Piecewise-type nonlinearities. such as clutch dampers in a torsional system. induce complex nonlinear dynamic behaviors that resemble super- and sub-harmonic responses. This study focuses on investigating the sub-harmonic responses induced by piecewise-type nonlinearities in the middle of various dynamic behaviors in a torsional vibratory system. https://shopredoners.shop/product-category/black-high-rise-wide-leg-crop/