Tail equivalent linearization method is based on first order reliability method, which obtains an equivalent linear system for the considered nonlinear problemwith equal tail probability related to a specified threshold and time. This method has been applied only to nonlinear non-degrading single- and multi-degrees of freedom shear beam twodimensional models and three-dimensional one-story rigid diaphragm supported by frames with in-plane uni-axial stiffness which is subjected to independent random excitation along the structural axes. To use TELM for more practical problems it is required to extend this method to cover more realistic material and excitation characteristics. In this paper, some of these developments have been presented. Application of TELM for bi-directional excitation with bi-axial material subjected to different incidence angles of excitation and using TELMfor degrading material which has been presented in the previous works of the authors have been reviewed briefly. In addition a new method for defining rotational dependent component of earthquake excitation in terms of independent translational components in the standard normal random variable space is proposed, and TELM has been used for this kind of excitation. Three examples related to these extensions have been presented; the comparison of the TELM results with Mote-Carlo simulation results shows good agreement.

In this study random vibration of a cantilever tapered beam under distributed stationary stochastic excitation with Gaussian probability density function is investigated. early free vibration analysis is performed to obtain the mode shapes of beam in form of Bessel functions, then the response is described in summation of mode shapes, and auto correlation of response is shaped by considering the mode shapes of tapered beam, also spectral density matrix of excitation is derived with cooperation of mode shapes and two dummy variables. in next step by means of frequency response and taking Fourier integral of autocorrelation of response, spectral density of displacement is computed and by using spectral density of displacement, variance of random displacements for various positions along the beam are achieved. Finally elasticity equation is applied to derive random strain and stress of beam. Comparing the variance of random stress with yield stress of beam leads to obtain probability of beam failure.

In this paper the behavior of a sensitive box to vibration, which connected to the vehicle by means of four undamped passive vibration isolator mounts, is studied using a ten DOF model to the road excitation. First, the governing differential equations are obtained using the Lagrange method, and the power spectral density (PSD) of the road profile is determined based on the ISO. The natural frequencies and mode shapes are extracted using modal analysis, and the response to the random base excitation is calculated based on random vibration theory. The root of mean square (RMS) is calculated by integration of PSD curve. The RMS of vertical acceleration of the box is plotted vs. the RMS of its relative vertical displacement by varying the stiffness of isolators in a real interval and the optimal stiffness is selected as the minimum point on the curve. The results indicate that the optimum isolator can be selected gardless of wheels and suspension system and the natural frequencies of isolator are between the natural frequencies of wheels and of suspension system. The RMS of acceleration the box has low sensitivity to the variation of the isolator stiffness around the optimum point.

In this paper, free vibration of a rotating beam is studied. Cross sectional area, elasticity modulus, moment of inertia, shear modulus, density and rotational speed are modeled as random variables. To study uncertainty, the stochastic [mite element method and second order perturbation technique are applied. The effects of rotational speed, setting angle and random property variances on "Coefficient of Variation" of first mode eigenvalue are investigated. This analysis is carried out for the Timoshenko and Bernoulli-Euler beam theories. Also, to determine the significance of random properties on response variation, a sensitivity analysis is applied. Results show that the first eigenvalue sensitivity curves for Bernoulli-Euler and Timoshenko beam models are approximately identical.

In the present paper, electrical energy harvesting from random vibrations of an Euler-Bernoulli nano-beam with two piezoelectric layers is investigated. The beam is composed of an aluminum layer together with two piezoelectric ceramic layers (PZT 5A) serving as energy harvesting sensors. In the proposed method, the equations governing the bimorph nano-beam will be analytically derived using classical beam theory with corresponding modification coefficients to the nano-structure applied. Then, the derived system of equations will be solved following Kantorovich method. Assumed boundary conditions for the nano-beam are as follows: a clamped end with the mass concentrated at the free end of the beam. Further, the input activation function of the system for energy harvesting was taken as being random. Since the objective of this research is to investigate the amount of harvested energy, the section on the results provides associated voltage and maximum output power curves with the bimorph nano-beam under random activation and input white noise, while also presenting the effects of characteristics and scale factor of the nano-particles on the amount of harvested energy.

The satellites on the ground during construction and transportation, in launching stage and operation in space are under various types of dynamic loads, including high and low frequency vibrational loads, acoustics, shock, impact, etc., each of which can be an important source in the creation of stress on the satellite. The satellite components should be designed in such a way that can continue to operate while facing these situations. Electronic boards, in particular their solder joints, are critical components of satellites. Therefore, investigation of damage in design process of boards have great importance. Loading pattern on the satellite during its operation is usually random which considered as quasi-static load. Improvement of the design of the satellite against the weaknesses shown while facing different loads is essential, and given the fact that it is time consuming and costly to carry out laboratory tests, the use of analytical methods for checking the strength and lifetime of the structure can be very useful. In this research, random vibrations environment is equivalent to pseudo-static loads, and using the multilayer plate theory, the stresses in solder joints and failure of joints under this loading will be investigated. Also, the effect of parameters such as electronic board width and the boundary condition of the printed circuit board on the solder joints' stress will be considered in analytical solution.

Random vibration analysis of tall structures faces multiple problems due to the large number of elements and high degrees of freedom; that is why this type of analysis is mostly used in simple structures and low degrees of freedom. In the past two decades, changes have been occurred in this type of analysis to be used in complex structures and the large number of elements. Pseudo-Excitation Method (PEM) presents a simple formulation for reducing the volume of operations. In this paper, a tall telecommunication tower is fully modeled as an example of such towers; it is analyzed by random vibration analysis with the help of the above method. Different conditions of the soil under the tower and different damping are used in modeling and analysis. The results show that structure response is strongly influenced by the soil conditions. In addition, higher modes have significant effects on the telecommunication tower response.

This study aims to investigate the influence of the plate uncertainty elastic modulus on free vibration response and buckling behavior. To this purpose, elastic modulus of plate is modeled as a random variable with a normal distribution. Spatial autocorrelation function is used for random fields. In this method, the correlation is dependent on the distance, as the points be far away from each other, the correlation is also reduced. Then, applying the powerful finite element method stochastic finite element relations were calculated using Monte Carlo simulation. To this purpose, a four-node Kirchhoff’ s element was used with twelve degrees of freedom. For the analysis, random variable is simulated 5, 000 times. At last, by numerical tests, the effects of uncertainty on elastic modulus are investigated on the natural frequencies and buckling loads of plate. The results of these tests show that the effect of uncertainty in elastic modulus of the plate has a different effect on the response of vibration and buckling of plate. So that these changes have low effect on free vibration responses of plate. But buckling loads are highly dependent on the elasticity coefficient.

The vehicle driving comfort has become one of the important factors of vehicle quality and receives increasing attention. In this paper, optimal points of vehicle suspension parameters are generated using modified non-dominated sorting genetic algorithm (NSGA-II) for Pareto optimization of 5-degree of freedom vehicle vibration model considering three conflicting functions simultaneously. In this way random profile is considered for the road excitation. Objective functions are vertical acceleration of seat, relative displacement between sprung mass and forward tire and relative displacement between sprung mass and rear tire. The results are compared with the previous works, where it indicates satisfactory behaviour of the optimum design points proposed in this work.