Rotating Systems

Rotating systems (rotors) are mechanical assemblies that spin about an axis and exhibit dynamics governed by mass, stiffness, damping, gyroscopic moments and any static or dynamic unbalance. Unbalance (mass offset) produces a synchronous rotating force proportional to m·e·Ω^2, which excites lateral vibration and can lead to resonance when the spin speed coincides with a natural frequency (critical speed). Gyroscopic effects from spinning inertia couple lateral motions and split forward/backward whirl frequencies. Damping controls resonance amplitudes and transmissibility. Engineering analysis uses equations of motion, modal properties, Campbell diagrams (frequency vs. spin speed) and balancing methods (single-/two-plane) to reduce synchronous response.

Governing FormulaKey equations and definitions (SI units): - Unbalance force (single mass): F_unb = m_u · e · Ω^2, where m_u = unbalanced mass (kg), e = eccentricity (m), Ω = spin speed (rad/s). (Ω = 2π·RPM/60) - Single-DOF lateral EOM (forced vibration): m x¨ + c x˙ + k x = F(t). - Natural frequency: ω_n = sqrt(k/m) (rad/s), f_n = ω_n/(2π) (Hz). - Damping ratio: ζ = c/(2·sqrt(m·k)). - Steady-state amplitude for harmonic force F0 cos(Ωt): |X| = F0 / sqrt((k - mΩ^2)^2 + (cΩ)^2). - Resonant amplitude for harmonic unbalance at Ω = ω_n: X_res = e/(2ζ) (for unbalance excitation m_u e Ω^2 with c = 2 m ζ ω_n). - Transmissibility for base excitation (displacement): TR = sqrt((1 + (2ζr)^2) / ((1 - r^2)^2 + (2ζr)^2)), where r = Ω/ω_n. - Gyroscopic moment: M_g = I_p · Ω · ˙θ (approx. for small precession rate ˙θ), where I_p is polar moment of inertia of the rotor. Assumptions: linear system, small vibrations (linearization valid), rigid-body discs where indicated, viscous damping.

Knowledge Check

10 Questions

1.A single small unbalanced mass m_u = 0.02 kg is located with eccentricity e = 1.0 mm on a rotor spinning at 3000 rpm. What is the magnitude of the synchronous unbalance force F_unb (N)? (Use Ω = 2π·RPM/60.)

2.A simple rotor model (single DOF lateral) has mass m = 10 kg and bearing stiffness k = 4.0×10^5 N/m. What is the natural frequency in Hz?

3.For the rotor in the previous question (m = 10 kg, k = 4.0×10^5 N/m), if viscous damping c = 400 N·s/m, what is the damping ratio ζ?

4.Using ω_n = 200 rad/s for a rotor, what is the corresponding critical speed in rpm (the speed at which spin equals this natural frequency)?

5.For base-excited transmissibility TR = sqrt((1+(2ζr)^2)/((1-r^2)^2+(2ζr)^2)), compute TR for r = Ω/ω_n = 1.5 and ζ = 0.05.

6.Which statement describes the typical effect of rotor spin (increasing Ω) caused by gyroscopic moments on forward and backward whirl frequencies for a disk-like rotor?

7.A rotor with eccentricity e = 1.0 mm and damping ratio ζ = 0.02 is operating at its resonant speed (Ω = ω_n). What is the steady-state lateral amplitude X_res (m) due to the unbalance?

8.If a shaft is modeled with 4 lumped-disc masses for lateral bending, how many natural bending modes (lateral natural frequencies) will this discretized system exhibit (neglecting rigid-body axial/ torsional DOF)?

9.A rotor disk has polar moment of inertia I_p = 0.20 kg·m^2, spins at 3000 rpm and is precessing slowly with ˙θ = 1.0 rad/s. What is the magnitude of the approximate gyroscopic moment M_g (N·m)?

10.An unbalance on a rotor produces a lateral forcing of a specific frequency relative to the shaft spin. What is the forcing frequency of the unbalance in the stationary frame?