Estimation of generalised frequency response functions

Volterra series theory has a wide application in the representation, analysis, design and control of nonlinear systems. A new method of estimating the Volterra kernels in the frequency domain is introduced based on a non-parametric algorithm. Unlike the traditional non-parametric methods using the DFT transformed input-output data, this new approach uses the time domain measurements directly to estimate the frequency domain response functions

Piecewise Volterra modelling of the Duffing oscillator in the frequency domain

When analysing the nonlinear Duffing oscillator, the weak nonlinearity is basically dependent on the amplitude range of the input excitation. The nonlinear differential equation models of such nonlinear oscillators, which can be transformed into the frequency domain, can generally only provide Volterra modelling and analysis in the frequency-domain over a fraction of the entire framework of weak nonlinearity. This paper discusses the problem of using a new non-parametric routine to extend the capability of Volterra analysis, in the frequency domain, to weakly nonlinear Duffing systems at a wider range of excitation amplitude range which the current underlying nonlinear differential equation models fail to address

The effects of cubic damping on vibration isolation

Vibration isolators are often assumed to possess linear viscous damping which has well known consequences for their performance. However, damping may be designed to be or prove to be nonlinear. This study investigates the effect of cubic damping, as an example of damping nonlinearity, in a single degree of freedom (SDOF) vibration isolation system. The response behaviour due to two excitation types, namely harmonic and broadband excitations, was examined.For harmonic excitation, the Harmonic Balance Method (HBM) was applied to yield approximate closed form solutions and simplified analytical expressions implicitly show the influence of cubic damping for particular frequency regions. The HBM solutions were verified using direct numerical integration. The presence of cubic damping proves to be beneficial for the force excited case. It reduces response amplitude around the resonance frequency and has similar response to an undamped system in the isolation region. In contrast, for base excitation, the cubic damping is detrimental at high excitation frequencies as the base excitation and isolated mass move almost together. The effect becomes more pronounced for larger excitation amplitudes.The case of base excitation was then considered for broadband excitation. The responses using direct numerical integration were presented using power spectral densities. In contrast to harmonic excitation, the amplitude of the response does not appear to approach that of the input. Instead, a higher effective cubic damping results in a higher vibration level of the isolated mass at frequencies below the resonance frequency. It also does not reduce explicitly the response amplitude around the resonance frequency unlike the linear viscous damping. For a constant displacement amplitude random excitation, the excitation frequency bandwidth is found to be a significant factor in the level of effective cubic damping. A broader excitation bandwidth results in a higher level of cubic damping force.The theoretical and numerical results for both harmonic and broadband excitation were validated experimentally. The experimental investigation was performed using a SDOF base excited vibration isolation system possessing a simple velocity feedback control active damper to reproduce the nonlinear damping force. The predictions were shown to be in good agreement with measurements thereby verifying the effects of cubic damping on a SDOF system undergoing harmonic and broadband base excitation

Identifying nonlinear wave interactions in plasmas using two-point measurements: a case study of Short Large Amplitude Magnetic Structures (SLAMS)

A framework is described for estimating Linear growth rates and spectral energy transfers in turbulent wave-fields using two-point measurements. This approach, which is based on Volterra series, is applied to dual satellite data gathered in the vicinity of the Earth's bow shock, where Short Large Amplitude Magnetic Structures (SLAMS) supposedly play a leading role. The analysis attests the dynamic evolution of the SLAMS and reveals an energy cascade toward high-frequency waves.Comment: 26 pages, 13 figure

Analysis of nonlinear oscillators using volterra series in the frequency domain Part I : convergence limits

The Volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems, both in the time and the frequency domain. The Volterra series is associated with the so-called weakly nonlinear systems, but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid Volterra series representation for a given nonlinear differential equation. Barrett(1965) proposed a time domain criterion to prove that the Volterra series converges with a given region for a class of nonlinear systems with cubic stiffness nonlinearity. In this paper this time-domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation

Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part \wt{\frak{g}}^+ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr\"odinger equation.Comment: 61 pg

Isospectral Flow and Liouville-Arnold Integration in Loop Algebras

A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger systems and the sine-Gordon equation. Each system has an associated invariant spectral curve and may be integrated via the Liouville-Arnold technique. The linearizing map is the Abel map to the associated Jacobi variety, which is deduced through separation of variables in hyperellipsoidal coordinates. More generally, a family of moment maps is derived, identifying certain finite dimensional symplectic manifolds with rational coadjoint orbits of loop algebras. Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps. The isospectral property follows from the Adler-Kostant-Symes theorem, and gives rise to invariant spectral curves. {\it Spectral Darboux coordinates} are introduced on rational coadjoint orbits, generalizing the hyperellipsoidal coordinates to higher rank cases. Applying the Liouville-Arnold integration technique, the Liouville generating function is expressed in completely separated form as an abelian integral, implying the Abel map linearization in the general case.Comment: 42 pages, 2 Figures, 1 Table. Lectures presented at the VIIIth Scheveningen Conference, held at Wassenaar, the Netherlands, Aug. 16-21, 199

A new frequency domain representation and analysis for subharmonic oscillation

For a weakly nonlinear oscillator, the frequency domain Volterra kernels, often called the generalised frequency response functions, can provide accurate analysis of the response in terms of amplitudes and frequencies, in a transparent algebraic way. However a Volterra series representation based analysis will become void for nonlinear oscillators that exhibit subharmonics, and the problem of finding a solution in this situation has been mainly treated by the traditional analytical approximation methods. In this paper a novel method is developed, by extending the frequency domain Volterra representation to the subharmonic situation, to allow the advantages and the benefits associated with the traditional generalised frequency response functions to be applied to severely nonlinear systems that exhibit subharmonic behaviour

Evaluation of output frequency responses of nonlinear systems under multiple inputs

In this paper, a new method for evaluating output frequency responses of nonlinear systems under multiple inputs, defined as a sum of sinusoids of different frequencies, is developed. The method circumvents difficulties associated with the existing “frequency-mix vector” based approaches and can easily be applied to investigate nonlinear behaviors of practical systems, including electronic circuits, at the system simulation and design stages. Application of the method to the analysis of nonlinear interference and distortion effects in communication receivers is studied, and specific procedures are proposed which can be directly used in practice for this analysi