1,720,998 research outputs found
Random response of linear hysteretic damping
No full textThe probabilistic characterization of the response of a single-degree-of-freedom (SDOF) oscillator with linear hysteretic damping excited by ground motion described by zero mean stationary Gaussian processes is achieved by profiting from a steady-state solution of the motion equation, valid when the excitation is given by the superposition of harmonics. The model of linear hysteretic damping has been introduced to fit damping mechanisms in which the dissipation rate is independent of frequency, and mathematically it is described by the Hilbert transform of the response.Though this model is debated since it violates the principle of causality, its intrinsic simplicity makes it preferable to other models. The steady-state solution of the motion equation proposed in this paper allows a closed form evaluation of the respone mean square value. However, the numerical examples show that this quantity is affected by the mechanism of energy dissipation only when this is large. On the contrary, for a low capacity of dissipation the response mean square value is rather insensitive to the dissipation mechanism
Numerics for the first-passage time of a Markov process
No full textThe first-time passage problem for a Fokker-Planck Markov process is considered. Both the differential approach of solution and the integral one are reviewed, but the latter is followed in the applications. These regard the Ornstein-Uhlenbeck process and the envelope of the response of an oscillator with nonlinear stiffness
Risposta deterministica ed aleatoria dell'oscillatore elastoviscoso invecchiante
No full textThe dynamic response of a SDoF linear oscillator with both linear damping of Kelvin type and aging viscous damping is analyzed. The integro-differential equation of motion is recast as a third order linear differential equation with variable exponentially decaying coefficients. As regards the deterministic excitations, the free vibrations caused by an impressed displacement, the unit step excitation, and the sinusoidal resonant excitation are considered. The stochastic excitation is a zero mean stationary Gaussian process output of a second order linear filter. In both cases the presence of the Kelvin damping is fundamental: if this is absent, the response diverges as the damping goes to zero. Moreover, the response is viscoplastic, that is an impressed displacement is not recovered, differently from the hereditary damping in which it is wholly recovered at infinite time
Equivalent non-linear potential systems: review of previous assumptions
No full textIn the field of stochastic dynamics there is a few exact solutions for the response of
dynamical systems. Thus, the methods of the equivalent linearization and of the equivalent nonlinearization
are often used. While the former yields a Gaussian response to a Gaussian excitation, the
latter gives a non Gaussian response, which is nearer to the exact unknown response of a non linear
system. Among the methods of equivalent non-linearization that based on the replacement of the
actual dynamic system by means of a potential system stands up. In fact, this method leads to a general
procedure differently from other non-linearization methods. The procedure is developed basing on the
assumption that the ratio is constant and equal to one, being the ratio of the cross moment of the
mechanical energy powered to j and the square of the velocity and the moment E[j1] . This
relationship lacks of an analytical demonstration. In this paper, numerical analysis are presented to
ascertain its validity without resorting to Monte Carlo simulation. It is found that in most cases it holds
true, but some others are doubtful. The effects of considering constant when it is not are ascertained
for a Duffing oscillator with linear plus cubic damping
Stochastic stability of the inverted pendulum subjected to delta- correlated base excitation
No full textThis paper is concerned with the stochastic stability of an inverted pendulum with a point mass at the top and a spring at the base; the bar is massless. The base is subjected at the base to a vertical acceleration A(t) that is supposed to be a white noise (delta-correlated) stochastic process. Both Gaussian and Poissonian white noises are considered. A line-like structure excited by a vertical ground motion can be idealized in this way. It is assumed that during the motion the angle of rotation θ remains small so that sin θ≅θ. In this way, the motion equation assumes the classical form of the second order oscillator, but the excitation is parametric so that there is a possibility of stochastic instability. The almost sure (sample) stability and the stability in the second moments are considered herein. It is found that the two stability criteria lead to notable differences in the stability boundaries and the almost sure stability is not conservative. The mean square stability under the Poisson white noise is determined only by the arrival rate of underlying Poisson counting process and by the mean square amplitude of the pulses: the cumulants beyond the second order do not affect it
Stochastic Stability Criteria for Second-Order Oscillator Parametrically Excited by Colored Noise
No full textA second order oscillator is considered having a random perturbation in its stiffness. This is given by a colored Gaussian or non-Gaussian process. In this way, the oscillator may stochastically stable or unstable according to the intensity of the excitation. The almost sure (sample) stochastic stability and the stability in the first three response statistical moments are compared for different excitation proc-esses: process with exponential autocorrelation, second order Gaussian process, bounded noise proc-ess. Notable differences in the stability boundaries are found either according to the stability criteria or to the type of excitation. These comparisons are lacking in literature
Reliability study of prestressed concrete beam: A comparison between the methods of third and second level
No full textThis paper concerns the safety of a simply supported prestressed concrete beam subject only to bending and shear loads: so the probability of failure Pf is calculated. This is achieved firstly at the so-called third probabilistic level that requires the knowledge of the probability density functions (p.d.f.) of both the stress S and the resistance R: attention is firstly focused on the construction of the latter. In order to do this the Monte Carlo Techniques have been chosen: they are based on the extraction of random numbers and are thus very suitable for simulating random events and quantities. Samples are obtained of the values of the moment capacity MR of the beam; the histograms of the samples are plotted and the statistical function that gives the best fitting is chosen as p.d.f. of MR. The p.d.f. of the stress is obtained from the p.d.f. of the loads provided in the literature: the cases of a single and of many repetitions are considered in calculating Pf.Pf is also found through the MFOSM level 2 Method evaluating the safety index β with an iterative procedure: a comparison is made between the two methods
Numeric solution of the Fokker-Planck-Kolmogorov equation
No full textThe solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested; 2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.
Keywords: Stochastic Differential Equations; Markov Vectors; Fokker-Planck-Kolmogorov Equation; Finite Element Numeric Solution; Modified Hermite Weighting Functions; Spline Interpolatio
Some remarks on the transformation of filtered gaussian processes: a useful tool for stochastic analysis
No full textThis paper is aimed to briefly present the state of the art regarding the memoryless nonlinear transformations of filtered Gaussian processes. First, filtering a Gaussian white noise produces a Gaussian colored process. Secondly, applying a memoryless nonlinear transformation to the process obtained in the first step, this is mapped to a non-Gaussian process. Markov methods of stochastic dynamics are applicable to a limited number of classes of non-Gaussian processes: the previously obtained transformations allow one to use all Markov methods including Itô's stochastic calculus
Exact determination of the peak factor for gust loading
No full textA new format for the gust factor G = 1 + g sigmaX/etaX and the peak factor g is proposed in such a way that these quantities have a given probability of exceedance. With reference to a SDoF model, the largest value CDF of diisplacement X is obtained by means of both Monte Carlo simulation and analytical tools. G and g are computed from the fractiles of this CDF
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