At-Speed Balancing: The Fundamentals

This is the fifth article in a series on the topic of rotor balancing, and the information presented up to this point has focused exclusively on low-speed balancing. Low-speed balancing is sufficient for rotors that perform as rigid bodies under all operating conditions. But rotors are mechanical structures, and when spun fast enough, they deflect elastically under their own unbalance forces and dynamic behavior. Such deflection relocates mass away from the axis of rotation, changing the balance state from what the low-speed balancing machine measured. This “flexible rotor” effect commonly occurs with long, slender, high-speed rotors — typically those of multistage steam turbines and centrifugal compressors. This article explains that phenomenon and how at-speed balancing resolves it:

• the basics of critical speeds and mode shapes,
• how amplitude and phase behave through resonance,
• why a two-plane low-speed balance is inadequate for flexible rotors, and
• what at-speed balancing accomplishes.

Critical Speeds and Mode Shapes

Every mechanical structure has natural frequencies, the frequencies at which it would vibrate on its own if deflected and released. When a force acts on a rotor at a natural frequency, even relatively small forces can produce large deflections — a condition called resonance. The once-per-revolution unbalance force can excite resonance when the rotor’s speed coincides with one of its natural frequencies. Such a speed is known as a critical speed.

Natural frequencies are properties of the entire mechanical system, not just the rotor alone. Though the rotor’s structure is typically the dominant influence, the rotor’s support structure also plays a role. The stiffnesses of the bearings and pedestals participate, so the same rotor can show different critical speeds on different supports.

Each natural frequency has a mode shape, the deflection pattern the rotor takes when vibrating at that frequency (Figure 1). Mode shapes are characterized by nodes, points of zero deflection, and anti-nodes, points of maximum deflection. At the first bending mode, a rotor supported on two bearings bows into a single arch with an anti-node near midspan and nodes near the bearings. At the second bending mode it takes an S-shape, with the two halves moving in opposition and a node near midspan. Higher modes add more arches and more nodes.

Amplitude and Phase Through Resonance

Run a flexible rotor up in speed while recording vibration amplitude and phase, and the record follows a characteristic pattern (Figure 2). Amplitude climbs into a sharp peak at its first critical speed, falls away, and climbs into another peak farther up the speed range at its second critical speed. Through each peak the phase of the vibration shifts by roughly 180 degrees.

Near a peak, the rotor’s deflected shape approximates the mode shape of that critical, so the vibration amplitude measured along the rotor follows it. The bowed shape rotates with the rotor, so a point on the shaft surface holds a nearly constant deflected position. A stationary probe reports the passing bow as once-per-revolution vibration.

The height and sharpness of each peak depend on damping. The oil films in the bearings, the seals, and the surrounding process fluid all contribute to damping by dissipating energy. The more they dissipate, the lower and broader the resonant peak. Lightly damped rotors show tall, narrow peaks, with the phase shift concentrated over a small speed range.

The phase shift through a critical carries physical meaning. Two angular locations describe the situation: the heavy spot, the angular location of the rotor’s mass eccentricity, and the high spot, the angular location of peak displacement as measured by a vibration proximity probe. See Figure 2 in the article on single-plane balancing.

Well below the first critical, the rotor deflection is nearly aligned with the unbalance force, so the high spot lies close to the heavy spot. As speed approaches a critical, the response lags the force, and at the critical the lag is roughly 90 degrees. Well above the critical the lag approaches 180 degrees, and the deflection settles opposite the heavy spot. This condition is called self-centering because the deflection causes the rotor’s mass center to move toward the axis of rotation.

The change in phase lag follows from what governs the response. Below a critical, the rotor’s elastic stiffness governs, and the deflection follows the force. Above it, inertia governs, and the deflection opposes the force.

As described in the article on two-plane balancing, the central principal axis of inertia is the axis about which the rotor’s mass is evenly distributed. When the rotor bows opposite the heavy spot, the principal axis of inertia moves toward the axis of rotation, which is fixed in place by the bearings. The rotor thus turns about its mass center, while its geometric center whirls about the axis of rotation at a radius approaching the mass-center eccentricity, creating the vibration a proximity probe observes.

Distributed Unbalance and Modal Unbalance

Real unbalance is distributed. Manufacturing tolerances, material density variation, fits, and assembly stack-up scatter small eccentricities along the rotor’s full length. For a rigid rotor, how the unbalance is distributed does not matter. As established in the article on two-plane balancing, any unbalance distribution in a rigid rotor may be reduced to two unbalance vectors in two correction planes. Correcting those two vectors remedies the effects of all the distributed unbalance as long as the rotor remains rigid.

For a flexible rotor the distribution is the whole problem. Each bending mode responds to the component of the unbalance distribution that matches its shape. Unbalance located at an anti-node drives that mode strongly. Unbalance located at a node drives it not at all. The component of the distribution that a given mode responds to is called the modal unbalance for that mode, and each mode has its own. Significant modal unbalance can remain after successful low-speed balancing, though its effects may not be apparent until the rotor is spun up to service speed.

A single case makes the failure concrete (Figure 3). Take a uniform rotor between two bearings with one concentrated unbalance at midspan, and balance it on a low-speed machine using correction planes near the two ends. The machine measures the rigid resultants, the net unbalance force and moment that a rigid rotor would transmit. The corrections cancel them, and the finished rotor runs clean at balancing speed. It would pass the residual unbalance verification test, as described in the article on the low speed balancing procedure. The midspan unbalance remains in place, opposed by two end corrections whose sum is equal to it in magnitude and opposite in angular location.

Now run the rotor toward its first bending critical. The first mode’s deflection is greatest at midspan and nearly zero at the ends, so the midspan unbalance drives the mode with full effect, while the end corrections push back with almost none from planes where the mode barely moves. The modal unbalance the low-speed balance never touched takes over, and the rotor bows into its first-mode arch. The bow relocates the rotor’s own mass off the axis of rotation, which raises the unbalance force, which deepens the bow. That feedback is the mechanism behind the sharp amplification near a critical. The result is a rotor that passes every low-speed measurement and still vibrates heavily near the critical, delivering once-per-revolution force into its bearings and reducing internal clearances at the point of greatest bow.

The case also explains why correction-plane position matters for flexible rotors. The end planes were adequate for the rigid resultants and nearly powerless over the first mode. A correction in a plane near midspan, where the mode moves most, would have addressed both. Selecting correction planes for their authority over the modes of concern is a defining feature of flexible-rotor balancing, and the number of planes grows with the number of modes.

What At-Speed Balancing Accomplishes

At-speed balancing measures the rotor’s response at speeds where the modes of concern participate, enabling correction of the modal unbalance. The essential change is the speed at which the measurement occurs. A low-speed machine can only report the rigid resultants because the mode shapes never manifest during low-speed balancing. But on a machine that can run the rotor up through a critical, the modal unbalance appears in the response as a magnitude and phase that a correction can take aim at.

At-speed balancing is not a substitute for low-speed balancing. On the contrary, it supplements it. Flexible rotors undergo low-speed balancing too. The rigid resultants must still be measured and corrected to tolerance on the low-speed machine, because their effects are present at every speed. What a low-speed machine cannot address is the modal unbalance, and that is the component uniquely targeted by at-speed balancing.

Next in the series: At-Speed Balancing: Machines and Procedure — at-speed balancing facilities, correction-plane selection for flexible rotors, the sequence of an at-speed balancing job, the plane-by-plane influence coefficient method and modal balancing, and specifications and acceptance.