

A Theoretical Method for Characterizing Nonlinear Effects in Paul Traps with Added Octopole Field.

In comparison with numerical methods, theoretical characterizations of ion motion in the nonlinear Paul traps always suffer from low accuracy and little applicability. To overcome the difficulties, the theoretical harmonic balance (HB) method was developed, and was validated by the numerical fourth-order Runge-Kutta (4th RK) method. Using the HB method, analytical ion trajectory and ion motion frequency in the superimposed octopole field, ε, were obtained by solving the nonlinear Mathieu equation (NME). The obtained accuracy of the HB method was comparable with that of the 4th RK method at the Mathieu parameter, q = 0.6, and the applicable q values could be extended to the entire first stability region with satisfactory accuracy. Two sorts of nonlinear effects of ion motion were studied, including ion frequency shift, Δβ, and ion amplitude variation, Δ(C(2n)/C0) (n ≠ 0). New phenomena regarding Δβ were observed, although extensive studies have been performed based on the pseudo-potential well (PW) model. For instance, the |Δβ| at ε = 0.1 and ε = -0.1 were found to be different, but they were the same in the PW model. This is the first time the nonlinear effects regarding Δ(C(2n)/C0) (n ≠ 0) are studied, and the associated study has been a challenge for both theoretical and numerical methods. The nonlinear effects of Δ(C(2n)/C0) (n ≠ 0) and Δβ were found to share some similarities at q < 0.6: both of them were proportional to ε, and the square of the initial ion displacement, z(0)(2).

The Effect of Temperature Dependency of Surface Emissivity in 1-D and 2-D Nonlinear Heat Radiation Equations

Knowledge of the temperature dependency of the physical properties such as surface emissivity, which controls the radiative problem, is fundamental for determining the thermal balance of many scientific and industrial processes. The surface emissivity generally depends on surface temperature, wavelength, surface material geometry (curvature, roughness), and direction of observation, and often changes with oxidation, melting, coating, and even surface pollution. Current work studies the influences of temperature dependency of surface emissivity on heat transfer in a lumped system. The problem was investigated for both one-dimensional (1-D) and two-dimensional (2-D) systems. For 1-D equations, two recent analytical methods called the homotopy perturbation method (HPM) and parameterized perturbation method (PPM) are presented. Unlike classic perturbation methods, these techniques do not need small parameters for nonlinear heat transfer equations. Thus, we can apply them for large values of surface emissivity. For the 2-D domain, a finite-element code is used to obtain the unsteady distribution of temperature. Three different functions were chosen to describe the thermal behavior of surface emissivity. The solutions of 1-D nonlinear equations are compared with the accurate numerical fourth-order Runge–Kutta method, and excellent agreement (maximum error of 0.0021%) was observed. The capabilities of employed analytical methods are discussed and it is shown that HPM needs fewer series terms in comparison with PPM. For both 1-D and 2-D equations, it is illustrated that the temperature gradient increased by adding to the order of emissivity variation versus temperature.

Analytical and Numerical Solutions of Vapor Flow in a Flat Plate Heat Pipe

In this paper, the optimal homotopy analysis method (OHAM) and differential transform method (DTM) were applied to solve the problem of 2D vapor flow in flat plate heat pipes. The governing partial differential equations for this problem were reduced to a non-linear ordinary differential equation, and then non-dimensional velocity profiles and axial pressure distributions along the entire length of the heat pipe were obtained using homotopy analysis, differential transform, and numerical fourth-order Runge-Kutta methods. The reliability of the two analytical methods was examined by comparing the analytical results with numerical ones. A brief discussion about the advantages of the two applied analytical methods relative to each other is presented. Furthermore, the effects of the Reynolds number and the ratio of condenser to evaporator lengths on the flow variables were discussed.Graphical abstract

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincare (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.