## 5 Conclusions  
Contact vibrations between relative moving components in contact systems, such as hinges, meshing gears, rolling bearings, and probe detection system, are often inevitable due to the external excitations or roughness and waviness between contact surfaces, where the power-form nonlinear load-deformation relationship \(F_{C}(x)=K_{C}\cdot x^{\alpha}\) is commonly satisfied in the contact interfaces. With the aid of the HB-AFT method and Floquet theory, we have investigated the dynamic characteristics of the hardening/softening nonlinear stiffness and hysteretic resonances for the typical continuous and one-sided contact power-form spring systems. First, it is found that the externally normal load can lead to hardening-to-softening (\(0<\alpha<1\)) or softening-to-hardening (\(\alpha>1\) dynamic stiffness characteristics to the continuous power-form system, unlike the purely softening (\(0<\alpha<1\)) or hardening (\(\alpha>1\)  
Figure 11: Phase portraits and their Poincare mappings of one-sided contact system for \(\omega=1\), (**a**) \(A=1.7798\), (**b**) \(A=1.78\), (**c**) \(A=1.785\), and (**d**) the enlargement of (**c**), where the red dots denote the Poincare mapping of unstable period-8 motion at \(A_{7}\) point (see Figure 8).  
Figure 10: Time series \(x(\tau)\) and their enlargements of one-sided contact system for \(\omega=1\), (**a**) \(A=1.78\), and (**b**) \(A=1.785\), where the cyan dots denote the period-1 mapping points of \(x(\tau)\) at intervals of excitation period \(2\pi\).  
dynamic stiffness characteristics without an externally normal load. Second, it is shown that the one-sided contact power-form spring model can exhibit remarkable softening spring characteristics, and the resonant response amplitude is dramatically affected by the damping coefficient in the case of loss of contact. Third, the location of the primary resonance moves to a higher frequency (\(\alpha>1\)) or lower frequency (\(0<\alpha<1\)) in either a continuous or one-sided system. Finally, it is indicated that multi-order super/sub-harmonic softening resonances and abundant bifurcation behaviors can also be excited beyond the loss of contact boundary, which can lead to multiple periodic solutions with different periods co-existing and can even excite type-I and crisis-induced intermittency chaos to the one-sided contact system. The results obtained in this paper could benefit to clarify the inherent correlations between dynamic stiffness characteristics and complex resonances in the vibrational contact systems.  
Conceptualization, Z.Z.; investigation, Z.Z., Q.W. and Y.Y.; methodology, Z.Z.; validation, Q.W., Z.Z. and Z.P.; supervision, Z.Z.; writing--original draft preparation, Q.W., Z.Z. and Y.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by China Postdoctoral Science Foundation (Grant No. 2019M661849), Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 2018K164C), Fundamental Research Funds for the Central Universities (Grant No. 30920021155), Open-end Research Fund of State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures (Grant No. KF2020-30), and National Natural Science Foundation of China under (Grant No. 11802130).  
Not applicable.  
Z.Z. gratefully acknowledges [PERSON] (Technische Universitat Ilmenau), [PERSON] and [PERSON] (Harbin Institute of Technology) for considerable help and valuable suggestions for the paper.  
The authors declare no conflict of interest.