Damped Simple Harmonic Oscillator Research Articles

Photometric detection of internal gravity waves in upper main-sequence stars

Context. Recent studies of massive stars using high-precision space photometry have revealed that they commonly exhibit stochastic low-frequency (SLF) variability. This has been interpreted as being caused by internal gravity waves excited at the interface of convective and radiative regions within stellar interiors, such as the convective core or sub-surface convection zones, or being caused by dynamic turbulence associated with sub-surface convection zones within the envelopes of main-sequence massive stars. Aims. We aim to compare the properties of SLF variability in massive main-sequence stars observed by the Transiting Exoplanet Survey Satellite (TESS) mission determined by different statistical methods, and confirm the correlation between the morphology of SLF variability and a star’s location in the Hertzsprung–Russell (HR) diagram. We also aim to quantify the impact of data quality on the inferred SLF morphologies using both fitting methodologies. Methods. From a sample of 30 previously observed and characterised galactic massive stars observed by TESS, we compare the resultant parameters of SLF variability, in particular the characteristic frequency, obtained from fitting the amplitude spectrum of the light curve with those inferred from fitting the covariance structure of the light curve using the CELERITE2 Gaussian process (GP) regression software and a damped simple harmonic oscillator (SHO) kernel. Results. We find a difference in the characteristic frequency obtained from the amplitude spectrum fitting and from fitting the covariance structure of the light curve using a GP regression with CELERITE2 for only a minority of the considered sample. However, the trends among mass, age, and the properties of SLF variability previously reported remain unaffected. We also find that the method of GP regression is more efficient in terms of computation time and, on average, more robust against the quality and noise properties of the input time series data in determining the properties of SLF variability. Conclusions. GP regression is a useful and novel methodology to efficiently characterise SLF variability in massive stars compared to previous techniques used in the literature. We conclude that the correlation between a star’s SLF variability, in particular the characteristic frequency, and its location in the HR diagram is robust for main-sequence massive stars. There also exists a distribution in the stochasticity of SLF variability in massive stars, which indicates that the coherency of SLF variability is also a function of mass and age in massive stars.

State-space Representation of Matérn and Damped Simple Harmonic Oscillator Gaussian Processes

Abstract Gaussian processes (GPs) are used widely in the analysis of astronomical time series. GPs with rational spectral densities have state-space representations which allow  ( n ) evaluation of the likelihood. We calculate analytic state space representations for the damped simple harmonic oscillator and the Matérn 1/2, 3/2 and 5/2 processes.

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”.

The stochastic dynamics of a nanobeam near an optomechanical resonator in a viscous fluid

We quantify the Brownian driven, stochastic dynamics of an elastic nanobeam immersed in a viscous fluid that is partially wrapped around a microdisk optical resonator. This configuration has been proposed as an optomechanical and nanoscale analog of the atomic force microscope [Srinivasan et al., Nano Lett. 11, 791 (2011)]. A small gap between the nanobeam and microdisk is necessary for the optomechanical transduction of the mechanical motion of the nanobeam. We compute the stochastic dynamics of the nanobeam in fluid for the precise conditions of the laboratory using deterministic finite element simulations and the fluctuation dissipation theorem. We investigate the dynamics of a nanobeam in water and in air and quantify the significance of the fluid-solid interaction between the nanobeam and the optical resonator. Our results in air show that, despite the complex geometry of the nanobeam, it can still be represented approximately as a damped simple harmonic oscillator. On the other hand, when the nanobeam is immersed in water there are significant deviations from the dynamics of a simple harmonic oscillator. The small gap between the nanobeam and the microdisk is found to be a significant source of additional dissipation. In air, the quality factor of the mechanical oscillation of the nanobeam is reduced by an order of magnitude due to the presence of the microdisk, however, the dynamics remain underdamped even in the presence of the microdisk. On the other hand, when placed in water, the dynamics without the microdisk is underdamped and with the microdisk the dynamics become strongly over damped.

Modes of growth in dynamic systems

Regardless of a system's complexity or scale, its growth can be considered to be a spontaneous thermodynamic response to a local convergence of down-gradient material flows. Here it is shown how system growth can be constrained to a few distinct modes that depend on the time integral of past flows and the current availability of material and energetic resources. These modes include a law of diminishing returns, logistic behaviour and, if resources are expanding very rapidly, super-exponential growth. For a case where a system has a resolved sink as well as a source, growth and decay can be characterized in terms of a slightly modified form of the predator–prey equations commonly employed in ecology, where the perturbation formulation of these equations is equivalent to a damped simple harmonic oscillator. Thus, the framework presented here suggests a common theoretical under-pinning for emergent behaviours in the physical and life sciences. Specific examples are described for phenomena as seemingly dissimilar as the development of rain and the evolution of fish stocks.

Experimental and theoretical studies on MEMS piezoelectric vibrational energy harvesters with mass loading

Experimental and theoretical investigations on micro-scale multi-morph cantilever piezoelectric vibrational energy harvesters (PZEHs) of the MicroElectroMechanical Systems (MEMS) are presented. The core body of a PZEH is a “multi-morph” cantilever, where one end is clamped to a base and the other end is free. This “fixed-free” cantilever system including a proof-mass (also called the end-mass) on the free-end that can oscillate with the multi-layer cantilever under continuous sinusoidal excitations of the base motion. A partial differential equation (PDE) describing the flexural wave propagating in the multi-morph cantilever is reviewed. The resonance frequencies of the lowest mode of a multi-morph cantilever PZEH for some ratios of the proof-mass to cantilever mass are calculated by either solving the PDE numerically or using a lumped-element model as a damped simple harmonic oscillator; their results are in good agreement (disparity≤0.5%). Experimentally, MEMS PZEHs were constructed using the standard micro-fabrication technique. Calculated fundamental resonance frequencies, output electric voltage amplitude V and output power amplitude P with an optimum load compared favorably with their corresponding measured values; the differences are all less than 4%. Furthermore, a MEMS PZEH prototype was shown resonating at 58.0±2.0Hz under 0.7g (g=9.81m/s2) external excitations, corresponding peak power reaches 63μW with an output load impedance Z of 85kΩ. This micro-power generator enabled successfully a wireless sensor node with the integrated sensor, radio frequency (RF) radio, power management electronics, and an advanced thin-film lithium-ion rechargeable battery for power storage at the 2011 Sensors Expo and Conference held in Chicago, IL. In addition, at 58Hz and 0.5, 1.0g excitations power levels of 32, and 128μW were also obtained, and all these three power levels demonstrated to be proportional to the square of the acceleration amplitude as predicted by the theory. The reported P at the fundamental resonance frequency f1 and acceleration G-level, reached the highest “Figure of Merit” [power density×(bandwidth/resonant frequency)] achieved amongst those reported in the up-to-date literature for high quality factor Qf MEMS PZEH devices.

Absorption Ångström coefficient, brown carbon, and aerosols: basic concepts, bulk matter, and spherical particles

Abstract. The concept of wavelength-dependent absorption Ångström coefficients (AACs) is discussed and clarified for both single and two-wavelengths AACs and guidance for their implementation with noisy absorption spectra is provided. This discussion is followed by application of the concept to models for brown carbon bulk absorption spectra including the damped simple harmonic oscillator model, its Lorentzian approximation, and the band-gap model with and without Urbach tail. We show that the band-gap model with Urbach tail always has an unphysical discontinuity in the first derivative of the AAC at the band-gap – Urbach-tail matching wavelength. Complex refractive indices obtained from the bulk damped simple harmonic oscillator model are used to calculate absorption spectra for spherical particles, followed by a discussion of their features. For bulk material and small particles, this model predicts a monotonic decrease of the AAC with wavelength well above the resonance wavelength; the model predicts a monotonic increase for large particles.

Investigation of phase transition in ferroelectric Pb0.70Sr0.30TiO3 thin films

We have carried out dielectric and Raman spectroscopy studies at the 298–623 K temperature range in polycrystalline Pb0.70Sr0.30TiO3 thin films grown by a soft chemical method. The diffuse phase-transition behavior of the thin films was observed by means of the dielectric constant versus temperature curves, which show a broad peak. Such behavior was confirmed later by Raman spectroscopy measurements up to 823 K, indicating that a diffuselike phase transition takes place at around 548–573 K. The damping factor of the E(1TO) soft mode was calculated using the damped simple harmonic oscillator model. On the other hand, Raman modes persist above the tetragonal to cubic phase transition temperature although all optical modes should be Raman inactive. The origin of these modes was interpreted in terms of a breakdown of the microscopic local cubic symmetry by chemical disorder. The lack of a well-defined transition temperature and the presence of broad bands at some temperature interval above the ferroelectric-paraelectric phase-transition temperature suggested a diffuse nature of the phase transition. This result corroborates the dielectric constant versus temperature data, which showed a broad ferroelectric phase transition in this thin film.

Low latitude geomagnetic field line resonance: Experiment and modeling

Geomagnetic field line resonances may be identified in ground magnetometer data by comparing the difference in amplitude and phase of signals recorded at two closely spaced sites or by examining the latitudinal variation in polarization properties across a more extended array. These two methods give comparable results for values of the resonant frequency and width at low latitudes (L < 3). We have also found an upper limit for the damping factor, γ∼0.07 at L=1.8, by applying a damped simple harmonic oscillator model. The field line resonance structure observed in 5 weeks of data showed only one resonant frequency at L=1.8 but up to four harmonies concurrently at L=2.8. An early local morning decrease in eigenfrequency was usually present at L=1.8. This is attributed to dynamic heavy ion mass loading effects in the ionosphere where the plasma density increases around dawn. The observed eigenfrequencies were used to evaluate two plasma density models. Calculations using a combined IRI‐90 and diffusive equilibrium (DE) model gave eigenfrequencies which are considerably smaller than the experimentally observed values at both L=1.8 and L=2.8. Furthermore, the calculated harmonic spacings at L=2.8 do not agree with the experimental values, although the diurnal trends were successfully modeled using the IRI‐DE plasma description. The low‐latitude plasma density model described by Bailey (1983) yields eigenfrequencies which show good agreement with the experimentally observed values at both latitudes.

Analytical solution to the master equation for a quantized cavity mode

We solve analytically the master equation for a quantized cavity mode, finding the time-dependent reduced density operator. The formalism applies to both amplifiers and attenuator, permitting the results to be converted between them by introducing a temperature that can be negative. We obtain a quasi-probability distribution function similar to that obtained by Glauber [ GlauberR. J., ed., Quantum Optics ( Academic, New York, 1969)] from our reduced density matrix and equate his parameters with the coefficients appearing in the master equation. We illustrate the formalism by calculating variances, expectation values, and photon statistics for the damped simple harmonic oscillator and the linear amplifier.

Discrete variable structure system with pseudo-sliding mode

AbstractVariable structure systems with sliding modes have been widely discussed and used in many different fields of applications. The precise behaviour at a switching surface is complicated because there the system is non-analytic. The damped simple harmonic oscillator with a nonlinear variable structure is discretised and analysed in detail, revealing the occurrence and structure of pseudo-sliding modes which give insight to the corresponding sliding modes for the continuous system. Necessary and sufficient conditions are obtained and the analysis illustrated with graphs from numerical solutions.

Thermal noise in simple electrical and mechanical devices

This presentation will provide an introduction to thermal noise as a process that limits the minimum detectable signal in a vibroacoustic sensor system. Starting with single degree‐of‐freedom systems, represented by a series RLC network and a damped simple harmonic oscillator, the equipartition theorem will be used to determine the mean‐square value of the noisy variable. Consideration of the dissipative element will then lead to the concept of noise spectral densities and the general fluctuation/dissipation relations such as the well‐known result for Johnson noise (v* = 4kBTR) in an electrical resistor. Generalization of this approach to three‐dimensional systems with a quasicontinuous distribution of normal modes will illustrate the connection between thermal noise and traditional concepts of “blackbody radiation” and introduce the concept of a quantum limit to the “white” noise spectrum. These results will be applied to the calculation of minimum detectable signals in interferometric fiber‐optic vibroa...

Lagrangians and systems they describe—how not to treat dissipation in quantum mechanics

A Lagrangian that yields equations of motion for a damped simple harmonic oscillator is shown not to describe this system, but a completely different physical system. Thus, we have a simple counterexample to the statement: ’’If a Lagrangian gives the correct equations of motion for a given system then the Lagrangian describes the system.’’ We also construct a physical system that the Lagrangian describes and derive some of its properties. The quantum theory based on the above Lagrangian has been used to attempt to discuss the quantum theory of dissipation. This is, however, incorrect since the system the Lagrangian describes has no dissipation.

Raman scattering study of the three structural phases of KMnF3

The Raman spectra of the three phases of KMnF3 have been measured and a detailed group theoretical analysis has been carried out in order to identify the observed peaks. The results for the intermediate temperature phase are consistent with a tetragonal structure of D4h18 symmetry but the results for the low-temperature phase do not unambiguously determine the crystal structure. Particular attention was paid to the temperature-dependent modes. The soft modes associated with the upper phase transition were fitted to the response functions for a damped simple harmonic oscillator and for a model including damping by phonon density fluctuations. The results favour the latter model and suggest that non-classical critical fluctuations play an important role near the upper phase transition.