Effect of Nonlinear Geometric Behavior on Random Response of Rotational Shell

Aimed at the large deformation problem of industry cooling tower shell, the mathematical model for the random response of rotational shell under the external random excitation is established by statistic perturbation method. The effect of nonlinear geometric behavior on the response of rotational shell is analyzed.

RANDOM RESPONSES OF A TWO-DEGREE-OF-FREEDOM STRUCTURE WITH STRONGLY NONLINEAR COUPLING AND PARAMETRIC INTERACTION

This paper investigates the random responses of a TDOF structure with strongly nonlinear coupling and parametric vibration. With the nonlinear cou- pling of inertia in the equations of motion of the system being removed by successive elimination, the non-Gaussian moment equation method (NGM) is applied and 69 moment equations are integrated with central cumulative truncation technique. The stochastic central difference-cum-statistical linearization method(SCD-SL) and the digital simulation method(DSM) are also used. A comparison of results by different methods are given and the SCD-SL method is the most efficient method.

INVESTIGATION OF RANDOM RESPONSE OF ROTATIONALSHELL WHEN CONSIDERING GEOMETRICNONLINEAR BEHAVIOUR

An iteration method of statistic linearization (IMSL) is presented. By this method, an equivalent linear term was formed in geometric relation and then an equivalent stiffness matrix for nonlinear term in vibration equation was established. Using the method to solve the statistic linear vibration equations, the effect of geometric nonlinearity on the random response of rotational shell is obtained.

Calculation of Random Response Spectral Moments of Elastic-viscoelastic Combined Systems

In random vibration analysis, the importance of spectral moments of the response stems from their relevance to system performance prediction. Usually,spectral moments are obtained by the frequency domain method. In present paper, the random response spectral moments of elastic-viscoelastic combined systems are calculated by complex modal analysis in the time domain. The analytical form results are obtained for random response spectral moments of an elastic-viscoelastic combined system to a stationary white noise excitation. The method presented is simple and easy to apply. It is hoped that this study would pave a way for the analysis of reliability of elastic-viscoelastic combined systems subjected to random excitations.

Analyzing Nonstationary Random Response of a SDOF System under the Evolutionary Random Excitation by Wavelet Transform

For evolutionary random excitations, a general method of analyzing nonstationary random responses of systems was presented in this paper. Firstly, for the evolutionary random excitation model, the evolutionary power spectrum density function （EPSD） of a random excitation was given by wavelet transform. Based on the EPSD, the nonstationary responses of a SDOF system subjected to evolutionary random excitations were studied. The application and validity of presented method were illustrated by numerical examples. In numerical examples, the recently developed stochastic models for El Centro （1934） and Mexico City （1985） earthquakes which preserve the nonstationary evolutions of amplitude and frequency content of ground accelerations were used as excitations. The nonstationary random mean-square responses of a SDOF system under these two excitations were evaluated and compared with simulated results.

Cross Validation Based Model Averaging for Varying-Coefficient Models with Response Missing at Random

In this paper, a model averaging method is proposed for varying-coefficient models with response missing at random by establishing a weight selection criterion based on cross-validation. Under certain regularity conditions, it is proved that the proposed method is asymptotically optimal in the sense of achieving the minimum squared error.

Random vortex induced vibration response of suspended flexible cable to fluctuating wind

A popular dynamical model for the vortex induced vibration(VIV)of a suspended flexible cable consists of two coupled equations.The first equation is a partial differential equation governing the cable vibration.The second equation is a wake oscillator that models the lift coefficient acting on the cable.The incoming wind acting on the cable is usually assumed as the uniform wind with a constant velocity,which makes the VIV model be a deterministic one.In the real world,however,the wind velocity is randomly fluctuant and makes the VIV of a suspended flexible cable be treated as a random vibration.In the present paper,the deterministic VIV model of a suspended flexible cable is modified to a random one by introducing the fluctuating wind.Using the normal mode approach,the random VIV system is transformed into an infinite-dimensional modal vibration system.Depending on whether a modal frequency is close to the aeolian frequency or not,the corresponding modal vibration is characterized as a resonant vibration or a non-resonant vibration.By applying the stochastic averaging method of quasi Hamiltonian systems,the response of modal vibrations in the case of resonance or non-resonance can be analytically predicted.Then,the random VIV response of the whole cable can be approximately calculated by superimposing the response of the most influential modal vibrations.Some numerical simulation results confirm the obtained analytical results.It is found that the intensity of the resonant modal vibration is much higher than that of the non-resonant modal vibration.Thus,the analytical results of the resonant modal vibration can be used as a rough estimation for the whole response of a cable.

RESPONSE ANALYSIS OF PIEZOELECTRIC SHELLS IN PLANE STRAIN UNDER RANDOM EXCITATIONS

The random response of a piezoelectric thick shell in plane strain state under boundary random excitations is studied and illustrated with a piezoelectric cylindrical shell. The differential equation for electric potential is integrated radially to obtain the electric potential as a function of displacement. The random stress boundary conditions are converted into homogeneous ones by transformation,which yields the electrical and mechanical coupling differential equation for displacement under random excitations. Then this partial differential equation is converted into ordinary differential equations using the Galerkin method and the Legendre polynomials,which represent a random multi-degree-of-freedom system with asymmetric stiffness matrix due to the electrical and mechanical coupling and the transformed boundary conditions. The frequency-response function matrix and response power spectral density matrix of the system are derived based on the theory of random vibration. The mean-square displacement and electric potential of the piezoelectric shell are finally obtained,and the frequency-response characteristics and the electrical and mechanical coupling properties are explored.

Random response of an oscillator with hysteresis damping

The response behaviour of an oscillator with Reid hysteresis damping under stationary Gauss white excitation is studied. By means of numerical simulation conducted on computer, the mean square response, probability density function and power spectral density function for the response are given. These response characteristics are compared with those predicted by equivalent linearization treatment.

Research on the random seismic response analysis for multi- and large-span structures to multi-support excitations

The pseudo excitation method(PEM) has been improved into a more practical form,on which the analytic formulae of seismic response power spectral density(PSD) of simplified large-span structural models have been derived.The analytic formulae and numerical computing results of seismic response PSD have been derived to study the mechanism of multi-support excitation effects,such as the wave-passage effect and incoherence effect,for the seismic response of multiand large-span structures.By using a multi-span truss as an example,the influence of multi-support excitation effects on the seismic response of such structures is studied.

An Application of Direct Ritz Method for Random Seismic Responses of Jacket Platforms

In this paper, the jacket platform is simulated by a non-uniform cantilever beam subjected to axial force. Based on the Hamilton theory, the equation of bending motion is developed and solved by the classical Ritz method combined with the pseudo-excitation method for random responses with non-classical damping. Usually, random responses of this continuous structure are obtained by orthogonality of modes, and some normal modes of the structure are needed, causing inconvenience for the analysis of the non-uniform beam whose normal modes are not easy to be obtained. However, if the pseudo-excitation method is extended to calculate random responses by combining it with the classical Ritz method, the responses of a non-uniform beam, such as auto-PSD function, cross-PSD and higher spectral moments, can be solved directly avoiding the calculation of normal modes. The numerical results show that the present method is effective and useful in aseismic design of platforms.

Parametric characteristic of the random vibration response of nonlinear systems

Volterra series is a powerful mathematical tool for nonlinear system analysis,and there is a wide range of nonlinear engineering systems and structures that can be represented by a Volterra series model.In the present study,the random vibration of nonlinear systems is investigated using Volterra series.Analytical expressions were derived for the calculation of the output power spectral density（PSD） and input-output cross-PSD for nonlinear systems subjected to Gaussian excitation.Based on these expressions,it was revealed that both the output PSD and the input-output crossPSD can be expressed as polynomial functions of the nonlinear characteristic parameters or the input intensity.Numerical studies were carried out to verify the theoretical analysis result and to demonstrate the effectiveness of the derived relationship.The results reached in this study are of significance to the analysis and design of the nonlinear engineering systems and structures which can be represented by a Volterra series model.

Nonlinear Stochastic Response of Rotational Shell Under the Action of Stationary Strong Wind Loads

Based on the theoretical analysis of nonlinear random response of structure, for the engineering practical problem, that is, the large deformation of industry cooling tower shell structure under the action of strong wind loads, the statistic perturbation method is also used to analyze some statistic characteristics of the nonlinear random response of rotational shell with geometric nonlinearity and stationary strong wind load considered. Through computation, some average values of nornal displacements and the nonlinear effect factor of the cooling tower shell are given.

CALCULATING RESPONSE SPECTRAL MOMENTS BY COMPLEX MODAL ANALYSIS

Response spectral moments are useful for system reliability analysis.Usually,spectral mo- ments are calculated by the frequency domain method.Based on the time domain modal analysis of random vibrations,the authors present a new method for calculating response spectral moments through response correlation functions.The method can be applied to both classical and non-classical damping cases and to three kinds of random excitations,i.e.,white noise,band-limited white noise, and filtered white noise.

A method for stochastic seismic response analysis of non-classically damped structures

A method to calculate the stationary random response of a non-classically damped structure is proposed that features clearly-defined physical meaning and simple expression. The method is developed in the frequency domain. The expression of the proposed method consists of three terms, i.e., modal velocity response, modal displacement response, and coupled (between modal velocity and modal displacement response). Numerical results from the parametric study and three example structures reveal that the modal velocity response term and the coupled term are important to structural response estimates only for a dynamic system with a tuned mass damper. In typical cases, the modal displacement term can provide response estimates with satisfactory accuracy by itself, so that the modal velocity term and coupled term may be ignored without loss of accuracy. This is used to simplify the response computation of non-classically damped structures. For the white noise excitation, three modal correlation coefficients in closed form are derived. To consider the modal velocity response term and the coupled term, a simplified approximation based on white noise excitation is developed for the case when the modal velocity response is important to the structural responses. Numerical results show that the approximate expression based on white noise excitation can provide structural responses with satisfactory accuracy.

Karhunen-loéve expansion for random earthquake excitations

This paper develops a trigonometric-basis-fimction based Karhunen-Loeve （KL） expansion for simulating random earthquake excitations with known covariance functions. The methods for determining the number of the KL terms and defining the involved random variables are described in detail. The simplified form of the KL expansion is given, whereby the relationship between the KL expansion and the spectral representation method is investigated and revealed. The KL expansion is of high efficiency for simulating long-term earthquake excitations in the sense that it needs a minimum number of random variables, as compared with the spectral representation method. Numerical examples demonstrate the convergence and accuracy of the KL expansion for simulating two commonly-used random earthquake excitation models and estimating linear and nonlinear random responses to the random excitations.

Seismic Response of Base-Isolated Structures underMulti-component Ground Motion Excitation

An analysis of a base-isolated structure for multi-component random ground motion is presented. The mean square respond of the system is Obtained under different parametric variations. The effectiveness of main parameters and the torsional component during an earthquake is quantified with the help of the response ratio and the root mean square response with and without base isolation. It is observed that the base isolation has considerable influence on the response and the effect of the torsional component is not ignored.

Analysis on Pseudo Excitation of Random Vibration for Structure of Time Flight Counter

Traditional computing method is inefficient for getting key dynamical parameters of complicated structure.Pseudo Excitation Method（PEM）is an effective method for calculation of random vibration.Due to complicated and coupling random vibration in rocket or shuttle launching,the new staging white noise mathematical model is deduced according to the practical launch environment.This deduced model is applied for PEM to calculate the specific structure of Time of Flight Counter（ToFC）.The responses of power spectral density and the relevant dynamic characteristic parameters of ToFC are obtained in terms of the flight acceptance test level.Considering stiffness of fixture structure,the random vibration experiments are conducted in three directions to compare with the revised PEM.The experimental results show the structure can bear the random vibration caused by launch without any damage and key dynamical parameters of ToFC are obtained.The revised PEM is similar with random vibration experiment in dynamical parameters and responses are proved by comparative results.The maximum error is within 9%.The reasons of errors are analyzed to improve reliability of calculation.This research provides an effective method for solutions of computing dynamical characteristic parameters of complicated structure in the process of rocket or shuttle launching.

Modal Correlation Based Analysis on Dynamic Characters of a Commercial Semi-Trailer's Chassis Frames

In order to evaluate two different schemes＇ structural dynamic characters, dynamic response analysis of a commercial truck＇s main chassis frames is carried out. On the basis of correlation study between the tested and calculated modal results, the assembled frames＇ finite element analysis （FEA） models with sufficient precision are built up. Random response analysis in frequency domain is carried out with these FEA models, RMS values of yon Mises and main principle stresses of these two frames are obtained. It shows that the analysis resuits of the distributing tendency and concrete value ranges are coincident very well with test results. And from the results, it could be concluded that frames of scheme A endures relative better loading conditions and should be adopted as the final scheme.

Behavioral Risk Profile of Men Who Have Sex with Men in Beijing, China： Results from a Cross-sectional Survey with Randomized Response Techniques

Background：Human immunodeficiency virus （HIV） is spreading rapidly among men who have sex with men （MSM） in China.Anonymous questionnaires or direct interviews have been frequently used to study their behavior.The aim of the study was to describe the behavioral risk profile of the MSM in Beijing using the randomized response techniques （RRTs）.Methods：A cross-sectional survey of sexual behavior among a sample of MSM was conducted in two HIV counseling and testing clinics in Beijing.The survey was carried out with an anonymous questionnaire containing sensitive questions on sexual behavior.To obtain the honest responses to the sensitive questions,three distinctive RRTs were used in the questionnaire：（1） Additive randomized response model for quantitative questions,（2） randomized response model for multiple choice questions,and （3） Simmons randomized response model for binomial questions.Formulae for the point estimate,variance,and confidence interval （CI） were provided for each specific model.Results：Using RRTs in a sample of 659 participants,the mean age at first homosexual encounter was estimated to be 21.7 years （95% CI：21.2-22.2）,and each had sex with about three （2.9,95% CI：2.4-3.4） male partners on average in the past month.The estimated rate for consistent condom use was 56.4% （95% CI：50.1-62.8%）.In addition,condom was estimated to be used among 80.0% （95% CI：74.1-85.9%） of the population during last anal sex with a male partner.Conclusions：Our study employed RRTs in a survey containing questions on sexual behavior among MSM,and the results showed that RRT might be a useful tool to obtain truthful feedback on sensitive information such as sexual behavior from the respondents,especially in traditional Chinese cultural settings.