Coupled Duffing Equations Research Articles

Applications of representation transformation and secular perturbation theory to coupled Duffing equations

In physics, the non-linear mode coupling is an important strategy to manipulate the mechanical properties of a vibrational system. Compared with the single-mode nonlinear system, the complex systems with two- or multi-mode nonlinear coupling have garnered considerable attention, among which the analytical solutions to the coupled Duffing equations are widely studied to solve nonlinear coupling. The fact is that the solving of the Duffing coupling equations generally starts with the eigenmodes solution of the linear equations. The trial solution of the coupled equations is the linear superposition of the eigenmodes. Under the secular perturbation theory and similar conditions, the Duffing coupling equation degenerates into two decoupled equations. However, thus far most of the solution methodologies are too complicated to unravel the underlying physical essence clearly. In this paper, first, by applying the representational transformation to the linear terms of the first-order coupled Duffing equations and the secular perturbation theory for the nonlinear terms, a decoupled expression of the first-order Duffing equations is derived, which can be solved more straightforwardly. Subsequently, in order to verify the correctness of the method, we design a coupled tuning fork mechanical vibration system, which consists of two experimental instruments to provide driving force and receive signals, two tuning forks and springs. The amplitude spectra are measured by an experimental instrument of forced vibration and resonance (HZDH4615), which provides a periodic driving signal for the tuning fork. The numerical fitting by software is employed to clarify the mechanism of the spectrum. Theoretically, the obtained fitting parameters can also evaluate some important attributes of the system. Most strikingly, due to the nonlinear coupling the splitting of the resonant peak and the phenomenon of “hysteresis loop” are clearly observed in the experiment. The research shows that the experimental results perfectly match the theoretical results obtained before. The method of solving coupled nonlinear equations in this article provides a solution and improvement of flexible adoption of nonlinear theory. On the other hand, it can be extended to coupled light and electricity systems, offer certain guidance for understanding the dynamic behavior of coupled systems, and will be conductive to the quantitative examination of numerous nonlinear coupling devices.

Analytical solution of the Duffing equation

PurposeThe 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.Design/methodology/approachThe 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.FindingsExpressions 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.Originality/valueThe 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.

A novel class of highly efficient and accurate time-integrators in nonlinear computational mechanics

A new class of time-integrators is presented for strongly nonlinear dynamical systems. These algorithms are far superior to the currently common time integrators in computational efficiency and accuracy. These three algorithms are based on a local variational iteration method applied over a finite interval of time. By using Chebyshev polynomials as trial functions and Dirac---Delta functions as the test functions over the finite time interval, the three algorithms are developed into three different discrete time-integrators through the collocation method. These time integrators are labeled as Chebyshev local iterative collocation methods. Through examples of the forced Duffing oscillator, the Lorenz system, and the multiple coupled Duffing equations (which arise as semi-discrete equations for beams, plates and shells undergoing large deformations), it is shown that the new algorithms are far superior to the 4th order Runge---Kutta and ODE45 of MATLAB, in predicting the chaotic responses of strongly nonlinear dynamical systems.

Existence of unpredictable solutions and chaos

Recently we introduced the concept of Poincaré chaos. In the present paper, by means of the Bebutov dynamical system, an unpredictable solution is considered as a generator of the chaos in a quasilinear system. The results can be easily extended to different types of differential equations. An example of an unpredictable function is provided. A proper irregular behavior in coupled Duffing equations is observed through simulations.

Nonlinear Analysis of Coupled Duffing Oscillators Based on Variable Step-Size Adomian Decomposition Method

In this paper, variable step-size Adomian Decomposition Method (VADM) is proposed to obtain approximate analytical solution of coupled Duffing oscillators. Using VADM, analytical expressions for two-degree-of-freedom coupled Duffing equations are derived for different amplitude values of external periodic excitation. The system stability is analyzed by analyzing time-domain graph and phase portraits. Compared with the MATLAB@ ODE45, simulation results reveal that VADM is effectively and well suited for nonlinear dynamic system.

Internal resonance in quartz crystal resonator and mass detection in nonlinear regime

This paper introduces the use of nonlinear jump phenomena in the frequency response of a quartz crystal resonator for mass detection. In contrast to recent studies that exploit parametric excitation for mass detection, our device exhibits nonlinear behavior modeled by the directly forced Duffing equation. In addition, internal resonance due to modal coupling is demonstrated for sufficiently large forcing amplitudes. A system of coupled Duffing equations based on a Galerkin expansion of the von Kármán plate equations is used to model these higher order effects.

Frequency tuning, nonlinearities and mode coupling in circular mechanical graphene resonators

We study circular nanomechanical graphene resonators by means of continuum elasticity theory, treating them as membranes. We derive dynamic equations for the flexural mode amplitudes. Due to the geometrical nonlinearity the mode dynamics can be modeled by coupled Duffing equations. By solving the Airy stress problem we obtain analytic expressions for the eigenfrequencies and nonlinear coefficients as functions of the radius, suspension height, initial tension, back-gate voltage and elastic constants, which we compare with finite element simulations. Using perturbation theory, we show that it is necessary to include the effects of the non-uniform stress distribution for finite deflections. This correctly reproduces the spectrum and frequency tuning of the resonator, including frequency crossings.

COALESCENCE OF THE TWO SECONDARY RESPONSES IN COUPLED DUFFING EQUATIONS

The novel coalescence of the secondary responses for the coupled Duffing equations are observed in this study. Two secondary responses that do not bifurcate from the primary responses merge into one due to saddle-node bifurcation generation within a specific parameter range. The frequency responses of the coupled Duffing equations are calculated using the harmonic balance method while the periodic orbits are detected by the shooting method. The stability of the periodic orbits is determined on utilizing Floquet theory. The parametric continuation algorithm is used to obtain the bifurcation points and bifurcation lines for a Duffing system with two varying parameters. The analytical results demonstrate the novel phenomenon that occurs in the Duffing equations.

A Nonlinear State-Space Model of a Resonating Single Fiber Scanner for Tracking Control: Theory and Experiment

A nonlinear state-space dynamic model of a resonating single fiber scanner is developed to understand scan distortion—jump, whirl, amplitude dependent amplitude and phase shifts—and as the basis for controllers to remove those distortions. The non-planar nonlinear continuum dynamics of a resonating base excited cantilever are reduced to a set of state-space coupled Duffing equations with centripetal acceleration. Methods for experimentally determining the model parameters are developed. The analytic frequency responses for raster, spiral and propeller scans are derived, and match experimental frequency responses for all three scan patterns, for various amplitudes, and using the same model parameters.

The role of higher modes in the chaotic motion of the buckled beam—II

Part II of this work completes the analysis of the necessary conditions for the chaos in the system of modal equations of motion of the buckled beam, i.e. a system of coupled Duffing equations. The direct calculation of the stable and unstable perturbed manifolds is used to establish the necessary condition for chaotic motion. It is shown that the higher components of the modified Melnikov vector do not change the critical condition obtained from consideration of only the first component. Analysis of the Lyapunov exponents of the system demonstrates the difference between necessary and sufficient conditions for steady-state chaos. Finally, the additional consideration of the non-hyperbolic modes shows that they can be neglected while developing the condition for the intersection of the stable and unstable manifolds.

Numerical computation of forced oscillations in coupled duffing equations

Oscillating phenomena in non-linear mechanical systems with two degrees of freedom described by coupled Duffing equations are studied from the computational view point. Galerkin approximations of order 7 are computed with a very high precision on an electronic computer by applying a numerical approximation method of Urabe for the Galerkin method. The existence of an exact isolated periodic solution in a small neighborhood of these Galerkin approximations is proved and the error bound of these Galerkin approximations is given. The stability of Stierel's integration method in combination with Galerkin approximations is shown.