The natural frequencies of U-tubes on multiple supports have been studied as a complement to the author’s previous work on the vibration of straight heat-exchanger tubes (reference [1]). A rapid estimation procedure for fundamental frequencies of tubes on symmetrical support spacings has been developed by expressing the frequency in the form  
f112πa11Ls2EIμ
where the square root contains the tube properties of Young’s modulus, cross-sectional second moment, and linear density; Ls is a characteristic length of support spacing; and a1 is a dimensionless number which is a strong function of the support geometry (as expressed by the ratio of bend radius to span lengths) and weak function of Poisson’s ratio and of tube axial moment of inertia. a1 has been plotted as a function of the ratio of the bend radius to the straight-span length, for usual values of Poisson’s ratio and small axial moment of inertia. The underlying assumptions for the use of such plots are examined and the theoretical basis for the statement of a lower bound is given, in order to show where the use of this method is applicable.
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