333 research outputs found
Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis
By using the Duffing oscillator as a case study, this paper shows that the harmonic components in the nonlinear system response to a sinusoidal input calculated using the Nonlinear Output Frequency Response Functions (NOFRFs) are one of the solutions obtained using the Harmonic Balance Method (HBM). A comparison of the performances of the two methods shows that the HBM can capture the well-known jump phenomenon, but is restricted by computational limits for some strongly nonlinear systems and can fail to provide accurate predictions for some harmonic components. Although the NOFRFs cannot capture the jump phenomenon, the method has few computational restrictions. For the nonlinear damping systems, the NOFRFs can give better predictions for all the harmonic components in the system response than the HBM even when the damping system is strongly nonlinear
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
Duffing revisited: Phase-shift control and internal resonance in self-sustained oscillators
We address two aspects of the dynamics of the forced Duffing oscillator which
are relevant to the technology of micromechanical devices and, at the same
time, have intrinsic significance to the field of nonlinear oscillating
systems. First, we study the stability of periodic motion when the phase shift
between the external force and the oscillation is controlled -contrary to the
standard case, where the control parameter is the frequency of the force.
Phase-shift control is the operational configuration under which self-sustained
oscillators -and, in particular, micromechanical oscillators- provide a
frequency reference useful for time keeping. We show that, contrary to the
standard forced Duffing oscillator, under phase-shift control oscillations are
stable over the whole resonance curve. Second, we analyze a model for the
internal resonance between the main Duffing oscillation mode and a
higher-harmonic mode of a vibrating solid bar clamped at its two ends. We focus
on the stabilization of the oscillation frequency when the resonance takes
place, and present preliminary experimental results that illustrate the
phenomenon. This synchronization process has been proposed to counteract the
undesirable frequency-amplitude interdependence in nonlinear time-keeping
micromechanical devices
A fast continuation scheme for accurate tracing of nonlinear oscillator frequency response functions
A new algorithm is proposed to combine the split-frequency harmonic balance method (SF-HBM) with arc-length continuation (ALC) for accurate tracing of the frequency response of oscillators with non-expansible nonlinearities. ALC is incorporated into the SF-HBM in a two-stage procedure: Stage I involves finding a reasonably accurate response frequency and solution using a relatively large number of low-frequency harmonics. This step is achieved using the SF-HBM in conjunction with ALC. Stage II uses the SF-HBM to obtain a very accurate solution at the frequency obtained in Stage I. To guarantee rapid path tracing, the frequency axis is appropriately subdivided. This gives high chance of success in finding a globally optimum set of harmonic coefficients. When approaching a turning point however, arc-lengths are adaptively reduced to obtain a very accurate solution. The combined procedure is tested on three hardening stiffness examples: a Duffing model; an oscillator with non-expansible stiffness and single harmonic forcing; and an oscillator with non-expansible stiffness and multiple-harmonic forcing. The results show that for non-expansible nonlinearities and multiple-harmonic forcing, the proposed algorithm is capable of tracing-out frequency response functions with high accuracy and efficiency
Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation
We develop performance criteria for the objective comparison of different
classes of single-degree-of-freedom oscillators under stochastic excitation.
For each family of oscillators, these objective criteria take into account the
maximum possible energy harvested for a given response level, which is a
quantity that is directly connected to the size of the harvesting
configuration. We prove that the derived criteria are invariant with respect to
magnitude or temporal rescaling of the input spectrum and they depend only on
the relative distribution of energy across different harmonics of the
excitation. We then compare three different classes of linear and nonlinear
oscillators and using stochastic analysis tools we illustrate that in all cases
of excitation spectra (monochromatic, broadband, white-noise) the optimal
performance of all designs cannot exceed the performance of the linear design.
Subsequently, we study the robustness of this optimal performance to small
perturbations of the input spectrum and illustrate the advantages of nonlinear
designs relative to linear ones.Comment: 24 pages, 12 figure
Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber
The nonlinear tuned vibration absorber (NLTVA) is a recently-developed
nonlinear absorber which generalizes Den Hartog's equal peak method to
nonlinear systems. If the purposeful introduction of nonlinearity can enhance
system performance, it can also give rise to adverse dynamical phenomena,
including detached resonance curves and quasiperiodic regimes of motion.
Through the combination of numerical continuation of periodic solutions,
bifurcation detection and tracking, and global analysis, the present study
identifies boundaries in the NLTVA parameter space delimiting safe, unsafe and
unacceptable operations. The sensitivity of these boundaries to uncertainty in
the NLTVA parameters is also investigated.Comment: Journal pape
Multistability and localization in forced cyclic symmetric structures modelled by weakly-coupled Duffing oscillators
Many engineering structures are composed of weakly coupled sectors assembled
in a cyclic and ideally symmetric configuration, which can be simplified as
forced Duffing oscillators. In this paper, we study the emergence of localized
states in the weakly nonlinear regime. We show that multiple spatially
localized solutions may exist, and the resulting bifurcation diagram strongly
resembles the snaking pattern observed in a variety of fields in physics, such
as optics and fluid dynamics. Moreover, in the transition from the linear to
the nonlinear behaviour isolated branches of solutions are identified.
Localization is caused by the hardening effect introduced by the nonlinear
stiffness, and occurs at large excitation levels. Contrary to the case of
mistuning, the presented localization mechanism is triggered by the
nonlinearities and arises in perfectly homogeneous systems
Building better oscillators using nonlinear dynamics and pattern formation
Frequency and time references play an essential role in modern technology and in living systems. The precision of self-sustained oscillations is limited by the effects of noise, which becomes evermore important as the sizes of the devices become smaller. In this paper, we review our recent theoretical results on using nonlinear dynamics and pattern formation to reduce the effects of noise and improve the frequency precision of oscillators, with particular reference to ongoing experiments on oscillators based on nanomechanical resonators. We discuss using resonator nonlinearity, novel oscillator architectures and the synchronization of arrays of oscillators, to improve the frequency precision
ANALYSIS OF THE NONLINEAR VIBRATIONS OF ELECTROSTATICALLY ACTUATED MICRO-CANTILEVERS IN HARMONIC DETECTION OF RESONANCE (HDR)
Micro- and nano-cantilevers have the potential to revolutionize physical, chemical, and biological sensing; however, an accurate and scalable detection method is required. In this work, a fully electrical actuation and detection method is presented, known as the Harmonic Detection of Resonance (HDR). In HDR, harmonic components of the current are measured to determine the cantilever\u27s resonance frequency. These harmonics exist as a result of nonlinearities in the system, principally in the electrostatic actuation force. In order to better understand this rich harmonic structure, a theoretical investigation of the micro-cantilever is undertaken. Both a lumped parameter model and a more accurate continuum model are used to derive the governing nonlinear modal equations of motion (EOM) of the cantilever. Various approximate solution methods applicable to nonlinear equations are then discussed including numerical integration, perturbation, and averaging. An averaging method known as the method of harmonic balance is then used to obtain steady state solutions to the micro-cantilever EOM. Low-order closed-form harmonic balance solutions are derived which explain many of the important features of the HDR results, such as the presence of parasitic capacitance in the first harmonic and super-harmonic resonance peaks in higher harmonics. Finally, higher-order computer generated harmonic balance solutions are presented which show good agreement with the experimental HDR results, validating both the modeling and the solution methods used
- …