Massespringdamper

Massespringdamper, or mass–spring–damper system, is a simple mechanical model used to study linear vibratory motion. It comprises a rigid mass m connected to a spring with stiffness k and a damper with damping coefficient c. An external force F(t) acts on the mass. For a single degree of freedom, the equation of motion is m x''(t) + c x'(t) + k x(t) = F(t). In the absence of forcing, the homogeneous equation governs the transient response.

Key characteristics include the natural frequency omega_n = sqrt(k/m) and the damping ratio zeta = c / (2 sqrt(k m)). The system behavior depends on zeta: underdamped (zeta < 1) with oscillatory decay at the damped frequency omega_d = omega_n sqrt(1 - zeta^2); critically damped (zeta = 1); overdamped (zeta > 1). The energy of the system consists of potential energy 1/2 k x^2 and kinetic energy 1/2 m x'^2, with damping dissipating energy as heat.

Steady-state and dynamic responses can be analyzed via the transfer function X(s)/F(s) = 1/(m s^2 + c s + k) and its frequency response X(jω)/F(jω) = 1/(k - m ω^2 + j c ω). A step input yields a transient that depends on zeta and a final displacement of F0/k for constant forcing. The impulse response, along with the step and frequency responses, provide a complete picture of the system’s dynamics.

Applications of the massespringdamper model include vibration isolation, vehicle suspensions, structural and mechanical vibration analysis, and various control contexts. In multi-degree-of-freedom systems, multiple masses, springs, and dampers yield a set of coupled equations described by mass and damping matrices, with extensions to nonlinear or nonviscous damping as needed.