Frequency Response

Frequency response describes how a dynamic system (typically modeled as a linear time-invariant system) responds to sinusoidal excitation as a function of excitation frequency. For vibration problems the frequency response is often expressed for single-degree-of-freedom (SDOF) or modal coordinates and is complex-valued, containing both magnitude and phase information. Key physical parameters that control the frequency response are mass m, stiffness k, damping c (or damping ratio ζ), and natural frequency ωn = sqrt(k/m). Resonance occurs near ω ≈ ωn where the response amplitude peaks; the peak amplitude and bandwidth are governed by damping. For multi-degree systems, modal superposition expresses the global frequency response as a sum of modal contributions, each with its own modal frequency and damping.

Governing FormulaGoverning and commonly used formulas (SI units): - Natural frequency: ωn = sqrt(k/m) [rad/s], fn = ωn/(2π) [Hz]. - Damping ratio: ζ = c/(2 m ωn). - Steady-state displacement amplitude for SDOF under harmonic force F(t)=F0 sin(ωt): X(ω) = F0 / sqrt((k - m ω^2)^2 + (c ω)^2) or expressed normalized: X(ω) = (F0/k) / sqrt((1 - r^2)^2 + (2 ζ r)^2), where r = ω/ωn. - Phase angle (displacement relative to force): φ(ω) = atan( (c ω) / (k - m ω^2) ). - Base-excitation transmissibility (ratio of mass displacement to base displacement): T(r,ζ) = sqrt( (1 + (2 ζ r)^2) / ((1 - r^2)^2 + (2 ζ r)^2) ). - Resonant steady-state amplitude for force excitation (r = 1): X_res = (F0/k) * (1/(2 ζ)) = F0/(c ωn). (Assumes linear SDOF, steady-state.) - Half-power (−3 dB) bandwidth relation (for small damping): ζ ≈ (ω2 - ω1) / (2 ωn), where ω1, ω2 are frequencies at which |X| = |X|max/√2. - Rotating unbalance force amplitude: F_u = m_e e ω^2 (radial harmonic force from eccentric mass), where m_e is unbalanced mass and e is eccentricity.

Knowledge Check

10 Questions

1.Which statement best defines the frequency response function (FRF) for a linear vibration system?

2.An SDOF system has mass m = 2.0 kg and stiffness k = 8000 N/m. What is the natural frequency fn in Hz?

3.A 1.0 kg SDOF oscillator has k = 10000 N/m and damping ratio ζ = 0.02. It is driven by F(t)=10 sin(2π·10 t) N (force amplitude F0 = 10 N at f = 10 Hz). What is the steady-state displacement amplitude (magnitude) approximately?

4.For base excitation of an SDOF isolator, transmissibility T is given by T = sqrt((1+(2ζr)^2)/((1−r^2)^2+(2ζr)^2)). If ζ = 0.05 and frequency ratio r = 2.0, what is T approximately?

5.For a lightly damped SDOF system under harmonic force excitation, the resonant (peak) displacement amplification relative to static deflection is approximately equal to which expression?

6.An experimental FRF of an SDOF shows a peak at ωn = 100 rad/s. The half-power frequencies are measured at ω1 = 95 rad/s and ω2 = 105 rad/s. Using the half-power bandwidth method, what is the estimated damping ratio ζ?

7.Consider the phase of the displacement FRF relative to an applied harmonic force for an SDOF. As excitation frequency ω → ∞ (much greater than ωn), what does the phase approach?

8.In a multi-degree-of-freedom system, when the excitation frequency is very close to the nth modal frequency, which modal property most directly controls the width and attenuation of the resonance peak in the frequency response?

9.A rotor has an unbalance mass m_e = 0.020 kg located with eccentricity e = 0.005 m and spins at f = 30 Hz. What is the amplitude of the resulting harmonic unbalance force F_u (N) acting on the bearing housing? (Use ω = 2πf.)

10.An SDOF system has m = 2.0 kg, k = 20000 N/m (so ωn = 100 rad/s), and damping ratio ζ = 0.10. If a harmonic force F0 = 50 N is applied at resonance (ω = ωn), what is the steady-state displacement amplitude X_res (m)?