#### Critical Speed Maps in Turbomachinery

For many years, one of the primary analysis techniques has been undamped critical speed analysis, and this technique is still performed today for the preliminary estimation of critical speeds and mode shape characteristics. First, let’s take a look at what this kind of analysis technique is and what it involves.

Critical speeds and their associated mode shapes are most influenced by the support (bearing and pedestal structure) stiffness magnitudes, the support locations, and the rotor’s mass and stiffness properties. Based on this, the following definition can be given. A critical speed map is a graph representing the effect of rotor support stiffness on the critical speed of the rotor. A general view of the critical speed map is shown in Figures 1-2.

With this definition in hand, the next question would be what is critical speed? Critical speed is the rotational speed that corresponds with a structure’s resonance frequency (or frequencies). A critical speed appears when the natural frequency is equal to the excitation frequency. The excitation may come from unbalance that is synchronous with the rotational velocity or from any asynchronous excitation.

To ensure stable vibration-free operation, the critical rotor speeds should not coincide with the operating speed, which must be considered when designing the machine. If such a case occurs during the design phase of the structure, engineers need to decide how to improve the rotor system to ensure stable vibration-free operation of the machine. For this, many analyses are required. One such analysis is the critical speed map.

Consider an example of a critical speed map as shown in Figure 3.

As shown on the map, the first mode is a cylindrical mode and the second mode is a conical mode. For these modes, as the stiffness of the support becomes zero, the vibration mode becomes a rigid mode with a natural frequency of zero. As the stiffness of the support becomes infinite, the vibration mode becomes a simply supported beam mode. On the other hand, for the third and fourth modes, when the stiffness of the support is zero, the vibration mode is an elastic mode with free boundary conditions. When the stiffness of the support is infinite, the vibration mode is a higher beam mode with simply supported boundary conditions.[2]

With the critical speed map defined, the final step in the analysis is to define what the actual support characteristics are in order to estimate the critical speeds. Using the results from bearing and support analysis techniques, speed-dependent total bearing/support principal stiffnesses (Kxx, Kyy) are cross-plotted on the map (Figure 2). Speeds where the support coefficient curves intersect with the critical speed curves are the potential critical speeds of the system.

Now that we’ve covered the basics of critical speeds and critical speed maps, let’s get into the more practical aspects. Specifically, what information can be gathered from a critical speed map to help an engineer understand how to improve the rotor system in order to ensure stable, vibration-free operation of the machine, and how that info is interpreted.

It is already known that in order to ensure stable vibration-free operation, the critical rotor speeds should not coincide with the operating speed. On the critical speed map, these are the places where the critical speed intersects with the operating speed or its range. These getting in the operating speed range will correspond to a certain stiffness of the supports. Therefore, the engineer will need to change the stiffness of the support to one in which the case described earlier will not occur. To do this, it will be enough to choose the stiffness of the supports, which is located to the left or to the right of the stiffness at which dangerous vibrations occur. It is important to note that with the newly selected stiffness, the case should not be repeated where the critical rotor speeds coincide with the operating speed.

The critical speed map is an important part of the overall analysis when designing rotors. It serves both as the preliminary estimation of critical speeds and mode shape characteristics, as well as a reference point of how the machine can be improved. When designing a rotor, it is important to understand the effect of the bearing stiffness on the critical speeds. The critical speed map can be used to show the evolution of the critical speeds of the rotor with respect to the bearing’s stiffness.

For the tasks described above, engineers need software that can help solve their problems and at the same time simplify their work. The good news is that there are already comprehensive standards and sophisticated engineering tools that make this task much easier for engineers. AxSTREAM RotorDynamics provides its users with comprehensive rotor simulation based on recognized approaches and API standards and allows engineers to take into account all the important dynamic effects of the rotor that affect the accuracy of the results.

To learn more about AxSTREAM RotorDynamics, schedule a meeting with our team at Info@SoftInWay.com.

Reference

1. API TR 684-1 1ST ED (2019). API Standard Paragraphs Rotordynamic Tutorial: Lateral Critical Speeds, Unbalance Response, Stability, Train Torsionals, and Rotor Balancing; First Edition
2. Kaneko Y., Kanki H., Kawashita R. Steam turbine rotor design and rotor dynamics analysis. Advances in Steam Turbines for Modern Power Plants. 2017. С. 127–151. URL: https://doi.org/10.1016/b978-0-08-100314-5.00007-5