This paper is investigated vibration of magneto-electro-elastic (MEE) composite conical shell on a nonlinear elastic foundation and under electric or magnetic potential while the influence of Geometrical nonlinearity is taken into account. The conical shell is modeled based on the von Karman approach while the influences of shear deformation and rotary inertia are heeded. Coupled relations of MEE material are utilized to derive the vectors of stress, electric displacement as well as magnetic induction. Quasi-static Maxwell equations, Gauss' laws as well as thin shell assumptions are used to determine electric and magnetic fields. The nonlinear ordinary differential equation of the shell is derived through the Lagrange approach. Lindstedt-Poincare method and modal analysis are hired in order to obtain nonlinear vibration responses of the MEE composite conical shell. For validation intention, some results of the literature are compared with some results of this study. The effects of several parameters including nonlinear and linear constants of foundation, electric and magnetic potentials, thickness as well as length on the values of fundamental linear frequency, nonlinear parameter, and the curves of nonlinear frequency ratio versus amplitude parameter are investigated. The results show that the increase of the nonlinear constant of elastic foundation or thickness causes the increase of the nonlinear frequency ratio. On the other hand, the nonlinear frequency ratio gets smaller values with an increase in the linear constants of the elastic foundation or length.

One of the most important issues in controller design and analysis of a nonlinear system is its degree of nonlinearity. Helbig et al. proposed a method in which the system, along with a linear reference model (which is the sum of a couple of first order transfer functions), is stimulated with a set of input patterns. The outputs of these systems are then compared and the nonlinearity measure of the system is obtained as the normalized difference between the outputs of these two systems. In this paper, the linear reference model is replaced with two simpler linear ones. The proposed method has been used for assessment of the nonlinearity measure of various nonlinear systems that are used as standard benchmarks by the nonlinear process control research community. The results show that, despite the use of this simpler model with less parameters, the calculated nonlinearity measure is almost similar to the one obtained by Helbig's method. Furthermore, the nonlinearity measure obtained by the proposed method can be calculated very much faster.

The purpose of this paper is to analyze in detail a special nonlinear partial differential equation (nPDE) of the second order which is important in physical, chemical and technical applications. The present nPDE describes nonlinear diffusion and is of interest in several parts of physics, chemistry and engineering problems alike. Since nature is not linear intrinsically the nonlinear case is therefore the general. We determine the classical Lie point symmetries including algebraic properties whereas similarity solutions are given as well as nonlinear transformations could derived. In addition, we discuss the nonclassical case which seems to be not solvable. Moreover we show how one can deduce approximate symmetries modeling the nonlinear part and we deduce new generalized symmetries of lower symmetry. The analysis allows one to deduce wider classes of solutions either of practical and theoretical usage in different domains of science and engineering.

The equations of nonlinear motion of clamped-hinged beam with an open crack were extracted and through solving them, the internal resonance in the cracked beam was studied. To this end, the crack was modeled as a torsional spring and the cracked beam was considered as two beam segments connected by a torsional spring. The equations of motion of the cracked beam were extracted considering the Geometrical nonlinearity. Then, using the Galerkin’ s method, these equations were changed to a set of nonlinear differential equations for vibration modes which were solved by the perturbation method. Since the mechanical energy of the beam in each mode depends on the instantaneous amplitude of vibration of the beam at the corresponding mode, so to analyze the influence of the crack on the energy exchange between the modes, the instantaneous amplitudes of the vibration modes were obtained. The results show that in the cracked beam the magnitude of the energy exchanged between the modes is less and the frequency is more than that in the intact beam. Also, by increasing the crack depth the frequency of energy exchange between the modes increases. The Vibration response obtained for the cracked beam with various amounts of the damping ratios shows that the frequency and the amplitude of energy exchange between the modes are independent of the system damping. To validate the results by the perturbation method, the equations of motions are also solved by a numerical method and the obtained results are in agreement with the results of the analytical method.

Because of difficulty in inspection and retrofit of foundation in comparison with other elements, the common design philosophy is to avoid any nonlinear deformation in the foundation. This paper shows that by employing controlled foundation nonlinearity, in predetermined sections with arrangements for inspection and retrofit, it is possible to reduce seismic demand on superstructure. Localizing nonlinear deformation to pre-specified zones in the foundation, it is possible to avoid wide spread nonlinear deformation across various members in the superstructure in the case of strong ground motions. To evaluate the efficiency of the proposed model, the response of steel braced frames is examined on rigid foundation, rocking elastic foundation and finally on rocking foundation with controlled nonlinear deformation. Results show that while rocking could be used to protect the superstructure elements from possible overloading during large earthquakes for low-rise structures; it has no remarkable effect on the response of high-rise structures. However, the proposed model with nonlinearity in the foundation could be used in both cases (low-rise and high-rise structures) to effectively control the response of structure.