  Apart from omnidirectional, a solid elastic sphere is a natural multimode and
multifrequency device for the detection of Gravitational Waves (GW). Motion
sensing in a spherical GW detector thus requires a multiple set of transducers
attached to it at suitable locations. Resonant transducers exert a significant
back action on the larger sphere, and as a consequence the joint dynamics of
the entire system has to be properly understood before reliable conclusions can
be drawn from its readout. In this paper, we present and develop a mathematical
formalism to analyse such dynamics, which generalises and enhances currently
existing ones, and which clarifies their actual range of validity, thereby
shedding light into the physics of the detector. In addition, the new formalism
has enabled us to discover a new resonator layout (we call it PHC) which only
requires five resonators per quadrupole mode sensed, and has mode channels,
i.e., linear combinations of the transducers' readouts which are directly
proportional to the GW amplitudes. The perturbative nature of our proposed
approach makes it also very well adapted to systematically assess the
consequences of small mistunings in the device parameters by robust analytic
methods. These prove to be very powerful, not only theoretically but also
practically: confrontation of our model's predictions with the experimental
data of the LSU prototype detector TIGA reveals an agreement between both
consistently reaching the fourth decimal place.
