rugómassza

Rugómassza is a basic mechanical model used in physics and engineering to study oscillatory motion. It consists of a point mass m attached to a linear spring with spring constant k, moving along one dimension. When displaced from equilibrium and released, it exhibits simple harmonic motion in the absence of damping or external forcing.

Governing equations: The equation of motion for a rugómassza can be written as m x'' + c x' + k x = F_ext(t), where c is the damping coefficient and F_ext represents an external force. In the undamped, free case (c = 0 and F_ext = 0), the equation reduces to x'' + (k/m) x = 0, with natural angular frequency ω0 = sqrt(k/m) and a period T = 2π/ω0. The mechanical energy is E = 1/2 k x^2 + 1/2 m v^2.

Damped motion: With c > 0, the rugómassza behaves as a damped harmonic oscillator. The damping ratio ζ = c / (2 sqrt(k m)) determines behavior. If ζ < 1, the system is underdamped and x(t) decays as e^{-ζ ω0 t} cos(ω_d t + φ), where ω_d = ω0 sqrt(1 - ζ^2). If ζ = 1, the motion is critically damped; if ζ > 1, it is overdamped.

Forced vibration: When F_ext(t) = F0 cos(ω t), the steady-state response has amplitude X = F0 / sqrt((k - m ω^2)^2 + (c ω)^2). Driving near the natural frequency ω0 can produce resonance in lightly damped systems.

Applications: The rugómassza model is a standard teaching tool and underpins understanding of simple harmonic motion, damping, and resonance. It informs design and analysis in mechanical engineering, vibration isolation, sensors, and materials testing.