Abstract：The nonlinear dynamics of the hinged-hinged shallow arches with torsional springs constrained at both ends in case of 1:1 internal resonance under external excitation are investigated. The dimensionless dynamic equations are achieved by introducing the basic assumptions of shallow arch. Then by removing the damper, external load and non-linear terms, the obtained linear equation and corresponding boundary conditions are used to determine the frequencies and modes which take the torsional springs into account. Two internal resonance types of crossing and veering are found when the torsional constraints adopt different stiffness values. Further, the dynamic equation was performed by a full-basis Galerkin discretization and the multiple scale method was used to study the internal resonances by perturbation analysis, which leads to both the polar- and Cartesian-form averaging equations whose coefficients have a one-to-one correspondence with the stiffness of torsional spring. The numerical results for 1:1 internal resonance between the two lowest modes show the nonlinear interactions can be excited by external excitation. Moreover, when the parameters are controlled in a certain range there have periodic, quasi-periodic and chaotic windows in the system, and it enters into chaos by the (inverse) period-doubling bifurcation.
2014, Vol. 35 
