Every component of a shell-and-tube heat exchanger vibrates at its own unique natural frequencies. Tubes are generally the most flexible parts and thus the most easily excited. Tubes can and do vibrate at different frequencies. The lowest natural frequency for any tube is called the fundamental or first mode. Higher natural frequencies are referred as the second mode, third mode, and so on. For conservative design purposes, only the fundamental natural frequency will be considered and will be called simply the natural frequency.

The natural frequency of a tube, like that of a simple beam, depends on the length of spans, how the ends are supported (clamped or simply), the type of intermediate supports (simply supported, pinned, or clamped), the cross-sectional geometry, the number of spans, and the materials of construction. Although the natural frequency of tubes can be measured experimentally, predictive methods to estimate approximate values are suitable for most engineering purposes.

Tubes in shell-and-tube heat exchangers are generally considered to be (1) rigidly fastened in the tube-sheets and (2) simply supported at intermediate points of contact with baffles and/or support plates. Some tubes in the center of the bundle may be supported by every baffle, whereas tubes that pass through the baffle windows may be supported only by every second or third baffle. Furthermore, the end spacings are often wider than central baffle spacings to accommodate nozzles and shell flanges. Thus, tubes in various parts of a bundle have different patterns of support and this results in tubes with different natural frequencies in the same heat exchanger.

A. Natural frequencies of straight tubes

For the vibration analysis of most shell-and-tube heat exchangers, determination of the lowest natural tube frequency is adequate. This can be done using a rigorous approach (Gorman, 1975), a graphical approach (Moretti, 1973), or a finite-clement computer program such as NASTRAN (1973). These provide higher-mode natural frequencies in addition to the lowest and the mode shapes for each. Figure 1 shows natural frequencies and mode shapes for the first five modes of vibration for a typical six-cross-pass heat exchanger. Notice the similarity in the natural frequencies for first modes of vibration and the differences in mode shape. Such information is useful in interpreting the behavior of tube vibration.