Resonance

Resonance is the condition in which a vibrating system is driven at a frequency close to one of its natural (undamped) frequencies, producing large steady-state amplitudes when damping is small. For linear systems the natural frequency (undamped) of a single-degree-of-freedom (SDOF) is wn = sqrt(k/m). Damping (characterized by the damping ratio zeta = c/(2*m*wn)) reduces peak amplitude and shifts the frequency of maximum response. In multi-degree-of-freedom systems, resonance occurs when the excitation frequency matches any modal natural frequency; the response shape follows the corresponding mode shape. Rotating unbalance generates a harmonic forcing of magnitude m*e*Omega^2 that can excite resonance when the rotation speed Omega equals a natural frequency. Practical vibration control aims to avoid operating at resonant speeds or to add sufficient damping or frequency separation.

Governing FormulaKey formulas (SI units): - Undamped natural frequency: wn = sqrt(k/m) [rad/s] - Damping ratio: zeta = c/(2*m*wn) [dimensionless] - Damped natural frequency: wd = wn*sqrt(1 - zeta^2) [rad/s] - Steady-state amplitude for harmonic force F0 sin(omega t): X = (F0/k)/sqrt((1 - r^2)^2 + (2*zeta*r)^2), where r = omega/wn - Amplitude at resonance (r = 1, for force input): X_res = (F0/k)/(2*zeta) (valid for small zeta) - Transmissibility for base excitation: TR = sqrt((1 + (2*zeta*r)^2)/((1 - r^2)^2 + (2*zeta*r)^2)) - Rotating unbalance force amplitude: F_unb = m*e*Omega^2 - Quality factor: Q = 1/(2*zeta) Assumptions: linear time-invariant SDOF unless otherwise stated; SI units (kg, N, m, s).

Knowledge Check

10 Questions

1.Which statement correctly defines resonance for a single-degree-of-freedom (SDOF) vibrating system?

2.For an undamped SDOF with mass m and stiffness k, which formula gives the natural frequency (rad/s)?

3.A mass m = 2.00 kg is attached to a spring of stiffness k = 800 N/m (undamped). What is the undamped natural frequency wn in rad/s? (Assume linear SDOF.)

4.An SDOF has m = 1.00 kg, k = 1000 N/m, and viscous damping ratio zeta = 0.10. What is the damped natural frequency wd (rad/s)?

5.For an SDOF under a harmonic force F(t)=F0*sin(omega*t) with F0 = 10 N, stiffness k = 1000 N/m, and damping ratio zeta = 0.05, what is the steady-state amplitude at resonance (omega = wn)?

6.For base excitation, transmissibility TR = sqrt((1 + (2*zeta*r)^2)/((1 - r^2)^2 + (2*zeta*r)^2)). For zeta = 0.20 and excitation frequency ratio r = 2.0, what is TR (approx)?

7.Which statement best describes resonance in a multi-degree-of-freedom (MDOF) system?

8.A rotor-bearing system has rotor mass m = 5.00 kg and unbalance eccentricity e = 0.001 m. The support stiffness is k = 5.0e4 N/m. Assuming negligible damping, what is the amplitude of the rotating (centrifugal) force F_unb at the rotation speed equal to the system natural frequency (i.e., at resonance)?

9.The quality factor Q is related to damping ratio zeta by Q = 1/(2*zeta). For zeta = 0.02, what is Q?

10.A linear SDOF has stiffness k = 200 N/m and initial mass m1 = 1.00 kg. Adding a tip mass delta m = 0.50 kg increases the mass to m2 = 1.50 kg. By approximately what percentage does the undamped natural frequency decrease? (Compute percent decrease = (wn1 - wn2)/wn1 * 100%)