Post-critical Behavior of Three-dimensional Composite Beams Near the Autoparametric Resonance under Flapwise Excitation

The bifurcation analysis of a symmetrically composite beam subjected to harmonic flapwise base excitation is studied when the ratio of flapwise and chordwise internal resonances is 1:2. results are obtained by the numerical solution of modulation equations. in previous works,the modulation equations did not exihibit any symmetric properties and the modified equations with symmetric properties are studied in this paper. chracteristics of the response are investigated in terms of time history, phase portraits diagrams and bifurcation diagram of poincare maps. it is observed that periodic and multi-periodic motions, jump and stable motion are appeared near the internal and external tunnig frequency parameters.