The rigid double-torus torsional oscillator (TO) is constructed to reduce any elastic effects inherent to complicated 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 He-4 with 0.6-ppb He-3 and 300-ppb He-3. 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 the 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 that of nonrigid TOs.