I have recently had the opportunity to work on a few steam turbine blade failure investigations and have found a whole new world of engineering simulation that I had not been exposed to. Analysis of turbine blades is its own animal and even though steam turbines have been in use in power generation for a very long time, the physics of power generation is so complex there are many areas that are still not well understood. Being new to turbine analysis I thought my experiences as I lean steam turbine analysis may be useful to others. The first topic I would like to describe has been fundamental to all of the turbine projects that I have been involved with: Modal analysis of a bladed disk row. In each investigation, modal analysis was used to calculate the natural frequencies and mode shapes for a particular stage of blades. This is a valuable tool in determining if the blades are operating near a resonant condition that could be responsible for a failure.
As you can find in any vibrations textbook, modal analysis is an eigenvalue procedure in which the eigenvalues of the equation of motion are the square of the natural frequencies and the eigenvectors are the mode shapes. For blade analysis, SimuTech Group uses an in-house developed code to run a modal Finite Element Analysis on turbine blades. This program is called BLADE.
A portion of the bladed disk is modeled in BLADE which usually consists of a 360°/N sector, where N is the number of blades in the row. The mass and stiffness matrices for the bladed disk sector are then reduced to a superelement containing selected fewer degrees of freedom. These selected degrees of freedom are called master degrees of freedom. They are selected in such a way as to be able to represent and predict the dynamic behavior of the bladed disk.
At operating speed, the rotating system stiffens because of the centrifugal effects. This stress stiffening causes the natural frequencies to be higher than their corresponding values calculated at zero RPM. The effect of the stress stiffening is evaluated and incorporated in the analysis.
If a rotating bladed disk is excited by a forcing which is fixed spatially, there are 3 conditions that need to be satisfied in order to produce a resonant condition:
1. The natural frequency of the blade row is equal to some per-rev forcing frequency
2. The number of nodal diameter of the natural mode equals the forcing harmonic number.
3. The excitation must be able to couple with the blade disk mode shape. For example, the forcing on the blade row must be in a direction that matches the mode shape deflection. If the forcing is along the axial direction of the turbine and the mode shape shows deflection only in the tangential direction, the mode can not be excited. Of course many modes contain components in both axial and tangential directions.
If all of the above conditions are met, resonance will occur. This natural frequency-forcing relationship is usually illustrated as an Interference diagram. In the Interference diagram, the natural frequency is plotted against the number of nodal diameters (harmonic content of a mode). An example of an interference diagram is shown above. The line through the origin is called the Impulse Line which corresponds to the operating speed. Whenever the Impulse Line intersects the natural frequency curves, a resonant condition may exist.
The Interference diagram is used to locate frequencies of interest for a more detailed stress analysis. Resonant stresses are calculated for these conditions and detuned as necessary to estimate the true dynamic stress in each blade. The dynamic stress will also depend on the damping present in the system as well as the stimulus ratio (the ratio of dynamic forcing to the steam bending force in the blade). Dynamic stress analysis is a discussion on its own and will likely be a topic of future entries.
Thanks for reading.