Nonlinear Oscillators

The seismic analysis and design of secondary attachments to buildings or industrial facilities is a topic of broad engineering interest, increasingly attracting the attention of researchers and practitioners. Examples of secondary systems include suspended ceilings and non-structural walls, piping systems and antennas, storage tanks, electrical transformers and glass façades. Although not part of the load bearing structure, their significance stems from the survivability requirement in the aftermath of a seismic event and their vast contribution to the overall construction costs. Nevertheless, past earthquakes have demonstrated that current methods for the seismic analysis and design of secondary structures lack the necessary rigor and robustness, resulting in expensive and often unreliable solutions. Secondary systems can be highly sensitive to accelerations and inter-story drifts, and their seismic performance is influenced by the primary-secondary dynamic interaction. In many situations however, the mass of the secondary system may be much lower than the mass of the floor at which it is connected and therefore a cascade approach is admissible. If the secondary system can be realistically modeled as a single-degree-of-freedom (SDoF) system, then the floor response spectra could be a powerful tool for quantifying its seismic response. In this study, the performance of light secondary systems is examined in presence of uncertainties in the seismic input. A set of principal axes of ground shaking is initially identified and an ensemble of bi-directional time series is generated. The response of a set of SDoF secondary oscillators (i.e. linear, Bouc-Wen, sliding and rocking) attached to a representative primary structure is then investigated and their design spectra are established. As demonstrated with Monte Carlo simulations for the selected case study, the angle of seismic incidence causes the highest variations in the engineering demand parameters for the sliding oscillators, while the elasto-plastic oscillator with the Bouc-Wen model experiences the least variations. Furthermore, investigations at different elevations show higher variations in the sliding and linear oscillators, depending on the seismic input. As expected, the viscous damping ratio is found to significantly influence the response of secondary systems vibrating close to the fundamental frequency of the primary structure. Moreover, the peak response of sliding oscillators is shown to be a smooth function of the sliding friction coefficient, while the rocking spectra, due to the strong nonlinear dynamics of the rocking blocks, are characterized by large values of the coefficient of variations.

A description in terms of phase and amplitude variables is given for nonlinear oscillators subject to white Gaussian noise described by Itô stochastic differential equations. The stochastic differential equations derived for the amplitude and the phase are rigorous, and their validity is not limited to the weak noise limit. It is shown that if Floquet vectors are used, then in the neighborhood of a limit cycle the phase variable coincides with the asymptotic phase defined through isochrons. Two techniques for the analysis of the phase and amplitude equations are discussed, that is, asymptotic expansion method and a phase reduction procedure based on projection operators technique. Formulas for the expected angular frequency, expected oscillation amplitude, and amplitude variance are derived using Itô calculus.

SILICON microelectromechanical system (MEMS)
resonator based oscillators are increasingly being adopted
for various applications in timing and frequency control [1, 2]
due to the reduced size, demonstrated compatibility with
semiconductor batch manufacturing and integration with IC
technologies. MEMS resonator based oscillators are also
essential building blocks for resonant sensors [3-5]. One of the
technical challenges limiting translation of resonant sensors has
historically been reliability of the vacuum package though a
number of technical solutions have been developed in recent
years [6-8]. A second challenge relates to the susceptibility of
MEMS resonators to non-linear effects limiting power handling
resulting in the device often operated at the critical point or just
prior to the onset of bifurcation in the response.

A tutorial discussion of negative resistance oscillators based on simple RLC circuits is presented. Two cases are based on parallel RLC circuits and two cases are based on series RLC circuits. Distortion is minimized by introducing symmetry in the movement of the complex pole-pair of the small signal model.

A detailed nonlinear analysis and design of a distributed voltage controlled oscillator (DVCO) is presented in this work. The analysis and design of the DVCO is performed using harmonic balance-based simulation techniques and envelope transient simulation. These techniques are used to make a complete nonlinear study of the behavior of the DVCO. Using the proposed nonlinear tools, a DVCO operating in the frequency tuning range between 1 and 2 GHz has been implemented in order to support the simulation results. The average output power and consumption along the frequency band for the designed DVCO are 5.2 dBm and 60.4 mW, respectively.