Uncertain Dynamic Response Research Articles

Numerical modeling and analysis of multistage gear transmission system in wind turbine gearbox under load uncertainty

A multistage gear transmission system (MGTS) is widely used in wind turbine gearboxes. Its dynamic responses are closely related to stochastic aerodynamic torque that has a wide interval range between cut-in and cut-out wind speeds. However, most of the dynamic models of MGTS lack attention to dynamic responses caused by uncertainties of multiple parameters under varied input torque. In this study, a dynamic modeling method in conjunction with interval uncertain analysis of MGTS caused by multiple load-related parameters due to varied input torques is proposed. A comprehensive dynamic model of MGTS is developed considering time-varying gear mesh transmission error, load-related gear mesh stiffness and bearing stiffness as well as flexible shafts. The reasonability of simulation is validated, and then the effects of multiple load-related parameters on uncertain dynamic responses are discussed in-depth. Results show that both gear mesh stiffness and bearing stiffness are enlarged with increasing input torque to cause uncertain dynamic responses. These uncertainties would propagate along the flexible shaft and be suppressed by system rigidity. The dominant transverse vibration modes of shafts are sensitive to bearing stiffness, thus, uncertain bearing stiffness has remarkable effects on dynamic responses than gear mesh stiffness.

Transient Dynamic Response of Generally Shaped Arches under Interval Uncertainties

This paper endeavors to investigate the characteristics of the transient dynamic response of a generally shaped arch when influenced by uncertain parameters while being subjected to specific external excitation. The equations of motion of the generally shaped arches are derived by the differential quadrature (DQ) method, and the deterministic dynamic responses are calculated using the Newmark-β method. By employing the Chebyshev inclusive function, an interval method based on a non-intrusive polynomial surrogate model is developed, and the uncertain dynamic responses are reckoned by enabling numerical simulations. The results of the proposed interval method are compared with those obtained from the scanning method for validation. The effects of various shapes and rise span ratios on the dynamic responses are investigated through a parametric study. The results suggest that the degree of fluctuation in the uncertain dynamic behavior is influenced by the type of parameter. Additionally, the responses of each shaped arch decrease with the increase in the rise span ratios, and with the same rise span ratio, the deterministic responses and corresponding uncertain responses are also affected by the shape of the arch, and they are considered to be at a minimum when the arch shape is parabolic. This study will enhance understanding of the dynamic properties of arches with uncertainties and provide some basis for the assessment and health monitoring of arch structures.

Dynamic Responses of Train-Symmetry-Bridge System Considering Concrete Creep and the Creep-Induced Track Irregularity

This work considers the influence of concrete creep on track irregularities and establishes the dynamic motion equation of the train-track-bridge coupling system. The track irregularity is obtained by superposition of the initial geometric irregularity and additional geometric irregularity of the steel rail caused by creep. When high-speed railway trains pass through bridges; the vertical acceleration and vertical displacement of continuous beam bridges are related to the train’s operating speed, and the influence of creep camber is relatively small. At the same time, considering the randomness of track irregularities, the dynamic responses of the train track bridge coupling system under the action of random track irregularities are analyzed, and the dynamic responses of trains at different operating speeds are obtained. The deterministic and uncertain dynamic responses of the train track bridge system were compared and analyzed to verify the accuracy of the Karhunen Loéve expansion (KLE)-Point estimate method (PEM) calculation results. The results indicate that the random characteristics of track irregularities have a significant impact on train dynamic response. Based on the random system vibration analysis and considering the safety and comfort indicators of high-speed railway trains, the creep deformation limit of a continuous beam bridge with a length of 48 m + 80 m + 48 m is obtained to be 19 mm. This is the first time that the dynamic responses of train-symmetry-bridge system are calculated by considering concrete creep and the creep-induced track irregularity, which has certain significance for understanding the dynamics of train -bridge system. In addition, the proposed creep threshold is also of great significance to ensure the safety of traveling.

Dynamic response analysis of a shaft-disk-drum rotor system with interval uncertainties

In this work, the dynamic responses of an uncertain flexible shaft-disk-drum rotor under elastic support and connection are investigated, where uncertain parameters are considered as unknown but bounded interval variables. The energy method is used to derive deterministic analytical model for the flexible shaft-disk-drum rotor, which is validated by modal tests. A non-intrusive interval method based on Chebyshev polynomial approximation is introduced to evaluate the uncertain dynamic response of the flexible shaft-disk-drum rotor system. Comparative studies of Monte Carlo simulation, scanning method and the interval method are conducted to illustrate the validity and accuracy of the approach proposed here.

A hybrid algorithm for studying the effects of various uncertainties on the nonlinear dynamic responses of steel storage tanks subjected to seismic base excitation

Uncertainty in the basic load and strength variables of a steel storage tank can significantly affect its structural performance and safety. Additionally, due to the inherent uncertainties affecting the various sources in practical situations, their handling in the nonlinear modeling methodology has nowadays become essential. However, it is still an issue to be investigated. Thus, this paper presents a numerical modeling methodology intended to determine the effects of various uncertainties on the dynamic responses of the liquid-shell systems of aboveground steel storage tanks under seismic base excitations. The uncertainties include mechanical properties of materials, geometry properties, liquid properties, and excitation parameters. The uncertain dynamic response of large steel aboveground storage tanks is investigated by using a hybrid algorithm. Eight tall liquid storage tanks with aspect ratios greater than unity are considered. The considered tanks have a similar radius but different heights. The effects of aspect ratio and liquid level on nonlinear dynamic responses are discussed. The point estimation method and the artificial neural network-based algorithm were used to simulate the sources of uncertainty and extract the statistical properties, including the probability density functions (PDF), and the cumulative distribution functions (CDF). An efficient approach with low computational cost compared to the conventional finite element method is proposed based on a combination of the boundary element method and the semi-analytical finite strip to analyze the flexible steel storage tanks subjected to seismic base excitation. Concentric annular rings are used to model the free surface of the liquid. The nonlinearities corresponding to both material properties and geometry deformations are included. According to uncertainty sources, a sensitivity analysis is also carried out. The results show that lognormal probability distribution is more suitable for predicting nonlinear dynamic responses of steel storage tanks under seismic base excitations.

Health Assessment of Infrastructures Using Limited Uncertain Dynamic Response Information following Earthquakes

ABSTRACT A health assessment technique for infrastructures is presented following earthquakes using uncertainty-filled information. A system identification (SI)-based concept is used to identify the stiffness of structural elements following earthquakes. The excitation information is not required, and responses need not be measured at all dynamic degrees of freedom. To consider uncertainty in representing a structure and dynamic responses, a significantly modified unscented Kalman filter (UKF)-based approach is used. To implement UKF, a two-stage SI approach is proposed. It is verified using three-dimensional truss-frame systems. The procedure is robust and not sensitive to the variances of initial stiffness and measurement noise, the two most uncertainty-prone parameters.

Uncertainty Analysis on Vehicle-Bridge System with Correlative Interval Variables Based on Multidimensional Parallelepiped Model

For vehicle-bridge system, structural uncertainties, especially the interval variables with correlation, have a great influence on dynamic response. Therefore, this paper proposes an effective uncertainty analysis method for vehicle-bridge system based on multidimensional parallelepiped (MP) model, which can reasonably deal with the correlation of interval variables. First, the vehicle-bridge system is simplified as a four degrees-of-freedom mass-spring vehicle model running on a simply supported beam. MP model is adopted to describe the uncertainties of all the interval variables. Second, via affine coordinate system transform, the interval variables with correlation are transformed as the independent variables, which is very convenient for uncertainty analysis. Finally, the uncertain dynamic response is approximated through the first-order Taylor interval expansion, and the upper and lower bounds can be calculated using the dynamic response at midpoints and the partial difference multiplied by interval radius. Because the correlation is sufficiently considered, the uncertainty analysis results on vehicle–bridge interaction system will be much more accurate than the traditional interval analysis method (IAM). Numerical example demonstrates the correctness and effectiveness of the proposed method.

Uncertain dynamic responses of imprecisely defined arbitrary order fractionally damped beam subject to various loads

PurposeRecently, fractional differential equations have been used to model various physical and engineering problems. One may need a reliable and efficient numerical technique for the solution of these types of differential equations, as sometimes it is not easy to get the analytical solution. However, in general, in the existing investigations, involved parameters and variables are defined exactly, whereas in actual practice it may contain uncertainty because of error in observations, maintenance induced error, etc. Therefore, the purpose of this paper is to find the dynamic response of fractionally damped beam approximately under fuzzy and interval uncertainty.Design/methodology/approachHere, a semi analytical approach, variational iteration method (VIM), has been considered for the solution. A newly developed form of fuzzy numbers known as double parametric form has been applied to model the uncertainty involved in the system parameters and variables.FindingsVIM has been successfully implemented along with double parametric form of fuzzy number to find the uncertain dynamic responses of the fractionally damped beam. The advantage of this approach is that the solution can be written in power series or compact form. Also, this method converges rapidly to have the accurate solution. The uncertain responses subject to impulse and step loads have also been computed and the behaviours of the responses are analysed. Applying the double parametric form, it reduces the computational cost without separating the fuzzy equation into coupled differential equations as done in traditional approaches.Originality/valueUncertain dynamic responses of fuzzy fractionally damped beam using the newly developed double parametric form of fuzzy numbers subject to unit step and impulse loads have been obtained. Gaussian fuzzy numbers are used to model the uncertainties. In the methodology using the alpha cut form, corresponding beam equation is first converted to an interval-based fuzzy equation. Next, it has been transformed to crisp form by applying double parametric form of fuzzy numbers. Finally, VIM has been applied to solve the same for the general fuzzy responses. Various numerical examples have been taken in to consideration.

Dynamic response analysis of an overhung rotor with interval uncertainties

Rotor system is an important part in rotating machineries. As a matter of fact, rotors are subject to uncertainties inevitably due to occasions such as assembling errors, material properties dispersion and variable working conditions. In order to obtain more reasonable evaluations of dynamic response of rotor systems, uncertainties are recommended to be taken into consideration. In this paper, the dynamic responses of an elastically supported uncertain overhung rotor are studied in which uncertain parameters are treated as unknown-but-bounded interval variables. The finite element method is used to derive the deterministic analysis model. A non-intrusive interval method based on Chebyshev polynomial approximation is proposed to evaluate the uncertain dynamic response of the rotor system. Comparative study of the interval method, the scanning method and the Monte Carlo simulation is carried out to illustrate the effectiveness and accuracy. Deflection upper bounds and lower bounds of the disc are obtained with respect to rotating speed in several typical uncertain cases. Results show that the uncertainties have significant effects on the dynamic behaviours of the rotor system and multiple source small uncertainties can lead to large fluctuations in the dynamic responses.

Inverse system method for dynamic loads identification via noisy measured dynamic responses

Purpose – The purpose of this paper is to focus on the influences of the uncertain dynamic responses on the reconstruction of loads. Design/methodology/approach – Based on the assumption of unknown-but-bounded (UBB) noise, a time-domain approach to estimate the uncertain time-dependent external loads is presented by combining the inverse system method in modern control theory and interval analysis in interval mathematics. Inspired by the concept of set membership identification in control theory, an interval analysis model of external loads time history, which is indeed a region or feasible set containing all possible loads being consistent with the bounded structural acceleration responses is established and further solved by two interval algorithms. Findings – Unlike traditional loads identification methods which only give a point estimation, an interval estimation of external loads time history, which is a region containing all the possible loads being consistent with the uncertain structural responses, is determined. The correlation characteristics among the responses of acceleration, velocity, and displacement are also discussed in consideration of the UBB uncertainty. Originality/value – For one hand, the solution of the inverse problem in original system is transformed to the solution of the direct problem in inverse system; for another, the authors deal with the uncertainty by use of interval analysis method, and the identified interval process, which contains any possible external loads time history being consistent with the bounded structural responses can be approximately obtained.

A modified hybrid uncertain analysis method for dynamic response field of the LSOAAC with random and interval parameters

For the prediction of dynamic response field of the luffing system of an automobile crane (LSOAAC) with random and interval parameters, a hybrid uncertain model is introduced. In the hybrid uncertain model, the parameters with certain probability distribution are modeled as random variables, whereas, the parameters with lower and upper bounds are modeled as interval variables instead of given precise values. Based on the hybrid uncertain model, the hybrid uncertain dynamic response equilibrium equation, in which different random and interval parameters are simultaneously included in input and output terms, is constructed. Then a modified hybrid uncertain analysis method (MHUAM) is proposed. In the MHUAM, based on random interval perturbation method, the first-order Taylor series expansion and the first-order Neumann series, the dynamic response expression of the LSOAAC is developed. Moreover, the mathematical characteristics of extrema of bounds of dynamic response are determined by random interval moment method and monotonic analysis technique. Compared with the hybrid Monte Carlo method (HMCM) and interval perturbation method (IPM), numerical results show the feasibility and efficiency of the MHUAM for solving the hybrid LSOAAC problems. The effects of different uncertain models and parameters on the LSOAAC response field are also investigated deeply, and numerical results indicate that the impact made by the randomness in the thrust of the luffing cylinder F is larger than that made by the gravity of the weight in suspension Q. In addition, the impact made by the uncertainty in the displacement between the lower end of the lifting arm and the luffing cylinder a is larger than that made by the length of the lifting arm L.

Uncertain Dynamic Responses of Fuzzy Arbitrary-Order Damped Beam

Uncertain dynamic responses of fuzzy fractionally damped beams have been studied using the newly developed double parametric form of fuzzy numbers subject to unit step and impulse loads. Uncertainties are assumed to be present in the initial conditions, and these are modeled through triangular convex normalized fuzzy sets. Using the alpha-cut form, the corresponding beam equation is first converted to an interval-based fuzzy equation. Next, it has been transformed to crisp form by applying a double-parametric form of fuzzy numbers. Finally, homotopy perturbation method (HPM) has been used to solve the same for obtaining the general fuzzy responses. Various numerical examples are taken into consideration. The results are plotted.

Uncertain dynamic responses of fuzzy fractionally damped spring-mass system

Uncertain dynamic responses of fuzzy fractionally damped spring-mass system

Collocation-based Stochastic Modeling of Uncertain Geometric Mistuning in Bladed Rotor

In this paper, a collocation–based stochastic model is proposed to analysis mistuned bladed–rotors with geometrical parameter uncertainties. The uncertain parameters are approximated by the Karhunen-Loe‘ve expansion with predefined correlation functions. The generalized polynomial chaos (gPC) expansion possessing random orthogonal basis is served as uncertain dynamic responses. A set of collocation points are generated from the roots of higher order random polynomial basis and the whole FEM model of the rotor is executed on each point to estimate the dynamic responses. These estimations then are used to calculate the coefficients of the gPC expansions. The most attractive feature of the method is that only multiple solutions of the original deterministic FEM model of the rotor are required. A numerical case study is presented in which the geometrical parameters of a blisk are considered as uncertain parameters with spatial variation. The probability distributions of the random natural frequencies are evaluated for rotational and non-rotational blisk. The results show high accuracy compared to the Monte Carlo simulations with 500 realizations. The impact of the parameter uncertainty on the Campbell diagram during the start up and start down are presented.

Two non-probabilistic set-theoretical models for dynamic response and buckling failure measures of bars with unknown-but-bounded initial imperfections

This paper is concerned with the problem of comparison of two non-probabilistic set-theoretical models for dynamic response and buckling failure measures of bars with unknown-but-bounded initial imperfections. Two kinds of non-probabilistic set-theoretical models are convex models and interval analysis models. In convex models and interval analysis models, the uncertain quantities are considered to be unknown except that they belong to given sets in an appropriate vector space. In this case, all information about the dynamic response and buckling failure measures of bars is provided by the set of dynamic responses and buckling failure measures consistent with the constraints on the uncertain quantities. The dynamic response estimate is actually a set in appropriate response space rather than a single vector. The set estimate is the smallest calculable set which contains the uncertain dynamic response, but it is usually impractical to calculate this set. Two set estimate methods are developed which can calculate the time varying box or hyperrectangle, i.e. interval vector in the response space that contains the true dynamic response. Comparison between convex models and interval analysis models in mathematical proofs and numerical calculations shows that, under the condition of the outer enclosed ellipsoid from a hyperrectangle or an interval vector, the set dynamic response predicted by interval analysis models is smaller than that yielded by convex models; under the condition of the outer enclosed hyperrectangle or an interval vector from an ellipsoid, the dynamic response set calculated by convex models is smaller than that obtained by interval analysis models.

Uncertain dynamic response of a deterministic elastic–plastic beam

The counterintuitive phenomenon of elastic–plastic beam dynamics was demonstrated by Symonds and Yu (ASME J Appl Mech 1985;52:517). An analytical model has been developed to explain this phenomenon from a deterministic viewpoint. However, experimental evidence in (Int J Impact Eng 1991;11(3):341; Int J Impact Eng 1991;11(4):445) showed that the response of this deterministic system is uncertain, which is studied qualitatively in the present paper based on parametric sensitive characteristics of the deterministic system and parametric uncertainty of the studied system. FEM and Monte Carlo method are applied to study this phenomenon.