Damped Harmonic Oscillator Research Articles

Solutions of the Mathieu–Hill Equation for a Trapped-Ion Harmonic Oscillator—A Qualitative Discussion

We investigate solutions of the classical Mathieu–Hill (MH) equation that characterizes the dynamics of trapped ions. The analytical model we introduce demonstrates the equations of motion are equivalent to those of a harmonic oscillator (HO). Two independent approaches are used, based on two classes of complex solutions of the MH equation. This paper addresses both a damped HO and parametric oscillator (PO) for an ion confined in an electrodynamic (Paul) trap, along with stability and instability regions for the associated periodic orbits.

Emergence of squeezed coherent states in Kaluza–Klein cosmology

In this work, we consider a propagating scalar field on Kaluza–Klein-type cosmological background. It is shown that this geometrical description of the Universe resembles – from a Hamiltonian standpoint – a damped harmonic oscillator with mass and frequency, both time-dependents. In this scenario, we construct the squeezed coherent states (SCSs) for the quantized scalar field by employing the invariant operator method of Lewis–Riesenfeld (non-Hermitian) in a non-unitary approach. The non-classicality of SCSs has been discussed by examining the quadrature squeezing properties from the uncertainty principle. Moreover, we compute the probability density, which allows us to investigate whether SCSs can be used to seek traces of extra dimensions. We then analyze the effects of the existence of supplementary space on cosmological particle production in SCSs by considering different cosmological eras.

Carbonyl stretching band in amides as Lorentz oscillator. Insights into anharmonicity and local environment in the liquid phase from NIR and MIR spectra of N-methylformamide and di-N,N-methylformamide

We investigated the anharmonicity and intermolecular interactions of N-methylformamide (NMF) and di-N,N-methylformamide (DMF) in the neat liquid phase with particular interest in the amide bands. The vibrational spectra, complex refractive index, and complex electric permittivity were determined in in the mid- (MIR) and near-infrared (NIR) regions (11,500–560 cm−1; 870–17857 nm). Dispersion analysis was based on the Classical Damped Harmonic Oscillator (CDHO) and simultaneous modelling of the real and imaginary components of the spectra. This data delivered insights into the vibrational energy dissipation and self-association in liquid amides. Identification of the MIR and NIR bands was based on anharmonic GVPT2//B3LYP/6-311++G(d,p) calculations. DMF and NMF follow distinct self-association, evidenced in the MIR fingerprint by the two components of the νCO, the analog of the Amide I band. These conclusions are supported by the structural information derived from the NIR spectra. Furthermore, the contribution of overtones and combination bands in the MIR spectra of amides was examined. The conclusions on molecular interactions and structural dynamics of NMF and DMF contribute to a deeper understanding of the effects of changes in the local environment of the amide group.

Multivariate Mittag-Leffler Solution for a Forced Fractional-Order Harmonic Oscillator

The harmonic oscillator is a fundamental physical–mathematical system that allows for the description of a variety of models in many fields of physics. Utilizing fractional derivatives instead of traditional derivatives enables the modeling of a more diverse array of behaviors. Furthermore, if the effect of the fractional derivative is applied to each of the terms of the differential equation, this will involve greater complexity in the description of the analytical solutions of the fractional differential equation. In this work, by using the Laplace method, the solutions to the multiple-term forced fractional harmonic oscillator are presented, described through multivariate Mittag-Leffler functions. Additionally, the cases of damped and undamped free fractional harmonic oscillators are addressed. Finally, through simulations, the effect of the fractional non-integer derivative is demonstrated, and the consistency of the result is verified when recovering the integer case.

Modelling plasmonic nanoparticles: Unveiling the optical torque potential with a dynamic driven damped harmonic oscillator model and width dependent effects

Linearly polarised light induces torque on anisotropic plasmonic nanoparticles, which allows for the manipulation of their orientation. In this study, we calculate the torque induced on a plasmonic nanocuboid by modelling the surface plasmon resonance (SPR) as a driven damped harmonic oscillator. Using numerical simulations, we demonstrate the model’s accuracy in predicting the torque induced for narrow nanocuboids. We also investigate the effects of varying the nanocuboid’s width on the torque and demonstrate that the torque behaviour exhibits drastic changes, with the formation and modification of additional modes, particularly at the transverse SPR wavelength. As the torque arises from the movement of charges, we utilize charge distribution profiles to explain this shift in torque behaviour.

Generalization of the Schrödinger Equation for Open Systems Based on the Quantum-Statistical Approach

Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.

Improved data analysis for molecular dynamics in liquid CCl4

Previously reported inelastic x-ray scattering spectra of a typical van der Waals molecular liquid CCl4 were reanalyzed using a generalized Langevin formalism with a memory function including a thermal and two viscoelastic relaxation processes and simple sparse modeling. The obtained excitations of longitudinal acoustic phonons show a largely positive deviation from the hydrodynamic value by about 57%, much larger than about 37% by the previously reported damped harmonic oscillator result. Such large values of fast sounds in molecular liquids larger than 15-20% of typical liquid metals are interpreted as extra energy losses for terahertz phonons by molecules' vibrational and rotational motions. The rates of the fast and slow viscoelastic relaxations in the memory function at low Q indicate large values, about 0.5 and 2 ps, which correspond to the vibrational and rotational motions of CCl4 molecules, respectively. These values are larger than those of the typical polar molecular liquid acetone, which may reflect the heavier atomic mass of CCl4. The Q dependences of the viscoelastic relaxation rates are discussed in terms of the lifetime and propagating length of the terahertz phonon oscillations. The microscopic kinematic longitudinal viscosity was obtained from the viscoelastic relaxation magnitudes and times, rapidly decreasing with Q from a macroscopic value at Q→0.

Canonical quantization of dissipative systems

A consistent quantization procedure for a dissipative system is introduced. To this end Hamilton's least action principle compatible with initial value problems is exploited to obtain the Hamiltonian for the dissipative systems. The derivation of the Schrodinger equation for the dissipative system is presented. The canonical quantization method introduced is simple as well as general for linear damping. The canonical quantization procedure is employed to quantize the damped harmonic oscillator.

Contact acoustic nonlinearity and local damage resonance for the detection of kissing bonds in structural adhesive joints

Adhesively bonded joints are susceptible to contamination of surfaces during manufacture and environmental deterioration in real operating conditions. These may cause the generation of so-called “kissing bonds” that can dramatically alter the strength of the joint leading to premature failure. Nonlinear acousto-ultrasonic (AU) techniques have shown great potential for monitoring kissing bonds with piezoelectric sensors permanently installed on the structural joint, thus enabling online and in-situ inspection. This paper investigated the combined contact acoustic nonlinearity (CAN) and local damage resonance (LDR) effects in the presence of kissing bonds in adhesive joints. Experimental nonlinear AU tests showed the formation of LDR frequency down-shifts and “jumps” of the fundamental damage resonance in adhesively bonded aluminium joints, with the kissing bond located internally to the overlapping region between the two adherends. These results were supported by a theoretical model based on the solution of the nonlinear Duffing's equation, under the assumption that the debonded region is a damped nonlinear harmonic oscillator subject to harmonic forcing. The Harmonic Balance Method was used to solve the nonlinear differential problem, showing the generation of frequency down-shifts via the dependence of the ratio between the excitation and LDR frequencies with the amplitude of the fundamental damage resonance. Additionally, two-dimensional finite element simulations using a reduced order model based on the Craig-Bampton technique were carried out to support experimental AU tests for the identification of the LDR frequency and the generation of nonlinear resonance effects. Good agreement between analytical, numerical, and experimental results revealed that a monitoring approach combining CAN and LDR is an extremely efficient and sensitive tool for ensuring integrity and safety of structural adhesive joints.

The only limit cycle that appears in damped harmonic oscillators affected by state-dependent impulses

This study is devoted to planar hybrid dynamical systems with state-dependent impulses to investigate phenomena related to sudden changes in the velocity of an object. In particular, the effect of the state-dependent impulses on the appearance of limit cycles is addressed, which has rarely been studied to date. When a toddler is placed on a swing and pushed gently from behind, the discontinuous changes in the direction and speed of movement of both the toddler and the swing can be described using such a hybrid dynamical system. The main theorem presented in this study reveals that, regardless of the amount of state-dependent impulses, they yield only one stable limit cycle within the region that is identified by the considered system. Moreover, state-dependent impulses lead to destabilization of the equilibrium point. In the proof of the main theorem, a Poincaré mapping is derived by examining the behavior of positive semi-orbits, and the characteristics of the sequence that is obtained from the Poincaré mapping are investigated in detail. Several simulations are performed to demonstrate the main theorem.

A quantitative model for spatio-temporal dynamics of root gravitropism.

Plant organs adapt their morphology according to environmental signals through growth-driven processes called tropisms. While much effort has been directed towards the development of mathematical models describing the tropic dynamics of aerial organs, these cannot provide a good description of roots due to intrinsic physiological differences. Here we present a mathematical model informed by gravitropic experiments on Arabidopsis thaliana roots, assuming a subapical growth profile and apical sensing. The model quantitatively recovers the full spatio-temporal dynamics observed in experiments. An analytical solution of the model enables us to evaluate the gravitropic and proprioceptive sensitivities of roots, while also allowing us to corroborate the requirement for proprioception in describing root dynamics. Lastly, we find that the dynamics are analogous to a damped harmonic oscillator, providing intuition regarding the source of the observed oscillatory behavior and the importance of proprioception for efficient gravitropic control. In all, the model provides not only a quantitative description of root tropic dynamics, but also a mathematical framework for the future investigation of roots in complex media.

A coupled damped harmonic oscillator model for arbitrary arrays of floating cylinders using homotopy methods

A coupled damped harmonic oscillator model is derived for the long-time motion of an array of floating cylinders moving in heave, forced by a prescribed initial displacement. Eigenmodes of the coupled damped harmonic oscillator model are set to be the complex resonances of the corresponding linear potential theory model of the array. Thus, the solution of the coupled damped harmonic oscillator model has the same form as the singularity expansion method. The derivation of the coupled damped harmonic oscillator model is underpinned by a homotopy method to calculate the complex resonances for arbitrary arrays. A recursive homotopy method is used to study the complex resonances as one of the geometrical parameters is varied. The coupled damped harmonic oscillator model is shown to give accurate approximations of the full linear solutions, except for exceptional cases in which additional complex resonances appear.

Collective modes in molten CsCl, CuCl and AgI observed by inelastic x-ray scattering

Inelastic x-ray scattering measurements have been carried out for molten CuCl and AgI and the results were compared with those in molten CsCl by Inui et al. (2007) [42]. The inelastic components arising from collective excitation were observed on both sides of the quasielastic component at the center in the dynamic structure factor. As a result of examining several model functions to fit the observed dynamic structure factor with convolution of the spectrometer resolution function, we adopted the conventional damped harmonic oscillator with a Lorentzian for the quasielastic component. From the apparent sound speed and the linewidth of the collective excitation, the ionic dynamics of molten AgI was found to be different from that in molten CsCl and CuCl, suggesting that AgI has the strongest covalent bonding nature among these molten salts. Meanwhile the relaxation times estimated from the width of the quasielastic components indicated that ionic dynamics in noble metal halides differs from that in molten CsCl.

Near-Field Coupling Effect Analysis of SMD Inductor Using 3D-EM Model

In this paper, we propose a methodology for analyzing the near-field coupling between two surface mount device (SMD) inductors using a 3-dimensional electromagnetic (3D-EM) model. To develop the 3D-EM model, we first constitute the equivalent circuit of the SMD inductor from the measured impedance and derive the loss tangent using circuit parameters. Secondly, the loss tangent using damped harmonic oscillator model is introduced to extract the effective permeability of core magnetic material in the SMD inductor. The optimization algorithm is used to compare the two loss tangents. Then the effective permeability is used in the magnetic material for the 3D-EM modeling of the SMD inductor. The validity of the proposed 3D-EM model is confirmed by comparing the impedance and S-parameters obtained from both measured and EM-simulated values for the two near-field coupled SMD inductors. Finally, the near-field coupling effects between the two adjacent SMD inductors are visualized in terms of coupling path visualization (CPV) using the proposed 3D-EM model, which demonstrates its usefulness for near-field coupling analysis.

Characterization of a forced damped harmonic oscillator in a magneto-optical trap of 133Cs atoms

We report the characterization of the forced damped harmonic oscillations of 133Cs atoms in a magneto-optical trap, which was realized by modulating the intensity of the lasers counterpropagating along the anti-Helmholtz coil axis. Trap parameters such as trap frequency, damping coefficient, and magnitude of the driving force were determined from the resonant vibrational amplitude of the 133Cs atomic cloud depending on the modulation frequency of the modulated laser intensity. The experimental results were compared with the theoretical ones based on the simple two-level and multi-level atom models, considering all possible transition lines used to trap the 133Cs atom, and were found to be consistent with their theoretical counterparts. Furthermore, we theoretically examined the effect of the repumping laser on trap parameters.

On the Arrow–Hurwicz differential system for linearly constrained convex minimization

ABSTRACT In a real Hilbert space setting, we reconsider the classical Arrow–Hurwicz differential system in view of solving linearly constrained convex minimization problems. We investigate the asymptotic properties of the differential system and provide conditions for which its solutions converge towards a saddle point of the Lagrangian associated with the convex minimization problem. Our convergence analysis mainly relies on a ‘Lagrangian identity’ which naturally extends on the well-known descent property of the classical continuous steepest descent method. In addition, we present asymptotic estimates on the decay of the solutions and the primal-dual gap function measured in terms of the Lagrangian. These estimates are further refined to the ones of the classical damped harmonic oscillator provided that second-order information on the objective function of the convex minimization problem is available. Finally, we show that our results directly translate to the case of solving structured convex minimization problems. Numerical experiments further illustrate our theoretical findings.

Fluidelastic modeling of a weathercock stabilization in a uniform flow

The relaxation dynamics of a weathercock free-to-rotate, in the presence of a uniform flow, as it aligns with the flow direction, is investigated experimentally in a wind-tunnel. The dynamics is observed to conveniently follow a damped harmonic oscillator behavior. At first order, the frequency is set by the aerodynamic coefficients. We show that a quasi static approach fails to precisely describe the relaxation dynamics and that non-stationary corrections are required to model the dynamics. A first strategy is to introduce added mass, added stiffness and added damping to the quasi-static approximation, following what is usually done in the context of vortex-induced vibrations. A second strategy is to introduce empirical corrections, whose scaling is obtained from the analysis of the experimental data. Finally, these two strategies are compared and we discuss the physical interpretations of the non-stationary corrections.

Effect of magnetic order on longitudinal Tomonaga-Luttinger liquid spin dynamics in weakly coupled spin- 12 chains

The quantum many-body interactions in one-dimensional spin-$\frac{1}{2}$ systems are subject to Tomonaga-Luttinger liquid (TLL) physics, which predict an array of multiparticle excitations that form continua in momentum-energy space. Here we use inelastic neutron spectroscopy to study the TLL spin dynamics in ${\mathrm{SrCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$, a compound which contains weakly coupled spin-$\frac{1}{2}$ chains of Co atoms, at 0.05 K and in a longitudinal magnetic field up to 9.0 T. The measurements were performed above 3.9 T, where the ground-state N\'eel antiferromagnetic (AFM) order is completely suppressed and the multiparticle excitations are exclusively of the TLL type. In this region and below 7.0 T, the longitudinal TLL mode---psinon/antipsinon (P/AP)---is unexpectedly well described by a damped harmonic oscillator (DHO) while approaching the zone center defining the static spin-spin correlations. A non-DHO-type, continuumlike signal is seen at higher fields, but deviations from the ideal one-dimensional TLL still remain. This change in the P/AP mode coincides with the phase transition between the longitudinal spin density wave (LSDW) and transverse AFM order. Inside the LSDW state, the DHO-type P/AP spectral weight increases and the linewidth broadens as the magnetic order parameter decreases. These results reveal the impact of three-dimensional magnetic order on the TLL spin dynamics; they call for beyond-mean-field treatment for the interchain exchange interactions.

Decoherence and Transition to Classicality for Time-Dependent Stochastic Quantum Systems with a General Environment

The emergence of classicality from a stochastic quantum system through decoherence is investigated. We consider the case where the parameters, such as mass, frequency, and the damping coefficient, vary with time. The invariant operator theory is employed in order to describe quantum evolution of the system. It is supposed that the system is in equilibrium with the environment at a finite temperature. The characteristics of decoherence, the classical correlation and the quantum coherence length are analyzed. The decoherence time is estimated in both position and momentum spaces. We verify from such analyses that the time dependence of the stochastic process affects the quantum-to-classical transition of the system. To promote the understanding of the results, we apply our development to a particular system which is the damped harmonic oscillator. Through this application, we confirm that the decoherence condition is satisfied in the limit of a sufficiently high temperature, whereas the classical correlation is not affected by the temperature.

Inertio-capillary rebound of a droplet impacting a fluid bath

The rebound of droplets impacting a deep fluid bath is studied both experimentally and theoretically. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. Measurements of the droplet trajectory and other rebound metrics are compared directly with the predictions of a linear quasipotential model, as well as fully resolved direct numerical simulations of the unsteady Navier–Stokes equations. Both models resolve the time-dependent bath and droplet shapes in addition to the droplet trajectory. In the quasipotential model, the droplet and bath shape are decomposed using orthogonal function decompositions leading to two sets of coupled damped linear harmonic oscillator equations solved using an implicit numerical method. The underdamped dynamics of the drop are directly coupled to the response of the bath through a single-point kinematic match condition which we demonstrate to be an effective and efficient model in our parameter regime of interest. Starting from the inertio-capillary limit in which both gravitational and viscous effects are negligible, increases in gravity or viscosity lead to a decrease in the coefficient of restitution and an increase in the contact time. The inertio-capillary limit defines an upper bound on the possible coefficient of restitution for droplet–bath impact, depending only on the Weber number. The quasipotential model is able to rationalize historical experimental measurements for the coefficient of restitution, first presented by Jayaratne & Mason (Proc. R. Soc. Lond.A, vol. 280, issue 1383, 1964, pp. 545–565).