Thomas D. Burton

On the multi-scale analysis of strongly non-linear forced oscillators

Abstract We consider the resonant response of strongly non-linear oscillators of the form u + 2ϵηu + mu + ϵƒ(u) = 2ϵpcosΩt, where ƒ(u) is an odd non-linearity, ϵ need not be small, and m = −1, 0, or + 1. Approximate solutions are obtained using a multiple-scale approach with two procedural steps which differ from the usual ones: (1) the detuning is introduced in the square of the excitation frequency Ω and as a deviation from the so called backbone curve and (2) a new expansion parameter α = α(ϵ) is defined, enabling accurate low order solutions to be obtained for the strongly non-linear case.

Chaotic dynamics of particle dispersion in fluids

An analysis of the Lagrangian motion for small particles denser than surrounding fluid in a two‐dimensional steady cellular flow is presented. The Stokes drag, fluid acceleration, and added mass effect are included in the particle equation of motion. Although the fluid motion is regular, the particle motion can be either chaotic or regular depending on the Stokes number and density ratio. The implications of chaotic motion to particle mixing and dispersion are discussed. Chaotic orbits lead to the dispersion of particle clouds which has many of the features of turbulent dispersion. The mixing process of particles is greatly enhanced since the chaotic advection has the property of ergodicity. However, a high dispersion rate was found to be correlated with low fractal dimension and low mixing efficiency. A similar correlation between dispersion and mixing was found for particles convected by a plane shear mixing layer.

A perturbation method for certain non-linear oscillators

Abstract We present a perturbation method for the analysis of single degree of freedom non-linear oscillation phenomena governed by an equation of motion containing a parameter ϵ which need not be small. The approach is to define a new parameter α = α ( ϵ ) in such a way that asymptotic solutions in power series in α converge more quickly than do the standard perturbation expansions in power series in ϵ. Phenomena considered are free vibration of strongly non-linear conservative oscillators and steady state response of strongly non-linear oscillators subject to weak harmonic excitation.

Non-linear oscillator limit cycle analysis using a time transformation approach

Abstract A method is presented for the analysis of limit cycle behavior of autonomous non-linear oscillators characterized by second order ordinary differential equations containing a small parameter. The method differs from the classical perturbation methods in that the dependent variable is not expanded in a power series in the small parameter. Rather, a new independent variable is sought such that in its domain the motion is simple harmonic. Use of this time transformation technique to generate limit cycle phase portrait, amplitude and period is presented. We show results of the application of the method to the van der Pol oscillator, to an oscillator with quadratic damping, and to a modified van der Pol oscillator which is statically unstable in the limit of small motion.

On higher order methods of multiple scales in non-linear oscillations-periodic steady state response

Abstract The application of the method of multiple scales (MMS) to determine higher order steady state responses of weakly non-linear, harmonically forced oscillators is considered. Two fundamentally different versions of MMS are currently in use in the study of such non-linear resonance phenomena. Our purpose is to describe the way in which these two versions lead to different results and to discuss each version from the standpoint of the basic ideas underlying the perturbation approach. The work is considered relevant because of the relative ease in making higher order calculations by using symbolic manipulation, which will probably render such higher order analyses routine in the future.

Large amplitude primary and superharmonic resonances in the Duffing oscillator

Second order perturbation analyses are presented of the order three superharmonic resonance in a Duffing oscillator, u+δu˙+u3 = P cos ϯt and of the primary resonance in a Duffing oscillator with softening non-linearity, u+δu˙-u3 = P cos ϯt. It is found in the former case that it is necessary to go to the second order to obtain realistic steady state stability results. In the latter problem, however, the second order solution gives little or no improvement on the first order result. Also presented, for the latter problem, is what is thought to be the first example of a “double jump” phenomenon for single degree of freedom, harmonically excited oscillators.

Analysis of non-linear autonomous conservative oscillators by a time transformation method

The problem considered is that of the free vibration of the class of strongly non-linear oscillators u + mu + αf(u) = 0, where m = 1, 0 or −1, f(u) is a non-linear function of the displacement u(t), and α is not small. Through a continuous stretching of the time t a new time T is found in whose domain the response is simple harmonic. This time transformation method allows the response u(t) and period τ to be calculated to any desired degree of accuracy; however, the primary objective is to develop reasonably simple, accurate approximations for the oscillators considered.

On the amplitude decay of strongly non-linear damped oscillators

The problem considered is that of the amplitude behavior of the strongly non-linear damped oscillator u + 2 ϵu + u + αf(u) = 0 where ϵ is small, u and f(u) are O(1), α may be arbitrarily large, and f(u) is an odd function. A simple approximation to the amplitude decay is obtained by first linearizing the static terms and then using the WKB approximation. Of particular interest is the influence of the non-linearity on the amplitude decay. Several examples which illustrate this influence are provided.

Quantification of chaotic dynamics for heavy particle dispersion in ABC flow

A six‐dimensional nonlinear dynamic system describing the Lagrangian motion of a heavy particle in the Arnold–Beltrami–Childress (ABC) flow was numerically studied. Lyapunov exponents and fractal dimension were used to quantify the chaotic motion. A single set of ABC flow parameters and a limited set of initial conditions were used. Given these restrictions, the following were found. (1) Attractor fractal dimension varies significantly with Stokes number, and, depending on inertia, periodic, quasiperiodic, and chaotic attractors may exist. (2) Particle drift reduces the fractal dimension when the drift is small. It can also cause irregular jumps when the drift parameter is close to one. (3) Quasiperiodic orbits on smooth two‐dimensional manifolds were shown to be the most common ultimate solutions of the system when either the inertia or the drift is relatively large. (4) Different initial conditions can lead to different attracting sets; however, most of them have the same dimension. (5) A direct measure...

Use of Random Excitation to Develop Pod Based Reduced Order Models for Nonlinear Structural Dynamics

This paper considers use of Proper Orthogonal Decomposition (POD), also known as the Karhunen-Loeve (K-L) method, to obtain reduced order dynamic models of nonlinear structural systems. The study applies POD to simulated time series data to extract dominant “modes” that describe the system behavior. The “POD modes” are used to formulate reduced order differential equation models (ROM’s) of the structure in which the dependent variables are the POD modal coordinates. Two example systems are considered: 1) a clamped beam whose tip is placed between attracting magnets; POD analysis of this system was done by Feeny and Kerschen [1] using harmonic excitation to excite chaotic motions, which were analyzed to develop the POD modes for reduced order modeling, and 2) a chain of oscillators having an isolated nonlinear Duffing element. Our approach is to generate the POD modes for model reduction by using strong, band limited random excitation to excite vibratory motions. The richness of this type of excitation is intended to provide responses whose POD-based reduced models can be used with reasonable accuracy for system parameters that differ from those used to generate the reduced order model (this is the central issue in using POD to generate reduced order nonlinear models of structures). Our results indicate that, due to the spectral richness of the random excitation, this type of excitation can be used with reasonable accuracy for conditions that differ from those used to generate the reduced order model. The method works well for the chain of oscillator system, but less well for the magnetic beam system, due to the presence of multiple stable equilibria in this system. A useful result of this work is the finding that the number of POD modes required to achieve accurate reduced order differential equation models may be considerably larger than the number of POD modes needed to accurately project the full model responses onto a subspace defined by dominant POD modes. Also, it is shown that for the clamped beam problem with multiple equilibria, we require more modes to develop a reduced order model than we require for a chain of oscillators.Copyright