Flow Induced Vibration Part 1: Using FEA to determine the natural frequency

In a recent project I had to design a spray lance to be inserted into a large vertical column. In an early design iteration, it was necessary to determine if the spray lance would suffer from flow induced vibration once the system was started. Since the system did not exist and there was no data about how it actually worked, a computer study of the system was performed to investigate the possibility of flow induced vibration. 

One of the useful tools in most FEA packages is the ability to determine the natural frequency of a part very quickly. Using FEA to determine the natural frequency rather than a using a hand calculation can help reduce the time and effort required to study the dynamics of a system. This is especially true if a part undergoes many design changes and the FEA package is linked to or part of the CAD package.

Knowing the natural frequency of a part is important when dealing with turbulent fluid flow or rotating machinery because if the natural frequency of a part happens to be close to the frequency of some forcing function that exists within a system the part is likely to fail at loads far below what a static analysis predicts. This occurs due to the fact that if the forcing function matches the natural frequency, the amplitude of the deflection due to the forcing function is magnified with each cycle, eventually leading to failure. This behavior is called resonance and in almost all engineering cases, is a very bad thing.

In the video example above, the first five calculated natural frequencies for the spray lance are 12.7 Hz, 14.7 Hz, 79.1 Hz, 90.4 Hz, and 218.4 Hz. 

As fluids flow across cylinders and other shapes, they can create a regular vortex pattern downstream of the shape. The frequency of these vortices can be expressed using the Strouhal Number which varies primarily as a function of shape and Reynolds number. In this case, the average fluid properties and velocity were used to calculate the Reynolds number across the spray lance to see if it falls within the zone (below Re<300,000) where vortex shedding occurs. In this case, the Reynolds number is approximately 3,600 which is in the vortex shedding zone. At this Reynolds number, the Strouhal Number is approximately 0.21. As a result, the critical velocity where the frequency of the vortex shedding equals the first two natural frequencies are 15.1 ft/s for the first natural frequency and 17.5 ft/s for the second natural frequency. The average velocity along the spray lance was found to be approximately 4.6 ft/s. At this velocity, the frequency of vortex shedding is approximately 3.9 Hz, which is well below the first natural frequency of 12.7 Hz. The fact that the average velocity is well below the first critical velocity suggests that the likelihood for vibration due to the vortex shedding is low.