Mass proportional damping system

If we take the same damped mass and spring system and, instead of driving the system with a motor, simply displace the mass and let it go, we find that it will oscillate with its natural resonance frequency and the amplitude of the oscillation will decay in time. The decay will be exponential with the amplitude proportional to exp(-t/T2p where the viscosity of the liquid will determine T. It is interesting to note that if you pluck the system, its oscillation and decay contains all the information of the system s behavior just as the frequency dependence of the amplitude in the driven simple harmonic oscillator experiment does. In the one case, the full width at half height of the resonance curve is 2/T, while in the other, the decay time constant is T. ... 

Trombetti, T, Silvestri, S. (2007). Novel schemes for inserting seismic dampers in shear type systems based upon the mass proportional component of Rayleigh damping matrix. Journal of Sound and Vibration, 302(3), 486-526. doi 10.1016/j. jsv.2006.11.030... 

The system is still comprised of the inertia force due to the mass and the spring force, but a new force is introduced. This force is referred to as the damping force and is proportional to the damping constant, or the coefficient of viscous damping, c. The damping force is also proportional to the velocity of the body and, as it is applied, it opposes the motion at each instant. 

Rayleigh s dissipation function is applied in order to properly handle the viscous component of the given dynamic system. In this case, it is assumed that the viscous nature of the dashpot will be linearly dependent upon the velocity of the mass. More specifically, the viscous damping force is considered to be proportional to the velocity of the mass and is given by the following relation. 

The analysis included a consistent mass matrix and viscous damping proportional to mass and stiffness. The system of nonlinear equations is solved by time-stepping direct... 

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