Tapered Thermowells Natural Frequency

It is also possible to calculate approximately the natural resonant frequency of a tapered thermowell based on the Rayleigh method which consists of assuming a deflected shape for the thermowell which is reasonable (i.e. consistent with the boundary conditions) then calculating the potential energy at maximum deflection (velocity everywhere equal to 0) and the maximum kinetic energy (deflection everywhere equal to 0) and equaling the two since by the principle of conservation of energy, the sum potential + kinetic = constant.

Notation:

k = D2/D1

d = D0/D1

E = Young’s Modulus (2. 1011 Pa for steel and stainless steel)

r = Density of the material (7800 kg/m3 for steel)

L, D0, D1, D2 are in millimetres.

The stiffness is a function of the inertia of the cross section. Because the cross section varies along the length, so does the inertia (2nd moment of area). The inertia is nominated as I(x) where x is the distance measured from the inner face of the flange (or end of the thread for screwed thermowells).

Eq. 2.1

where D(x) is the diameter at x.

Eq. 2.2

hence we get for I(x):

Eq. 2.3

The simplest deflected shape which we can use is:

Eq. 2.4

as it is consistent with the cantilevered condition at x=0 & x=L since it is the deflected shape of a cantilever of uniform section subjected to a uniform load (l is an arbitrary constant).

The instantaneous speed is: Eq. 2.5

the maximum kinetic energy occurs when cos(wt)=1 and is given by:

Eq.2.6

or:

Eq. 2.7

Eq. 2.8

The internal energy in the thermowell at maximum deflection is:

Eq. 2.9

where M(x) is the bending moment at x.

The bending moment, M(x), is a function of the stiffness and curvature of the deflected thermowell:

Eq. 2.10

Since we assumed the shape, we have:

Eq. 2.11

We get for the internal energy:

Eq. 2.12

Replacing I(x) by its expression in equation 2.3 and integrating we finally obtain:

Eq. 2.13

and finally we get by equating Ui and Uk since according to the principle of conservation of energy, when the kinetic energy is maximum, the potential (internal) energy is zero and vice-versa.

Eq. 2.14

The natural frequency is to a close approximation:

Eq. 2.15