*2.2. Mechanical Model of the Host System with the Absorber*

We now attach a DVA to the host system. The basis of the model is the same as the one mentioned in the previous subsection. The only additional element is the absorber mass *m*2, which is attached to the primary system through a spring and a linear damper. The schematic depiction of this two-DoF system is provided in Figure 2. ✻✻ ✻✼ ✻✽

**Figure 2.** The host system with the attached DVA.

The absorber's spring encompasses a linear term and a cubic term. The differential equations describing the dynamics of the system are

$$\begin{aligned} m\_1 \ddot{\mathbf{x}}\_1 + c\_1 \dot{\mathbf{x}}\_1 + k\_1 \mathbf{x}\_1 + c\_2 \left( \dot{\mathbf{x}}\_1 - \dot{\mathbf{x}}\_2 \right) + k\_2 \left( \mathbf{x}\_1 - \mathbf{x}\_2 \right) + k\_{\text{n2}} \left( \mathbf{x}\_1 - \mathbf{x}\_2 \right)^3 &= F\_\text{R} \\ m\_2 \ddot{\mathbf{x}}\_2 + c\_2 \left( \dot{\mathbf{x}}\_2 - \dot{\mathbf{x}}\_1 \right) + k\_2 \left( \mathbf{x}\_2 - \mathbf{x}\_1 \right) + k\_{\text{n2}} \left( \mathbf{x}\_2 - \mathbf{x}\_1 \right)^3 &= \mathbf{0} \end{aligned} \tag{6}$$

where *k*<sup>2</sup> and *k*nl2 are the linear and cubic coefficients of the absorber stiffness, respectively, while *c*<sup>2</sup> is the linear coefficient of the absorber damping. Let us introduce the following expressions

$$\zeta\_2 = \frac{c\_2}{2\sqrt{m\_2k\_2}} ; \,\omega\_{\mathbf{n}2} = \sqrt{\frac{k\_2}{m\_2}} ; \,\varepsilon = \frac{m\_2}{m\_1} ; \,\gamma = \frac{\omega\_{\mathbf{n}2}}{\omega\_{\mathbf{n}1}} . \tag{7}$$

By diving Equation (6) by *m*1, applying the expressions from (7) and (3), dividing by *ω*<sup>2</sup> n1 and utilizing the dimensionless time *τ* and dimensionless displacements *x* ˜<sup>1</sup> and *x* ˜<sup>2</sup> = *x*2/*x*0, we obtain

$$\begin{aligned} &\mathbf{\tilde{x}}\_{1}^{\prime\prime} + 2\boldsymbol{\tilde{\zeta}}\_{1}\mathbf{\tilde{x}}\_{1}^{\prime} + \mathbf{\tilde{x}}\_{1} + 2\boldsymbol{\varepsilon}\boldsymbol{\tilde{\zeta}}\_{2}\boldsymbol{\gamma}\left(\mathbf{\tilde{x}}\_{1}^{\prime} - \mathbf{\tilde{x}}\_{2}^{\prime}\right) + \boldsymbol{\varepsilon}\boldsymbol{\gamma}^{2}\left(\mathbf{\tilde{x}}\_{1} - \mathbf{\tilde{x}}\_{2}\right) + \boldsymbol{\varepsilon}\mathbf{\kappa}\_{\text{n}\mathbf{l}2}\left(\mathbf{\tilde{x}}\_{1} - \mathbf{\tilde{x}}\_{2}\right)^{\mathbf{3}} = \boldsymbol{\tilde{F}}\_{\text{R}} \\ &\boldsymbol{\varepsilon}\mathbf{\tilde{x}}\_{2}^{\prime\prime} + 2\boldsymbol{\varepsilon}\boldsymbol{\zeta}\_{2}\boldsymbol{\gamma}\left(\mathbf{\tilde{x}}\_{2}^{\prime} - \mathbf{\tilde{x}}\_{1}^{\prime}\right) + \boldsymbol{\varepsilon}\boldsymbol{\gamma}^{2}\left(\mathbf{\tilde{x}}\_{2} - \mathbf{\tilde{x}}\_{1}\right) + \boldsymbol{\varepsilon}\mathbf{\kappa}\_{\text{n}\mathbf{l}2}\left(\mathbf{\tilde{x}}\_{2} - \mathbf{\tilde{x}}\_{1}\right)^{\mathbf{3}} = \mathbf{0}, \end{aligned} \tag{8}$$

where *κ*nl2 = *k*nl2*x* 2 0 / (*k*1*ε*). A variable change is performed, where *x* ˜<sup>3</sup> = *x* ˜ <sup>1</sup> − *x* ˜<sup>2</sup> (relative displacement of *m*2), hence Equation (8) transforms into

$$\begin{aligned} \mathfrak{x}\_{1}^{\prime\prime} + 2\mathfrak{z}\_{1}\mathfrak{x}\_{1}^{\prime} + \mathfrak{x}\_{1} + 2\mathfrak{e}\mathfrak{z}\_{2}\gamma\mathfrak{x}\_{3}^{\prime} + \mathfrak{e}\gamma^{2}\mathfrak{x}\_{3} + \mathfrak{e}\mathfrak{x}\_{\text{nl2}}\mathfrak{x}\_{3}^{3} &= \mathfrak{F}\_{\mathbb{R}} \\ \mathfrak{e}\left( \left( \mathfrak{x}\_{1}^{\prime\prime} - \mathfrak{x}\_{3}^{\prime\prime} \right) - 2\mathfrak{z}\_{2}\gamma\mathfrak{x}\_{3}^{\prime} - \gamma^{2}\mathfrak{x}\_{3} - \mathfrak{x}\_{\text{nl2}}\mathfrak{x}\_{3}^{3} \right) &= 0 \end{aligned} \tag{9}$$
