Undamped Duffing Oscillator Research Articles

Analytical solution of the cantilevered elastica subjected to a normal uniformly distributed follower load

We report the full analytical solution of the large deformations of a cantilevered elastica loaded by a uniformly distributed follower pressure. We consider an unshearable, inextensible and linear elastic rod. We obtain a spatial nonlinear differential equation in the curvatures, analogous to the undamped Duffing oscillator with a constant driving force. We solve such differential equation, obtaining the curvature, although in implicit form, for arbitrarily large values of the load. We are then able to obtain the rotations owing to a change of variables from the curvilinear abscissa to the curvature. This step is somewhat mandatory due to the implicit nature of the solution. Finally, with the same change of variables, it is possible to obtain a closed-form solution for the deformation in Cartesian coordinates. The solutions show that the rod deforms into drop-like shapes. The number of drops is equal to the number of spatial periods of the solution, which goes with q∗1/3, with q∗ a dimensionless load normalised to the bending stiffness. Interestingly, we find that for q∗⩾3094.2, the number of drop-like shapes does not increase, but remains three.

Universal energy scaling law for optimally excited nonlinear oscillators.

We compute the optimal temporal profile for an external driving force F(t) that can maximize the energy absorption of any driven nonlinear oscillator. The technique is based on constraining the maximum amplitude of the force field such that optimal control theory can provide quasianalytical solutions. We illustrate this computational technique for the undamped Duffing oscillator as well as for a driven quantum mechanical two-level system. We find that under optimal force conditions the asymptotic time-dependence of the maximum amplitude growth is given by a power law X(t)∼t^{2/α}, where the (possibly noninteger) exponent is determined by the highest degree of the oscillator's nonlinearity α. As a universal result, this predicts that the maximal energy absorption of any nonlinear oscillator grows (under an optimized force field) quadratically in time. We also find for the two-level system that-even under optimized excitation conditions-the maximally achievable inversion does not monotonically increase with the force amplitude. It is characterized by an interesting sequence of n-cycle thresholds as well as a self-termination of the growth.

The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential

We examine the role of long--range interactions on the dynamical and statistical properties of two 1D lattices with on--site potentials that are known to support discrete breathers: the Klein--Gordon (KG) lattice which includes linear dispersion and the Gorbach--Flach (GF) lattice, which shares the same on--site potential but its dispersion is purely nonlinear. In both models under the implementation of long--range interactions (LRI) we find that single--site excitations lead to special low--dimensional solutions, which are well described by the undamped Duffing oscillator. For random initial conditions we observe that the maximal Lyapunov exponent $\lambda $ %: (a) tends to a positive value for KG and grows like $\varepsilon^(.25)$ for GF as the energy density $\varepsilon=E/N$ increases; (b) saturates to a positive value as the number of particles $N$ increase, scales as $N^{-0.12}$ in the KG model and as $N^{-0.27}$ in the GF with LRI, suggesting in that case an approach to integrable behavior towards the thermodynamic limit. Furthermore, under LRI, their non-Gaussian momentum distributions are distinctly different from those of the FPU model.

Exact analytical solutions for forced undamped Duffing oscillator

The steady-state response of an undamped Duffing oscillator to periodic external forces is studied. The forcing functions are chosen such that the time course of the displacement can be described by exact analytical expressions. The displacement and forcing functions are governed by Jacobi elliptic functions and thus are periodic but generally non-harmonic. The parameter of the elliptic functions is deliberately chosen. For certain parameter choices, exact analytical expressions are found for the frequency–amplitude relation.

Harmonic Balance Analysis of Snap-Through Orbits in an Undamped Duffing Oscillator

A simple nonlinear undamped Duffing oscillator has been studied for its snap-through behavior at large-amplitude vibrations. We present an algorithm that uses the harmonic balance (HB) method to find amplitude and frequency relationships in two- and three-term approximations for solutions that lie outside the separatrix in the phase space. Trends of the approximate solution properties are examined with reference to an analysis of the limit as the trajectory approaches the separatrix.

Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Duffing equation

With non-linearities, the frequency spectrum of an undamped Duffing oscillator should be composed of odd multiples of the driving frequency which can be interpreted as resonance driving terms. It is expected that the frequency spectrum of the corresponding numerical solution with high accurateness should contain nearly the same components. Hence, to contain these Fourier components and to calculate the amplitudes of these components in a more accurate and efficient way is the key to develop a new numerical method with high stability, accuracy and efficiency for the Duffing equation. To explore the possibility of using trigonometrically-fitting technique to build a numerical method with resonance spectrum, we design four types of Numerov methods, in which the first one is the traditional Numerov method, which contains no Fourier component, the second one contains only the first resonance term, the third one contains the first two resonance terms, and the last one contains the first three resonance terms, and apply them to the well-known undamped Duffing equation with Dooren's parameters. The numerical results demonstrate that the Numerov method fitted with the Fourier components is much more stable, accurate and efficient than the one with no Fourier component. The accuracy of the fitted method with the first three Fourier components can attain 10 −9 for a remarkable range of step sizes, including nearly infinite, except individual small range of instability, which is much higher than the one of the traditional Numerov method, with eight orders for step size of π / 2.011 .

Frequency of a Vanderpol oscillator with large cubic restoring nonlinearity

In [2], a modified Poincare method has been presented and applied to evaluate the period (frequency) of free oscillation in an undamped Duffing oscillator with large cubic restoring nonlinearity. There, the system examined is supposed to be near certain linear one with unknown (to be evaluated) frequency.

ANALYSIS OF POTENTIAL ENERGY IN BIREFRINGENT FIBER THROUGH UNDAMPED DUFFING OSCILLATOR EQUATION

In this paper, we consider the pulse propagation through the birefringent fiber with linear coupling effects. Using the Stokes parameter formalism and conserved densities, we construct the undamped Duffing oscillator equation from integro-differential equations. We also discuss the variation of potential well for different choices of physical parameters. From the linear stability analysis and numerical methods, we also discuss the stability, instability and bifurcation phenomenon in the birefringent fiber through Duffing oscillator type equation.

Evolution of polarization of a nonlinear pulse in birefringent fiber with quintic effects

We investigate the evolution of polarization of a soliton pulse with quintic self and cross phase modulation effects in birefringent nonlinear fiber. To characterize the state of polarization of a soliton pulse, we derive a system of integro-differential equations. The solitary wave solutions are obtained to discuss the different state of polarization. Moreover, after constructing suitable conserved densities, we derive the undamped Duffing oscillator equation from the evolution of the Stokes parameters. The stability analysis is carried out by both analytical and numerical methods.

ANALYSIS OF DUFFING'S OSCILLATOR EQUATION WITH TIME-DEPENDENT PARAMETERS

In this study, an undamped Duffing's oscillator equation with time-dependent parameters has been considered. The time-varying part is expanded in a series of ultraspherical polynomials in the spirit of Sinha and Chou and only the constant part is retained. The non-linearity parameter is assumed to be small so that the number of iterations required is only two. The results compare well with those obtained by the Runge–Kutta fourth order method.

ANALYTICAL SOLUTION OF THE FORCED DUFFING'S OSCILLATOR

This paper presents an analytical approach based on the power series method for determining the periodic solutions of the forced undamped Duffing's oscillator. The time variable is first transformed into a new harmonically oscillating time which transforms the governing differential equation into a form suitable for power series analysis. Hamilton's principle for non-conservative systems is then used to determine the frequency of the oscillating time. The formation is applied to a number of periodic solutions and excellent agreement is obtained with numerical solutions. The potential of the method is highlighted by comparing the results for a periodic response with perturbation and harmonic balance solutions.