Latin Hypercube Sampling Technique Research Articles

Deterministic and probabilistic prediction of maximum wall facing displacement of geosynthetic-reinforced soil segmental walls using multivariate adaptive regression splines

A surrogate model-based response surface method is proposed to predict the maximum wall facing displacement of geosynthetic-reinforced soil (GRS) segmental walls using the multivariate adaptive regression splines (MARS). The synthetic datasets are generated using the Latin hypercube sampling (LHS) technique for both training and testing purposes. For each input dataset, the maximum wall facing displacement is evaluated using the finite difference software FLAC. An expression for the maximum wall facing displacement is proposed in terms of basis functions (piecewise linear). The performance of the proposed model is assessed in terms of the ratio of predicted and simulated maximum wall facing displacement, and the results show that the performance is satisfactory. Moreover, the study identifies that the effect of soil friction angle is most apparent in predicting the maximum wall facing displacement. The efficiency of the proposed model is compared with other soft computing techniques, and the result indicates that the proposed model gives minimum error. Further, a probabilistic study is performed in terms of the normalized maximum wall facing displacement. The results show that the probabilistic prediction of the maximum wall facing displacement using the proposed model is within the range of specified limit.

Single and Multi Objective Optimization for Injection Molding Using Numerical Simulation with Surrogate Models and Genetic Algorithms

Abstract The objective of this study is to develop an integrated computer-aided engineering (CAE) optimization system that can quickly and intelligently determine the optimal process conditions for injection molding. This study employs support vector regression (SVR) to establish the surrogate model based on executions of three-dimensional (3D) simulation for a selected dataset using the latin hypercube sampling (LHS) technique. Once the surrogate model can satisfactorily capture the characteristics of simulations with much less computing resources, a hybrid optimization genetic algorithm (GA) or a multi-objective optimization GA is then used to evaluate the surrogate model to search the global optimal solutions for the single or multiple objectives, respectively. The performance and capabilities of other surrogate modeling approaches, such as polynomial regression (PR) and artificial neural network (ANN), are also investigated in terms of accuracy, robustness, efficiency, and requirements for training samples. Experimental validations and applications of this work for process optimization of a special box mold and a precision optical lens are presented.

A New Method for Predicting Site-Specific Wave-Induced Hull Girder Loads Acting on Ship-Shaped Offshore Installations in Benign Conditions

The aim of this study is to establish an innovative method for predicting wave-induced hull girder loads acting on ship-shaped offshore installations. Unlike in trading ships, the determination of hull girder loads of ship-shaped offshore installations must consider site-specific environmental conditions. However, the accurate determination of wave-induced loads is not straightforward owing to uncertainties in the ocean environmental conditions and location-dependent complexities, among other factors. This study proposes a probabilistic approach to determine the site-specific design values of wave-induced loads acting on ship-shaped offshore installations that also accounts for the effects of mooring system type. Six target regions – the North Sea, Gulf of Mexico, western coast of Africa, the eastern coast of South America, southeastern coast of Asia and northwestern coast of Australia – were studied to compare the results corresponding to various sea states. A set of wave scenarios representing all possible wave events for each target region were selected using the Latin hypercube sampling technique. To demonstrate this method, the design values of the wave-induced vertical bending moments were determined for a very large crude oil carrier (VLCC)-class structure with a hypothetical floating, production, storage and offloading (FPSO) unit. The effects of the mooring system type (e.g., single-point mooring versus spread mooring) on the wave-induced hull girder loads of the ship-shaped offshore installations were also evaluated. The validation of the developed method was made by comparison with existing results in the literature, its novelty is associated with a new methodology that can accurately determine wave-induced hull girder loads of ship-shaped offshore installations taking into account the effects of site-specific ocean environmental conditions and mooring system type, and its main contribution to the industry is to provide core technology for the safe and economical design of ship-shaped offshore hull structures.

Effect of Architectural Building Design Parameters on Thermal Comfort and Energy Consumption in Higher Education Buildings

It has been challenging for designers to identify the appropriate design parameters that would reduce building energy consumption while achieving thermal comfort for building occupants. This study aims to determine the most important architectural building design parameters (ABDPs) that can increase thermal comfort and reduce energy use in educational buildings. The effect of 15 ABDPs in an Australian educational lecture theatre and their variabilities on energy consumption and students’ thermal comfort for each parameter were analysed using Monte Carlo (MC) techniques. Two thousand simulations for every input parameter were performed based on the selected distribution using the Latin hypercube sampling (LHS) technique. Sensitivity analyses (SA) and uncertainty analyses (UA) were used to assess the most important ABDPs in terms of thermal discomfort hours and energy consumption. The study found that the ABDPs, such as cooling set-point temperatures and roof construction, significantly reduce the operative temperature by up to 14.2% and 20.0%, respectively. Consequently, these reductions could significantly shorten the thermal discomfort hours, thereby reducing energy consumption by 43.7% and 41.0%, respectively. The findings of this study enable building designers to identify which ABDPs have a substantial impact on thermal comfort and energy consumption.

Multiple Earthquake Effects on Vulnerability of Horizontally Curved RC Bridges

ABSTRACT Bridges subjected to mainshock–aftershock sequences experience significant damage as a result of repeated shaking. Therefore, neglecting the effects of multiple earthquakes may lead to a considerable underestimation of bridges seismic vulnerability. The uncertainties in modeling bridge structural systems as well as the representation of seismic loads significantly affect the bridge vulnerability, especially for horizontally curved bridges, where complex loading conditions exist. In this study, fragility curves for horizontally curved bridges are developed for both mainshock only and mainshock–aftershock sequence cases. The seismic response of a typical four-span horizontally curved reinforcement concrete box-girder bridge is evaluated using the cloud analysis method and the effects of uncertainties on the probability of failure of the bridge components are examined and quantified using the Latin hypercube sampling technique. The results from this investigation indicate that considering the mainshock only could significantly underestimate the damage level even by around 50% and that incorporating uncertainties could change the bridge capacity by more than 45% of its estimated value.

Towards application of uncertainty quantification procedure combined with experimental procedure for assessment of the accuracy of the DEM approach dedicated for granular flow modeling

There is a high demand for accurate and fast numerical models for dense granular flows found in many industrial applications. Nevertheless, before numerical model can be used its need to be always validated against experimental data. During the validation, it is important to consider how the measurement data sets, as well as the numerical models, are affected by errors and uncertainties. In this study, the uncertainty quantification for the Discrete Element Method (DEM) model was performed based on selected quantities of interest (QOI), which were measured at a test rig. The uncertainty quantification was performed with open-source Dakota code, and the Latin hypercube sampling technique was used to determine test points. Various correlations between the input and output data were investigated to assess the impact of the possible input data errors on the output values determined by the solver. The results were validated against the measurement data from a novel in-house experimental test rig. The novel character of this work is developed procedure for study the impact of the uncertainties related with the input data on numerical results delivered by DEM model. In-house algorithm written using OpenCV libraries for determining particle motion characteristics from image dataset was used for results determination and postprocessing.

Sensitivity analysis of capital and energy production cost for off-grid building integrated photovoltaic systems

This paper presents a methodology for performing sensitivity analysis on the capital and energy production costs of off-grid Building Integrated Photovoltaic (BIPV) systems. The different influencing parameters that are considered in this study are the daily electric load on the system, available sunshine hours, per unit costs of components, maintenance and replacement costs, several design and rating factors, interest rates, and carbon credits. The probabilistic-based Latin hypercube sampling (LHS) technique and log-log regression were used to develop the correlations for the capital cost and energy production cost of the system. The results were validated by comparing them with case studies available in the published literature. Then, a standardized version of the dataset was used to derive the elasticity constants associated with each parameter, which quantified the change in costs when the parameter value was increased by a unit percentage. The effects on the costs of increasing the parameters by 10% were quantified, ranked and compared. For the capital cost, the three most sensitive parameters were the daily electric load on system, the fudge factor and the available sunshine hours. In contrast, for the energy production cost, the three most sensitive parameters were the annual maintenance cost factor, the fudge factor and the interest rate. The correlation trends between these parameters and the costs were also plotted. Precautions and potential stakeholder actions for reducing the costs of off-grid BIPV systems are suggested.

Multi-Variate Optimization of Polymer Electrolyte Membrane Fuel Cells in Consideration of Effects of GDL Compression and Intrusion

This study applies, tests, and compares comprehensive surrogate-based optimization techniques to optimize the performance of polymer electrolyte membrane fuel cells (PEMFCs). Moreover, parametric cases considering four important design variables, i.e., gas diffusion layer thickness (), channel depth (), channel width (), and land width (), are defined using the latin hypercube sampling technique under reasonable constraint conditions. Multidimensional, two-phase PEMFC model simulations are performed to generate the training and test data under these design conditions. Three famous surrogate models, i.e., response surface approximation (RSA), radial basis neural network (RBNN), and kriging (KRG), are employed to construct objective functions for the PEMFC cell voltages operating in the galvanostatic mode, and their accuracies are tested and compared using root mean square error and adjusted R-square. The surrogates are then linked to stochastic optimization algorithms, i.e., genetic algorithm and particle swarm optimization, to determine the optimal design points. A comparative study of these surrogates reveals that the KRG model outperforms the RBNN and RSA models in terms of prediction capability. Furthermore, the PEMFC model simulations at the optimal design points clearly demonstrate that performance improvements of around 56–69 mV at 2.0 A cm−2 are achieved with the optimum design values compared to typical PEMFC design conditions.

Dynamics and optimal control of an online game addiction model with considering family education

<abstract><p>The problem of online game addiction among teenagers is becoming more and more serious in many parts of the world. Many of them are addicted to online games due to the lack of family education, which is an important factor that can not be ignored. To explore the optimal strategy for controlling the spread of game addiction, a new dynamic model of teenagers' online game addiction with considering family education is developed. Firstly, we perform a qualitative dynamic analysis of the model. We study the nonnegativity and boundedness of solutions, the basic reproduction number $ R_{0} $, and the existence and stability of equilibria. We then consider a model with control measures of family education, isolation and treatment, and obtain the expression of optimal control. In the numerical simulation, we study the global sensitivity analysis by the combination of Latin Hypercube Sampling (LHS) and partial rank correlation coefficient (PRCC) techniques, and show the relationship between $ R_{0} $ and each parameter. Then the forward backward sweep method with fourth order Runge-Kutta is used to simulate the control strategy in each scenario. Finally, the optimal control strategy is obtained by comparing incremental cost-effectiveness ratio (ICER) and infection averted ratio (IAR) under all strategies. The results show that with sufficient financial resources, adding the family education measures can help more teenagers avoid being addicted to games and control the spread of game addiction more effectively.</p></abstract>

Structural dynamic reliability analysis of super large-scale lattice domes during earthquakes using the stochastic finite element method

For large-scale dome structures subjected to earthquake ground motions, the need for accurate and efficient approaches that account for uncertainties in design, material properties, loads, damping, and manufacturing processes has grown significantly. In uncertainty analyses, the theory of probability, uncertainty quantification, and reliability analysis approaches are commonly used. However, because of computing efficiency and capability, the space scale of dome structures is severely limited. As a result, the computational issues of the complex dynamic reliability that super-large-scale dome structures encounter during earthquakes are challenging. The uncertainties of the dynamic demands of a super-large-scale lattice dome with multiple variables are quantified using a stochastic finite element method (SFEM) with nonlinear time-history analysis developed in this paper, and this method can consider the randomness of variables in more detail for a structure. The dynamic reliability is investigated in depth using the efficient Latin hypercube sampling (LHS) technique to reduce the number of repetitive time-consuming calculations. To improve the efficiency and accuracy of the analysis, an optimization approach based on the genetic algorithm (GA) is used to obtain the probability of structural failure and the reliability index. The results show that these methods are highly efficient for super-large-scale structures. Finally, a sensitivity analysis based on variable randomness is conducted to further evaluate the effects of variables on structural performance using various evaluation methods. This paper provides an efficient solution for structural dynamic reliability problems of super-large-scale structures in the SFEM framework.

Seismic fragility of base-isolated single-storey unreinforced masonry buildings equipped with classical and recycled rubber bearings in Himalayan regions

Unreinforced masonry (URM) buildings are one of the most common constructions all around the world, but they are significantly prone to structural damage in case of earthquake. Due to the high vulnerability of URM buildings to seismic loading, a quantitative risk assessment along with identification of additional protection measures has become necessary. Even though base isolation is a suitable seismic protection option, its conventional cost may prevent a wide implementation in developing countries. In this context, this study presents a comparative seismic fragility analysis of a single-storey URM building before and after base isolation using both classical laminated rubber bearings (LRBs) and recycled-rubber fibre-reinforced elastomeric isolators (RR-FREIs). The main aim of the study is to assess the effectiveness of RR-FREIs as alternative to LRBs for seismic vulnerability reduction of low-resistance URM buildings located in Himalayan regions. Fragility analysis accounted for uncertainties associated with both material properties and earthquake records. The Latin hypercube sampling (LHS) technique was used to reduce the computational work while maintaining acceptable accuracy. The building response was obtained using nonlinear layered shell element models, through nonlinear time history analysis of fixed-base (FB) building models and fast nonlinear analysis of base-isolated (BI) building models. Fragility curves show a significant reduction in the vulnerability of BI buildings in comparison to their FB counterpart, as well as a satisfactory performance of RR-FREIs in comparison with LRBs. A parametric study is also discussed to identify the influence of different intensity measures, fragility analysis method, and variability in isolator properties.

Blind-Kriging based natural frequency modeling of industrial Robot

High-precision assembly conditions tend to necessitate consideration of the vibration modes of industrial robots. The modal characteristics of complex systems such as industrial robots are highly nonlinear. It means that mechanics experiments and finite element methods (FEM) to evaluate such features are usually expensive. Surrogate models combined with simulation-based design are widely used in engineering issues. However, few investigations apply surrogate models to industrial robots' modal analysis. We propose a practical scheme, i.e., the Blind-Kriging (KRG-B) based natural frequency prediction of the industrial robot, utilizing the Latin Hypercube Sampling (LHS) technique. A reliable dataset with 120 samples is generated for surrogate models based on the FEM. Then, the fourteen surrogate models with different optimization algorithms are evaluated to identify the optimal model for the natural frequency. In addition, the accuracy and robustness of the optimal surrogate model are investigated under different training samples. KRG-B model has better robustness (good fitting accuracy for both higher and lower order modes) and higher computational efficiency (1.133 s, the shortest time among all models). The proposed scheme mapping robot's joint angle and the natural frequency offers a valuable basis for further studying dynamic characteristics in industrial robotics.

A Systematic Design Optimization Approach for Multiphysics MEMS Devices Based on Combined Computer Experiments and Gaussian Process Modelling.

This paper presents a systematic and efficient design approach for the two degree-of-freedom (2-DoF) capacitive microelectromechanical systems (MEMS) accelerometer by using combined design and analysis of computer experiments (DACE) and Gaussian process (GP) modelling. Multiple output responses of the MEMS accelerometer including natural frequency, proof mass displacement, pull-in voltage, capacitance change, and Brownian noise equivalent acceleration (BNEA) are optimized simultaneously with respect to the geometric design parameters, environmental conditions, and microfabrication process constraints. The sampling design space is created using DACE based Latin hypercube sampling (LHS) technique and corresponding output responses are obtained using multiphysics coupled field electro–thermal–structural interaction based finite element method (FEM) simulations. The metamodels for the individual output responses are obtained using statistical GP analysis. The developed metamodels not only allowed to analyze the effect of individual design parameters on an output response, but to also study the interaction of the design parameters. An objective function, considering the performance requirements of the MEMS accelerometer, is defined and simultaneous multi-objective optimization of the output responses, with respect to the design parameters, is carried out by using a combined gradient descent algorithm and desirability function approach. The accuracy of the optimization prediction is validated using FEM simulations. The behavioral model of the final optimized MEMS accelerometer design is integrated with the readout electronics in the simulation environment and voltage sensitivity is obtained. The results show that the combined DACE and GP based design methodology can be an efficient technique for the design space exploration and optimization of multiphysics MEMS devices at the design phase of their development cycle.

Optimal design of large-scale solar-aided hydrogen production process via machine learning based optimisation framework

Hydrogen is an important energy carrier in the transportation sector and an essential industrial feedstock for petroleum refineries, methanol, and ammonia production. Renewable energy sources, especially solar energy have been investigated for large-scale hydrogen production in thermochemical, electrochemical, or photochemical manners due to considerable greenhouse gas emissions from the conventional steam reforming of natural gas and oil-based feedstock. The solar steam methane reforming using molten salt (SSMR-MS) is superior due to its unlimited operation hours and lower total annualized cost (TAC). In this work, we extend the existing optimisation framework for optimal design of SSMR-MS in which machine learning techniques are employed to describe the relationship between solar-related cost and molten salt heat duty and establish relationships of TAC, hydrogen production rate and molten salt heat duty with independent input variables in the whole flowsheet based on 18,619 sample points generated using the Latin hypercube sampling technique. A hybrid global optimisation algorithm is adopted to optimise the developed model and generate the optimal design, which is validated in SAM and Aspen Plus V8.8. The computational results demonstrate that a significant reduction in TAC by 14.9 % ~ 15.1 %, and CO2 emissions by 4.4 % ~ 5.2 % can be achieved compared to the existing SSMR-MS. The lowest Levelized cost of Hydrogen Production is 2.4 $ kg−1 which is reduced by around 17.2 % compared to the existing process with levelized cost of 2.9 $ kg−1.

Optimization of a vacuum cleaner fan suction and shaft power using Kriging surrogate model and MIGA

Optimization of vacuum cleaner fan components is a low-cost and time-saving solution to satisfy the increasing requirement for compact energy-efficient cleaners. In this study, surrogate-based optimization technique is used and for the first time it is focused on maximization of Airwatt parameter, which describes the fan suction power, as an objective function (Case II). Besides, the shaft power is minimized (Case I) as another optimization target in order to reduce the power consumption of the vacuum cleaner. 11 geometrical variables of 3 fan components including impeller, diffuser and return channel are selected as the optimization design variables. 80 training points are distributed in the sample space using Advanced Latin Hypercube Sampling (ALHS) technique and the outputs of sample points are calculated by means of CFD simulations. Kriging and RSA surrogate models have been fitted to the outputs of the sample space. Through coupling of constructed Kriging models and Multi-Island Genetic Algorithm (MIGA), the optimal design for each of the optimization cases is presented and evaluated using numerical simulations. A 20.22% reduction in shaft power in Case I and an improvement of 27.73% in Airwatt in Case II have been achieved as the overall results of this study. Despite achieving goals in both optimization cases, a slight decrease in Airwatt in Case I (−6.20%) and a slight increase in shaft power in Case II (+4.82%) are observed relative to primary fan. Furthermore, the Analysis of Variance (ANOVA) determines the importance level of design variables and their 2-way interactions on the objective functions. It was concluded that geometrical parameters related to all of the fan components must be considered simultaneously to conduct a comprehensive optimization. The reasons of enhancement in optimal cases compared with the reference design have been further investigated by analysis of the fan internal flow field. Post-processing of the CFD results demonstrates that the applied geometrical modifications cause a more uniform flow through the flow passages of the optimal fan components.

PAIR COPULA TECHNIQUE OF SEISMIC VULNERABILITY ASSESSMENT OF CABLE-STAYED BRIDGE SYSTEM

As a high-order statically indeterminate structure, long-span cable-stayed bridges usually consist of many components such as main towers, stay cables, main beams, auxiliary piers, and connecting piers. Due to the mutual influence between components under earthquakes, the accurate simulations of the correlations among component seismic responses are critical to the vulnerability assessment of the overall system of a cable-stayed bridge. Pair Copula can simulate the correlation between two components. It is theoretically feasible to use Pair Copula in a hierarchically iterated way to simulate the whole system of a cable-stayed bridge. Therefore, a new seismic vulnerability assessment method of a cable-stayed bridge system is proposed based on Pair Copula iterative model. Based on structural uncertainty parameters and ground motion uncertainty, Latin hypercube sampling technique is used to establish bridge-ground motion probabilistic seismic response analysis samples. The correlations among the seismic responses of components are quantified through nonlinear dynamic time history and correlation analysis. Pair Copula models are fitted with maximum likelihood estimation and optimized based on AIC and BIC criteria. Through the hierarchical iteration of Pair Copulas, the overall model of a cable-stayed bridge is established, and its seismic vulnerability is evaluated. The engineering example shows that the correlation among multiple components can be accurately simulated based on the technology of hierarchical iteration of Pair Copula. With the assumption that the seismic responses of components are completely unrelated, the seismic vulnerability of a whole cable-stayed bridge system will be significantly overestimated.

A Human-Bovine Schistosomiasis Mathematical Model with Treatment and Mollusciciding.

To mitigate the spread of schistosomiasis, a deterministic human-bovine mathematical model of its transmission dynamics accounting for contaminated water reservoirs, including treatment of bovines and humans and mollusciciding is formulated and theoretically analyzed. The disease-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number [Formula: see text], while global stability of the endemic equilibrium is investigated by constructing a suitable Lyapunov function. To support the analytical results, parameter values from published literature are used for numerical simulations and where applicable, uncertainty analysis on the non-dimensional system parameters is performed using the Latin Hypercube Sampling and Partial Rank Correlation Coefficient techniques. Sensitivity analysis to determine the relative importance of model parameters to disease transmission shows that the environment-related parameters namely, [Formula: see text] (snails shedding rate of cercariae), [Formula: see text] (probability that cercariae shed by snails survive), c (fraction of the contaminated environment sprayed by molluscicides) and [Formula: see text] (mortality rate of cercariae) are the most significant to mitigate the spread of schistosomiasis. Mollusciciding, which directly impacts the contaminated environment as a single control strategy is more effective compared to treatment. However, concurrently applying mollusciciding and treatment will yield a better outcome.

Device uncertainty propagation in low‐ductility RC frames retrofitted with BRBs for seismic risk mitigation

AbstractPassive control systems, such as buckling‐restrained braces (BRBs), have emerged as efficient tools for seismic response control of new and existing structures by imparting strength and stiffness to buildings, while providing additional high and stable energy dissipation capacity. Systems equipped with BRBs have been widely investigated in literature; however, only a deterministic description of the BRBs’ properties is typically considered. These properties are provided by the manufacturer and are successively validated by qualification control tests according to code‐based tolerance limits. Therefore, the device properties introduced within the structure could differ from their nominal design estimates, potentially leading to an undesired seismic performance. This study proposes a probabilistic assessment framework to evaluate the influence of BRBs’ uncertainty on the seismic response of a retrofitted RC frame. For the case study, a benchmark three‐story RC moment‐resisting frame is considered where BRBs’ uncertainty is defined compatible to the standardized tolerance limits of devices’ quality control tests. This uncertainty is implemented through a two‐level factorial design strategy and Latin hypercube sampling technique. Cloud analysis and probabilistic seismic demand models are used to develop fragility functions for the bare and retrofitted frame for four damage states while also accounting for the uncertainty in the property of BRBs. Risk estimates are successively evaluated for three case study regions. The results show that, for the considered case study structure, these uncertainties could lead to an increase of fragility up to 21% and a variation in seismic risk estimates up to 56%.

Prediction analysis of deflection in the construction of composite box-girder bridge with corrugated steel webs based on MEC-BP neural networks

This paper presents an efficient computational framework addressing the deflection prediction problem for prestressed concrete (PC) rigid frame bridge with corrugated steel webs (CSWs) erected by the balanced cantilever method. The framework consists of three stages of analysis: (1) obtain the datasets for establishing the prediction model, in which the input samples and output response are generated by the Latin hypercube sampling (LHS) technique and high-fidelity three-dimensional (3D) finite element (FE) model, respectively; (2) based on the datasets, an optimized neural network model that integrates with mind evolutionary computation (MEC) algorithm and back-propagation (BP) algorithm is established, then the model performance is evaluated by the statistical criteria; and (3) the well-trained MEC-BP model is used to predict the deflection for CSWs-PC bridge construction. The applicability and efficiency of the framework are demonstrated with a real CSWs-PC bridge. Results show that the proposed method achieves much better prediction performance than the conventional BP model, and can deliver an efficient prediction tendency for the deflection control of a CSWs-PC bridge during the complex construction process.

Multi-fidelity analysis and uncertainty quantification of beam vibration using co-kriging interpolation method

In this paper, a multi-fidelity surrogate model is created using co-kriging methodology to determine the natural frequencies of beams by combining the fidelities of Euler-Bernoulli (low-fidelity) and Timoshenko (high-fidelity) beam finite element models. This study of free vibration of beams involves uncertainties in material properties. The sampling space for the co-kriging surrogate model is created using an optimal latin hypercube sampling technique. It is shown that the co-kriging surrogate model predicts the natural frequencies with high computational efficiency and accuracy, in the entire design space. The computational efficiency and utility of the multi-fidelity model is demonstrated through its application to a problem of quantifying uncertainties in the natural frequencies of a tapered beam using Monte Carlo Simulation.