  Designing tunable resonators is of practical importance in active/adaptive
sound generation, noise control, vibration isolation and damping. In this
paper, we propose to exploit the biasing fields (induced by internal pressure
and radial electric voltage) to tune the three-dimensional and small-amplitude
free vibration of a thick-walled soft electro-active (SEA) spherical balloon.
The incompressible isotropic SEA balloon is characterized by both neo-Hookean
and Gent ideal dielectric models. The equations governing small-amplitude
vibrations under inhomogeneous biasing fields can be linearized and solved in
spherical coordinates using the state-space formalism, which establishes two
separate transfer relations correlating the state vectors at the inner surface
with those at the outer surface of the SEA balloon. By imposing the mechanical
and electric boundary conditions, two separate analytical frequency equations
are derived, which characterize two independent classes of vibration for
torsional and spheroidal modes, respectively. Numerical examples are finally
conducted to validate the theoretical derivation as well as to investigate the
effects of both radial electric voltage and internal pressure on the resonant
frequency of the SEA balloon. The reported analytical solution is truly and
fully three-dimensional, covering from the purely radial breathing mode to
torsional mode to any general spheroidal mode, and hence provides a more
accurate prediction of the vibration characteristics of tunable resonant
devices that incorporate the SEA spherical balloon as the tuning element.
