Simple Oscillator Research Articles

Optimization of a viscous-elastic damper coupling two simple oscillators based on nonlinear stochastic response

Abstract Among the techniques used to control or mitigate the structural vibrations induced by dynamic input, such as wind and earthquake, the dissipative coupling is one of the most applied, especially for its ease of implementation. Indeed, for example in large urban areas, it is common to find adjacent structures where the space between the buildings becomes smaller. To optimally select the visco-elastic features of the dissipative device to be used, the paper retraces the path followed by the previous scientific works proposing new design criteria. Such criteria are based on the nonlinear stochastic response of two simple oscillators linked by a damper whose hysteretic behavior is represented by a Bouc-Wen model. A state-space formulation of the equations of motion has been adopted to facilitate the analysis of the dynamic response. At the same time, the loading is hypothesized as a zero-mean Gaussian excitation. Consequently, the nonlinear response has been approximately evaluated by the equivalent linearized standard deviations for both displacements and accelerations. Subsequently, formulations of objective functions, based on the Minimax and Total Energy of the equivalent linearized stochastic response, have been applied to determine the optimal configurations of the coupled system. The influence of both noise power amplitude and soil typology on the designed systems has been also investigated. Suggestions related to the path to achieve pre-fixed targets (as balancing of displacements and accelerations) are provided.

Analytical Modeling of Effluent Discharges Spreading by Tidal Oscillation Currents on a Sloping Beach

ABSTRACTAn analytical solution of the two‐dimensional advection–diffusion equation is introduced that models the spreading of effluent discharges into coastal waters affected by tidal oscillation currents on a sloping beach. It is assumed that the advection part may be expressed as a vector of fluid velocity with two components. The first component presents the longshore flow, in the direction, by a simple tidal oscillation model. The weaker second component, in the direction, reflects the cross‐shore depth‐variation velocity. By separating the variables, the equation in two dimensions is reduced to the product of two time‐dependent equations in one dimension that can be solved analytically. Some analysis and graphical plotting with discussion related to the product solution are given. Comparisons with an available analytical solution for a steady‐state case and numerical simulations are also addressed.

Mixed-mode oscillations and chaos in a complex chemical reaction network involving heterogeneous catalysis.

The nonlinear dynamical behavior in a complex isothermal reaction network involving heterogeneous catalysis is studied. The method first determines the multiple steady states in the reaction network. This is followed by an analysis of bifurcation continuations to identify several kinds of bifurcations, including limit point, Bogdanov-Takens, generalized Hopf, period doubling, and generalized period doubling. Numerical simulations are performed around the period doubling and generalized period doubling bifurcations. Rich nonlinear behaviors are observed, including simple sustained oscillations, mixed-mode oscillations, non-mixed-mode chaotic oscillations, and mixed-mode chaotic oscillations. Concentration-time plots, 2D phase portraits, Poincaré maps, maximum Lyapunov exponents, frequency spectra, and cascade of bifurcations are reported. Period-doubling and period-adding routes leading to chaos are observed. Maximum Lyapunov exponents are positive for all the chaotic cases, but they are also positive for some non-chaotic orbits. This result diminishes the reliability of using maximum Lyapunov exponents as a tool for determining chaos in the network under study.

Extreme multi-stability and circuit implementation for a two-ReLU-memristor-based jerk oscillator

Memristor-based oscillation circuits are prone to produce coexisting infinite attractors depending on the initial conditions of memristors, leading to the appearance of extreme multi-stability. In this paper, we propose a novel memristive jerk oscillator by bringing two ReLU-type memristors in a simple jerk oscillator and investigate its dynamical behaviors associated with the coupling parameters using bifurcation plots and Lyapunov exponent plots. Further, we discuss the planar equilibrium state and its stability, and then numerically explore the coexisting infinite attractors driven by the initial conditions of two ReLU-type memristors. Because of the intervention of the two ReLU-type memristors, the memristive jerk oscillator has a planar equilibrium state whose stability closely relies on the initial conditions of two ReLU-type memristors, and different initial conditions cause different attractors to coexist, resulting in bidirectional extreme multi-stability. Finally, the memristive jerk oscillator is implemented by analog circuit and digital hardware platform, and the numerical results are confirmed by circuit simulations and hardware experiments.

Experiential learning with online vibro-acoustic demonstrators

After 25 years of delivering lecture-based learning at a university with mixed success I now supplement my short courses with online experiential learning demonstrators for visualizing and interacting with vibro-acoustic systems. The demonstrators use simple javascript coding and calls to the plotly free libraries. No special software is required (but stronger computers obviously perform faster). Learners can quickly adjust parameters like stiffnesses, masses, and damping using sliders and immediately visualize changes to vibration response and sound fields. Students may exercise demonstrators for one- and two-dimensional acoustic waves; vibrations and mode shapes of simple oscillators, beams, and plates; and sound radiated by and transmitted through panels. Some of the demonstrators are freely available at hambricacoustics.com (others are only available to my short course students).

Snap-through and indirect reduced-order modelling

Snap-through is a nonlinear jump phenomenon that is increasingly being designed into engineering structures. The rich and varied dynamics of snap-through present a significant bottleneck in the design process as costly dynamic simulations are required to ensure predictable behaviour. The purpose of this research is to investigate the use of a recent projection-based, reduced-order modelling technique, namely, implicit condensation and expansion with inertial compensation, to reproduce the snap-through behaviours found in bistable finite element systems in analogous models with minimal degrees of freedom. In this work, the free response of a simple mass-spring oscillator is used to differentiate different types of snap-through. This provides a framework for discussing the dynamics of a family of buckled beams. The ability of the reduced-order models to faithfully reproduce these dynamics is investigated by testing the existence and stability of predicted periodic orbits. It is shown that the minimum projection basis required to capture snap-through accurately can be found through considering the minimum energy required for different types of snap-through. For bistable beams, this can be further simplified to just requiring an eigenvalue analysis at the most unstable equilibrium.

Complex dynamics in nonlinear small time-delayed optoelectronic oscillator and application in fast reservoir computing and pulse generation

We investigated a time-delayed optoelectronic oscillator (OEO) that displays a wide range of complex dynamic behavior under small time delay. The phase-space trajectory distributions in different dynamic regimes were compared which brings a new perspective on the underlying mechanism of the transition process. It was found that bifurcation is always possible no matter how small the time delay is even if the universal adiabatic approximation model is invalid. Hereby we proposed a versatile simple oscillator which has a potential capacity as memory carrier and high-dimensional state spatial mapping ability that brings 1000 times computing-efficiency improvements of reservoir computing over the large time delay one. Furthermore, we demonstrated a new approach for a tunable optoelectronic pulse generator (repetition rate at 0.2 MHz and 0.25 GHz) which depends critically on time-delayed input electrical pulse. The proposed oscillator is also a promising system for the applications of fast chaos-based communication.

Biorealistic Neuronal Temperature-Sensitive Dynamics within Threshold Switching Memristors: Toward Neuromorphic Thermosensation.

Neuromorphic nanoelectronic devices that can emulate the temperature-sensitive dynamics of biological neurons are of great interest for bioinspired robotics and advanced applications such as in silico neuroscience. In this work, we demonstrate the biomimetic thermosensitive properties of two-terminal V3O5 memristive devices and showcase their similarity to the firing characteristics of thermosensitive biological neurons. The temperature-dependent electrical characteristics of V3O5-based memristors are used to understand the spiking response of a simple relaxation oscillator. The temperature-dependent dynamics of these oscillators are then compared with those of biological neurons through numerical simulations of a conductance-based neuron model, the Morris-Lecar neuron model. Finally, we demonstrate a robust neuromorphic thermosensation system inspired by biological thermoreceptors for bioinspired thermal perception and representation. These results not only demonstrate the biorealistic emulative potential of threshold-switching memristors but also establish V3O5 as a functional material for realizing solid-state neurons for neuromorphic computing and sensing applications.

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces. The multiple scales method (MSM) is applied to solve the proposed problem. Several types of rotational pendulum oscillators are studied and talked about in detail. These include the forced damped rotating pendulum oscillator with gallows, the damped standard simple pendulum oscillator, and the damped rotating pendulum oscillator without gallows. The MSM first-order approximations for all the cases mentioned are derived in detail. The obtained results are illustrated with concrete numerical examples. The first-order MSM approximations are compared to the fourth-order Runge–Kutta (RK4) numerical approximations. Additionally, the maximum error is estimated for the first-order approximations obtained through the MSM, compared to the numerical approximations obtained by the RK4 method. Furthermore, we conducted a comparative analysis of the outcomes obtained by the used method (MSM) and He-MSM to ascertain their respective levels of precision. The proposed method can be applied to analyze many strong nonlinear oscillatory equations.

Study on the variable length simple pendulum oscillation based on the relative mode transfer method.

In this study, we employed the principle of Relative Mode Transfer Method (RMTM) to establish a model for a single pendulum subjected to sudden changes in its length. An experimental platform for image processing was constructed to accurately track the position of a moving ball, enabling experimental verification of the pendulum's motion under specific operating conditions. The experimental data demonstrated excellent agreement with simulated numerical results, validating the effectiveness of the proposed methodology. Furthermore, we performed simulations of a double obstacle pendulum system, investigating the effects of different parameters, including obstacle pin positions, quantities, and initial release angles, on the pendulum's motion through numerical simulations. This research provides novel insights into addressing the challenges associated with abrupt and continuous changes in pendulum length.

Improving quantitative literacy through a general education acoustics course

Most Colleges and Universities have some form of quantitative reasoning requirement for graduation. A narrow view of this requirement is that students need to do some mathematics in the course to demonstrate some facility at calculating quantities. Additionally, students often discuss their discomfort with mathematics when discussing their anxiety about registering in a non-majors physics course. In this presentation, I will discuss strategies used in a musical acoustics course for non-science majors to address proportionalities, graphical representations, and approximations to help students get past equations to think about the underlying physical relationships. Often, I find that students actually are much better at using mathematics than their self-assessments. By giving students different contexts for mathematical thinking, the quantitative elements make more sense to them and calculations become less intimidating. The examples I will discuss focus on using classroom-based experiments to identify relationships between dependent and independent variables on topics such as musical scales, simple oscillators, and fourier analysis.

The El Niño Southern Oscillation Recharge Oscillator with the Stochastic Forcing of Long-Term Memory

The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero very slowly with a negative power law. The corresponding non-zero ensemble mean during the integration period can exert a pronounced influence on the ensemble-mean dynamics of the RO model. The state-dependent noise, also called the multiplicative noise, can present its influence by reducing the relaxation coefficient and by introducing periodic external forcing. The decreasing relaxation coefficient can enhance the oscillation amplitude and shorten the oscillation period. The forced frequency is close to the natural frequency. The two mechanisms together can further amplify the amplitude and shorten the period, compared with the state-independent noise or additive noise, which only exhibits its influence by introducing non-periodic external forcing. These two mechanisms explicitly elucidate the influence of the stochastic forcing on the ensemble-mean dynamics of the RO model. It provides comprehensive knowledge to better understand the interaction between the fast-varying stochastic forcing and the slow-varying deterministic system and deserves further investigation.

Soil-structure interaction effects on out-of-plane seismic response and damage of masonry buildings with shallow foundations

A significant amount of damage and casualties induced by several strong-motion earthquakes which recently stroke South-East Mediterranean area is due to the major seismic vulnerability of residential buildings. In small villages and mid-size towns, those buildings very often consist of two- to four-story, unreinforced masonry (URM) structures not designed for earthquake resistance, with direct foundations usually corresponding to an in-depth extension of load-bearing walls. For such structures, especially when founded on soft soils, site amplification and soil-foundation-structure interaction (SFSI) can significantly affect the seismic performance; conversely, such phenomena should be investigated through methods that allow a trade-off between accuracy and computational effort, hence encouraging their implementation in engineering practice. This paper provides a comprehensive updated description of the studies carried out in the last years by the authors, which are based on both linear and nonlinear, parametric, dynamic analyses of complete soil-foundation-structure (SFS) models representative of existing residential building configurations on different soils. Specifically, the parametric study investigated SFS models with different masonry types, aspect ratios, and code-conforming homogeneous and heterogeneous soil profiles. The methodology and analysis results allowed for reaching the following objectives: (i) predicting the elongation of the fundamental period and the variation of equivalent damping of the SFS system with respect to fixed-base conditions, through a simplified approach based on an equivalent simple oscillator; and (ii) estimating the probability of exceeding increasing damage levels associated with out-of-plane overturning of URM walls, through fragility functions that take into account SFS interaction. The effectiveness of these simplified tools was successfully validated against well-documented case studies, at the scales of both single instrumented buildings and urban area.

Modelling Predator–Prey Interactions: A Trade-Off between Seasonality and Wind Speed

Predator–prey interactions do not solely depend on biotic factors: rather, they depend on many other abiotic factors also. One such abiotic factor is wind speed, which can crucially change the predation efficiency of the predator population. In this article, the impact of wind speed along with seasonality on various parameters has been investigated. Here, we present two continuous-time models with specialist and generalist type predators incorporating the effect of wind and the seasonality on the model parameters. It has been observed that wind speed plays a significant role in controlling the system dynamics for both systems. It makes the systems stable for both of the seasonally unperturbed systems. However, it controls the chaotic dynamics that occur in case of no wind for the seasonally perturbed system with the predator as a specialist. On the other hand, for the seasonally perturbed system with a generalist predator, it controls period-four oscillations (which occur considering no wind speed) to simple limit-cycle oscillations. Furthermore, the wind parameter has a huge impact on the survival of predator species. The survival of predator species may be achieved by ensuring a suitable range of wind speeds in the ecosystem. Therefore, we observe that seasonality introduces chaos, but wind reduces it. These results may be very useful for adopting necessary management for the conservation of endangered species that are massively affected by wind speed in an ecosystem.

3D stellar motion in the axisymmetric Galactic potential and the e–z resonances

Context. The full phase-space information on the kinematics of a huge number of stars provided by Gaia Data Release 3 increases the demand for a better understanding of the 3D stellar dynamics. Aims. In this paper, we investigate the possible regimes of motion of stars in the axisymmetric approximation of the Galactic potential, applying a 3D observation-based model developed elsewhere. The model consists of three components: the axisymmetric disc, the central spheroidal bulge, and the spherical halo of dark matter. The axisymmetric disc model is divided into thin and thick stellar discs and H I and H2 gaseous disc subcomponents, by combining three Miyamoto-Nagai disc profiles of any model order (1, 2, or 3) for each disc subcomponent, to reproduce a radially exponential mass distribution. The physical and structural parameters of the Galaxy components are adjusted by observational kinematic constraints. Methods. The phase space of the two-degrees-of-freedom model was studied by means of the Poincaré and dynamical mapping, the dynamical spectrum method, and the direct numerical integrations of the Hamiltonian equations of motion. Results. For the chosen physical parameters, the nearly circular (close to the rotation curve) and low-altitude stellar behaviour is composed of two weakly coupled simple oscillations, radial and vertical motions. The amplitudes of the vertical oscillations of these orbits gradually increase with the growing Galactocentric distances, in concordance with the exponential mass decay assumed. However, for increasing planar eccentricities, e, and the altitudes over the equatorial disc, z, new regimes of stellar motion emerge as a result of the beating between the radial and vertical oscillation frequencies, which we refer to as e–z resonances. The corresponding resonant motion produces the characteristic sudden increase or decrease in the amplitude of the vertical oscillation, bifurcations in the dynamical spectra, and the chains of islands of stable motion in the phase space. Conclusions. The results obtained can be useful in understanding and interpreting the features observed in the stellar 3D distribution around the Sun.

Evaluation of the Basic Designs of a Micro Device that Provides Vibrational Stimulation to Cells

It is known that the cells responds to external mechanical stimulations. Although the effectiveness of vibrational stimulation for the osteoanagenesis has been reported, the clarification of detailed mechanism for this phenomenon is insufficient. In this study, a micro device has been developed to evaluate the cell dynamics and responses to vibrations. The micro device has an array of moving micro stages which have transparent 5 µm thick thin film to enable them to observe the cell responses to vibrational stimulations by using an optical microscope. The moving micro stages are moved with a needle actuated by piezo actuator. Microfabrication processes, such as conventional photolithography, lift-off, and sacrificial layer etching, were used to fabricate the micro device. We designed two types of concepts for supporting and vibrating moving micro stages. Prototypes were fabricated and evaluated under vibrational conditions. Proposed design with the moving micro stages vibrating perpendicular to the beams generated simple linear oscillation without rotation. It was verified that the fabricated micro stage could be vibrated at the acceleration amplitude of 0.1 and 0.2 G with frequency 15, 45, and 90 Hz.

Analysis and Design of Injection-Locked Oscillators Coupled to an External Resonator

This work investigates the nonlinear dynamics of an injection-locked power oscillator inductively coupled to an external resonator. This allows a high-efficiency power transfer while ensuring a constant oscillation frequency versus the coupling factor, unlike free-running implementations. The investigation focuses on the impact of the external-resonator elements on the locking range, output power, efficiency, and phase noise. The aim is to derive a strategy for an optimum selection of these elements. Initially, the effect of the coupled resonator is theoretically studied using a simple oscillator model, based on a cubic nonlinearity. For practical oscillators, two kinds of analysis methods, compatible with the use of commercial harmonic-balance (HB) simulators, are presented. The first one is semianalytical and is based on the extraction of a phase-dependent nonlinear admittance function from HB simulations. The system response is predicted in a flexible and computationally efficient manner, but coupling effects are considered at the fundamental frequency only. The second set of methods is fully based on HB and relies on the combination of a nonlinear immittance function and a Thevenin/Norton equivalent. The impact of the external resonator on the stability properties is analyzed through bifurcation detection. The phase-noise spectrum is predicted with a semianalytical formulation that demonstrates the benefit of the injection-locked operation. For validation, the methods have been applied to a Class-E oscillator at 13.56 MHz.

Sliding bursting oscillations related to transcritical bifurcation delay in an excited vector field with frequency switching

The purpose of this paper is to report sliding bursting dynamics related to transcritical bifurcation delay, a common route to bursting dynamics. As an example, a simple nonlinear oscillator with a slow parametric excitation is considered, in which bursting oscillations resulted from transcritical bifurcation delay can be observed. Typically, such oscillations exhibit prolonged rest phases, characterized by the oscillations that moves along the unstable origin of the system. We show that interesting dynamical characteristics can be observed in the bursting oscillations if the excitation frequency switches between two different values. In particular, additional rest phases, i.e., the newly created non-zero sliding behaviors, characterized by that the trajectory slides along the frequency switching threshold for some time, can be exhibited in the presence of frequency switching. This forms the so-called sliding bursting oscillations. Then, we investigate the dynamical mechanisms of these oscillations and explore their transitions in relation to the variation of frequency switching threshold. As a result, several different patterns of sliding bursting oscillations are obtained. In particular, we find that sliding bursting oscillations may diverge to infinity for some threshold windows, and this can be understood by the analysis of attracting basins.

Experimental and numerical investigations of forced response of multi-element lean-premixed hydrogen flames

Understanding the distinctive thermoacoustic properties of multi-element lean-premixed hydrogen flames is crucial for the development of hydrogen-capable gas turbines. Extending our earlier investigations of the self-excited dynamics of a cluster of lean-premixed pure hydrogen-air flames, here we explore the dynamic response to harmonic velocity perturbations, using an integrative approach combining loudspeaker-forced direct measurements and LES-based numerical simulations. Electronically-excited OH intensity distribution, generally assumed to be equivalent to the flame's heat release rate, shows an anchored conical reaction zone. Numerical simulations of heat release rate contours, however, reveal the formation of a more concentrated thin annulus region resembling an inverted V shaped layer, created by preferential diffusion of hydrogen molecules. This difference between OH-intensity distribution and heat release rate contours suggests consequential uncertainties in surrogate-dependent flame transfer function evaluations; the transfer function gains from measured data are well defined by a fast-decaying low-pass filter behavior, but the simulation result exhibits slowly-decaying large gain with slight overshoot over the entire range of frequency. Because the systematic uncertainties associated with the estimation of the flame's heat release rate cannot be accurately quantified, we rely on a reduced-order network modeling framework combined with direct measurement to test the accuracy of the prediction of the growth of self-sustained pressure fluctuations. We demonstrate that LES-based transfer functions predict the system's stability more accurately, and this suggests that the calculated heat release rate data are closer to the hydrogen flame's true heat release response than the wavelength-specific light intensity measurement. This assessment enables us to identify structurally simple moderate transverse oscillations of local regions containing atypically concentrated energy as pivotal processes controlling high-frequency hydrogen combustion dynamics.

Non-linear oscillators with Kuramoto-like local coupling: Complexity analysis and spatiotemporal pattern generation

Can a simple oscillator system, as in cellular automata, sustain complex nature upon discretization in time and space? The answer is by no means trivial as even the most simple, two-state, nearest neighbours cellular automata can lead to Universal Turing Machine (UTM) computing power. This study analyses a recently proposed model consisting of a ring of identical excitable Adler-type oscillators with local Kuramoto-like coupling in terms of its complexity. Regions with non-trivial, complex behaviour have been identified, where spatiotemporal maps closely resemble those found in elementary cellular automata, not only from the visual perspective but also from entropic measures characterization. Also, the possibility of enhanced computation at the edge of chaos is explored by monitoring the effective complexity measure, entropy density and informational distance, following previous approaches. Distance matrix, and the corresponding dendrograms, show that in the complex regions, a non-trivial set of hierarchies emerges where local communities can appear and sustain themself in time. The fact that coupled oscillators can be realized through dedicated electronic circuitry or the result of natural processes can make finding such complex dynamics, with potential computational power, an important one in a broad scope of applications and implications.