First order homogeneous dynamical systems 1: theoretical formulation

Of late, the attention of the dynamicists has increasingly been focused on the multi-degree of freedom (MDOF) nonlinear dynamical systems. In the present paper, a new class of conservative two-DOF nonlinear dynamical systems - first order homogeneous dynamical (FOHD) systems - has been proposed. This investigation is motivated by two-DOF cracked concrete beams undergoing small deformations. For these mechanical systems, the nodal forces are functions homogeneous of order one of the nodal displacements and vice-versa. Under assumptions of lumped nodal masses and classical damping, the equations of motion have been derived in the paper. The nodal displacement space has been partitioned into four elastically-distinct regions. Within the two nonlinear elastic regions, the stiffness and damping coefficients as well as the modal frequencies have been shown to vary continuously but remain constant within the two linear regions. Peculiar characteristics distinguishing the FOHD systems from other known MDOF nonlinear dynamical systems have been identified. Theoretical significance of the proposed FOHD systems in the general nonlinear dynamical systems theory has been brought out. The issues such as empirical validation of the predicted dynamical response and the practical relevance of the work done for the concrete beams under working loads have also been discussed.