Use of the Hilbert transform in modal analysis of linear and non-linear structures

An initial study into the application of the Hilbert transform in modal analysis procedures is presented. It is shown that typical structural non-linearities such as non-linear damping and stiffness can be detected and identified directly without the need to generate explicit models. No assumptions regarding the degree of non-linearity are made, which is a restriction in the classical methods for dealing with non-linearities. The properties of the Hilbert transform are discussed with respect to linear and non-linear dynamical systems, and a discrete transform, developed from the continuous functions, is derived in the frequency domain and adapted to modal analysis data in the form of mobility transfer functions. Truncation effects arising from limited frequency ranges of the mobility transfer functions are accounted for by employing correction terms in the frequency domain. Several examples are studied of single and multi-mode systems with non-linearities such as friction, clearance and non-linear stiffness. These examples indicate that the Hilbert transform offers a new method for extending modal analysis to the domains of non-linear systems.     

Frequency response characteristics of structures with single and multiple clearance-type non-linearity

The forced frequency response characteristics of systems which can be considered to exhibit single or normal mode characteristics and which incorporate spatially localized clearance-type non-linearities are analyzed by using both analytical and digital simulation methods. The non-linearities are considered to have symmetrical/asymmetrical characteristics and may be subject to pre-load conditions. It is shown that with a pre-loaded asymmetrical clearance element information from frequency response tests can be gainfully employed in estimating the clearance levels. Further, the effect of a displaced dynamic equilibrium position, resulting from asymmetry, is shown to be an important factor in predicting the resonant behaviour of such systems.     

Self-interaction and gauge invariance

A simple unified closed form derivation of the non-linearities of the Einstein, Yang-Mills and spinless (e.g. chiral) meson systems is given. For the first two, the non-linearities are required by locality and consistency; in all cases, they are determined by the conserved currents associated with the initial (linear) gauge invariance of the first kind. Use of first-order formalism leads uniformly to a simple cubic self-interaction.Supported by USAF OAR under Grant AFOSR 70-1864.     

Toy models and stylized realities

I discuss the role of toy models as theoretical tools for understanding complex systems of interacting agents. I review some concrete examples, in order to illustrate how this approach is able to capture, though in an admittedly stylized way, the interactions and non-linearities which are responsible for the rich phenomenology observed in reality. This allows one to interpret the system's behavior in terms of phase transitions and critical phenomena.     

Effective non-linear coefficients of optical waveguides

Using the weakly guiding approximation, and with the assumption that the non-linearities are sufficiently weak so as not to change the transverse mode profile, perturbation theory is used to derive simple expressions for evaluating the modal, or effective, non-linear coefficients of optical waveguide structures based on the non-linear coefficients of the constituent materials. In particular, the third-order and fifth-order optical non-linearities of refraction and absorption are discussed. The expressions are used to estimate the effective modal non-linear refraction coefficients for various types of slab waveguides, focusing primarily on femtosecond refractive non-linearities in semiconductor amplifiers, including quantum-well structures. The results are of particular relevance to the design of all-optical switches based on these effects.     

EXPERIMENTAL VALIDATION OF THE CONSTANT LEVEL METHOD FOR IDENTIFICATION OF NON-LINEAR MULTI-DEGREE-OF-FREEDOM SYSTEMS

System identification for non-linear dynamical systems could find use in many applications such as condition monitoring, finite element model validation and determination of stability. The effectiveness of existing non-linear system identification techniques is limited by various factors such as the complexity of the system under investigation and the type of non-linearities present. In this work, the constant level identification approach, which can identify multi-degree-of-freedom systems featuring any type of non-linear function, including discontinuous functions, is validated experimentally. The method is shown to identify accurately an experimental dynamical system featuring two types of stiffness non-linearity. The full equations of motion are also extracted accurately, even in the presence of a discontinuous non-linearity.     

An overview of MIMO-FRF excitation/averaging/processing techniques

The use of characterized excitation and choice of averaging techniques are fundamental to the estimation of multiple input, multiple output (MIMO) frequency response function (FRF) data. The characteristics of the excitation and averaging selected greatly influence the quality of the resulting MIMO-FRF measurements. Presented is an overview of the basic excitation methods, such as random, periodic random, pseudorandom, and burst random (random transient) as well as more advanced excitation methods, such as burst-cyclic random, slow random, MOOZ random, and periodic chirps. The application of these excitation and averaging methods is discussed relative to lightly or heavily damped systems, systems with small non-linearities, FRF models, and peak to RMS (crest factor) as well as signal-to-noise ratio (SNR) issues. Experimental examples are given to demonstrate the important issues.     

Rheology and aging: A simple approach

We introduce a rheological model to describe the low-frequency mechanical properties of systems near a fluid/paste transition. We propose a Landau-like expansion for the vicinity of this transition, treating the stress relaxation rate as an order parameter. This leads to a formally simple model that allows us to describe the interplay between aging and non-linearities in the mechanical response of the system. We focus here on systems prepared by fluidification under a strong shear, on which mechanical measurments are performed (oscillatory rheology, stress relaxation, response to a steady shear rate), after a waiting time during which the system evolves on its own. Received 14 June 2000     

Fiber designs for high figure of merit and high slope dispersion compensating fibers

When the first dispersion compensating fiber modules were introduced to the market in the mid '90s, the only requirement to the fiber was that it should have a negative dispersion. As the bit rate and the complexity of the optical communication systems have increased, several other requirements have been added such as low loss, low non-linearities and the ability of broadband dispersion compensation.     

The response of two-degree-of-freedom systems with quadratic and cubic non-linearities to multifrequency parametric excitations

The response of two d.o.f. systems with quadratic and cubic non-linearities to multi-frequency parametric excitations is determined by using the method of multiple scales. Four first-order ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when principal parametric resonances of both modes and combination resonances of the additive and difference type occur simultaneously. In all cases the steady state solutions and their stability are determined. Numerical results depicting the various resonances are presented.     

Stochastic perturbation of non-Kerr law optical solitons

The dynamics of optical solitons with non-Kerr law non-linearities, in presence of stochastic perturbation terms, is studied in this paper. The Langevin equations are derived and it is proved that the solitons travel through a fiber with a fixed mean velocity. The non-linearities that are considered here are the power law, parabolic law and the dual-power law types.     

Towards 5G: Performance evaluation of 60 GHz UWB OFDM communications under both channel and RF impairments

Detailed analysis on the impact of RF and channel impairments on the performance of Ultra-Wideband (UWB) wireless Orthogonal Frequency Division Multiplexing (OFDM) systems based on the IEEE 802.15.3c standard, for high data-rate applications using the 60 GHz millimetre frequency band is presented in this paper. This frequency band, due to the large available bandwidth is very attractive for future and 5G wireless communication systems. The usage of OFDM at millimetre-wave (mmWaves) frequencies is severely affected by non-linearities of the Radio Frequency (RF) front-ends. The impact of impairments is evaluated, in terms of some of the most important key performance indicators, including spectral efficiency, power efficiency, required coding overhead and system complexity, Out-Of-Band Emissions (OOBEs), Bit Error Rate (BER) target and Peak Signal-to-Noise Ratio (PSNR). Additionally, joint distortion effects of coexisting Phase-Noise (PN), mixer IQ imbalances and Power Amplifier (PA) non-linearities, on the performance degradation of a mmWave radio transceiver, combined with various multipath fading channels, are investigated. Subsequently, the power efficiency of the system is evaluated by estimating values of the PA Output-Power-Backoff (OBO) needed to meet the requirements for the Transmit Spectrum Mask (TSM) and BER target. Finally, a comparison of the system overall performance between uncoded and coded OFDM systems combined with Quadrature Amplitude Modulations (16 and 64 QAM) and its maximum operable range are evaluated by transmitting a Full HD uncompressed video frame under five different RF impairment conditions over a typical LOS kiosk 60 GHz IEEE channel model.     

Two-degree-of-freedom systems with quadratic non-linearities subjected to parametric and self excitation

The method of multiple scales is used to study the response of two-degree-of-freedom systems with quadratic non-linearities under the simultaneous effects of a harmonic parametric excitation and self excitation. The principal parametric resonance of the first mode and a three-to-one internal resonance is considered, followed by the case of internal and parametric resonance of the second mode. In both cases the stability of the system is also studied. Amplitude and frequency response curves are presented for both cases. The character of stability and the mode in which the system loses its stability is also discussed.     

A theoretical study of limit cycle oscillations of plenum air cushions

Air cushion vehicles (ACV) are prone to the occurrence of dynamic instabilities which frequently appear as stable finite amplitude oscillations. The aim of this work is to ascertain if the non-linearities characteristics of ACV dynamics generate limit cycle oscillations for cushion systems operating at conditions for which a linear theory predicts instability. The types of non-linearity that can occur are discussed, and an analysis is presented for a single cell flexible skirted plenum chamber constrained to move in pure heave only. Two cushion feed cases are considered: a plenum box supply and a duct. The results obtained by a Galerkin/describing function analysis are compared with those generated by a full numerical simulation. For the plenum box supply system, it is shown that the limit cycles can be suppressed by using a piston to introduce high frequency small amplitude volume oscillations into the plenum chamber.     

Electronic structure of two-dimensional semiconductor systems

The motion of two-dimensional carriers in semiconductor heterojunctions, quantum wells and superlattices is discussed, with emphasis on subband dispersion parallel to the interfaces and on quantization in a perpendicular magnetic field. The envelope function approximation is shown to provide an efficient and accurate method for the calculation of electronic states. It is shown that for states derived from degenerate or coupled bulk bands (valence bands, k . p-coupled conduction and valence bands in narrow-gap semiconductors) striking non-parabolicities of the subband dispersion and non-linearities of the Landau levels versus magnetic field occur. Results for GaAs-AlGaAs and InAs-GaSb systems are presented and compared with experiment.     

High-frequency power spectra of systems described by non-linear Langevin equations

We show that, for systems described by non-linear first- and second-order Langevin equations, the high-frequency correlations always have respectively, an ω^?2 and ω^?4 asymptotic behavior, irrespective of the shape and strength of the non-linearities, and whether detailed balance is obeyed in the stationary state or not.     

In situ calibration of a Michelson type, speckle-shearing interferometer: Wobbling mirror effect

In situ calibration of piezoelectric transducers in a phase stepping, Michelson type, speckle-shearing interferometer can be affected by mirror wobbling introduced by the transducers. When this happens, a modulation of the curves employed for calibration can be readily detected in imaging systems with large apertures. A calibration method that takes into account the modulation of these curves is, therefore, essential to insure the required regularity of the phase steps. A computer model of speckle-shearing interferometry is used to deal with piezoelectric non-linearities and mirror wobbling in the interferometer, and to show how the calibration curves are affected when both phenomena are present. Computer simulated and experimental results validate the calibration technique advanced here.     

The response of one-degree-of-freedom systems with cubic and quartic non-linearities to a harmonic excitation

The method of multiple scales is used to analyze the response of single-degree-of freedom systems with cubic and quartic non-linearities to a harmonic excitation. Two first-order ordinary differential equations describing the evolution of the amplitude and the phase are derived for superharmonic resonances of order two and four, subharmonic resonances of order one-half and one-fourth, and the supersubharmonic resonances of order $\text{3}\text{2}$ and $\text{2}\text{3}$. In all cases, the steady state solutions and their stability are determined and representative numerical results are included.