  The rigid double-torus torsional oscillator (TO) is constructed to reduce any
elastic effects in-herent to complicate TO structures, allowing explicit
probing for a genuine supersolid signature. We investigated the frequency- and
temperature-dependent response of the rigid double-torus TO containing solid
4He with 0.6 ppb 3He and 300 ppb 3He. We did not find evidence to support the
frequency-independent contribution proposed to be a property of supersolid
helium. The frequency-dependent contribution which comes from the simple
elastic effect of solid helium coupled to TO is essentially responsible for the
entire response. The magnitude of the period drop is linearly proportional to
${f}^{2}$, indicating that the responses observed in this TO are mostly caused
by the overshoot of `soft' solid helium against the wall of the torus.
Dissipation of the rigid TO is vastly suppressed compared to those of non-rigid
TOs.
