A combination of Newton's second law, a transformation using three consecutive Euler angles, and Taylor expansions is used to develop three nonlinear integro-differential equations describing the flexural-flexural-torsional vibration of metallic and composite beams. The twisting curvature is used to define a physical twisting variable which makes the equations of motion unique and independent of the rotation sequence of the Euler angles.

A numerical-perturbation approach is used to analyze the response of metallic and composite beams to parametric and external excitations. First, the linear eigenfunctions and natural frequencies are calculated using a combination of the state-space concept and the fundamental-matrix method. Then, the method of multiple scales is used to construct a set of nonlinear autonomous first-order ordinary-differential equations describing the slow-time modulation of the amplitudes and phases of the interacting modes in the presence of one-to-one and/or two-to-one internal resonances. The inversion symmetry, D, symmetry, and 0(2) symmetry of the system are studied using the modulation equations. The solutions of the modulation equations may be fixed points, limit cycles, or chaotic solutions.