Random Natural Frequencies Research Articles

Uncertain Dynamic Characteristic Analysis for Structures with Spatially Dependent Random System Parameters.

This work presents a robust non-deterministic free vibration analysis for engineering structures with random field parameters in the frame of stochastic finite element method. For this, considering the randomness and spatial correlation of structural physical parameters, a parameter setting model based on random field theory is proposed to represent the random uncertainty of parameters, and the stochastic dynamic characteristics of different structural systems are then analyzed by incorporating the presented parameter setting model with finite element method. First, Gauss random field theory is used to describe the uncertainty of structural material parameters, the random parameters are then characterized as the standard deviation and correlation length of the random field, and the random field parameters are then discretized with the Karhunen-Loeve expansion method. Moreover, based on the discretized random parameters and finite element method, structural dynamic characteristics analysis is addressed, and the probability distribution density function of the random natural frequency is estimated based on multi-dimensional kernel density estimation method. Finally, the random field parameters of the structures are quantified by using the maximum likelihood estimation method to verify the effectiveness of the proposed method and the applicability of the constructed model. The results indicate that (1) for the perspective of maximum likelihood estimation, the parameter setting at the maximum value point is highly similar to the input parameters; (2) the random field considering more parameters reflects a more realistic structure.

Synchronization in disordered oscillator lattices: Nonequilibrium phase transition for driven-dissipative bosons

We show that a lattice of phase oscillators with random natural frequencies, described by a generalization of the nearest-neighbor Kuramoto model with an additional cosine coupling term, undergoes a phase transition from a desynchronized to a synchronized state. This model may be derived from the complex Ginzburg-Landau equations describing a disordered lattice of driven-dissipative Bose-Einstein condensates of exciton polaritons. We derive phase diagrams that classify the desynchronized and synchronized states that exist in both one and two dimensions. This is achieved by outlining the connection of the oscillator model to the quantum description of localization of a particle in a random potential through a mapping to a modified Kardar-Parisi-Zhang equation. Our results indicate that long-range order in polariton condensates, and other systems of coupled oscillators, is not destroyed by randomness in their natural frequencies.

Stochastic eigenfrequency and buckling analyses of plates subjected to random temperature distributions

Many studies previously reported in the literature have demonstrated, both theoretically and experimentally, the influence of thermally-induced stresses on the static and dynamic behavior of structures, due to the so-called stress-stiffening effect. In most cases of practical interest, temperature variations associated to environmental and operational conditions are governed by rather complex combinations of conduction, convection and radiation mechanisms. As a result, the temperature values at different points of a structure are very difficult to control and can rationally be considered as random quantities. In this context, the present paper addresses the stochastic modeling and characterization of the influence of thermal stresses on the natural frequencies of thin rectangular plates, assuming space-dependent temperature fluctuations modeled as stationary two-dimensional Gaussian random fields. For this purpose, based on the hypotheses of the classical Kirchhoff plate theory, a Rayleigh-Ritz-based dynamic model is first derived for the bending vibrations of plates, accounting for the presence of thermal stresses. This model is combined with the Karhunen-Loève expansion (KL), which is used to discretize the temperature random field, after which the statistics of the random natural frequencies are estimated by Monte Carlo sampling. Numerical simulations are performed for plates under free boundary conditions. Simulation results, which encompass sampling-based statistics for the thermal stresses and the first six natural frequencies of the plate, are presented and discussed. In addition, since thermal stresses can induce buckling, reliability has also been estimated considering this type of failure. Results enable to conclude that space-dependent temperature uncertainty can be significant upon the vibration and buckling behavior of plates, which justifies its consideration.

Optimal global synchronization of partially forced Kuramoto oscillators.

We consider the problem of global synchronization in a large random network of Kuramoto oscillators where some of them are subject to an external periodically driven force. We explore a recently proposed dimensional reduction approach and introduce an effective two-dimensional description for the problem. From the dimensionally reduced model, we obtain analytical predictions for some critical parameters necessary for the onset of a globally synchronized state in the system. Moreover, the low dimensional model also allows us to introduce an optimization scheme for the problem. Our main conclusion, which has been corroborated by exhaustive numerical simulations, is that for a given large random network of Kuramoto oscillators, with random natural frequencies ωi, such that a fraction of them is subject to an external periodic force with frequency Ω, the best global synchronization properties correspond to the case where the fraction of the forced oscillators is chosen to be those ones such that |ωi-Ω| is maximal. Our results might shed some light on the structure and evolution of natural systems for which the presence or the absence of global synchronization is a desired property. Some properties of the optimal forced networks and their relation to recent results in the literature are also discussed.

Investigation of Soil Layers Stochasticity Effects on the Spatially Varying Seismic Response Spectra

The present study investigates the contribution of the stochasticity of a layered soil deposit on the seismic ground motion spatial variability via the random vibration theory. The seismic response evaluation is based on the 1-D shear wave propagation through random media. The soil layers stochasticity is introduced as spatial random fields of mass density and shear modulus of the soil deposit. The stochastic variation of these two parameters results in random natural frequency and damping ratio of an equivalent single degree of freedom oscillator modelling the soil deposit. Under the assumption of stationary Gaussian random process of the incident motion, the ground surface response spectra are obtained from the cross power spectral density of the total ground surface motion. The efficiency and the validity of the present approach are conducted via some applications. Numerical results showed that, the shape of the global coherency is controlled by that of the incident motion. The stochastic aspect of the soil layers causes disturbances in the surface motion and affects the coherency and the variability of the ground surface response spectra.

Time-variant random interval natural frequency analysis of structures

This paper presents a new robust method namely, unified interval Chebyshev-based random perturbation method, to tackle hybrid random interval structural natural frequency problem. In the proposed approach, random perturbation method is implemented to furnish the statistical features (i.e., mean and standard deviation) and Chebyshev surrogate model strategy is incorporated to formulate the statistical information of natural frequency with regards to the interval inputs. The comprehensive analysis framework combines the superiority of both methods in a way that computational cost is dramatically reduced. This presented method is thus capable of investigating the day-to-day based time-variant natural frequency of structures accurately and efficiently under concrete intrinsic creep effect with probabilistic and interval uncertain variables. The extreme bounds of the mean and standard deviation of natural frequency are captured through the embedded optimization strategy within the analysis procedure. Three particularly motivated numerical examples with progressive relationship in perspective of both structure type and uncertainty variables are demonstrated to justify the computational applicability, accuracy and efficiency of the proposed method.

Comparing the locking threshold for rings and chains of oscillators.

We present a case study of how topology can affect synchronization. Specifically, we consider arrays of phase oscillators coupled in a ring or a chain topology. Each ring is perfectly matched to a chain with the same initial conditions and the same random natural frequencies. The only difference is their boundary conditions: periodic for a ring and open for a chain. For both topologies, stable phase-locked states exist if and only if the spread or "width" of the natural frequencies is smaller than a critical value called the locking threshold (which depends on the boundary conditions and the particular realization of the frequencies). The central question is whether a ring synchronizes more readily than a chain. We show that it usually does, but not always. Rigorous bounds are derived for the ratio between the locking thresholds of a ring and its matched chain, for a variant of the Kuramoto model that also includes a wider family of models.

Rotational and ply-level uncertainty in response of composite shallow conical shells

This paper presents the quantification of rotational and ply level uncertainty of random natural frequency for laminated composite conical shells by using surrogate modeling approach. The stochastic eigenvalue problem is solved by using QR iteration algorithm. Sensitivity analysis is carried out to address the influence of different input parameters on the output natural frequencies. The sampling size and computational cost is reduced by employing the present approach compared to direct Monte Carlo simulation. The stochastic mode shapes are also depicted for a typical laminate configuration. Statistical analysis is presented to illustrate the results and its performance.

Stochastic Response of Laminated Composite Shell Panel in Hygrothermal Environment

In this article, the effect of random system properties is accounted to estimate the free vibration of multilayered composite shell panel in hygrothermal environment. The majority of previous investigations assumed that the material properties are independent of temperature and moisture. Establishing margin on material properties for design, analysis, and conceptualizing the material is highly difficult and uncertain. To perturb that, a first-order perturbation technique is adopted to obtain the response statistics of the structure by obtaining the mean and variance of random natural frequencies. The higher-order shear deformation theory with 9 degrees of freedom per node with the von-Karman sense of nonlinear kinematics is employed for generating basic formulation. The analysis is carried out by using quadratic C0 eight-noded isoparametric element. The governing equation for free vibration of laminated composite panel is derived using variational principle, which is a generalization of the principle of virtual displacement. The solution methodology is validated with published results. Mean and variance are obtained for different cross-ply of spherical and cylindrical laminates with different stacking sequence and boundary conditions.

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.

Stochastic Response of Laminated Composite Shell Panel in Hygrothermal Environment

In this article, the effect of random system properties is accounted to estimate the free vibration of multilayered composite shell panel in hygrothermal environment, instead the majority of previous investigations assumed that the material properties are independent of temperature and moisture. Establishing margin on material properties for design, analysis, and conceptualizing the material is highly difficult and uncertain. To perturb that, a first-order perturbation technique is adopted to obtain the response statistics of the structure by obtaining the mean and variance of the random natural frequencies. The higher-order shear deformation theory with nine degrees of freedom (DOFs) per node with von-Karman sense of nonlinear kinematics is employed for generating basic formulation. The analysis is carried out by using quadratic C°eight-noded isoparametric element. The governing equation for free vibration of laminated composite panel is derived using variational principle, which is generalization of the principle of virtual displacement. The solution methodology is validated with published results. Mean and variance is obtained for different cross-ply of spherical and cylindrical laminates with different stacking sequence and boundary conditions.

Coherence Stability and Effect of Random Natural Frequencies in Populations of Coupled Oscillators

We consider the (noisy) Kuramoto model, that is a population of \(N\) oscillators, or rotators, with mean-field interaction. Each oscillator has its own randomly chosen natural frequency (quenched disorder) and it is stirred by Brownian motion. In the limit \(N \rightarrow \infty \) this model is accurately described by a (deterministic) Fokker–Planck equation. We study this equation and obtain quantitatively sharp results in the limit of weak disorder. We show that, in general, even when the natural frequencies have zero mean the oscillators synchronize (for sufficiently strong interaction) around a common rotating phase, whose frequency is sharply estimated. We also establish the stability properties of these solutions (in fact, limit cycles). These results are obtained by identifying the stable hyperbolic manifold of stationary solutions of an associated non disordered model and by exploiting the robustness of hyperbolic structures under suitable perturbations. When the disorder distribution is symmetric the speed vanishes and there is a one parameter family of stationary solutions, as pointed out by Sakaguchi (Prog Theor Phys 79:39–46, 1988): in this case we provide more precise stability estimates. The methods we use apply beyond the Kuramoto model and we develop here the case of active rotator models, that is the case in which the dynamics of each rotator in absence of interaction and noise is not simply a rotation.

The asymptotic behavior of the order parameter for the infinite-N Kuramoto model

The Kuramoto model, first proposed in 1975, consists of a population of sinusoidally coupled oscillators with random natural frequencies. It has served as an idealized model for coupled oscillator systems in physics, chemistry, and biology. This paper addresses a long-standing problem about the infinite-N Kuramoto model, which is to describe the asymptotic behavior of the order parameter for this system. For coupling below a critical level, Kuramoto predicted that the order parameter would decay to 0. We use Fourier transform methods to prove that for general initial conditions, this decay is not exponential; in fact, exponential decay to 0 can only occur if the initial condition satisfies a fairly strong regularity condition that we describe. Our theorem is a partial converse to the recent results of Ott and Antonsen, who proved that for a special class of initial conditions, the order parameter does converge exponentially to its limiting value. Consequently, our result shows that the Ott-Antonsen ansatz does not completely capture all the possible asymptotic behavior in the full Kuramoto system.

Geometrically Nonlinear Free Vibration of Laminated Composite Plate Embedded With Piezoelectric Layers Having Uncertain Material Properties

In this paper, the nonlinear free vibration stochastic characteristic of a smart laminated composite plate having random system properties is presented. The transverse shear effects have been included in the system equation in the frame work of higher order shear deformation theory. The analysis uses the Green-Lagrange nonlinear strain displacement relationship to model geometric nonlinearity. The direct iteration approach is used to handle deterministic geometric nonlinearity, and the perturbation approach is employed to handle the randomness in the system properties. Mean and variance of the random natural frequencies have been obtained by employing a C0 isoparametric nonlinear finite element model. Comparisons with the published results show the accuracy of the proposed procedure. A few results covering various features have been presented for a laminated composite plate with different boundary conditions.

Reliable dynamic analysis of an uncertain compound bridge under traffic loads

In this paper the dynamic response of compound bridges under moving loads is analyzed. The main structure is modeled as a two-layer beam consisting of a steel girder connected elastically to the concrete deck. The governing sixth-order partial differential equation of motion of the homogenized beam is extended to include uncertainty in the mechanical property of the interface. The slip modulus is considered to be a random variable, whose probability density function is considered to be zero outside of a given interval, that is, it is a member of a fuzzy set with bounded uncertainty. The analysis of the illustrative problem is based on the interval representation with a triangular membership function of the slip modulus assigned. Membership functions of the random natural frequencies are determined. Both, fuzzy peak deflection and acceleration are derived for a single-span compound bridge subjected to a single moving load as well as to a series of repetitive moving loads traveling with constant speed.

MAXIMUM RESPONSE ESTIMATION BY STOCHASTIC DYNAMIC RESPONSE ANALYSIS CONSIDERING UNCERTAINTIES OF NATURAL CIRCULAR FREQUENCY AND DAMPING FACTOR

This paper aims at estimating the maximum dynamic response of the structure with time-invariant random natural circular frequency and damping factor subjected to random excitations. In the stochastic response analyses, the λ-PDF probability density function and the Gegengauer polynominal approximation is used to deal with the bounded parameter uncertainties. In order to estimate maximum responses, the 4th moment and the beta distribution model are used. Two types of probability density function for parameter uncertainties are studied, and evaluation method of high-order moment is introduced in each case. A series of numerical analyses show that the proposed method estimates the maximum response quite accurately. The method is applied to calculating response spectra considering parameter uncertainties to investigate the required accuracy of the damping factor in the seismic design.

Synchronization transition of limit-cycle system with homogeneous phase shifts

The behaviors of coupled oscillators, each of which has periodic motion with random natural frequency in the absence of coupling, are investigated when phase shifts are considered. In the system of coupled oscillators, phase shifts are the same between different oscillators. Synchronization and synchronization transition are revealed with different phase shifts. Phase shifts play an important role for this kind of system. When the phase shift α < 0.5π, the synchronization state can be attained by increasing the coupling, and the system cannot reach the synchronization state while α α 0.5π. A clear scaling between complete synchronization critical coupling strength Kpc and α - 0.5π is found.

Second-order statistics of large-amplitude free vibration of a random composite plate

This article addresses the non-linear free vibration stochastic characteristic of a laminated composite plate having random material properties. The transverse shear effects have been included in the system equation in the frame work of higher-order shear deformation theory. The analysis uses Green—Lagrange non-linear strain—displacement relationship to model geometric non-linearity. Direct iteration approach is used to handle deterministic system non-linearity and perturbation approach is presented to handle the randomness in the system properties. Mean and variance of the random natural frequencies have been obtained by employing a C0 isoparametric non-linear finite-element model. Comparisons with the published results show the accuracy of the proposed procedure. A few results covering various features have been presented for a laminated composite plate with different boundary conditions.

Higher Period Stochastic Bifurcation of Nonlinear Airfoil Fluid‐Structure Interaction

The higher period stochastic bifurcation of a nonlinear airfoil fluid‐structure interaction system is analyzed using an efficient and robust uncertainty quantification method for unsteady problems. The computationally efficient numerical approach achieves a constant error with a constant number of samples in time. The robustness of the method is assured by the extrema diminishing concept in probability space. The numerical results demonstrate that the system is even more sensitive to randomness at the higher period bifurcation than in the first bifurcation point. In this isolated point in parameter space the clear hierarchy of increasing importance of the random nonlinearity parameter, initial condition, and natural frequency ratio, respectively, even suddenly reverses. Disregarding seemingly less important random parameters based on a preliminary analysis can, therefore, be an unreliable approach for reducing the number of relevant random input parameters.

The Comparison of the Vibration Response Variance of a Plate Subjected to Uniformly Distributed Excitation and Diffused Sound Field Excitation

The response of a structure subjected to diffused sound field excitation at high frequencies can be predicted by Statistical Energy Analysis (SEA), in which the mean energy level of the structure can be obtained. In addition to the mean energy level, the variance of the response is necessary in order to predict the tolerance limit of the response for a conservative design purpose. The theoretical variance prediction model for the case of the diffuse sound field excitation has been developed based on the assumption that the diffused sound field excitation can be regarded as a uniformly distributed point force excitation (rain-on-the-roof excitation). This model yields the ensemble variance of the vibration response taken over the structures with random parameters. In this paper, the group of plate structures with random natural frequencies is generated by Gaussian random Young's moduli of the plates, of which responses are numerically calculated by modal analyses for diffused sound field excitation and rain-on-the-roof excitation. The ensemble variances of the response to these two excitations are compared to the variance value predicted by the theoretical model. The result from the comparison shows good agreement between the prediction method and the numerical results, and the applicability of the prediction method to diffuse sound field excitation case is also discussed.