A generalized framework for studying the vibrations of lumped parameter systems

In this short note we show that the classical approach for obtaining the governing equation for the vibration of lumped parameter systems is not sufficiently general enough to incorporate well accepted mechanisms for the dissipation and for the storage of energy that are constituents of the vibrating system. We discuss a generalization that reduces to a coupled system of differential-algebraic equations that need to be solved simultaneously. We show that the generalization allows us to study vibrating systems wherein one has an implicit relationship between the relevant forces and kinematical quantities with regard to the mechanisms for the storage and the dissipation of energy.