This study aims to design two types of vibration isolators, with different spring mechanisms, using a foldable structure that is based on a cylinder torsional buckling pattern, and evaluate the vibration isolation performance of each design. Vibration isolation is achieved through nonlinear spring characteristics of the isolators, which have zero spring stiffness, achieved by attaching a linear spring to the foldable isolator structure. The two vibration isolators differ in the mechanical elements that constitute the foldable structure, which undergo tensile forces as the structure folds. For the first isolator, the mechanical elements are represented only by tension springs, which appropriately undergo tension. For the second isolator, the mechanical elements are designed so that embedded compression springs within the elements compress under tension, thus enabling the elements to work as tension springs. The excitation experiment results revealed that the different spring mechanisms produced equivalent resonance frequencies but different damping effects at the resonance and higher frequencies. When oscillations with multiple amplitudes were input, larger input amplitudes were found to correspond to lower resonance frequencies for both isolators. This trend contradicts that described in the nonlinear vibration theory modeled by the Duffing equation, and was determined to be caused by hysteresis of the spring phenomena in the vibration isolators.