## 4 Results
Therefore, the system exhibits softening dynamic stiffness characteristics in the loss contact case, which is caused by the average contact time decreasing as the vibration amplitude increases [24]. Similarly, for \(\alpha>1\) or \(0<\alpha<1\), as shown in Figure 5b,c, dramatically softening dynamic stiffness characteristics are exhibited in the loss contact case (compared with the continuous system). Especially, for \(\alpha>1\), the one-sided contact system can have monotonic softening stiffness characteristics (see Figure 5b). This softening dynamic support has been found in the ball bearing-rotor system, where the nonlinearity of Hertzian point contact exists between the balls and raceways of the support ball bearings [27]. Figure 4: Influence of normal load \(W\) on \(\alpha\) the frequency-response peak-to-peak curves of continuous system, (**a**) \(\alpha=1.5\) and (**b**) \(\alpha=0.6\). As shown in Figures 6 and 7, with the increase in the externally normal constant load \(W\), the location of the primary resonance moves to a high frequency (\(\alpha>1\)) or low frequency (\(0<\alpha<1\)), which agrees with the rule of the continuous system in Section 4.1. In addition, for the one-sided contact system, it is obvious that the amplitude boundary of loss contact increases as normal load \(W\) increases, which means that there is less chance of losing contact. Moreover, the damping coefficient \(\zeta\) has a significant effect on the response amplitude of a one-sided contact system in the case of loss contact (see Figures 6a and 7a). Therefore, the identification and control of the damping coefficient are very important for the quality control of the practical one-sided contact system with high accuracy requirement, such as the micro-cantilever probe system in the atomic force microscope [28]. Figure 5: Comparisons of frequency-response peak-to-peak curves between continuous system and one-sided contact system, (**a**) \(\alpha=1\), (**b**) \(\alpha=1.5\), and (**c**) \(\alpha=0.6\), where the magenta dash line denotes the loss contact boundaries. Excitation Characteristics of One-Sided Contact System (\(\zeta\) = 0.032, \(\alpha\) = 1, \(W\) = 1)  
Next, the excitation characteristics of the one-sided contact system will be considered in detail. As shown in Figure 8a, for \(\omega\) = 1, with the increase in the excitation amplitude \(A\), the stable response of period-1 solution branch undergoes period-doubling bifurcation at \(A_{1}\), and the coming period-2 solution branch undergoes hysteresis and jumping at the cyclic folding bifurcation points \(A_{2}\) and \(A_{3}\), which agrees well with the numerical bifurcation diagrams shown in Figure 8b. The period-doubling bifurcation leads coexistence of the stable period-1 solution and period-2 solution in the system. For example, the period-1 solution \(P_{1}\) and period-2 solution \(P_{2}\) coexist when \(A\) = 1. On the other hand, for \(\omega\) = 1 and \(A\) = 1, the corresponding period-1 solution \(P_{1}\) and period-2 solution \(P_{2}\) are located on the primary resonance and 1/2-order sub-harmonic resonance (see Figure 9a), respectively. It is indicated that the intrinsic triggering mechanism of period-doubling bifurcation at \(A_{1}\) is the softening dynamic stiffness characteristics of the one-sided contact system, where the loss of contact nonlinearity leads to the softening 1/2-order sub-harmonic resonance and the corresponding period-2 motions excited. In addition, the system also has higher-order stable solutions such as period-3, period-4, and period-5 solution branches coexisting (see Figure 9b), and they are excited by softening 1/3-order, 1/4-order, and 1/5-order sub-harmonic resonances, respectively. It is clear that the coexisting solution characteristics of a one-sided contact system are complex due to the loss of contact nonlinearity.