External Harmonic Force Research Articles

Comparison of standard Bayesian methods and PINNs for the reconstruction of acoustic fields in 'in situ' combustion chambers

Acoustic measurements, utilizing sensors such as microphones strategically placed, are commonly employed to monitor acoustic systems and extract quantities of interest, such as acoustic reflection coefficients. Typically, a specialized test rig must be constructed to allow for external harmonic acoustic forcing via loudspeakers. In this study, we propose two different methods: physics-informed neural networks (PINNs) and the extended Kalman filter, to reconstruct the acoustic field of a system and assess the acoustic reflection coefficient at a defined boundary. We demonstrate numerically that a single external broadband forcing signal suffices for these methods. This implies that these methods can be applied to 'in situ' configurations, such as industrial combustion chambers, where the broadband nature of combustion noise serves as the acoustic forcing element. Furthermore, we investigate the performance of the two aforementioned methods in reconstructing the acoustic systems and corresponding acoustic boundaries.

Modal and non-modal stability for Hagen–Poiseuille flow with non-ideal fluid

Modal and non-modal stability analyses are applied to Hagen–Poiseuille flow with a non-ideal fluid. The non-ideal fluid is defined as a fluid close to its vapor–liquid critical point. In this region, properties of the fluid deviate significantly from the assumptions of the ideal gas model. In this paper, the specific example of CO2 near the critical point is taken as a non-ideal fluid. We studied fluids at supercritical pressure and different wall temperatures so that the centerline temperatures can be lower, equal, and higher than the pseudo-critical temperature. Flow instability is characterized by the Reynolds number, and the product of the Prandtl and Eckert numbers. In modal stability analysis, we observe that there is no unstable mode in Hagen–Poiseuille flow with a non-ideal fluid. Regarding the growth rate, as the axial wavenumber increases, another mode becomes the least stable. The non-modal theory is employed to investigate the optimal response to harmonic external force and transient energy growth. The influence of axial and azimuthal wave numbers, Prandtl and Eckert numbers, and thermodynamic states are also taken into account. In this study, we identify an generalized inflection point in the transcritical base profile, causing the transcritical state to be the most unstable. In non-modal instability, we observe that the optimal response mainly occurs at time invariant axisymmetric disturbance. This suggests that the axisymmetric disturbance could potentially initiate the transition to turbulence.

Nonlinear vibration and acoustic radiation of an internally resonant buckled beam

Although the nonlinear vibration of buckled beam has been thoroughly studied, its acoustic radiation does not get much attention. This paper deals with the nonlinear vibro-acoustic problem of an internally resonant buckled beam immersed in an infinite fluid. A finite element method based on an arbitrary Lagrangian-Eulerian framework is adopted to analyze the nonlinear interaction between the large-deformed buckled beam and the finite-amplitude acoustic waves in the fluid. The effects of acoustic pressure excitation, external harmonic excitation, and internal resonance on the vibro-acoustic responses of the buckled beam are discussed in the first and second primary resonances. It is found that the amplitude of the acoustic pressure can be comparable to the amplitude of the external harmonic force. This leads to an increase in the critical harmonic excitation amplitude for the occurrence of quasi-periodic responses and an increase in the second mode frequency component of quasi-periodic responses compared to the case neglecting the acoustic pressure excitation on the beam. Saturation phenomena are discovered in the second mode frequency components of beam displacement and acoustic pressure. Due to the 2:1 internal resonance, different time-frequency characteristics, quadrupole acoustic directivity patterns with low radiation efficiency, and dipole acoustic directivity patterns with high radiation efficiency can be observed at different harmonic excitation amplitudes and frequencies. Understanding these phenomena will help design loudspeakers and recognize sound sources.

Spectral proper orthogonal decomposition of harmonically forced turbulent flows

Many turbulent flows exhibit time-periodic statistics. These include turbomachinery flows, flows with external harmonic forcing and the wakes of bluff bodies. Many existing techniques for identifying turbulent coherent structures, however, assume the statistics are statistically stationary. In this paper, we leverage cyclostationary analysis, an extension of the statistically stationary framework to processes with periodically varying statistics, to generalize the spectral proper orthogonal decomposition (SPOD) to the cyclostationary case. The resulting properties of the cyclostationary SPOD (CS-SPOD for short) are explored, a theoretical connection between CS-SPOD and the harmonic resolvent analysis is provided, simplifications for the low and high forcing frequency limits are discussed, and an efficient algorithm to compute CS-SPOD with SPOD-like cost is presented. We illustrate the utility of CS-SPOD using two example problems: a modified complex linearized Ginzburg–Landau model and a high-Reynolds-number turbulent jet.

Theoretical study on dynamic response stability of reinforced concrete structures with a negative skeleton curve gradient under steady-state oscillation

The dynamic stability of reinforced concrete structures in regions of reduced restoring force was theoretically investigated according to the ductility and negative gradient in the skeleton curve of their restoring force - displacement relationship, using the frequency response function derived by Caughey (1960). This method derives the frequency response function, which is a relational expression between input and response, for a single-degree-of-freedom system subjected to a harmonic external force in the form of a cosine wave; it also derives two conditions that make the response solution of a nonlinear one-degree-of-freedom system unstable. The results of the investigation showed that the behavioral range in which the structure was stable in the region of reduced restoring force could be determined using the ductility and negative stiffness indicated by the skeleton curve of the restoring force - displacement relationship. Parametric studies varying the ductility and negative stiffness also suggested the possibility that dynamic stability cannot be ensured in the region of reduced restoring force when the ductility is excessive.

Anishchenko–Astakhov Quasiperiodic Generator Excited by External Harmonic Force

Anishchenko–Astakhov Quasiperiodic Generator Excited by External Harmonic Force

Configuration of a gravel-rubber geotechnical seismic isolation system from laboratory and field tests

We present the outcomes derived from controlled laboratory experiments on utilizing gravel rubber mixtures (GRM) as a geotechnical seismic isolation (GSI) stratum beneath the foundational structure of the EuroProteas prototype. The study encompassed three distinct GRM compositions characterized by varying rubber proportions relative to the total mixture weight (0%, 10%, and 30%) employed as the foundation soil. These GSI-structure systems were subjected to external harmonic forces applied atop the structure of EuroProteas across a broad spectrum of frequencies and force magnitudes. Preceding the commencement of field experiments, resonant column and cyclic triaxial compression assessments were conducted on specimens featuring identical rubber fractions and mean grain size ratios. Our laboratory tests revealed that an increase in rubber content resulted in a reduction of the shear modulus (Go) and an augmentation of the damping ratio (Do) while concurrently inducing a more linear character in the G/Go-logγ-D profiles. The findings derived from the laboratory examinations align closely with those ascertained from field investigations. Expressly, it was confirmed that a GSI layer with a thickness of 0.50 meters, composed of a GRM incorporating 30% rubber content, exhibited a discernible reduction in the overall stiffness of the GSI-structure system, accompanied by a corresponding alteration in its fundamental vibrational frequency. The enhanced damping within the system manifested through observable reductions in acceleration recordings and diminished strain development at the base of the GRM layer with 30% rubber content, encompassing a wide range of frequencies.

Prospects for mathematical modeling in mining system development: accounting for global oscillations and seismic waves

The article discusses the potential of mathematical modeling in understanding the impact of vibrations and seismic waves, aiming at enhancing the sustainability of systems within the mining industry. It explores the dynamic response of a tall, elastic structure with a uniform cross-section and a fixed cylindrical fluid reservoir, subject to various complex boundary conditions. The study delves into the vibrational behavior of the structure when exposed to seismic and harmonic forces, calculating frequency, vibration periods, and deriving formulas for stress, tension, deformation, bending moments, and shear forces in different parts of the structure through both theoretical and experimental approaches. Additionally, the article derives the differential equation for the free oscillation of a tall hydraulic structure in pure bending with an incorporated mass load under appropriate boundary conditions, identifying specific vibration frequencies and periods. The forced vibration scenario is also examined, focusing on the structure's foundation movement due to external harmonic forces. Numerical computation technology is utilized to analyze the change laws of principal quantities that describe both free and forced vibrational movements of the hydraulic structure, showcasing the applicability of these models in predicting and mitigating the effects of seismic activities on mining infrastructure.

Vibration-induced friction modulation for a general frequency of excitation

Applying an oscillatory load is one of the most efficient ways to alter friction forces. Several theoretical and experimental studies on the influence of oscillatory loads on friction have been conducted, investigating the effect of both in-plane and out-of-plane oscillations for different tribological pairings. However, in the literature, the effect of an oscillatory load on the friction force has been studied with an emphasis on dynamic loads characterized by a high-frequency content, while a clear statement as to what is considered “high-frequency” is missing. Moreover, the effect of a combination of load directions on the friction reduction is not accounted for. Therefore, this study aims to determine the vibration-induced effect on friction regardless of the frequency range and direction of harmonic force for a single and multi-degree-of-freedom system. Analytical methods are used to obtain the friction modulation due to harmonic loads, considering a classical mass–spring–dashpot system on a moving belt and the Amontons–Coulomb law. It is found that, in the case of continuous slip, a general relation for the vibration-induced friction modulation is obtained utilizing the velocity response function of the investigated system. The latter is used to highlight a threshold from which the high-frequency regime starts and to determine the stick–slip boundaries. Moreover, through the velocity response function, the influence of different external harmonic forces is investigated and discussed. This includes considerations of phase, excitation frequency, system characteristics, and the choice of the normal contact force expression.

Integral Resonant Controller for Suppressing Car’s Oscillations and Eliminating its Inherent Jump Phenomenon

This work deals with modeling and controlling the oscillations of a horizontally-supported car under the effect of a nonlinear spring, a damper, and a harmonic excitation external force. The proposed controller is the integral resonant controller (IRC) which is a first order oscillator coupled the car via a linear variable differential transformer (LVDT) and a servo-controlled linear actuator(SCLA). The multiple scales perturbation method is used to obtain an approximate solution and stability analyses are carried out. Based on the stability analysis, the optimum values of the controller parameters are recommended in this work for a better control operation. Several response curves are plotted for clarifying the concept of the proposed integral resonant control algorithm.A numerical simulation of the car’s motion is achieved by numerically integrating the model equations with the 4th order Rung-Kutta algorithm. The applied controller can treat the severe jumps in the oscillation amplitude where it is suppressed at different conditions of forward and backward sweeping of the amplitude and frequency of the external excitation force.

Modeling and analysis of a piezoelectric transducer embedded in a nonlinear damped dynamical system

This paper focuses on the dynamical analysis of the motion of a new three-degree-of-freedom (DOF) system consisting of two segments that are attached together. External harmonic forces energize this system. The equations of motion (EOM) are derived utilizing Lagrangian equations, and the approximate solutions up to the third order are investigated using the methodology of multiple scales. A comparison between these solutions and numerical ones is constructed to confirm the validity of the analytic solutions. The modulation equations (ME) are acquired from the investigation of the resonance cases and the solvability conditions. The bifurcation diagrams and spectrums of Lyapunov exponent are presented to reveal the different types of the system’s motion and to represent Poincaré maps. The piezoelectric transducer is connected to the dynamical system to convert the vibrational motion into electricity; it is one of the energy harvesting devices which have various applications in our practical life like environmental and structural monitoring, medical remote sensing, military applications, and aerospace. The influences of excitation amplitude, natural frequency, coupling coefficient, damping coefficient, capacitance, and load resistance on the output voltage and power are performed graphically. The steady-state solutions and stability analysis are discussed through the resonance curves.

Vertical and horizontal dynamic response of suction caisson foundations

Abstract In this article, the dynamic response of suction caisson foundations is studied using a three-dimensional finite element model with an absorbing boundary. The adopted formulation is based on the substructuring method. This formulation has been applied to analyze the effect of soil–structure interaction on the dynamic response of the suction foundation as a function of the kind of load. The suction caisson foundations are embedded in viscoelastic homogenous soils and subjected to external harmonic forces. For each frequency, the dynamic impedance connects the applied forces to the resulting displacement. The constitutive elements of the system are modeled using the finite element volumes and shell elements. The numerical results for the dynamic response of the suction foundations are presented in terms of vertical and horizontal displacements as well as vertical and horizontal dynamic impedances. The results indicated that the overall dynamic response is highly affected by the suction caisson diameter, the soil stiffness variation, and the suction caisson length.

On nonlinear vibrations of an elastic plate on a fractional viscoelastic foundation in a viscoelastic medium in the presence of the one-to-one internal resonance

In the present paper, the dynamic response of a nonlinear Kirchhoff–Love plate resting on a viscoelastic foundation in a viscoelastic medium, damping features of which are described by the Kelvin–Voigt fractional derivative model is studied by the generalized method of multiple time scales. The viscoelastic features of the foundation are modelled via the two fractional derivative models: Kelvin–Voigt or standard linear solid model with fractional derivatives. Resolving equations are obtained for the case of the one-to-one internal resonance for determining nonlinear amplitudes and phases in the case of free oscillations, when the eigen frequencies of the two dominant oscillation modes are close to each other, and for the case of forced oscillations, when the frequency of the external harmonic force is close in value to the eigen frequencies of the interacting eigen modes. The resulting system of equations allows one to control the damping properties of the environment and the base by changing the parameters of fractionality, which expands the range of problems of applicability of this solution.

A chaotic oscillation generator based on mixed dynamics of adaptively coupled Kuramoto oscillators

A chaotic oscillation generator based on the concept of mixed dynamics is implemented on the field programmable gate array (FPGA). This generator reproduces the dynamics of two adaptively coupled Kuramoto phase oscillators. It is experimentally shown that the generator exhibits oscillations corresponding to a chaotic attractor and a chaotic repeller (attractor in reversed time). We have shown that the chaotic attractor and the chaotic repeller intersect on a closed invariant set - a reversible core, which confirms the existence of the so-called mixed dynamics. It was found that for different values of the parameters of the generator, either ordinary dissipative chaos or mixed dynamics can be realized. It is shown that in the case of mixed dynamics the behavior of the trajectories in phase space becomes more complex and the spectral characteristics change towards a more uniform power distribution over the spectrum frequencies. The action of harmonic external force on the mixed dynamics of the generator is also investigated and it is shown that this leads to additional complexity of the dynamical behavior of the generator’s output signals.

Effect of external fields on pattern formation of charged grains in plasma

AbstractTwo‐dimensional cluster and pattern formation by charged grains in plasma under external harmonic force and transverse magnetic field is investigated using molecular dynamics simulation. It is found that, as a result of the Yukawa‐like grain‐grain interaction and the external fields, the asymptotic quasistationary state of an initially homogeneous and Maxwellian velocity distributed grain system can be a disk cluster with the grain trajectories having flowery zonal patterns.

A passive self-tuning vibration neutraliser using nonlinear coupling between the degrees of freedom

Vibration neutralisers are widely used to suppress vibration of a host structure subject to external excitation at a specific frequency. When attached to a structure, they create a notch filter in the frequency response, reducing the vibration levels considerably. An inherent limitation is related to the narrow frequency range in which they are effective. Over the years, there have been different attempts to make the vibration absorber more robust. These include using active and semi-active control strategies to change the tuning frequency, using piezoelectric elements with shunt circuits, electromagnetic actuators, servo motors, and even exploring nonlinear effects. In many cases, there is a need for an external power source to modify the characteristics of the system. This paper concerns a passive vibration neutraliser that can adapt automatically to one of two frequencies corresponding to the frequency of an external harmonic force. Importantly, it does this without the need for an external power source. The device consists of a beam-like neutraliser that is attached to the host structure at its centre through a roller bearing. The stiffness element is a rectangular beam that can rotate in the bearing, changing the stiffness it presents to the direction of excitation. The paper describes an experimental study into such a device, and an analytical model is proposed that qualitatively captures the time-domain behaviour of the experimental device. A possible mechanism by which the device self-tunes to either of the two frequencies of excitation is discussed. Supplementary material is also provided to show the device in operation.

Most probable escape paths in periodically driven nonlinear oscillators.

The dynamics of mechanical systems, such as turbomachinery with multiple blades, are often modeled by arrays of periodically driven coupled nonlinear oscillators. It is known that such systems may have multiple stable vibrational modes, and transitions between them may occur under the influence of random factors. A methodology for finding most probable escape paths and estimating the transition rates in the small noise limit is developed and applied to a collection of arrays of coupled monostable oscillators with cubic nonlinearity, small damping, and harmonic external forcing. The methodology is built upon the action plot method [Beri et al., Phys. Rev. E 72, 036131 (2005)] and relies on the large deviation theory, the optimal control theory, and the Floquet theory. The action plot method is promoted to non-autonomous high-dimensional systems, and a method for solving the arising optimization problem with a discontinuous objective function restricted to a certain manifold is proposed. The most probable escape paths between stable vibrational modes in arrays of up to five oscillators and the corresponding quasipotential barriers are computed and visualized. The dependence of the quasipotential barrier on the parameters of the system is discussed.

Self-consistent description of Bose-Bose droplets: Harmonically trapped quasi-two-dimensional droplets

We describe a quantum droplet of a Bose-Bose mixture squeezed by an external harmonic forces in one spatial direction. Our approach is based on the self-consistent method formulated in [1]. The true spatial droplet profile in the direction of confinement is accounted for, however local density approximation is assumed in the free directions. We define a numerical approach to find the beyond-mean-field contribution to the chemical potential (Lee-Huang-Yang chemical potential) -- the quantity that determines the droplet's profile. In addition to the numerical approach, we find the Lee-Huang-Yang potential in the analytic form in two limiting cases: a perturbative result for a strong confinement and a semiclassical expression when confinement is very weak.

Collocation approaches to the mathematical model of an Euler–Bernoulli beam vibrations

Taylor-Matrix and Hermite-Matrix Collocation methods are presented to obtain the modal vibration behavior of an Euler–Bernoulli beam. The methods provide approximate solutions in the truncated Taylor series and the truncated Hermite series by using Chebyshev–Lobatto collocation points and operational matrices. The approaches are applied to the well-known transverse vibration model of a simply-supported Euler–Bernoulli type beam. The beam is assumed under the effect of external harmonic force with spatially varying amplitude. Firstly, the model problem is transformed into a system of linear algebraic equations. Then the approximate solution is computed by solving the obtained algebraic system. The proposed methods are compared with the exact solutions. Mode shapes of the first three fundamental frequencies are obtained for each approach. The convergence behaviors of transverse displacements and absolute errors are calculated for each mode. The numerical results are shown and compared. Based on the given figures and tables, one can state that the methods are remarkable, dependable, and accurate for approaching the mentioned model problems.

Periodically driven spheroid in a viscous fluid at low Reynolds numbers

In this paper, we study the motion of a spheroid of a moderate aspect ratio in a viscous fluid under the action of an external harmonic force. We first derive the dynamics equation of the particle oscillating along one of its axes and subject to damping, Basset memory, and second history integral forces at small Reynolds numbers, and then, we proceed to obtain an analytical solution of this equation at resonance. With graphical representation, we observe that for a prolate spheroid, the conventional Q-curves show a greater variation with respect to the particle aspect ratio, particle–fluid density ratio, and natural frequency; the variation is significantly larger for the curve corresponding to the second history force. Furthermore, we find that all three forces affect the amplitude of motion: the amplitude increases with the strength of damping as well as the second history integral forces, whereas the presence of Basset memory decreases it. Remarkably, Basset memory causes a phase-shift in the oscillations, while the other two forces have no effect on the phase. Since our solutions are analytical, they may have valuable application in experiments involving more complex systems, in particular, to understand the effect of external force on the transport of micro-particles.