Amplitude Probability Density Function Research Articles

Fokker-Planck dynamics of two coupled van der Pol oscillators subjected to stochastic forcing.

We present probabilistic solutions to a pair of mutually coupled Van der Pol oscillators subjected to stochastic forcing. We consider three different types of coupling: reactive, dissipative, and nonlinear coupling. Using stochastic averaging, we derive the stationary Fokker-Planck equationfor each oscillator, yielding a probability density function for the fluctuation amplitude. For each coupling type, we numerically validate the Fokker-Planck solutions for different noise levels and coupling strengths, with a focus on the stochastic and bifurcation characteristics. The validated analytical expressions derived in this study could serve to improve the prediction and control of a generic class of coupled oscillator systems operating near the Hopf point in the presence of noise.

Gaussianity Evaluation of HD-sEMG Signals with Aging and Sex During Low and Moderate Isometric Contractions of the Biceps Brachii

IntroductionAging is associated with muscle decline, which alters both functional and anatomical properties of the neuromuscular system. These modifications can be reflected in high-density surface electromyography (HD-sEMG) signals. This study examines how age and sex impact the shape of the amplitude Probability Density Function (PDF) of HD-sEMG signals. Materials and MethodsMonopolar HD-sEMG signals were collected from the Biceps Brachii in a cohort of 17 individuals: 10 women (mean age: 22.9 ± 3.6 years) and 7 men (mean age: 24.4 ± 2.5 years) in the younger group, and 10 women (mean age: 69.8 ± 4.8 years) and 7 men (mean age: 72.8 ± 2.7 years) in the elderly group. The recordings were conducted during an elbow flexion at both 20% and 40% maximum voluntary contraction. The signal amplitude was evaluated using root means square amplitude (RMSA) and the PDF shape of each HD-sEMG signal was assessed through skewness, excess Kurtosis, and robust functional statistics. These shape distance metrics evaluate the departure from Gaussianity related to muscle aging. a) We conducted a comparison study of the HD-sEMG PDF shapes between younger and elderly individuals. b) Evaluating differences between men and women. c) Considering monopolar and Laplacian electrode configurations that are sensitive to different muscle regions. ResultsA) The HD-sEMG PDFs of elderly subjects demonstrated a lower departure from Gaussianity than their younger counterparts. B) Women exhibited lower RMSA values than men, and, on average, a lower departure from Gaussianity whatever the age and contraction level C) Trends of departure from Gaussianity with contraction level, seems to be influenced by the electrode configuration. In fact, a decrease in Gaussianity departure is observed with monopolar recordings where an increase is observed with Laplacian one, clearly indicating different muscle region assessment. DiscussionThe findings highlight the influence of factors such aging, sex, contraction level and electrode montage on the shape of the HD-sEMG PDF, emphasizing the significance of using this descriptor for monitoring and better assessment of muscle aging.

Probability density function of roll amplitude for parametric rolling using moment equation

For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of the roll amplitude. In this study, first, we derive the moment values of roll amplitude by combining the moment equations and the linearity of expectation. With this, we propose a method to estimate the probability density function of the roll amplitude by using the obtained moment values. The results of our proposed method are compared to those obtained from Monte Carlo simulations. When the higher-order cumulant neglect closure method is used, the moment values resulting from the moment equations are close to the results of corresponding Monte Carlo simulations. In addition, our proposed method for deriving the probability density function of the roll amplitude is validated by comparison with Monte Carlo simulation results. In conclusion, we can state that the proposed methods for deriving the moments and the probability density function of the roll amplitude are available for practical use cases.

An amplitude probability density function model under broadband multimodal stochastic vibration fatigue response

In this paper, a fatigue life prediction model based on the amplitude probability density function under broadband multimodal stochastic vibration response is presented. An analysis method is proposed to address the dispersion of the third- and fourth-order normalized moments of rain-flow amplitude distributions. The unified relationships between the first four-order normalized moments of rain-flow amplitude distributions and the spectral parameters are established to determine the model parameters. Through comparison with other frequency-domain methods and based on the tail probability density distribution of rain-flow amplitude, the proposed model offers more precise and stable fitting results under broadband multimodal response power spectra.

Stochastic response and stability analysis of nonlinear vehicle energy-regenerative suspension system with time delay

With the rapid development of new energy vehicles, a novel vehicle energy-regenerative suspension for simultaneous vibration suppression and energy harvesting has become one of the key development directions of the next generation of vehicle design. In order to improve the performance of suspension system, the time-delayed feedback control is applied to the novel suspension system subjected to different external excitation. At the same time, there are few studies related to the stochastic dynamics of nonlinear vehicle energy-regenerative suspension system with time delay. This paper studies stochastic response and stability of nonlinear vehicle energy-regenerative suspension with time-delayed feedback control under the excitation of Gaussian white noise. First of all, a stochastic dynamic model of the nonlinear vehicle energy-regenerative suspension with time delay system is established. Secondly, using the stochastic averaging method based on generalized harmonic function, the steady-state probability density function (PDF) of amplitude is obtained. The accuracy of the presented approach is verified through the comparison of analytical and numerical solutions. In addition, through combining the stochastic averaging method with the Khasminskii method, the top Lyapunov exponent (TLE) of the system is obtained, and then the influence of time delay and system parameters on the asymptotic Lyapunov stability is discussed. Finally, the effects of time delay and system parameters on system performance are also studied.

A probability density function of stress amplitude applicable to power spectral density curves of structural random vibration response

The power spectral density curves of the random vibration response of a plate specimen with different spectral width coefficients were obtained by selecting different frequency ranges, and the probability distributions of the stress amplitudes were obtained by applying Monte Carlo method and rainflow counting method. Each probability distribution was divided into three segments by the corresponding zero-order spectral moment value, and each segment was fitted using either a Rayleigh function or an exponential function. Finally, a probability density function of stress amplitude was proposed by combining two Rayleigh functions and an exponential function. The proposed probability density function can be used to obtain the stress amplitude probability distribution for the power spectral density curves of structural random vibration response with any bandwidth, and the accuracy of the proposed function was verified based on the response data of a specimen of 7075-T651 aluminum alloy material under the action of six typical power spectral curves.

Bifurcation Regulations Induced by Joint Noise in a Tri-Rhythmic Van Der Pol System

Tri-rhythmical nature has attracted extensive attention from scholars in describing the dynamical behaviors of self-sustained systems. In this paper, we consider a tri-rhythmic van der Pol system and give a bifurcation analysis of a stochastic tri-rhythmic self-sustained system under joint noise perturbation. Based on an approximate approach, we give the stationary probability density function of amplitudes, and we find that the noise and time-delay feedback, regulating the velocity time-delay feedback strength parameter, may not cause transitions among unimodal, bimodal and trimodal in the tri-rhythmic system. More stochastic bifurcations appear by regulating the time delay in this system. The system, surprisingly, undergoes five times of stochastic bifurcations when the time delay is monotonically increased. It is shown that the time delay is more sensitive to the tri-rhythmic system and a much stronger dependence on the system. From a biological point of view, the reaction rate of biological molecules can be enhanced or diminished by the change of the noise intensity or correlation time of Gaussian colored noise. More surprisingly, an increase in displacement feedback will delay the reaction rate; however, the effect of an increase in velocity feedback on the reaction rate depends on the time delay. A detailed research on the parameter space indicates that time delay and colored noise parameters can effectively control the multirhythmicity of the system. Finally, this paper verifies the effectiveness of the theoretical solution through the numerical results of Monte Carlo simulation of the original system. These results may help to further explore forking bifurcations in real-world applications. This research provides new insight into the understanding of nontrivial effects of joint noise on the tri-rhythmic system.

Human-exoskeleton cooperation for reducing the musculoskeletal load of manual handling tasks in orchid farms

Work-related musculoskeletal diseases of the lower back and shoulders are frequently caused by lifting, carrying and trunk bending. Therefore, a hybrid exoskeleton (HExo) that provides external torque to the upper extremities and back may reduce discomfort in the lower back and shoulder tendinopathies. In this study, we designed and fabricated HExo, a novel upper-limb and lower-back combined exoskeleton, and evaluated its impact in reducing the musculoskeletal load on the shoulder and back during agricultural work in orchid farms. The human motion tracking technique was utilised to collect the joint angles of orchid farmers during their work, and the joint angle data were subsequently used for the design of HExo. Electromyography (EMG) and centre of pressure (CoP) measurements were taken for 5 participants performing farm tasks and 16 participants performing laboratory-standardised tasks both with and without the HExo. The amplitude probability density function and Wilcoxon signed rank test were utilised in the data analysis to verify the advantage in wearing HExo. The use of HExo resulted in a reduction in erector spinae, latissimus dorsi and anterior deltoid muscle activation of up to 36.92 % (p = 0.05), 31.43 % (p = 0.017), and 57.59 % (p = 0.012), respectively. Furthermore, its use did not affect the ability of participants to maintain balance. Hence, HExo could be a promising for reducing solution musculoskeletal load.

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.

Analysis of the stationary probability density of a generalized and bistable van der Pol system excited by colored noise

The stochastic P-bifurcation behavior of bi-stability in a generalized van der Pol oscillator with the fractional damping under colored noise and thermal excitation is investigated. Firstly, using the principle of minimal mean square error and lin-earization method, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order van der Pol system. Secondly, the system amplitude stationary probability density function is obtained by the stochastic averaging, and then based on the singularity theory, the critical parametric con-ditions for the system amplitude stochastic P-bifurcation are found. Finally, the types of the stationary probability density function of the system amplitude are qualitatively analyzed in each area divided by the transition set curves. The con-sistency between the analytical results and the numerical results acquired from Monte-Carlo simulation also testifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

Higher-order statistical moments of predicted sonic boom waveforms through turbulence

Sonic booms generated by supersonic aircraft are affected by turbulence in the atmospheric boundary layer through which they propagate. Turbulence effects lead to random variability of the sonic boom waveforms measured on the ground, complicating the prediction of such waveforms. As an initial effort to predict the waveform variability, the solution of the coherent or mean sonic boom waveform has previously been formulated by the author. The current paper extends the formulation to derive an expression of the second-order statistical moment necessary to calculate the variance of the spectral amplitudes. Since the derivation uses a full wave equation, results using the derived expression will be compared with those obtained using a parabolic approximation. In order to fully quantify the uncertainty of our predictions, the higher order statistical moments are also formulated and are shown to be approximately zero. Consequently, the probability density function (pdf) of the spectral amplitudes is predicted to be Gaussian. The formulation is further extended to determine the pdf of the loudness of sonic booms and quantify the uncertainties associated with the loudness prediction. Results from the formulation are compared with available flight test data.

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.

An Exposimetric Electromagnetic Comparison of Mobile Phone Emissions: 5G versus 4G Signals Analyses by Means of Statistics and Convolutional Neural Networks Classification

To gain a deeper understanding of the hotly contested topic of the non-thermal biological effects of microwaves, new metrics and methodologies need to be adopted. The direction proposed in the current work, which includes peak exposure analysis and not just time-averaged analysis, aligns well with this objective. The proposed methodology is not intended to facilitate a comparison of the general characteristics between 4G and 5G mobile communication signals. Instead, its purpose is to provide a means for analyzing specific real-life exposure conditions that may vary based on multiple parameters. A differentiation based on amplitude-time features of the 4G versus 5G signals is followed, with the aim of describing the peculiarities of a user’s exposure when he runs four types of mobile applications on his mobile phone on either of the two mobile networks. To achieve the goals, we used signal and spectrum analyzers with adequate real-time analysis bandwidths and statistical descriptions provided by the amplitude probability density (APD) function, the complementary cumulative distribution function (CCDF), channel power measurements, and recorded spectrogram databases. We compared the exposimetric descriptors of emissions specific to file download, file upload, Internet video streaming, and video call usage in both 4G and 5G networks based on the specific modulation and coding schemes. The highest and lowest electric field strengths measured in the air at a 10 cm distance from the phone during emissions are indicated. The power distribution functions with the highest prevalence are highlighted and commented on. Afterwards, the capability of a convolutional neural network that belongs to the family of single-shot detectors is proven to recognize and classify the emissions with a very high degree of accuracy, enabling traceability of the dynamics of human exposure.

Extracting nonlinear aerodynamic damping of crosswind-excited tall buildings based on probability density function of displacement amplitude

Extracting nonlinear aerodynamic damping of crosswind-excited tall buildings based on probability density function of displacement amplitude

EMG probability density function: a new way to look at EMG signal filling from single motor unit potential to full interference pattern.

An analytical derivation of the EMG signal's amplitude probability density function (EMG PDF) is presented and used to study how an EMG signal builds-up, or fills, as the degree of muscle contraction increases. The EMG PDF is found to change from a semi-degenerate distribution to a Laplacian-like distribution and finally to a Gaussian-like distribution.We present a measure, the EMG filling factor, to quantify the degree to which an EMG signal has been built-up. This factor is calculated from the ratio of two non-central moments of the rectified EMG signal. The curve of the EMG filling factor as a function of the mean rectified amplitude shows a progressive and mostly linear increase during early recruitment, and saturation is observed when the EMG signal distribution becomes approximately Gaussian. Having presented the analytical tools used to derive the EMG PDF, we demonstrate the usefulness of the EMG filling factor and curve in studies with both simulated signals and real signals obtained from the tibialis anterior muscle of 10 subjects. Both simulated and real EMG filling curves start within the 0.2 to 0.35 range and rapidly rise towards 0.5 (Laplacian) before stabilizing at around 0.637 (Gaussian). Filling curves for the real signals consistently followed this pattern (100% repeatability within trials in 100% of the subjects). The theory of EMG signal filling derived in this work provides (a) an analytically consistent derivation of the EMG PDF as a function of motor unit potentials and motor unit firing patterns; (b) an explanation of the change in the EMG PDF according to degree of muscle contraction; and (c) a way (the EMG filling factor) to quantify the degree to which an EMG signal has been built-up.

STOCHASTIC STABILITY AND PARAMETRIC CONTROL IN A GENERALIZED AND TRI-STABLE VAN DER POL SYSTEM WITH FRACTIONAL ELEMENT DRIVEN BY MULTIPLICATIVE NOISE

The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic [Formula: see text]-bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior.

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.

Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace’s method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary response amplitude PDF. Next, the PI technique is employed, which is based on a discretized version of the CK equation solved in short time steps. This is done relying on the Laplace’s method of integration which yields an approximate analytical solution of the integral involved in the CK equation. In this manner, the repetitive integrations generally required in the classical numerical implementation of the procedure are avoided. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. Notably, the technique can also account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. Applications to oscillators with Van der Pol and Duffing type nonlinear restoring force models, and Preisach hysteretic models, are presented. Appropriate comparisons with Monte Carlo simulation data are shown, demonstrating the efficiency and accuracy of the proposed approach.

Vibration fatigue life prediction method for needled C/SiC composite based on frequency response curve with low signal strength

The experimental results show that needled C/SiC composite exhibits nonlinear stiffness characteristics in the process of vibration. In order to consider the effect of the number of fiber layers on the fatigue properties, the SN curve related to the number of 0° layer fibers for needled C/SiC composite is proposed in this investigation. According to the relationship between the frequency response curve and the signal strength, an amplitude probability density function under narrow band random vibrations was proposed for predicting the fatigue life for the notched plate specimen. The results showed that the proposed method has a good prediction effect.

Enhancement of direct waves based on the probability density function of seismic wave amplitudes

SUMMARY The enhancement of direct waves in seismic records is desirable in various seismological situations, where the accurate picking of seismic phases is necessary for hypocentre location, spectral inversion, waveform inversion and seismic tomography. In this study, we developed a new method to enhance direct waves in seismic waveforms based on the statistical distribution (probability density function, PDF) of the amplitudes of waveforms. The seismic waveform changes from a direct wave to a coda wave as time proceeds, which also causes the PDF of its amplitude to vary with time. The PDF becomes similar to a Gaussian or Rayleigh distribution with time, and hence, direct waves can be enhanced using the dissimilarity between the PDF of the amplitude of the seismic waveform and the Gaussian or Rayleigh distribution. In the proposed method, the Kullback–Leibler divergence is used to measure dissimilarity and it is applied as a weight. We confirmed that the method can effectively enhance direct P and S waves by applying it to seismic records of the 2016 Kumamoto earthquake, synthetic waveforms with noise and volcanic events. The method was also applied to waveforms of the 2008 Iwate–Miyagi Nairiku earthquake, which revealed that direct waves were enhanced, and an overview of the wave field and rupture process was obtained. The proposed method can detect direct waves even from subevents, thereby producing detailed information on the earthquake rupture process. We detected two significant pulses and determined their source location and rupture times. The estimated pulse sources were consistent with parts of the rupture scenario proposed by the previous studies. In addition, for local and teleseismic event records, the proposed method was compared with other methods to study its merits and demerits. The present method can enhance seismic phases on records of local and teleseismic events. Our method detects or enhances direct waves based neither on their amplitudes nor on their temporal variations but on the statistical distribution of their amplitudes, which has both advantages and disadvantages. Although polarization filtering is a powerful method that can enhance seismic phases, it requires three-component waveforms. The proposed method can be applied even to a single component record, which is a significant advantage.