Theoretical analysis of damping properties of two-degree-of-freedom linear vibrational systems

Analytical solutions for complete two-degree-of-freedom (2 DOF) vibrational systems comprising primary and secondary resilient and damping elements and subjected to a harmonic base excitation are obtained for the first time and presented in terms of transmissibility of primary and secondary mass and resonance and invariant frequencies of primary mass, which is considered to be the object of isolation. Analytical expressions for optimum values of main system parameters including primary and secondary dampings, mass ratio and secondary and primary springs stiffness ratios are also newly presented. Damping properties of systems are analysed for the case of a constant optimum damping, which minimizes the primary mass transmissibility at the lowest resonance frequency, and for the case of frequency-variable optimum damping, providing the minimization over the entire range of forced frequency. The expressions for optimum primary and secondary dampings are given in the form of quadratic algebraic equations for the case of complete 2 DOF systems. The influence of the mass and spring stiffness ratios on the system response is investigated as well. The obtained results, being large of a general nature and obtained for large ranges of main parameters, can be applied to different classes of vibration isolation problems, such as isolation of a large item of machinery, a light piece of instrumentation or a ground vehicle.