Linear Oscillator Research Articles

Directionality and polarization of response spectral ordinates in the 2023 Kahramanmaras, Türkiye earthquake doublet

Until recently, the orientation of maximum horizontal spectral response was generally believed to not have a predominant orientation at rupture distances greater than 5 km. However, a recent study found that the orientation of maximum spectral response for strike-slip earthquakes in the NGA-West2 database tends to occur close to the epicentral transverse orientation, that is, an orientation perpendicular to a line connecting the epicenter to the station. This article investigates directionality in the 6 February 2023 Türkiye doublet earthquakes ( Mw 7.8 and 7.5) with strike-slip faulting. The orientation of the maximum response of 5%-damped linear elastic oscillators was studied. The spatial distribution of the level of polarization, which in this article refers to the amount of directionality, and intensities at specific orientations were also studied. The maximum spectral response was found to occur systematically close to the epicentral transverse orientation, consistent with previous observations for other strike-slip earthquakes. For the Mw 7.8 event where the location of maximum slip was relatively far from the epicenter, it was found that the orientation of maximum spectral response is, on average, closer to the maximum slip transverse orientation (i.e. perpendicular to a line connecting the station to the surface projection of the point of maximum slip) when compared to the epicentral transverse orientation over most period ranges. This suggests that the maximum slip transverse orientation may be a better estimator for determining the orientation of maximum spectral response in large-magnitude strike-slip earthquakes, although further study using more events is warranted. Polarized motions were observed over large geographical areas, and the orientation of maximum spectral response was found to be close to the epicentral or maximum slip transverse for Joyner–Boore distances up to the farthest studied (400 km). These findings further support the case for the development of orientation-dependent ground motion models for strike-slip earthquakes.

The level-crossing problem of a weakly damped particle in quadratic potential well under harmonic excitation

Escape and level-crossing are fundamental and closely related problems in transient dynamics. Often, when a particle reaches a critical displacement, its escape becomes inevitable. Therefore, escape models based on truncated potentials are often used, resulting in similar problems to level-crossing formulations. Two different types of dynamics can be identified, leading to different kinds of level-crossing depending on the relationship between the damping and the excitation level. The first one (“fast escape”) is mainly governed by the initial energy of the system, which is determined through the initial conditions. The second one (“slow escape”) is governed by the beatings determined through the relationship between external excitation and damping. An analytic approach for estimating the size and location of the safe basins (SBs) in the plane of the initial conditions (ICs) of a 1-DOF externally excited oscillator is suggested. It enables the identification of the set of ICs where the particle never reaches a certain threshold under the given excitation. The SBs depend on the damping coefficient and the excitation’s amplitude, frequency, and phase. Nonetheless, one can describe the essential properties of an SBs in the case of the almost resonant excitation using only two parameters: the forced response amplitude and the damping coefficient ratio to the difference between the natural and the excitation frequencies. Although the analysis is performed for a linear oscillator, it provides insight into the rush erosion process of the SBs (“Dover cliff” phenomenon), described previously only for nonlinear systems. The analysis reveals that the “Dover cliff” phenomenon is related to the decay rate of the transient motion and that it can occur even in linear systems too. From the engineering point of view, the rush erosion of the SBs is critical in noisy environments where devices operating in regions close to the “Dover cliff” are unsafe. Due to its simplicity, the proposed mechanical model might be generic for further analysis of the escape and level-crossing problems considering various nonlinearities (e.g., Coulomb friction, small polynomial-type nonlinearities of the restoring force, or constant restoring force). Possible applications include but are not limited to avoiding collisions for systems with clearances and durability analysis of brittle materials subjected to noisy loads.

Characterizing affect dynamics with a damped linear oscillator model: Theoretical considerations and recommendations for individual-level applications.

People show stable differences in the way their affect fluctuates over time. Within the general framework of dynamical systems, the damped linear oscillator (DLO) model has been proposed as a useful approach to study affect dynamics. The DLO model can be applied to repeated measures provided by a single individual, and the resulting parameters can capture relevant features of the person's affect dynamics. Focusing on negative affect, we provide an accessible interpretation of the DLO model parameters in terms of emotional lability, resilience, and vulnerability. We conducted a Monte Carlo study to test the DLO model performance under different empirically relevant conditions in terms of individual characteristics and sampling scheme. We used state-space models in continuous time. The results show that, under certain conditions, the DLO model is able to accurately and efficiently recover the parameters underlying the affective dynamics of a single individual. We discuss the results and the theoretical and practical implications of using this model, illustrate how to use it for studying psychological phenomena at the individual level, and provide specific recommendations on how to collect data for this purpose. We also provide a tutorial website and computer code in R to implement this approach. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Model reduction of a periodically forced slow–fast continuous piecewise linear system

AbstractIn this paper, singular perturbation theory is exploited to obtain a reduced-order model of a slow–fast piecewise linear 2-DOF oscillator subjected to harmonic excitation. The nonsmooth nonlinearity of piecewise linear nature is studied in the case of bilinear damping as well as with bilinear stiffness characteristics. We propose a continuous matching of the locally invariant slow manifolds obtained in each subregion of the state space, which yields a reduced-order model of the same nature as the full dynamics. The frequency-response curves obtained from the full system and the reduced-order models suggest that the proposed reduction method can capture nonlinear behaviors such as super- and subharmonic resonances.

On weakly turbulent solutions to the perturbed linear harmonic oscillator

abstract: We introduce specific solutions to the linear harmonic oscillator, named bubbles. They form resonant families of invariant tori of the linear dynamics, with arbitrarily large Sobolev norms. We use these modulated bubbles of energy to construct a class of potentials which are real, smooth, time dependent and uniformly decaying to zero with respect to time, such that the corresponding perturbed quantum harmonic oscillator admits solutions which exhibit a logarithmic growth of Sobolev norms. The resonance mechanism is explicit in space variables and produces highly oscillatory solutions. We then give several recipes to construct similar examples using more specific tools based on the continuous resonant (CR) equation in dimension two.

Grazing, Homoclinic Orbits and Chaos in a Single-Loop Feedback System with a Discontinuous Function

Bifurcations and chaos of a three-dimensional single-loop feedback system with a discontinuous piecewise linear feedback function are examined. Chaotic attractors are generated at the same time of the destabilization of foci accompanied with grazing. Multiple periodic solutions are connected with homoclinic orbits based at a pseudo saddle-focus, which satisfies the condition of Shil’nikov chaos formally. The generation of chaotic oscillations is shown in a circuit experiment on a linear ring oscillator with a comparator. The homoclinic bifurcations and chaos are also shown in a ring neural network with a nonmonotonic neuron.

Correlation between seismic intensity measures and engineering demand parameters of reinforced concrete frame buildings through nonlinear time history analysis

In this paper, the correlation between different seismic intensity measures (IM) and engineering demand parameters (EDP) of reinforced concrete (RC) buildings was evaluated by means of nonlinear dynamic analyses (NLDA). The Costa Rican ground motion database and four RC buildings (4-, 6-, 8- and 10-stories) were considered in this research. Estimations of conventional IMs were derived directly from the ground motion records as well as from the response of a single degree of freedom (SDOF) linear oscillator, with a period equal to that of the fundamental period of the buildings. NLDAs were performed using unscaled and scaled records to account for different intensity levels, and the performance of the buildings was characterized in terms of three EDPs: the average inter-story drift ratio, the maximum inter-story drift ratio, and the Park & Ang damage index. Results demonstrate a significant improvement in the correlation between IMs and EDPs when the dynamic characteristics of the structure are accounted for through the response of the SDOF oscillator. On the other hand, the peak ground velocity turned out to be an effective IM, independent of the dynamic characteristics of the buildings. This is very useful when assessing potential seismic damage both for immediate decision-making and for the characterization of the seismic hazard of a region.

Response moments of parametrically excited linear oscillators based on the Wiener path integral

Closed-form expressions are derived for the moments of response of linear stochastic oscillators by making use of the Wiener path integrals. A novel approach based on the characteristic functionals of stochastic processes is used. An extension to linear stochastic oscillators with fractional derivatives is also presented.

Floor Seismic Response Spectrum for the Building Equipped with Visco-Elastic Decouplers

By installing the visco-elastic (VE) decouplers between the floor slab and the beam, the floors of a building are isolated from the primary structure. Such a structural system, termed VE decoupled floors, has an increased damping and a reduced inertial force for mitigating the floor seismic response. These features lead to a significantly improved seismic performance of the important facilities resting (or anchored) on floors. This paper focuses on the floor seismic response spectrum (FSRS) of the building equipped with VE decoupled floors. First, the frequency-dependent property of the structure is dealt with, whereas the time-variant stiffness and damping of the VE decoupler are replaced by a couple of stabilized substitutions, and the appropriation of this substitution is numerically validated. Secondly, given the stabilized substitutive model, the FSRS is derived based on the classic complex modal decomposition process and the stochastic dynamics of linear oscillators. In this process, the input seismic excitation is considered via a specific model that accounts for its non-stationary nature. Lastly, a parametrical analysis is conducted to validate the proposed FSRS model and to reveal some special features of the VE decoupled flooring system. It is shown that the FSRS can be better mitigated if a more flexible VE decoupler and a larger mass (isolation) ratio are incorporated. A VE decoupled floor installed at the level of the roof is more effective than the ones installed at the lower levels. Basically, the peak of FSRS can be reduced by 70% if the VE decoupling system is properly designed.

Raychaudhuri equation and bouncing cosmology

This work deals with an exhaustive study of bouncing cosmology in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker spacetime. The geometry of the bouncing point has been studied extensively and used as a tool to classify the models from the point of view of cosmology. Raychaudhuri equation (RE) has been furnished in these models to classify the bouncing point as regular point or singular point. Behavior of time-like geodesic congruence in the neighborhood of the bouncing point has been discussed using the Focusing Theorem which follows as a consequence of the RE. An analogy of the RE with the evolution equation for a linear harmonic oscillator has been made and an oscillatory bouncing model has been discussed in this context.

The synergistic effect of the multiple parameters of vibro-impact nonlinear energy sink

This article studies the dynamics and efficiency of a vibro-impact damper (single-sided vibro-impact nonlinear energy sink—SSVI NES) depending on the exciting force parameters. The damper is coupled with a linear oscillator—the primary structure. It is shown that the damper is quite effective in a wide range of the exciting force amplitude and in the range of its frequency, which are higher than the resonant frequency; damper efficiency in these regions is fairly stable. The dynamics of the vibro-impact system “primary structure—SSVI NES” is rich and complex, which, however, does not impair the damper efficiency. In complex oscillatory regimes, the damper makes bilateral impacts: it hits both an obstacle and directly against the primary structure, which actually turns a single-sided NES into a double‐sided one. The optimization procedure and the choice of optimal damper parameters play a very important role in damper design. Optimizing multiple damper parameters instead of three shows a synergistic effect and provides better results.

A new perspective on the dynamic forced 2-DOF system with the non-perturbative approach

An elegant and straightforward method based on the equivalent linearized method is offered for the analysis of nonlinear vibrations of forced 2-DOF dynamical systems, in search of a simple good approximate solution with the non-perturbative approach. In the present approach, the 2-DOF system is transformed into a decoupled linear oscillator with instantaneous frequency identification. In this approach, we avoid the restriction on the periodic force coefficient, which was considered to be well below unity in various perturbation methods. This liberation of the amplitude of the periodic force led to the emergence of new patterns of the formation of periodic solution waves. It is found that the existence of the periodic force has an inhibitory effect on the scattering factors unless their frequency approaches the natural frequency of the dynamic system. The validation of the scheme is shown by comparing the numerical findings with the existing results. The supremacy of the proposed scheme is demonstrated by making numerical comparisons for each pattern of the obtained solution. The analytical solutions show high consistency proving the accuracy of the non-perturbative approach. This motivates us to extend the scheme for solving free and forced nonlinear vibrations of more coupled structures. It is noteworthy as a new technique with significant usefulness.

On a principle for mass sensing using self-excited template dynamics of coupled oscillators and root-finding algorithms

In this work, we analyze a mass sensing mechanism that relies on self-excited template dynamics of a simple network of two coupled oscillators, one linear and one nonlinear, to detect changes in the mass of the linear oscillator with tunable sensitivity. The resultant shift in the ratio of oscillation amplitudes is predicted using perturbation analysis and verified against numerical results obtained via parameter continuation. A physical realization of the mass sensing mechanism is proposed, in which the linear oscillator is represented by a periodically excited actuator/microcantilever assembly, the nonlinear oscillator is simulated in silico on a finite interval of time while driven by the periodic steady-state response of the linear oscillator, and a nonlinear root-finding algorithm is used to identify the input to the actuator that reproduces the sought, coupled, self-excited dynamics. The analysis shows that the sensor gain can be adjusted by appropriate tuning of the coupling stiffness without modifications to the physical components. Numerical results for two different actuator models demonstrate rapid convergence of the root-finding algorithm from a trivial initial solution guess over ranges of values of model parameters and added mass ratios.

Late-time evolution of the flat Friedmann universe over non-Archimedean space

The paper discusses the classical and quantum aspects of the dynamics of the flat Friedmann model of the universe in the late phase of its evolution, predominantly determined by the cosmological constant [Formula: see text]. In the classical part, the Lagrangian of the model is expressed in the Minkowski space and in the Euclidean space in the form of the Lagrangian of the inverted oscillator and the Lagrangian of the standard linear harmonic oscillator, respectively. The classic actions and Feynman propagators were determined as well as the dependence of the scale factor on time in the case of the Minkowski-type metric. Within the quantum part, the asymptotic form of the Wheeler–DeWitt wave function of the model in the considered expanding phase was first determined, and then the vacuum [Formula: see text]-adic states with the specific conditions of their existence. In the generalized adelic approach, a vacuum adelic wave function is constructed, whose form implies minisuperspace discretization.

A tristable nonlinear energy sink to suppress strong excitation vibration

As well known, the vibration reduction efficiency of the nonlinear energy sink (NES) is poor under strong excitation. In this paper, a tristable NES (TNES) is proposed. The TNES can degenerate into bistable and mono-stable NES by adjusting the geometric parameters of the springs. The governing equations of a linear oscillator coupled with the TNES under harmonic excitation are derived. The approximate analytical solution of the coupled system is obtained by using the harmonic balance method and verified numerically. The vibration suppression efficiency of TNES and NES is compared. The dynamic behavior of TNES under strong excitation is demonstrated. The results show that the nonlinear restoring force is softened due to the wide distribution of the three stable points of TNES. Therefore, compared with NES, the proposed TNES can suppress stronger excitation vibration. In addition, the low side barrier depth is conducive to TNES to perform chaotic inter-well oscillation, which can effectively suppress the strong excitation vibration and obtain good vibration reduction performance. As a result, the proposed TNES can eliminate the detached resonance curve and enlarge the effective range of the NES. Under relatively weak excitation, the vibration suppression efficiency of TNES is slightly lower than that of NES, although the TNES is also relatively significant. Therefore, this paper reveals the vibration suppression mechanism of TNES, and provides a way to effectively solve the problem of low vibration reduction efficiency of NES under strong excitation.

Nonlinear Perturbation of Random Matrix Theory.

We consider a system of linear oscillators, or quantum states, described by random matrix theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values, and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Owing to universal features of random matrix theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.

On the generalized Bézier-based integration approach for co-simulation applications

This paper investigates the efficiency of a recently developed numerical approach for the integration of the dynamics of co-simulation subsystems. In this regard, a robust Bézier-based multi-step integration approach is implemented in the subsystem solvers instead of conventional approaches, like the Euler method. Although such simplistic integration formulas proved to be accurate and easy to incorporate, they often suffer from instability when using large step sizes, particularly in solving stiff differential equations. On the other side, multistep methods can show better accuracy and convergence properties, which make them proper candidates for the solution of co-simulated dynamics. This paper puts forward the use of the Bézier integration formulas in the integration of Multibody system (MBS) dynamics, with a particular focus on co-simulation algorithms. To evaluate the proposed method efficiency, some straightforward benchmark problems are considered as case studies, such as a linear oscillator and two nonlinear hydraulic cranes. This study reveals the ability of the Bézier formula to keep the mechanical energy balance of the overall system. As a consequence, the stability of co-simulation results is improved. Furthermore, implementing the Bézier formulas minimizes the residual energy, which consequently reduces the energy errors in the co-simulation environment.

ENSO phase space dynamics with an improved estimate of the thermocline depth

The recharge oscillator model of the El Niño Southern Oscillation (ENSO) describes the ENSO dynamics as an interaction between the eastern tropical Pacific sea surface temperatures (T) and subsurface heat content (thermocline depth; h), defining a dynamical cycle with different phases. h is often approximated on the basis of the depth of the 20 °C isotherm (Z20). In this study we will address how the estimation of h affects the representation of ENSO dynamics. We will compare the ENSO phase space with h estimated based on Z20 and based on the maximum gradient in the temperature profile (Zmxg). The results illustrate that the ENSO phase space is much closer to the idealised recharge oscillator model if based on Zmxg than if based on Z20. Using linear and non-linear recharge oscillator models fitted to the observed data illustrates that the Z20 estimate leads to artificial phase dependent structures in the ENSO phase space, which result from an in-phase correlation between h and T. Based on the Zmxg estimate the ENSO phase space diagram show very clear non-linear aspects in growth rates and phase speeds. Based on this estimate we can describe the ENSO cycle dynamics as a non-linear cycle that grows during the recharge and El Nino state, and decays during the La Nina states. The most extreme ENSO states are during the El Nino and discharge states, while the La Nina and recharge states do not have extreme states. It further has faster phase speeds after the El Nino state and slower phase speeds during and after the La Nina states. The analysis suggests that the ENSO phase speed is significantly positive in all phases, suggesting that ENSO is indeed a cycle. However, the phase speeds are closest to zero during and after the La Nina state, indicating that the ENSO cycle is most likely to stall in these states.

Synchronization measurement based on Poincaré’s sphere

Synchronization is an important phenomenon occurring in many complex systems, such as neural networks, power grids, or social networks. Various methods have been proposed to measure synchronization, each with its advantages and limitations. In this work, we introduce the Poincaré sphere as a generalized measure of synchronization, which provides a useful tool to visualize the similarities between oscillations. We propose the notion of “total synchronization,” which considers amplitude, frequency, and phase synchronization, all of which can be measured and visualized simultaneously on the Poincaré sphere. We show that visualizing oscillatory trajectories on the Poincaré sphere has advantages over the more typical two-dimensional representation, particularly in cases where a projection onto a two-dimensional space cannot capture the similarities between different oscillations. Furthermore, we apply this new synchronization measurement to linear and nonlinear oscillators and provide two concrete examples of its application in visualizing phase jitter in noisy oscillations and quantizing solutions of oscillator-based Ising machines. Our work demonstrates the usefulness of the Poincaré sphere as a generalized measure of synchronization and provides a new perspective on studying synchronization in complex systems.

Monocular underwater measurement of structured light by scanning with vibrating mirrors

As a kind of visual detection technology, 3D structured light measurement technology has the advantages of high accuracy, rapid measurement, and active control, which is widely used in the research work of underwater surveys. Since light is refracted as it propagates underwater, the information of the measured object cannot be obtained directly from the captured image, which affects measurement accuracy. Therefore, it is necessary to correct and compensate for the refraction. In this paper, the structure of the combination of linear structured light and optical oscillator is adopted. In order to verify the function and accuracy of the system, experiments were conducted on object measurement in the air and underwater, respectively. The measurement experiments were performed on standard workpieces in air and the measurement results were compared with the real sizes of the workpieces. The relative error of the measurement was within 2%, which verifies the correctness of the construction of the system model and the accuracy of the calibration results. The results of the underwater experiments can approximate the measurement accuracy in air, which proves the correctness of the underwater model construction and the method of optical surface refraction compensation.