Uncertain Natural Frequencies Research Articles

A study of the natural frequencies of the functionally graded curved beams including porosities: Deterministic and stochastic modeling

In this research article, an investigation into the natural frequencies of functionally graded porous curved beams is presented, covering both deterministic and stochastic domains. The stochastic finite element analysis is based on a three-node, C0 continuous element, and higher-order shear deformation theory, utilizing the first-order perturbation technique (FOPT). The study includes a comparative analysis of three porosity distribution models, including a recently proposed sinusoidal model. Additionally, the study examines the influence of various factors, such as porosity distributions, boundary conditions, volume fraction indices, and geometric configurations, on the natural frequencies. The analysis encompasses uncertain natural frequencies in functionally graded curved beams, accounting for material stochasticity through FOPT. Monte Carlo simulation (MCS) is used to validate the accuracy of the FOPT model, increasing the approach’s reliability. Potential future research in this area could investigate the effects of stochastic variations in geometric parameters, including dimensions and boundary conditions.

Uncertain natural characteristics analysis of laminated composite plates considering geometric nonlinearity

In order to achieve the fine and high reliability design of laminated composite plates, the randomness of geometric and material parameters, along with the nonlinear mechanical characteristics is taken into deliberately account. A novel method for solving the uncertain natural characteristics is proposed, in which the uncertain natural frequencies first cover effects of the geometric nonlinearity and are obtained with a multiscale feature extraction and fusion network (MFEFN). Theoretical analytical solutions of the nonlinear natural frequencies and nonlinear frequency ratios are derived and then the statistical properties of the two uncertain natural characteristics are obtained. Influences of the random geometric parameters and material parameters on the two natural characteristics are further studied based on the sensitivity analysis. The accuracy of the present uncertain natural characteristics analyses is demonstrated by comparing linear natural frequencies, nonlinear natural frequencies and their statistical properties numerical results with those provided using other models available in the open literatures. The solving efficiency advantages of the proposed MFEFN-based procedure are clearly shown by comparing with MCS. The present research results provide theoretical basis and technical means for accurate analysis and optimal design of the laminated structures.

Research on NNBoost-based Uncertain Natural Frequency of Composite Laminates for Satellite Structures

Research on NNBoost-based Uncertain Natural Frequency of Composite Laminates for Satellite Structures

Stochastic prediction of natural frequencies of laminated composite beams by using a high-order statistical moment based approach

Statistical Moment (SM) based modelling is a quite straightforward approach in stochastic modelling of uncertain structures. However, the method still has deficiencies including determination of SMs of natural frequencies of vibratory structures, and it has not been tested yet for realistic structures. This study aims such verification by employing high-degree statistical moments in stochastic equations. In this respect, SM approach is applied for two different uncertainty cases. In the first case, uncertain parameters are experimentally reproduced from the batch of laminated composite beams. Then, those uncertainties are fed to SM equations used in finite element model to obtain descriptive statistical quantities (mean, variance, skewness, and kurtosis) of stochastic natural frequencies. Next, Pearson model is utilized to obtain probability density function of the natural frequency by using standardized SMs. Beside this, uncertain fundamental natural frequency of fifty samples of composite beams is measured by experimental modal tests. All SM based predictions and modal test results are also compared with numerical Monte Carlo Simulations. The latter case examines composite beams having nonnormal uncertain thickness. Since the results are in good harmony with each other, it is concluded that high-order SM based approach may effectively be used in uncertainty modelling of realistic structures.

An effective approach for stochastic natural frequency analysis of circular beams with radially varying material inhomogeneities

The natural frequency of functionally graded material beams is closely related to the spatially varying material model. Specially when the model is stochastic due to the variability of observational data. In this regard, this paper presents an effective approach for stochastic natural frequency analysis of circular composite beams with radially varied material properties. The higher-order shear deformation based spectral element model is first setup for structural natural frequency analysis. The random field theory is introduced to represent the radially varied stochastic material properties. To improve numerical efficiency of the brute-force simulation method, a multivariate Gegenbauer polynomial based sparse regression model is proposed to mimic the structural uncertain natural frequency function. Several numerical examples are presented to examine the performance of this proposed approach. Results have illustrated the high efficiency and numerical accuracy of utilizing the sparse regression method for uncertain natural frequency analysis of composite beams with radially varied stochastic material properties.

Natural frequency analysis of functionally graded material beams with axially varying stochastic properties

• Functionally graded material properties of beams are modelled as random fields. • A generalized eigenvalue equation is derived for stochastic free vibration analysis. • Effective surrogate models are developed based on the polynomial chaos expansion. • Several examples are presented to demonstrate the engineering applications. The functionally graded material (FGM) has a potential to replace ordinary ones in engineering reality due to its superior thermal and dynamical characteristics. In this regard, the paper presents an effective approach for uncertain natural frequency analysis of composite beams with axially varying material properties. Rather than simply assuming the material model as a deterministic function, we further extend the FGM property as a random field, which is able to account for spatial variability in laboratory observations and in-field data. Due to the axially varying input uncertainty, natural frequencies of the stochastically FGM (S-FGM) beam become random variables . To this end, the Karhunen–Loève expansion is first introduced to represent the composite material random field as the summation of a finite number of random variables. Then, a generalized eigenvalue function is derived for stochastic natural frequency analysis of the composite beam. Once the mechanistic model is available, the brutal Monte-Carlo simulation (MCS) similar to the design of experiment can be used to estimate statistical characteristics of the uncertain natural frequency response. To alleviate the computational cost of the MCS method, a generalized polynomial chaos expansion model developed based on a rather small number of training samples is used to mimic the true natural frequency function. Case studies have demonstrated the effectiveness of the proposed approach for uncertain natural frequency analysis of functionally graded material beams with axially varying stochastic properties .

Hybrid uncertain natural frequency analysis for structures with random and interval fields

This paper presents a robust non-deterministic free vibration analysis for engineering structures involving hybrid, yet spatially dependent, uncertain system parameters. Distinguished from the conventional hybrid uncertain eigenvalue problem, the concept of interval field is enclosed with random field model such that, both the stochastic and non-stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non-deterministic free vibration analysis. In order to determine the probabilistic characteristics (i.e., means and standard deviations) of the extremities of structural natural frequencies, an extended unified interval stochastic sampling (X-UISS) method is implemented for the purpose of effective hybrid uncertain free vibration analysis. By meticulously blending sharpness-promised interval eigenvalue analysis with stochastic sampling techniques, the stochastic profiles (i.e., probability density functions (PDFs) and the cumulative distribution functions (CDFs)) of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods. The applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures.

Uncertainty propagation in vibrational characteristics of functionally graded carbon nanotube-reinforced composite shell panels

Understanding the effect of mechanical uncertainties can play a significant role in design of the nanocomposites. The uncertain natural frequencies of moderately thick doubly-curved functionally graded composite panels reinforced by carbon nanotube (CNT) are investigated. Specifically, doubly-curved shell panels, including spherical, cylindrical and hyperbolic paraboloid panels are examined. To evaluate uncertainty propagation, uncertainty resources including distribution of the CNT through the thickness as well as the mechanical properties of the CNT and polymer matrix are taken into consideration. To assess the propagated uncertainties in the vibrational characteristics of nanocomposite panels, the interval analysis method is employed while the mechanical properties of nanocomposite panels are predicted using the modified rule of mixture method. Based on the comparison between the results of the present study and those reported in the literature, the accuracy of the results is validated. The sensitivity analysis is performed to distinguish the most prominent uncertain variables. Furthermore, numerical results reveal the influences of various uncertainty resources on the upper and lower bounds of uncertain frequencies and uncertainty propagation percent.

Gaussian process model-based approach for uncertainty quantification of natural frequencies of bridge

Uncertainty in bridge structural parameters will unavoidably lead to uncertainty in natural frequencies of bridge. Uncertainty quantification of natural frequencies of bridge enables to provide more accuarte information of strucutral dynamic characteristics for the bridge design. Mean and variance, two importance order statistics, can be used to characterize the uncertainty of natural frequencies. This paper aims at developing an efficient and reliable appraoch to compute the mean and varaince of natural frequencies of bridge. Bridge is usually greatly complex and large scale, so the traditional Monte Carlo simulation method is infeasible due to the issue of high computational cost. In this paper, Gaussian process model (GPM) is adopted to replace the complex FEM of bridge, aiming at mapping the relationship between the uncertain parameters and natural frequencies. Within GPM framework, the mean and variance can be efficiently obtained. The present GPM-based method is suitable for calculating mean and variance of physical system responses with arbitrary parameter probability distributions. And its feasibility is verified through a benchmark function with analytical mean and variance. Finally, the present GPM-based approach is applied to compute statistics of natural frequencies of Anqing Railway Yangtze River Bridge.

Uncertain natural frequency analysis of composite plates including effect of noise – A polynomial neural network approach

This paper presents the quantification of uncertain natural frequency for laminated composite plates by using a novel surrogate model. A group method of data handling in conjunction to polynomial neural network (PNN) is employed as surrogate for numerical model and is trained by using Latin hypercube sampling. Subsequently the effect of noise on a PNN based uncertainty quantification algorithm is explored in this study. The convergence of the proposed algorithm for stochastic natural frequency analysis of composite plates is verified and validated with original finite element method (FEM). Both individual and combined variation of stochastic input parameters are considered to address the influence on the output of interest. The sample size and computational cost are reduced by employing the present approach compared to direct Monte Carlo simulation (MCS).

LMI-based design of a robust direct adaptive attitude control for a satellite with uncertain parameters

Three axes attitude control of CNES microsatellite is considered. The satellite has uncertain inertia, natural frequency and damping. Satellite dynamics are modelized using descriptor form, with matrices affinely dependent on the uncertain parameters. LMI-based methods using S-variables allow to design a robust adaptive controller with tunable gain adaptation parameters. Illustrated simulations show that the controller stabilizes the satellite with any inertia physically acceptable. Theoretical results can be applied to any descriptor system rational in the uncertainties.

Vibration control of high-rise buildings for wind: a robust passive and active tuned mass damper

Tuned mass dampers (TMDs) have been installed in many high-rise buildings, to improve their resiliency under dynamic loads. However, high-rise buildings may experience natural frequency changes under ambient temperature fluctuations, extreme wind loads and relative humidity variations. This makes the design of a TMD challenging and may lead to a detuned scenario, which can reduce significantly the performance. To alleviate this problem, the current paper presents a proposed approach for the design of a robust and efficient TMD. The approach accounts for the uncertain natural frequency, the optimization objective and the input excitation. The study shows that robust design parameters can be different from the optimal parameters. Nevertheless, predetermined optimal parameters are useful to attain design robustness. A case study of a high-rise building is executed. The TMD designed with the proposed approach showed its robustness and effectiveness in reducing the responses of high-rise buildings under multidirectional wind. The case study represents an engineered design that is instructive. The results show that shear buildings may be controlled with less effort than cantilever buildings. Structural control performance in high-rise buildings may depend on the shape of the building, hence the flow patterns, as well as the wind direction angle. To further increase the performance of the robust TMD in one lateral direction, active control using LQG and fuzzy logic controllers was carried out. The performance of the controllers is remarkable in enhancing the response reduction. In addition, the fuzzy logic controller may be more robust than the LQG controller.

Enhancing Robustness of Floor Vibration Control by Using Asymmetric Tuned Mass Damper

When floor vibration problems occur in existing buildings, TMD (tuned mass damper) can be a viable alternative to resolving the problem. Only when TMD has been exactly tuned to the natural frequency of the floor, it can control the vibration as intended in design. However, TMD gets inefficient in the situation where the natural frequency changes as a result of the uncontrollable variation of the floor mass weight. This physical phenomenon is often called as TMD-off-tuning. This study proposes asymmetric TMD for enhancing the robustness of floor vibration control against uncertain natural frequencies. The proposed TMD features two asymmetric linear springs such that the floor vibrational energy can be dissipated through both the translational and rotational motion. An easy-to-use graphical optimization method was developed in this study. The asymmetric TMD proposed outperformed in vibration control by 28% compared to that of conventional TMD. The robustness of asymmetric TMD of this study was two times higher than that of conventional TMD.

Proposed robust tuned mass damper for response mitigation in buildings exposed to multidirectional wind

The most common device for control of tall buildings under wind loads is the tuned mass damper (TMD). However, during their lifetimes, high-rise and slender buildings may experience natural frequency changes under wind speed, ambient temperatures and relative humidity variations, among other factors, which make the TMD design challenging. In this paper, a proposed approach for the design of robust TMDs is presented and investigated. The approach accounts for structural uncertainties, optimization objectives and input excitation (wind or earthquake). For the use of TMDs in buildings, practical design parameters can be different from the optimum ones. Nevertheless, predetermined optimal parameters for a primary structure with uncertainties are useful to attain design robustness. To illustrate the applicability of the proposed approach, an example of a very slender building with uncertain natural frequencies is presented. The building represents a case study of an engineered design that is instructive. Basically, due to its geometry, the building behaves differently in one lateral direction (cantilever building) than the other (shear building). The proposed approach shows its robustness and effectiveness in reducing the response of tall buildings under multidirectional wind loads. In addition, linear-quadratic Gaussian and fuzzy logic controllers enhanced the performance of the TMD. Copyright © 2012 John Wiley & Sons, Ltd.

Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations

The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and non-Gaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply.

An Analysis Method for Fuzzy Natural Frequency using Response Surface Methodology

In this paper, we propose an efficient procedure for finding the possible ranges of uncertain natural frequencies of a finite element (FE) model with uncertain parameters. Uncertain model parameters are described by input fuzzy numbers and then output fuzzy numbers are calculated by fuzzy arithmetic. The variation range of natural frequency is obtained from the membership function of output fuzzy number. The fuzzy arithmetic used is based on the α-cut concept and interval analysis. To reduce the computational effort for the interval analysis, the global optimization technique using response surface methodology (RSM) is introduced. Furthermore, we propose an approach for finding the feasible ranges of uncertain model parameters when the ranges of natural frequencies are specified. This is an inverse problem. A simple frame model is used to demonstrate the effectiveness of the proposed procedure and approach. The examples show that the membership functions of uncertain natural frequencies are efficiently and accurately estimated by the proposed procedure. In addition, the ranges of uncertain model parameters are properly evaluated by the proposed approach.

Fixed-structure robust controller synthesis via decentralized static output feedback

This paper describes and illustrates a unified methodology for robust, fixed-structure controller synthesis. The approach is based upon direct fixed-structure controller synthesis using a decentralized static output feedback formulation as a general framework for representing a large class of controller structures. Scaled Popov bounds for the real structured singular value are used to account for real parameter uncertainty and provide the means for optimizing a worst-case ℋ︁2 cost bound with respect to the free parameters of the controller. Quasi-Newton optimization algorithms are used to solve the resulting numerical optimization problem. Initial stability multiplier and scaling matrices needed in scaled Popov synthesis are obtained by solving an LMI feasibility problem. Using both centralized and decentralized controller structures, numerical results are obtained for a 16th-order acoustic duct model with uncertain damped natural frequencies and for a two-dimensional beam-spring model with uncertain actuator locations. © 1998 John Wiley & Sons, Ltd.

MMAE/MMAC control for bending with multiple uncertain parameters

A moving-bank multiple model estimator/controller (MMAE/MMAC) based on linear system, quadratic cost, and Gaussian noise (LQG) assumptions is used to quell unwanted vibrations in a simulated large flexible space structure. The structure, known as the Space Integrated Controls Experiment (SPICE), exists at Phillips Laboratory, Kirtland Air Force Base, New Mexico. The structure consists of a large platform and a smaller platform connected by a tripod of flexible legs. The purpose of the control system is to maintain a very precise line-of-sight (LOS) vector through the center of the spacecraft. Kalman filtering, used to estimate the position and velocity of the bending modes of the structure, and LQG control techniques are the primary design tools used in the MMAE/MMAC algorithms. Implementing a parallel bank of filters increases robustness when uncertainties exist in the system model, here specifically allowing adaptation to uncertain and changing undamped natural frequencies of the bending modes of the structure. A moving-bank algorithm is utilized to reduce the computational loading. The MMAE/MMAC design provides a well-suited method of estimating variations in the vector of undamped natural frequencies and quelling vibrations in the structure. The MMAE/MMAC was able to track numerous parameter changes and jumps while providing adequate control for the structure.

Maximum-entropy-type Lyapunov functions for robust stability and performance analysis

We present two Lyapunov functions that ensure the unconditional stability and robust performance of a modal system with uncertain damped natural frequency. Each Lyapunov function involves the sum of two matrices, the first being the solution to the so-called maximum-entropy equation and the second being a constant auxiliary portion. The significant feature of these Lyapunov functions is that the guaranteed robust stability region is independent of the weighting matrix, while the performance bounds are relatively tight compared to alternative approaches. Thus, these Lyapunov functions are less conservative than standard bounds that tend to be highly sensitive to the choice of state space basis.

Fluctuation of Structural Response, Why and How

The uncertainties involved in a structural system are discussed in conjunction with the resultant fluctuation of structural response. The methodology studied is a stochastic finite element method which is based on the mean-centered second order perturbation technique. Small probabilistic variables are assigned to finite elements in order to represent the uncertainties, and the rates of change of the finite element solution with respect to the probabilistic variables are evaluated by means of the perturbation technique. A way is described to estimate the variance of structural response in relation to the variance of the uncertainties. Numerical examples are given in regard to inply stresses of an FRP laminated plate with uncertain stacking parameters and natural frequencies of spatial pipe whose restraint locations are uncertain.