Nonlinear Duffing Research Articles

The application of inerter in tuned mass absorber

In this paper we investigate the dynamics of tuned mass absorber with additional viscous damper and inerter attached to the pendulum. The devices are used to damp out oscillations of non-linear Duffing oscillator. Analysis of how these devices influence the dynamics of tuned mass absorber and its damping properties is shown. We calculate the detailed bifurcation diagrams and show how by changing the parameters of damper and inerter one can eliminate dangerous dynamic instabilities from the systems. Finally, in the last section we present an optimization of TMA׳s parameters in order to achieve best efficiency in damping of the main body vibrations.

A multiple scale time domain collocation method for solving non-linear dynamical system

Recently, a simple time domain collocation (TDC) method was proposed by researchers including the first author of the present paper, and has been successfully applied to obtain the harmonic, subharmonic, and superharmonic responses of the nonlinear Duffing oscillator. The TDC method is based on the point-collocation method performing within an appropriate period of the periodic solution, wherein the approximate solution is assumed as a Fourier series. Upon using the TDC method, the ordinary differential equation is transformed into a system of non-linear algebraic equations (NAEs), which can be readily solved by an NAE solver. In this study, using the Duffing oscillator as the prototype, we develop a multiple scale time domain collocation (MSTDC) method, by introducing a series of optimal multiple scales to the Fourier series of the approximate solution, to alleviate the ill-posedness of the system of collocation-resulting NAEs due to the inclusion of very high order harmonics in the approximate solution. Besides the MSTDC method, a multiple scale differential transformation (MSDT) method is proposed by introducing the multiple scales to the classical differential transformation method. Finally, numerical experiments are carried out to verify the accuracy and efficiency of the MSTDC and the MSDT methods.

Determination of effective mechanical properties of a double-layer beam by means of a nano-electromechanical transducer

We investigate the mechanical properties of a doubly clamped, double-layer nanobeam embedded into an electromechanical system. The nanobeam consists of a highly pre-stressed silicon nitride and a superconducting niobium layer. By measuring the mechanical displacement spectral density both in the linear and the nonlinear Duffing regime, we determine the pre-stress and the effective Young's modulus of the nanobeam. An analytical double-layer model quantitatively corroborates the measured values. This suggests that this model can be used to design mechanical multilayer systems for electro- and optomechanical devices, including materials controllable by external parameters such as piezoelectric, magnetostrictive, or in more general multiferroic materials.

Pseudo almost periodic solutions for a class of nonlinear Duffing system with a deviating argument

This paper is concerned with a nonlinear Duffing system with pseudo almost periodic coefficients and delay. Under proper conditions, by using theory of exponential dichotomies and contraction mapping principle, some sufficient conditions are established to ensure the existence and uniqueness of pseudo almost periodic solutions, which improve and extend some known relevant results. Moreover, an example is given to illustrate the theoretical results.

Circuit electromechanics with a non-metallized nanobeam

We have realized a nano-electromechanical hybrid system consisting of a silicon nitride beam dielectrically coupled to a superconducting microwave resonator. We characterize the sample by making use of the Duffing nonlinearity of the strongly driven beam. In particular, we calibrate the amplitude spectrum of the mechanical motion and determine the electromechanical vacuum coupling. A high quality factor of 480 000 at a resonance frequency of 14 MHz is achieved at 0.5 K. The experimentally determined electromechanical vacuum coupling of 11.5 mHz is quantitatively compared with finite element based model calculations.

Parameter Influence on the Harmonically Excited Duffing Oscillator

In this paper the influence of the initial conditions and the interaction of the parameters on the motion of the strong nonlinear Duffing oscillator are investigated. The initial conditions are arbitrary and need not be zero. An analytical procedure for solving the strong nonlinear differential equation with excitation term is developed. The obtained solutions give the physical explanation of the excited vibrations caused by the excitation force and non-zero initial conditions. The analytical results are compared with numerical results and show good agreement.

Enhancement of the performance of a hybrid nonlinear vibration energy harvester based on piezoelectric and electromagnetic transductions

A multiphysics model of a hybrid piezoelectric–electromagnetic vibration energy harvester (VEH), including the main sources of nonlinearities, is developed. The continuum problem is derived on the basis of the extended Hamilton principle, and the modal Galerkin decomposition method is used in order to obtain a reduced-order model consisting of a nonlinear Duffing equation of motion coupled with two transduction equations. The resulting system is solved analytically using the method of multiple time scales and numerically by means of the harmonic balance method coupled with the asymptotic numerical continuation technique. Closed-form expressions for the moving magnet critical amplitude and the critical load resistance are provided in order to allow evaluation of the linear dynamic range of the proposed device. Several numerical simulations have been performed to highlight the performance of the hybrid VEH. In particular, the power density and the frequency bandwidth can be boosted, by up to 60% and 29% respectively, compared to those for a VEH with pure magnetic levitation thanks to the nonlinear elastic guidance. Moreover, the hybrid transduction permits enhancement of the power density by up to 84%.

A pseudospectral method for nonlinear Duffing equation involving both integral and non‐integral forcing terms

AbstractThe Legendre pseudospectral method is developed for the numerical solution of nonlinear Duffing equation involving both integral and non‐integral forcing terms. By using differentiation matrix, the problem is reduced to the solution of a system of algebraic equations. The method is general, easy to implement, and yields very accurate results. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. Copyright © 2014 John Wiley & Sons, Ltd.

Optimal scale polynomial interpolation technique for obtaining periodic solutions to the Duffing oscillator

A novel optimal scale polynomial interpolation method (OSPIM) is proposed to attack the Duffing oscillator. This method is based on the ideas of multi-scaling and equilibrated matrix, such that the condition number of the coefficient matrix of the polynomial interpolation is minimized. The OSPIM can eliminate the Runge phenomenon, which occurs in the conventional polynomial interpolation, and is well suited for solving nonlinear oscillatory systems. In addition, we further alleviate the ill-posedness of polynomial interpolation by proposing a half-order technique, with which one can use a \(m\)-order polynomial to interpolate as many as \(2m+1\) points. We then employ the half-order OSPIM, i.e., OSPIM(H), as a trial function in conjunction with the simple point-collocation method, to solve the nonlinear Duffing equation. Moreover, the differential transformation method is used for the first time to solve a forced Duffing oscillator to compare with the present method. Finally, illustrative examples verify the accuracy and efficiency of the present method.

Disclosing the Complexity of Nonlinear Ship Rolling andDuffing Oscillators by a Signum Function

In this paper we study the nonlinear dynamical system ˙ x= f(x;t) from a newly developed theory, viewing the time-varying function of sign(kfk 2 kxk 2 2(f x) 2 )= sign(cos 2q) as a key factor, where q is the intersection angle between x and f. It together with sign(cosq) can reveal the complexity of nonlinear Duffing oscillator and a quadratic ship rolling oscillator. The barcode is formed by plotting sign(kfk 2 kxk 2 2(f x) 2 ) with respect to time. We analyze the barcode to point out the bifurcation of subharmonic motions and the range of chaos in the parameter space. The bifurcation diagram obtained by plotting the percentage of the first set of dis-connectivity A 1 :=fsign(cosq)=+1 and sign(cos 2q)=+1g with respect to the amplitude of harmonic loading leads to a finer structure of a devil staircase for the ship rolling oscillator, as well as a cascade of subharmonic motions to chaos for the Duffing oscillator.

The Consistency between MLP Method and Modified Method of Multiple Scales for Strong Nonlinear Primary Resonance of Duffing Equation Subject to Harmonic Excitation in Complex Number Field

With changing strong nonlinear Duffing equation subject to harmonic excitation with damping in complex number field as an object, the amplitude frequency response equation of primary resonance of the system is obtained through parametric transformation with application of MLP method and modified method of multiple scales. In different approximation solution forms and different time scales, the two methods lead to the same amplitude frequency response equation. Thus, the two methods are mutually verifiable. Numerical analysis shows that for the strong nonlinear Duffing equation with damping in complex number field, the nonlinear stiffness coefficient is more than zero and the amplitude frequency response curve of primary resonance leans to the left, which is different from the weak nonlinear Duffing equation.Chinese books catalog: O321

Nonlinear dynamics in aluminum nitride contour-mode resonators

In this work, we discuss the self-heating nonlinear dynamics of aluminum nitride contour mode resonators. The self-heating introduces a quadratic coupling between the resonator motion and its temperature. This coupling produces a nonlinear frequency response identical to the weakly nonlinear Duffing resonator in steady-state. However, since the thermal relaxation rate is much smaller than the mechanical relaxation rate, the dynamics of the resonator differ from the conventional Duffing resonator.

Application of Variational Iteration Method for Dropping Damage Evaluation of the Suspension Spring Packaging System

The dropping damage evaluation for packaging system is essential for safe transportation and storage. A dynamic model of nonlinear cubic-quintic Duffing oscillator for the suspension spring packaging system was proposed. Then, a first-order approximate solution was obtained by applying He’s variable iteration method. Based on the results, a damage evaluation equation was derived, which reveals the main controlling physical parameters for damage potential of drop to packaged products concretely. Finally, the dropping damage boundary curves and surfaces for the system were discussed. It was found that decreasing the suspension angle can improve the safe region of the system.

Highly accurate computation of finite-time Lyapunov exponent

Aiming at the shortcomings of current method of calculating finite-time Lyapunov exponent (FTLE), such as low accuracy, inability to obtain boundary values, etc., a method of highly accurately computing FTLE is proposed based on dual number theory. Firstly, the weakness and disadvantages of the finite difference method used widely for computing FTLE are described. Secondly, the dual number theory is introduced to evaluate the derivatives accurately and efficiently, and its distinct virtues are also presented. The computation of Cauchy-Green deformation tensors for a dynamical system is transformed into a numerical integration problem of solving the nonlinear ordinary differential equation in dual number space by the new method. Finally, the proposed method is applied to typical pendulum system and nonlinear Duffing oscillator separately. The results of simulation experiments indicate that the new method is effective, convenient and accurate for computing the field of FTLE, from which Lagrangian coherent structures can be identified successfully.

Nonlinear dynamics of stochastic resonance and its application in the method of weak signal detection

According to the exited stochastic resonance theory, we cannot obtain the dynamic behavior of a stochastic resonance (SR) system intuitively. In order to reveal the dynamic mechanism of SR, a kind of first-order Duffing equation attractor is analyzed at first, and then the property of nonlinear Duffing equation is studied, based on which the nonautonomous Duffing equation attractor curve is deduced. The output of SR system can be obtained by mapping the input signal on the attractor curve, and the dynamic mechanism of SR is explained by using the mapping method. Analysis of the result indicates that the intrinsic signal can push the system to move along the attractor curve, and the noise can evoke a transition response of the system under the given conditions. Some exited SR weak signal detection methods, such as the parameter-adjustment and damping-adjustment are extended by the proposed dynamic mechanism.

Asymptotic solution to the generalized Duffing equation for disturbed oscillator in stochastic resonance

A class of nonlinear generalized Duffing equation for disturbed oscillator is considered. Firstly, the typical Duffing equation is solved. Then approximate solutions to the nonlinear Duffing equation for disturbed oscillators in stochastic resonance is obtained using the generalized functional variation principle, and the uniform validity is proved.

Antiperiodic Solutions for a Kind of Nonlinear Duffing Equations with a Deviating Argument and Time-Varying Delay

This paper deals with a kind of nonlinear Duffing equation with a deviating argument and time-varying delay. By using differential inequality techniques, some very verifiable criteria on the existence and exponential stability of antiperiodic solutions for the equation are obtained. Our results are new and complementary to previously known results. An example is given to illustrate the feasibility and effectiveness of our main results.

Micromanipulation transfer of membrane resonators for circuit optomechanics

A capacitive coupling between mechanical resonator and a microwave cavity enables readout and manipulation of the vibrations. We present a set-up to carry out such experiments with aluminum membranes fabricated as stamps and transferred in place with micromanipulation. The membrane is held in place by van der Waals forces, and is supported by three microscopic points. We measure the lowest mechanical modes, and conclude the membrane vibrates as an essentially free–free resonator. Sliding clamping conditions are identified via a softening Duffing nonlinearity. The method will enable reduction of clamping losses, while maintaining a narrow vacuum gap for strong capacitive coupling.

A new sliding control strategy for nonlinear system solved by the Lie-group differential algebraic equation method

For a control problem of nonlinear system ẋ=f(x,u,t), the optimal control by minimizing a performance index is reformulated to be a set of differential algebraic equations (DAEs) with the Lagrangian being partially replaced by an exponentially time-decaying constraint: L1(x,t)=A0e-αt, and meanwhile the control force is bounded by |u|⩽umax. Then, we develop an implicit GL(n,R) Lie-group DAE (LGDAE) method to find u(t) by solving the DAEs: ẋ=f(x,u,t) and L1(x,t)-A0e-αt=0. Similarly, we propose a new sliding mode control (SMC) strategy by using the LGDAE to solve the control force, where in addition to the equivalent control force we add a compensated control force which is used to quickly steer and continuously enforce the state trajectory on the sliding surface. This novel SMC is robust and is chattering-free for regulator problem and finite-time tracking problem of nonlinear systems. Furthermore, we combine the above two methods as being a two-stage controller for the forced nonlinear Duffing oscillator by stabilizing it to an equilibrium point. The present SMC together with the LGDAE is also used to stabilize the state trajectory of some uncertain chaotic systems to a desired state point. Its robustness against uncertainty is obvious.

The Study of a Nonlinear Duffing – Type Oscillator Driven by Two Voltage Sources

In the present work, a detailed study of a nonlinear electrical oscillator with damping and external excitation is presented. The system under study consists of a Duffing-type circuit driven by two sinusoidal voltage sources having different frequencies. The dynamical behavior of the proposed system is investigated numerically, by solving the system of state equations and simulating its behavior as a circuit using MultiSim. The tools of the theoretical approach are the bifurcation diagrams, the Poincare sections, the phase portraits, and the maximum Lyapunov exponent. The numerical investigation showed that the system has rich complex dynamics including phenomena such as quasiperiodicity, 3-tori, and chaos.