Rotordynamics and critical speeds

The purchasing department has found a cheaper forging for the TM-4 multistage pump shaft. The production shaft is 65 mm in diameter at the impellers; the proposed one is 55 mm, lighter and easier to machine, and the stress calculations from MEC-360 still pass. Farah is not convinced, and she asks Owen to check the effect on the rotor's first critical speed, the speed at which the shaft's own bending resonance is excited by its unbalance. A machine that runs near a critical speed can vibrate violently even when well balanced, and it may rub its wear rings and seize. Rotordynamics, the study of vibrating rotating shafts, is a specialized field, but its basic ideas follow directly from Lessons 4, 6 and 10.

The Jeffcott rotor

The simplest rotor model, studied by Föppl and by Jeffcott in the late nineteenth and early twentieth centuries, is a disk of mass m at the middle of a massless flexible shaft of lateral stiffness k, running in rigid bearings. The disk's centre of mass is offset from the shaft's geometric centre by a small eccentricity e. When the shaft spins at ω, unbalance makes the shaft bow and its centre whirl around the bearing axis in a circle of radius r_w. For steady synchronous whirl with damping ζ, the whirl amplitude is

This is the rotating-unbalance curve of Lesson 4. The speed at which the whirl peaks, ω_n, is the critical speed: the natural frequency of the shaft's lateral bending, excited by its own unbalance. The program's table shows the behaviour. Below the critical speed, the shaft bows toward the heavy spot (the phase lag is small). At the critical speed the whirl is limited only by damping, ten times the eccentricity for ζ = 0.05, and the phase lag is 90°. Above it, the phase lag approaches 180°: the heavy spot moves toward the inside of the whirl orbit, and at high speeds the disk spins about its own centre of mass, with the shaft centre whirling at a radius approaching e. This self-centring above the critical speed is why some machines, such as steam turbines, are designed to run above their first critical speed, passing through it quickly during run-up.

Stiff and flexible rotors

A rotor running well below its first critical speed is called stiff, or rigid: its bending deflection under unbalance is small, and a low-speed shop balance holds at full speed. A rotor running above one or more critical speeds is flexible: its deflected shape changes with speed, and it may need multi-plane balancing at several speeds (Lesson 6). Most industrial pumps are designed as stiff rotors, with the first dry critical speed above the running speed. The TP-80, with its short overhung shaft, has a first critical speed of several thousand rpm, comfortably above 1,760 rpm. Multistage pumps like the TM-4, with long bearing spans carrying several impellers, are where critical speeds become a design problem.

Estimating the first critical speed

For a shaft carrying several masses, Rayleigh's method of Lesson 2 extends naturally. Using the static deflection curve under the weights of the masses as the assumed mode shape gives

where δ_i is the static deflection at mass i. Like all Rayleigh estimates, it is slightly high. A second classical formula, due to Dunkerley, combines the frequencies the shaft would have with each mass alone, \(1/\omega_1^2 \approx \sum 1/\omega_i^2\), and gives a lower bound. The true value lies between the two, and when they agree within a few percent, either is good enough for a first check. Detailed rotordynamic analysis, required by industry standards for critical-service pumps, uses finite element rotor models with bearing and seal stiffness and damping coefficients, and computes damped natural frequencies and responses as functions of speed.

# Rotordynamics: the Jeffcott rotor and the first critical speed of the TM-4 multistage pump
import math
# 1. Jeffcott rotor: disk with eccentricity e on a massless shaft, synchronous whirl amplitude
e = 20e-6                                  # mass eccentricity, m
zeta = 0.05
print(" speed/critical   whirl amplitude/e   phase deg")
for r in (0.5, 0.8, 0.95, 1.0, 1.05, 1.2, 2.0, 4.0):
    amp = r**2 / math.sqrt((1 - r**2)**2 + (2 * zeta * r)**2)
    ph = math.degrees(math.atan2(2 * zeta * r, 1 - r**2))
    print(f"{r:14.2f} {amp:19.2f} {ph:11.0f}")
print("Well above the critical speed the disk spins about its own centre of mass (amplitude -> e).")

# 2. TM-4 rotor: first lateral critical speed by Rayleigh's method and by Dunkerley's formula
g = 9.81; E = 200e9; rho = 7850
L = 1.10                                   # bearing span, m
imps = [(0.36, 4.2), (0.48, 4.2), (0.62, 4.2), (0.74, 4.2)]   # impeller positions (m) and masses (kg)
def critical(d, n_el=44):
    I = math.pi * d**4 / 64
    masses = list(imps) + [((j + 0.5) * L / n_el, rho * math.pi * d**2 / 4 * L / n_el) for j in range(n_el)]
    def defl(x):                           # static deflection at x from all weights (simply supported)
        tot = 0.0
        for a, m in masses:
            P = m * g; b = L - a
            if x <= a: tot += P * b * x * (L**2 - b**2 - x**2) / (6 * E * I * L)
            else:      tot += P * a * (L - x) * (2 * L * x - x**2 - a**2) / (6 * E * I * L)
        return tot
    num = sum(m * defl(a) for a, m in masses); den = sum(m * defl(a)**2 for a, m in masses)
    w_ray = math.sqrt(g * num / den)
    inv = 0.0                              # Dunkerley: 1/w^2 = sum of 1/w_i^2 for each mass alone
    for a, m in masses:
        b = L - a; k = 3 * E * I * L / (a**2 * b**2)
        inv += m / k
    return w_ray * 60 / (2 * math.pi), 60 / (2 * math.pi) / math.sqrt(inv)
n_run = 3560
for d in (0.050, 0.055, 0.060, 0.065, 0.070):
    nr, nd = critical(d)
    print(f"Shaft {d*1000:.0f} mm: Rayleigh {nr:6.0f} rpm, Dunkerley {nd:6.0f} rpm (lower bound); "
          f"margin above running speed {(nr - n_run)/n_run:6.1%}")
# 3. Wet critical speed: wear-ring and balance-drum fluid stiffness (Lomakin effect) adds support stiffness
d = 0.065; nr, _ = critical(d)         # the production shaft
k_shaft = (2 * math.pi * nr / 60)**2 * (sum(m for _, m in imps) + 0.5 * rho * math.pi * d**2 / 4 * L)
for k_fluid in (0.0, 2e6, 5e6):
    m_eff = k_shaft / (2 * math.pi * nr / 60)**2
    n_wet = math.sqrt((k_shaft + k_fluid) / m_eff) * 60 / (2 * math.pi)
    print(f"  added fluid stiffness {k_fluid/1e6:3.0f} MN/m: first critical about {n_wet:5.0f} rpm")
# 4. Oil whirl in the journal bearings appears just below half running speed
print(f"\nOil whirl would show at about 0.42-0.48 x running speed: {0.42*n_run/60:.1f}-{0.48*n_run/60:.1f} Hz;"
      f" it becomes whip if running speed exceeds twice the first critical ({2*nr:.0f} rpm)")

 speed/critical   whirl amplitude/e   phase deg
          0.50                0.33           4
          0.80                1.74          13
          0.95                6.63          44
          1.00               10.00          90
          1.05                7.51         134
          1.20                3.16         165
          2.00                1.33         176
          4.00                1.07         178
Well above the critical speed the disk spins about its own centre of mass (amplitude -> e).
Shaft 50 mm: Rayleigh   3002 rpm, Dunkerley   2919 rpm (lower bound); margin above running speed -15.7%
Shaft 55 mm: Rayleigh   3498 rpm, Dunkerley   3398 rpm (lower bound); margin above running speed  -1.7%
Shaft 60 mm: Rayleigh   4007 rpm, Dunkerley   3889 rpm (lower bound); margin above running speed  12.6%
Shaft 65 mm: Rayleigh   4525 rpm, Dunkerley   4388 rpm (lower bound); margin above running speed  27.1%
Shaft 70 mm: Rayleigh   5050 rpm, Dunkerley   4894 rpm (lower bound); margin above running speed  41.9%
  added fluid stiffness   0 MN/m: first critical about  4525 rpm
  added fluid stiffness   2 MN/m: first critical about  5132 rpm
  added fluid stiffness   5 MN/m: first critical about  5927 rpm

Oil whirl would show at about 0.42-0.48 x running speed: 24.9-28.5 Hz; it becomes whip if running speed exceeds twice the first critical (9050 rpm)

The program models the TM-4 rotor as a simply supported shaft with a 1.10 m bearing span, four 4.2 kg impellers and the shaft's own mass in 44 elements. For the production 65 mm shaft, Rayleigh's method gives a first dry critical speed of 4,525 rpm and Dunkerley's formula 4,388 rpm, so the true value lies between them: 23 to 27 percent above the 3,560 rpm running speed. For the proposed 55 mm shaft, the estimate falls to about 3,500 rpm, within 2 percent of running speed: the pump would run on its critical speed. Shaft stiffness scales with d⁴, while the rotor mass falls much less because the impellers do not change, so the 15 percent smaller diameter lowers the critical speed by about 23 percent. Farah's suspicion is confirmed, and the cheaper forging is rejected.

Wet and dry critical speeds

The calculation above gives the dry critical speed, with the pump empty and the rotor supported only by its bearings. In a running pump, the close clearances at the wear rings, the balance drum and the interstage bushings carry a pressure drop, and the flow through them produces a centring force proportional to displacement, called the Lomakin effect. These annular seals act as extra bearings with significant stiffness and damping. The program shows the effect of adding a fluid stiffness of 2 or 5 MN/m at the impellers: the first critical rises from 4,525 rpm to about 5,130 and 5,930 rpm. The wet critical speed of a multistage pump can therefore be well above the dry one. The effect weakens as the clearances wear, because the stiffness falls with increasing clearance. A pump designed to rely on the Lomakin effect for its separation margin can drift toward its critical speed as it wears, which is why pump standards ask for both dry and wet analyses and consider worn clearances. The program's fluid stiffness values are illustrative; real values come from seal analysis or testing.

Separation margins

Standards for rotating machinery specify a separation margin between the operating speed range and any critical speed, so that the machine does not run in the region of high amplification. API 610, for centrifugal pumps in petroleum, petrochemical and natural gas service, and its companion guidance on rotordynamics, set margins that depend on the amplification factor of the mode; a well-damped mode needs less margin than a sharply peaked one. Margins of 15 to 25 percent are typical in practice, and a value of 20 percent is a common design target. Owen's report quotes the margin for each shaft diameter: 27 percent for the 65 mm shaft against the dry critical, more against the wet; −2 percent for the 55 mm shaft. For the TM-4, which is sold on variable-speed drives up to 3,800 rpm for some boiler applications, the 65 mm shaft's margin against 3,800 rpm is still 19 percent dry, which is why Farah keeps it.

Oil whirl and other self-excited instabilities

Not all rotor vibration is forced by unbalance. A shaft in a lightly loaded plain journal bearing can be driven into a self-excited whirl by the bearing's own oil film. The oil in the clearance travels at about half the journal's surface speed, and when the journal is displaced, the pressure field produces a cross-coupled force perpendicular to the displacement, which can feed energy into a forward whirl at a frequency of about 0.42 to 0.48 times running speed. This oil whirl is often stable in amplitude, but if running speed exceeds about twice the first critical speed, the whirl frequency locks onto the critical speed and becomes oil whip, a violent instability that does not go away when speed is raised further. For the TM-4, whose journal bearings were designed in MEC-360, whirl would appear at 25 to 28 Hz, and whip would require speeds above about 9,000 rpm, far above its operating range. Pressure-dam, tilting-pad and other bearing designs are used where whirl is a risk. Other self-excited mechanisms, such as internal friction and aerodynamic cross-coupling in compressors, are beyond this course.

Gyroscopic effects and Campbell diagrams

A spinning disk that tilts as the shaft bends generates gyroscopic moments, which stiffen forward whirl modes and soften backward ones, so the natural frequencies of a rotor change with speed. A Campbell diagram plots the natural frequencies of the rotor against speed, together with lines for the excitation frequencies, 1×, 2× and vane pass. Intersections of the 1× line with natural frequency curves are the critical speeds. For the TM-4, the impellers are narrow and close to the bearings, so gyroscopic effects are small, and the simple model is adequate. On overhung fans and turbines with large disks far from the bearings, they can be large, and a Campbell diagram from a rotor model is essential.