ABSTRACT. This paper is concerned with the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of an elastic beam which is acted upon by axial compression, lateral forces and is in contact with a semi-infinite medium acting as a foundation For certain ranges of the acting axial compression force, the solvability of the equations follows from the coerciVity of their linear parts. Beyond these ranges this coercivity is lost It is shown here that the coercivity which ensures the global solvability can be generated by the nonlinear parts ofthe equations for a certain type offoundation.KEY WORDS AND PHRASES: Global solvability, fourth-order nonlinear boundary value problems, homogeneous nonlinearity, Leray-Schauder fixed point theorem, coercivity 1991 AMS SUBJECT CLASSIFICATION CODES: 49G99, 73H05, 73K15.
A lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic stiffness constant is considered for modeling the free and forced axial vibrations of a graphene sheet with one fixed end and one free end with a mass attached. It is demonstrated through this simple model that, in free vibration, within certain initial energy level and depending upon its length and the nonlinear elastic constants, that there exist bounded periodic solutions which are non-sinusoidal, and that for each fixed energy level, there is a bifurcation point depending upon material constants, beyond which the periodic solutions disappear. The amplitude, frequency, and the corresponding wave solutions for both free and forced harmonic vibrations are calculated analytically and numerically. Energy sweep is also performed for resonance applications.
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