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.

DIRECT SPECTURM ANALYSIS OF THE STATE SPACE FOR RANDOM VIBRATION OF MDOF LINEAR DAMPED SYSTEM

This paper proceeds from the general case of the unsymmetric linearized multi-degrees of free- dom(MDOF)systems.By adopting the general complex modal theory of the state space,the response analysis for a sys- tem subjected to random excitation of the same source is carried out using as a kind of direct spectrum analysis method in frequency domain.With the input of power spectral density function given,the explicit expression of the power spectral density function matrix of the output response can be obtained.By taking Fourier inverse transform,the integrated expres- sions of the correlation function matrix and of the spectrum moment matrix are obtained.Comparing with the time domain method,this method enjoys the merit of visualization and avoids the procedure of transformation from the obtained re- sponse correlation function to be solved for the output spectrum utilizing Fourier transform.This paper has extended the application range of the traditional frequency domain analysis method.The mean square values and variety of statistical val- ues can be obtained conveniently.This method and the time domain method are different in approach but equally satisfac- tory in their results.

Random Vibration Analysis of the Electronic Equipment Cabinet

In order to understand the vibration characteristic of system structure of electronic equipment cabinet within the particular vibration frequency,the finite element analysis software-ANSYS is used to simulate the tests of random vibrations of the cabinet system and obtain the isopleths graph of deformation and stress of the cabinet.It can confirm maximum of deformation and stress of the cabinet and position happened.Through more analysis of the frequency response curve,which can confirm harm- ful consequences random vibrations caused and weak link of the cabinet structure.The numerical simulation results are in good a- greement with the experimental results.It shows that this research provides an efficient method for the anti-seismic design and the dynamic optimization design.

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.

Random Vibration Analysis of Electronic Equipment Based on the Substructure Method

To improve the computational efficiency in large-scale problems,the substructure method was proposed in the finite element analysis(FEA)of constructions,electronics devices,auto parts,etc.The effect of random vibration on an electronic chassis was analyzed by the substructure method via the ANSYS FEA.The basic principles of the substructure methods were briefly discussed,and then modeling and FEA,including model analysis and power spectral density(PSD)analysis,of the electronic chassis was conducted with substructure method via the ANSYS.Finally vibration experiment was done to verify the accuracy of FEA with substructure method,which proved that the substructure method,with adequate accuracy,could improve the computational efficiency for large-scale electronic chassis.Based on the stress distribution from FEA and experimental results,a comprehensive structural optimization was proposed.

Adaptive inverse control of random vibration based on the filtered-X LMS algorithm

Random vibration control is aimed at reproducing the power spectral density （PSD） at specified control points. The classical frequency-spectrum equalization algorithm needs to compute the average of the multiple frequency response functions （FRFs）, which lengthens the control loop time in the equalization process. Likewise, the feedback control algorithm has a very slow convergence rate due to the small value of the feedback gain parameter to ensure stability of the system. To overcome these limitations, an adaptive inverse control of random vibrations based on the filtered-X least mean-square （LMS） algorithm is proposed. Furthermore, according to the description and iteration characteristics of random vibration tests in the frequency domain, the frequency domain LMS algorithm is adopted to refine the inverse characteristics of the FRF instead of the traditional time domain LMS algorithm. This inverse characteristic, which is called the impedance function of the system under control, is used to update the drive PSD directly. The test results indicated that in addition to successfully avoiding the instability problem that occurs during the iteration process, the adaptive control strategy minimizes the amount of time needed to obtain a short control loop and achieve equalization.

Improvement for design of beam structures in large vibrating screen considering bending and random vibration

Demand for large vibrating screen is huge in the mineral processing industry. As bending and random vibration are not considered in a traditional design method for beam structures of a large vibrating screen, fatigue damage occurs frequently to affect the screening performance. This work aims to conduct a systematic mechanics analysis of the beam structures and improve the design method. Total motion of a beam structure in screening process can be decomposed into the traditional followed rigid translation(FRT), bending vibration(BV) and axial linear-distributed random rigid translation(ALRRT) excited by the side-plates. When treated as a generalized single-degree-of-freedom(SDOF) elastic system analytically, the BV can be solved by the Rayleigh's method. Stochastic analysis for random process is conducted for the detailed ALRRT calculation. Expressions for the mechanics property, namely, the shearing force and bending-moment with respect to BV and ALRRT, are derived, respectively. Experimental and numerical investigations demonstrate that the largest BV exists at the beam center and can be nearly ignored in comparison with the FRT during a simplified engineering design. With the BV and FRT considered, the mechanics property accords well with the practical situation with the maximum error of 6.33%, which is less than that obtained by traditional method.

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.

Non-random vibration analysis for general viscous damping systems

The authors recently developed a kind of non-probabilistic analysis method, named as‘non-random vibration analysis’, to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to be an effective supplement of the traditional random vibration theory. In this paper, we further extend the nonrandom vibration analysis into the general viscous damping system, and formulate a method to calculate the dynamic response bounds of a viscous damping vibration system under uncertain excitations. Firstly, the unit impulse response matrix of the system is obtained by using a complex mode superposition method. Secondly, an analytic formulation of the system dynamic response middle point and radius under uncertain excitations is derived based on the Duhamel’s integral, and thus the upper and lower response bounds of the system can be obtained. Finally, two numerical examples are investigated to demonstrate the effectiveness of the proposed method.

General Response Surface Reliability Analysis for Fuzzy-Random Uncertainty both in Basic Variables and in State Variables

A general response surface（RS） method is presented for reliability analysis of complex structure/mechanism with fuzzy-random uncertainty both in basic variables and in failure state variables. On the basis of equivalent transformation from fuzzy basic variable to random basic variable, the fuzziness and randomness in the basic variables are considered simultaneously in the presented general RS method. Once the fuzzy basic variables are transformed into the random basic variables, the conventional RS method is employed to establish the general RS for the complex structure/mechanism with implicit limit state equation by finite element numerical simulation. Furthermore, the general failure probability is defined according to the probability formula for fuzzy-random event by taking the fuzziness and randomness in the failure-safety state into consideration, and an appropriate fuzzy operator is adopted to calculate the general failure probability for the complex structure/mechanism with multiple implicit failure modes. Finally, a general reliability analysis of an elastic linkage mechanism is introduced to illustrate the present method.

Reliability analysis for aeroengine turbine disc fatigue life with multiple random variables based on distributed collaborative response surface method

The fatigue life of aeroengine turbine disc presents great dispersion due to the randomness of the basic variables,such as applied load,working temperature,geometrical dimensions and material properties.In order to ameliorate reliability analysis efficiency without loss of reliability,the distributed collaborative response surface method(DCRSM) was proposed,and its basic theories were established in this work.Considering the failure dependency among the failure modes,the distributed response surface was constructed to establish the relationship between the failure mode and the relevant random variables.Then,the failure modes were considered as the random variables of system response to obtain the distributed collaborative response surface model based on structure failure criterion.Finally,the given turbine disc structure was employed to illustrate the feasibility and validity of the presented method.Through the comparison of DCRSM,Monte Carlo method(MCM) and the traditional response surface method(RSM),the results show that the computational precision for DCRSM is more consistent with MCM than RSM,while DCRSM needs far less computing time than MCM and RSM under the same simulation conditions.Thus,DCRSM is demonstrated to be a feasible and valid approach for improving the computational efficiency of reliability analysis for aeroengine turbine disc fatigue life with multiple random variables,and has great potential value for the complicated mechanical structure with multi-component and multi-failure mode.

Vibration reliability analysis for aeroengine compressor blade based on support vector machine response surface method

To ameliorate reliability analysis efficiency for aeroengine components, such as compressor blade, support vector machine response surface method(SRSM) is proposed. SRSM integrates the advantages of support vector machine(SVM) and traditional response surface method(RSM), and utilizes experimental samples to construct a suitable response surface function(RSF) to replace the complicated and abstract finite element model. Moreover, the randomness of material parameters, structural dimension and operating condition are considered during extracting data so that the response surface function is more agreeable to the practical model. The results indicate that based on the same experimental data, SRSM has come closer than RSM reliability to approximating Monte Carlo method(MCM); while SRSM(17.296 s) needs far less running time than MCM(10958 s) and RSM(9840 s). Therefore,under the same simulation conditions, SRSM has the largest analysis efficiency, and can be considered a feasible and valid method to analyze structural reliability.

Vibration Response and Stress Analysis of Planar Elastic Tube Bundle Induced by Fluid Flow

Flow?induced vibration plays a positive role on heat transfer enhancement. Meanwhile, it is also a negative factor for fatigue strength. Satisfying the fatigue strength is the primary prerequisite for heat transfer enhancement. This paper numerically studied the flow?induced vibration of planar elastic tube bundle based on a two?way fluid–structure interaction(FSI) calculation. The numerical calculation involved the unsteady, three?dimensional incompressible governing equations solved with finite volume approach and the dynamic balance equation of planar elastic tube bundle solved with finite element method combined with dynamic mesh scheme. The numerical approach was verified by comparing with the published experimental results. Then the vibration trajectory, deformation and stress contour of planar elastic tube bundle were all studied. Results show that the combined movement of planar elastic tube bundle represents the agitation from inside to outside. The vibration of out?of?plane is the main vibration form with the typically sinusoidal behavior because the magnitude of displacement along the out?of?plane direction is the 100 times than the value of in?plane direction. The dangerous point locates in the innermost tube where the equivalent stress can be utilized to study the multiaxial fatigue of planar elastic tube bundle due to the alternating stress concentration. In the velocity range of 0.2-3 m/s, it is inferred that the vibration amplitude plays a role on the stress response and the stress amplitude is susceptible to the fluid velocity. This research paves a way for studying the fatigue strength of planar elastic tube bundle by flow?induced vibration.

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.

Dynamic Response Analysis of A 5 MW NREL Wind Turbine Blade under Flap-Wise and Edge-Wise Vibrations

A wind turbine is subjected to a regime of varying loads.For example,each rotor revolution causes a complete gravity stress reversal in the low-speed shaft,and there are varying stresses from the out-of-plane loading cycle due to fluctuating wind load.Consequently,wind turbine blade design is governed by fatigue rather than ultimate load considerations.Previous studies have adopted many different beam theories,using different techniques and codes,to model the National Renewable Energy Laboratory(NREL)5MWoffshore wind turbine blade.There are differences,from study to study,in the free vibration results and the dynamic response.The contribution of this study is to apply the code written by the authors to the different beam theories used with the aim of comparing the different beam theories presented in the literature and that developed by the authors.This paper reports the investigation of the effects of deformation parameters on the dynamic characteristics of the NREL 5 MW offshore wind turbine blades predicted by the different beam theories.The investigation of free vibrations is a fundamental step in the analysis of structural dynamics,and this study compares different computational structural methods and investigates their effect on the predicted dynamic response.The modal characteristics of every model examined have been combined with strip theory to determine the dynamic response of the blade.

VIBRATION NUMERICAL ANALYSIS OF COUNTER-ROTATING TURBINE WITH WAKE-FLOW USING FLUID-STRUCTURE INTERACTION METHOD

The design of counter-rotating turbine is one of new techniques to improve the thrust-weight ratio of jet propulsion engines.Numerical analysis of a low pressure（LP）counter-rotating turbine rotor blade is presented by using ANSYS/CFX software.Interaction of aerodynamics and solid mechanics coupling in the computation is applied.In some rating of turbine,stress distribution and vibration characteristics of low pressure turbine（LPT）blade are computed.The wake aerodynamic forces and LPT blade vibration are transformed in frequency domain using fast Fourier transform（FFT）method.The results show that under wake aerodynamic force excitation,the first order modal vibration is more easily aroused and the higher order response cannot be ignored.Moreover,with different temperature fields,the vibration responses of blade are also different.

Vibration analysis of large bulb tubular pump house under pressure pulsations

A 3D finite element model of the Huaiyin third pumping station of the Eastern Route of the South-to-North Water Transfer is described in this paper. Two methods were used in the calculation and vibration analysis of the pumping station in both the time domain and the frequency domain. The pressure pulsation field of the whole flow passage was structured on the basis of pressure pulsations recorded at some locations of the physical model test. Dynamic time-history analysis of the pump house under pressure pulsations was carried out. At the same time, according to spectrum characteristics of the pressure pulsations at measuring points and results of free vibration characteristics analysis of the pump house, the spectrum analysis method of random vibration was used to calculate dynamic responses of the pump house. Results from both methods are consistent, which indicates that they are both reasonable. The results can be used for reference in anti-vibration safety evaluation of the Huaiyin third pumping station.

Seismic spatial effects on dynamic response of long-span bridges in stationary inhomogeneous random fields

The seismic analysis of long-span bridges subjected to multiple ground excitations is an important problem. The conventional response spectrum method neglects the spatial effects of ground motion,and therefore may result in questionable conclusions.The random vibration approach has been regarded as more reliable.Unfortunately,so far, computational difficulties have not yet been satisfactorily resolved.In this paper,an accurate and efficient random vibration approach—pseudo excitation method (PEM),by which the above difficulties are overcome,is presented.It has been successfully used in the three dimensional seismic analysis of a number of long-span bridges with thousands of degrees of freedom and dozens of supports.The numerical results of a typical bridge show that the seismic spatial effects~ particularly the wave passage effect,are sometimes quite important in evaluating the safety of long-span bridges.

Dynamic Characteristics Analysis and Vibration Experiment of Upper-time of Flight Counter (U-ToFC)

The dynamic characteristic parameters of Up-time of Flight Counter （U-ToFC） are important for research of structure optimization and reliability. However, the current simulation is performed based on homogenous material and simplified constraint model, the correct and reliability of results are difficult to be guaranteed. The finite element method based on identification of material parameters is proposed for this investigation on dynamic analysis, simulation and vibration experiment of the U-ToFC. The structure of the U-ToFC is complicated. Its＇ outside is made of aluminum alloy and inside contains electronic components such as capacitors, resistors, inductors, and integrated circuits. The accurate material parameters of model are identified difficultly. Hence, the parameters identification tests are performed to obtain the material parameters of this structure. On the basis of the above parameters, the experiment and FEA are conducted to the U-ToFC. In terms of the flight acceptance test level, and two kinds of joints condition between the U-ToFC and fixture are considered. The natural frequencies, vibration shapes and the response of the power spectral density of the U-ToFC are obtained. The results show simulation which is based on parameters identification is similar with vibration experiment in natural frequencies and responses. The errors are less than 10%. The vibration modes of simulation and experiment are consistent. The paper provides a more reliable computing method for the dynamic characteristic analysis of large complicated structure.

AN EQUIVALENT NONLINEARIZATION METHOD FOR ANALYSINGRESPONSE OF NONLINEAR SYSTEMS TO RANDOM EXCITATIONS

In this paper, a new equivalent nonlinearization method is developed and used in analysing the response of nonlinear systems to Gaussian while noise excitation. Its basic idea and calculation method are expounded. With the help of the presented method, several kinds of usual nonlinear random vibration systems are analyzed. The numerical results show that the mean square responses of the proposed approach are much closer to the exact solutions or Monte Carlo solutions, than that obtained from equivalent linearization method.