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.

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.