Critical Speed

Critical speed in rotating machinery is the shaft rotational speed at which the forcing frequency of rotation (spin speed) coincides with one of the natural frequencies of the rotor–shaft system, producing resonance and large lateral vibration (whirl). In practice, critical speeds are determined by the lateral bending natural frequencies (bending modes) of the shaft with attached disks, and they can be excited by static or dynamic unbalance, periodic forces, or internal coupling (gyroscopic/damping effects). Simple rotor models (Jeffcott rotor) capture the first critical speed; more complex distributed-parameter models (Euler–Bernoulli beams with disks) give multiple modal critical speeds. Important practical concepts: avoid operating at or near critical speeds, use run-up/run-down procedures, increase stiffness or reduce mass to raise the critical speed, or add damping to limit resonant amplitudes. For high-speed rotors, gyroscopic moments split forward and backward whirl frequencies, modifying critical speeds.

Governing FormulaKey formulas and principles (SI units): 1) Relation between natural frequency and stiffness/mass (single-degree-of-freedom/Jeffcott rotor): ω_n = sqrt(k/m) (rad/s) f_n = ω_n / (2π) (Hz) n_crit = f_n * 60 (rpm) where k is lateral stiffness (N/m), m is disk mass (kg). 2) Rotating unbalance (synchronous) force: F_unbalance = m_u * e * ω^2 (N) where m_u is unbalance mass (kg), e is eccentricity (m), ω is angular speed (rad/s). 3) Damped natural frequency (single-DOF): ω_d = ω_n * sqrt(1 - ζ^2) where ζ = c / (2 * sqrt(k m)) is damping ratio. 4) Beam (Euler–Bernoulli) scaling for first bending frequency (order-of-magnitude): ω ∝ sqrt(E I / (m' L^4)) where E is Young's modulus (Pa), I is area moment of inertia (m^4), m' is mass per unit length (kg/m), L is span (m). 5) Equivalent stiffness for a beam with a concentrated disk: For a simply supported beam with a concentrated load at midspan, deflection δ = P L^3/(48 E I) => k_eq = 48 E I / L^3. For a cantilever with an end load, δ = P L^3/(3 E I) => k_eq = 3 E I / L^3. 6) Rotordynamic equation (linearized, including gyroscopic effects): M ẍ + (C + Ω G) ẋ + K x = F(t) where M is mass matrix, C is damping matrix, G is gyroscopic matrix, Ω is spin speed, K is stiffness matrix. Gyroscopic terms cause forward/backward frequency split.

Knowledge Check

10 Questions

1.Which statement correctly defines the (first) critical speed of a rotating shaft with a disk?

2.For a simple Jeffcott rotor modeled as a single disk of mass m on a massless elastic shaft with lateral stiffness k, which expression gives the first critical speed in radians per second?

3.Numerical: A disk of mass m = 50.0 kg is mounted on a flexible shaft with effective lateral stiffness k = 2.00×10^5 N/m. Estimate the first critical speed in rpm. (Use ω_n = sqrt(k/m), f = ω_n/(2π), rpm = 60 f.)

4.How does increasing the lateral stiffness k of a rotor–shaft system (constant disk mass) generally affect the first critical speed?

5.Which scaling law best describes the dependence of the first bending natural frequency ω of a uniform Euler–Bernoulli beam (length L, flexural rigidity E I, mass per unit length m') in SI units (order-of-magnitude)?

6.Numerical: A rotor has an unbalance mass m_u = 0.020 kg at eccentricity e = 0.010 m. If the shaft spins at 3000 rpm, what is the magnitude of the synchronous unbalance force F = m_u e ω^2? (Use ω = 2π * rpm/60.)

7.What is the primary effect of structural damping on the response of a rotor operated at its critical speed (resonance)?

8.Modal property question: Which statement about higher bending modes and critical speeds is correct for a rotor–shaft system?

9.Numerical: A simply supported shaft of length L = 1.00 m with diameter d = 0.020 m (E = 210 GPa) carries a concentrated disk mass m = 10.0 kg at midspan. Using the equivalent stiffness for a simply supported beam with a midspan load k_eq = 48 E I / L^3, compute the approximate first critical speed in rpm. (Take I = π d^4 / 64.)

10.How do gyroscopic effects influence critical speeds of a moderately high-speed rotor?