## 1 Introduction  
Modern vibration isolators leverage intrinsic kinematic nonlinearities to enhance performance. The need for vibration isolation arises in applications such as building safety or retrofitting sensitive devices to mitigate structural damage. For systems with continuous nonlinear restoring force and viscous damping, theoretical foundations are well established, especially in the field of negative and quasi-zero-stiffness structures [1, 2, 3, 4, 5], where the concept proposed by [PERSON] [6] and [PERSON] [7] has been exploited to reduce the dynamic stiffness of suspensions without compromising the load-bearing capacity, leading to the development of High-Static-Low-Dynamic Stiffness (HSLDS) isolators. [PERSON] and [PERSON] [8] demonstrated that piecewise restoring forces create complex dynamic scenarios, commonly encountered in systems with impacts [9], backlash and multiple contacts [10], motion-limiting constraints [11], and breathing cracks [12], just to name a few. However, as noted by [PERSON] and [PERSON] [13], the literature primarily coversoscillators with the piecewise-linear characteristic, which may not always represent real systems accurately. Experimental studies by [PERSON] et al. [14] showed that depending on the mechanism layout, piecewise nonlinear-linear restoring forces must be considered due to the limitations of compressive springs unable to carry tractive forces. [PERSON] [15] studied the periodic response of nonlinear oscillators with bilinear stiffness and performed parametric analyses. Results on isolated resonance curves (IRCs) have highlighted sets of parameters that lead to abrupt resonance behavior, which could have detrimental structural effects. This approach was also applied by [PERSON] et al. [16] in analyzing HSLDS cam-roller-spring mechanisms with piecewise stiffness, where the theoretical formulation was developed, and experimental results were presented, though no direct comparisons were made. Nonlinear isolators often exhibit saddle-node bifurcations and hysteresis, which leads to sudden jumps in the oscillation amplitude [17]. The possibility of mitigating these phenomena has been addressed in Refs. [18,19]. Specifically, by leveraging the \"dynamic cancellation\" effect [20], introducing tuned inerters, it is possible to retain the beneficial effects of the HSLDS characteristic while canceling the hysteresis induced by the nonlinear restoring force, resulting in a saddle-node free frequency response curve. Dissipation mechanisms have long posed challenges in accurately modeling experimental observations [21]. [PERSON] [22] provided closed-form solutions for the time response and stick-slip boundaries of oscillators with linear restoring force and dry friction and its work has been later extended by several authors [23, 24, 25, 26, 27]. [PERSON] and [PERSON] [28] analyzed the steady-state response of a Duffing oscillator with pure cubic stiffness and combined viscous and Coulomb damping. Using the Harmonic Balance Method (HBM), the analysis pointed out that in the case of high viscous damping, where the primary solution branch exhibits reduced transmissibility, stable IRCs may arise [29,30]. Closely related to the present research, [PERSON] et al. [31] presented a study incorporating a more sophisticated damping formulation to investigate the nonlinear dynamics of QZS systems in the presence of hysteretic dry friction. Moreover, to accurately model the dissipation of a pneumatic HSLDS isolator, [PERSON] and [PERSON] [32] included linear and cubic viscous damping, combined with Coulomb friction. Capturing the behavior of dynamical systems through simple model is challenging [33]. To derive approximate solutions, various methods treat dissipative and nonlinear restoring forces as perturbations to the underlying linear system. Among these, averaging methods exploit the time-scale separation between fast and slow dynamics [34].