Natural Frequency

Natural frequency is the frequency at which a mechanical system vibrates freely (no forcing) about an equilibrium position when disturbed. For linear single-degree-of-freedom (SDOF) systems the undamped natural frequency depends only on mass and stiffness. Damping reduces oscillation amplitude and slightly shifts the oscillation frequency to the damped natural frequency. In forced vibration, resonance occurs when the excitation frequency equals the system's natural frequency, producing maximum steady-state response for low damping. Modal natural frequencies are intrinsic properties of continuous or multi-degree systems and are independent of the amplitude of motion (linear theory).

Governing FormulaCommon formulas (SI units): - Undamped natural angular frequency: ω_n = sqrt(k / m) (rad/s) - Undamped natural frequency: f_n = ω_n / (2π) (Hz) - Damped natural frequency: ω_d = ω_n * sqrt(1 - ζ^2) (rad/s), f_d = ω_d / (2π) - Damping ratio: ζ = c / (2 * m * ω_n) (dimensionless) - Forced rotating unbalance force magnitude: F(t) = m_u * e * ω^2 (N) - Transmissibility (base excitation, displacement ratio): T(r, ζ) = sqrt((1 + (2 ζ r)^2) / ((1 - r^2)^2 + (2 ζ r)^2)), where r = ω / ω_n - Resonance condition (undamped/forced): excitation frequency ω = ω_n (rad/s) Assumptions: linear system, small displacements, SI units (kg, N/m, s, Hz).

Knowledge Check

10 Questions

1.Which definition correctly describes the natural frequency of a linear single-degree-of-freedom (SDOF) system?

2.A mass m = 2.00 kg is attached to a spring of stiffness k = 2000 N/m (undamped SDOF). What is the natural frequency f_n in Hz?

3.For an SDOF with m = 1.00 kg, k = 1000 N/m and damping ratio ζ = 0.10, what is the damped natural frequency f_d in Hz?

4.A rotor system has an undamped natural angular frequency ω_n = 314 rad/s. At what rotational speed (in revolutions per minute, rpm) will resonance occur due to rotating unbalance?

5.If the stiffness k of an SDOF system is increased by a factor of 4 while mass m is unchanged, how does the undamped natural frequency f_n change?

6.Which statement about the effect of viscous damping on natural frequency is correct?

7.For base excitation of an SDOF system with damping ratio ζ = 0.05, what is the displacement transmissibility T at resonance (excitation frequency equal to natural frequency, r = 1)?

8.Two springs, each k = 1000 N/m, are attached in parallel to a mass m = 1.00 kg (SDOF). What is the undamped natural frequency f_n (Hz)?

9.If an SDOF system's mass is increased by 20% (m_new = 1.20 m_old) while stiffness remains constant, approximately by what percent does the undamped natural frequency f_n change?

10.For a lightly damped SDOF under harmonic forcing, the frequency at which the steady-state response amplitude peaks (peak frequency ω_p) is given by which relation (valid when ζ < 1/√2)?