Damped free vibration: viscous damping and logarithmic decrement

Why it matters

Damping decides how quickly a disturbed structure settles, how large a resonance can grow and, in aeroelasticity, whether a wing mode stays stable as speed rises: flutter is the speed at which a mode's total damping falls to zero. Damping cannot be calculated reliably from drawings, so it is measured, most often from the decay of free vibration through the logarithmic decrement. This topic gives you the model and the measurement.

Key ideas

Viscous damping model. A dashpot produces a force opposing the velocity, F_d = −c·ẋ, with c the damping coefficient (N·s/m). The equation of motion of a damped SDOF system in free vibration is

m·ẍ + c·ẋ + k·x = 0

Viscous damping is linear, which makes the equation easy to solve and lets modal analysis carry over. Real structural damping comes from material hysteresis, friction in joints and fasteners, and air; for light damping it is represented by an equivalent viscous damping ratio.

Characteristic equation. Trying x = e^(s·t) gives m·s² + c·s + k = 0, so s = −ζ·ω_n ± ω_n·√(ζ² − 1). The nature of the motion depends on the damping ratio ζ = c/c_c, where c_c = 2·√(k·m) = 2·m·ω_n is the critical damping coefficient.

• ζ < 1, underdamped: the roots are complex, the motion oscillates at the damped natural frequency ω_d = ω_n·√(1 − ζ²) inside an exponentially decaying envelope e^(−ζ·ω_n·t). Almost all aircraft structures fall here, with ζ typically 0.01–0.05.
• ζ = 1, critically damped: two equal real roots; the system returns to equilibrium fastest without overshooting more than once.
• ζ > 1, overdamped: two real negative roots; slow, non-oscillatory return.

Underdamped response. x(t) = e^(−ζ·ω_n·t)·[x₀·cos(ω_d·t) + ((v₀ + ζ·ω_n·x₀)/ω_d)·sin(ω_d·t)]. Note that ω_d < ω_n, but for ζ = 0.05 the difference is only 0.13 %.

Logarithmic decrement. The ratio of any two successive peaks (one damped period T_d = 2π/ω_d apart) is constant: x_i/x_(i+1) = e^(ζ·ω_n·T_d). Its natural logarithm is the logarithmic decrement δ = 2π·ζ/√(1 − ζ²). For small damping δ ≈ 2π·ζ. Using peaks n cycles apart averages out measurement noise: δ = (1/n)·ln(x₀/x_n). Count cycles carefully: x₀ to x_n spans n periods.

Energy view. In one cycle a lightly damped system loses a fraction ≈ 2δ of its energy, since energy goes as amplitude squared. A useful rule: the number of cycles for the amplitude to halve is ln 2/δ ≈ 0.11/ζ.

Other damping types (for awareness). Coulomb (dry friction) damping gives a linear, not exponential, decay of amplitude (each half cycle the amplitude drops by 2·F_f/k). Structural (hysteretic) damping is written as a complex stiffness k·(1 + i·g); at resonance g ≈ 2ζ. If a measured decay is a straight line rather than an exponential, the damping is not viscous and the log decrement is not constant.

Link to aeroelasticity. In flight, aerodynamic forces add their own damping (positive or negative) to each mode. The V–g method and flight flutter testing track the damping of each mode as speed increases; the flutter speed is where it crosses zero.

Formulas

m·ẍ + c·ẋ + k·x = 0 — damped free vibration; c damping coefficient (N·s/m).

c_c = 2·√(k·m) = 2·m·ω_n — critical damping coefficient (N·s/m).

ζ = c/c_c = c/(2·m·ω_n) — damping ratio (dimensionless).

ω_d = ω_n·√(1 − ζ²) — damped natural frequency (rad/s), underdamped only (ζ < 1). T_d = 2π/ω_d (s).

x(t) = X·e^(−ζ·ω_n·t)·sin(ω_d·t + φ) — underdamped free response; X and φ from initial conditions.

δ = ln(x_i/x_(i+1)) = (1/n)·ln(x₀/x_n) — logarithmic decrement (dimensionless), from peaks one or n cycles apart, same side of equilibrium.

δ = 2π·ζ/√(1 − ζ²) and inversely ζ = δ/√(4π² + δ²); for ζ < about 0.1, δ ≈ 2π·ζ.

n_half = ln 2/δ — cycles for the amplitude to halve.

Worked examples

Example 1 (standard). m = 10 kg, k = 4000 N/m, c = 40 N·s/m. Find ω_n, ζ, ω_d, δ and the amplitude ratio after 5 cycles.

1. ω_n = √(k/m) = √(4000/10) = 20 rad/s.
2. c_c = 2·√(k·m) = 2·√(40 000) = 400 N·s/m, so ζ = c/c_c = 40/400 = 0.10.
3. ω_d = ω_n·√(1 − ζ²) = 20·√0.99 = 19.90 rad/s.
4. δ = 2π·ζ/√(1 − ζ²) = 0.6283/0.99499 = 0.6315.
5. After 5 cycles: x₅/x₀ = e^(−5δ) = e^(−3.157) = 0.0425.

Answer: ω_n = 20 rad/s, ζ = 0.10, ω_d = 19.90 rad/s, δ = 0.631; the amplitude falls to 4.25 % in 5 cycles.

Example 2 (GATE level). In a ground test a 5 kg lumped-mass model is plucked. Its peak displacement falls from 10 mm to 2.5 mm in 6 cycles and the measured period is 0.20 s. Find ζ, ω_n, the damping coefficient and the stiffness.

1. δ = (1/n)·ln(x₀/x_n) = (1/6)·ln(10/2.5) = (1/6)·ln 4 = 1.3863/6 = 0.2310.
2. ζ = δ/√(4π² + δ²) = 0.2310/√(39.478 + 0.0534) = 0.2310/6.2874 = 0.0367.
3. ω_d = 2π/T_d = 2π/0.20 = 31.42 rad/s; ω_n = ω_d/√(1 − ζ²) = 31.42/0.99932 = 31.44 rad/s.
4. c = 2·ζ·m·ω_n = 2 × 0.0367 × 5 × 31.44 = 11.55 N·s/m.
5. k = m·ω_n² = 5 × 31.44² = 4941 N/m.

Answer: ζ ≈ 0.037, ω_n ≈ 31.4 rad/s, c ≈ 11.6 N·s/m, k ≈ 4.94 kN/m. The amplitude halves every ln 2/δ = 3.0 cycles, consistent with 10 → 5 → 2.5 mm in 6 cycles.

Common mistakes

• Using ln(x₀/x_n) without dividing by n when the peaks are several cycles apart, or counting peaks instead of intervals.
• Taking one peak above and the next below equilibrium; that is half a cycle and gives δ/2.
• Writing ζ = c/(2·√(k/m)); the critical coefficient is 2·√(k·m) = 2·m·ω_n.
• Applying ω_d = ω_n·√(1 − ζ²) to an overdamped system, which does not oscillate.
• Assuming damping noticeably lowers the oscillation frequency of a lightly damped aircraft structure; it mainly controls decay and resonant amplitude.
• Treating friction (linear) decay as viscous; the log decrement then varies from cycle to cycle.

For GATE AE

Typical questions: find ζ or c from the decay of two peaks; find ω_d from ω_n and ζ; identify under-, critically or overdamped behaviour from m, c and k; compute the number of cycles for the amplitude to fall to a given fraction; compare the exact and small-damping forms of δ. Practise both directions (δ → ζ and ζ → δ) and keep track of whether a given frequency is ω_n or ω_d.

Quick check

1. m = 2 kg, k = 800 N/m. What is the critical damping coefficient?
2. ζ = 0.1. What is δ (exact)?
3. Two successive peaks are 8 mm and 6 mm. Estimate ζ.
4. Is a system with m = 1 kg, k = 100 N/m and c = 25 N·s/m underdamped?
5. For ζ = 0.05 and ω_n = 10 rad/s, what is ω_d?

Answers: 1. 2·√(800 × 2) = 80 N·s/m; 2. 0.631; 3. δ = ln(8/6) = 0.288, ζ ≈ 0.0458; 4. No, c_c = 20 N·s/m so ζ = 1.25, overdamped; 5. 9.987 rad/s.