Analyzing the Nonlinear vibration of beams is one of the important issues in structural engineering. According to this, an impressive analytical method which is called modified iteration perturbation method (MIPM) is used to obtain the behavior and frequency of a cantilever beam with geometric Nonlinear. This new method is combined by the Mickens and iteration methods. Moreover, this method does not require small parameter in the equation which is difficult to be found for Nonlinear oscillation. The accuracy of the solution that is obtained by the use of MIPM has been shown graphically and compared with exact solution. Comparison shows that good adaptation is obtained and MIPM as a powerful method for solving the vibrational behavior of structures analytically.

In this paper, Nonlinear vibration control of a plate is investigated using a Nonlinear modified positive position feedback method that is applied through a piezoelectric layer on the plate. Based on the classical theory of displacement and strain relations with von Karman, intended equations of motion for the smart plate have been obtained. In this model, transverse vibrations are studied and stimulations are performed for the primary resonance. Boundary conditions of the smart plate are simply supported. The plate is thickness symmetrical. Using the Galerkin method the temporal Nonlinear equations governing the system have been found. Then, the free and forced vibrations of the structure with the Nonlinear modified positive position feedback controller have been solved using the Method of Multiple Scales to obtain an analytical solution. Results show that this controller reduces the amplitude of the vibration by inducing an increase in the damping coefficient. In addition, this provides a higher level of suppression in the overall frequency domain response by increasing the compensator gain. Finally, the results of the analytical solution for the closed-loop Nonlinear modified positive position feedback controller are presented and compared with the result of the conventional positive position feedback controller and Nonlinear integral resonant controller. The results show that the performance of the Nonlinear modified positive position feedback controller is better than other controllers and significantly reduces the vibration amplitude.

Recently, a large amount of studies have been related to Nonlinear systems with multi-degrees of freedom as well as continuous systems. The purpose of this paper is to optimize passive vibration absorbers in linear and Nonlinear states for an Euler-Bernoulli beam with a Nonlinear vibratory behavior under concentrated moving load. The goal parameter in the optimization is maximum deflection of the beam. The large deformation for beam modeling is considered, i.e. the relation between strains and deflections is Nonlinear. The force magnitude and beam length are two effective factors for the beam deflection. vibration absorber with linear damping and linear or Nonlinear stiffness is also considered in this manuscript. The results show that, for normal forces and short beams, linear and Nonlinear models have similar behaviors, while surveying Nonlinear behavior is necessary by increasing the force and length of the beam, i.e. large deflections. Moreover, the difference between linear and Nonlinear beam models for regular force magnitudes and beam lengths is negligible. For higher loads and longer beams, beam model Nonlinearity can be important. Results demonstrate that, in the presented numerical values (train bridge application) for cubic Nonlinear vibration absorber, there are two optimal locations for vibration absorber installation: one inclined from the middle of the beam to the direction of moving loads and the second which is more interestingly inclined from the middle of the beam to moving loads in the opposite direction. Moreover, depending on the model’s numerical parameters, for short beams, linear vibration absorber is more effective, while for long beams, cubic Nonlinear beam behaves better than the linear one.

This study aims to investigate the effect of functionally graded materials (FGMs) on the internal resonances of nanorods in torsional vibration. The von-Kármán type Nonlinearity is considered and the governing equation of motion is derived using Hamilton's principle based on the surface elasticity theory. It is assumed that the properties of the functionally graded (FG) nanorod vary through the radius direction based on power-law distribution. Then, the multi-mode Galerkin method is implemented to convert the partial differential equation to an ordinary differential one. In the next step, the method of multiple scales is used to derive the natural frequencies as well as the conditions in which the internal resonances occur. The results are presented for two types of end conditions, fixed-fixed and fixed-free, and the effects of variations of various parameters like length, radius, and amplitude of vibration on natural frequencies are investigated. This research shows that functionally graded materials differ in the state of happening the internal resonances in the presence of the surface energy.

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.

This paper is focused on the study of Nonlinear vibration of rotating laminated composite cross-ply cylindrical shells on a Nonlinear rotating elastic foundation. In this study, FSDT is employed while the geometrical Nonlinearity of the cylindrical shell is modeled considering the von Karman approach. It should be mentioned that this study is accomplished considering the influences of initial hoop tension as well as Coriolis and Centrifugal accelerations. The Nonlinear equation of the rotating laminated composite cross-ply cylindrical shell is extracted via the Ritz method and then is written in the state space form. Then, modal analysis and the multiple scales method are applied to the Nonlinear vibration equation in the state space form to obtain relations for Nonlinear forward and backward frequency ratios. Validation of the results of this study is investigated considering some results published in the literature and good agreement is observed. Finally, the effects of the Nonlinear and linear constants of the rotating foundation, radius, total thickness, length, and rotation speed on the linear frequencies, Nonlinear parameters, and the curves of Nonlinear frequency ratios versus amplitude parameters are acquired. The results show that the increase of the Nonlinear constant of the rotating foundation doesn’t influence the linear frequencies. Besides, linear frequencies increase with increase of the linear constants of the rotating elastic foundation and decrease with increase of the radius or total thickness. Furthermore, the increase of the Nonlinear constant of the rotating elastic foundation or total thickness leads to an increase in Nonlinear parameters and Nonlinear frequency ratios. Conversely, the increase of the linear constants of the rotating foundation or the radius leads to a decrease in Nonlinear parameters and frequency ratios. Moreover, the increase in amplitude parameters leads to an increase in the Nonlinear frequency ratios.

In this paper, Nonlinear vibration analysis of functionally graded piezoelectric (FGP) beam with porous materials is investigated based on the Timoshenko beam theory. Material properties of FG porous beam are described according to the rule of mixture which is modified to approximate material properties with porosity phases. Ritz method is used to obtain the governing equation which is then solved by a direct iterative method to determine the Nonlinear vibration frequencies of FGP porous beam subjected to different boundary conditions. The effects of external electric voltage, material distribution profile, porosity volume fraction, slenderness ratios and boundary conditions on the Nonlinear vibration characteristics of the FGP porous beam are discussed in detail. The results indicate that piezoelectric layers have a significant effect on the Nonlinear frequencies. Also, it is found that porosity has a considerable influence on the Nonlinear frequency, especially when the electric voltage is applied.