Forced Harmonic Oscillator Research Articles

Assessing the structure of classical molecular optimal control landscapes

We consider laser-driven optimal control landscape of a molecule from a classical mechanical perspective. The goal of optimal control in the present work is to steer the molecule from an initial state to a target state, denoted by two distinct points in phase space. Thus, a particular control objective is given as the difference between the final achieved phase space point and the target. The corresponding control landscape is defined as the latter control objective as a functional of the control field. While previous examination of the landscape critical points (i.e., a suboptimal point on the landscape where there is a zero gradient) has shown that the landscape topology is generally trap-free, the structure of the landscape away from these critical points is not well understood. We explore the landscape structure by examining an underlying metric defined as the ratio R of the gradient-based optimization path length of the control field evolution to the Euclidean distance between a given initial control field and the resultant optimal control field, where the latter field corresponds to a point at the top of the landscape. We analyze the path length-to-distance ratio R analytically for a linear forced harmonic oscillator and numerically for a nonlinear forced Morse oscillator. For the linear forced harmonic oscillator, we find that R⩽2 and reaches its minimum value of 1 (i.e., corresponding to “a straight shot” through control space) in the large target time limit, as well as at special finite target times. The ratio R is similarly small for Morse oscillator simulations when following a steepest-ascent path to the top of the landscape, implying that the landscape is quite smooth and devoid of gnarled features. This conclusion is exemplified for a path discovered with R≃1.0 where simply following the initial gradient direction takes the climb very close to the top of the landscape. These findings are consistent with a variety of previous like simulations examining R in quantum control scenarios.

Temperature perturbation related to the invisible ink vibrationally excited nitric oxide monitoring (VENOM) technique: a simulation study.

The limits of applicability of the invisible ink variant of the vibrationally excited nitric oxide monitoring (VENOM) technique for three distinct flow fields is reported in this work. This technique involves the generation of a grid of vibrationally excited NO (X,Π2) by exciting the NO A-X electronic transition at 226 nm, which subsequently relaxes via fluorescence and collisional quenching to produce vibrationally excited NO (X,Π2). This grid is then probed by two laser sheets tuned to distinct rotational states. The resulting images allow for the simultaneous measurement of temperature and velocity. The flow fields presented in this work provide a range of NO concentrations, vibrational lifetimes, pressures, temperatures, and collisional quenching, which explore the applicability of the invisible ink variant to a wide range of conditions. We have modelled the initial NO, O2, and N2 vibrational and rotational energy distribution resulting from the combination of fluorescence and quenching of electronically excited NO. The subsequent rethermalization of the sample, in particular the long-time vibrational relaxation, has been modelled using a forced harmonic oscillator model. The time-dependent temperature perturbation due to the invisible ink technique is evaluated for two distinct timescales: a short-timescale temperature rise resulting from collisional quenching and rotational/translational thermalization and a long-timescale temperature rise caused by vibrational thermalization. Under low pressures where fluorescence dominates quenching, there is minimal temperature perturbation of the flow field on the timescale of a VENOM measurement, and the short-timescale temperature perturbation only becomes significant at high NO seed concentrations. The predicted signal-to-noise ratio of the invisible ink method is unaffected for low-pressure, low-temperature flow fields. However, preserving signal-to-noise ratio for a high-temperature, high-pressure flow field could prove challenging due to the impact of quenching and self-absorption. Overall, we find that the invisible ink method is predicted to be a viable laser-based diagnostic for velocimetry and thermometry over a wide range of experimental conditions.

Nonlinear effect of forced harmonic oscillator subject to sliding friction and simulation by a simple nonlinear circuit

We study the nonlinear behaviors of mass-spring systems damped by dry friction using simulation by a nonlinear LC circuit damped by anti-parallel diodes. We show that the differential equation for the electric oscillator is equivalent to that of the mechanical system when a piecewise linear model is used to simplify the diodes' I–V curve. We derive series solutions to the differential equation under weak nonlinear approximation which can describe the resonant response as well as amplitudes of superharmonic components. The experimental results are consistent with the series solutions. We also present the phenomenon of hysteresis. A theoretical analysis along with numerical simulations is conducted to explore the stick-slip boundary. The correspondence between the mechanical and electric oscillators makes it easy to demonstrate the behaviors of this nonlinear oscillator on a digital oscilloscope. It can be used to extend the linear RLC experiment at the undergraduate level.

Driving a dissipative quantum oscillator

The open driven quantum harmonic oscillator is studied for three different possible mechanisms of driving; a generalization of the forced oscillator, driving due to a time-dependent coupling to a two-level system, and driving due to an adiabatic modulation of the frequency. The first case encompasses all possible configurations going from the standard forced harmonic oscillator to the classical pumping type of driving typically used in quantum optics. In the second case, an additional two-level system, coupled to the oscillator at a time-varying rate, is assumed and the reduced dynamics of the oscillator and the qubit is examined. In the driving due to adiabatic modulation, the dynamics of a harmonic oscillator whose frequency slowly varies in time is studied. The study done in this paper is focused on a harmonic type of driving, although the solutions are derived for an arbitrary type of driving. In all the cases, the oscillator is assumed to be in contact with a zero- or finite-temperature environment, which provides dissipation to the system. The treatment done in this paper is fully analytic and the derivation of the solutions is based on the employment of the double Fourier transform of the Wigner function.

Decoupled Direct State Transition Matrix Calculation with Runge-Kutta Methods

A myriad of methods, including many optimization and estimation algorithms, require the sensitivities of a dynamic system governed by a set of ordinary differential equations (ODEs). In this paper, the decoupled direct method (DDM) is derived for the calculation of first- and second-order state transition matrices (STMs) using fixed-step-size Runge-Kutta (RK) ODE solvers. Implications for both explicit and implicit RK methods are discussed. Emphasis is placed on the implicit class because of recent interest in the astrodynamics community and because the decoupling of the state propagation and STM calculation eliminates the need to evaluate the Jacobian and Hessian iteratively at each propagation step. Thus, significant computational savings are achievable compared to the traditional propagation of the variational equations, particularly for systems with expensive dynamics. In the current work, two implementations of the DDM are extended both to second-order STMs and to the double-integrator form of the RK equations. One of the strategies presented in this paper, newly derived to include second-order sensitivities, takes advantage of the structure of the system, allowing for the STMs of nonlinear systems to be calculated using efficient linear-algebra techniques rather than iterative methods. This linear-algebra-based approach is shown to be more efficient than the second alternative, and is also more efficient than the propagation of the variational equations for computationally intensive dynamical systems. The effects of important implicit RK customizations, such as variable-fidelity dynamics models and parallelization, are also examined in two examples: a simple, proof-of-concept forced harmonic oscillator, and an Earth-orbiting space object subject to either a low- or high-computational-cost geopotential model.

Hydrodynamic Coefficients for a 3-D Uniform Flexible Barge Using Weakly Compressible Smoothed Particle Hydrodynamics

The numerical modelling of the interactions between water waves and floating structures is significant for different areas of the marine sector, especially seakeeping and prediction of wave-induced loads. Seakeeping analysis involving severe flow fluctuations is still quite challenging even for the conventional RANS method. Particle method has been viewed as alternative for such analysis especially those involving deformable boundary, wave breaking and fluid fragmentation around hull shapes. In this paper, the weakly compressible smoothed particle hydrodynamics (WCSPH), a fully Lagrangian particle method, is applied to simulate the symmetric radiation problem for a stationary barge treated as a flexible body. This is carried out by imposing prescribed forced simple harmonic oscillations in heave, pitch and the two- and three-node distortion modes. The resultant, radiation force predictions, namely added mass and fluid damping coefficients, are compared with results from 3-D potential flow boundary element method and 3-D RANS CFD predictions, in order to verify the adopted modelling techniques for WCSPH. WCSPH were found to be in agreement with most results and could predict the fluid actions equally well in most cases.

Quantum corrections of work statistics in closed quantum systems

We investigate quantum corrections to the classical work characteristic function (CF) as a semiclassical approximation to the full quantum work CF. In addition to explicitly establishing the quantum-classical correspondence of the Feynman-Kac formula, we find that these quantum corrections must be in even powers of ℏ. Exact formulas of the lowest corrections (ℏ^{2}) are proposed, and their physical origins are clarified. We calculate the work CFs for a forced harmonic oscillator and a forced quartic oscillator respectively to illustrate our results.

Longitudinal and Transverse Oscillations of an Elastically Fixed Wall of a Wedge-Shaped Channel Installed on a Vibrating Foundation

The problem of mathematical modeling of longitudinal and transverse oscillations of an elastically fixed wall of a narrow wedge-shaped channel filled with a viscous incompressible liquid and installed on vibrating foundation was stated and solved analytically. The problem was considered in a flat definition for the mode of steady-state forced harmonic oscillations. The mathematical model developed includes the Navier-Stokes equations and the equation of continuity for a thin layer of viscous incompressible liquid and the equations of dynamics of an elastically fixed channel wall. The conditions of adhesion of the liquid to the channel walls and the conditions of a free outflow of liquid at the ends were chosen as the boundary conditions. For the stated problem, the system of dimensionless variables was proposed. Analytical expressions for the wall displacements and the hydrodynamic parameter of the layer of liquid were obtained. The calculations, which made it possible to estimate the influence of inclination of the channel wall and the viscosity of the liquid to damping of oscillations of the elastically fixed wall, were performed.

Energy-preserving variational integrators for forced Lagrangian systems

The goal of this paper is to develop energy-preserving variational integrators for time-dependent mechanical systems with forcing. We first present the Lagrange-d’Alembert principle in the extended Lagrangian mechanics framework and derive the extended forced Euler-Lagrange equations in continuous-time. We then obtain the extended forced discrete Euler-Lagrange equations using the extended discrete mechanics framework and derive adaptive time step variational integrators for time-dependent Lagrangian systems with forcing. We consider three numerical examples to study the numerical performance of energy-preserving variational integrators. First, we consider the example of a nonlinear conservative system to illustrate the advantages of using adaptive time-stepping in variational integrators. In addition, we demonstrate how the implicit equations become more ill-conditioned as the adaptive time step decreases through a condition number analysis. As a second example, we numerically simulate the time-dependent example of a forced harmonic oscillator to demonstrate the superior energy performance of energy-preserving integrators for mechanical systems with explicit time-dependent forcing. Finally, we consider a damped harmonic oscillator using the adaptive time step variational integrator framework. The adaptive time step increases monotonically for the dissipative system leading to unexpected energy behavior.

Frequency–amplitude behavior in the incipient movement of grains under vibration

The onset of the movement of particles placed on a horizontal rough surface subject to a vertical sinusoidal vibration is investigated through tracking experiments, theoretical analysis, and numerical simulations. The frequency of vibration needed to move particles decays exponentially with the amplitude of the oscillatory input. This behavior is explained through a simple mechanism in which a forced damped harmonic oscillator with a spring constant represents all the interactions between the particle and the surface. The numerical results compare well with experimental data, demonstrating that the forces included in the numerical calculations suitably account for the main particle response, even though the complexity of the surface is not fully taken into account. Describing the way in which frequency varies with amplitude could be relevant to technological applications such as cleaning of material surfaces.

Unsteady aerodynamics of a diamond wing configuration

The damping derivatives associated with the pitching, yawing, and rolling motion of the SAGITTA flying wing configuration at low Mach number conditions are presented. A tailless variant of the configuration and a variant with attached double vertical tail are investigated. The damping derivatives are determined by means of the aerodynamic response to forced harmonic oscillations. The required data for the determination of the damping derivatives are obtained from time-accurate Reynolds-averaged Navier–Stokes computations. The calculation methodology for the pitch-, yaw-, and roll-damping derivatives for arbitrary freestream conditions is described and a short evaluation of the approach is presented. Angle of attack and sideslip angle trends as well as the effect of the double vertical tail on the dynamic stability are investigated. The damping derivative of every considered type of motion exhibits significant non-linearities with respect to the freestream condition for angles of attack larger than $$8^\circ$$ . The pitch-damping and roll-damping derivative indicate a dynamically stable behavior at all considered freestream conditions for the configuration with and without vertical tail. The yaw-damping characteristic is more critical. Both configurations exhibit unstable behavior at several freestream conditions. The vertical tail, however, considerably improves the yaw-damping characteristic of the SAGITTA configuration.

Application of the 3D Forced Harmonic Oscillator Model in the DSMC Method

An implementation of the three-dimensional forced harmonic oscillator model of vibration–translation energy transfer in atom–diatom and diatom–diatom collisions is proposed. The implementation employs precalculated lookup tables for transition probabilities and is suitable for the direct simulation Monte Carlo method. It takes into account the microscopic reversibility between the excitation and deexcitation processes, and it satisfies the detailed balance requirement at equilibrium. The implementation is verified for oxygen and nitrogen thermal heat baths and validated for aftershock relaxation of oxygen in argon. Comparison with the one-dimensional forced harmonic oscillator model shows large differences in macroparameters for high-temperature bath relaxation and moderate-to-large differences in realistic oxygen shock conditions.

A new approach using lattice Boltzmann method to simulate fluid structure interaction

A numerical approach based on the Lattice Boltzmann method (LBM) and volume penalization technique is proposed to tackle the problem of the interaction of a moving rigid cylinder with an incompressible fluid flow. The present methodology was applied to an oscillating rigid cylinder in a fluid flow. Two cases were considered: flow around a cylinder undergoing forced harmonic oscillations, and flow around a cylinder in free oscillations due to the fluid forces at Reynolds number 20. Our results were validated and compared with those obtained from Code_Saturne. To compare the results of the LBM and Code_Saturne, and to ensure that the resolution of the structure would be the same in both simulations, we used the Shiels et al. [12] structure adimensionalisation. The simulations of LBM were implemented for two level refined grid around the cylinder, where the grid refinement technique proposed by Lagrava et al. [11] based on multi-domain approach was chosen. The flow characteristics, the oscillation frequency, the lift and drag coefficients values associated were investigated. A good agreement was found between the compared results.

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form $$W(x,x')=0$$ (which is called here the equivalent system of the 3rd type). It is important that the function $$W(x,x')$$ can be built a priori. Once $$W(x,x')$$ is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition $$W(x,x')=0$$ ; (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.

Harmonic generation by nonlinear self-interaction of a single internal wave mode

Weakly nonlinear theory is used to explore the dynamics of a single-mode internal tide in variable stratification with rotation. Nonlinear self-interaction in variable stratification generates a perturbation which is forced with double the original frequency and wavenumber. The dynamics of the perturbation is analogous to a forced harmonic oscillator, with the steady-state solution corresponding to a bound harmonic matching the forcing frequency and wavenumber. When the forcing frequency is near a natural frequency of the system, even a small-amplitude (nearly linear) internal tide may induce a significant harmonic response. Idealized stratification profiles are utilized to explore the relevance of this effect for oceanic$M_{2}$baroclinic internal tides, and the results indicate that a rapidly growing harmonic may occur in some environments near the Equator, but is unlikely at higher latitudes. The results are relevant to recent observations of$M_{4}$(harmonic) internal tides in the South China Sea and elsewhere. More generally, nonlinear self-interaction may contribute to the transfer of energy to smaller scales and the dissipation of baroclinic internal tides, especially in equatorial waters.

State-specific simulation of oxygen vibrational excitation and dissociation behind a normal shock

A complete vibration–vibration–translation (V–V–T) bound–bound transitions rates database for O2–O2 collisions is generated using the forced harmonic oscillator (FHO) model with temperature range between 1000 and 20,000K. The postshock process of oxygen vibrational excitation and dissociation is simulated by solving the system of master equations coupled with the one-dimensional compressible flow equations. The results agree well with existing shock tube data. The state-specific calculations show that inclusion of pure vibration–vibration (V–V) processes could reduce the discrepancy with experimental data and the quasi-steady-state (QSS) approximation is valid when dissociation starts to dominate.

Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies

This paper is concerned with the existence of quasi-periodic response solutions for a class of reversible forced harmonic oscillators with two basic frequencies ω=(1,α), α is an irrational number. Since we do not impose usual Diophantine or Brjuno conditions on ω, it can also be Liouvillean. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for reversible systems.

State-to-state models of vibrational relaxation in Direct Simulation Monte Carlo (DSMC)

In the present work, the application of state-to-state models of vibrational energy exchanges to the Direct Simulation Monte Carlo (DSMC) is considered. A state-to-state model for VT transitions of vibrational energy in nitrogen and oxygen, based on the application of the inverse Laplace transform to results of quasiclassical trajectory calculations (QCT) of vibrational energy transitions, along with the Forced Harmonic Oscillator (FHO) state-to-state model is implemented in DSMC code and applied to flows around blunt bodies. Comparisons are made with the widely used Larsen–Borgnakke model and the in uence of multi-quantum VT transitions is assessed.

Análisis de potencia de la membrana basilar en la cóclea por resonancia mecánica

This paper presents the power analysis to the mechanical model of the basilar membrane in the cochlea as a system of forced damped harmonic oscillators without lateral coupling proposed by Lesser and Berkeley. The Lagrange’s equation for dissipative mechanical systems and the energy method are used to obtain the general equation of the system. Next a solution by complex exponential is proposed using the resonance analysis considering only excitations of pure tones to obtain the equation of displacement, and with its derived the equation of velocity. The power in the system is the multiplication between the equations of the velocity and the excitation force. Finally the equation of the average power in the system is obtained. This new solution has the advantage of determining the relationship between the excitation frequency of the system and the position along the basilar membrane where the average power is maximum. This implies that the distance where there is a maximum transfer of energy between the wave propagating in the perilymph and the mechanical displacement of the basilar membrane on the hair cells in the organ of Corti is known. The power analysis is successfully compared with the two-dimensional model of the cochlea developed by Neely using finite differences and with the experimental results of Békésy. In both experiments are used the same mechanical parameters of the basilar membrane and the same set of frequencies of evaluation proposed in the original papers in order to compare the different methodologies.

ПРО НЕПЕРІОДИЧНІ КОЛИВАННЯ ПРИ ДИНАМІЧНИХ НАВАНТАЖЕННЯХ ВИКОНАВЧОГО МЕХАНІЗМУ МОБІЛЬНИХ ВЕРСТАТІВ З ПАРАЛЕЛЬНОЮ КІНЕМАТИКОЮ

Urgency of the research. A special place in the development of mechano-assembly production begins to occupy equipment based on mechanisms with parallel kinematics. Their advantages are manifested in their low metal capacity, energy intensity, dynamism, and also in kinematic possibilities. However, one of the important tasks in constructing such machines is a dynamic analysis of the executive branch; therefore research in this direction is relevant.Target setting. In forecasting the operational characteristics of mobile robot machines on the basis of mechanisms with parallel kinematics, the dynamic analysis of the executive part of the machine tool system is an important task, the solution of which is expedient to perform on the basis of attracting funds simulation behavior of the spatial structures of the spatial structure.Actual scientific researches and issues analysis. On the basis of the analysis of recent studies, we can conclude that the design of machines and mechanisms crucial role in choosing the parameters of the future product has a reasonable prediction of its performance indicators, based on strength and strength calculations. At the same time, the indicators of the expected accuracy of the executive motions of the product are determined based on the chosen calculation schemes, which are based on a number of assumptions that allow them to be reduced to solvable mathematical constructions – the equation of statics and dynamics.Uninvestigated parts of general matters defining. In some cases, assumptions in calculations in the design of machines, allow you to find common solutions that are based on the established modes of operation of the product, but drop some cases that are of practical interest and are not trivial. This applies, first of all, to the dynamic processes that can accompany the projected product.The research objective. Conduct an analysis of the executive part of the machine system based on the attraction of the simulation behavior of spatial structures.The statement of basic materials. In the course of the study, a calculation scheme was proposed that allowed an analysis of the dynamics of the design features, and is represented by a spatial mass system m suspended in space on flexible elements with rigidity parameters cx, cy, cz and damping hx, hy, hz.. For this system, within the limits of small amplitudes, it has been stated that the frequency of the proper oscillations f0 can be calculated from the dependence of the frequency on mass and stiffness. The scheme for dynamic analysis of the executive branch of the machine with parallel kinematics consisting of a body of mass m, suspended on two springs with rigidity c1 and c2, is presented. For this scheme a mathematical description of the harmonic oscillator, which is the executive link of the mobile machine, is taken, if we take into account the perturbing effect. To study the peculiarities of the dynamics of the loaded executive part of the mobile machine, an imitation model was created in the Simulink computer environment. At given values, the reaction of the model to forced harmonic oscillations is obtained. The magnitude of the amplitudes in such a model does not exceed 2 mm, however, deserves attention to the quasi-random nature of oscillations, although the equilibrium effect has a sinusoidal form.Conclusions. The obtained result suggests that oscillations with a wide frequency and amplitude spectrum arise in the executive link of the mobile machine, which can cause resonance of the elements of the technological system over a wide range of frequencies. Such a mode of operation of mobile machines with mechanisms of parallel kinematics should be taken into account in the design of technological processes and in the technological preparation of production in order to ensure the productivity and accuracy of equipment throughout its lifetime.