Optimizing gas turbine rotordynamics: a robust design approach for vibration analysis

When engineering cutting edge products, not every design parameter can be padded with a deterministic factor of safety. This can result in an over-engineered product with heavier and more expensive components. On the other hand, the designs that function well in a deterministic setup (e.g., spherical cows in a vacuum), could break down the moment real-world chaos hits them.

Even with a relatively simple beam-model of a gas turbine, determining the non-linear relation between a combination of the factors and the respective response can be complex. Therefore, instead of relying on one approach for design conception, engineers can wrap the structural skeleton of a deterministic, physics-based design with stochastic methods in a probabilistic skin.

Beyond deterministic critical speeds: designing for real-world uncertainties

Campbell diagrams are used to determine the ideal operation ranges for which the natural frequencies of a gas-turbine system do not match the operating speeds or excitation orders. The points where the matches happen are known as the critical speeds or frequencies which are to be avoided in an ideal design and operation of the turbine.

However, in the real world these theoretically ideal designs are additionally subjected to vastly different temperatures, material microstructures, machining tolerances, and other, possibly unaccounted, operating conditions. Accounting for uncertainties results in designs that are more resilient to the alteration or loss of functionality and performance under most circumstances. Once these uncertainties are adopted in vibrational design by a Rotordynamics Engineer, non-domain experts in NVH/vibrational design such as a Design Engineer could turn the results from this exercise into safer design choices about rotor diameters and turbine operating ranges. In this scenario, a simulation process and data management (SPDM) framework like VOLTA could be useful for cross-functional engineering teams: SPDM stores the rotordynamics model and automates the related simulations so that design engineers can quickly and reliably choose rotor geometries and operating ranges that are statistically safe, with full traceability of how those decisions were made.

Simulating rotor variations and creating campbell diagrams with PERMAS

PERMAS is a high-performance finite element analysis (FEA) software optimized for large-scale models and offers fast solution times through efficient solvers and advanced parallelization techniques. Since it supports a wide range of physical analyses within a single unified environment, including vibrations and acoustics (including fluid-structure interaction) and linear and nonlinear dynamic analysis, it’s particularly suitable for complex engineering simulations.
We used a beam-model of a 5-kN, single spool, thrust turbine design created in PERMAS. The geometry, frequency loads, constraints and the bearing stiffness matrices can be quickly set up in PERMAS’ pre-processor VisPER.

In this study, the outer diameters of the compressor and turbine rotors are varied while keeping the rest of the boundary and initial conditions unchanged.

The results from the PERMAS simulation are the rotor-shaft system’s natural frequencies which are then plotted on the Campbell diagram against the excitation frequency of the system using Python.

These natural frequencies can be classified as forward and backward whirls (FW and BW), depending on their precession direction. Then, the synchronous excitation line is overlaid on the Campbell diagram and the intersection points where the rotor-shaft system’s rotational frequency matches the natural frequencies are identified as the critical speeds. This post-processing step is also undertaken via Python.

When approaching the above steps manually or through a script, the Rotordynamics Engineer will need to do the following for each design iteration:

1. change the diameters of the compressor and the turbine rotors,
2. carry out the simulations of the updated geometries,
3. run the Python script to: calculate the intersection of the synchronous excitation line with the natural frequencies; plot the critical speeds on the Campbell diagram.

This workflow results in a total of six critical speeds denoted as 1FW, 1BW, 2FW, 2BW, 3BW, and 4BW at the respective whirl modes for the rotor-shaft system. The numbering denotes different vibration modes, while FW and BW indicate forward and backward whirl behaviour respectively.

Accelerating vibration simulation analysis with ESTECO software

1. Automate PERMAS simulations into the modeFRONTIER workflow

The above steps can be actionized in modeFRONTIER’s simulation workflow using the PERMAS connector. VisPER’s sampling wizard is used to create a PERMAS control file with the task loop that will be run by modeFRONTIER after changing the input values of the rotor diameters in the model. In VisPER’s sampling wizard, the desired parametrization of the Design Elements must be defined by the Design Variables. Optionally, the default values and the upper and lower bounds can be defined, which can then be directly read into modeFRONTIER from the model files.
Such workflow automation can support multidisciplinary design optimization (MDO) studies where multiple objectives, constraints and engineering domains must be evaluated simultaneously.

During the introspection of the PERMAS model in modeFRONTIER, data from the control file, the model files and the results are translated into the input and output parameters. From the results, a matrix with all the natural frequencies can be selected to be transferred to the Python node for operation by the subsequent nodes in the process. This Python node is automatically executed as part of the simulation workflow and creates a Campbell diagram for the given set of rotor diameters.
After the main input and output parameters, their flow and the process is developed in modeFRONTIER by a Rotordynamics Engineer, the workflow is ready to be uploaded to VOLTA digital engineering platform.

2. Scale and democratize PERMAS simulations with VOLTA

The modeFRONTIER’s simulation workflow can be shared with teams or individuals and executed on selected computational resources using VOLTA digital engineering platform. The simulation workflow model, now part of a VOLTA project, also points to all the files necessary for it to run (its dependencies), such as the control file, the model files and the Python script. Any execution of the simulation process from VOLTA creates a snapshot of all of these dependencies. This way the Rotordynamics Engineer is always able to tell which versions of the dependencies and workflow were used to generate which results. This level of traceability is critical in today’s engineering world and especially important in a collaborative environment.

To determine the general behaviour of the system, an initial design of experiments (DOE) study is carried out in VOLTA with the rotor-diameter Design Variables in meters (m) as input parameters and the critical speeds in Hertz (Hz) as output parameters. The diameter values are sampled in the entire design range, resulting in a dataset that can be analyzed to gain initial understanding of the system.

Using the results of this initial DoE, a sensitivity analysis between the input rotor diameters and the output critical speeds at respective modes is undertaken in VOLTA. The influences on critical speeds whose dependencies are spread over several rotor diameters is hence quantified in a correlation matrix.

The correlation matrix chart shows that the critical speeds for the 1BW and the 2BW modes are dependent on almost all the rotor diameters. Hypothesisation of the influence of a single or a combination of rotor-diameter variations on these critical speeds is difficult, if not impossible. Hence, these critical speeds are selected for the Robust Design Explorations.

3. Conduct robust design exploration with VOLTA

Robustness is the ability of a system's responses to remain insensitive to variations of input parameters and variations in their operating environment with minimal alteration or loss of functionality. This technique additionally accounts for the robustness of the design combinations defined by the rotor diameters in the parametric input region of interest. By employing the robustness and reliability domain in VOLTA where the designs defined by the combinations of the rotor diameters are the nominal designs and can additionally be checked for robustness in the critical speeds at the 1BW and the 2BW modes. Under this analysis, VOLTA samples additional points in the vicinity of the nominal design point as per a selected distribution.

These additional points can be sampled using the Latin Hypercube-Monte Carlo sampling. This mode of constrained sampling splits the distribution into several intervals with the required probability and samples one design point per interval.
For example, for a normal distribution around a nominal design point, the mean of the distribution is just the nominal design point defined by the diameter values. The standard deviations of the diameter values around the nominal design points are determined by:

• manufacturing tolerances,
• thermal effects,
• material properties.

Constraints on the output parameters are added to as well to identify the designs violating requirements due to the deviations in the inputs. Robustness of the nominal design can be inferred from the number of unfeasible designs in yellow. Occurrence of a number of yellow designs would mean that, given a standard deviation in the distribution of the input diameter values with the nominal design as the mean, the critical speed fluctuates more than the acceptable tolerance.
In this manner, any of the designs in the region of interest can be checked for robustness. The robust designs are those for which none of the designs from their extended region of uncertainty violates the respective constraints.

Key takeaways

• Anyone on the engineering team (for example, the Design Engineer) can run the workflow on VOLTA to: evaluate safer operating ranges, identify sensitivities and explore new design variations.
• Team permissions and access are carefully controlled. Design Engineers can run the workflow, view and download results, but cannot modify the workflow. Rotordynamics engineers can see who used their workflow, when and how. This can help identify misuse of the model (e.g. outside of its intended domain), problems with the process and possible avenues for improvement.
• In the process of Robust Design, we leave the comfort of a “guaranteed” safety factor and venture into the transparency of calculated risks. The team embraces uncertainty, designs around it and shares cross‑disciplinary knowledge of the experts from several aspects of engineering design.
• Stochastic models are always based on data and an event that has never happened before can still occur (Black Swan). The question is: if it occurs, would you rather trust just the physics analysed in the model, or the physics in the model validated for robustness under uncertainty?