What is critical speed in rotor dynamics?

Critical speed is the rotational speed at which the excitation frequency (1× per revolution) matches the rotor's natural frequency. Near this speed, residual unbalance forces drive large-amplitude whirling. It is one of the most critical design parameters for turbines, compressors, and electric motors.

What is the Jeffcott rotor model?

Proposed by H.H. Jeffcott in 1919, it represents a rotor as a lumped mass m at mid-span on a massless flexible shaft with stiffness k and viscous damping c. The critical speed is Nc = 60/(2π)×√(k/m) rpm. Despite its simplicity, it captures the key resonance physics and serves as a first approximation for real multistage rotors.

How does damping ratio affect the response?

Higher damping ratio ζ reduces the peak amplitude at resonance (amplitude = e/(2ζ) at critical speed). In practice, ζ = 0.05–0.2 is typical, provided by journal bearings, squeeze-film dampers, or other damping elements. The rotor never fully avoids resonance, but adequate damping makes rapid traversal through critical speed safe.

How can I move the critical speed away from the operating range?

Increase shaft stiffness k (larger shaft diameter, shorter bearing span) to raise Nc, or reduce rotor mass m. API standards require at least ±20% separation margin between critical speed and the operating speed range. For super-critical operation (running above Nc), rapid traverse through the critical speed during startup and shutdown is essential.

› Rotor Dynamics Critical Speed Calculator Back

Rotor Dynamics · Jeffcott Model

Rotor Dynamics Critical Speed Calculator

Enter rotor mass, shaft stiffness, eccentricity, and damping ratio to compute critical speed Nc. Visualize the unbalance response curve and animated whirl orbit to understand resonance behavior.

Parameters

Rotor Mass m 10.0 kg

Shaft Stiffness k 1.0×10⁶

×10⁵ N/m (slider × 10⁵)

Eccentricity e 0.10 mm

Damping Ratio ζ 0.05

Operating Speed n 1000 rpm

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Critical Speed Nc (rpm)

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ωc (rad/s)

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Peak Amplitude (mm)

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Op. Amplitude (mm)

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Speed Ratio n/Nc

Jeffcott Rotor Equations

$$N_c = \frac{60}{2\pi}\sqrt{\frac{k}{m}}\text{ rpm}$$

Unbalance response amplitude:

$$A = \frac{e \cdot r^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$$

$r = \omega/\omega_c$ (speed ratio)

Design note: API standards require critical speeds to be at least ±20% away from the operating speed range. For super-critical rotors, rapid traversal through the critical speed during start/stop is key.

What is Rotor Critical Speed?

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What exactly is "critical speed" for a rotor? Is it just the maximum speed it can spin?

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Not quite! It's the speed at which the rotor vibrates most violently. Basically, every rotor has a natural frequency. When the spinning speed matches this frequency, even a tiny unbalance causes huge vibrations. Try moving the "Operating Speed" slider in the simulator up and down—you'll see the vibration amplitude spike at one specific speed.

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Wait, really? So the vibration gets huge only at that one speed? What determines where that spike happens?

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Exactly! The location of that spike—the critical speed—is set by the rotor's mass and the shaft's stiffness. A heavier rotor or a floppier (less stiff) shaft lowers the critical speed. In the simulator, you can test this: increase the "Rotor Mass" or decrease the "Shaft Stiffness" and watch the peak on the graph shift left to a lower speed.

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Okay, but the graph doesn't go to infinity. What flattens the peak? And what's the "eccentricity" parameter for?

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Great observation! Real systems have damping—like friction in bearings—which limits the vibration. That's the "Damping Ratio" slider. The "Eccentricity" is the distance between the rotor's center of mass and its geometric center; it's the source of the unbalance force. A larger 'e' means a bigger shaking force. Play with the damping slider to see how it tames the peak, and increase eccentricity to see how it raises the entire response curve.

Physical Model & Key Equations

The simulator uses the Jeffcott rotor model—a point mass on a flexible, massless shaft. The fundamental critical speed is the rotational speed at which the system's natural frequency is excited.

$$N_c = \frac{60}{2\pi}\sqrt{\frac{k}{m}}\ \text{rpm}$$

$N_c$: Critical speed (revolutions per minute). $k$: Shaft stiffness (N/m). $m$: Rotor mass (kg). This shows the critical speed increases with stiffness and decreases with mass.

When running at any speed, the vibration amplitude due to mass unbalance is calculated. The response depends on the speed ratio, eccentricity, and damping.

$$A = \frac{e \cdot r^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}\quad \text{where}\quad r = \frac{n}{N_c}$$

$A$: Vibration amplitude (m). $e$: Mass eccentricity (m). $r$: Speed ratio. $\zeta$: Damping ratio. The denominator causes the amplitude to peak near $r=1$ (critical speed), but damping ($\zeta$) controls the peak's height.

Real-World Applications

Turbomachinery Design (Gas Turbines, Compressors): Engineers must calculate critical speeds to ensure the machine's operating range is safely away from them. For instance, a jet engine's high-pressure spool might operate at 15,000 rpm, so designers adjust bearing stiffness and rotor dimensions to push its first critical speed well above or below this range to avoid catastrophic vibration during flight.

API Standard Compliance for Pumps & Fans: Industry standards like API 610 for pumps mandate that critical speeds be at least 20% away from the operating speed. This simulator helps designers quickly check if a proposed rotor design (with its specific mass and stiffness) meets this margin of safety before detailed and expensive prototyping.

Super-Critical Rotor Operation (Large Steam Turbines): Some large rotors must operate above their first critical speed. The key is to accelerate through the critical speed quickly during startup to minimize vibration exposure. The damping and unbalance response calculated here are crucial for planning these transient maneuvers.

Diagnosing Vibration Problems in Maintenance: If a motor or fan develops high vibration at a specific speed, maintenance teams can use this principle to diagnose the issue. By comparing the problem speed to the calculated critical speed, they can determine if they are dealing with a resonance problem (requiring a balance or stiffness change) or another fault like misalignment.

Common Misconceptions and Points to Note

First, the misconception that "the critical speed is a dangerous speed that must never be passed." While it is indeed dangerous to operate steadily at that speed, many high-speed rotating machines are designed with "supercritical operation" in mind. The key is to properly design the damping and to pass through quickly during startup and shutdown. For example, a certain turbo-molecular pump has its first critical speed around 10,000 rpm, but its operating speed is 80,000 rpm. During shutdown, vibration increases momentarily in this 10,000 rpm range, but it can be passed safely thanks to damping design.

Next, setting the "equivalent mass" or "equivalent stiffness" used in simulators. This is often the most challenging part in practice. For instance, when considering a mass (m) concentrated at the center of a shaft, you don't simply substitute the actual rotor mass. You need to add a portion of the shaft's own mass (typically 1/3 to 1/2). For stiffness (k), it's often not just the shaft stiffness but the stiffness of the bearings and support structure that dominates. Be mindful as the type of bearing (ball bearing vs. journal bearing) and oil film stiffness can have a significant impact.

Finally, the idealistic notion that "vibration will disappear if the eccentricity (e) is reduced to zero." In reality, unbalance inevitably occurs due to machining/assembly errors, material inhomogeneity, thermal deformation, and wear during operation. Using the tool to see how amplitude changes with e is for predicting the effect of balance correction. For example, you can qualitatively understand that reducing eccentricity from 10μm to 5μm (balance correction) can nearly halve the amplitude at resonance. Rather than aiming for perfect zero, a cost-effective sense of keeping it within the allowable amplitude limit is crucial.

Related Engineering Fields

The concept of this critical speed calculation is, from a different perspective, essentially the same as "resonance problems" in various engineering fields. The first that comes to mind is structural dynamics. The vibrations buildings and bridges experience from wind or earthquakes are also analyzed by the relationship between natural frequency (equivalent to critical speed) and forced vibration force (equivalent to unbalance force). Modal analysis, a technique used for this, is also employed to solve complex vibration modes of rotors.

Another is control engineering, particularly the field of "vibration control." You saw in the simulator that increasing the damping ratio (ζ) suppresses the resonance peak. This is similar to the operation of "increasing feedback gain for stabilization" in active vibration control. In fact, for rotors using magnetic bearings, the design of this control system directly determines the damping characteristics and influences how easily the critical speed can be passed.

Furthermore, it is deeply related to material mechanics and fatigue analysis. The large vibration amplitude at resonance generates repeated stress in the shaft, significantly reducing the material's fatigue life. For instance, cases where a pump shaft operated for extended periods near its critical speed fractured earlier than its design life are not uncommon. The stress amplitude and cycle count obtained from vibration analysis become direct input data for fatigue life prediction.

For Further Learning

Once you are comfortable with the Jeffcott rotor model, the next step is extending it to a multi-degree-of-freedom system. Actual machine rotors are closer to "continuous bodies" with multiple disks. These are modeled with multiple masses and springs, and the equations of motion are set up using matrices ($[M]\{\ddot{x}\} + [C]\{\dot{x}\} + [K]\{x\} = \{F\}$). Solving this yields the 1st, 2nd, 3rd... multiple critical speeds and their corresponding vibration modes (deflection shapes). This forms the basis for creating a "Campbell diagram."

Mathematically, it's important to follow the derivation process of the frequency response function we dealt with. This is the forced vibration solution of a second-order linear ordinary differential equation. Try checking the flow in a textbook: starting from the solution without damping, then adding the damping term, and further adding the unbalance forcing term... With this understanding, the shape of the graph output by the tool (why the peak rises at r=1 and becomes rounded with damping) will make intuitive sense.

The next topic directly relevant to practical work is the mechanism of "unstable vibration". Resonance like critical speed is not called "unstable." There are phenomena unique to rotating bodies, such as "oil whip" or "high-speed instability," where vibration grows divergently upon reaching a certain speed range. These are self-excited phenomena where damping changes from "absorbing" to "supplying" vibration energy, representing one of the greatest challenges in supercritical operation design. It is recommended to start by learning the concept of "Rayleigh's stability criterion" for judging stability.