### 2.1 Description of the System

The considered nonlinear vibrating mechanical system, illustrated in Figure 1, consists of two RFs (i.e., \(m_{1}\) and \(m_{2}\)) linked by the shear rubber springs and the gap-activated compression rubber spring (see *k*_{1y} and Δ*k*_{1y} in Figure 1, respectively), where the former is symmetrically installed, while the latter is asymmetrically arranged on a single side with the average gap *μ*.

Here \(m_{1}\) and \(m_{2}\) are the main vibrating and isolative RFs, respectively, driven by a pair of identical vibrators mounted on both sides of the isolative RF 2. As illustrated in Figure 1, two vibrators can rotate with the same or opposite directions and generate certain exciting forces. The centroid of the RF 1 coincides with that of RF 2, which is also the equilibrium point of the system, marked as *o*, and the coordinate system *oxy* is established with *o* as the origin.

It should be pointed that the shear rubber springs are constrained to generate stiffness only in the horizontal direction and thus the relative motion between two RFs is restricted in *y*-direction. The motion of the system is considered as the plane motion, and the system has four degrees of freedom: the displacements of two RFs in *x*-, *y*-, and *ψ*-directions, denoted by *x*, *y*_{1}, *y*_{2}, and *ψ*, respectively. The angular positions of both vibrators are denoted by *φ*_{01} and *φ*_{02}.

To derive the differential equations of motion for the system, the nonlinear damping force \(F_{1} (y_{1} ,y_{2} ,\dot{y}_{1} ,\dot{y}_{2} )\) and the nonlinear restoring force \(F_{2} (y_{1} ,y_{2} )\) are firstly given by Eq. (1):

$$\left\{ \begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} F_{1} (y_{1} ,y_{2} ,\dot{y}_{1} ,\dot{y}_{2} ) \hfill \\ = \left\{ {\begin{array}{*{20}c} {f_{1y} (\dot{y}_{1} - \dot{y}_{2} ),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (y_{1} - y_{2} ) \ge - \mu ,} \\ {(f_{1y} + \Delta f_{1y} )(\dot{y}_{1} - \dot{y}_{2} ),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (y_{1} - y_{2} ) < - \mu ,{\kern 1pt} {\kern 1pt} } \\ \end{array} } \right. \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} F_{1} (y_{1} ,y_{2} ) \hfill \\ = \left\{ {\begin{array}{*{20}c} {k_{1y} (y_{1} - y_{2} ),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (y_{1} - y_{2} ) \ge - \mu ,} \\ {k_{1y} (y_{1} - y_{2} ) + \Delta k_{1y} (y_{1} - y_{2} + \mu ),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (y_{1} - y_{2} ) < - \mu ,{\kern 1pt} {\kern 1pt} } \\ \end{array} } \right. \hfill \\ \end{gathered} \right.$$

(1)

where *f*_{1y} and Δ*f*_{1y} are the damping constant of relative motion between two RFs for the shear rubber spring and gap-activated compression rubber spring, respectively; *k*_{1y} and Δ*k*_{1y} are the stiffness of the shear rubber spring and gap-activated compression rubber spring, respectively; *μ* is the gap between the gap-activated compression rubber spring Δ*k*_{1y} and RF 1.

### 2.2 Motion Differential Equations of the System

#### 2.2.1 Absolute Motion Differential Equation

The equivalent stiffness and the equivalent damping constant of the relative motion between two RFs in *y*-direction are assumed to be \(k^{\prime}_{1y}\) and \(f^{\prime}_{1y}\), respectively. The absolute motion differential equations for the considered system, formulated by using the Lagrange’s principle, are as follows:

$${\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ \begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} M_{1} \ddot{y}_{1} + f^{\prime}_{1y} (\dot{y}_{1} - \dot{y}_{2} ) + k^{\prime}_{1y} (y_{1} - y_{2} ) = 0, \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} M_{{2}} \ddot{y}_{{2}} - f^{\prime}_{{{1}y}} \dot{y}_{{1}} + (f^{\prime}_{{{1}y}} + f_{{{2}y}} )\dot{y}_{{2}} - k^{\prime}_{{{1}y}} y_{{1}} + (k^{\prime}_{{{1}y}} + k_{{{2}y}} )y_{{2}} \hfill \\ = m_{{0}} r[(\dot{\varphi }_{01}^{2} {\text{sin}}\varphi_{01} - \ddot{\varphi }_{01} {\text{cos}}\varphi_{01} ) + (\dot{\varphi }_{02}^{2} {\text{sin}}\varphi_{02} - \ddot{\varphi }_{02} {\text{cos}}\varphi_{02} )], \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} M\ddot{x} + f_{x} \dot{x} + k_{x} x = m_{0} r[\sigma_{1} (\dot{\varphi }_{01}^{2} \cos \varphi_{01} + \ddot{\varphi }_{01} \sin \varphi_{01} ) \hfill \\ + \sigma_{2} (\dot{\varphi }_{02}^{2} \cos \varphi_{02} + \ddot{\varphi }_{02} \sin \varphi_{02} )], \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} J\ddot{\psi } + f_{\psi } \dot{\psi } + k_{\psi } \psi \hfill \\ = m_{0} l_{0} r\{ [\dot{\varphi }_{01}^{2} \sin (\varphi_{01} - \sigma_{1} \beta_{1} ) - \;\ddot{\varphi }_{01} \cos (\varphi_{01} - \sigma_{1} \beta_{1} )] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + [\dot{\varphi }_{02}^{2} \sin (\varphi_{02} - \sigma_{2} \beta_{2} ) - \;\ddot{\varphi }_{02} \cos (\varphi_{02} - \sigma_{2} \beta_{2} )]\} , \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} J_{0i} \ddot{\varphi }_{0i} + f_{0i} \dot{\varphi }_{0i} \hfill \\ = T_{{{\text{e}}i}} - m_{0} r[\ddot{y}_{2} \cos \varphi_{0i} - \sigma_{i} \ddot{x}\sin \varphi_{0i} + l_{0} \ddot{\psi }\cos (\varphi_{0i} - \sigma_{i} \beta_{i} )], \hfill \\ \end{gathered} \right.$$

(2)

with

\(M_{1} = m_{1}\), \(M_{2} = m_{2} + m_{01} + m_{02}\), \(M = M_{1} + M_{2}\), \(m_{01} = m_{02} = m_{0}\), \(J_{0i} = m_{0i} r^{2}\), \(J = Ml_{{\text{e}}}^{{2}},\)

$$\sigma_{i} = \left\{ {\begin{array}{*{20}c} {( - 1)^{i + 1} ,} & {{\text{two}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{counter - rotating}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{vibrators,}}} \\ {1,} & {{\text{two}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{co - rotating}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{vibrators,}}} \\ \end{array} i=1,2}, \right.$$

where *M*_{1} is the mass of RF 1, while *M*_{2} is that of RF 2 (including two vibrators); *M* is the mass of the total vibrating system; *m*_{01} and *m*_{02} are the mass of the vibrator 1 and 2, respectively; *m*_{0} is the mass of the standard vibrator; *r* is the eccentric radius of each vibrator; *J*_{0i} is the moment of inertia of the vibrator *i* (*i*=1, 2); *J* is the moment of inertia of the total vibrating system about its mass center; *l*_{0} is the distance between the rotation center of each vibrator and the mass center of the total vibrating system; *l*_{e} is the equivalent rotary radius of the total vibrating system about its mass center; *f*_{0i} is the damping coefficient of axis of the induction motor *i* (*i*=1, 2); *f*_{x}, *f*_{2y}, and *f*_{ψ} are the damping constant of the isolative RF in *x*-, *y*-, and *ψ*-directions, while *k*_{x}, *k*_{2y}, and *k*_{ψ} are the stiffness parameters; *β*_{i} is the angle between the line from the rotation center of the vibrator *i* to the mass center of the total vibrating system and *x*-axis; *T*_{ei} is electromagnetic torque of the motor *i*.

#### 2.2.2 Relative Motion Differential Equation

Since the vibrating system considered in Figure 1 consists of two RFs, it is necessary to derive the relative motion differential equation of the system in *y*-direction.

To facilitate the study, some parameters and omissions are set as follows: (i) The average phase and the phase difference of the two vibrators are set as *φ* and 2*α*, respectively, i.e., *φ*_{01}=*φ+α*, *φ*_{02}=*φ−α*; (ii) When the system is operated synchronously, the average angular velocity of the two vibrators is denoted by *ω*_{m0}; (iii) Based on Ref. [15], in the steady state, the relationships between acceleration and displacement satisfy the fact of \(\ddot{y}_{i} = - \omega_{{{\text{m0}}}}^{{2}} y_{i} (i = 1,2)\), and the coil isolative springs are relatively very soft, generally satisfy the fact of \(k_{x} ({\text{or}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k_{2y} ) < < k_{1y}\) in engineering; (iv) \(\;\ddot{\varphi }_{01}\) and \(\;\ddot{\varphi }_{02}\) in the second formula of Eq. (2) can be neglected when the system is operated synchronously.

Based on the above parameter setting and omissions, the formulae with respect to *y*_{1} and *y*_{2} in Eq. (2), can be rearranged as follows:

$$M_{1} \ddot{y}_{1} + f^{\prime}_{1y} (\dot{y}_{1} - \dot{y}_{2} ) + k^{\prime}_{1y} (y_{1} - y_{2} ) = 0,$$

(3)

$$\begin{aligned} & M^{\prime}_{2} \ddot{y}_{2} - f^{\prime}_{1y} (\dot{y}_{1} - \dot{y}_{2} ) - k^{\prime}_{1y} (y_{1} - y_{2} ) \\ & \quad = m_{0} r\omega_{{{\text{m0}}}}^{{2}} [\sin (\varphi + \alpha ) + \sin (\varphi - \alpha )], \\ \end{aligned}$$

(4)

where \(M^{\prime}_{1} = M_{1}\), and \(M^{\prime}_{2}\)=\(M_{2}\)−(*k*_{2y}/\(\omega_{{{\text{m0}}}}^{{2}}\))≈\(M_{2}\) after ignoring *k*_{2y}.

To obtain the solution of the relative motion between two RFs in *y*-direction, by the procedure of Eq. (3)×\(M^{\prime}_{2}\)/(\(M^{\prime}_{1}\)+\(M^{\prime}_{2}\))−Eq. (4)×\(M^{\prime}_{1}\)/(\(M^{\prime}_{1}\)+\(M^{\prime}_{2}\)), the motion differential equation described by the relative displacement, relative velocity, and relative acceleration, are obtained, see Eq. (5):

$$m\ddot{y}_{12} + f^{\prime}_{1y} \dot{y}_{12} + k^{\prime}_{1y} y_{12} = - mr_{{{\text{m2}}}} r\omega_{{{\text{m0}}}}^{{2}} \left[ {\sin (\varphi + \alpha ) + \sin (\varphi - \alpha )} \right],$$

(5)

with

\(m = \frac{{M^{\prime}_{1} M^{\prime}_{2} }}{{M^{\prime}_{1} + M^{\prime}_{2} }}\), \(r_{{{\text{m2}}}} = \frac{{m_{0} }}{{M^{\prime}_{2} }} \approx \frac{{m_{0} }}{{M_{2} }}\), \(y_{12} = y_{1} - y_{2}\), where *m* is called the induced mass of the vibrating system.

### 2.3 Motion Responses of the System

#### 2.3.1 Relative Motion Responses

Based on Eq. (5), the natural frequency of the relative motion between two RFs in *y*-direction, can be deduced as Eq. (6):

$$\omega_{0} = \sqrt {\frac{{k^{\prime}_{1y} }}{m}} = \sqrt {\frac{{k^{\prime}_{1y} (M^{\prime}_{1} + M^{\prime}_{2} )}}{{M^{\prime}M^{\prime}_{2} }}} .$$

(6)

Besides, based on the method of solving responses in Ref. [31], the relative motion response, i.e., the solution of \(y_{12}\), can also be obtained, see Eq. (7):

$$\begin{gathered} y_{12} = F_{12} [\sin (\varphi + \alpha - \gamma_{12} ) + \sin (\varphi - \alpha - \gamma_{12} )] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = 2F_{12} \cos \alpha \sin (\varphi - \gamma_{12} ), \hfill \\ \end{gathered}$$

(7)

with

\(F_{12} = - \frac{{r_{{{\text{m2}}}} z_{0}^{2} r}}{{\sqrt {(1 - z_{0}^{2} )^{2} + (2\xi^{\prime}_{1y} z_{0} )^{2} } }}\), \(\gamma_{12} = \arctan \left( {\frac{{2\xi^{\prime}_{1y} z_{0} }}{{1 - z_{0}^{2} }}} \right)\),

\(z_{0} = \frac{{\omega_{{{\text{m0}}}} }}{{\omega_{{0}} }}\), \(\xi^{\prime}_{1y} = \frac{{f^{\prime}_{1y} }}{{2\sqrt {k^{\prime}_{1y} m} }}\), where *F*_{12} is the coefficient that the single harmonic excitation contributes to the effective vibration amplitude of the relative motion between two RFs; *γ*_{12} is the relative phase lag angle; *z*_{0} is the ratio between the operating frequency and the natural frequency; \(\xi^{\prime}_{1y}\) is the equivalent critical damping ratio of the relative motion between two RFs in *y*-direction.

According to Eq. (7), the relative vibration amplitude, denoted by *δ*_{12} here, is calculated by \(\delta_{12} = 2\left| {F_{12} } \right|\cos \alpha\), where *δ*_{12} is the absolute value of the sum of amplitude vectors.

#### 2.3.2 Absolute Motion Response

Using the transfer function method and the superposition theorem [31], the absolute responses in *x*-, *y*-, and *ψ*-directions are presented by:

$$\left\{ \begin{gathered} x = F_{x} [\sigma_{1} \cos (\varphi + \alpha + \gamma_{x} ) + \sigma_{2} \cos (\varphi - \alpha + \gamma_{x} )], \hfill \\ y_{1} = F_{1y} [\sin (\varphi + \alpha - \gamma_{1y} ) + \sin (\varphi - \alpha - \gamma_{1y} )], \hfill \\ y_{2} = F_{2y} [\sin (\varphi + \alpha - \gamma_{2y} ) + \sin (\varphi - \alpha - \gamma_{2y} )], \hfill \\ \psi = F_{\psi } [\sin (\varphi + \alpha - \sigma_{1} \beta_{1} + \gamma_{\psi } ) + \sin (\varphi - \alpha - \sigma_{2} \beta_{2} + \gamma_{\psi } )], \hfill \\ \end{gathered} \right.$$

(8)

with

\(F_{x} = - \frac{{r_{{\text{m}}} r}}{{\rho_{x} }}\), \(F_{1y} = m_{0} r\omega_{{{\text{m0}}}}^{{2}} \sqrt {\frac{{\eta_{{{\text{c1}}}}^{2} + \eta_{{{\text{d1}}}}^{{2}} }}{{\eta_{{{\text{a1}}}}^{{2}} + \eta_{{{\text{b1}}}}^{2} }}}\),

\(F_{\psi } = - \frac{{r_{{\text{m}}} rr_{l} }}{{\rho_{\psi } l_{{\text{e}}} }}\), \(F_{2y} = m_{0} r\omega_{{{\text{m0}}}}^{{2}} \sqrt {\frac{{\eta_{{{\text{g1}}}}^{{2}} + \eta_{{{\text{d1}}}}^{{2}} }}{{\eta_{{{\text{a1}}}}^{{2}} + \eta_{{{\text{b1}}}}^{{2}} }}}\),

$$\gamma_{1y} = \left\{ {\begin{array}{*{20}c} {\arctan \left( {\frac{{\eta_{{{\text{b1}}}} \eta_{{{\text{c1}}}} - \eta_{{{\text{a1}}}} \eta_{{{\text{d1}}}} }}{{\eta_{{{\text{a1}}}} \eta_{{{\text{c1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} }}} \right),} & {\eta_{{{\text{a1}}}} \eta_{{{\text{c1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} > 0,} \\ {{\uppi } + \arctan \left( {\frac{{\eta_{{{\text{b1}}}} \eta_{{{\text{c1}}}} - \eta_{{{\text{a1}}}} \eta_{{{\text{d1}}}} }}{{\eta_{{{\text{a1}}}} \eta_{{{\text{c1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} }}} \right),} & {\eta_{{{\text{a1}}}} \eta_{{{\text{c1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} < 0,} \\ \end{array} } \right.\gamma_{2y} = \left\{ {\begin{array}{*{20}c} {\arctan \left( {\frac{{\eta_{{{\text{b1}}}} \eta_{{{\text{g1}}}} - \eta_{{{\text{a1}}}} \eta_{{{\text{d1}}}} }}{{\eta_{{{\text{a1}}}} \eta_{{{\text{g1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} }}} \right),} & {\eta_{{{\text{a1}}}} \eta_{{{\text{g1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} > 0,} \\ {{\uppi } + \arctan \left( {\frac{{\eta_{{{\text{b1}}}} \eta_{{{\text{g1}}}} - \eta_{{{\text{a1}}}} \eta_{{{\text{d1}}}} }}{{\eta_{{{\text{a1}}}} \eta_{{{\text{g1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} }}} \right),} & {\eta_{{{\text{a1}}}} \eta_{{{\text{g1}}}} + \eta_{{{\text{b1}}}} \eta_{{{\text{d1}}}} < 0,} \\ \end{array} } \right.$$

\(\gamma_{g} = \arctan \frac{{2\xi_{{{\text{n}}g}} (\omega_{{{\text{n}}g}} /\omega_{{{\text{m}}0}} )}}{{1 - (\omega_{{{\text{n}}g}} /\omega_{{{\text{m}}0}} )^{2} }},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g = x,\psi\), \(r_{{\text{m}}} = \frac{{m_{0} }}{M}\),

\(r_{l} = \frac{{l_{0} }}{{l_{{\text{e}}} }}\), \(\rho_{g} = 1 - \omega_{{{\text{ng}}}}^{{2}} /\omega_{{{\text{m0}}}}^{{2}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g = x,\psi\), \(\xi_{{{\text{n}}x}} = \frac{{f_{x} }}{{2\sqrt {k_{x} M} }}\),

\(\xi_{{{\text{n}}\psi }} = \frac{{f_{\psi } }}{{2\sqrt {k_{\psi } J} }}\), \(\omega_{{{\text{n}}x}} = \sqrt {k_{x} /M}\), \(\omega_{{{\text{n}}\psi }} = \sqrt {k_{\psi } /J}\),

$$\begin{gathered} \eta_{{{\text{a1}}}} = \omega_{{{\text{m0}}}}^{{4}} M_{1} M_{2} - \omega_{{{\text{m0}}}}^{{2}} M_{1} (k^{\prime}_{1y} + k_{2y} ) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \omega_{{{\text{m0}}}}^{2} k^{\prime}_{1y} M_{2} - \omega_{{{\text{m0}}}}^{{2}} f^{\prime}_{1y} f_{2y} + k^{\prime}_{1y} k_{2y} , \hfill \\ \end{gathered}$$

\(\eta_{{{\text{b1}}}} = - \omega_{{{\text{m0}}}}^{{3}} [f^{\prime}_{1y} M_{2} + M_{1} (f^{\prime}_{1y} + f_{2y} )] + \omega_{{{\text{m0}}}} (f^{\prime}_{1y} k_{2y} + f_{2y} k^{\prime}_{1y} )\),

\(\eta_{{{\text{c1}}}} = k^{\prime}_{1y}\), \(\eta_{{{\text{d1}}}} = \omega_{{{\text{m0}}}} f^{\prime}_{1y}\), \(\eta_{{{\text{g1}}}} = k^{\prime}_{1y} - \omega_{{{\text{m0}}}}^{{2}} M_{1}\).

### 2.4 Derivation of the Equivalent Stiffness Using the Asymptotic Method

Since the damping of vibrating machines is relatively very small, the nonlinear damping force \(F_{1} (y_{1} ,y_{2} ,\dot{y}_{1} ,\dot{y}_{2} )\) in Eq. (1) can be directly replaced by \(f^{\prime}_{1y} (\dot{y}_{1} - \dot{y}_{2} )\). The nonlinear restoring force \(F_{2} (y_{1} ,y_{2} )\), however, should be equivalently linearized, which will be addressed in the following.

Based on the asymptotic method [15], the first approximate solution of Eq. (5) is obtained as:

$$y_{12} = \delta_{12} \cos (\omega_{{{\text{m0}}}} t + \theta ) = \delta_{12} \cos \kappa ,$$

(9)

with

$$\theta = {\text{arccot}}\left[ {\frac{{f^{\prime}_{1y} }}{2m}\frac{1}{{(\omega_{0}^{2} - \omega_{{{\text{m0}}}}^{{2}} )}}} \right],$$

where *θ* is the phase lag angle of the response for the relative motion.

Therefore, the nonlinear restoring force \(F_{2} (y_{1} ,y_{2} )\) can be rewritten as:

$$\begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} F_{2} (y_{1} ,y_{2} ) = F_{2} (y_{12} ) = \varepsilon f_{2} (y_{12} ) = \varepsilon f_{2} (\delta_{12} \cos \kappa ) \hfill \\ {\kern 1pt} = \left\{ {\begin{array}{*{20}c} {k_{1y} \delta_{12} \cos \kappa ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \le \kappa \le {\kern 1pt} {\kern 1pt} {\uppi } - \kappa_{0} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{and}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\uppi } + \kappa_{0} \le \kappa \le 2{\pi ,}{\kern 1pt} } \\ {k_{1y} \delta_{12} \cos \kappa + \Delta k_{1y} (\delta_{12} \cos \kappa + \mu ),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\uppi } - \kappa_{0} \le \kappa \le {\uppi } + \kappa_{0} ,{\kern 1pt} {\kern 1pt} } \\ \end{array} } \right. \hfill \\ \end{gathered}$$

(10)

where *ε* is a small parameter derived from the asymptotic method, \(\kappa_{0} = \arccos (\mu /\delta_{12} )\).

The equivalent stiffness is obtained directly as:

$$k^{\prime}_{1y} = \frac{1}{{{\uppi }\delta_{12} }}\int_{0}^{{2{\uppi }}} {\varepsilon f_{2} (\delta_{12} \cos \kappa )} \cos \kappa {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{d}}\kappa .$$

(11)

Substituting Eq. (10) into Eq. (11), and based on the subsection integral method, the expression of the equivalent stiffness under the condition of \(\delta_{12} \ge \mu\), is deduced as:

$$k^{\prime}_{1y} = k_{1y} + \Delta k_{1y} \left[ {\frac{2}{{\uppi }}\arccos \left( {\frac{\mu }{{\delta_{12} }}} \right) - \frac{2\mu }{{{\uppi }\delta_{12} }}\sqrt {1 - \left( {\frac{\mu }{{\delta_{12} }}} \right)^{2} } } \right].$$

(12)