Attenuation Process of a Single-Degree-of-Freedom Mass-Spring System under Damping

A single-degree-of-freedom mass-spring system represents one of the most fundamental models in structural dynamics, comprising mass, spring, and damping components. When subjected to initial disturbance, the system exhibits free vibration with gradually decreasing amplitude under damping effects, known as decay vibration.

The presence of damping dissipates system energy, causing vibration amplitude to decay exponentially over time. The decay rate depends critically on the damping ratio parameter. When the damping ratio is less than 1 (underdamped condition), the system shows oscillatory decay; when equal to 1 (critically damped), it returns to equilibrium position fastest without oscillation; when greater than 1 (overdamped), it slowly returns to equilibrium position.

MATLAB enables efficient simulation of this physical process through these implementation steps: Establish the equation of motion including inertial force, damping force, and elastic restoring force Numerically solve using ODE solvers (e.g., ode45) with appropriate initial conditions Visualize displacement-time curves to observe characteristic exponential envelope patterns The simulation typically involves defining system parameters (mass, stiffness, damping coefficient), implementing the differential equation function, and using time-stepping algorithms for numerical integration.

Such simulations find wide engineering applications, including seismic performance evaluation of building structures and vehicle suspension system design. By adjusting damping parameters, engineers can optimize dynamic response characteristics for specific performance requirements.