10.1 Introduction to Rotor Dynamics
Rotor dynamics is the branch of mechanical engineering concerned with the lateral and torsional vibration of rotating shafts. Every rotating machine — turbines, compressors, electric motors, spindles, turbochargers — has one or more critical speeds at which the rotor's natural frequency coincides with the excitation frequency (typically the running speed), producing dangerously large amplitudes.
The consequences of operating at or near a critical speed include:
- Excessive bearing loads and premature bearing failure.
- Rubbing between rotor and stator (seal damage, blade tip rub).
- Shaft fatigue from cyclic bending stress.
- Catastrophic failure from shaft fracture or bearing seizure.
Rotor dynamics is not optional analysis for high-speed machinery — it is a design requirement. This chapter develops the Jeffcott rotor model, critical speed calculation, forced response, balancing, and the Campbell diagram framework for multi-mode systems.
10.2 The Jeffcott Rotor Model
The Jeffcott rotor (also called the de Laval rotor) is the simplest model that captures the essential physics of rotor critical speeds. It consists of:
- A rigid disk of mass $m$ mounted at the midpoint of a massless elastic shaft.
- The shaft is supported at its ends by bearings modeled as linear springs of stiffness $k$.
- The disk has an eccentricity $e$ (distance between the geometric center and the center of mass).
10.2.1 Equations of Motion
In a rotating reference frame at angular velocity $\omega$, the lateral displacement $z$ of the disk center of mass satisfies:
$$m\ddot{z} + c\dot{z} + kz = m e \omega^2 \cos(\omega t)$$
where $c$ is the viscous damping coefficient. The forcing term $m e \omega^2$ arises from the centrifugal force of the unbalanced mass.
10.2.2 Undamped Natural Frequency
Setting $c = 0$ and $F = 0$ (free vibration):
$$\omega_n = \sqrt{\frac{k}{m}}$$
This is the critical speed in rad/s. In RPM:
$$N_{cr} = \frac{\omega_n \times 60}{2\pi} = \frac{30}{\pi}\sqrt{\frac{k}{m}}$$
10.3 Critical Speeds and Resonance
A critical speed is any rotational speed at which the excitation frequency equals a natural frequency of the rotor-bearing system. For the Jeffcott model, there is one critical speed at $\omega = \omega_n$.
10.3.1 Physical Mechanism
Below critical speed ($\omega < \omega_n$): The shaft deflects in the direction of the unbalance force. The orbit is small.
At critical speed ($\omega = \omega_n$): The forcing frequency matches the natural frequency. Amplitude grows without bound (undamped) or to a large finite value (damped). The shaft whirls — the deflected shape rotates at the shaft speed.
Above critical speed ($\omega > \omega_n$): The rotor tends to rotate about its center of mass rather than its geometric center. This is called self-centering or supercritical operation. The orbit amplitude decreases with further speed increase.
10.3.2 Supercritical vs. Subcritical Operation
| Regime | Speed range | Behavior |
|---|---|---|
| Subcritical | $\omega < \omega_n$ | Shaft bends toward unbalance; amplitude increases with speed |
| Critical | $\omega \approx \omega_n$ | Resonance; maximum amplitude |
| Supercritical | $\omega > \omega_n$ | Self-centering; amplitude decreases |
Most high-speed machinery (turbines, compressors) operates supercritically — the rotor must pass through the critical speed during start-up and shut-down.
10.4 Forced Response and Magnification Factor
The steady-state amplitude of the Jeffcott rotor under unbalance excitation is:
$$X = \frac{e \cdot r^2}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$$
where:
- $r = \omega / \omega_n$ is the frequency ratio
- $\zeta = c / (2m\omega_n)$ is the damping ratio
- $e$ is the eccentricity (unbalance)
10.4.1 Magnification Factor
The magnification factor (MF) normalizes the response relative to the static deflection $e$:
$$\mathrm{MF} = \frac{X}{e} = \frac{r^2}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$$
At resonance ($r = 1$):
$$\mathrm{MF}_{res} = \frac{1}{2\zeta}$$
For $\zeta = 0.05$ (typical for lightly damped rotors): $\mathrm{MF}_{res} = 10$. The orbit amplitude is 10× the unbalance eccentricity.
10.4.2 Phase Relationship
The phase angle between the unbalance force and the response is:
$$\phi = \arctan\left(\frac{2\zeta r}{1 - r^2}\right)$$
- Below critical: $\phi \approx 0°$ (response in phase with force).
- At critical: $\phi = 90°$.
- Above critical: $\phi \to 180°$ (response opposes force — self-centering).
The phase shift through 90° at resonance is the most reliable experimental indicator of a critical speed.
10.5 Damping in Rotor Systems
Damping in rotors arises from multiple sources:
10.5.1 Bearing Damping
- Fluid-film bearings: Viscous damping from the oil film. Moderate $\zeta$ (0.02–0.10).
- Rolling element bearings: Very low damping ($\zeta < 0.01$). Dangerous near critical speeds.
- Squeeze-film dampers: Deliberately added oil film between bearing housing and support. $\zeta$ = 0.1–0.3.
10.5.2 Structural Damping
- Shaft material internal friction (hysteretic damping).
- Foundation and support structure compliance.
- Typically $\zeta$ = 0.001–0.01 for steel shafts.
10.5.3 Aerodynamic Damping
- Blade and disk windage.
- Seal rub forces (negative damping — destabilizing).
- Significant in turbomachinery.
10.5.4 Design Implication
Low damping means high amplification at critical speeds. Never operate continuously at $r \approx 1$ with $\zeta < 0.02$ — bearing loads and shaft stress will exceed design limits.
10.6 Balancing
Balancing reduces the unbalance eccentricity $e$, thereby reducing the forcing amplitude and the response at all speeds (especially critical).
10.6.1 Static vs. Dynamic Balancing
- Static balance: Center of mass lies on the rotation axis. Achieved by single-plane correction. Sufficient for thin disks ($L/D < 0.5$).
- Dynamic balance: Eliminates both static unbalance and couple unbalance (two-plane correction). Required for long rotors ($L/D > 0.5$).
10.6.2 Balance Quality Grades (ISO 1940)
The permissible residual unbalance is specified by balance quality grade $G$:
$$e_{per} = \frac{1000 G}{\omega}$$
where $e_{per}$ is in μm and $\omega$ is in rad/s. Common grades:
| Grade | Application |
|---|---|
| G 40 | Crankshaft drives, agricultural machinery |
| G 6.3 | General machinery, electric motors |
| G 2.5 | Gas/steam turbines, machine tool spindles |
| G 1.0 | Precision grinding spindles, gyroscopes |
10.6.3 Field Balancing
For installed rotors, in-situ balancing (modal balancing, influence coefficient method) corrects unbalance without disassembly. Requires vibration measurements at bearing locations and trial weight placement.
10.7 Multi-Mode Systems and the Campbell Diagram
Real rotors are not Jeffcott rotors — they have distributed mass, multiple disks, flexible bearings, and gyroscopic effects. A real rotor has multiple natural frequencies (modes), each with its own critical speed.
10.7.1 The Campbell Diagram
The Campbell diagram plots natural frequency vs. rotational speed. Each mode appears as a curve. Critical speeds occur where a mode curve intersects the synchronous excitation line ($\omega_{excitation} = \omega_{rotation}$).
For a simple two-support rotor with a single disk:
- First bending mode: The Jeffcott critical speed.
- Second bending mode: Higher frequency, typically 3–5× the first critical.
- Rigid body modes: Very low frequency (foundation motion).
10.7.2 Gyroscopic Effects
A spinning disk has angular momentum $\mathbf{L} = I_p \omega \mathbf{k}$. When the shaft bends, the gyroscopic moment opposes the bending, stiffening the rotor. This causes:
- Split criticals: The forward and backward whirl modes have different frequencies.
- Speed-dependent natural frequencies: Campbell curves are not horizontal lines but increase with speed.
For a disk on a massless shaft with polar moment $I_p$ and transverse moment $I_d$:
$$\omega_{n,forward}^2 = \frac{k}{m}\left(1 + \frac{I_p \omega}{I_d \omega_n}\right)$$
10.8 Pass-Through Critical Speeds
Every supercritical machine must pass through at least one critical speed during start-up and shut-down. The transient response during this passage can exceed the steady-state resonant amplitude.
10.8.1 Slow Passage
If the acceleration rate is slow compared to the decay rate of the transient (sweep rate $q = \alpha / \omega_n^2 \ll 1$), the rotor reaches near-steady-state amplitude at resonance:
$$X_{trans} \approx X_{res,ss} = \frac{e}{2\zeta}$$
10.8.2 Fast Passage
If the sweep is rapid ($q \gg 1$), the amplitude at resonance is reduced:
$$X_{trans} \approx X_{res,ss} \left(1 - e^{-2\pi\zeta/\sqrt{q}}\right)$$
10.8.3 Design Rules for Pass-Through
- Accelerate quickly through the critical (but not so fast as to excite higher modes).
- Add damping (squeeze-film dampers) to limit resonant amplitude.
- Balance to tight grade to minimize forcing.
- Verify bearing clearance accommodates the resonant orbit.
- Monitor vibration during first start-up (Bode plot: amplitude and phase vs. speed).
10.9 Instabilities in Rotor Systems
Beyond critical speeds, rotors can experience self-excited instabilities:
10.9.1 Oil Whirl and Oil Whip
In fluid-film journal bearings, the oil film can exert a tangential force that drives forward whirl at approximately half the shaft speed (oil whirl, $\omega_{whirl} \approx 0.5\omega$). If the whirl frequency locks onto a natural frequency, oil whip occurs — a destructive self-excited vibration.
Prevention: Short bearings ($L/D < 0.5$), preloaded bearings, tilting pads, or squeeze-film dampers.
10.9.2 Dry Whirl and Rub
Contact between rotor and stator (seal, bearing, blade tip) creates friction that can induce dry whirl — a full annular rub at sub-synchronous frequency.
10.9.3 Aerodynamic Cross-Coupling
In high-pressure turbomachinery, the fluid forces on impeller/blade stages can have negative damping (energy input from the flow), leading to flutter above a threshold speed.
10.10 Rotor Design Checklist
- Calculate first critical speed from shaft stiffness and disk mass.
- Plot Campbell diagram including gyroscopic effects for all significant modes.
- Identify all critical speeds within the operating speed range and pass-through range.
- Calculate forced response at each critical with actual unbalance and damping.
- Verify bearing loads at critical speeds (dynamic load = static load + unbalance force).
- Specify balance grade per ISO 1940.
- Design pass-through strategy (acceleration rate, damping, monitoring).
- Check stability margins for oil whirl, rub, and aerodynamic instabilities.