Nonlinear dynamics of a tubular beam considering distortion of the cross sections and internal resonances

The nonlinear dynamics of a thin tube under the action of a harmonic external load is addressed in the paper. Use is made of a beam-like model which extends the Timoshenko beam model with further kinematic descriptors, related to the change in shape of the cross section. The external load is applied in half cap of the pipe, directly triggering both the bending of the axis-line and the flattening of the cross sections. The equations of motion are projected on a reduced basis constituted by the first three linear modes, and then the solutions are sought via the multiple scale method, for two different external resonance conditions. Internal resonances among the modes are considered as well. The outcomes, compared with pure numerical solutions, highlight the possible energy exchange between local modes, i.e., those describing flattening and warping of the cross sections, and global modes, i.e., those related to bending of the axis-line and rotation of the cross section of the pipe.