Suppose that the fluid is flowing into the Ushaped tube at velocity V and the tube is vibrating at angular velocity . Consider a small section of the fluid that is on the inlet side away from the point of flexture at distance r.
The Coriolis force on the small fluid section m is
During the down cycle, the tube applies an upward resisting force to the fluid or the fluid pushes the tube down. On the outlet side, the Coriolis force has the opposite direction.
To simply the problem, we assume that the tube has a perfect U shape with a cross section area of A. The length and width are l, d, respectively. The opposite directions of Coriolis forces on inlet and outlet sides result in a twisting moment T_{c}
A K factor can be introduced to compensate for the more generalized Ushape.
where Q_{m} = AV is the mass flow rate.
The governing equation of twisting is
where I_{u} is the inertia of the Ushaped tube, C_{u} is the damping coefficient, K_{u} is the stiffness, is the twist angle, and t is time.
Recall that the Coriolis flowmeters are vibrating the Ushaped tube to generate the rotation, the real angular velocity is function of vibrating frequency :
Assuming that the damping term C_{u} is negligible, the equation of twisting becomes
The particular solution (steadystate solution) of the twist angle is
Furthermore, the velocity of the turning corners of the Ushaped tube are and the displacement difference between these two corners is d/2. Therefore, the time lag between these two corners is
By measuring the time lag , the mass flow rate can be obtained
In vibration analysis, it is custom to use the natural frequency as a basis and normalize frequency terms against it. The natural frequency of the Ushaped tube system is (note that I_{u} includes the mass of the fluid in the tube)
The mass flow rate then becomes
