Probability Density Function Of Displacement Research Articles

Closed-form solution of the peak factor of hardening non-Gaussian cross-wind response with limited time history samples

The cross-wind response generally follows an un-skewed hardening non-Gaussian distribution around vortex-resonance wind speed, a great amount of samples are required to ensure the accuracy of the peak factor. This study aims to propose a robust methodology for estimating the peak factor, even when dealing with limited samples. The accurate description of the probability distribution for cross-wind response is essential to achieve a reliable peak factor. Since both the harmonic self-excited and Gaussian buffeting components contribute to the non-Gaussianity of the cross-wind response, this study explicates the probability density function (PDF) of the composite process combining the harmonic and Gaussian elements. Subsequently, the PDF of displacement is ascertained by the energy ratio between self-excited vibration and random buffeting, also a kurtosis-based function that can be derived from limited samples. Then the cumulative distribution function (CDF) of the extreme is derived based on the displacement PDF, and a closed-form solution determined by kurtosis for peak factor is derived. Finally, the validity of the proposed method is verified through both Monte Carlo simulations and field measurements. This method offers a practical mean to estimate the extreme values of hardening non-Gaussian responses in highly flexible structures using a limited number of samples.

Transient Response Analysis of Nonlinear Oscillators With Fractional Derivative Elements Under Gaussian White Noise Using Complex Fractional Moments

Abstract Complex fractional moment (CFM), which is defined as the Mellin transform of a probability density function (PDF), has been successfully employed to find the response PDF of a wide variety of integer-order nonlinear oscillators. In this paper, a CFM-based analysis is performed to determine the transient response PDF of nonlinear oscillators with fractional derivative elements under Gaussian white noise. First, an equivalent linear system is introduced for the purpose of deriving the Fokker–Planck (FP) equation for response amplitude. The equivalent natural frequency and equivalent damping coefficient of the system need to be determined, taking into account both the nonlinear and fractional derivative elements of the original oscillator. Moreover, to convert the FP equation into the governing equation of CFMs, these equivalent coefficients must be given in polynomial form of amplitude. This paper proposes formulas for appropriately determining the equivalent coefficients, based on an equivalent linearization technique. Then, applying stochastic averaging, the FP equation is derived from the equivalent linear system. Next, the Mellin transform converts the FP equation into coupled linear ordinary differential equations for amplitude CFMs, which are solved with a constraint corresponding to the normalization condition for a PDF. Finally, the inverse Mellin transform of the CFMs yields the amplitude PDF. The joint PDF of displacement and velocity is also obtained from the amplitude PDF. Three linear and nonlinear fractional oscillators are considered in numerical examples. For all cases, the analytical results are in good agreement with the pertinent Monte Carlo simulation results.

Design and analysis of a galloping energy harvester with V-shape spring structure under Gaussian white noise

This paper presents the design of an innovative galloping energy harvester (GEH) that incorporates an elastic structure to reconstructed the traditional GEH. The mathematical model of the GEH is derived based on the Euler–Bernoulli beam theory and Kirchhoff’s law. Due to the nonlinear properties of the model, the partial linearization technique and Fokker–Planck–Kolmogorov equation are used to further explore the joint probability density function (PDF) and the stationary PDF of displacement and velocity. Subsequently, Monte Carlo simulation is used to illustrate the behavior of the theoretical results. The simulation results show that, under different conditions, the stationary PDF exhibits double-peak or single-peak shape, which is caused by the compression or elongation state. Meanwhile, higher wind speeds correspond to larger displacement amplitudes. Then, the stationary PDF of displacement amplitude is obtained using the stochastic average method, and the expressions for mean square voltage and mean output power are derived. Numerical results indicate that when the wind speed is below the critical wind speed for galloping phenomenon, the vibration of the GEH is mainly driven by noise. When the wind speed exceeds the critical wind speed, the GEH is mainly affected by the wind speed. Higher wind speeds correspond to higher mean output power. An optimal electromechanical coupling coefficient and the ratio of the mechanical and electrical can be derived based on the GEH parameters when galloping occurs.

Load–displacement relation and gap distribution between rough surfaces: Partial differential equations approach

We develop a theoretical model to predict the load–displacement relation and probability density function for gaps between contacting rough surfaces. We derive partial differential equations from the previous model (Joe et al., 2018), and extend them to the non-adhesive contact problem. The predictions of the present theory are compared with numerical results using the Green’s function molecular dynamics algorithm, and a good agreement is obtained.

Random vibration and reliability analysis of fluid-conveying pipe under white noise excitations

Due to the random excitation in the external environment, the fluid-conveying pipe would inevitably undergo random vibration. This paper investigates the random vibration and reliability of simply-supported fluid-conveying pipes under white noise excitations. Dynamic equations of the nonlinear pipe conveying fluid under white noise excitations are modeled by the generalized Hamilton principle. Then probability density functions (PDFs) of the amplitude and energy of the system are determined by using the modified stochastic averaging method. The PDFs of displacement and velocity of the system are also discussed. The accuracy of results is verified by the Monte Carlo method. In addition, the first-passage problem of the pipe system is investigated for the first time. In order to obtain the PDFs of reliability and first failure time of the system, the forward finite difference method is adopted to solve the backward Kolmogorov equation of the system. The effects of different parameters on the amplitude, energy, reliability, and first failure time are investigated. The results show that the amplitude of the system becomes large and the first-passage time is advanced with the increase of excitation intensity or fluid speed. The amplitude of the system is decreased and the first-passage time is delayed with the increase of the damping coefficient.

Particle Image micro-Rheology (PIR) using displacement probability density function

We present a novel approach to perform passive microrheology. A method to measure the rheological properties of fluids from the Brownian motion of suspended particles. Rheological properties are found from the particles' mean square displacements (MSDs) as a function of measurement time lag. Current state-of-the-art approaches find the MSD by tracking multiple particles' trajectories. However, particle tracking approaches face many limitations, including low accuracy and high computational cost, and they are only applicable to low particle seeding densities. Here, we present a novel method, termed particle image rheometry (PIR), for estimating the particle ensemble MSD from the temporal evolution of the probability density function of the displacement as a function of measurement time lag. First, the probability density function (PDF) of the particle displacements for each time lag is found using a generalized ensemble image cross-correlation approach that eliminates the need for particle tracking. Then, PDFs are used to calculate the MSD from which the complex viscosity of the solution is measured. We evaluate the performance of PIR using synthetic datasets and show that it can achieve an error of less than 1% in passive microrheology measurements, which corresponds to a twofold lower error than existing methods. Finally, we compare the measured complex viscosity from PIR with bulk rheometry for a polymeric solution and show agreement between the two measurements.

Anomalous Dynamical Scaling Determines Universal Critical Singularities.

Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at a large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.

Investigation of Synechocystis sp. CPCC 534 Motility during Different Stages of the Growth Period in Active Fluids

The motility behavior of suspended microorganisms plays an essential role in the properties of active fluids. Despite the important progress in our understanding of microorganisms’ motility in recent years, there are still several open questions about the dynamics of cell motility in active suspensions. Of special interest is the relationship between cell motility and age. In this study, cyanobacterium Synechocystis sp. CPCC 534 was used as the model microorganism, and the cell trajectories were tracked for 78 days during the cell growth period. Results showed that the length of cell trajectories had substantially increased from the exponential growth phase to the stationary phase and had declined at the end of the stationary phase. Similar trends were observed for the cells’ mean squared displacement (MSD), the time-dependent diffusion coefficient of cell suspensions, and the cell displacement probability density function (PDF). These results suggest that the cellular age of microorganisms has a significant effect on various metrics of cell motility and, therefore, can impact the transport properties of active suspensions.

Control Mechanisms for Mobile Devices

In this paper we consider control mechanisms for mobile devices in a stochastic environment. In particular, we consider a device in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional space subject to Brownian perturbations where a control mechanism moves the device towards its target location at a speed which is a function of its displacement. For this scenario, we construct stochastic differential equations for the mobility process and solve for the steady state probability density function of displacement. From this we are able to give general solutions to key metrics such as displacement outage (the long term probability of exceeding a given distance from the target), connectivity probability (derived from the SNR distribution in a Rayleigh channel with pathloss), the mean time at which the device first exceeds a given distance from the target, and the mean kinetic energy required by the control mechanism. We evaluate these metrics for important special cases of the control mechanism and also study the optimization problem of minimizing kinetic energy over the parameters of the control function.

A generalised diffusion equation corresponding to continuous time random walks with coupling between the waiting time and jump length distributions

This work outlines a transiently coupled continuous time random walk framework. The coupling is between the displacement probability density function (PDF) and the elapsed waiting time, and is of the form . Coupling of this kind generates larger displacements for longer waiting times, however, decouples on longer timescales. Such coupling is proposed to be physically relevant to systems in which diffusion is driven by the development of internal stresses which release and develop cyclically. This article outlines the associated generalised diffusion equation (GDE), which describes the time evolution of the position PDF, . The solution for is obtained using the properties of the Fox H function, both in terms of its transform properties but also its expansion theorems. The second moment and the asymptotic features of the solution are extracted. The relaxation of back to the solution of a decoupled-type GDE is highlighted.

Conditional sea surface statistics and their impact on geophysical sea surface parameters retrieved from SAR altimetry signals

In this study we present an analytical formulation of synthetic-aperture radar (SAR) altimetry signals including narrow banded nonlinear wave fields and conditional statistics between wave elevation displacements, horizontal wave slopes and vertical wave particle velocities. Considering the wave elevation displacements coskewness with respect to horizontal slopes leads to an analytical formulation of the electromagnetic bias within a SAR-mode altimeter stack. This formulation can be either parametrized by the significant wave height (SWH) and mean wave steepness, or in terms of the variance of vertical wave velocities. The effect of conditional vertical wave particle velocity variances with respect to the observed horizontal wave slopes close to nadir incidence angles leads to an effective reduction of the azimuth blurring of SAR-mode stacks. We present here a formulation of this effect by examining JONSWAP ocean wave spectra. In most cases this effect reduces the azimuth blurring by 10% to 30%. Additionally we investigate the effect of a nonlinear wave elevation displacement probability density function (PDF) on estimated geophysical parameters. We were able to show that including an elevation displacement skewness of 0.13 improves significantly the SWH consistency between altimetry and ECMWF Reanalysis v5 ERA5 results.All of these effects are validated with respect to ERA5 model data in the North East Atlantic region and in situ data located in the German Bight and Baltic Sea.The developed model can be used in both SAR and conventional altimetry retrackers.

The Fokker-Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm.

By collecting from literature data experimental evidence of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live Escherichia coli cells, we obtain the probability density function of molecules' displacement and we derive the corresponding Fokker-Planck equation. Molecules' distribution emerges to be related to the Krätzel function and its Fokker-Planck equation to be a fractional diffusion equation in the Erdélyi-Kober sense. The irreducibility of the derived Fokker-Planck equation to those of other literature models is also discussed.

Random diffusivity processes in an external force field

Brownian yet non-Gaussian processes have recently been observed in numerous biological systems, and corresponding theories have been constructed based on random diffusivity models. Considering the particularity of random diffusivity, this paper studies the effect of an external force acting on two kinds of random diffusivity models whose difference is embodied in whether the fluctuation-dissipation theorem is valid. Based on the two random diffusivity models, we derive the Fokker-Planck equationswith an arbitrary external force, and we analyze various observables in the case with a constant force, including the Einstein relation, the moments, the kurtosis, and the asymptotic behaviors of the probability density function of particle displacement at different timescales. Both the theoretical results and numerical simulations of these observables show a significant difference between the two kinds of random diffusivity models, which implies the important role of the fluctuation-dissipation theorem in random diffusivity systems.

Uncertainty estimation for ensemble particle image velocimetry

We present a novel approach to estimate the uncertainty in ensemble particle image velocimetry (PIV) measurements. The ensemble PIV technique is widely used when the cross-correlation signal-to-noise ratio is insufficient to perform a reliable instantaneous velocity measurement. Despite the utility of ensemble PIV, uncertainty quantification for this type of measurement has not been studied. Here, we propose a method for estimating the uncertainty directly from the probability density function of displacements found by deconvolving the ensemble cross-correlation from the ensemble autocorrelation. We then find the second moment of the probability density function and apply a scaling factor to report the uncertainty in the velocity measurement. We call this method the moment of probability of displacement (MPD). We assess MPD’s performance with synthetic and experimental images. We show that predicted uncertainties agree well with the expected root mean square (RMS) of the error in the velocity measurements over a wide range of image and flow conditions. MPD shows good sensitivity to various PIV error sources with around 86% accuracy in matching the RMS of the error in the baseline data sets. So, MPD establishes itself as a reliable uncertainty quantification algorithm for ensemble PIV. We compared the results of MPD against one of the existing instantaneous PIV uncertainty approaches, moment of correlation (MC). We adapted the MC approach for ensemble PIV, however, its primary limitations remain the assumption of the Gaussian probability density function of displacements and the Gaussian particles’ intensity profile. In addition, our analysis shows that ensemble MC consistently underestimates the uncertainty, while MPD outperforms that and removes the limiting Gaussian assumption for the particle and probability density function, thus overcoming the limitations of MC.

Stochastic Bifurcation and Energy Transfer in an Inerter-Based Pendulum Vibration Absorber

Abstract In this theoretical study, the vibration suppression, energy transfer, and bifurcation characteristics, as a function of dimensionless parameters, are investigated for an inerter-based pendulum vibration absorber (IPVA) attached to a linear single-degree-of-freedom spring-mass-damper system (primary structure), subject to white noise excitation. A perturbation method is introduced to detect and track the bifurcation points of the system. It is shown that the marginal probability density function of the pendulum angular displacement undergoes a P-bifurcation at critical parameter values, transitioning from monomodality to bi-modality. A cumulant-neglect technique is used to predict the mean squares of the system, which are compared to the response of a linear system without the IPVA to quantify the vibration suppression. It is shown that the IPVA leads to effective vibration mitigation of the structure in the neighborhood of the P-bifurcation, which is achieved by transferring the kinetic energy of the structure to the pendulum. The results are validated by a Monte Carlo simulation that is used to numerically approximate the marginal probability density function for the pendulum angle as well as the mean squares.

Imaging and characterization of transitions in biofilm morphology via anomalous diffusion following environmental perturbation.

Microorganisms form macroscopic structures for the purpose of environmental adaptation. Sudden environmental perturbations induce dynamics that cause bacterial biofilm morphology to transit to another equilibrium state, thought to be related to anomalous diffusion processes. Here, detecting the super-diffusion characteristics would offer a long-sought goal for a rapid detection method of biofilm phenotypes based on their dynamics, such as growth or dispersal. In this paper, phase-sensitive Doppler optical coherence tomography (OCT) and dynamic light scattering (DLS) are combined to demonstrate wide field-of-view and label-free internal dynamic imaging of biofilms. The probability density functions (PDFs) of phase displacement of the backscattered light and the dynamic characteristics of the PDFs are estimated by a simplified mixed Cauchy and Gaussian model. This model can quantify the super-diffusion state and estimate the dynamic characteristics and macroscopic responses in biofilms that may further describe dispersion and growth in biofilm models.

Ergodic property of random diffusivity system with trapping events.

A Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous environment. This paper considers a Langevin system containing a random diffusivity and an α-stable subordinator with α<1. This model describes the particle's motion in complex media where both the long trapping events and random diffusivity exist. We derive the general expressions of ensemble- and time-averaged mean-squared displacements which only contain the values of the inverse subordinator and diffusivity. Further taking specific time-dependent diffusivity, we obtain the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement. The results imply the nonergodicity of the random diffusivity model with any kind of diffusivity, including the critical case where the model presents normal diffusion.

The stationary response of piezoelectric cantilever beam model excited by colored noise

This article provides a method to predict the steady-state response of a piezoelectric cantilever under the excitation of Gaussian colored noise which is a better approximation to the ambient random excitation than the white noise. In this paper, a cantilever energy harvester model, which is fully-paved with piezoelectric ceramic and excited by Gaussian colored noise, is built by the Lagrange’s equations and is nondimensionalized. Then the dynamic equation and piezoelectric equation are transformed into a pair of stochastic differential equations about transient equivalent amplitude and transient phase. After that, the stochastic averaging method is applied to simplify the stochastic differential equations into an Itô type equation about the equivalent amplitude, the drift coefficient and the diffusion coefficient are obtained. This method provides another approach which is different from the stochastic linearization method and moment method. On this basis, some crucial outputs such as the steady-state probability density function (PDF) of the equivalent amplitude, the beam’s displacement, the transient output voltage, the joint probability density function of displacement and velocity as well as the mean square value of output voltage are all obtained. With different noise intensity values and delay factors, the Monte Carlo numerical simulation has verified the correctness of the theoretical prediction.

2D and 2.5D Numerical Simulations for Vortex-Induced Vibration (VIV) of a 1.5:1 Rectangular Cylinder

The semi-periodic vortex-shedding phenomenon caused by flow separation at the windward corners of a rectangular cylinder would result in significant vortex-induced vibrations (VIVs). Based on the aeroelastic experiment of a rectangular cylinder with side ratio of 1.5:1, 2-dimensional (2D) and 2.5-dimensional (2.5D) numerical simulations of the VIV of a rectangular cylinder were comprehensively validated. The mechanism of VIV of the rectangular cylinder was in detail discussed in terms of vortex-induced forces, aeroelastic response, work analysis, aerodynamic damping ratio and flow visualization. The outcomes showed that the numerical results of aeroelastic displacement in the cross-wind direction and the vortex-shedding procedure around the rectangular cylinder were in general consistence with the experimental results by 2.5D numerical simulation. In both simulations, the phase difference between the lift and displacement response increased with the reduced wind speed and the vortex-induced resonance (VIR) disappeared at the phase difference of approximately 180∘. The work done by lift force shows a close relationship with vibration amplitudes at different reduced wind speeds. In 2.5D simulations, the lift force of the rectangular cylinder under different wind speeds would be affected by the presence of small-scale vortices in the turbulence flow field. Similarly, the phase difference between lift force and displacement response was not a constant with the same upstream wind speed. Aerodynamic damping identified from the VIV was mainly dependent on the reduced wind speed and negative damping ratios were revealed at the lock-in regime, which also greatly influenced the probability density function (PDF) of wind-induced displacement.

Non-Gaussian, transiently anomalous, and ergodic self-diffusion of flexible dumbbells in crowded two-dimensional environments: Coupled translational and rotational motions.

We employ Langevin-dynamics simulations to unveil non-Brownian and non-Gaussian center-of-mass self-diffusion of massive flexible dumbbell-shaped particles in crowded two-dimensional solutions. We study the intradumbbell dynamics of the relative motion of the two constituent elastically coupled disks. Our main focus is on effects of the crowding fraction ϕ and of the particle structure on the diffusion characteristics. We evaluate the time-averaged mean-squared displacement (TAMSD), the displacement probability-density function (PDF), and the displacement autocorrelation function (ACF) of the dimers. For the TAMSD at highly crowded conditions of dumbbells, e.g., we observe a transition from the short-time ballistic behavior, via an intermediate subdiffusive regime, to long-time Brownian-like spreading dynamics. The crowded system of dimers exhibits two distinct diffusion regimes distinguished by the scaling exponent of the TAMSD, the dependence of the diffusivity on ϕ, and the features of the displacement-ACF. We attribute these regimes to a crowding-induced transition from viscous to viscoelastic diffusion upon growing ϕ. We also analyze the relative motion in the dimers, finding that larger ϕ suppress their vibrations and yield strongly non-Gaussian PDFs of rotational displacements. For the diffusion coefficients D(ϕ) of translational and rotational motion of the dumbbells an exponential decay with ϕ for weak and a power-law variation D(ϕ)∝(ϕ-ϕ^{★})^{2.4} for strong crowding is found. A comparison of simulation results with theoretical predictions for D(ϕ) is discussed and some relevant experimental systems are overviewed.