The nonlinear planar response of a hinged-clamped beam to a principal param
etric resonance of either its first or second mode or a combination paramet
ric resonance of the additive type of its first two modes is investigated.
The analysis accounts for mid-plane stretching, a static axial load, a rest
raining spring at one end, and modal damping. The natural frequency of the
second mode is approximately three times the natural frequency of the first
mode for a range of static axial loads, resulting in a three-to-one intern
al resonance. The method of multiple scales is used to attack directly the
governing nonlinear integral-partial-differential equation and associated b
oundary conditions and derive three sets of four first-order nonlinear ordi
nary-differential equations describing the modulation of the amplitudes and
phases of the first two modes in the cases of (a) principal parametric res
onance of either the first or the second mode, and (b) a combination parame
tric resonance of the additive type of these modes. Periodic motions and pe
riodically and chaotically modulated motions of the beam are determined by
investigating the equilibrium and dynamic solutions of the modulation equat
ions. For the case of principal parametric resonance of the first mode or c
ombination parametric resonance of the additive type, trivial and two-mode
solutions are possible, whereas for the case of parametric resonance of the
second mode, trivial, single, and two-mode solutions are possible. The tri
vial and two-mode equilibrium solutions of the modulation equations may und
ergo either a supercritical or a subcritical Hopf bifurcation, depending on
the magnitude of the axial load. For some excitation parameters, we found
complex responses including period-doubling bifurcations and blue-sky catas
trophes.