Nonlinear Single Mode Research Articles

Modal analysis of non-conservative systems with friction-induced strong nonlinear damping by using response-controlled testing

In recent years, the response-controlled stepped-sine testing (RCT) method has made important contributions to the evolution of the single nonlinear mode (SNM) method from a postulate to a well-established theory with the accurate modal identification of various nonlinear mechanical systems ranging from simple benchmark beams to real missiles. Since the RCT method resides on the SNM theory which is based on Rosenberg’s periodic motion definition of nonlinear modes, it has been regarded as limited to weakly damped mechanical systems that mainly exhibit strong conservative nonlinearities. An important contribution of this current paper is extending the RCT method to non-conservative mechanical systems with complex nonlinear modes. The extension is first numerically validated on a strongly friction-damped cantilever beam and then experimentally validated on the actuation mechanism of a real control fin that exhibits strong softening-stiffening nonlinearity (up to 50% shift of natural frequency) due to friction/backlash and very high and nonlinear damping (up to 30% modal damping ratio) due to friction. Another important contribution of this paper is to demonstrate the applicability of the superposition principle in nonlinear dissipative systems with closely spaced modes at an acceptable accuracy if the nonlinear modes are similar in shape to their corresponding linear modes and the modal coupling terms in the equation of motion can be ignored. The assumption of similar nonlinear modes is reasonable for most practical applications that exhibit dissipative nonlinearities due to mechanical joints. Although the assumption of ignorable coupling terms does not reside on a solid theoretical framework, numerical studies on a friction-damped lumped system with closely spaced modes give promising results, pointing to the significant potential of the modal superposition in nonlinear systems.

Generation of stable temporal doublet by a single-mode silicon core optical fiber

A highly nonlinear single mode anomalous dispersion silicon-core fiber is designed and optimized at the operating wavelength of 2.2 µm for the purpose of generating stable temporal pulse doublets. To designate the output pulse pair as a perfect Gaussian doublet, two new parameters, dissimilarity factor () and co-similarity index (μ cs) are introduced. Different input pulse parameters such as power, pulse width and chirp are optimized to obtain Gaussian temporal doublets at the shortest optimum length (∼few cm) which is sufficiently smaller in comparison to silica fibers reported earlier. The output pulse pairs remain as a doublet for quite a good stability length. In view of serving practical purposes, the possibilities of fluctuations of input power and pulse width are included to investigate the changes in stability length and effective repetition rate (ERR). The change in ERR along the fiber length produces a remarkable change in free carrier concentration in core, which has also been taken into account for the first time as per our knowledge to obtain the temporal pulse doublet in the so designed Si-core fiber.

Effects of internal resonances in the pitch glide of Chinese gongs.

The framework of nonlinear normal modes gives a remarkable insight into the dynamics of nonlinear vibratory systems exhibiting distributed nonlinearities. In the case of Chinese opera gongs, geometrical nonlinearities lead to a pitch glide of several vibration modes in playing situation. This study investigates the relationship between the nonlinear normal modes formalism and the ascendant pitch glide of the fundamental mode of a xiaoluo gong. In particular, the limits of a single nonlinear mode modeling for describing the pitch glide in playing situation are examined. For this purpose, the amplitude-frequency relationship (backbone curve) and the frequency-time dependency (pitch glide) of the fundamental nonlinear mode is measured with two excitation types, in free vibration regime: first, only the fundamental nonlinear mode is excited by an experimental appropriation method resorting to a phase-locked loop; second, all the nonlinear modes of the instrument are excited with a mallet impact (playing situation). The results show that a single nonlinear mode modeling fails at describing the pitch glide of the instrument when played because of the presence of 1:2 internal resonances implying the nonlinear fundamental mode and other nonlinear modes. Simulations of two nonlinear modes in 1:2 internal resonance confirm qualitatively the experimental results.

Generation of Four-Wave Mixing in a Highly Non-Linear Optical Fiber Using a Tunable Dual Wavelength Fiber Laser Source

In this paper, we demonstrate a Four Wave Mixing (FWM) effect in a highly nonlinear single mode optical fiber. A tunable dual-wavelength fiber laser is used to provide the pump and signal probe to generate partially degenerate FWM. Calculations using coupled differential equations and experimental results in the generation of FWM were analyzed. We observed that FWM conversion efficiency depends more significantly on the wavelength detuning between pump and signal wavelength as compared to zero-dispersion wavelength. The tunability is achieved using an arrayed waveguide grating which provides different wavelength combinations of the dual-wavelength fiber laser output and can be tuned from 1530.47nm to 1548.61nm with a spacing of 100GHz. The highly nonlinear fiber has a nonlinear coefficient, γ of 10.8 W-1km-1. Hence, to achieve higher FWM output power the pump power of the dual wavelength fiber laser has to be higher than that of the signal probe. The optimum measured values of pump and signal power are +13.3dBm and +5dBm respectively, limited buy the current experimental setup.

Synthesis of nonlinear frequency responses with experimentally extracted nonlinear modes

Determining frequency response curves is a common task in the vibration analysis of nonlinear systems. Measuring nonlinear frequency responses is often challenging and time consuming due to, e.g., coexisting stable or unstable vibration responses and structure-exciter-interaction. The aim of the current paper is to develop a method for the synthesis of nonlinear frequency responses near an isolated resonance, based on data that can be easily and automatically obtained experimentally. The proposed purely experimental approach relies on (a) a standard linear modal analysis carried out at low vibration levels and (b) a phase-controlled tracking of the backbone curve of the considered forced resonance. From (b), the natural frequency and vibrational deflection shape are directly obtained as a function of the vibration level. Moreover, a damping measure can be extracted by power considerations or from the linear modal analysis. In accordance with the single nonlinear mode assumption, the near-resonant frequency response can then be synthesized using this data. The method is applied to a benchmark structure consisting of a cantilevered beam attached to a leaf spring undergoing large deflections. The results are compared with direct measurements of the frequency response. The proposed approach is fast, robust and provides a good estimate for the frequency response. It is also found that direct frequency response measurement is less robust due to bifurcations and using a sine sweep excitation with a conventional force controller leads to underestimation of maximum vibration response.

Nonlinear hybrid modal synthesis based on branch modes for dynamic analysis of assembled structure

Abstract This paper describes a simple and fast numerical procedure to study the steady state responses of assembled structures with nonlinearities along continuous interfaces. The proposed strategy is based on a generalized nonlinear modal superposition approach supplemented by a double modal synthesis strategy. The reduced nonlinear modes are derived by combining a single nonlinear mode method with reduction techniques relying on branch modes. The modal parameters containing essential nonlinear information are determined and then employed to calculate the stationary responses of the nonlinear system subjected to various types of excitation. The advantages of the proposed nonlinear modal synthesis are mainly derived in three ways: (1) computational costs are considerably reduced, when analyzing large assembled systems with weak nonlinearities, through the use of reduced nonlinear modes; (2) based on the interpolation models of nonlinear modal parameters, the nonlinear modes introduced during the first step can be employed to analyze the same system under various external loads without having to reanalyze the entire system; and (3) the nonlinear effects can be investigated from a modal point of view by analyzing these nonlinear modal parameters. The proposed strategy is applied to an assembled system composed of plates and nonlinear rubber interfaces. Simulation results have proven the efficiency of this hybrid nonlinear modal synthesis, and the computation time has also been significantly reduced.

Active control of Chinese gongs

Active control provides one of the most promising ways of modifying instruments' sound. However, among the various control techniques covered by this discipline, most of the experimental applications have so far been limited to instruments keeping a linear behavior in normal playing conditions. This study explores the possibility of applying active control to Chinese gongs. These instruments exhibit geometric nonlinearities in their dynamical behavior such as a very characteristic pitch glide of the fundamental mode. The implementation of a nonlinear control of this pitch glide is introduced following a two-steps process. First, a reduced nonlinear model of the instrument dynamics is determined from the von Karman and normal form theories. This formulation allows for describing the nonlinear dynamics of the fundamental mode by one single nonlinear mode governed by a Duffing equation. The experimental identification of the Duffing model parameters is performed by measuring the backbone curve of the fundame...

Nonlinear Analysis of a Gyroklystron Amplifier with Misaligned Electron Beam

In this paper, a nonlinear single mode analysis has been developed for studying the beam-wave interaction behavior of gyroklystron amplifiers. The normalized parameters: field amplitude, length and detuning parameter, are optimized for the efficient device operation through the contour plots, and have been extended here in terms of a mathematical model. Further, in the present work, the momentum and phase for each cavity of the gyroklystron amplifier have been obtained by considering the actual device parameters instead of arbitrarily chosen parameters to estimate the efficiency in individual cavities to get the overall efficiency of the device. The bandwidth of the device is thus obtained by estimating the variations of RF power output with frequency. Moreover, the effects of practical problems of the electron beam misalignment and axial velocity spread on the performance of gyroklystron amplifier have also been analyzed in the presented model. It has been found that the RF output power is more sensitive in comparison to the device bandwidth to the electron beam misalignment. To validate the developed mathematical model, an experimental 35 GHz four-cavity gyroklystron amplifier, reported in the literature, has been selected. The results from the nonlinear formulation predicted a stable RF output power of 180 kW at 34.9 GHz, with ~56 dB gain, 26 % efficiency and a bandwidth of 0.54 % for a 72 kV, 9.6 A gyrating electron beam. The analytically obtained values of RF output power and gain are found to be in agreement with the experimental results within ~5 %. The analytical result has also been validated through the 3D PIC simulation using a commercially available code within ~5 %.

Nonlinear single-mode and multi-mode panel flutter oscillations at low supersonic speeds

In this study we numerically analyse nonlinear flutter oscillations of elastic plate in a gas flow. In contrast to many other studies, we use inviscid flow model instead of piston theory or other simplified aerodynamic theories. This study aims to investigate the region of low supersonic Mach numbers, 1<M<2, where several plate eigenmodes can be simultaneously unstable, and resulting oscillations are governed by nonlinear interaction of growing modes. Three types of unstable plate behaviour have been obtained. First, at 0.76<M<1, the plate diverges. Second, at 1<M≤1.67, single-mode flutter occurs in three distinct forms: limit cycle in the first mode (1<M<1.33 and 1.5<M≤1.67) or higher modes; limit cycle in the first and second modes being in internal 1:2 resonance (1.12<M<1.33 and 1.42<M<1.5); and non-periodic oscillations with several dominating frequencies being in more complex ratio (1.33<M<1.42). Third, at M=1.82 and increased dynamic pressure, coupled-mode flutter appears. Amplitudes and spectra of each limit cycle type are analysed. The role of aerodynamic nonlinearity in the formation of limit cycle oscillations is discussed.

Prediction of modal dispersion in nonlinear single mode optical fiber using simple Chebyshev technique

We have predicted the normalized group delay and modal dispersion in Kerr nonlinear single mode fibers using a recently proposed simple and efficient Chebyshev method in presence of optical nonlinearity. It is seen that the proposed technique estimates the dispersion parameters accurately since results obtained by our method agree excellently with those from rigorous techniques over a large range of normalized frequency. Although we’ve considered step index fiber as a typical example, our analysis is also applicable to graded index fibers as well.

A simple and effective method to analyze the propagation characteristics of nonlinear single mode fiber using Chebyshev method

The Chebyshev method is very well known for its simplicity as an approximate analytical formalism to study propagation characteristics of single mode fibers with linear optical media. The method is inapplicable in the context of nonlinear fiber for deeply involved multiplication of various power coefficients. In this article, we propose and develop a simple iterative method based on Chebyshev technique to investigate propagation characteristics of single mode nonlinear fibers and show that our results match with available rigorous results reliably. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:787–790, 2014

Steady state response of fractionally damped nonlinear viscoelastic arches by residue harmonic homotopy

The nonlinear single-mode dynamic behaviour of the viscoelastic arch whose damping is governed by fractional derivatives is considered. A set of ordinary differential equations is derived for the primary resonance and is solved by the residue harmonic homotopy to obtain all the steady state solutions. A parametric study is carried out to determine the influence of the viscoelastic damping characteristics of the material on the responses. Both the fractional order and material modulus ratio are effective in enhancing the stability of the structure. Multiple bifurcation solutions, jump phenomenon, saddle-node are observed analytically without numerical integration and are confirmed by numerical integration.

All-solid highly nonlinear singlemode fibers with a tailored dispersion profile

We investigate a novel approach to obtain highly nonlinear fibers with a tailored group velocity dispersion around a desired wavelength region of interest. Rather than exploiting longitudinal holes to control the average refractive index of the cladding and hence the fiber's waveguide dispersion, as in holey fibers, we propose using an all-solid cladding with a suitably chosen refractive index difference relative to the core. We demonstrate numerically that this solution allows a large freedom in the manipulation of the overall fiber dispersive properties, while enabling, in practice, a much more accurate control of the fiber's structural properties during fabrication. Effectively single mode guidance over a broad wavelength range can be achieved through the use of a second outer cladding forming a W-type index profile. We derive simple design rules for dispersion controlled fibers, based on which an algorithm for the automatic dispersion optimization is proposed, implemented and used to design various nonlinear fibers for all-optical processing and supercontinuum generation. Fabrication of a lead silicate fiber with flattened dispersion at telecoms wavelengths confirms the potential of these new fibers.

Nonlinear Rayleigh–Taylor instabilities in fast Z pinches

A simplified analytic model is presented to describe the implosion of a plasma column by an azimuthal magnetic field of sufficient magnitude to drive a strong shock wave into the plasma. This model is employed together with buoyancy-drag-based models of nonlinear single-mode and turbulent multimode Rayleigh–Taylor growth to investigate the mixing process in such fast Z pinches. These models give predictions that characterize limitations the instability can impose on the implosion in terms of maximum convergence ratios attainable for an axially coherent pinch. Both the implosion and instability models are validated with results from high-resolution numerical simulations.

Influence of background plasma on operation efficiency of high-power continuous wave (cw) gyrotrons

A slow wave non-linear single mode stationary theory describing influence of background plasma which occurs in the gyrotron cavity on gyrotron operation has been developed. It is shown that the main effects of a low density background plasma on efficiency are concerned with decreasing the mismatch between operational and cyclotron frequencies and decreasing the beam-wave coupling coefficient. Numerical calculations are performed for the 140 GHz 1 MW CW gyrotron developed in Karlsruhe for fusion applications. Interpretation of some experimental effects which are not yet clearly understood is addressed.

Anomalous resistivity of current-driven isothermal plasmas due to phase space structuring

The anomalous electric resistivity of collisionless plasmas is an important issue in the physics of hot plasmas, e.g., in the context of auroral particle acceleration and of reconnection in the solar corona. The linear stability theory of isothermal current driven space plasmas predicts an ion-acoustic instability if the relative drift velocity of the current carrying particles exceeds a certain threshold, which, generally, depends on the plasma parameters. The spectrum of waves, excited by a marginal instability, is very narrow. Hence, the wave power at saturation and the corresponding electric resistivity due to wave-particle interaction cannot be obtained by means of a quasilinear, weak turbulence approach and the nonlinear single mode theory provides too small saturation amplitudes. To solve the nonlinear problem a newly developed unsplit conservative Eulerian Vlasov code is applied to simulate a strongly magnetized current driven plasma, which can be considered in 1D1V (one spatial, one velocity space direction). Instead of periodic boundary conditions, usually used as they are simpler to treat, open boundaries are implemented which allow to maintain a constant current flow. Simulated is a typical almost isothermal (Te=2Ti) hot (κTi=1keV) space plasma for the real mass ratio mi∕me=1836. The initial spontaneous instability is followed by a three-stage nonlinear evolution: First electron trapping leads to the formation of electron phase space holes. Due to a steepening of the leading edges of the potential wells the electron phase space holes gradually become asymmetric, they grow in size and deepen. The phase space holes accelerate until they move much faster than the initial ion-acoustic waves. The interaction of the current carriers with the asymmetric potential wells and causes a nonvanishing net momentum transfer between the particles and the self-generated electric field. After a few ion plasma periods ion trapping starts until, finally, an electrostatic double layer arises. It is found that the nonlinear saturated state of the system is dominated by the particle interaction with coherent phase space structures. The corresponding anomalous resistivity is slightly modulated with an oscillation period τ≈ωpi−1). For a macroscopic description its major part can be parameterized by means of an effective collision rate νeff of the order of 10−2ωpe≈0.5ωpi, where ωpe is the electron and ωpi the ion plasma frequency.

Linear and nonlinear Rayleigh-Taylor growth at strongly convergent spherical interfaces

Recent attention has focused on the effect of spherical convergence on the nonlinear phase of Rayleigh-Taylor growth. For instability growth on spherically converging interfaces, modifications to the predictions of the Layzer model for the secular growth of a single, nonlinear mode have been reported [D. S. Clark and M. Tabak, Phys. Rev. E 72, 056308 (2005)]. However, this model is limited in assuming a self-similar background implosion history as well as only addressing growth from a perturbation of already nonlinearly large amplitude. Additionally, only the case of single mode growth was considered and not the multimode growth of interest in applications. Here, these deficiencies are remedied. First, the connection of the recent nonlinear results (including convergence) to the well-known results for the linear regime of growth is demonstrated. Second, the applicability of the model to more general implosion histories (i.e., not self-similar) is shown. Finally, to address the case of multimode growth with convergence, the recent nonlinear single mode results are combined with the Haan model formulation for weakly nonlinear multimode growth. Remarkably, convergence in the nonlinear regime is found not to modify substantially the multimode predictions of Haan’s original model.

Multimode Rayleigh-Taylor growth at strongly convergent spherical interfaces

Recently, attention has focused on the effects of spherical convergence on the nonlinear phase of Rayleigh-Taylor growth. In particular, for instability growth on spherically converging interfaces, modifications to the predictions of the Layzer model for the secular growth of a single, nonlinear mode have been reported. On the other hand, applications of interest involve surface perturbations which include the superposition of many unstable modes growing simultaneously. Such weakly nonlinear, multimode growth has previously been studied in the context of the well-known Haan model. Here, we combine the most recent results for enhanced nonlinear single mode growth on spherical interfaces with the Haan model formulation for multimode growth. Remarkably, the multimode results are found not to be substantially modified by including the effects of convergence.

Efficiencies for the single-mode operation of a quantum optical nonlinear shift gate

We investigate the single mode operation of a quantum optical nonlinear \pi phase shift gate implemented by a single two-level atom in one-dimensional free space. Since the single mode property of the input photons at the atom is not preserved in the interaction at the atom, we analyze the effeciency of single mode operation that can still be achieved. We show how the input pulse shape can be optimized to obtain high efficiencies for the nonlinear single mode operation. With this analysis, we obtain an optimal single mode transmittance per photon of 78% for the successful nonliner \pi phase shift operation.

Improvement of Nonlinear Single Resonant Mode Method

Presently, the Nonlinear Single Resonant Mode (NLSRM) method is the most efficient method for extending standard linear modal analysis concept to nonlinear systems. In this method, it is assumed that the mode of vibration in the resonant condition is close to the nonlinear normal mode and only one mode has nonlinear behavior. Therefore, predicting which one of the modes will exhibit nonlinear behavior is very important especially for a large system. The main object of this paper is twofold: (1) to improve the NLSRM method, (2) to use the sensitivity analysis for the prediction of nonlinear mode. The results are compared to the results of harmonic balance (HB) and time domain (TD) methods. It is shown that there is a very good agreement between HB and TD results with those of improved NLSRM methods and also the methodology for prediction of nonlinear mode is well suited.