APPLICATION OF NON-SMOOTH MODELLING TECHNIQUES TO THE DYNAMICS OF A FLEXIBLE IMPACTING BEAM

Non-smooth modelling techniques have been successfully applied to lumped mass-type structures for modelling phenomena such as vibro-impact and friction oscillators. In this paper, the application of these techniques to continuous elements using the example of a cantilever beam is considered. Employing a Galerkin reduction to form an N -degree-of-freedom modal model, a technique for modelling impact phenomena using a non-smooth dynamics approach is demonstrated. Numerical simulations computed using the non-smooth model are compared with experimentally recorded data for a flexible beam constrained to impact on one side. A method for dealing with sticking motions when numerically simulating the beam motion is presented. In addition, choosing the dimension of the model based on power spectra of experimentally recorded time series is discussed.     

Grazing bifurcations in an elastic structure excited by harmonic impactor motions

In this article, non-smooth dynamics of an elastic structure excited by a harmonic impactor motion is studied through a combination of experimental, numerical, and analytical efforts. The test apparatus consists of a stainless steel cantilever structure with a tip mass that is impacted by a shaker. Soft impact between the impactor and the structure is considered, and bifurcations with respect to quasi-static variation of the shaker excitation frequency are examined. In the experiments, qualitative changes that can be associated with grazing and corner-collision bifurcations are observed. Aperiodic motions are also observed in the vicinity of the non-smooth bifurcation points. Assuming the system response to be dominated by the structure’s fundamental mode, a non-autonomous, single degree-of-freedom model is developed and used for local analysis and numerical simulations. The predicted grazing and corner-collision bifurcations are in agreement with the experimental results. To study the local bifurcation behavior at the corner-collision point and explore the mechanism responsible for the aperiodic motions, a derivation is carried out to construct local Poincaré maps of periodic orbits at a corner-collision point such as the one observed in the soft-impact oscillator.     

A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems

The high-order purely frequency-based harmonic balance method (HBM) presented by Cochelin and Vergez (2009) [1] and extended by Karkar et al. (2013) [2] now allows to follow the periodic solutions of regularized non-smooth systems (stiff systems). This paper compares its convergence property to a reference method in applied mathematics: orthogonal collocation with piecewise polynomials. A first test is conducted on a nonlinear smooth 2 degree-of-freedom spring mass system, showing better convergence of the HBM. The second test is conducted on a one degree-of-freedom vibro-impact system with a very stiff regularization of the impact law. The HBM continuation of the nonlinear mode was found to be very robust, even with a very large number of harmonics. Surprisingly, the HBM was found to have a better convergence than the collocation method for this vibro-impact system.     

Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation

A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the help of the modified quasi-conservative averaging, the total energy of the new system can be approximated as a Markov process, and the stationary probability density function （PDF） of the total energy is derived. The response PDFs of the original system are obtained using the analytical solution of the stationary PDF of the total energy. The validity of the theoretical results is tested through comparison with the corresponding simulation results. Moreover, stochastic bifurcations are also explored.     

High excitation energy structures in208Pb

Level-crossing signals of impact radiation of He I have been analyzed with regard to the determination of excitation matrices. The signals have been registered at various electricfield strengths by sweeping a magnetic field strength ?_z through the positions ofΔm=2 crossings of Zeeman substates ofn ³ D ₂ andn ³ D ₃ levels (n=4 and 5) of He I excited by 35keV-He⁺-He collisions. We analyzed the signal amplitudes and signal shapes with regard to a spin-dependence of the excitation matrix usually discarded by theoretical arguments. To this end a parametrization of spin-dependent excitation matrices has been performed, and the dependence of the level-crossing amplitudes on the 19 parameters obtained have been studied. It turned out that the observed relative signal amplitudes can satisfactorily be explained by only two parameters, the two multipole components of orderk=2 andk=4 of the alignment of the orbital angular momentum. Thus in agreement with theoretical estimations no spin-dependence must be taken into account.     

Microwave heating of electrons of a dense plasma column at frequencies higher than electron cyclotron frequency

In the overdense collisionless plasma column inserted through the narrow sides of a rectangular waveguide, the excited electron cyclotron harmonic waves (CHWs) are studied by means of two movable probes and a phase interferometer in the range of parameters of 2 >ω/ω _ce >1; 0·5 < (ω _pe /ω)² < 15. Two kinds of CH waves have been found in the dipole mode:

1. Stable backward CH waves with the phase velocity in the direction from the axis to the periphery.
2. Unstable backward CH waves with the phase velocity in the opposite direction.

From the correlation measurements and amplitude distribution of the CH waves at the applied frequency and its second harmonic frequency it has been found that in the range of 2 >ω/ω _ce > >1·5 efficient nonlinear resonant interaction of CH waves takes place. The position of the loci of resonant interaction inside the plasma column has been found both experimentally and by a simple new graphical procedure for the resonant conditions of the formω ₂=2ω ₁; k₂=2¦k₁¦. In absence of this condition, no effective generation of the second harmonic frequency has been found. The resonant interaction of CHWs atω/ω _ce =1·85 is the cause of self-trapping of CH waves between the zones of resonant interaction in radial direction and of the anomalous heating rate of electrons.     

Global dynamics of a vibro-impacting linear oscillator

The steady state, vibro-impacting responses of one dimensional, harmonically excited, linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation. The steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction. Unlike the free, harmonically excited oscillator, the phase space of a vibro-impacting system may be inhabited by many attracting sets. For example, there are sub-harmonic, multi-impact, periodic orbits and chaotic, steady state responses. In order to build a qualitative understanding of vibro-impact response, an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges. Use is made of the Poincaré section or stroboscopic mapping technique, essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer. The qualitative understanding gained from the analysis and simulations is discussed in an engineering context.     

Stochastic responses of vibro-impact duffing oscillator excited by additive Gaussian noise

The stationary responses of vibro-impact Duffing oscillator excited by additive Gaussian white noise are studied by using the quasi-conservative averaging method. With the help of a non-smooth variable transformation and the Dirac delta function, the response probability density functions (PDFs) are formulated analytically. Meanwhile, the results are validated numerically using Monte Carlo simulations.     

Small-angle scattering during formation of periodic structures in light-sensitive films subjected to laser radiation

Patterns of small-angle scattering that appear in thin AgCl films containing granular silver upon the formation of spontaneous periodic structures by p-polarized and circularly polarized laser beams (He-Ne laser, P=8 mW, λ = 632.8 nm) are studied. It is found that, at angles of incidence exceeding 4° regardless of the type of polarization, the diffraction of the waveguide modes excited by the dominant C gratings from neighboring microgratings with the vectors K ≠ K c dominantly contributes to the small-angle scattering. In the case of circularly polarized light, the pattern of small-angle scattering becomes more complicated as a result of the formation of S _? gratings and related secondary regular gratings.     

Transverse (lateral) instantaneous force of an acoustical first-order Bessel vortex beam centered on a rigid sphere

In a recent report [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719-724], it has been found that the instantaneous axial force (i.e. acting along the axis of wave propagation) of a Bessel acoustic beam centered on a sphere is only determined for the fundamental order (i.e. m = 0) but vanishes when the beam is of vortex type (i.e. m > 0, where m is the order (or helicity) of the beam). It has also been recognized that for circularly symmetric beams (such as Bessel beams of integer order), the transverse (lateral) instantaneous force should vanish as required by symmetry. Nevertheless, in this commentary, the present analysis unexpectedly reveals the existence of a transverse instantaneous force on a rigid sphere centered on the axis of a Bessel vortex beam of unit magnitude order (i.e. |m| = 1) not reported in [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719-724]. The presence of the transverse instantaneous force components of a first-order Bessel vortex beam results from mathematical anti-symmetry in the surface integrals, but vanishes for the fundamental (m = 0) and higher-order Bessel (vortex) beams (i.e. |m| > 1). Here, closed-form solutions for the instantaneous force components are obtained and examples for the transverse components for progressive waves are computed for a fixed and a movable rigid sphere. The results show that only the dipole (n = 1) mode in the scattering contributes to the instantaneous force components, as well as how the transverse instantaneous force per unit cross-sectional surface varies versus the dimensionless frequency ka (k is the wave number in the fluid medium and a is the sphere’s radius), and the half-cone angle β of the beam. Moreover, the velocity of the movable sphere is evaluated based on the concept of mechanical impedance. The proposed analysis may be of interest in the analysis of transverse instantaneous forces on spherical particles for particle manipulation and rotation in drug delivery and other biomedical or industrial applications.     

Controlling chaos in dynamic-mode atomic force microscope

We successfully demonstrated the first experimental stabilization of irregular and non-periodic cantilever oscillation in the amplitude modulation atomic force microscopy using the time-delayed feedback control. A perturbation to cantilever excitation force stabilized an unstable periodic orbit associated with nonlinear cantilever dynamics. Instead of the typical piezoelectric excitation, the magnetic excitation was used for directly applying control force to the cantilever. The control force also suppressed the cantilever's occasional bouncing motions that caused artifacts on a surface image.     

Perturbative analysis of cross-talk of two pieces of photorefractive phase gratings stored in one point by angle multiplexing

In this paper, we have applied a perturbative expanding method to the hopping model, and studied the cross-talk between two pieces of phase gratings stored in one point of photorefractive material by angle multiplexing. The coupling equations and their steady solution have been derived. It has been found that the fundamental harmonic of the first grating is coupled with the second-order harmonic of the second grating and the second-order harmonic of the first gating is coupled with the fundamental harmonic of the second grating. The positions of maximums of the plots about the normalised fundamental and second-order harmonics of the space-charge field vs. the normalised grating vector (k/k₀) are at k/k₀=1.414 and k/k₀=1.01 for Δk/k₀=0, respectively, which are at k/k₀=1 and for the case of storing single grating in one point. This kind of deviation has been confirmed by our two-wave coupling experiment with Ce:KNSBN crystal. At the same time, we have found the degree of deviation of the second-order harmonic of the space-charge field strongly depends on the difference of normalised grating vector (k/k₀)between the two pieces of gratings, but that of fundamental harmonic weakly depends on it. In addition, it has been found that the degrees of deviation of both harmonics do not depend on the modulation depth.     

Four-wave mixing and hyper-Raman scattering in CuCl

Non-degenerate four-wave mixing using two non-collinear laser beams with frequencies (wavevectors) ω_p, ω_t (k_p, k_t) respectively is studied in CuCl. Two emission lines at frequencies ω⁽¹⁾=2ω_t-ω_p, and ω⁽²⁾=2ω_p-ω_t are observed. Their excitation spectrum is sharply peaked if the phase-match condition k⁽¹⁾=2k_t-k_p is fulfilled. This is the case, if ω_p coincides with the hyper-Raman lines (R⁺_T, R^-_T) of the laser labelled (t) in a well-defined geometrical configuration.     

Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations

There have been many contributions concerned with non-smooth dynamics. The purpose of this study is focused on the global stochastic dynamics of a kind of vibro-impact oscillator under the multiple harmonic and bounded noisy excitations. The well-known cell-to-cell mapping method is firstly developed to investigate the incursive fractal boundaries between the attracting domains of different random attractors, and a specific Poincaré map is then set up to explore the noise-contaminated dynamical transitions in the system. Lastly, the leading Lyapunov exponents and the surrogate tests are used to identify the noise-contaminated dynamics. It is shown that several random attractors will coexist in the phase space of the randomly driven system by adjusting the parameters’ values, and fractal boundaries may also arise between the attracting domains of different random attractors. Under the joint action of the harmonic excitation and the weak bounded noise excitation, the noisy period-doubling process, similar to a deterministic one, can appear in the Poincaré’s global cross-section by increasing the strength of the bounded noisy excitation. Moreover, the noisy periodic, the noisy chaotic, and the random-dominant dynamics are also distinguished from the noise-contaminated signals.     

Impulsive control of a class of vibro-impact systems

In this Letter, the impulsive control method is developed to stabilize the chaotic motions in a class of vibro-impact systems. The strategy of the control is to implement the pulses just when the impact occurs. As applications of this method, we present the numerical simulations of two impact oscillators. Our numerical results indicate that the method used here could suppress chaos into periodic orbits which embedded in the chaotic attractor effectively, and also show that the method is robust even for high levels of multiplicative noise or additive noise.     

Harmonic balance/Galerkin method for non-smooth dynamic systems

Models of non-linear systems frequently introduce forces with bounded continuity resulting in non-smooth (even discontinuous) flow. Examples include systems with clearances, backlash, friction, and impulses. Asymptotic methods require smooth (differentiable) flow and are therefore ill-suited for analyzing non-smooth systems. In these cases, the traditional harmonic balance method may be used to obtain approximate periodic solutions, but the method suffers from extremely slow convergence in general. Generalizations of the traditional harmonic balance method are introduced in this paper that result in superior convergence rates and superior modes of convergence. These improvements derive from the introduction of one or more expansion functions that possesses the same degree of continuity as the exact solution. In particular, forming an infinite series of such functions results in an expansion in the same function space of the exact solution. This expansion converges pointwise to the exact solution and to all derivatives thereof. These improvements are illustrated by example upon re-evaluating a classical single degree-of-freedom model for friction-induced vibration.     

On effective molecular electronic charge densities and vibrational potential energy functions

Empirical relationships connecting equilibrium internuclear distances, harmonic force constants and atomic numbers for diatomic molecules which were presented in a recent communication by Anderson and Parr (Chem. Phys. Lett.10, 293 (1971)) are discussed in detail for 4b-6b and 1a-7b molecules as sample cases. It is shown how to estimate cubic and quartic force constants for these diatomic molecules using these relationships. Methods for estimating stretching force constants in polyatomic and excited state diatomic molecules are presented; sample cases CO₂, CS₂, OCS and the states of CO are treated.     

Nonlinear spring model for frictional stick-slip motion

Frictional stick-slip dynamics is discussed using a model of one oscillator pulled by a nonlinear spring force. We focus our attention on the nonlinear spring parameter k₀. The dynamics of the model is carefully studied, both numerically and analytically. Our numerical investigation, which involves bifurcation diagrams, shows a rich spectrum of dynamical behavior including periodic, quasi-periodic and chaotic states. On the other hand, and for a good selection of parameters , the motion of the particle involves periodic stick-slip, erratic and intermittent motions, characterized by force fluctuations, and sliding. This study suggests that the transition between each of motion strongly depends on the nonlinear parameter k₀. The system also displays resonance at fractional frequencies of the oscillator.