## 2 Theoretical Formulation  
The proposed HSLDS isolator consists of a mass \(m\) suspended by a vertical linear spring of stiffness \(k_{v}\) and \(\lambda\) linear horizontal springs of stiffness \(k_{h}\); the displacement \(\Delta_{h}\) denotes the horizontal spring preload, \(L\) is the length of the oscillating rod, and \(\theta\) is the angle between the rod and the horizontal direction. Dissipation phenomena are modelled considering a viscous damper \(c\) and a constant dry friction force \(F_{c}\); the absolute displacement of the mass is given by \(x\), and the mass is subject to a harmonic forcing of amplitude \(F\) and frequency \(\omega\); \(t\) is the time variable. A schematic of the system is shown in Figure 1a. In real applications, compression springs are often used, but they cannot sustain tensile loads. Therefore, when the oscillation amplitude exceeds a critical value \(x_{cr}\), it is necessary to account for the detachment of the horizontal springs from the pushrod using a piecewise function to represent the restoring force. The equation of motion for the isolator shown in Figure 1a incorporating nonlinear stiffness and damping effects is expressed as [39]  
\[m\ddot{x}+c\dot{x}+k_{v}x+F_{c}\;\text{sign}\big{(}\dot{x}\big{)}+F_{k}(x)=F \text{cos}(\omega t) \tag{1}\]  
where dots denote time derivatives and \(F_{k}(x)\) is the nonlinear restoring force with piecewise characteristic. In the case of static equilibrium position with the oscillating rod parallel to the ground (\(\theta=0\) rad), \(F_{k}(x)\) is given by [14]  
\[F_{k}(x)=\left\{ \begin{array}{cc}\lambda\;k_{h}\Big{(}L-\Delta_{h}-\sqrt{L^{ 2}-x^{2}}\Big{)}&\frac{x}{\sqrt{L^{2}-x^{2}}}&\left|x\right|\leq\left|x_{cr} \right|x_{cr}=\sqrt{L^{2}-\left(L-\Delta_{h}\right)^{2}}\\ 0&\left|x\right|>\left|x_{cr}\right|\end{array} \right. \tag{2}\]  
Figure 1: HSLDS isolator with piecewise nonlinear–linear stiffness, viscous damper and Coulomb friction. (**a**) Schematic and (**b**) static characteristics. Dissipation is modeled considering viscous damping and Coulomb dry friction, without possibility of having stick-slip transitions. Therefore, the present formulation is suitable to analyze the system dynamic only in the case of slip conditions, i.e., continuous motion. To study the system's behavior under stick-slip transitions, enhanced friction models are needed [40,41]. Using a dimensionless notation, Equation (1) is reduced to  
\[u^{\prime\prime}+2\overline{\xi}u^{\prime}+u+\overline{\mu}\,\text{sign}(u^{ \prime})+F_{k}(u)=F_{0}\text{cos}(\Omega\tau) \tag{3}\]  
where  
\[ \begin{array}{c}\omega_{n}^{2}=\frac{k_{n}}{m},\ \ \tau=\omega_{n}t,\ \ (\cdot)^{ \prime}=\frac{d}{d\tau},\ \ \Omega=\frac{\omega_{n}}{\omega_{n}},\ \ u=\frac{\tau}{L},\\ \delta_{h}=\frac{\Lambda_{h}}{L},\ \ \overline{\xi}=\frac{c}{2\sqrt{k_{n}m}},\ \ \overline{\mu}=\frac{F}{k_{n}L},\ \ \overline{F}_{0}=\frac{F}{k_{n}L},\ \ r=\frac{k_{h}}{k_{n}}\end{array}  \tag{4}\]  
and the dimensionless nonlinear piecewise restoring force reads as follows:  
\[ \begin{array}{c}\overline{F}_{k}(u)=\frac{\overline{F}_{k}(x)}{k_{n}L}= \left\{\begin{array}{cc}\lambda\ r\Big{(}1-\delta_{h}-\sqrt{1-u^{2}}\Big{)} \ \frac{u}{\sqrt{1-u^{2}}}&|u|\leq|u_{cr}|\\ 0&|u|>|u_{cr}|\end{array}\right.\\ u_{cr}=\sqrt{1-\left(1-\delta_{h}\right)^{2}}\end{array}  \tag{5}\]  
Using [PERSON]'s expansion, the nonlinear restoring force can be approximated as  
\[\overline{F}_{k}(u)=\left\{ \begin{array}{cc}(\alpha-1)u+\beta u^{3}+\gamma u ^{5}&|u|\leq|u_{cr}|\\ 0&|u|>|u_{cr}|\end{array} \right. \tag{6}\]  
with coefficients derived up to the fifth order  
\[\alpha=1-\lambda r\delta_{h},\ \ \beta=\frac{\lambda}{2}r(1-\delta_{h}),\ \ \gamma=\frac{3}{8}\lambda r(1-\delta_{h}) \tag{7}\]  
Due to the symmetry of the system, the restoring force is an odd function [42]; therefore, only odd terms appear in (6). Moreover, the critical deflection \(u_{cr}\) is null when the preload \(\delta_{h}=0\).