1. Service Tube Vibration Analysis Method Comparison: NUMERICAL VS
Service Tube Vibration Analysis Method Comparison: NUMERICAL VS. CLOSED FORM SOLUTION
L:\Engineering Library\Analysis\Lifing\Cyclic Fatigue\HCF\Vibration\Natural Frequency of Beams (vibration).xls
For simple cases, close correlation between FEA and closed form solution.

2. For complicated tube configurations, it’s not so straight-forward.
Note that on this part there are two seal areas, It may be reasonable to assume that the pair of O-ring seals holds the thin end in a fixed position changing boundary condition at that end to fixed-fixed and changing the ‘C’ in the equation from 2.45 to This is relevant in either the equation or FEA problem.
The tube thickens, and the natural frequency of the thick part is a little lower in the first part, and higher in the second part and it’s not easy to predict the resulting frequency.
Conclusion: For simple tubes, use closed form solution, for more complicated configurations, an estimation can be made, but you may require a numeric solution.