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The purpose of this paper is to present an analytical approximate solution of the nonlinear mathematical model of the bifilar pendulum.

First, the equation of motion derived based on the classical dynamics law by only an angular oscillation assumption and vertical oscillation is neglected. The energy balance method is applied to solve an established model and an analytical formulation has been obtained for the nonlinear frequency of the bifilar pendulum.

A comparison of results with those obtained by a numerical solution of the exact model (without any simplifications) shows the precise accuracy even for a large amplitude of oscillation.

The proposed model and solution are relatively simple and can be applied instead to a linear model for achieving accurate results.

An iteration technique has been developed based on the Mickens iteration method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, coupled mass–spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced.

In this paper, the truncated terms of the Fourier series have been used and utilized in every step of the iteration.

The obtained results are valid for whole ranges of vibration amplitude of the oscillations. The approximated results are compared with existing and corresponding numerical (considered to be exact) results which show excellent agreement. The error analysis has been carried out and shown acceptable results for the proposed iteration technique.

Effectiveness of the proposed iteration technique is found in comparison with other existing methods. The method is demonstrated by examples.

The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.

The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.

Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.

The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

The purpose of this study is to address the nonlinear oscillations of single-crystal silicon micro-electromechanical systems (MEMS) accelerometers subjected to mechanical excitation.

The nonlinear behavior was detected and analyzed by using experimental, analytical and numerical approaches. Piezoelectric shaker as a source of mechanical excitation and differential laser Doppler vibrometer in combination with a micro system analyzer were used in the experimental effort. Two types of devices considered included nonencapsulated samples and samples encapsulated in nitrogen gas compressed between two glasses. Numerical and analytical investigations were conducted to analyze the nonlinear response. A novel method has been suggested to calculate the nonlinear parameters. The obtained experimental, numerical and analytical results are in good agreement.

It has been found that the nonlinearity leads to a shift in frequencies and generates higher harmonics, but, most importantly, reveals new phenomena, such as the jump and instability of the vibration amplitudes and phases.

It has been shown that under the constant excitation force, the MEMS device can work in both linear and nonlinear regions. The role of the beat phenomenon has been also addressed and discussed. It has been found that the attributes of the nonlinear response are strongly dependent on the level and duration of the excitation. It is concluded that the nonlinear response of the systems is strongly dependent on the level of the excitation energy. It has been also concluded that larger quality factors are able to enhance dramatically the nonlinear effects and vice versa.

An effective practical algorithm based on array oriented computations, suitable for use with vector computational architectures, is described. The algorithm allows for deteremination of self‐oscillations with unknown (in advance) periods, yet also includes the situation when the period is predetermined, particularly in the case of forced oscillations.

The purpose of this paper is to evaluate the accuracy of linear and quasi-steady aerodynamic models of aircraft aerodynamic models when a small unmanned aerial system flies in the presence of strong wind and gust at a high angle of attack and a high sideslip angle.

Compatibility analysis were done to improve the quality of recorded flight test data. A robust method called fuzzy logic modeling is used to set up the aerodynamic models. The reduced frequency is used to represent the unsteadiness of the flow field according to Theodorsen’s theory. The work done by the aerodynamic moments on the motions is used as the criteria of stability.

In portions of flight, aircraft’s stability and control derivatives were unstable and nonlinear functions of airflow angles and angular rates. The roll angle had an important effect on unsteadiness of directional oscillatory damping derivatives. The pilot-induced oscillation and wing rock possibilities were investigated and dismissed so that the lateral directional oscillatory motion was classified as a nonlinear Dutch roll oscillation. Major modeling enhancements or real-time parameter identification are required for the control of a small unmanned aerial system in off-nominal conditions. The robustness tests of all-weather autopilot systems must be done with consideration of sign change.

Oscillatory damping derivatives were reconstructed using flight test data and the inadequacy of engineering level software in predicting this type of instability observed and demonstrated for a flight in the presence of wind shear and external disturbances.

The purpose of this paper is to evaluate the performance of a numerical method for the solution to shallow-water equations on a staggered grid, in simulations for shear instabilities at two convective Froude numbers.

The simulations start from a small perturbation to a base flow with a hyperbolic-tangent velocity profile. The subsequent development of the shear instabilities is studied from the simulations using a number of flux-limiting schemes, including the second-order MINMOD, the third-order ULTRA-QUICK and the fifth-order WENO schemes for the spatial interpolation of the nonlinear fluxes. The fourth-order Runge-Kutta method advances the simulation in time.

The simulations determine two parameters: the fractional growth rate of the linear instabilities; and the vorticity thickness of the first nonlinear peak. Grid refinement using 32, 64, 128, 256 and 512 nodes over one wave length determines the exact values by extrapolation and the computational error for the parameters. It also determines the overall order of convergence for each of the flux-limiting schemes used in the numerical simulations.

The four-digit accuracy of the numerical simulations presented in this paper are comparable to analytical solutions. The development of this reliable numerical simulation method has paved the way for further study of the instabilities in shear flows that radiate waves.

This paper aims to study the oscillation phenomenon before chaos as well as its mechanism of occurrence in the energy-saving and emission-reduction system.

The system dynamics analysis, phase portrait analyses, equilibrium point analysis and bifurcation curve were applied to this paper.

First, the authors find an oscillation phenomenon previous to chaos. Second, on the one hand, the existence of two unstable saddles is the reason for the occurrence of oscillation phenomenon. On the other hand, the increasing of carbon emissions can arouse oscillation phenomenon.

This paper finds an oscillation phenomenon previous to chaos in the energy-saving and emission-reduction system. The mechanism of occurrence of oscillation phenomenon is studied. The existence of two unstable saddles is the reason for the occurrence of such oscillation phenomenon. The oscillation is related with fold bifurcation. The study also provides a theoretical basis for the further study of chaos control.

In this paper, the important aspects of vibration analysis of carbon nanotubes (CNTs) as nano-resonators are studied. This study has covered the important nonlinear phenomena such as jump super-harmonic and chaotic behavior. CNT is modeled by using the modified nonlocal theory (MNT).

In previous research studies, the effects of CNT’s rotary inertia, stiffness and shear modulus of the medium were neglected. So by considering these terms in MNT, a comprehensive model of vibrational behavior of carbon nanotube as a nanosensor is presented. The nanotube is modeled as a nonlocal nonlinear beam. The first eigenmode of an undamped simply supported beam is used to extract the nonlinear equation of CNT. Harmonic balance method is used to solve the equation, while to study its super-harmonic behavior, higher-order harmonic terms were used.

In light of frequency response equation, jump phenomenon and chaotic behavior of the nanotube with respect to the amplitude of excitation are investigated. Also in each section of the study, the effects of elastic medium and nonlocal parameters on the vibration behavior of nanotube are investigated. Furthermore, parts of the results in linear and nonlinear cases were compared with results of other references.

The present modification of the nonlocal theory is so important and useful for accurate investigation of the vibrational behavior of nano structures such as a nano-resonator.

The nonlinear friction disturbance of the moving joint surface of the feed system can lead to the residual vibration of the system, prolong the stability time of the system and reduce the motion precision and machining precision of the machine tool. This paper aims to concern the vibration between joint surfaces caused by nonlinear friction.

The model is established from the micro and macro scale based on the LuGre model. The friction characteristics of the moving joint surface are explored. The friction experiment of GCr15 pin and 45 steel disk is designed and the influence of lubrication condition, speed, acceleration and normal load on friction characteristics are studied.

Among the drive speed, damping and stiffness, the negative gradient effect of friction, which is characterized by the difference of static and dynamic friction coefficient Δµ, is the main cause of friction vibration between moving joint surfaces. Sufficient lubrication, a proper increase of speed and acceleration, a reasonable reduction of normal load can reduce the negative gradient effect, which can weaken the vibration caused by the nonlinear friction and improve the friction characteristics of the moving joint surface.

In the past studies, more attention has been paid to revealing the relationship between the relative speed and friction, while the acceleration is often ignored. The negative gradient effect of friction is improved in this paper by changing the contact conditions. Research findings of this paper effectively improve the friction characteristics of the moving interface and provide the basis for restraining the nonlinear vibration between the moving interfaces.

The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-11-2019-0476/