Peak Probability Density Function Research Articles

Response prediction and fatigue life estimation of thin-walled structures under random excitation using the modified equivalent linearization method

This paper proposes a fast estimation method for the high-cycle fatigue of geometrically nonlinear thin-walled structures under random loading. The core idea of this method is utilizing a modified equivalent linearization method to obtain the statistical properties of the modal displacement response of the thin-walled structures, which is used to generate samples and then transform them into stress samples to determine the peak probability density function of the von Mises stress for fatigue life estimation. There are three critical issues addressed in implementing the method: (1) Using the implicit condensation technique to introduce the membrane softening effect in the regulated equivalent linearization method to improve the computational accuracy of the displacement statistical properties and convert the higher order moments into easily computable second order moments by separating the displacement mean values, so the regulated equations of motion for non-zero mean systems are derived, and fix the problem that the regulated equivalent linearization method does not apply to the prediction of the dynamic response of general thin-walled structures. (2) A nonlinear relationship between modal displacement and stress is derived with the implicit condensation and expansion method to transform the generated displacement samples to stress samples, which are used to determine probability density functions of stress and their derivatives. (3) Simplifying the probability density function of the three-dimensional joint distribution to the probability density function of the two-dimensional joint distribution by taking advantage of the Bayes rule, thus deriving a fast formula for the probability density function of the peak Mises stress.

Suppression of extreme events and chaos in a velocity-dependent potential system with time-delay feedback

The foremost aim of this study is to investigate the influence of time-delayed feedback on extreme events in a non-polynomial system with velocity dependent potential. To begin, we investigate the effect of this feedback on extreme events for four different values of the external forcing parameter. Among these four values, in the absence of time-delayed feedback, for two values, the system does not exhibit extreme events and for the other two values, the system exhibits extreme events. On the introduction of time-delayed feedback and varying the feedback strength, we found that extreme events get suppressed as well as get induced. When the feedback is positive, suppression occurs for a larger parameter region whereas in the case of negative feedback it is restricted to the limited parameter region. We confirm our results through Lyapunov exponents, probability density function of peaks, dmax plot and two parameter probability plot. Finally, we analyse the changes in the overall dynamics of this system under the influence of time-delayed feedback. We notice that complete suppression of chaos occurs in the considered system for higher values of the time-delayed feedback.

Closed-form solution based on peak probability density function for statistical fatigue life prediction under Broad-Band random loading

In this study, an analytical method for fast spectral fatigue-life prediction under Broad-Band random loading is proposed. It is based on the power spectral density of stress in the structure and the Rice peak stress distribution of a stationary Gaussian random process. The novel simple closed-form solution employs special HyperGeometric function which analytically describes the transcendent error weighting function between the Gaussian and Rayleigh distributions inherent in the Rice model. HyperGeometric-based solution outperforms other currently existing approximate methods based on the Rice peak distribution, e.g. Chaudhury, Kam, Dover, Chow and Lee, Lü et al. etc. Straightforward computational applicability and fast engineering implementation of obtained general closed-form results is accented throughout the investigation.

A frequency-domain model assessing random loading damage by the strain energy density parameter

The paper presents a new fatigue damage assessment model defined for the strain energy density parameter. The model is defined for calculations in the frequency domain for the case of narrowband loading. Analytical expressions are derived for the probability density function of peaks, the level crossing spectrum, and for the expected damage that is directly linked to the power spectral density of the strain energy parameter. The paper presents a theoretical proof which shows that the damage obtained for the energy model is equal to the damage obtained for the stress model. A numerical example is used to verify the correctness of the proposed frequency-domain damage estimation formula by comparison with experimental and time-domain fatigue life estimation results for two simulated random time-histories for the case of narrowband and broadband loading. The comparison between experimental and calculation results confirm the very good agreement between frequency and time domain approaches.

Turbulent flame structure characteristics of hydrogen enriched natural gas with CO2 dilution

This paper investigated the hydrogen enriched methane/air flames diluted with CO2. The turbulent premixed flame was stabilized on a Bunsen type burner and the two dimensional instantaneous OH profile was measured by Planar Laser Induced Fluorescence (PLIF). The flame front structure characteristics were obtained by extracting the flame front from OH-PLIF images. And the turbulence-flame interaction was analyzed through the statistic parameters. The role of hydrogen addition as well as CO2 dilution on the features of turbulent flame were revealed by those parameters. In this work, hydrogen fractions of 0, 0.2 and CO2 dilution ratios of 0, 0.05 and 0.1 were studied. Results showed that hydrogen addition can enhance turbulent burning velocity ST/SL through decreasing the scale of the finer structure of the wrinkled flame front, caused by the smaller flame instability scale. In contrast, CO2 dilution decreased turbulent burning velocity ST/SL due to its inactive response to turbulence perturbation and larger flame wrinkles. For all flames, the probability density function (PDF) profile of the local curvature radius R shows a bias to positive value, resulted from the flame intrinsic instability. The PDF profile of R decreases with CO2 dilution, while the value of local curvature radius corresponding to the peak PDF is larger. This indicates that larger wrinkles structure was generated due to CO2 dilution, which leads to the decrease in ST/SL as a consequence. Hydrogen addition increases the flame volume and results in more intense combustion. CO2 dilution has a decrease effect on flame volume for both XH2 = 0 and XH2 = 0.2 while the decrease is obvious at XH2 = 0.2, ZCO2 = 0.1. In all, hydrogen enrichment improves the combustion while CO2 can moderate combustion. Therefore, adding hydrogen and CO2 in natural gas can be a potential method for adjusting the combustion intensity in combustion chamber during the combustor design.

Acoustic fatigue properties investigation of plain weave C/SiC composite plate

In this paper, an acoustic fatigue model is proposed to obtain the residual strength of plain weave C/SiC composite plate subjected to acoustic loading. The constant amplitude normalized residual strength model and the S-N curves are used in new model, in which the probability density function of stress peaks are adopted to evaluate the effects of random loading on the degradation of residual strength. The degradation of residual strength in a time increment is calculated via an integration of the decrease of residual strength due to the stress peaks in the definitional domain of the peak probability density function. The evolutions of the residual strength under acoustic loading are determined via a double integral method. The residual strengths of C/SiC composite plate predicted by proposed model are in good agreement with the experimental results.

An investigation on residual strength and failure probability prediction for plain weave composite under random fatigue loading

The residual strength and the failure probability of the plain weave carbon-fiber reinforced epoxy composite subjected to random fatigue loading are investigated. According to the constant amplitude residual model and the experimental S-N curves, a new model is proposed, in which both the probability density function of stress peaks and the power spectrum density of stochastic processes are employed. The degradation of residual strength (dSr) in a time increment (dt) is calculated via an integration of the decrease of residual strength due to the stress peaks (dSrp) in the definitional domain of the peak probability density function. A linear damage accumulation of residual strength is used in this time increment. The evolutions of the residual strength with time are determined via a double integral method. The new model’s prediction results are in good agreement with the experiment results and it can give the curves of the variation of fatigue life with failure probability.

Characteristics of base pressure fluctuations of a typical missile configuration with a propulsive jet

Experiments were carried out to investigate the pressure fluctuations in the base region of a typical missile configuration at a freestream Mach number of 0.7 in the presence and absence of a propulsive jet. The objective was to characterize these pressure fluctuations and identify the underlying mechanisms responsible for their origin. Experiments included unsteady pressure measurements at six azimuthal locations and schlieren visualization. Substantial variation in the characteristics of pressure fluctuations is noticed along the azimuthal direction due to the asymmetry of the model. The probability density function (PDF) of pressure fluctuations indicates a significant drop in the peak PDF value and a substantial broadening at the highest nozzle pressure ratio (NPR). With an increase in NPR, the skewness decreases and approaches a Gaussian value while the kurtosis decreases and is seen to go below that of a Gaussian. Spectra of the pressure fluctuations indicate that the narrowband tones can be classified into three types based on the Strouhal number. Two types of tones among the three aforementioned are seen to be suppressed in the presence of the jet. Cross-correlation plots in the absence of the jet exhibit a large amplitude, low frequency oscillation which exists even after large separation times. On the other hand, in the presence of the jet, the oscillations dampen rapidly and vanish as the separation time increases. Finally, an attempt has been made to identify the possible primary mechanisms that are driving the base pressure fluctuations based on the observed characteristics of the acquired unsteady pressure signal.

Extreme-value analysis for ML-PMHT, Part 1: threshold determination

The Maximum Likelihood Probabilistic Multi-Hypothesis Tracker (ML-PMHT) can be used as a powerful multisensor, low-observable, multitarget active sonar tracker. It is a non-Bayesian algorithm that uses a generalized likelihood ratio test to differentiate between clutter and targets. Prior to this paper, the detection threshold used for target discrimination was determined either through trial and error or with lengthy Monte Carlo simulations.We present a new method for determining this threshold by assuming that clutter is uniformly distributed in the search space (which is one of the basic assumptions of the ML-PMHT algorithm) and then treating the log-likelihood ratio (LLR) as a random variable transformation. In this manner we can obtain an expression for the value of any random point on the likelihood surface caused by clutter. We then use extreme value theory to obtain an expression for the probability density function (PDF) of the peak point of the LLR surface due to clutter. From this peak PDF, we can then calculate a threshold based on some desired (small) false track acceptance probability.

Estimation of Random Sonic Fatigue Life Based on Peak Probability Density of Von Mises Stress

As to the random sonic fatigue problem of thin walled structure of aerospace flight vehicle, estimation method for fatigue life based on Von Mises stress peak probability density is investigated. Assuming that Von Mises stress process satisfies three-parameter Weibull distribution, the peak probability density function of Von Mises stress is derived through the threshold crossing analysis and the peak distribution analysis of stationary random process. According to the Miner linear cumulative damage theory, the method for estimating the fatigue life in the frequency domain is established with the Von Mises stress peak probability density function applied. As an example, an aero-engine Combustor liner is considered, using coupled Finite Element Method (FEM) and Boundary Element Method (BEM) method, the structure vibration response to limited bandwidth Gaussian white noise is calculated. Based on the results, fatigue life of the structure is estimated by using the proposed method. Further more, the influences of the probability density function which is characterized by three-parameter Weibull distribution and two-parameter Weibull distribution respectivily for Von Mises stress response of the Combustor liner structure subjected to random accoustic loadings are discussed.

On double peak probability density functions of duffing oscillator to combined deterministic and random excitations

The principal resonance of Duffing oscillator to combined deterministic and random external excitation was investigated. The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.

The effect of autocorrelation length on the real area of contact and friction behavior of rough surfaces

Autocorrelation length (ACL) is a surface roughness parameter that provides spatial information of surface topography that is not included in amplitude parameters such as root-mean-square roughness. This paper presents a relationship between ACL and the friction behavior of a rough surface. The influence of ACL on the peak distribution of a profile is studied based on Whitehouse and Archard’s classical analysis [Whitehouse and ArchardProc. R. Soc. London, Ser. A 316, 97 (1970)] and their results are extended to compare profiles from different surfaces. The probability density function of peaks and the mean peak height of a profile are given as functions of its ACL. These results are used to estimate the number of contact points when a rough surface comes into contact with a flat surface, and it is shown that the larger the ACL of the rough surface, the less the number of contact points. Based on Hertzian contact mechanics, it is shown that the real area of contact increases with increasing of number of contact points. Since adhesive friction force is proportional to the real area of contact, this suggests that the adhesive friction behavior of a surface will be inversely proportional to its ACL. Results from microscale friction experiments on polished and etched silicon surfaces are presented to verify the analysis.

On double-peak probability density functions of a Duffing oscillator under narrow-band random excitations

The principal resonance of Duffing oscillator under the narrow-band random exter nal excitation is investigated. The method of multiple scales is used to determi ne the equations of modulation of amplitude and phase. The one-peak probability density function of each of the two stable stationary solutions is then calculat ed by perturbation analysis in the resonant case. These two functions are combin ed by using the probability of realization of the two stable stationary solutions to calculate the double_peak probability density function approximately. Theor etical analyses are verified by numerical results.

Sigma Limiting Effects on the Response of a Ceramic Matrix Beam

High sigma events were studied in the narrowband random response of a cantilevered beam excited by an electrodynamic shaker. The effects of truncating the input signal to the shaker with a Blackglas beam are discussed. Tests were conducted with the shaker option of the shaker controller sigma limiting turned off, sigma limited to 2.5, and sigma limited to 3.5. Large data files were recorded and analyzed. The results were compared with similar previous tests with an aluminum alloy beam. The sigma values calculated varied considerably with the record lengths. When a fatigue failure was assumed, the cumulative fatigue damage was computed as a function of the sigma value of the response peak probability density function. Very little damage was found for the high sigma peaks compared to the damage that occurred with the many smaller peaks.

A statistical model for fatigue crack growth under random loads including retardation effects

A model for the statistical analysis of crack growth under random loading that includes the loading sequence effect is presented. The model defines and incorporates an equivalent closure stress that is included in the fatigue crack growth law via the effective stress intensity factor. The equivalent closure stress for each loading process is obtained from the probability density function of peaks p( S) in the random loading process, the properties of the material and the specimen geometry. The model was applied to the analysis of crack growth life under random loading on sheets of two different aluminum alloys: 2024-T351 and 2219-T851. The crack-growth lifetimes thus obtained were consistent with experimental data and with the results obtained by using a cycle-by-cycle simulation scheme.

SEABOTTOM BACKSCATTER STUDIES IN THE WESTERN CONTINENTAL SHELF OF INDIA

In this paper, a study is initiated to observe the interaction effect of the sound signal with three different sediment bottoms in the shelf area between Cochin and Mangalore in the western continental shelf of India. An echo signal acquisition system has been designed and interfaced with the 12 kHz echosounder installed onboard ORV Sagar Kanya. The reflection coefficients including attenuation at the seawater/bottom interface are computed in the three different sediment areas based on the sediment mean grain size. The experimental coherent reflection coefficients are calculated using the attenuation corrected reflection coefficients and the normalized cross-correlation between successive backscatter echo signal waveforms in those areas. Further, analyses conducted by determining the echo peak Probability Density Function (PDF) and matching them with the experimental echo peak histograms provide root mean square (rms) roughness amplitude in the three different survey areas. The rms roughness values are used to compute the coherent reflection coefficients. An attempt to establish concurrence between the coherent reflection coefficients based upon the rms roughness amplitude and the experimental coherent reflection coefficients using the backscatter echo signals, reveals the importance of seawater/bottom interface roughness in the coarse grained sediment bottoms like sand and silty sand. The existence of microtopographic features are responsible for the seawater/bottom interface roughness. However, in the fine grained sediment area, the bottom does not contain any such feature.

Probability Density Functions of Acoustically Induced Strains in Experiments with Composite Plates

Composite structures, which are widely used in the design of flight vehicles, exhibit nonlinear behavior under in-service acoustic excitations typical of jet engine effluxes. Therefore, the responses of such structures can be non-Gaussian random processes. The dynamic strains in carbon fiber-reinforced plastic plates with clamped boundaries have been investigated experimentally in a progressive wave tube. Computer analysis concentrated on evaluation of the probability density function and its skewness and kurtosis parameters, which describe deviations from the Gaussian (normal) law model. The results show that a great lack of symmetry exists in the distribution plots of strain instantaneous values, and this influences peak probability density functions. The latter characteristic is the focus of attention for flight vehicles because it is associated directly with fatigue damage accumulation, which may be the cause of failure. The analytical description of these phenomena is also discussed. Approximation of instantaneous value distributions is made in the piecewise-Gaussain form and yields peak distribution formulas that are similarly constructed from Rayleigh law sections. The main advantage of both these solutions for experimental data fitting is that the probability density expressions obtained are convenient for use in subsequent theoretical modeling. Accuracy of the proposed method is demonstrated via a number of examples.

High cycle fatigue of austenitic stainless steels under random loading

The effect of high cycle random loading at 330°C and at 550°C on the fatigue resistance of austenitic stainless steels Type 316LN has been investigated. The aim was to improve the predictions of the consequences of thermal fluctuations occurring in some reactor components. The experimental means used for this purpose was a minicomputer, which generated random stationary Gaussian load sequences in a Markov matrix, and several servo-hydraulic fatigue testing machines. This method has the advantage of taking into account the mean load for each cycle. The specimens tested were taken from two plates which were at two different heats, one 45 mm thick and slightly cold worked, the other 26.5 mm thick. Random fatigue tests were mainly performed in the endurance region of the fatigue curve, using the stair-case method to determine the scatter. The experimental data were analyzed in terms of the test temperature and the irregularity factor, with the aim of establishing design curves for reactor components. The results obtained have shown that the random loading is more damaging than the constant amplitude loading when compared on the basis of the mean square root conventional fatigue limit. They have also shown that the linear damage accumulation, according to the Miner's rule, yields conservative results at least when used in conjunction with the probability density function of peaks for a case where the irregularity factor is near 100%. Based on these observations a method is proposed for predicting the fatigue life from the design curves, which includes random loading effects with the irregularity factor as the indexing parameter.

Density and Size Estimates of Cisco (Coregonus artedii) Using Analysis of Echo Peak PDF from a Single-Transducer Sonar

We estimated size and density of fish in three Wisconsin lakes from echo peak probability density functions (PDFs) obtained at night with a single-transducer 70-kHz echosounder. At night, cisco (Coregonus artedii) dominated the pelagic zone in all three lakes. The beam pattern effect was removed with a deconvolving filter technique. Fish size was estimated by fitting a combination of Rice PDFs to the deconvolved fish scattering PDF. Vertical density profiles and size estimates obtained acoustically corresponded to distributions and lengths of fish caught in vertical gill nets. The proportion of different size classes caught in gill nets agreed fairly well with the proportions determined acoustically. This analysis can be applied to signals from noncalibrated sonars and can be used to calibrate simultaneously obtained echo squared integration values. With calibrated sonars, target strength can be estimated in situ. For Cisco, TS = 21.9 log10L − 67.2, where TS is target strength in (decibels) and L is fish length (centimetres). The average number of Cisco in the three lakes ranged from 89 to 1551 fish/ha, corresponding to a biomass of 2–223 kg/ha. Maximum densities range from 12 to 49 fish/1000 m3.

Volume scattering: Echo peak PDF

The statistical properties of the backscattered acoustic echo from clouds of scatterers are examined. These clouds could be plankton, nekton, bubbles, etc., or a combination. The clouds of interest are sufficiently dense so that echoes from the individual scatterers overlap. The scatterers are assumed to be randomly distributed in space so that, at high enough acoustic frequencies, the individual echoes are not correlated (random phase). The probability density function (PDF) of the maximum echo value achieved in a depth gate has been analytically derived, computer simulated, and measured in the field. It is shown to be quite different than the Rayleigh PDF of the envelope, even for gate durations smaller than one transmission ping duration. An approximate analytical expression for the echo peak PDF is derived by first modeling the echo as a discrete ‘‘stepped’’ waveform where the ‘‘steps’’ are statistically independent of each other. The problem is then solved by straightforward extremal statistics. There is excellent agreement between the theoretical PDF, computer simulations, and field data. While we are concerned with gate durations several times one ping duration, the analytical expression for the PDF is shown to accurately describe the PDF over ranges of the gate duration less than the ping duration to much greater than the ping duration. From the echo peak PDF one can (1) estimate the abundance of occasional large scatterers that are within the clouds (for example large fish feeding on plankton) since their occurrence is differentiable from the rest of the cloud in the PDF, and (2) clearly differentiate between conditions where the echoes overlap and do not overlap. Once it is determined whether or not they overlap, lower or upper bounds in scatterer density (number per unit volume) may be calculated. Both of these applications can be performed in situ.