Forced vibration amplitude and resonance

Free and forced vibrations

Theory (differential equation)

Mass m, spring constant k, damping force −b v, driving force F0 cos pt:

m x'' + b x' + k x = F0 cos pt
x'' + 2γ x' + ω0² x = f0 cos pt

with 2γ = b/m, ω0² = k/m, f0 = F0/m.

Amplitude

Steady state: x = A cos(pt − φ). Substituting and comparing the coefficients of cos(pt − φ) and sin(pt − φ):

A (ω0² − p²) = f0 cosφ
A (2γ p)     = f0 sinφ

A = f0 / √[(ω0² − p²)² + 4γ²p²] ,   tanφ = 2γp/(ω0² − p²)

Amplitude resonance

A is maximum when D = (ω0² − p²)² + 4γ²p² is minimum. dD/d(p²) = −2(ω0² − p²) + 4γ² = 0:

p_r² = ω0² − 2γ²      (p_r < ω0)
A_max = f0 / (2γ √(ω0² − γ²))

Resonance condition: the driving frequency equals √(ω0² − 2γ²), very nearly the natural frequency if damping is small. For γ → 0, A becomes very large at p = ω0.

Sharpness of resonance

It is the rate at which the amplitude falls as p departs from p_r.

 A |      /\  small damping (sharp)
   |     /  \
   |    / __ \___ large damping (flat)
   |___/_/____\\___ p
        p_r

• Small γ gives a tall, narrow peak (sharp); large γ gives a low, broad peak.
• It is measured by the quality factor Q = ω0/2γ = ω0/Δω, where Δω is the half-power bandwidth.