Introduction / Context
Critical or whirling speed is the rotational speed at which a rotating shaft–disc system experiences resonance due to coincidence of the running speed with a natural frequency, leading to large lateral vibrations. Designers must understand the variables that set this speed.
Given Data / Assumptions
- Uniform shaft with a mounted disc.
- Small eccentricity between mass center and geometric center.
- Linear, classical whirling theory applies (small deflections).
Concept / Approach
The lateral natural frequency depends on shaft stiffness (a function of length, diameter, and boundary conditions) and the attached mass distribution. Eccentricity introduces a rotating unbalance force that excites the mode strongly near the natural frequency.
Step-by-Step Solution
1) Shaft span/length enters stiffness k ∝ EI/L³ for typical beam-like behavior; longer spans reduce stiffness and lower critical speed.2) Disc mass (often related to its diameter and thickness) changes the effective mass m at the critical section; higher mass lowers natural frequency and critical speed.3) Eccentricity e creates an unbalance force m·e·ω² that peaks near resonance; while e does not change the natural frequency itself, it determines the response amplitude and practical safe speed, effectively influencing usable critical speed considerations.
Verification / Alternative check
Rayleigh or Dunkerley estimates show explicit dependence on stiffness (length) and mass; design charts account for unbalance via eccentricity.
Why Other Options Are Wrong
- Single-parameter answers ignore coupled dependence on stiffness (length), mass (disc), and excitation via eccentricity.
- None of these contradicts theory and practice.
Common Pitfalls
- Confusing effect on natural frequency (stiffness, mass) with excitation amplitude (eccentricity); both matter to practical critical speed operation.
Final Answer
All of these
Discussion & Comments