First order homogeneous dynamical systems 2: application to cracked concrete beams

In this two-part paper, a new class - first order homogeneous dynamical (FOHD) systems - of two-DOF conservative nonlinear dynamical systems has been explored. Theoretical formulation and significance of proposed theory have been presented in part-I. Using the proposed theory, the dynamical behaviour of the two-DOF cracked concrete beam has been predicted here. A new type of phase plot for MDOF dynamical systems has been proposed. Depending upon the loading details and system parameters, the vibration response of these essentially nonlinear systems can be linear, bilinear or nonlinear. Forced vibrations about the passive state have been predicted to resemble linear vibrations in some respects. Like other nonlinear dynamical systems, concrete beam response has also been found to be quite sensitive to initial conditions and system parameters, and to exhibit sub-harmonics and combination sub-harmonics. Feasibility of a nonlinear tuned mass vibration absorber has also been explored. Empirical validation and practical relevance of the proposed theory have been discussed.