Augmented perpetual manifolds and perpetual mechanical systems-part II: theorem for dissipative mechanical systems behaving as perpetual machines of third kind

Abstract

Perpetual points in mathematics defined recently, and their significance in nonlinear dynamics and their application in mechanical systems is currently ongoing research. The perpetual points significance relevant to mechanics so far is that they form the perpetual manifolds of rigid body motions of unforced mechanical systems, which lead to the definition of perpetual mechanical systems. The perpetual mechanical systems admit as perpetual points rigid body motions which are forming the perpetual manifolds. The concept of perpetual manifolds extended to the definition of augmented perpetual manifolds that an externally excited multi-degree of freedom mechanical system is moving as a rigid body, and may exhibit particle-wave motion. This article is complementary to the work done so far applied to natural perpetual dissipative mechanical systems with motion defined by the exact augmented perpetual manifolds, whereas the internal forces, and individual energies are examined, to understand further the mechanics of these systems while their motion is in the exact augmented perpetual manifolds. A theorem is proved stating that under conditions when the motion of a perpetual natural dissipative mechanical system is in the exact augmented perpetual manifolds, all the internal forces are zero, which is rather significant in the mechanics of these systems since the operation on augmented perpetual manifolds leads to zero internal degradation. Moreover, the theorem is stating that the potential energy is constant, and there is no dissipation of energy, therefore the process is internally isentropic, and there is no energy loss within the perpetual mechanical system. Also in this theorem is proved that the external work done is equal to the changes of the kinetic energy, therefore the motion in the exact augmented perpetual manifolds is driven only by the changes of the kinetic energy. This is also a significant outcome to understand the mechanics of perpetual mechanical systems while it is in particle-wave motion which is guided by kinetic energy changes. In the final statement of the theorem is stated and proved that the perpetual dissipative mechanical system can behave as a perpetual machine of third kind which is rather significant in mechanical engineering. Noting that the perpetual mechanical system apart of the augmented perpetual manifolds solutions is having other solutions too, e.g., in higher normal modes and in these solutions the theorem is not valid. The developed theory is applied in the only two possible configurations that a mechanical system can have. The first configuration is a perpetual mechanical system without any connection through structural elements with the environment. In the second configuration, the perpetual mechanical system is a subsystem, connected with structural elements with the environment. In both examples, the motion in the exact augmented perpetual manifolds is examined with the view of mechanics defined by the theorem, resulting in excellent agreement between theory and numerical simulations. The outcome of this article is significant in physics to understand the mechanics of the motion of perpetual mechanical systems in the exact augmented perpetual manifolds, which is described through the kinetic energy changes and this gives further insight into the mechanics of particle-wave motions. Also, in mechanical engineering the outcome of this article is significant, because it is shown that the motion of the perpetual mechanical systems in the exact augmented perpetual manifolds is the ultimate, in the sense that there are no internal forces which lead to degradation of the internal structural elements, and there is no energy loss due to dissipation.

Appendices

Appendix-A

The solutions in exact augmented perpetual manifolds for two types of external forces provided in [21] in the following Table

Table 10 Forms of external forces with explicit equations defining the velocities, the displacements and the wave velocities in the exact augmented perpetual manifolds

10 are presented.

See Table 10

The kinetic energies for these two type external forces are given by,

• The kinetic energy $\left( {E_{T,a}^{\left( 1 \right)} } \right)$ associated with a velocity given by equation (A.2a) for the first type of external forces $\left( {F_{k}^{\left( 1 \right)} \left( t \right)} \right)$ in Table 10, using Eq. (7c) is given by,

  $$ \begin{aligned} E_{T,a}^{\left( 1 \right)} \left( t \right) = & \frac{1}{2} \cdot \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} M_{i,j} \cdot \left( {\frac{{\eta^{2} }}{{4 \cdot \left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} \cdot t^{4} + \frac{\eta \cdot c}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} \cdot t^{3} - } \right. \\ & \left( {\frac{{\eta^{2} \cdot t_{0}^{2} }}{{2 \cdot \left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \frac{{\eta \cdot c \cdot t_{0} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} - \frac{{c^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} - \frac{{\eta \cdot \dot{x}_{a,1} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }}} \right) \cdot t^{2} \\ & - \left( {\frac{{\eta \cdot c \cdot t_{0}^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \frac{{2 \cdot c^{2} \cdot t_{0} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} - \frac{{2 \cdot c \cdot \dot{x}_{a,1} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }}} \right) \cdot t + \frac{{\eta^{2} \cdot t_{0}^{4} }}{{4 \cdot \left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} \\ & \left. { + \frac{{\eta \cdot c \cdot t_{0}^{3} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \left( {\frac{{c^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} - \frac{{\eta \cdot \dot{x}_{a,1} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }}} \right) \cdot t_{0}^{2} - \frac{{2 \cdot c \cdot \dot{x}_{a,1} \left( {t_{0} } \right) \cdot t_{0} }}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }} + \dot{x}_{a,1}^{2} \left( {t_{0} } \right)} \right). \\ \end{aligned} $$

  (A.5a)

• The kinetic energy $\left( {E_{T,a}^{\left( 2 \right)} } \right)$ associated with a velocity given by equation (A.2b) for the second type of external forces $\left( {F_{k}^{\left( 2 \right)} \left( t \right)} \right)$ in Table 10, using Eq. (7c) is given by,

  $$ \begin{aligned} E_{T,a}^{\left( 2 \right)} \left( t \right) = & \frac{1}{2} \cdot \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} M_{i,j} \cdot \left( {\left( {\frac{{A_{{{\text{ex}}}}^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}}^{2} }} \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t + \theta_{{{\text{ex}}}} } \right)} \right.} \right. \\ & \left. { - \frac{{2 \cdot A_{{{\text{ex}}}} \cdot \dot{x}_{a,2} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} \cdot \omega_{{{\text{ex}}}} }} - \frac{{2 \cdot A_{ex}^{2} \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t_{0} + \theta_{{{\text{ex}}}} } \right)}}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}}^{2} }}} \right) \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t + \theta_{{{\text{ex}}}} } \right) \\ & + \left. {\left( {\frac{{A_{ex}^{2} \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t_{0} + \theta_{{{\text{ex}}}} } \right)}}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}}^{2} }} + \frac{{2 \cdot A_{{{\text{ex}}}} \cdot \dot{x}_{a,2} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} \cdot \omega_{{{\text{ex}}}} }}} \right) \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t_{0} + \theta_{{{\text{ex}}}} } \right) + \dot{x}_{a,2}^{2} \left( {t_{0} } \right)} \right). \\ \end{aligned} $$

  (A.5b)

The power associated with the two types of the external forces, given in Table 10, is,

• The power $\left( {P_{{{\text{ex}},a}}^{\left( 1 \right)} } \right)$ of all the external forces, associated with the first type of external forces $\left( {F_{k}^{\left( 1 \right)} \left( t \right)} \right)$ given by equation (A.1a) in Table 10, and velocity given by equation (A.2a) in Table 10, in explicit form is,

  $$ \begin{aligned} P_{{{\text{ex}},a}}^{\left( 1 \right)} \left( t \right) = & \left\{ {F_{i}^{\left( 1 \right)} \left( t \right)} \right\}^{T} \times \left\{ {\dot{x}_{a,1} \left( t \right)} \right\} \\ = & \frac{{\mathop \sum \nolimits_{i = 1}^{N} \mathop \sum \nolimits_{j = 1}^{N} M_{i,j} }}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }} \cdot \\ & \left( {\frac{{\eta^{2} }}{{2 \cdot \mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }} \cdot \left( {t^{2} - t_{0}^{2} } \right) \cdot {\text{t}} + \frac{{c^{2} }}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }} \cdot \left( {t - t_{0} } \right) + \frac{\eta \cdot c}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }} \cdot \left( {t - t_{0} } \right) \cdot t} \right. \\ & \left. { + \frac{\eta \cdot c}{{2 \cdot \mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }} \cdot \left( {t^{2} - t_{0}^{2} } \right) + \eta \cdot \dot{x}_{a,1} \left( {t_{0} } \right) \cdot t + c \cdot \dot{x}_{a,1} \left( {t_{0} } \right)} \right) \\ \end{aligned} $$

  (A.6a)

• The power $\left( {P_{ex,a}^{\left( 2 \right)} } \right)$ for the second type of external forces $\left( {F_{k}^{\left( 2 \right)} \left( t \right)} \right)$ that are given by equation (A.1b) in Table 10, and velocity given by equation (A.2b) in Table 10, in explicit form is,

  $$ \begin{aligned} P_{{{\text{ex}},a}}^{\left( 2 \right)} \left( t \right) = & \left\{ {F_{i}^{\left( 2 \right)} \left( t \right)} \right\}^{T} \times \left\{ {\dot{x}_{a,2} \left( t \right)} \right\} \\ = & \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} M_{i,j} \cdot \left( { - \frac{{A_{{{\text{ex}}}}^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}} }} \cdot \sin \left( {\omega_{{{\text{ex}}}} \cdot t + \theta_{{{\text{ex}}}} } \right) \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t + \theta_{{{\text{ex}}}} } \right)} \right. \\ & \left. { + \left( {\frac{{A_{ex}^{2} \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t_{0} + \theta_{{{\text{ex}}}} } \right)}}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}} }} + \frac{{A_{ex} \cdot \dot{x}_{a,2} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }}} \right) \cdot \sin \left( {\omega_{{{\text{ex}}}} \cdot t + \theta_{{{\text{ex}}}} } \right)} \right) \\ \end{aligned} $$

  (A.6b)

The work done $\left( {W_{{{\text{ex}},a}} } \right)$ by the external forces in the exact augmented perpetual manifolds is given with integration of the external forces power in time using Eq. (7b). The explicit form of the external work done by the two types of the external forces, given in Table 10, is:

• The work $\left( {W_{{{\text{ex}},a}}^{\left( 1 \right)} } \right)$ done by all the external forces of the first type $\left( {F_{k}^{\left( 1 \right)} } \right)$ defined by equation (A.1a), with integration of equation (A.6a) is obtained as follows,

  $$ \begin{aligned} W_{{{\text{ex}},a}}^{\left( 1 \right)} \left( t \right) = & \frac{1}{2}\mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} M_{i,j} \cdot \left( {\frac{{\eta^{2} }}{{4 \cdot \left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} \cdot t^{4} } \right. + \frac{\eta \cdot c}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} \cdot t^{3} + \\ & - \left( {\frac{{\eta^{2} \cdot t_{0}^{2} }}{{2 \cdot \left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \frac{{\eta \cdot c \cdot t_{0} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} - \frac{{c^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} - \frac{{\eta \cdot \dot{x}_{a,1} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }}} \right) \cdot t^{2} + \\ & - \left( {\frac{{\eta \cdot c \cdot t_{0}^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \frac{{2 \cdot c^{2} \cdot t_{0} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} - \frac{{2 \cdot c \cdot \dot{x}_{a,1} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }}} \right) \cdot t \\ & + \frac{{\eta^{2} \cdot t_{0}^{4} }}{{4 \cdot \left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \frac{{c^{2} \cdot t_{0}^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \frac{{\eta \cdot c \cdot t_{0}^{3} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} }} + \\ & \left. { - \frac{{\eta \cdot \dot{x}_{a,1} \left( {t_{0} } \right) \cdot t_{0}^{2} }}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }} - \frac{{2 \cdot c \cdot \dot{x}_{a,1} \left( {t_{0} } \right) \cdot t_{0} }}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} }}} \right) = T_{a}^{\left( 1 \right)} \left( t \right) - T_{a}^{\left( 1 \right)} \left( {t_{0} } \right) \\ \end{aligned} $$

  (A.7a)

The last term obtained using Eq. (A.5a) for any time instant $-t$ and for the time instant $- t_{0}$ that the motion with the first type of external forces is starting, and certifies Eq. (23) that is the e-part of the theorem.

• The work $\left( {W_{{{\text{ex}},a}}^{\left( 2 \right)} } \right)$ done by all the external forces of the second type $\left( {F_{k}^{\left( 2 \right)} } \right)$ defined by equation (A.1b), with integration of equation (A.6b) is obtained as follows,

  $$ \begin{aligned} W_{{{\text{ex}},a}}^{\left( 2 \right)} \left( t \right) = & \frac{1}{2} \cdot \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} M_{i,j} \cdot \left( {\frac{{A_{{{\text{ex}}}}^{2} }}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}}^{2} }} \cdot \cos^{2} \left( {\omega_{{{\text{ex}}}} \cdot t + \theta_{{{\text{ex}}}} } \right)} \right. \\ & - 2 \cdot \left( {\frac{{A_{{{\text{ex}}}}^{2} \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t_{0} + \theta_{{{\text{ex}}}} } \right)}}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}}^{2} }} + \frac{{A_{{{\text{ex}}}} \cdot \dot{x}_{a,2} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} \cdot \omega_{{{\text{ex}}}} }}} \right) \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t + \theta_{{{\text{ex}}}} } \right) \\ & + \left. {\frac{{A_{ex}^{2} \cdot \cos^{2} \left( {\omega_{{{\text{ex}}}} \cdot t_{0} + \theta_{{{\text{ex}}}} } \right)}}{{\left( {\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} } \right)^{2} \cdot \omega_{{{\text{ex}}}}^{2} }} + \frac{{2 \cdot A_{{{\text{ex}}}} \cdot \cos \left( {\omega_{{{\text{ex}}}} \cdot t_{0} + \theta_{{{\text{ex}}}} } \right) \cdot \dot{x}_{a,2} \left( {t_{0} } \right)}}{{\mathop \sum \nolimits_{j = 1}^{N} M_{k,j} \cdot \omega_{{{\text{ex}}}} }}} \right) \\ = & T_{a}^{\left( 2 \right)} \left( t \right) - T_{a}^{\left( 2 \right)} \left( {t_{0} } \right) \\ \end{aligned} $$

  (A.7b)

The last term of equation (A.4b) is obtained using equation (A.5b) for the two time instants, the time instant $- t_{0}$ that the application of the second type of the external force is starting, and any other time instant-$t$. The equality of the work done by the external forces with the difference of the kinetic energies, certify analytically, Eq. (23) for the second type of forces, and therefore the part-e of the theorem.

Appendix-B

In this appendix, both sets of the nonsmooth equations of motion of the mechanical system in Sect. 3.2 are written in a certain form for their numerical solution. Since they have differential inclusion, the algorithm for two switch hypersurface boundary functions of [26] based on the switch model developed in [37], is used.

Considering the equations of motion (30a-e) with the following change of variables,

$$ s_{1} = r_{1} ,s_{2} = r_{2} ,s_{4} = \dot{r}_{2} $$

(B.1)

lead to the following first-order system of differential equations,

$$ \left\{ {\dot{s}_{i} } \right\} = \left\{ {\begin{array}{*{20}c} {S_{1} } \\ {S_{2} } \\ {S_{3} } \\ {S_{4} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {s_{3} } \\ {s_{4} } \\ {G_{1} + F_{{ns,r_{1} }} } \\ {G_{2} + F_{{ns,r_{2} }} } \\ \end{array} } \right\},{\text{with}}\quad i = 1,2, \ldots ,4, $$

(B.2)

with,

$$ \begin{aligned} G_{1} = & \frac{1}{{m_{r,1} }} \cdot \left( {k_{r} \cdot \left( {s_{2} - s_{1} } \right) + k_{nl,r} \cdot \left( {s_{2} - s_{1} } \right)^{3} + c_{r} \cdot \left( {s_{4} - s_{3} } \right)} \right. \\ & + c_{nl,r} \cdot \tanh \left( {b \cdot \left( {s_{4} - s_{3} } \right)} \right) + g_{1} \cdot \left( {s_{2} - s_{1} } \right)^{2} \cdot \left( {s_{4} - s_{3} } \right) \\ & - \left. {\alpha \cdot s_{3} - \mu \cdot s_{1} - \mu_{nl} \cdot s_{1}^{3} + A_{ex} \cdot \sin \left( {\omega_{{{\text{ex}}}} \cdot t} \right)} \right) \\ \end{aligned} $$

(B.3a)

$$ \begin{aligned} G_{2} = & \frac{1}{{m_{r,2} }} \cdot \left( {k_{r} \cdot \left( {s_{1} - s_{2} } \right) + k_{nl,r} \cdot \left( {s_{1} - s_{2} } \right)^{3} + c_{r} \cdot \left( {s_{3} - s_{4} } \right)} \right. \\ & + c_{nl,r} \cdot \tanh \left( {b \cdot \left( {s_{3} - s_{4} } \right)} \right) + g_{1} \cdot \left( {s_{2} - s_{1} } \right)^{2} \cdot \left( {s_{3} - s_{4} } \right) \\ & - \left. {\frac{{m_{r,2} }}{{m_{r,1} }} \cdot \alpha \cdot s_{4} - \frac{{m_{r,2} }}{{m_{r,1} }} \cdot \mu \cdot s_{2} - \frac{{m_{r,2} }}{{m_{r,1} }} \cdot \mu_{nl} \cdot s_{2}^{3} + \frac{{m_{r,2} }}{{m_{r,1} }} \cdot A_{ex} \cdot \sin \left( {\omega_{{{\text{ex}}}} \cdot t} \right)} \right) \\ \end{aligned} $$

(B.3b)

and,

$$ F_{{ns,r_{1} }} \in \left\{ {\begin{array}{*{20}l} { - \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 + \delta s_{3} }},} \hfill & {s_{3} > 0} \hfill \\ {\mu_{1}^{s,r} F_{1}^{N,r} \cdot \left[ { - 1,1} \right],} \hfill & {s_{3} = 0} \hfill \\ {\frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 - \delta \cdot s_{3} }}{ },{ }} \hfill & {s_{3} < 0} \hfill \\ \end{array} } \right. $$

(B.4a)

and,

$$ F_{{ns,r_{2} }} \in \left\{ {\begin{array}{*{20}l} { - \frac{{m_{r,2} }}{{m_{r,1} }} \cdot \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 + \delta \cdot s_{4} }},} \hfill & {s_{4} > 0} \hfill \\ {\frac{{m_{r,2} }}{{m_{r,1} }} \cdot \mu_{1}^{s,r} \cdot F_{1}^{N,r} \cdot \left[ { - 1,1} \right],} \hfill & {s_{4} = 0} \hfill \\ {\frac{{m_{r,2} }}{{m_{r,1} }} \cdot { }\frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 - \delta \cdot s_{4} }},} \hfill & {s_{4} < 0} \hfill \\ \end{array} } \right. $$

(B.4b)

Considering Eqs. (31d-e) that define the two switch hypersurface boundary functions (HBF), with the variables defined by Eq. (B.1) the two HBF are given by,

$$ h_{1} \left( {s_{1} ,s_{2} ,s_{3} ,s_{4} } \right) = s_{3} ,{\text{for}}\,{\Sigma }_{1} , $$

(B.5)

$$ h_{2} \left( {s_{1} ,s_{2} ,s_{3} ,s_{4} } \right) = s_{4} ,{\text{for}}\,{\Sigma }_{2} . $$

(B.6)

Convexification of the friction force $\left({F}_{ns{,r}_{1}}\right)$ lead to,

$$ F_{{ns,r_{1} }} \in \left\{ {\begin{array}{*{20}l} { - \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 + \delta \cdot s_{3} }},} \hfill & {s_{3} > 0} \hfill \\ {\mu_{1}^{s,r} \cdot F_{1}^{N,r} \cdot \left[ { - 1,1} \right],} \hfill & {s_{3} = 0} \hfill \\ {\frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 - \delta \cdot s_{3} }}{ },{ }} \hfill & {s_{3} < 0} \hfill \\ \end{array} } \right. $$

(B.7)

therefore,

$$ S_{3, + } = G_{1} - \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{m_{r,1} \cdot \left( {1 + \delta \cdot s_{3} } \right)}} $$

(B.8)

and,

$$ S_{3, - } = G_{1} + \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{m_{r,1} \cdot \left( {1 - \delta \cdot s_{3} } \right)}}{ } $$

(B.9)

The convexification of the other friction force leads to,

$$ F_{{ns,r_{2} }} \in \left\{ {\begin{array}{*{20}l} { - \frac{{m_{r,2} }}{{m_{r,1} }} \cdot \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 + \delta \cdot s_{4} }},} \hfill & {s_{4} > 0} \hfill \\ {\frac{{m_{r,2} }}{{m_{r,1} }} \cdot \mu_{1}^{s,r} \cdot F_{1}^{N,r} \cdot \left[ { - 1,1} \right],} \hfill & {s_{4} = 0} \hfill \\ {\frac{{m_{r,2} }}{{m_{r,1} }} \cdot \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{1 - \delta \cdot s_{4} }},} \hfill & {s_{4} < 0} \hfill \\ \end{array} } \right. $$

(B.10)

therefore,

$$ S_{4, + } = G_{2} - \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{m_{r,1} \cdot \left( {1 + \delta \cdot s_{4} } \right)}} $$

(B.11)

$$ S_{4, - } = G_{2} + { }\frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{m_{r,1} \cdot \left( {1 - \delta \cdot s_{4} } \right)}} $$

(B.12)

Finally the two normal vectors to each one of the switch hypersurfaces are given by,

$$ n_{1}^{T} = \left[ {\begin{array}{*{20}c} {\frac{{\partial h_{1} }}{{\partial s_{1} }}} & {\frac{{\partial h_{1} }}{{\partial s_{2} }}} & {\frac{{\partial h_{1} }}{{\partial s_{3} }}} & {\frac{{\partial h_{1} }}{{\partial s_{4} }}} \\ \end{array} } \right]^{T} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 \\ \end{array} } \right]^{T} ,{\text{for}}\,{\Sigma }_{1} $$

(B.13)

$$ n_{2}^{T} = \left[ {\begin{array}{*{20}c} {\frac{{\partial h_{2} }}{{\partial s_{1} }}} & {\frac{{\partial h_{2} }}{{\partial s_{2} }}} & {\frac{{\partial h_{2} }}{{\partial s_{3} }}} & {\frac{{\partial h_{2} }}{{\partial s_{4} }}} \\ \end{array} } \right]^{T} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 1 \\ \end{array} } \right]^{T} ,{\text{for}}\,{\Sigma }_{2} . $$

(B.14)

In switch model the identification of regions that the system belongs is found with the multiplication of the normal vectors with the vector fields. Simplifying things the explicit form is,

$$ n_{1}^{T} \times \left\{ {\begin{array}{*{20}c} {S_{1} } \\ {S_{2} } \\ {S_{3} } \\ {S_{4} } \\ \end{array} } \right\} = S_{3} $$

(B.15)

Therefore, convexification of the vector field using the switch model near the surface ${\Sigma }_{1}$ should be done only in the ${S}_{3}$ term of the vector field using the parameter,

$$ a_{1} = \frac{{S_{3, - } + \tau^{ - 1} \cdot h_{1} }}{{S_{3, - } - S_{3, + } }} $$

(B.16)

Also, for the other hypersurface boundary,

$$ n_{2}^{T} \times \left\{ {\begin{array}{*{20}c} {S_{1} } \\ {S_{2} } \\ {S_{3} } \\ {S_{4} } \\ \end{array} } \right\} = S_{4} $$

(B.17)

Therefore, the convexification of the vector field using the switch model near the surface ${\Sigma }_{2}$ must be done only in ${S}_{4}$ term of the vector field using the parameter,

$$ a_{2} = \frac{{S_{4, - } + \tau^{ - 1} \cdot h_{2} }}{{S_{4, - } - S_{4, + } }} $$

(B.18)

The aforementioned definitions are sufficient for the numerical solution of the 2-dof nonsmooth system with the application of the double switch algorithm given in Fig. A.1 of Appendix A in [26].

Considering equation (34) with the following change of variables,

$$ q_{1} = r_{{{\text{ROM}}}} ,\,q_{2} = \dot{r}_{{{\text{ROM}}}} $$

(B.19)

lead to the following first-order system of differential equations,

$$ \begin{aligned} \left\{ {\dot{q}_{i} } \right\} = \left\{ {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ \end{array} } \right\} = & \left\{ {\begin{array}{*{20}c} {q_{2} } \\ { - \frac{\alpha }{{m_{r,1} }} \cdot q_{2} - \frac{\mu }{{m_{r,1} }} \cdot q_{1} - \frac{{\mu_{nl} }}{{m_{r,1} }} \cdot q_{1}^{3} + \frac{{f_{r} \left( t \right)}}{{m_{r,1} }} - \frac{{\mu_{1}^{s,r} \cdot F_{1}^{N,r} }}{{m_{r,1} \cdot \left( {1 + \delta \cdot \left| {q_{2} } \right|} \right)}} \cdot {\text{sign}}\left( {q_{2} } \right)} \\ \end{array} } \right\} \\ & \quad {\text{with}}\,{\text{i = 1,2,}} \ldots \\ \end{aligned} $$

(B.20)

and after convexification of the vector field leads [37],

$${Q}_{2}\in \left\{\begin{array}{c}-\frac{\alpha }{{m}_{r,1}} \cdot {q}_{2}-\frac{\mu }{{m}_{r,1}} \cdot {q}_{1}-\frac{{\mu }_{nl}}{{m}_{r,1}} \cdot {q}_{1}^{3}+\frac{{f}_{r}\left(t\right)}{{m}_{r,1}}-\frac{{\mu }_{1}^{s,r} \cdot {F}_{1}^{N,r}}{{m}_{r,1} \cdot \left(1+\delta \cdot {q}_{2}\right)}, {q}_{2}>0\\ -\frac{\alpha }{{m}_{r,1}} \cdot {q}_{2}-\frac{\mu }{{m}_{r,1}} \cdot {q}_{1}-\frac{{\mu }_{nl}}{{m}_{r,1}} \cdot {q}_{1}^{3}+\frac{{f}_{r}\left(t\right)}{{m}_{r,1}}+\left\{\frac{{\mu }_{1}^{s,r} \cdot {F}_{1}^{N,r}}{{m}_{r,1}} \cdot \left[-\mathrm{1,1}\right]\right\},{q}_{2}=0 \\ -\frac{\alpha }{{m}_{r,1}} \cdot {q}_{2}-\frac{\mu }{{m}_{r,1}} \cdot {q}_{1}-\frac{{\mu }_{nl}}{{m}_{r,1}} \cdot {q}_{1}^{3}+\frac{{f}_{r}\left(t\right)}{{m}_{r,1}}+\frac{{\mu }_{1}^{s,r} \cdot {F}_{1}^{N,r}}{{m}_{r,1} \cdot \left(1-\delta \cdot {q}_{2}\right)}, {q}_{2}<0\end{array}\right\}$$

(B.21)

And the switch algorithm developed in [37] can be applied for the numerical solution of Eq. (34) using the vector fields defined by the equations (B.20–21).