Damped Harmonic Oscillator Research Articles

A validated robust and automatic procedure for vibration analysis of bridge structures using MEMS accelerometers

Abstract Today, short- and long-term structural health monitoring (SHM) of bridge infrastructures and their safe, reliable and cost-effective maintenance has received considerable attention. From a surveying or civil engineer’s point of view, vibration-based SHM can be conducted by inspecting the changes in the global dynamic behaviour of a structure, such as natural frequencies (i. e. eigenfrequencies), mode shapes (i. e. eigenforms) and modal damping, which are known as modal parameters. This research work aims to propose a robust and automatic vibration analysis procedure that is so-called robust time domain modal parameter identification (RT-MPI) technique. It is novel in the sense of automatic and reliable identification of initial eigenfrequencies even closely spaced ones as well as robustly and accurately estimating the modal parameters of a bridge structure using low numbers of cost-effective micro-electro-mechanical systems (MEMS) accelerometers. To estimate amplitude, frequency, phase shift and damping ratio coefficients, an observation model consisting of: (1) a damped harmonic oscillation model, (2) an autoregressive model of coloured measurement noise and (3) a stochastic model in the form of the heavy-tailed family of scaled t-distributions is employed and jointly adjusted by means of a generalised expectation maximisation algorithm. Multiple MEMS as part of a geo-sensor network were mounted at different positions of a bridge structure which is precalculated by means of a finite element model (FEM) analysis. At the end, the estimated eigenfrequencies and eigenforms are compared and validated by the estimated parameters obtained from acceleration measurements of high-end accelerometers of type PCB ICP quartz, velocity measurements from a geophone and the FEM analysis. Additionally, the estimated eigenfrequencies and modal damping are compared with a well-known covariance driven stochastic subspace identification approach, which reveals the superiority of our proposed approach. We performed an experiment in two case studies with simulated data and real applications of a footbridge structure and a synthetic bridge. The results show that MEMS accelerometers are suitable for detecting all occurring eigenfrequencies depending on a sampling frequency specified. Moreover, the vibration analysis procedure demonstrates that amplitudes can be estimated in submillimetre range accuracy, frequencies with an accuracy better than 0.1 Hz and damping ratio coefficients with an accuracy better than 0.1 and 0.2 % for modal and system damping, respectively.

Optimized auxiliary oscillators for the simulation of general open quantum systems

A method for the systematic construction of few-body damped harmonic oscillator networks accurately reproducing the effect of general bosonic environments in open quantum systems is presented. Under the sole assumptions of a Gaussian environment and regardless of the system coupled to it, an algorithm to determine the parameters of an equivalent set of interacting damped oscillators obeying a Markovian quantum master equation is introduced. By choosing a suitable coupling to the system and minimizing an appropriate distance between the two-time correlation function of this effective bath and that of the target environment, the error induced in the reduced dynamics of the system is brought under rigorous control. The interactions among the effective modes provide remarkable flexibility in replicating non-Markovian effects on the system even with a small number of oscillators, and the resulting Lindblad equation may therefore be integrated at a very reasonable computational cost using standard methods for Markovian problems, even in strongly non-perturbative coupling regimes and at arbitrary temperatures including zero. We apply the method to an exactly solvable problem in order to demonstrate its accuracy, and present a study based on current research in the context of coherent transport in biological aggregates as a more realistic example of its use; performance and versatility are highlighted, and theoretical and numerical advantages over existing methods, as well as possible future improvements, are discussed.

Peculiar temperature dependence of dynamical sound speed in liquid Se50Te50 by inelastic x-ray scattering

Inelastic x-ray scattering (IXS) measurements were carried out on liquid Se50Te50 at temperatures up to 1323 K to investigate how the atomic dynamics changes with temperature through the semiconductor-metal transition. The acoustic mode was clearly observed in the dynamic structure factor and its energy was determined using a damped harmonic oscillator model. The dynamical sound speed obtained by IXS for nm−1 was found to behave similarly to the temperature dependence of ultrasonic sound velocity with a minimum near the semiconductor metal transition. The results can be explained by taking liquid Se50Te50 to be an inhomogeneous liquid consisting of metallic and semiconducting domains whose size is approximately 3 nm. The relaxation time obtained from frequency dependence of the dynamical sound speed was on a picosecond time scale.

Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian

An approach to quantization of the damped harmonic oscillator (DHO) is developed on the basis of a modified Bateman Lagrangian (MBL); thereby some quantum mechanical aspects of the DHO are clarified. We treat the energy operator for the DHO, in addition to the Hamiltonian operator that is determined from the MBL and corresponds to the total energy of the system. It is demonstrated that the energy eigenvalues of the DHO exponentially decrease with time and that transitions between the energy eigenstates occur in accordance with the Schr\"{o}dinger equation. Also, it is pointed out that a new critical parameter discriminates different behaviours of transition probabilities.

Some remarks on few recent results on the damped quantum harmonic oscillator

In a recent paper, Deguchi et al. (2019), the authors proposed an analysis of the damped quantum harmonic oscillator in terms of ladder operators. This approach was shown to be partly incorrect in Bagarello et al. (2019), via a simple no-go theorem. More recently, (Deguchi and Fujiwara, 2019), Deguchi and Fujiwara claimed that our results in Bagarello et al. (2019) are wrong, and compute what they claim is the square integrable vacuum of their annihilation operators. In this brief note, we show that their vacuum is indeed not a vacuum, and we try to explain what is behind their mistakes in Deguchi et al. (2019) and Deguchi and Fujiwara (2019). We also propose a very simple example clarifying the core of the problem.

WSGF — Time for a change part 1: Damped Harmonic Oscillators and the Vibrator/Ground model

In land seismic exploration a vibrator is used to generate a controlled wave-train of seismic energy into the earth by the forced oscillations of the ground surface over several seconds in what is known as a vibroseis sweep. Usually, these oscillations are sinusoidal in character and are generated by a hydraulic force acting on two masses, a baseplate (BP) that is in firm contact with the ground and a reaction mass (RM) positioned above the baseplate via a hydraulic actuator connecting both masses. The oscillations generated by the vibrator mechanism propagate away from the vibrator location through the subsurface in the form of seismic waves and are reflected/refracted back to the surface and recorded electronically by multiple sensors laid out on the surface or in a borehole. The energy received at each sensor is the result of the seismic signal generated by the vibrator (its signature) modified by the earth’s geology and the travel paths taken by the propagating wave. In mathematical terms, the signal received at the sensor is the seismic signal convolved by the earth. Unlike impulsive seismic sources, such as dynamite, the raw vibroseis field records are generally uninterpretable because of the relatively low energy output of the source and the multiple overlapping seismic signals that are reflected and refracted during the transmission of the sweep. In addition, the vibroseis signal is contaminated with undesirable environmental and source-generated noise and must be ‘deconvolved’ with the sweep signal to produce an interpretable record that resembles the wavelet generated by an impulsive source. Obviously, the more accurate the deconvolution operator is in replicating the source signature, the greater the resolution obtained in the processing of the seismic data.

Decoherence of a Damped Anisotropic Harmonic Oscillator under Magnetic Field Effects in a Two-Dimensional Noncommutative Phase-Space

In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.

On twin peak quasi-periodic oscillations resulting from the interaction between discoseismic modes and turbulence in accretion discs around black holes

ABSTRACT Given the peculiar and (in spite of many efforts) unexplained quasi-periodic oscillation twin peak phenomena in accretion disc power spectral density observations, the present exploratory analytical article tries to inquire deeper into the relationship between discoseismic modes and the underlying driving turbulence in order to assess its importance. We employ a toy model in the form of a Gaussian white noise-driven damped harmonic oscillator with stochastic frequency. This oscillator represents the discoseismic mode. (Stochastic damping was also considered, but interestingly it was found to be less relevant for the case at hand.) In the context of this model, we find that turbulence interacts with disc oscillations in interesting ways. In particular, the stochastic part in the oscillator frequency behaves as a separate driving agent. This gives rise to 3:2 twin peaks for some values of the physical parameters, which we find. We conclude with the suggestion that the study of turbulence be brought to the forefront of disc oscillation dynamics, as opposed to being a mere background feature. This change of perspective carries immediate observable consequences, such as considerably shifting the values of the (discoseismic) oscillator frequencies.

A complete damped harmonic oscillator using an Arduino and an Excel spreadsheet

The recent works of oscillators are mainly focused on underdamped oscillation. Therefore, the writer proposes an experimental apparatus to demonstrate all types of damped harmonic oscillation. The apparatus utilizes an Arduino for data acquisition and an Excel spreadsheet for data analysis. By using this apparatus, the type of damped harmonic oscillation can be determined, the equation of motion can be achieved, and the motion can be visualized in a graph on an Excel spreadsheet. The quality of fitted data is very good with R2 being close to 1 for every type of motion. The flexibility of customization and automation make this media provide real-time visualization and become more practical to be applied in a class.

On the mechanism of open-loop control of thermoacoustic instability in a laminar premixed combustor

We identify mechanisms through which open-loop control of thermoacoustic instability is achieved in a laminar combustor and characterize them using synchronization theory. The thermoacoustic system comprises two nonlinearly coupled damped harmonic oscillators – acoustic and unsteady heat release rate (HRR) field – each possessing different eigenfrequencies. The frequency of the preferred mode of HRR oscillations is less than the third acoustic eigenfrequency where thermoacoustic instability develops. We systematically subject the limit-cycle oscillations to an external harmonic forcing at different frequencies and amplitudes. We observe that forcing at a frequency near the preferred mode of the HRR oscillator leads to a greater than 90 % decrease in the amplitude of the limit-cycle oscillations through the phenomenon of asynchronous quenching. Concurrently, there is a resonant amplification in the amplitude of HRR oscillations. Further, we show that the flame dynamics plays a key role in controlling the frequency at which quenching is observed. Most importantly, we show that forcing can cause asynchronous quenching either by imposing out-of-phase relation between pressure and HRR oscillations or by inducing period-2 dynamics in pressure oscillations while period-1 in HRR oscillations, thereby causing phase drifting between the two subsystems. In each of the two cases, acoustic driving is very low and hence thermoacoustic instability is suppressed. We show that the characteristics of forced synchronization of the pressure and HRR oscillations are significantly different. Thus, we find that the simultaneous characterization of the two subsystems is necessary to quantify completely the nonlinear response of the forced thermoacoustic system.

Multiple stability switches and Hopf bifurcation in a damped harmonic oscillator with delayed feedback

This paper takes into consideration a damped harmonic oscillator model with delayed feedback. After transforming the model into a system of first-order delayed differential equations with a single discrete delay, the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation. Particularly, in view of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given. In order to check the rationality of our theoretical results, numerical simulations for some specific examples are also carried out by means of the MATLAB software package.

Free Oscillations of a Damped Simple Pendulum: An Analog Simulation Experiment

The frequency of free oscillation of a damped simple pendulum with large amplitude depends on its amplitude unlike the amplitude-independent frequency of oscillation of a damped simple harmonic oscillator. This aspect is not adequately emphasized in the undergraduate courses due to experimental and theoretical difficulties. We propose an analog simulation experiment to study the free oscillations of a simple pendulum that could be performed in an undergraduate laboratory. The needed sinusoidal potential is obtained approximately by using the available AD534 IC by suitably augmenting the electronic circuitry. To keep the circuit simple enough we restrict the initial angular amplitude of the simple pendulum to a maximum of [Formula: see text]. The results compare well qualitatively with the theoretical results. The small quantitative discrepancy is attributed to the inexact nature of the used “sinusoidal potential”.

MEMS BASED BRIDGE MONITORING SUPPORTED BY IMAGE-ASSISTED TOTAL STATION

Abstract. In this study, the feasibility of Micro-Electro-Mechanical System (MEMS) accelerometers and an image-assisted total station (IATS) for short- and long-term deformation monitoring of bridge structures is investigated. The MEMS sensors of type BNO055 from Bosch as part of a geo-sensor network are mounted at different positions of the bridge structure. In order to degrade the impact of systematic errors on the acceleration measurements, the deterministic calibration parameters are determined for fixed positions using a KUKA youBot in a climate chamber over certain temperature ranges. The measured acceleration data, with a sampling frequency of 100 Hz, yields accurate estimates of the modal parameters over short time intervals but suffer from accuracy degradation for absolute position estimates with time. To overcome this problem, video frames of a passive target, attached in the vicinity of one of the MEMS sensors, are captured from an embedded on-axis telescope camera of the IATS of type Leica Nova MS50 MultiStation with a practical sampling frequency of 10 Hz. To identify the modal parameters such as eigenfrequencies and modal damping for both acceleration and displacement time series, a damped harmonic oscillation model is employed together with an autoregressive (AR) model of coloured measurement noise. The AR model is solved by means of a generalized expectation maximization (GEM) algorithm. Subsequently, the estimated model parameters from the IATS are used for coordinate updates of the MEMS sensor within a Kalman filter approach. The experiment was performed for a synthetic bridge and the analysis shows an accuracy level of sub-millimetre for amplitudes and much better than 0.1 Hz for the frequencies.

Mechanical harmonic chaos game

We consider an analogue in classical mechanics to Barnsley’s chaos game in which a particle acts as a (possibly damped) harmonic oscillator whose centre of attraction randomly changes at fixed time intervals. This system exhibits a rich behaviour depending on the relationship between the strength of the attractive force and the drag force. We show how this mechanical system includes Barnsley’s (discrete) chaos game, producing for example the Sierpinski gasket, and in addition we observe evolutions which have interesting symmetries not exhibited by the Barnsley game. The system also provides a classical analogue of the quantum chaos game previously investigated by the current authors.

Bridging Dynamical Systems and Optimal Trajectory Approaches to Speech Motor Control With Dynamic Movement Primitives.

Current models of speech motor control rely on either trajectory-based control (DIVA, GEPPETO, ACT) or a dynamical systems approach based on feedback control (Task Dynamics, FACTS). While both approaches have provided insights into the speech motor system, it is difficult to connect these findings across models given the distinct theoretical and computational bases of the two approaches. We propose a new extension of the most widely used dynamical systems approach, Task Dynamics, that incorporates many of the strengths of trajectory-based approaches, providing a way to bridge the theoretical divide between what have been two separate approaches to understanding speech motor control. The Task Dynamics (TD) model posits that speech gestures are governed by point attractor dynamics consistent with a critically damped harmonic oscillator. Kinematic trajectories associated with such gestures should therefore be consistent with a second-order dynamical system, possibly modified by blending with temporally overlapping gestures or altering oscillator parameters. This account of observed kinematics is powerful and theoretically appealing, but may be insufficient to account for deviations from predicted kinematics—i.e., changes produced in response to some external perturbations to the jaw, changes in control during acquisition and development, or effects of word/syllable frequency. Optimization, such as would be needed to minimize articulatory effort, is also incompatible with the current TD model, though the idea that the speech production systems economizes effort has a long history and, importantly, also plays a critical role in current theories of domain-general human motor control. To address these issues, we use Dynamic Movement Primitives (DMPs) to expand a dynamical systems framework for speech motor control to allow modification of kinematic trajectories by incorporating a simple, learnable forcing term into existing point attractor dynamics. We show that integration of DMPs with task-based point-attractor dynamics enhances the potential explanatory power of TD in a number of critical ways, including the ability to account for external forces in planning and optimizing both kinematic and dynamic movement costs. At the same time, this approach preserves the successes of Task Dynamics in handling multi-gesture planning and coordination.

Reconstruction approach to quantum dynamics of bosonic systems

We propose an approach to analytically solve the quantum dynamics of bosonic systems. The method is based on reconstructing the quantum state of the system from the moments of its annihilation operators, dynamics of which is solved in the Heisenberg picture. The proposed method is general in the sense that it does not assume anything on the initial conditions of the system such as separability, or the structure of the system such as linearity. It is an alternative to the standard master equation approaches, which are analytically demanding especially for large multipartite quantum systems. To demonstrate the proposed technique, we apply it to a system consisting of two coupled damped quantum harmonic oscillators.

A damped harmonic oscillator in the GUP framework

نوسانگرهای هماهنگ به خاطر ویژگی‌های دینامیکی در گرانش کوانتومی از اهمیت زیادی برخوردارند. می‌توان دینامیک این نوسانگر را از طریق رابطه عدم قطعیت تعمیم‌یافته، که در مدل‌های مختلف کیهان‌شناسی مثل نظریه ریسمان و گرانش کوانتومی حلقه پیشنهاد شده است مورد اصلاح قرار داد. در این مقاله معادله نوسانگر هماهنگ میرای کوانتومی با توجه به اصل عدم قطعیت تعمیم‌یافته حل شده است، و اثر این اصلاحات در دستگاه نوسانگر میرا مورد بررسی قرار گرفته است.

A geometric description for simple and damped harmonic oscillators

In this work we consider the Riemannian geometry associated with the differential equations of one dimensional simple and damped linear harmonic oscillators. We show that the sectional curvatures are completely determined by the oscillation frequency and the friction coefficient and these physical constants can be thought as obstructions for the manifold to be flat. Moreover, equations of simple and damped harmonic oscillators describe nonisomorphic solvable Lie groups with nonpositive scalar curvature.

Microdrop impact on soft substrates at low Weber numbers

In this work, we experimentally investigate the impact of microdrops on diverse soft substrates at low Weber numbers. We show that the dynamic behaviors of the impinging microdrops are very different from that of macroscopic droplets reported in the literature. Whereas various impact phenomena have been identified for millimeter-sized water droplets in previous studies, we only observed regular deposition for droplets of a few tens of micrometers in the experiment. In contrast to the significant slowdown of the receding millimeter-sized droplets, no effects of substrate stiffness on the impact process have been identified at microscale. These different dynamics are explained by the different dynamic response of soft viscoelastic materials during droplet impact over different timescales. Finally, the analysis of the post-impact droplet oscillation with the theory for damped harmonic oscillators reveals that the damping coefficient is also size-dependent while the spring constant is not. A simple scaling argument was proposed to understand this finding.

Comment on ``The one-dimensional harmonic oscillator damped with Caldirola--Kanai Hamiltonian''

Several conceptual errors in a recently published paper ({\it Rev.\ Mex.\ F\'{\i}s.\ E} {\bf 64} (2018) 47) dealing with the damped one-dimensional harmonic oscillator are pointed out.