## 4 Results  
Frequency-Response Characteristics of Continuous System (H(x) = 1, \(\zeta\) = 0.01, \(A\) = 0.1)  
For \(\alpha\) = 1.5 and \(W\) = 0, the continuous system is power-form nonlinear without externally normal load. Its primary resonance exhibits hardening spring characteristics since the period-1 frequency-response curve bends to the right (see Figure 2a), and cyclic folding bifurcations at turning points \(A_{1}\) and \(A_{4}\) can bring jumping phenomena to the system. For \(\alpha\) = 0.6 and \(W\) = 0, as shown in Figure 2b, the primary resonance of the system exhibits softening spring characteristics, and the 2-order, 3-order, and 5-order super-harmonic resonances are excited obviously. Herein, transcritical bifurcations at \(A_{5}\) and \(A_{6}\) lead period-1 solution unstable and 2-order superharmonic resonance emerging. Overall, as \(\alpha\) > 1 (or 0 \(<\alpha<\) 1), the resonant skeleton line of the primary resonant frequency-response curve has no inflection point, i.e., the system has a single monotonic softening (or hardening) dynamic stiffness characteristic, respectively, but the hysteretic degree can be adjusted by \(\alpha\). As shown in Figure 3, the resonant curves gradually slope more to the right and the resonant amplitudes decreases as \(\alpha\) increases, i.e., the hardening spring characteristics get stronger. The above results agree well with the theoretical studies of the literature [19]. Figure 3: For \(W\) = 0, influence of \(\alpha\) on the frequency-response peak-to-peak curves of continuous system, (**a**) \(\alpha\) > 1 and (**b**) 0 \(<\alpha<\) 1. Figure 2: For \(W\) = 0, stable (solid) and unstable (dashed) periodic frequency-response peak-to-peak curves of continuous system, (**a**) \(\alpha\) = 1.5 and (**b**) \(\alpha\) = 0.6. As shown in Figure 4, when a constant load is applied (i.e., \(W>0\)), an inflection point appears and the skeleton line of the primary resonant frequency-response curve bends from left to right (for \(\alpha>1\)) or right to left (for \(0<\alpha<1\)), which changes the dynamic stiffness characteristics of the continuous system. For example, when \(\alpha=1.5\) and \(W=5\), the response amplitude is lower than the inflection point of the skeleton line, which makes the system have a softening spring (i.e., softening dynamic stiffness) characteristics (see Figure 4a). However, if the response amplitude is higher than the inflection point of the skeleton line (when \(\alpha=1.5\) and \(W=1\)), the system has a softening-to-hardening spring (i.e., softening-to-hardening dynamic stiffness) characteristics (see Figure 4a), and in this case, cyclic folding bifurcations occur at four turning points of the resonant frequency-response curve. On the other hand, when \(0<\alpha<1\), as shown in Figure 4b, the frequency-response skeleton lines of the primary resonances have the hardening-to-softening characteristics, which makes the system have a hardening dynamic stiffness characteristic when the response amplitude is lower than the inflection point of the skeleton line (such as \(B_{3}\) point when \(W=1\)). Besides, for \(\alpha>1\) (or \(0<\alpha<1\)), the locations of the primary resonances move to high frequency (or low frequency) as the externally normal constant load \(W\) increases. Frequency-Response Characteristics of One-Sided Contact System (H(\(x\)) = Heaviside(\(x\)), \(\zeta=0.032\), \(A=0.1\), \(W=1\))  
For \(\alpha=1\), the continuous system is linear, so its dynamic stiffness is consistent with the static stiffness of the system. There are no hardening/softening stiffness characteristics, and the skeleton of the primary resonant frequency-response curve is a straight line \(\omega=1\) (see Figure 5a). In contrast, for a power-form \(\alpha=1\) one-sided contact system, an inflection point appears at the boundary of loss contact and the skeleton line of the primary resonant frequency-response curve bends to the left.