⟿ m ẍ + c ẋ + k x = 0 // 2ND ORDER LINEAR SYSTEM

Mass·Spring·Damper

EIGENVALUE · DAMPING RATIO · REAL-TIME RESPONSE

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MASS (m) 1.0kg

SPRING CONSTANT (k) 20N/m

DAMPING COEFFICIENT (c) 2.0N·s/m

INITIAL DISPLACEMENT (x₀) 1.0m

PHYSICAL SYSTEM

TIME RESPONSE x(t)

CHARACTERISTIC PROPERTIES

NATURAL FREQUENCY ωn

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— Hz

DAMPING RATIO ζ

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DAMPED FREQ. ωd

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T = — s

CURRENT STATE

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t = 0.00 s

EIGENVALUE & EIGENVECTOR ANALYSIS

EIGENVALUES (s₁, s₂) — A = [[0,1],[−k/m,−c/m]]

s₁ = —

s₂ = —

EIGENVECTORS (v = [1, s]ᵀ)

v₁ = —

v₂ = —

PHASE PORTRAIT (x — ẋ)

PHASE PLANE + EIGENVECTOR DIRECTIONS

ⓘ System: m·ẍ + c·ẋ + k·x = 0 (free vibration, ẋ₀ = 0). ωn = √(k/m), ζ = c / (2√(km)). Eigenvalues s = −ζωn ± ωn√(ζ²−1); ζ<1 underdamped (complex conjugate, oscillatory), ζ=1 critically damped, ζ>1 overdamped (two real roots). The solution is calculated analytically; the animation is in real-time (1 s = 1 s). Real eigenvector directions are plotted on the phase portrait when eigenvalues are real.