IPMC beam dynamics in fluid

The IPMC beam dynamics is governed by [Clough and Penzien (1993)] 

Here Y, I, C, pm, and A denote the effective Young s modulus, the area moment of inertia, the internal damping ratio, the density, and the cross-sectional area of the IPMC beam, respectively, and f z, t) is the distributed force density acting on the beam. 

Performing the Laplace transform of rc(2, f)on the time variable t, we convert (9.5) into the Laplace domain  

The force term F z, s) can be attributed to several sources, including the hydrodynamic force Fhydro(-, ) on the IPMC beam, actuation-induced bending moment Mipmc(-J) s), and the bending moment Mfl (2 , s) induced by the hydrodynamic loading from the passive fin. Thydro( , s) can be expressed as [Sader (1998)]  

Here pv and Cy are the equivalent mass per unit length and the equivalent damping coefficient of IPMC in the fluid, respectively  

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