MasseFederDämpferSystem

MasseFederDämpferSystem (mass–spring–damper system) is a classical mechanical model used to describe the dynamic behavior of a single-degree-of-freedom body connected to a linear spring and a viscous damper. It consists of a rigid mass m, a linear spring with stiffness k, and a damper with viscous coefficient c. The mass can be driven by an external force F(t) and may exhibit free motion when F(t)=0.

Mathematically, the motion is governed by the second-order differential equation: m x¨ + c x˙ + k x = F(t), where x(t) is the displacement of the mass. The natural frequency is ω_n = sqrt(k/m) and the damping ratio is ζ = c/(2 sqrt(k m)). The system is linear and time-invariant under the assumptions of small displacements and linear elements.

In the time domain, solutions describe free vibration (F(t)=0) with behavior determined by ζ: underdamped (ζ<1) with oscillatory decay, critically damped (ζ=1) with fastest non-oscillatory return to equilibrium, and overdamped (ζ>1) with slow return. In the frequency domain, the transfer function from force to displacement is H(s) = X(s)/F(s) = 1 / (m s^2 + c s + k).

Common applications include vibration isolation, automotive and aerospace suspensions, structural damping analysis, and control system modeling. The model is idealized: it assumes a lumped mass, linear springs, viscous damping, and small displacements. Energy in the system comprises kinetic energy (½ m v^2) and potential energy (½ k x^2), with dissipated energy by the damper.