Simple Harmonic Oscillator Research Articles

Masing Manifolds: Reconciling the Masing Conditions with Real Hysteresis in Jointed Structures

The Masing conditions establish a criterion to relate the loading curve of a hysteretic system (e.g., systems with friction or plasticity) to its complete hysteresis loop. For the field of joint mechanics, where hysteretic models are often used to describe the dissipative, tangential behavior within an interface, the Masing conditions allow for significant computational savings when the normal load is constant. In practice, though, jointed systems experience time varying normal forces that modify the tangential behavior of the system. Consequently, the hysteretic behavior of jointed structures do not adhere to the Masing conditions. In this work, this discrepancy between the Masing conditions and behavior exhibited by jointed structures is explored, and it is hypothesized that if the Masing conditions accounted for variations in normal force, then they would more accurately represent jointed structures. A new set of conditions is introduced to the original set of Masing conditions, yielding a « Masing manifold » that spans the tangential displacement-tangential force-normal force space. Both a simple harmonic oscillator and a built-up structure are investigated for the case of elastic dry friction, and the results show that the hysteresis of both of these systems conforms to the three dimensional Masing manifold exactly, provided that a set of constraints are satisfied, even though the hysteresis does not conform with the original Masing conditions.

Solution for the simple harmonic oscillator problem via unilateral Fourier transform

Solution for the simple harmonic oscillator problem via unilateral Fourier transform

Proportional and Derivative Coupling: a Way to Achieve Synchronization for Coupled Oscillators

In order to achieve synchronization, the concept of proportional coupling and derivative coupling, as analogue to PD control theory, is proposed. The author analyzes from the coupled simple harmonic oscillators to Kuromoto model, and to the coupled Van der Pol oscillators. It is conjectured that only proportional coupling for two or multiple linear and nonlinear coupled oscillators is not sufficient most of the time to achieve synchronization, whereas derivative coupling is the dominant factor for two or multiple linear and nonlinear coupled oscillators to get synchronization. In addition, it is noticed that if there is no coupling, there is no synchronization if starting from different initial conditions. In order to achieve synchronization, coupling is required. However, coupling does not necessary mean it will achieve synchronization, it has to be coupled in a particular way, and otherwise it will not get synchronization. As a result of this study, two major questions for the linear and nonlinear coupled oscillators to get synchronization are proposed. Generally, only proportional coupling (i.e. spring coupling) is earlier to analyze as it is a simple physics problem. Once we couple them using the derivative coupling approach, it will get major complications.

Studies of Putative Human Bioenergy Using a Torsion Pendulum with a Metal Chain-Link Support instead of a Nylon Filament Support

Earlier experiments have established that a torsion pendulum consisting of a steel mesh hemisphere suspended by a nylon filament above a human subject is capable of detecting and collecting data that may represent human bioenergy. Among the effects observed are a wide range of subject-induced rotational frequencies of the pendulum and displacement of the pendulum from its natural center of oscillation. The nylon filament support confers the properties of a simple harmonic oscillator (sho) to the pendulum. The resulting sho swings of the pendulum obscure smaller swings that correspond to subject effects on the pendulum. For this work, it was desired to modify the pendulum to reduce or eliminate the sho oscillatory swings. A proposed modification was to replace the nylon filament with a length of chain. The idea was that the chain would confer properties of an ‘energy well’ to the pendulum. Rotational forces exerted on the chain-supported pendulum would cause the chain to twist and thus rise up the side of the energy well. Ideally, the only restoring force would be the tendency of the twisted links to untwist, so as to drop back down to the bottom of the energy well. This strategy was successful. Control experiments in which the chain-supported pendulum was stimulated to rotational motion by air puffs showed a rapid deflection of the pendulum, which represented traveling up the side of the energy well. The deflected pendulum then immediately returned to its original position and remained there, with no sho oscillations being evident. This is consistent with the chain-link pendulum conforming to the characteristics of an energy well. The chain-supported pendulum was used to assess subject effects on the pendulum. The basic pattern of the subject effects resembled those obtained with a nylon filament-supported pendulum, except the large sho swings that are a feature of the filament-supported pendulum, were absent from the effects observed by the chain-supported pendulum. Thus, for the first time, the pattern of subject-induced effects could be observed, measured, and analyzed without being obscured by sho oscillations. Soon after the subject was seated under the pendulum, its rotational motion drove the pendulum rotational position up one side of the energy well, where it stayed throughout the experiment. This establishes that the force is a spiral force, which conforms to our earlier observations. The now-visible underlying subject-induced oscillations are nuanced and sophisticated and are reminiscent of the complexity of music or speech. Moreover, these oscillations encompass a range of very low frequencies, from less than 0.01 Hz to 0.15 Hz. The frequency amplitudes rise dramatically at the lowest frequencies, suggesting the existence of additional even lower frequencies; but if they exist, the current pendulum cannot detect them. The frequency amplitudes toward the upper end do not fade away completely, suggesting that what is observed represents what there is. These frequencies are much lower than what is detected by electrodes attached to the scalps of subjects which measure minute voltage fluctuations called ‘brain waves.’ These voltage-related brain waves range from 0.5-42 Hz. Low frequency humpback whale sounds, often attributed to being a means of long-range communication, range from 30-8,000 Hz. As far as we know, the low frequencies observed with the chain-supported pendulum are unprecedented. As of now, the nature and origin of the subject effects measured by the nylon filament and chain-supported pendulums are not known. Accumulating evidence argues that they are not merely subject-induced thermal convection currents, but some other form of energy that originates from the subject. As an energy of very low frequency, it could be expected to propagate over long distances, reminiscent of low frequency whale sounds. This article addresses these and other issues.

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.

Novel slow–fast behaviour in an oscillator driven by a frequency-switching force

When an oscillator switches abruptly between different frequencies, there is some ambiguity in deciding how the system should be modelled at the switch. Here we describe two seemingly natural models of a switch in a simple periodically-forced harmonic oscillator, which disagree starkly in their predictions of its long time behaviour. Attempting to resolve the disagreement by ‘regularizing’ the switch not only preserves the disagreement, but shows it increases with time. One of the models corresponds to a conventional ‘Filippov’ description of a nonsmooth system, while the second exhibits a structure that irreversibly ages, developing a number of novel multi-scale behaviours that we believe have not been reported before. These include slow–fast staircases, novel mixed-mode oscillations, and a synchronized canard explosion. These features are proven to exist using asymptotic analysis, but as they involve a slow–fast time-scale separation that increases with time, they lie beyond the reach of numerical methods.

Adaptive SVSF-KF estimation strategies based on the normalized innovation square metric and IMM strategy

The smooth variable structure filter (SVSF) is highly robust to modeling uncertainty and unknown disturbances. Recent developments to the SVSF have allowed for the creation of an adaptive estimation scheme termed the SVSF-KF, which balances the optimality of the Kalman filter (KF) with the robustness of the SVSF. The approach utilizes the KF estimate during normal operation, and utilizes the robust SVSF gain to estimate the states during the presence of a fault. However, the gain adaptation involved in detecting a fault and switching between the KF and SVSF estimates suffers from several limitations, including unwanted chattering. In this work, we review the original SVSF-KF approach and present two novel SVSF-KF strategies based on the normalized innovation square metric and the interacting multiple model strategy to address these limitations. Experimental simulations involving a simple harmonic oscillator subject to a fault condition are conducted, which verify the effectiveness of our proposed approaches.

Collapse-revival of entanglement in a non-commutative harmonic oscillator revealed via coherent states and path integral

Abstract We develop an approach using coherent states and path integral to investigate the dynamics of entanglement in a simple two-dimensional non-commutative harmonic oscillator. We start by employing a Bopp shift to convert the Hamiltonian describing the system into a commutative equivalent one. This allows us to construct coherent states and calculate the propagator in standard way. By deriving the explicit expression of the time-dependent coherent states and considering its connection with the number states, we provide exact results for evaluating the degree of entanglement between the ground state and any excited state through the purity function. The interesting emerging result is that, as long as the non-commutativity parameter is non-zero, our system exhibits the phenomenon of collapse and revival of entanglement.

Quantum Monte Carlo: Reduction of time‐step error with the wave function correction method

AbstractWe compare the accuracy and precision of the wave function correction method to that of traditional diffusion quantum Monte Carlo (traditional DQMC) simulations. Direct calculation of corrections to trial wave functions reduces time step error for Be and Ne electronic ground state energies when compared to traditional DQMC results. The fixed‐node method is utilized for these fermionic systems, so error due to incorrect node locations is not addressed. This correction DQMC method achieves more accurate energies at larger time steps, promising decreased computational cost to reach a given accuracy. For an optimized Be trial wave function, the statistical uncertainty per unit of computational effort is also reduced relative to traditional DQMC. Correction DQMC utilizes walker feed integrals and growth factors, as reported previously. We illustrate the method in a simple harmonic oscillator example, and we report high accuracy energies for Be and Ne at several time steps.

The bound-state solutions of the one-dimensional pseudoharmonic oscillator

We study the bound states of a quantum mechanical system consisting of a simple harmonic oscillator with an inverse square interaction, whose interaction strength is governed by a constant $\alpha$. The singular form of this potential has doubly-degenerate bound states for $-1/4\leq\alpha<0$ and $\alpha>0$; since the potential is symmetric, these consist of even and odd-parity states. In addition we consider a regularized form of this potential with a constant cutoff near the origin. For this regularized potential, there are also even and odd-parity eigenfunctions for $\alpha\geq-1/4$. For attractive potentials within the range $-1/4\leq\alpha<0$, there is an even-parity ground state with increasingly negative energy and a probability density that approaches a Dirac delta function as the cutoff parameter becomes zero. These properties are analogous to a similar ground state present in the regularized one-dimensional hydrogen atom. We solve this problem both analytically and numerically, and show how the regularized excited states approach their unregularized counterparts.

Favorite problems on acoustic transducers and linear systems theory

Practitioners of acoustics sometimes forget how lucky we are that the vast majority of acoustic phenomena and devices we encounter in daily life are linear and solutions using linear systems theory work very well for predicting behavior and designing devices. Typical examples are microphones, loudspeakers and acoustic propagation at sufficiently low excitation levels. One might contrast this to a viscous dashpot, which is also commonly modeled as a linear device. The sliding seals in all viscous dashpots render them essentially-nonlinear from the start. One favorite problem I give year after year, in various forms, is reported here. A generic acoustic transducer, based on a linear simple harmonic oscillator is presented and often the system response function is given in the problem statement. Derivation of the transfer function could be a problem from earlier or later in the class. The students are asked to predict the time domain output of the transducer, subject to a periodic forcing composed of a sawtooth wave, using frequency-domain convolution and Fourier series. Various sub-questions explore the effect of transducer bandwidth and damping on the output signal. A variation of this problem using time-domain convolution is also a favorite. Predictions can be compared to measurements.

Numerical methods, energy conservation, and a new method for particle motion in magnetic fields

This article mathematically examines the energy conservation of explicit Runge–Kutta methods and basic symplectic integration methods for a simple harmonic oscillator and a particle in a magnetic field. With the demonstration of the failure of these methods to conserve the energy of a particle in a magnetic field, a new integration method is presented for this type of system where energy conservation is dictated by the type of floating point variable used, not the size of the time step or the method order. Higher order versions of the method improve the accuracy in the calculation of the particle position and allow for more accurate calculations of the velocity in a spatially varying field. Furthermore, the new method is demonstrated to have improved phase accuracy when compared to several common implicit methods.

Misalignment effects on the performance and stability of x-ray free electron laser oscillator

We report on the effects of transverse spatial misalignment on the performance and stability of x-ray free electron laser oscillators (XFELOs). For this study, we adopt the FEL driven paraxial resonator model in which the transverse profile of the radiation field is represented by Gauss-Hermite mode expansion. Then, we apply Gaussian optics for transporting the radiation field along with the misalignment in the optical cavity. We divide the misalignment into relevant categories of static and periodic types to understand their effects on FEL gain, lasing, and saturation. After that, we associate the first order moments of the radiation field to that of driven simple harmonic oscillator models to identify regions of instability. Throughout this report, we validate our models and analyses via theoretical calculations, analytical approximations, and simulations.

Fixing the rotating-wave approximation for strongly detuned quantum oscillators

Periodically-driven oscillators are commonly described in a frame co-rotating with the drive and using the rotating-wave approximation (RWA). This description, however, is known to induce errors for off-resonant driving. Here we show that the standard quantum description, using creation and annihilation of particles with the oscillator's natural frequency, necessarily leads to incorrect results when combined with the RWA. We demonstrate this on the simple harmonic oscillator and present an alternative operator basis which reconciles the RWA with off-resonant driving. The approach is also applicable to more complex models, where it accounts for known discrepancies. As an example, we demonstrate the advantage of our scheme on the driven quantum Duffing oscillator.

Generalized para-Bose states

In this paper, we construct integrals of motion in a para-Bose formulation for a general time-dependent quadratic Hamiltonian, which, in its turn, commutes with the reflection operator. In this context, we obtain generalizations for the squeezed vacuum states (SVS) and coherent states (CS) in terms of the Wigner parameter. Furthermore, we show that there is a completeness relation for the generalized SVS owing to the Wigner parameter. In the study of the probability transition, we found that the displacement parameter acts as a transition parameter by allowing access to odd states, while the Wigner parameter controls the dispersion of the distribution. We show that the Wigner parameter is quantized by imposing that the vacuum state has even parity. We apply the general results to the case of the time-independent para-Bose oscillator and find that the mean values of the coordinate and momentum have an oscillatory behavior similarly to the simple harmonic oscillator, while the standard deviation presents corrections in terms of the squeeze, displacement, and Wigner parameters.

Circuit Complexity from Supersymmetric Quantum Field Theory with Morse Function

Computation of circuit complexity has gained much attention in the theoretical physics community in recent times, to gain insights into the chaotic features and random fluctuations of fields in the quantum regime. Recent studies of circuit complexity take inspiration from Nielsen’s geometric approach, which is based on the idea of optimal quantum control in which a cost function is introduced for the various possible path to determine the optimum circuit. In this paper, we study the relationship between the circuit complexity and Morse theory within the framework of algebraic topology, which will then help us study circuit complexity in supersymmetric quantum field theory describing both simple and inverted harmonic oscillators up to higher orders of quantum corrections. We will restrict ourselves to N=1 supersymmetry with one fermionic generator Qα. The expression of circuit complexity in quantum regime would then be given by the Hessian of the Morse function in supersymmetric quantum field theory. We also provide technical proof of the well known universal connecting relation between quantum chaos and circuit complexity of the supersymmetric quantum field theories, using the general description of Morse theory.

Conservative dynamics in a novel class of 3D generalized thermostatted systems.

This paper reports a method to derive a novel class of 3D generalized thermostatted oscillators from a simple damped harmonic oscillator. Its detailed procedure is obtained through a mathematical derivation. Then, we propose an example system to show the effectiveness of the method. Furthermore, the numerical analysis is performed to investigate its rich conservative dynamics, including chaotic sea, hierarchical invariant tori, and the coexistence of chaotic seas and invariant tori with islands-around-islands hierarchy, even if there are three dissipative terms in the example system. To verify the existence of conservative chaos at the physical level, an analog circuit is presented to observe the existing conservative chaotic flows in National Instruments multisim. We finally provide two systems, which can produce conservative chaotic flows with more complicated topologies, and draw our conclusions.

Coastal Trapped Waves: Normal Modes, Evolution Equations, and Topographic Generation

Abstract Coastal trapped waves (CTWs) transport energy along coastlines and drive coastal currents and upwelling. CTW modes are nonorthogonal when frequency is treated as the eigenvalue, preventing the separation of modal energy fluxes and quantification of longshore topographic scattering. Here, CTW modes are shown to be orthogonal with respect to energy flux (but not energy) when the longshore wavenumber is the eigenvalue. The modal evolution equation is a simple harmonic oscillator forced by longshore bathymetric variability, where downstream distance is treated like time. The energy equation includes an expression for modal topographic scattering. The eigenvalue problem is carefully discretized to produce numerically orthogonal modes, allowing CTW amplitudes, energy fluxes, and generation to be precisely quantified in numerical simulations. First, a spatially uniform K1 longshore velocity is applied to a continental slope with a Gaussian bump in the coastline. Mode-1 CTW generation increases quadratically with the amplitude of the bump and is maximum when the bump’s length of coastline matches the natural wavelength of the CTW mode, as predicted by theory. Next, a realistic K1 barotropic tide is applied to the Oregon coast. The forcing generates mode-1 and mode-2 CTWs with energy fluxes of 6 and 2 MW, respectively, which are much smaller than the 80 MW of M2 internal-tide generation in this region. CTWs also produce 1-cm sea surface displacements along the coast, potentially complicating the interpretation of future satellite altimetry. Prospects and challenges for quantifying the global geography of CTWs are discussed.

A non-trivial PT-symmetric continuum Hamiltonian and its eigenstates and eigenvalues

In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, C, for this system. All calculations are performed analytically and without approximation.

Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden-type equations

The nonlinear generalized modified Emden-type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of the nonlocal transformation and the linearizability into HO or DHO determines the nature/structure of the dynamical forces involved (hence, determine the structure of the dynamical equation). Yet, a reverse engineering strategy is used so that the exact solutions of the emerging GMEE are nonlocally transformed to find the exact solutions of the HO and DHO dynamical equations. Consequently, whilst the exact solution for the HO remains a textbook one, the exact solution for the DHO (never reported elsewhere, to the best of our knowledge) turns out to be manifestly the most explicit and general solution that offers consistency and comprehensive coverage for the associated under-damping, critical-damping, and over-damping cases (i.e. no complex settings for the coordinates and/or the velocities are eminent/feasible). Moreover, for all emerging dynamical system, we report illustrative figures for each solution as well as the corresponding phase-space trajectories as they evolve in time.