Response Spectra for Nonlinear Oscillators

1975 ◽  
Vol 97 (4) ◽  
pp. 1223-1226 ◽  
Author(s):  
J. E. Manning
Keyword(s):  
Nonlinear Oscillator ◽  
Response Spectrum ◽  
Duffing Oscillator ◽  
Response Spectra ◽  
Nonlinear Oscillators ◽  
Random Excitation ◽  
Heuristic Approach ◽  
Perturbation Approach ◽  
Spectral Bandwidth ◽  
Nonlinear Spring

Methods are presented for calculating the response spectrum of a nonlinear oscillator with broadband random excitation. A perturbation approach is used for oscillators with slightly nonlinear spring elements to show that the effect of the nonlinearity is a slight shift in the spectrum peak. A new heuristic approach is used for oscillators with large nonlinearities and small damping to show that in addition to a shift in the peak an increase in spectral bandwidth also occurs. The Duffing oscillator is studied as a specific example.

Response Envelope Statistics for Nonlinear Oscillators With Random Excitation

1978 ◽  
Vol 45 (1) ◽  
pp. 170-174 ◽  
Author(s):  
W. D. Iwan ◽  
P.-T. Spanos
Keyword(s):  
Computer Simulation ◽  
Linear System ◽  
Analytical Method ◽  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Nonlinear Oscillators ◽  
Random Excitation ◽  
Weakly Nonlinear ◽  
Response Envelope

An approximate analytical method is presented for determining both the stationary and nonstationary amplitude or envelope response statistics of a lightly damped and weakly nonlinear oscillator subject to Gaussian white noise. The method is based on the solution of an equivalent linear system whose parameters are functions of the response itself. The solution derived by the approximate method is compared with that obtained by computer simulation for a Duffing oscillator.

Laplace–Pade Parametric Homotopy Perturbation Method for Uncertain Nonlinear Oscillators

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
S. Chakraverty ◽  
N. R. Mahato
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Nonlinear Oscillator ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Parametric Form ◽  
Homotopy Perturbation

Nonlinear oscillators have wide applicability in science and engineering problems. In this paper, nonlinear oscillator having initial conditions varying over fuzzy numbers has been initially taken into consideration. Here, the fuzziness in the uncertain nonlinear oscillators has been handled using parametric form. Using parametric form in terms of r-cut, the nonlinear uncertain differential equations are reduced to parametric differential equations. Then, based on classical homotopy perturbation method (HPM), a parametric homotopy perturbation method (PHPM) is proposed to compute solution enclosure of such uncertain nonlinear differential equations. A sufficient convergence condition of parametric solution obtained using PHPM is also proved. Further, a parametric Laplace–Pade approximation is incorporated in PHPM for retaining the periodic characteristic of nonlinear oscillators throughout the domain. The efficiency of Laplace–Pade PHPM has been verified for uncertain Duffing oscillator. Finally, Laplace–Pade PHPM is also applied to solve other uncertain nonlinear oscillator, viz., Rayleigh oscillator, with respect to fuzzy parameters.

ANALYTICAL ESTIMATES OF THE EFFECT OF NONLINEAR DAMPING IN SOME NONLINEAR OSCILLATORS

2000 ◽  
Vol 10 (09) ◽  
pp. 2257-2267 ◽  
Author(s):  
JOSÉ L. TRUEBA ◽  
JOAQUÍN RAMS ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Nonlinear Damping ◽  
Damping Coefficient ◽  
Critical Parameters ◽  
Nonlinear Dissipation ◽  
Simple Pendulum ◽  
Melnikov Theory ◽  
Damped Systems

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

Lagging Motion of Forced Nonlinear Oscillators

2003 ◽  
Author(s):  
Sand Woo Karng ◽  
Ki Young Kim ◽  
Ho-Young Kwak
Keyword(s):  
Nonlinear Systems ◽  
Applied Field ◽  
Nonlinear Oscillator ◽  
Phase Plane ◽  
Inverted Pendulum ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Expansion Ratio ◽  
Amplitude Response ◽  
Calculation Results

The lagging motion of nonlinear oscillators with respect to the externally driven field was treated analytically and the calculation results were compared with observed results. Such lagging problem may occur in nonlinear systems whose behavior crucially depends on the frequency of the applied force. The lagging motion of the nonlinear oscillator with respect to the harmonically driven field made the oscillator respond in a way that reduced the effect of the applied field. The calculation considering the lagging motion yielded proper results in the expansion ratio of the bubble under ultrasound and trajectories in phase plane, the frequency spectrum for a forced inverted pendulum, and the amplitude response of the Duffing oscillator.

scholarly journals A Multi-Objective Ground Motion Selection Approach Matching the Acceleration and Displacement Response Spectra

2018 ◽  
Vol 10 (12) ◽  
pp. 4659 ◽  
Author(s):  
Yabin Chen ◽  
Longjun Xu ◽  
Xingji Zhu ◽  
Hao Liu
Keyword(s):  
Ground Motion ◽  
Response Spectrum ◽  
Structural Response ◽  
Response Spectra ◽  
Computationally Efficient ◽  
Ground Motion Selection ◽  
Displacement Response ◽  
Rc Frames ◽  
Selection Approach ◽  
Displacement Response Spectra

For seismic resilience-based design (RBD), a selection of recorded time histories for dynamic structural analysis is usually required. In order to make individual structures and communities regain their target functions as promptly as possible, uncertainty of the structural response estimates is in great need of reduction. The ground motion (GM) selection based on a single target response spectrum, such as acceleration or displacement response spectrum, would bias structural response estimates leading significant uncertainty, even though response spectrum variance is taken into account. In addition, resilience of an individual structure is not governed by its own performance, but depends severely on the performance of other systems in the same community. Thus, evaluation of resilience of a community using records matching target spectrum at whole periods would be reasonable because the fundamental periods of systems in the community may be varied. This paper presents a GM selection approach based on a probabilistic framework to find an optimal set of records to match multiple target spectra, including acceleration and displacement response spectra. Two major steps are included in that framework. Generation of multiple sub-spectra from target displacement response spectrum for selecting sets of GMs was proposed as the first step. Likewise, the process as genetic algorithm (GA), evolvement of individuals previously generated, is the second step, rather than using crossover and mutation techniques. A novel technique improving the match between acceleration response spectra of samples and targets is proposed as the second evolvement step. It is proved computationally efficient for the proposed algorithm by comparing with two developed GM selection algorithms. Finally, the proposed algorithm is applied to select GM records according to seismic codes for analysis of four archetype reinforced concrete (RC) frames aiming to evaluate the influence of GM selection considering two design response spectra on structural responses. The implications of design response spectra especially the displacement response spectrum and GM selection algorithm are summarized.

Sequences of Global Bifurcations and the Related Outcomes after Crisis of the Resonant Attractor in a Nonlinear Oscillator

1997 ◽  
Vol 07 (11) ◽  
pp. 2437-2457 ◽  
Author(s):  
W. Szemplińska-Stupnicka ◽  
E. Tyrkiel
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Resonance ◽  
Basins Of Attraction ◽  
Global Bifurcations ◽  
System Behavior ◽  
Coexisting Attractors ◽  
System Parameters

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

scholarly journals Preliminary results of ground-motion characteristics

2012 ◽  
Vol 55 (4) ◽  
Author(s):  
Francesca Bozzoni ◽  
Carlo Giovanni Lai ◽  
Laura Scandella
Keyword(s):  
Ground Motion ◽  
Site Response ◽  
Response Spectrum ◽  
Field Tests ◽  
Response Spectra ◽  
Ground Acceleration ◽  
Preliminary Results ◽  
Emilia Earthquake ◽  
Particle Orbit ◽  
2012 Emilia Earthquake

The preliminary results are presented herein for the engineering applications of the characteristics of the ground motion induced by the May 20, 2012, Emilia earthquake. Shake maps are computed to provide estimates of the spatial distribution of the induced ground motion. The signals recorded at the Mirandola (MRN) station, the closest to the epicenter, have been processed to obtain acceleration, velocity and displacement response spectra. Ground-motion parameters from the MRN recordings are compared with the corresponding estimates from recent ground-motion prediction equations, and with the spectra prescribed by the current Italian Building Code for different return periods. The records from the MRN station are used to plot the particle orbit (hodogram) described by the waveform. The availability of results from geotechnical field tests that were performed at a few sites in the Municipality of Mirandola prior to this earthquake of May 2012 has allowed preliminary assessment of the ground response. The amplification effects at Mirandola are estimated using fully stochastic site-response analyses. The seismic input comprises seven actual records that are compatible with the Italian code-based spectrum that refers to a 475-year return period. The computed acceleration response spectrum and the associated dispersion are compared to the spectra calculated from the recordings of the MRN station. Good agreement is obtained for periods up to 1 s, especially for the peak ground acceleration. For the other periods, the spectral acceleration of the MRN recordings exceeds that of the computed spectra.<br />

Response spectral matching of horizontal ground motion components to an orientation-independent spectrum (RotDnn)

2020 ◽  
pp. 875529302097098
Author(s):  
Luis A Montejo
Keyword(s):  
Ground Motion ◽  
Response Spectrum ◽  
Time History ◽  
Response Spectra ◽  
Major Axis ◽  
Component Level ◽  
Nonlinear Characteristics ◽  
Target Spectrum ◽  
Horizontal Ground Motion ◽  
Median Spectrum

This article presents a methodology to spectrally match two horizontal ground motion components to an orientation-independent target spectrum (RotDnn). The algorithm is based on the continuous wavelet transform decomposition and iterative manipulation of the two horizontal components of a seed record. The numerical examples presented follow current ASCE/SEI 7 specifications and therefore maximum-direction spectra (RotD100) are used as target for the match. However, the proposed methodology can be used to match other RotDnn spectra, like the median spectrum (RotD50). It is shown that with the proposed methodology the resulting RotDnn from the modified horizontal components closely match the smooth target RotDnn spectrum, while the response spectrum for each horizontal component continue to exhibit a realistic jagged behavior. The response spectra variability at the component level within suites of spectrally matched motions was found to be of the same order than the variability measured in suites composed of amplitude scaled records. Moreover, the spectrally matched records generated preserved most of the characteristics of the seed records, including the nonlinear characteristics of the time history traces and the period-dependent major axis orientations.

Computation of the adiabatic invariant of the symmetric nonlinear oscillator $$d^2y \over dt^2+Q(\epsilon t)y^n=R(\epsilon t)y^m$$

1998 ◽  
Vol 9 (2) ◽  
pp. 187-194
Author(s):  
J. HU
Keyword(s):  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Adiabatic Invariant ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

In a recent paper, the author showed that for certain symmetric bisuperlinear equations, cosine-like boundary behaviours will not yield symmetric solutions [1]. In this paper, we attack the adiabatic invariant problem by showing that, for these strongly nonlinear oscillators, the adiabatic invariant is intimately related to z′(0;∈) for a family of solutions.

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