Non-Linear Free Vibrations of Beams By the Finite Element and Continuation Methods

Hierarchical Finite Element Analyses of Geometrically Non-Linear Vibration of Beams and Plane Frames

“…Often, particularly in analyses of slender beams, this force is approximated by its mean value over the element [14].…”

Non-Linear Steady State Vibrations of Beams Excited by Vortex Shedding

“…This approach to considered particular problem is more adequate than very popular modal ones because the influence of more then one mode of vibration can be taken into account in an easy way. As shown in many papers (see, for example references [19][20][21][22]), the interaction of two or more modes of vibration could be very important if nonlinear systems are considered. Moreover, beams with the non-uniform spatial distribution of the beam bending stiffness, the mass per unit length or the cross-section characteristic dimension can be modelled more accurately.…”

On Interactions of Oscillation Modes for a Weakly Non-Linear Undamped Elastic Beam With an External Force

“…As stated above, terms that give rise to secular terms may occur on the right-hand side of equation (19). To eliminate these terms we introduce two time-scales, t "t and t " t, and assume that q L can be expanded in a formal power series in , that is,…”

“…We substitute this into equation (19) WEAKLY NON-LINEAR ELASTIC BEAM and collect equal powers in . The O( )-problem becomes…”

“…The interaction between the di!erent oscillation modes is studied and a justi"cation is given in which cases mode truncation is valid. For fourth order strongly non-linear PDEs numerical "nite element methods can be used, as is done for example in reference [19].…”