Nonlinear Random Response Research Articles

Stochastic Analysis for the Embedded Single-Walled Carbon Nanotube Under Random Vibrations

It is inevitable that carbon nanotubes (CNTs) are subjected to random vibrations due to environmental noise, which may impair their mechanical properties. The nonlinear dynamics associated with single-walled carbon nanotubes (SWCNTs) under random noise has been investigated in this work. First, based on the nonlocal theory and the Euler–Bernoulli beam model with fixed supports at both ends, the governed equations and boundary conditions are established according to Hamilton’s principle. Second, the Galerkin method is used to approximate the differential equations of motion and the stochastic averaging method (SAM) is applied to predict the approximate response of the original system. Subsequently, a detailed parametric study is conducted to analyze the nonlinear random vibration response of the SWCNT with nonlocal effects, and the effectiveness of the proposed method is verified by comparing the analytical results with those from the Runge–Kutta method. Finally, by solving the backward Kolmogorov (BK) equation and the generalized Pontryagin (GP) equation simultaneously, the conditional reliability function (CRF) and mean first-passage time (MFPT) for determining the reliability of SWCNT systems are obtained.

Geometrically Nonlinear Random Response of Stiffened Laminated Plates by Proper-Orthogonal-Decomposition-Based Reduced-Order Modeling

The geometrically nonlinear random vibration response of stiffened laminated plates under acoustic excitation is carried out by using the equivalent linearization (EL) technique combined with the reduced-order model and the finite element method. The nonlinear equations of motion of stiffened laminated plates are obtained based on large deflection finite element formulation. The proper orthogonal decomposition technique is used to select the modal basis, and a reduced-order model of the structures is established. Based on the force error minimization approach, the EL formulation for solving nonlinear random vibration response is derived. Then the EL technique is incorporated into finite element software NASTRAN by the direct matrix abstraction program, and the direct matrix abstraction program also realizes the iterative process of determining the modal equivalent linear stiffness matrix. Then, the statistical dynamic response results obtained by EL and linear method are compared and analyzed. It is verified that the proposed method has high accuracy and efficiency, and it can be applied to the geometric nonlinear random vibration response analysis of stiffened laminated plates. The results show that geometric nonlinearity plays an important role in the vibration response of stiffened laminated plates, especially under high sound pressure loads.

Nonlinear random seismic response analysis of the double-trough aqueduct based on fiber beam element model

As an important lifeline project, large reinforced concrete aqueduct structures sometimes pass through areas with high seismic intensity. Therefore, the aqueduct structure is susceptible to earthquake damage or destruction during the water conveyance process, resulting in interruption of water conveyance and major safety accidents. In fact, because concrete is a multiphase composite material, it has obvious nonlinear and random characteristics. At the same time, seismic excitation is also naturally random due to differences in frequency spectrum and duration. Based on this, this paper uses the concrete elastoplastic random damage constitutive relationship and the random seismic excitation model to carry out the random nonlinear seismic response analysis of the double-trough aqueduct structure. The thin-walled characteristics of the girder section of the aqueduct structure is significantly different from that of the general bridge structure, and there is a certain error in the simulation of the seismic response of the aqueduct with traditional beam elements. Therefore, this paper combines the refined fiber beam element model, compiles the TCL language program suitable for the OpenSEES open analysis platform, and establishes the refined fiber beam finite element model of the aqueduct structure based on the random damage constitutive relationship of the concrete. The study of the response law and failure mechanism of aqueduct structures considering the coupling of concrete nonlinearity and seismic excitation randomness provides an important theoretical basis for the seismic optimization design of large-scale aqueduct structures.

Nonlinear flutter and random response of composite panel embedded in shape memory alloy in thermal-aero-acoustic coupled field

An analytical model for the nonlinear aeroelastic, flutter postponement and random response of the composite panel with shape memory alloy (SMA) in the coupled multi-fields is presented. One-dimensional thermodynamic model is adopted to describe the recovery stress generated by the SMA. Then the recovery stress is introduced into the stress-strain relation, and the von-Karman's large deformation theory, third-order nonlinear piston theory are employed to establish the motion equation of the panel subjected to the combined thermal, aerodynamic and noise loads, in which the noise excitation is simulated by the Gaussian White Noise. The distributions of the stress cycle block are described by using the rain-flow cycle matrix. A few comparisons for the recovery stress of SMA, buckling temperature and flutter pressure are performed to validate the established model. The suppression effect of the SMA on the flutter boundary, post-buckling and response of the system are highlighted. Considering the Thermal-Aero-Acoustic coupled field, the nonlinear dynamic characteristics and post-flutter behavior of the system are analyzed. Many new phenomena, mode jumping, snap-through response and intermittent jumping motion have been observed.

Nonlinear random responses and fatigue prediction of elastically restrained laminated composite panels in thermo-acoustic environments

This paper presents a formulation for predicting the nonlinear random response of the elastically restrained laminated composite panel subjected to thermo-acoustic loads. Based on the laminated plate theory and Von Kármán large deflection and classical thin plate theories, the natural characteristics are obtained via Rayleigh-Ritz method and then the governing equations of the panel subjected to combined acoustic and thermal loads are formulated. The nonlinear partial differential equations of motion are transformed to a set of coupled nonlinear ordinary differential equations in truncated modal coordinates. A numerical example where the acoustic load is considered as the Gaussian band-limited white noise is given to perform the process of obtaining the mode and responses of the panel. Taking the natural frequency obtained from the finite element method as a reference value, the process of obtaining the natural frequencies is validated by comparing the frequency results. Numerical results show that the buckling, snap-through, and nonlinear random vibrations of the thermal-elastic restrained panel can be predicted accurately. Comparing stress PSD distributions with fatigue damage distributions, the first-order mode is proved to be valid for determining the most dangerous area for fatigue life prediction.

Response analysis of non-linear compound random vibration of a high-speed elevator

This study presents a non-linear constitutive equation for the rolling guide-shoes associated with a high-speed elevator system. This was done to accurately evaluate the dynamic behavior of a high-speed elevator car and analyze the action mechanism of random factors during manufacturing and installation on the dispersion of vibration acceleration. Through the combination of the Hertz contact theory and the Bouc-Wen hysteretic model, the non-linear vibration model of the elevator car system was founded. This model was equivalent to a linear system by least squares technique, and then the random parameters and the random excitation were converted by random perturbation method and pseudo excitation method. The acceleration response sensitivities of each random parameters, the means and standard deviations of transverse vibration acceleration responses at the observation point were obtained. In the case, the transverse vibration acceleration responses of the car system were calculated. The elevator car’s vibration instance was analyzed under the different degrees of variation of the random parameters and the random excitation. The results showed that the randomness of geometric parameters has the greatest influence on transverse acceleration. The variability of parameters affects the dispersion degree of the transverse vibration responses while the variability of the excitation mainly affects the amplitude of the vibration response. This study provides an effective method for the analysis of non-linear compound random vibration responses of high-speed elevator car system, and provides a reference for the vibration control design and safety assessment.

Geometrically Nonlinear Random Vibration Responses of Laminated Plates Subjected to Acoustic Excitation

An equivalent linearization technique has been incorporated into NASTRAN to predict the geometrically nonlinear random vibration responses of laminated plates subjected to acoustic excitation. The nonlinear equations of motion of laminated plates are obtained based on the classical laminated plate theory and large deflection finite element formulation. The existing internal variables in the modal frequency response solution sequence of NASTRAN are called directly by the direct matrix abstraction program, and the direct matrix abstraction program implements the iterative procedure for determining the modal equivalent linear stiffness matrix. In this way, the usual manner of computing variables is retained; that is, these variables are determined internally within NASTRAN using information provided in the bulk data file written by PATRAN. The numerical results for a thin, simply supported, aluminum plate are in good agreement with previously reported data, which confirms the proposed method has reasonable precision and high efficiency. Then, the proposed method is applied for analyzing the geometrically nonlinear random responses of thin, simply supported, laminated plates. It is shown that the geometric nonlinearity plays an important role in the vibration response of laminated plates, particularly at high acoustic pressure loading.

Nonlinear Random Response Analyses of Panels Considering Transverse Shear Deformations under Combined Thermal and Acoustic Loads

The panel structures of flight vehicles at supersonic or hypersonic speeds are subjected to combined thermal, acoustic, and aerodynamic loads. Because of the combined thermal and acoustic loads, the panel structure may exhibit nonlinear random vibration responses, such as the snap‐through phenomenon and random vibrations. These unique dynamic behaviors of the panel structure under combined thermal and acoustic loads can result in serious damage or fatigue failure of the panel structures of high‐speed flight vehicles. This study investigates the nonlinear random responses of thin and thick panels under combined thermal and acoustic loads. The panels are modeled based on the first‐order shear deformation theory (FSDT) to account for transverse shear deformations. The von‐Karman nonlinear strain–displacement relationship is used for geometric nonlinearity in the out‐of‐plane direction of the panel. The thermal load distribution is assumed to be constant in the thickness direction of the panel. The random acoustic load is represented as stationary White–Gaussian random pressure with zero mean and uniform magnitude over the panels. Static and dynamic equations are derived using the principle of virtual work and the nonlinear finite element method. A thermal postbuckling analysis is conducted using the Newton–Raphson method, and the dynamic nonlinear equations are solved using the Newmark‐β time integration method. In the present numerical analyses, the snap‐through responses for both the thin and thick panels are investigated, and the results indicate that the loading conditions that cause snap‐through are different for thin and thick panels.

Random Vibrations of Nonlinear Continua Endowed with Fractional Derivative Elements

In this paper, two techniques are proposed for determining the large displacement statistics of random exciting continua endowed with fractional derivative elements: Boundary Element Method (BEM) based Monte Carlo simulation; and Statistical Linearization (SL). The techniques are applied to the problem of nonlinear beam and plate random response determination in the case of colored random external load. The BEM is implemented in conjunction with a Newmark scheme for estimating the system response in the time domain in conjunction with repeated simulations, while SL is used for estimating efficiently and directly, albeit iteratively, the response statistics.

Karhunen-Loéve expansion for random earthquake excitations

This paper develops a trigonometric-basis-function 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.

Nonlinear Response of Carbon-Carbon Laminated Panels to Random Acoustic Excitation under a Gradient Temperature Field

Future missions for aircraft will expose structures to severe thermal and acoustic loads. Efficient analysis methods for predicting nonlinear random response and fatigue life are urgently important. This paper presents a finite element model for analyzing nonlinear random dynamic behaviors of Carbon-Carbon composite panels under temperature gradients and Guassian excitations. The temperature distribution over the plate follows double sine curve. A finite element formulation combined with the equivalent linearization approach and normal mode method is established. The global system of equations is reduced to a set of nonlinear, coupled modal equations. Examples are given for an orthotropic C/C composite laminated panel at various combinations of temperatures gradients and sound pressure levels. Numerical results include RMS values of maximum deflection, time histories of deflection response and stress response, power spectrum densities, probability distribution functions and higher statistical moments. Numerical results verified all three types of panel motions for a simply supported orthotropic laminated plate: small-deflection random vibration about the initial equilibrium positions, snap-through motions between the two buckled positions, and nonlinear random response about new equilibrium positions after post-buckling. Numerical results will provide the important reference basis to aero engine structural integrity design and improving the structure dynamic strength and working life.

Compliant liquid column damper modified by shape memory alloy device for seismic vibration control

Liquid column dampers (LCDs) have long been used for the seismic vibration control of flexible structures. In contrast, tuning LCDs to short-period structures poses difficulty. Various modifications have been proposed on the original LCD configuration for improving its performance in relatively stiff structures. One such system, referred to as a compliant-LCD has been proposed recently by connecting the LCD to the structure with a spring. In this study, an improvement is attempted in compliant LCDs by replacing the linear spring with a spring made of shape memory alloy (SMA). Considering the dissipative, super-elastic, force-deformation hysteresis of SMA triggered by stress-induced micro-structural phase transition, the performance is expected to improve further. The optimum parameters for the SMA-compliant LCD are obtained through design optimization, which is based on a nonlinear random vibration response analysis via stochastic linearization of the force-deformation hysteresis of SMA and dissipation by liquid motion through an orifice. Substantially enhanced performance of the SMA–LCD over a conventional compliant LCD is demonstrated, the consistency of which is further verified under recorded ground motions. The robustness of the improved performance is also validated by parametric study concerning the anticipated variations in system parameters as well as variability in seismic loading.

Reference-free damage detection for truss bridge structures by continuous relative wavelet entropy method

This article proposes a continuous relative wavelet entropy–based reference-free damage detection algorithm for truss bridge structures. Advantages of the proposed method are that (1) there is no need to measure dynamic response of pristine structures, in other words, the method is reference-free; (2) it is suitable for highly nonlinear and nonstationary random response data due to the multiresolution signal analysis feature of the continuous wavelet transform; and (3) it is sensitive to slight damage extents (i.e. 5%–10%) for the tested damage type (i.e. loosening of bolts). In order to demonstrate consistency and sensitivity of the proposed method, multiple experimental tests using a laboratory-size truss structure were mainly conducted for various damage scenarios and progressive damage states. The proposed continuous relative wavelet entropy–based reference-free damage detection algorithm showed reliable damage localization capabilities, and it is proven as an effective method compared to other damage detection methods that are dependent on the measurement signals from pristine structures. Due to the generality of the proposed method, applications to identify other types of damage based on different types of signals can be expected.

Nonlinear vibration of FGM plates under random excitation

In this study, the nonlinear vibration of clamped FGM (Functionally Graded Material) plates under random excitation is presented. The mechanical properties such as the Young’s modulus, the Poisson’s ratio, the thermal expansion coefficients and the density of materials are assumed to vary through thickness direction according to the volume fraction of the constituents using a simple power law distribution. Temperature dependent properties of the constituents of the FGM plate are considered for the random vibration in thermal environment. Temperature changes are assumed to be either uniform throughout plate or vary linearly only in thickness direction. Nonlinearities are considered to be von Karman type due to in-plane stretching. Excitation is considered as a stationary random process with a zero mean and Gaussian distributed. Random pressure is simulated by using Monte Carlo method. Nonlinear random responses are computed for different sound pressure levels, temperatures and material mixture parameters. Thermal-buckling and snap-through type behavior under random excitation is examined.

Alternative modal basis selection procedures for reduced-order nonlinear random response simulation

Three procedures to guide selection of an efficient modal basis in a nonlinear random response analysis are examined. One method is based only on proper orthogonal decomposition, while the other two additionally involve smooth orthogonal decomposition. Acoustic random response problems are employed to assess the performance of the three modal basis selection approaches. A thermally post-buckled beam exhibiting snap-through behavior, a shallowly curved arch in the auto-parametric response regime and a plate structure are used as numerical test articles. The results of a computationally taxing full-order analysis in physical degrees of freedom are taken as the benchmark for comparison with the results from the three reduced-order analyses. For the cases considered, all three methods are shown to produce modal bases resulting in accurate and computationally efficient reduced-order nonlinear simulations.

Thermo-acoustic random response of temperature-dependent functionally graded material panels

A nonlinear finite element model is provided for the nonlinear random response of functionally graded material panels subject to combined thermal and random acoustic loads. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are derived using the first-order shear-deformable plate theory with von Karman geometric nonlinearity and the principle of virtual work. The thermal load is assumed to be steady state constant temperature distribution, and the acoustic excitation is considered to be a stationary white-Gaussian random pressure with zero mean and uniform magnitude over the plate surface. The governing equations are transformed to modal coordinates to reduce the computational efforts. Newton–Raphson iteration method is employed to obtain the dynamic response at each time step of the Newmark implicit scheme for numerical integration. Finally, numerical results are provided to study the effects of volume fraction exponent, temperature rise, and the sound pressure level on the panel response.

Analysis of SMA Hybrid Composite Structures in MSC.Nastran and ABAQUS

A thermoelastic constitutive model for shape memory alloy (SMA) actuators embedded in a composite structure, termed an SMA hybrid composite (SMAHC), was recently implemented in the commercial finite element codes MSC.Nastran and ABAQUS. The model can be easily implemented in any code that has the capability for analysis of laminated composite structures with temperature-dependent material properties. The model is also relatively easy to use, and requires input of only fundamental engineering properties. A brief description of the model is presented, followed by a discussion of implementation and usage in commercial codes. Results are presented from static and dynamic analysis of SMAHC beams of two types: a beam clamped at each end and a cantilever beam. Nonlinear static (post-buckling) and random response analyses are demonstrated for the first specimen. Static deflection (shape) control is demonstrated for the cantilever beam. Approaches for modeling SMAHC material systems with embedded SMA in ribbon and small round wire product forms are demonstrated and compared. The results from commercial codes are compared with those from a research code as validation of commercial implementations; excellent correlation is achieved in all the cases.

Non-linear random response of laminated composite shallow shells using finite element modal method

This paper investigates the large-amplitude multi-mode random response of thin shallow shells with rectangular planform at elevated temperatures using a finite element non-linear modal formulation. A thin laminated composite shallow shell element and the system equations of motion are developed. The system equations in structural node degrees-of-freedom (DOF) are transformed into modal co-ordinates, and the non-linear stiffness matrices are transformed into non-linear modal stiffness matrices. The number of modal equations is much smaller than the number of equations in structural node DOF. A numerical integration is employed to determine the random response. Thermal buckling deflections are obtained to explain the intermittent snap-through phenomenon. The natural frequencies of the infinitesimal vibration about the thermally buckled equilibrium positions (BEPs) are studied, and it is found that there is great difference between the frequencies about the primary (positive) and the secondary (negative) BEPs. All three types of motion: (i) linear random vibration about the primary BEP, (ii) intermittent snap-through between the two BEPs, and (iii) non-linear large-amplitude random vibration over the two BEPs, can be predicted. Copyright © 2006 John Wiley & Sons, Ltd.

Nonlinear random responses of a structure parametrically coupled with liquid sloshing in a cylindrical tank

The nonlinear random interaction of an elastic structure with liquid sloshing dynamics in a cylindrical tank is investigated in the neighborhood of 2:1 internal resonance. Such internal resonance takes place when the natural frequency of the elastic structure is close to twice the natural frequency of the anti-symmetric sloshing mode (1,1). The excitation is generated from the response of a linear shaping filter subjected to a Gaussian white noise. The analytical model involves three sloshing modes (1,1), (0,1) and (2,1). The system response statistics and stability boundaries are numerically estimated using Monte Carlo simulation. The influence of the excitation center frequency, its bandwidth, and the liquid level on the system responses is studied. It is found that there is an irregular energy exchange between the structure and the liquid free surface motion when the center frequency is close to the structure natural frequency. Depending on the excitation power spectral density, the liquid free surface experiences zero motion, uncertain motion (intermittency), partially developed motion, and fully developed random motion. The structure response probability density function is almost Gaussian, while the liquid elevation deviates from normality. The unstable region, where the liquid motion occurs, becomes wider as the excitation intensity increases or as the bandwidth decreases. As the liquid depth or the structure spring stiffness decreases, the region of nonlinear interaction shrinks and is associated with a shift of the peak of the structure mean square response toward the left side of the frequency axis.

Nonlinear Random Response of Panels in an Elevated Thermal-Acoustic Environment

Sonic fatigue is generally considered as being one of the major design areas for the newest generation of high-speed flight vehicles. Efficient analysis methods for predicting nonlinear random response and fatigue life are urgently needed. This paper presents a finite element formulation for the prediction of nonlinear random response of thin isotropic/composite panels subjected simultaneously to high acoustic loads and elevated temperatures. Laminated plate theory and von Karman large displacement relations are used to derive the nonlinear equations of motion for an arbitrarily laminated composite panel subjected to combined acoustic and thermal loads. The nonlinear equations of motion in physical degrees of freedom are transformed to a set of coupled nonlinear equations in truncated modal coordinates, retaining fewer degrees of freedom. Numerical integration is employed to obtain the panel response to simulated Gaussian band-limited white noise. To validate the formulation, results are compared with existing linear and nonlinear solutions to assess the accuracy of nonlinear modal stiffness matrices and simulated random loads. Examples are given for an isotropic panel at various combinations of sound pressure levels and temperatures. Numerical results include rms values of maximum deflection and strain, time histories of deflection and strain response, probability distribution functions, power spectrum densities, and higher statistical moments. Numerical results predicted all three types of panel motions for a thermal buckled simply supported isotropic plate: linear random vibration about one of the buckled equilibrium position, snap-through motions between the two buckled positions, and nonlinear random response over both buckled positions.