Good Rotordynamics Performance for Industrial Rotating Machinery

(*) Written by Minhui He (Ph.D., BRG Machinery Consulting)

Industrial rotating machinery typically faces two types of vibration problems: unbalance response and stability. A rotor always has some unbalance. When rotating, the unbalance weight generates a dynamic force that rotates at the same frequency as the rotor’s rotational frequency. Therefore, the unbalance force is synchronous, leading to synchronous or 1X vibration. 

The main concern for unbalance response is resonance. When the excitation frequency matches the system’s natural frequency, a small excitation will cause large vibration. Figure 1 shows the Bode plot of a typical rotating machine. From low speed, the vibration amplitude increases since the unbalance force is proportional to the square of the speed (ω2). The vibration reaches the first peak when the speed matches the first natural frequency of the system (ωn1). In other words, the machine is running at its first critical speed when ω=ωn1. Passing the first critical speed, the vibration decreases as the system is moving away from this resonance situation. The vibration remains relatively low until the rotor speed starts to approach the second critical speed. A rotor system with good unbalance response should have the following characteristics:

• Machine should have low vibration within its operating speed range. 
• Machine’s operating speed range should stay sufficiently away from any critical speeds. This situation is quantified by the separation margin (SM) in Figure 1. 
• The critical clearances (e.g. bearings and annular seals) should not be rubbed when the machine passes through the critical speeds. There are some subtleties in this requirement, but basically, we want the peaks to be smooth, which means the associated modes are well damped. 

Figure 1. Bode plot of unbalance response

America Petroleum Institute (API) has specific requirements in each of those three aspects in order to mitigate the resonance concern due to rotor unbalance.

The other vibration concern is rotor instability. Unlike the unbalance response, this type of vibration is self-excited, which means there is a feedback loop and the excitation force is controlled by the vibration itself. Figure 2 (a) shows the stable vibration of a single degree of freedom (SDOF) oscillator. As shown in this figure, the vibration amplitude decreases in time, and the rate of decrease is dependent on the damping. Conversely, if the system is unstable, the vibration will increases in time, which is shown in (b).

Figure 2a. Comparison of stable and unstable single degree of freedom oscillator [Stable]

Figure 2. Comparison of stable and unstable single degree of freedom oscillator [Unstable]

Rotor instability usually manifest itself as subsynchronous vibration. As shown in Figure 3, the compressor always has some 1X vibration due to the unbalance. However, some subsynchronous vibration starts to show up above 7000 rpm. Keep increasing the speed, the subsynchronous component becomes very high at approximately 9500 rpm. The frequency of this subsynchronous vibration is the natural frequency of a rotor mode, which is typically its first forward mode as in this case.

Figure 3. Waterfall plot of a compressor becoming unstable

The destabilizing force may come from various components in turbomachinery, including fixed geometry fluid film bearings, annular seals, internal friction with shrunk-on sleeves, etc. Meanwhile, damping is the mechanism to keep the system stable. The rotor system becomes unstable if the destabilizing force exceeds the damping force, leading to negative net damping.

The stability level is usually represented by the logarithmic decrement (δ), which indicates the net damping. A positive δ indicates a stable rotor while a negative value indicates an unstable one. API 617 (compressor) requires the logarithmic decrement be no less than 0.1 after all relevant components are included in the calculation.

Widely used in rotating machinery, fluid film journal bearings have strong influence on a machine’s overall rotordynamics and reliability. To achieve good rotordynamic performance, the following bearing design guidelines can be applied: 

• The bearing should have proper stiffness so that the critical speeds are placed sufficiently away from the operating speed range (see Figure 1). 
• The bearing should provide adequate damping since damping is the mechanism to suppress vibration, and the bearing is the main source of damping in the rotor system.
• To make the damping effective, the bearing cannot be overly stiff. Damping dissipates the vibrational energy by squeezing the oil film. An overly stiff bearing would “pin” the journal and prevent the necessary squeeze motion across the bearing clearance.
• Compared to fixed geometry bearings, tilting pad bearings have negligible destabilizing force. Therefore, they are often used in high speed and/or light load applications where rotor instability is a concern. 
• To optimize the bearing dynamics, many design parameters can be fine-tuned, including bearing clearance, length, preload and offset, etc. 
• For better stability, asymmetry is preferred, which means the bearing’s stiffness in one direction (Kxx) is noticeably different from that in the orthogonal direction (Kyy). However, symmetry (Kxx=Kyy) is often preferred to achieve better unbalance response. 

In conclusion, a machine with good rotordynamic performance should be operating sufficiently away from its critical speeds with low vibration; it should have adequate damping so that the rotor is stable and the relevant critical speeds are well damped. As a critical component, the fluid film bearings can be optimized in design to help achieve such characteristics.