Stochastic Modelling and Analysis of Rotating Bladed Discs

This study focusses on probabilistic modelling of the bladed disc system and numerical estimation of the distributions of the response quantities of the system. Stochastic finite element model of the system consisting of all the assemblies and the hub is developed and reported. The spatial inhomogeneity of mistuned structures is modelled as non-Gaussian random field. Experimentally, the system parameters can be measured at the specified locations of the bladed disk structure. In this analysis, a synthetic data is generated which represent this measured data set. Further, Nataf transformation is implemented to each component of the data set to get the polynomial chaos expansion framework of the system parameters. Since, the random field of the system parameter is approximated as correlated random variables, Spearman’s rank correlation coefficient is used in this manuscript to obtain that correlation among the random parameters across the domain. The approximated probability density function obtained through the aforementioned methodology is compared with the target probability density function of the parameter using Kullback–Liebler (KL) entropy as a metric. Also, the same KL entropy is used as a metric to check the convergence of polynomial chaos terms in the expansion. Next, the proposed polynomial chaos method is integrated with commercial finite element software to quantify the propagation of randomness associated with system parameters into the response quantities. Subsequently, the statistical processing helps in estimating the probabilistic measure of the required response quantities. The results obtained through the conventional Monte Carlo (MC) simulations have been used as the benchmark to compare the response characteristics obtained through the proposed algorithm.

Estimating Design Loads for Floating Structures Using Environmental Contours

To ensure acceptable operation and/or survival of floating structures in extreme conditions, nonlinear time-domain simulations are often used to predict the structural response at the design stage. An environmental contour (EC) is commonly employed to identify critical sea states that serve as input for numerical simulations to assess the safety and performance of marine structures. In many studies, marginal and conditional distributions are defined to construct bivariate joint probability distributions for variables such as significant wave height and zero-crossing period; then, environmental contours can be constructed using the inverse first-order reliability method (IFORM). This study adopts alternative models to describe the generalized dependence structure between the environmental variables using copulas; the Nataf transformation is also discussed as a special case. Environmental contours are constructed, making use of measured wave data from moored buoys. Derived design loads are applied on a semi-submersible platform to assess possible differences. In addition, the long-term extremes of the tension of the mooring lines are estimated, considering uncertainties in the structural response using a 3D model (that includes response variability, ignored with the EC approach) to help establish more accurate design loads using Monte Carlo simulation. Results offer a clear indication of the extreme response of the floating structure based on the different models.

Comparative Study of Extreme Value Estimation Approaches for Mooring Lines Top Tension Responses

It is known that the mooring system response of floating production units subjected to environmental loads is nonlinear. Even though wave elevations can be assumed as Gaussian processes for short-term periods, corresponding line tension responses generally are not, due to second-order slow-drift floater motions and intrinsic nonlinearities of the system. In this work, short-term extreme responses are estimated based on two different approaches. In the first one, a number of probability distributions are fitted to the tension time histories’ peaks samples and classic order statistics is applied to determine the most probable extreme line tension corresponding to a short-time period (3-h) in order to identify the one with best performance. The effect of correlation between consecutive peaks in the extremes estimation is investigated through the one-step Markov chain condition by using a Nataf transformation-based model. In the second approach, a more robust and recently developed method named average conditional exceedance rate (ACER) is investigated, where dependencies between maxima can be easily taken into account. Additionally, effects of major parameters in dynamic analyses, such as simulation length and discretization level of the wave spectrum, are evaluated. All time-series-based extreme estimates are compared with the estimates directly obtained from a sample of epochal maxima (Gumbel method). Numerical examples cover two study cases for mooring lines belonging to Floating Production Storage and Offloading (FPSO) units installed offshore Brazil. It is shown that the consideration of dependence between peaks leads to lower extreme estimates and that both approaches return accurate results.

Improvements of Rackwitz–Fiessler Method for Correlated Structural Reliability Analysis

Rackwitz–Fiessler (RF) method is well accepted as an efficient way to solve the uncorrelated non-Normal reliability problems by transforming original non-Normal variables into equivalent Normal variables based on the equivalent Normal conditions. However, this traditional RF method is often abandoned when correlated reliability problems are involved, because the point-by-point implementation property of equivalent Normal conditions makes the RF method hard to clearly describe the correlations of transformed variables. To this end, some improvements on the traditional RF method are presented from the isoprobabilistic transformation and copula theory viewpoints. First of all, the forward transformation process of RF method from the original space to the standard Normal space is interpreted as the isoprobabilistic transformation from the geometric point of view. This viewpoint makes us reasonably describe the stochastic dependence of transformed variables same as that in Nataf transformation (NATAF). Thus, a corresponding enhanced RF (EnRF) method is proposed to deal with the correlated reliability problems described by Pearson linear correlation. Further, we uncover the implicit Gaussian copula hypothesis of RF method according to the invariant theorem of copula and the strictly increasing isoprobabilistic transformation. Meanwhile, based on the copula-only rank correlations such as the Spearman and Kendall correlations, two improved RF (IRF) methods are introduced to overcome the potential pitfalls of Pearson correlation in EnRF. Later, taking NATAF as a reference, the computational cost and efficiency of above three proposed RF methods are also discussed in Hasofer–Lind reliability algorithm. Finally, four illustrative structure reliability examples are demonstrated to validate the availability and advantages of the new proposed RF methods.

Resonance Analysis of Train–Track–Bridge Interaction Systems with Correlated Uncertainties

Over the last decades, the resonance-related dynamics for bridge systems subjected to a moving train has been researched and discussed from mechanics, physics and mathematics. In the current work, new perspectives of train-induced resonance analysis are investigated through introducing random propagation process into the train–bridge dynamic interactions. Besides, the Nataf-transformation-based point estimation method is applied to generate pseudorandom variables following arbitrarily correlated probability distributions. A three-dimensional (3D) nonlinear train-ballasted track–bridge interaction model founded on fundamental physical and mechanical principles is employed to convey and depict train–bridge interactions with random properties considered. After that, extensive applications are illustrated in detail for revealing the statistical characteristics of the so-called “random resonance”. Numerical results show that the critical train speeds associated with resonance and cancelation are random in essence owing to the variability of system parameters; the correlation between parameters exerts obvious influences on system dynamic behaviors; the last vehicle of a train will be in more violent vibrations compared to the front vehicles; the influences of track irregularities on the wheel–rail interactions are significantly greater than those of resonance.

Probabilistic Energy Flow Calculation through the Nataf Transformation and Point Estimation

With the increasing capacity of renewable energy sources, uncertainties regarding renewable energy and other dynamic loads in integrated energy systems (IESs) are increasing. Thus, it is necessary to study the probabilistic energy flow (PEF) of IESs. However, existing PEF calculation methods such as the point estimate method (PEM) are computationally inefficient when there are many random variables and estimated points; moreover, relatively large errors can occur when the estimated points are outside their limits. Hence, this paper presents a calculation method that addresses these problems. Because there are correlations among the variables, the Nataf transformation is employed to control the correlation quickly and effectively. A model for an IES that is interconnected with natural gas and electricity systems and accounts for the uncertainties of wind plants, photovoltaic power plants, and dynamic gas loads is presented. Correlations between wind plants and photovoltaic power plants are handled using the Nataf transformation. Finally, a modified PEM is developed to solve the PEF. For situations in which the estimated points exceed their boundaries, the power transformation and equal constraint transformation methods are used. The results of time-domain simulations demonstrate the effectiveness of the proposed approach.