# New limit cycles of dry friction oscillators under harmonic load

Madeleine Pascal

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## To cite this version:

Madeleine Pascal. New limit cycles of dry friction oscillators under harmonic load. Nonlinear Dynamics, 2012, 70 (2), pp.1435–1443. 10.1007/s11071-012-0545-5. hal-00747517

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Submitted on 6 Jun 2016

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Distributed under a Creative Commons Attribution 4.0 International License
# New limit cycles of dry friction oscillators under harmonic load

Madeleine Pascal

**Abstract** We consider a system composed of two masses connected by linear springs. One of the masses is in contact with a driving belt moving at a constant velocity. Friction force, with Coulomb's characteristics, acts between the mass and the belt. Moreover, the mass is also subjected to a harmonic external force. Several periodic orbits including stick phases and slip phases are obtained. In particular, the existence of periodic orbits including a part where the mass in contact with the belt moves in the same direction at a higher speed than the belt itself is proved. Non-sticking orbits are also found for a non-moving belt. We prove that this kind of solution is symmetric in space and in time.

**Keywords** Dry friction · Periodic orbits · Coupled oscillator · Stick-slip motion

## 1 Introduction

This paper is a continuation of several investigations [5–7], related to vibrating systems excited by dry friction. These systems are frequently encountered in many industrial applications. One of the most popular models of stick-slip oscillators consists of several

masses connected by linear springs, one (or more) of the masses is in contact with a driving belt moving at a constant velocity. In the past, several authors investigated the behavior of this system, with different friction laws and with or without external actions and damping [1, 4], mainly via numerical approach.

However, assuming Coulomb's laws of dry friction, the corresponding dynamical model is a piecewise linear system, and even for multi-degree-of-freedom cases, some analytical results about the existence and the stability of periodic orbits including stick-slip phases have been obtained [5–7].

One interesting phenomenon is the existence, inside periodic orbits with stick and slip parts, of an “overshooting” slip phase. During this part of the orbit, the mass in contact with the belt moves in the same direction at a higher speed than the belt itself.

Up to now [9], this phenomenon has been observed only for more complex friction characteristics than Coulomb's ones. In [7], a self-excited stick-slip oscillator including two degrees of freedom is considered. Assuming Coulomb's laws of dry friction, a set of periodic orbits including an overshooting part is obtained using analytical methods.

In this work, we consider the same model of dry friction oscillator subjected to a harmonic external force. Several periodic orbits containing stick phases and slip phases are obtained. In particular, the existence of periodic orbits including an overshooting part is proved.
<figure><img src="image_3.png" /><figcaption>Fig. 1 Dry friction oscillator</figcaption></figure>

## 2 Problem formulation

The system (Fig. 1) is composed of two masses $m_1, m_2$ connected by two linear springs of stiffness $k_1, k_2$. The second mass is in contact with a belt moving at a constant velocity $v_0$. A friction force $\tilde{F}$ acts between the mass and the belt. Moreover, the second mass is also subjected to an external force $\tilde{R}$ given by

$$ \tilde{R} = \tilde{p} \cos(\tilde{\omega} t' + \varphi) \\ (\tilde{p}, \tilde{\omega}, \varphi \text{ are constant parameters}) \qquad (1) $$

The equations of motion related to this system are written as

$$ \begin{aligned} x_1'' + x_1 - \chi x_2 &= 0, \\ x_2'' + \chi \eta (x_2 - x_1) &= \eta u + p \cos(\omega t + \varphi) \end{aligned} \qquad (2) $$

$x_1, x_2$ are the displacements of the masses,

$$ \begin{aligned} \eta &= \frac{m_1}{m_2}, & \chi &= \frac{k_2}{k_1 + k_2}, & u &= \frac{\tilde{F}}{k_1 + k_2}, \\ p &= \eta \frac{\tilde{p}}{k_1 + k_2}, & t &= \Omega t', & \Omega &= \sqrt{\frac{k_1 + k_2}{m_1}}, \\ (O)' &= \frac{d(O)}{dt}, & \omega &= \frac{\tilde{\omega}}{\Omega} \end{aligned} \qquad (3) $$

The dry friction force $u$ is obtained from Coulomb's laws:

$$ u = \begin{cases} u_s \operatorname{sign}(V - x_2'), & \text{if } V - x_2' \neq 0, \\ \chi(x_2 - x_1) - \frac{p}{\eta} \cos(\omega t + \varphi), & \text{if } V - x_2' = 0, \quad |\chi(x_2 - x_1) - \frac{p}{\eta} \cos(\omega t + \varphi)| < u_r, \\ u_s, & \text{if } V - x_2' = 0, \quad \chi(x_2 - x_1) - \frac{p}{\eta} \cos(\omega t + \varphi) > u_r, \\ -u_s, & \text{if } V - x_2' = 0, \quad \chi(x_2 - x_1) - \frac{p}{\eta} \cos(\omega t + \varphi) < -u_r, \quad 0 < u_s < u_r, \quad V = \frac{v_0}{\Omega}, \end{cases} \qquad (4) $$

$u_r$ is the friction force at rest (sticking), $u_s$ is the slipping friction force.

## 3 Prediction of the oscillations exhibited by the system

The dynamical behavior of this oscillator includes several phases of slip and stick motion of $m_2$. For each kind of motion, a close form solution is available.

### 3.1 Slip motion of $m_2$ with $x_2' < V$

The solution is obtained from a modal analysis of (2) where $u = u_s$

$$ \begin{aligned} Z(t) &= H(t)(Z_0 - F_0) + F(t), \\ F(t) &= \begin{pmatrix} R(t) \\ R'(t) \end{pmatrix}, \\ R(t) &= Q \cos(\omega t + \varphi), \quad F_0 = F(0), \end{aligned} \qquad (5) $$

$$ \begin{aligned} Z &= \begin{pmatrix} z \\ z' \end{pmatrix}, & Z_0 &= Z(0), \\ z &= X - d_0, & X &= (x_1, x_2)^t, \\ H(t) &= \begin{pmatrix} H_1(t) & H_2(t) \\ H_3(t) & H_1(t) \end{pmatrix}, & d_0 &= (d_{01}, d_{02})^t, \\ d_{01} &= \frac{u_s}{1 - \chi}, & d_{02} &= \frac{d_{01}}{\chi}, \\ Q &= (q_1, q_2)^t, & q_1 &= \frac{p\chi}{(\omega^2 - \omega_1^2)(\omega^2 - \omega_2^2)}, \\ q_2 &= q_1 \frac{(1 - \omega^2)}{\chi} \end{aligned} \qquad (6) $$

The two by two matrices $H_i(t)$ ($i = 1, 2, 3$) and the natural frequencies ($\omega_1, \omega_2$) are obtained in analytical form (see Appendix 1).

### 3.2 Slip motion of $m_2$ with $x_2' > V$ (overshooting)

The solution is obtained from (2) where $u = -u_s$

$$ \begin{aligned} Z(t) &= H(t)(Z_0 - F_0) + F(t) + 2L(t)d_0, \\ L(t) &= \begin{pmatrix} H_1(t) - I \\ H_3(t) \end{pmatrix}, & I &= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \end{aligned} \qquad (8) $$
### 3.3 Stick motion of $m_2$ ($x'_2 = V$)

This motion is related to the following dynamical system:

$$x_1'' + x_1 - \chi x_2 = 0, \quad x_2'' = 0 \quad (9)$$

The solution [5] is given by

$$Z(t) = \Gamma(t)Z_0, \quad \Gamma(t) = \begin{pmatrix} \Gamma_1(t) & \Gamma_2(t) \\ \Gamma_3(t) & \Gamma_1(t) \end{pmatrix} \quad (10)$$

The two by two matrices $\Gamma_i(t)$ ($i=1,2,3$) are given in Appendix 1. Moreover, during all this kind of motion, the following constraint holds:

$$|\chi\eta(x_2 - x_1) - p \cos(\omega t + \varphi)| < \eta u_r \quad (11)$$

## 4 Symmetrical periodic solutions

A first set of periodic orbits of period $\Theta = 2\pi/\omega$ is obtained. These motions involve for each period first a slip motion of $m_2$ with $x'_2 < V$ followed by a phase of stick motion of the mass ($x'_2 = V$). At the beginning ($t=0$) of the slip motion, the initial conditions are given by

$$z_2'(0) = V, \\ \chi\eta(z_2(0) - z_1(0)) = p \cos\varphi + \eta(u_r - u_s) \quad (12)$$

For $0 < t < \tau$, the system undergoes a slip motion defined by (5).

At the end ($t = \tau$) of the slip motion, the following condition is assumed:

$$Z(\tau) = E Z_0, \quad E = \begin{pmatrix} -I & 0 \\ 0 & I \end{pmatrix} \quad (13)$$

The time lag $\varphi$ of the external force is given by $\varphi = (\pi - \omega\tau)/2$. From (12) and (13), we deduce:

$$-\eta(u_r + u_s) < \chi\eta(z_{2c} - z_{1c}) - p \cos(\omega\tau + \varphi) \\ = -\eta(u_r - u_s) < \eta(u_r - u_s) \quad (14)$$

For $\tau < t < \tau + T$, the system motion is a sticking motion given by

$$Z(t) = \Gamma(t - \tau)E Z_0$$

A periodic solution of period $\Theta$ is obtained if the following relation is fulfilled:

$$Z_0 = \Gamma(T)E Z_0, \quad (T = \Theta - \tau) \quad (15)$$

Taking into account the properties (Appendix 1) of the matrices $H(t)$, $\Gamma(t)$, the conditions for the existence

of this periodic solution are given by the following system of four scalar equations:

$$(H_1 + I)(z_0 - Qs_0) + H_2(z'_0 + Q\omega c_0) = 0, \\ H_i = H_i(\tau) \\ (\Gamma_1 + I)z_0 - \Gamma_2 z'_0 = 0, \quad \Gamma_i = \Gamma_i(T), \quad (16) \\ (i = 1, 2) \\ s_0 = \sin(\omega\tau/2), \quad c_0 = \cos(\omega\tau/2)$$

$\tau$ and hence the time duration $T = \Theta - \tau$ of the stick motion, together with the initial conditions $z(0)$, $z'(0)$ are computed. As in the case of the self-excited dry friction oscillator considered in [5], an interesting property of symmetry is proved for these orbits (see Appendix 2):

$$Z(t) = EZ(\tau - t), \quad 0 < t < \tau/2, \quad \text{(slip motion)} \\ Z(t_1) = EZ(T - t_1), \quad 0 < t_1 < T/2, \quad (17) \\ \text{(stick motion),} \quad t_1 = t - \tau$$

A numerical validation is made for the following set of data:

$$\chi = 0.5, \quad \eta = 3.8, \quad u_s = 0.02, \quad u_r = 0.1607, \\ V = 1, \quad \omega = 0.6, \quad p = 0.05$$

The other parameters related to this orbit are computed:

$$\tau = 7.749, \quad T = 2.723, \quad \Theta = 10.472, \\ z_1(0) = 1.0609, \quad z_2(0) = 1.3615, \quad z'_1(0) = 0.581$$

The corresponding phase portraits ($z_i, z'_i$), $i = (1, 2)$ of the two masses are shown on Fig. 2: these curves are symmetrical with respect to the vertical line $z=0$. (The thick parts of the curves are related to the slip motion while the thin parts show the stick motion.)

The constraints derived from (11) during the stick motion:

$$(t < t_1 < T, \quad t_1 = t - \tau) \\ f_1 \equiv \chi\eta(z_2 - z_1) + p \sin(\omega(t_1 + \tau/2)) \\ \qquad - \eta(u_r - u_s) \le 0 \\ f_2 \equiv \chi\eta(z_2 - z_1) + p \sin(\omega(t_1 + \tau/2)) \\ \qquad + \eta(u_r + u_s) > 0$$

are fulfilled.
<figure><img src="image_4.png" /><figcaption>Fig. 2 Phase portraits of the symmetrical solution</figcaption></figure>

## 5 Periodic orbits including an overshooting part

A second set of periodic orbits of period $\Theta = 2\pi/\omega$ is obtained. For each period, the motion is composed of three parts. The first one is a slip motion of $m_2$ with $x'_2 < V$ for $0 < t < \tau$; the next part ($0 < t - \tau < \tau_1$) is an overshooting slip motion of the mass ($x'_2 > V$); the last part $0 < t - \tau - \tau_1 < T$

($T = \Theta - \tau - \tau_1$) is a stick motion of $m_2$.

At the beginning of the motion for $t = 0$, the conditions (12) are fulfilled. If at $t = \tau$, instead of (13), the following conditions:

$$z'_{2}(\tau) = V, \\ \chi \eta (z_{2}(\tau) - z_{1}(\tau)) - p \cos(\omega \tau + \varphi) + \eta(u_{r} + u_{s}) < 0 \quad (18)$$

are fulfilled, we get an overshooting motion for $t > \tau$. This motion ends at $t = \tau + \tau_1$ if, at this time:

$$z'_{2c} \equiv z'_{2}(\tau + \tau_{1}) = V, \\ -\eta(u_{r} + u_{s}) < \chi(z_{2c} - z_{1c}) \\ -p \cos(\omega(\tau + \tau_{2}) + \varphi) < \eta(u_{r} - u_{s}) \\ z_{ic} \equiv z_{ic}(\tau + \tau_{1}), \quad (i = 1, 2) \quad (19)$$

For $\tau + \tau_1 < t < \tau + \tau_1 + T$, the system undergoes a sticking motion. A periodic solution of period $\tau + \tau_1 + T = \Theta$ is obtained if:

$$Z(\Theta) = Z_0 \quad (20)$$

Taking into account the constraints deduced from (18), (19) and (20), the solution is defined by five linear equations with respect to ($z_{10}$, $z_{20}$, $z'_{10}$):

$$a_{i}z_{10} + b_{i}z_{20} + c_{i}z'_{10} + d_{i} = 0, \quad (i = 1, \dots, 5) \quad (21)$$

$(a_i, b_i, c_i, d_i)$ are given in Appendix 3.

Assuming that $\Delta = \det(a_j, b_j, c_j) \neq 0$, $(j = 1, 2, 3)$, the values of ($z_{10}$, $z_{20}$, $z'_{10}$) are obtained:

$$z_{10} = \frac{\Delta_1}{\Delta}, \quad z_{20} = \frac{\Delta_2}{\Delta}, \quad z'_{10} = \frac{\Delta_3}{\Delta} \\ \Delta_1 = \det \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{vmatrix}, \quad (22) \\ \Delta_2 = \det \begin{vmatrix} d_1 & a_1 & c_1 \\ d_2 & a_2 & c_2 \\ d_3 & a_3 & c_3 \end{vmatrix}, \\ \Delta_3 = \det \begin{vmatrix} d_1 & b_1 & a_1 \\ d_2 & b_2 & a_2 \\ d_3 & b_3 & a_3 \end{vmatrix}$$

The parameters ($\tau, \tau_1$) are the roots of the compatibility conditions:

$$F_k(\tau, \tau_1) = a_{k+3}\Delta_1 + b_{k+3}\Delta_2 + c_{k+3}\Delta_3 + d_{k+3}\Delta_4 \\ = 0, \quad (k = 1, 2) \quad (23)$$

Assuming that $(\chi, \eta, V, u_s)$ are given data, $u_r$ is deduced from the relation:

$$u_r = u_s + f, \quad f = \chi(z_{20} - z_{10}) - \frac{p}{\eta} \cos \varphi \quad (24)$$

An example of overshooting orbit is obtained for the values:

$$\chi = 0.2, \quad \eta = 4, \quad u_s = 0.05, \quad V = 1, \\ \omega = 0.6289, \quad p = 0.05, \quad \varphi = 0$$

The other parameters related to this solution are computed:

$$\tau = 4.909, \quad \tau_1 = 2.8275, \quad T = 2.2543, \\ \Theta = 9.9908, \quad u_r = 0.4585, \quad z_1(0) = 1.2809, \\ z_2(0) = 3.3861, \quad z'_1(0) = -0.4117$$

The phase portraits ($z_i, z'_i$), $i = (1, 2)$ of the two masses are shown on Fig. 3. (The heavy thick part of
<figure><img src="image_5.png" /><figcaption>Fig. 3 Phase portraits of the overshooting periodic solution</figcaption></figure>

the curves shows the overshooting motion while the thin one is related to the stick motion.)

## 6 Non-sticking periodic solutions

In industrial applications, avoiding sticking phases of motion is sometimes necessary. In the past, several authors [2, 3, 8] investigated the existence of periodic non-sticking solutions of a one-degree-of-freedom oscillator subjected to simple harmonic loading. The mass is in contact with a fixed surface and a dry friction force acts between the mass and the surface. The aim of these works is to obtain some estimates about the minimum external force amplitude needed to prevent this sticking motion. The non-sticking orbit involves for each period a slipping motion with a negative mass velocity, and a slipping motion with a positive mass velocity (overshooting motion). Moreover, the authors assumed that the motion is symmetric in space [2, 3, 8] and time [3, 8].

In the following, this problem is revisited for the two-degree-of-freedom oscillator considered in this work.

Let us consider the system described in Fig. 1, and assume the following initial conditions:

$$z'_{20} = V, \quad \chi \eta(z_{20} - z_{10}) > p \cos \varphi + \eta(u_r - u_s) \quad (25)$$

For $0 < t < \tau$, the system undergoes a slipping motion with $z'_{2} < V$, given by (5).

This motion ends at $t = \tau$ if $z'_{2B} \equiv z'_{2}(\tau) = V$.

Let us assume that $\tau = \pi/\omega = \Theta/2$ and that

$$\begin{aligned} \chi \eta(z_{2B} - z_{1B}) &< p \cos(\omega\tau + \varphi) - \eta(u_r + u_s) \\ &\equiv -p \cos \varphi - \eta(u_r + u_s), \end{aligned} \quad (26)$$

For $t > \tau$ the system undergoes an overshooting slipping motion ($z'_{2} > V$). This motion is given by

$$\begin{aligned} Z(t) = H(t - \tau)(Z_B - F_B) \\ + 2L(t - \tau)d_0 + F(t) \end{aligned} \quad (27)$$

A periodic motion of period $\Theta = 2\pi/\omega$ is obtained if

$$\begin{aligned} Z_0 &= Z(\Theta) \equiv H(\tau)(Z_B - F_B) + 2L(\tau)d_0 + F_0, \\ Z_B &= Z(\tau) = H(\tau)(Z_0 - F_0) + F_B, \end{aligned} \quad (28)$$

$$F_B = F(\tau) = -F_0$$

From (28) we deduce:

$$\begin{aligned} [H(\tau) - H(-\tau)][Z_B - F_B] + 2L(\tau)d_0 &= 0, \\ H(\tau) - H(-\tau) &= 2 \begin{vmatrix} 0 & H_2 \\ H_3 & 0 \end{vmatrix}, \end{aligned} \quad (29)$$

From (29), we obtain

$$\begin{aligned} H_2(z'_B - Q\omega \sin \varphi) + (H_1 - I)d_0 &= 0 \\ H_3(z_B + Q \cos \varphi + d_0) &= 0 \end{aligned} \quad (30)$$

From (30), if $\det(H_3) \neq 0$, the following relations hold:

$$\begin{aligned} z_B &= -Q \cos \varphi - d_0 \quad \text{i.e. } x_B = -Q \cos \varphi, \\ z'_B &= Q \omega \sin \varphi + H_2^{-1}(I - H_1)d_0 \end{aligned} \quad (31)$$

From (28), (31) and the relation:

$$H_1^2 - I = H_2 H_3, \quad \text{hence } H_3 = H_2^{-1} H_1^2 - H_2^{-1} \quad (32)$$

the following results are obtained

$$\begin{aligned} z_0 &= Q \cos \varphi - d_0, \quad \text{i.e. } x_0 = -x_B = Q \cos \varphi \\ z'_0 &= -Q \omega \sin \varphi + H_2^{-1}(I - I)d_0 = -z'_B \end{aligned} \quad (33)$$

This last condition and the relation $z'_{20} \equiv z'_{2B} = V$ lead to $V = 0$.

In conclusion, non-sticking periodic orbits are obtained only for $V = 0$, and the motion is symmetric in space and time (see Appendix 4).

The initial conditions and the time lag $\varphi$ of the external force are deduced from (33):

$$\begin{aligned} z_{10} &= q_1 \cos \varphi - d_{01}, \quad z_{20} = q_2 \cos \varphi - d_{02}, \\ z'_{10} &= -q_1 \omega \sin \varphi - a_1 d_{01} - a_2 d_{02} \end{aligned} \quad (34)$$
$$
\begin{align}
\sin \varphi &= -(b_1 d_{01} + b_2 d_{02})/q_2 \omega, \nonumber \\
|(b_1 d_{01} + b_2 d_{02})/q_2 \omega| &< 1, \nonumber \\
b_1 &= d(\varphi_1 - \varphi_2)\lambda_1 \lambda_2, \quad b_2 = d(\lambda_2 \varphi_2 - \lambda_1 \varphi_1) \tag{35} \\
\varphi_i &= \omega_i \tan g(\omega_i \pi/2\omega), \nonumber \\
\lambda_i &= \frac{1 - \omega_i^2}{\chi} \quad (i=1,2), \quad d = \chi/(\omega_1^2 - \omega_2^2) \nonumber
\end{align}
$$

The constraints deduced from (25) and (26) lead to the
same condition:

$$
\chi \eta (q_2 - q_1) \cos \varphi - p \cos \varphi - \eta u_r > 0 \quad (36)
$$

and from (36), a condition about the minimum value of
the external force amplitude needed to avoid a sticking
motion is obtained:

$$
p > \left| D \frac{b_1 d_{01} + b_2 d_{02}}{(1 - \omega^2)\omega} \right|, \quad D = (\omega^2 - \omega_1^2)(\omega^2 - \omega_2^2) \tag{37}
$$

A numerical validation is performed for the following
values of the parameters:

$$
\begin{array}{llll}
\chi = 0.3, & \eta = 4, & \omega = 0.6, & p = 1, \\
u_s = 0.1, & u_r = 0.2996, & \Theta = 10.472 &
\end{array}
$$

The corresponding values of the initial conditions and
of the time lag $\varphi$ are obtained:

$$
\begin{align*}
x_{10} &= 1.5608, & x_{20} &= 3.3295, \\
x'_{10} &= 0.1523, & \varphi &= 0.3925
\end{align*}
$$

The phase portraits (xᵢ, x'ᵢ), i = (1, 2) of the two
masses are shown on Fig. 4 (the thick parts of the
curves are related to the overshooting motion). These
curves are symmetrical with respect to the origin 0.

Under the assumption that there are only two transi-
tions of motion during one period, we prove that non-
sticking orbits are symmetric in space and time for al-
most all values of ω (see Appendix 5).

7 Conclusion

In this work, the steady state response of a two-degree-
of-freedom oscillator subjected to dry friction and har-
monic load is considered. Assuming Coulomb’s laws
of dry friction, the existence of several interesting peri-
odic orbits, including stick and slip phases, is proved.
In particular, periodic solutions with a phase during
which the mass in contact with the belt moves faster

<figure><img src="image_6.png" /><figcaption>Fig. 4 Phase portrait of the non-sticking solution</figcaption></figure>

than the belt are obtained. Moreover, in the case of a
non-moving belt, a set of non-sticking periodic solu-
tions is obtained, and we prove that these orbits are
symmetrical in space and in time.

**Appendix 1**

$$
\begin{align}
H_i(t) &= \Lambda B_i(t) \Lambda^{-1}, \quad (i = 1, 2, 3), \notag \\
B_2(t) &= \begin{pmatrix} s_1/\omega_1 & 0 \\ 0 & s_2/\omega_2 \end{pmatrix}, \tag{38} \\
s_j &= \sin(\omega_j t) \quad (j = 1, 2), \notag \\
B_1(t) &= B'_2(t), \quad B_3(t) = B''_2(t) \notag
\end{align}
$$

The natural frequencies ($\omega_1, \omega_2$) are the roots of the
characteristic equation:

$$
\begin{equation}
\begin{aligned}
D(s^2) &= \det(K - Is^2) = 0, \\
K &= \begin{pmatrix} 1 & -\chi \\ -\chi\eta & \chi\eta \end{pmatrix}
\end{aligned}
\tag{39}
\end{equation}
$$

The eigenvectors $\psi_j = (\begin{smallmatrix} 1 \\ \lambda_j \end{smallmatrix})$, ($j = 1, 2$) are defined by
$(K - I\omega_j^2)\psi_j = 0$.

These matrices fulfil the following property:

$$
\begin{align}
H_1^2(t) - H_2(t)H_3(t) &= 0, \tag{40} \\
\Gamma_i(t) &= \Sigma \gamma_i(t) \Sigma^{-1}, \quad (i=1,2,3), \nonumber \\
\gamma_2(t) &= \begin{pmatrix} \sin t & 0 \\ 0 & t \end{pmatrix}, \quad \Sigma = \begin{pmatrix} 1 & \chi \\ 0 & 1 \end{pmatrix}, \tag{41} \\
\gamma_1(t) &= \gamma'_2(t), \quad \gamma_3(t) = \gamma''_2(t) \nonumber
\end{align}
$$

The matrices $\Gamma_i(t)$ fulfil also the property:

$$
\Gamma_1^2(t) - \Gamma_2(t)\Gamma_3(t) = 0 \quad (42)
$$
## Appendix 2

For $\tau/2 < t < \tau$, the periodic solution is defined by

$$Z(\tau - t) = H(\tau - t)(Z_0 - F_0) + F(\tau - t) \quad (43)$$

From the identities:

$$ 
\begin{aligned}
F(\tau - t) &= EF(t), \quad (\text{i.e. } F(\tau) = EF_0) \\
H(\tau - t) &= H(\tau)H(-t), \quad H(-t)E = EH(t)
\end{aligned} 
\quad (44) $$

the first relation (17) is deduced.

For $T/2 < t_1 < T$, the solution is defined by

$$Z(t) = \Gamma(T - t_1)EZ_0 \quad (45)$$

From the identities:

$$ 
\begin{aligned}
\Gamma(T - t_1) &= \Gamma(T)\Gamma(-t_1), \\
E\Gamma(-t_1) &= \Gamma(t_1)E, \\
Z(\Theta) &= \Gamma(T)EZ_0 = Z_0
\end{aligned} 
\quad (46) $$

the last relation (17) follows.

## Appendix 3

$$ 
\begin{aligned}
a_1 &= \tilde{H}_{11} - \Gamma_{11}, & b_1 &= \tilde{H}_{12} - \Gamma_{12}, \\
c_1 &= \tilde{H}_{13} - \Gamma_{13} \\
d_1 &= -(\tilde{H}_{11}q_1 + \tilde{H}_{12}q_2)\cos\varphi \\
    &\quad + \omega(\tilde{H}_{13}q_1 + \tilde{H}_{14}q_2)\sin\varphi + q_1\cos\varphi_C \\
    &\quad + 2(h_{11}-1)d_{01} + 2h_{12}d_{02} + (\tilde{H}_{14} + \Gamma_{14})V, \\
a_2 &= \tilde{H}_{21}, & b_2 &= \tilde{H}_{22} - 1, & c_2 &= \tilde{H}_{23} \\
d_2 &= -(\tilde{H}_{21}q_1 + \tilde{H}_{22}q_2)\cos\varphi \\
    &\quad + \omega(\tilde{H}_{23}q_1 + \tilde{H}_{24}q_2)\sin\varphi + q_2\cos\varphi_C \\
    &\quad + 2h_{21}d_{01} + 2(h_{22}-1)d_{02} + (\tilde{H}_{24} + T)V, \\
a_3 &= \tilde{H}_{31} + \Gamma_{31}, & b_3 &= \tilde{H}_{32} + \Gamma_{32}, \\
c_3 &= \tilde{H}_{11} - \Gamma_{11} \\
d_3 &= -(\tilde{H}_{31}q_1 + \tilde{H}_{32}q_2)\cos\varphi \\
    &\quad + \omega(\tilde{H}_{11}q_1 + \tilde{H}_{12}q_2)\sin\varphi - q_1\sin\varphi_C \\
    &\quad + 2h_{31}d_{01} + 2h_{32}d_{02} + (\tilde{H}_{12} - \Gamma_{12})V, \\
a_4 &= H_{41}, & b_4 &= H_{42}, & c_4 &= H_{21}, \\
d_4 &= (H_{22}-1)V - q_2\omega\sin(\varphi_B) \\
    &\quad - (H_{41}q_1 + H_{42}q_2)\cos\varphi \\
    &\quad + (H_{21}q_1 + H_{22}q_2)\omega\sin\varphi,
\end{aligned} 
\quad (47) 
$$

$$ 
\begin{aligned}
a_5 &= \tilde{H}_{41}, & b_5 &= \tilde{H}_{42}, & c_5 &= \tilde{H}_{21}, \\
d_5 &= (\tilde{H}_{22} - 1)V - q_2\omega\sin(\varphi_C) \\
    &\quad - (\tilde{H}_{41}q_1 + \tilde{H}_{42}q_2)\cos\varphi \\
    &\quad + (\tilde{H}_{21}q_1 + \tilde{H}_{22}q_2)\omega\sin\varphi \\
    &\quad + 2(h_{41}d_{01} + h_{42}d_{02}) \\
H_{ij} &= H_{ij}(\tau), & h_{ij} &= H_{ij}(\tau_1), & \tilde{H}_{ij} &= H_{ij}(\tau + \tau_1) \\
\varphi_B &= \omega\tau + \varphi, & \varphi_C &= \varphi_B + \omega\tau_1
\end{aligned} 
\quad (51) $$

## Appendix 4

From (5), (19) and (30), the non-sticking periodic orbit obtained with the assumption that $\tau = \pi/\omega = \Theta/2$ is described by

$$ 
\begin{aligned}
X(t) &= H(t)(X_0 - F_0 - \Delta_0) + F(t) + \Delta_0, \\
X(t + \tau) &= H(t)(X_B - F_B + \Delta_0) \\
            &\quad + F(t + \tau) - \Delta_0,
\end{aligned} 
\quad (52) \\
0 \le t \le \tau, \quad \Delta_0 = \begin{pmatrix} d_0 \\ 0 \end{pmatrix} $$

Taking into account the properties

$$F(t + \tau) = -F(t), \quad 0 \le t \le \tau, \quad X_B = -X_0 \quad (53)$$

the symmetry of the motion is deduced:

$$X(t) = -X(t + \tau), \quad 0 \le t \le \tau \quad (54)$$

## Appendix 5

Let us consider the same kind of non-sticking periodic orbits as in Sect. 6, with $V = 0$ but without the assumption $\tau = \Theta/2$.

For $0 < t < \tau$, the motion is given by (5), while for $\tau < t < \Theta$, the motion is described by

$$ 
\begin{aligned}
\bar{Z}(t) &= H(t - \tau)(\bar{Z}_B - F_B) + F(t), \\
\bar{Z}(t) &= \begin{pmatrix} \bar{z}(t) \\ z'(t) \end{pmatrix}, \quad \bar{z}(t) = z(t) + 2d_0, \\
\bar{Z}_B &= \bar{Z}(\tau)
\end{aligned} 
\quad (55) $$

A periodic motion of period $\Theta = 2\pi/\omega$ is obtained if $Z(\Theta) = Z_0$ or:

$$ 
\begin{aligned}
\bar{Z}_0 &= H(\tau_1)(\bar{Z}_B - F_B) + F_0, \\
Z_B &= Z(\tau) = H(\tau)(Z_0 - F_0) + F_B, \\
F_B &= F(\tau), \quad \tau_1 = \Theta - \tau, \quad \bar{Z}_0 = \bar{Z}(0)
\end{aligned} 
\quad (56) $$
From (56), we deduce:

$$
\begin{align}
\xi_1 &\equiv Z_B - F_B + E(Z_0 - F_0) \nonumber \\
&= (H(\tau) + E)(Z_0 - F_0), \tag{57} \\
\xi_2 &\equiv \bar{Z}_B - F_B + E(\bar{Z}_0 - F_0) \nonumber \\
&= (H(-\tau_1) + E)(\bar{Z}_0 - F_0) \nonumber
\end{align}
$$

From the relations:

$$
H(t)E = E(H(t))^{-1} = EH(-t) \quad (58)
$$

it follows:

$$
\begin{align}
(H(-t) - E)(H(t) + E) &= 0, \tag{59} \\
(H(t) - E)(H(-t) + E) &= 0 \notag
\end{align}
$$

From (57) and (59), we deduce

$$
(H(-\tau) - E)\xi_1 = 0, \quad (H(\tau_1) - E)\xi_2 = 0 \quad (60)
$$

On the other hand, it is not difficult to show that

$$
\xi_1 - \xi_2 = Z_B - \bar{Z}_B + E(Z_0 - \bar{Z}_0) = 0 \quad (61)
$$

Let us introduce the following notations:

$$
\begin{align*}
\xi_i &= \begin{pmatrix} X_i \\ Y_i \end{pmatrix}, \quad (i=1,2) \\
X_1 &= (H_1 - I)(z_0 - Q \cos \varphi) \\
    &\quad + H_2(z'_0 + Q \omega \sin \varphi), \\
Y_1 &= H_3(z_0 - Q \cos \varphi) \\
    &\quad + (H_1 + I)(z'_0 + Q \omega \sin \varphi), \\
X_2 &= (h_1 - I)(\bar{z}_0 - Q \cos \varphi) \\
    &\quad - h_2(z'_0 + Q \omega \sin \varphi), \\
Y_2 &= -h_3(\bar{z}_0 - Q \cos \varphi) \\
    &\quad + (h_1 + I)(z'_0 + Q \omega \sin \varphi), \\
H_j &= H_j(\tau), \quad h_j = h_j(\tau_1), \quad (j=1,2,3)
\end{align*}
$$

From (60), it results:

$$
\begin{align}
Y_1 &= P_1 X_1, \quad P_1 = H_2^{-1} (H_1 + I), \tag{63} \\
Y_2 &= P_2 X_2, \quad P_2 = -h_2^{-1} (h_1 + I) \notag
\end{align}
$$

The relations $X_1 = X_2$, $Y_1 = Y_2$, deduced from (61)
are equivalent to:

$$
X_1 = X_2, \quad P X_1 = 0, \quad P = P_1 - P_2 \quad (64)
$$

From (64), two cases are obtained:

1. $\det(P) \neq 0$ leads to $X_1 = 0$, hence

$$
X_2 = 0, \quad Y_1 = 0, \quad Y_2 = 0 \tag{65}
$$

From (56) and (64), we deduce

$$
\begin{align}
z_B - Q \cos \varphi_B &= z_0 - Q \cos \varphi, \notag \\
z'_B + Q \omega \sin \varphi_B &= -(z'_0 + Q \omega \sin \varphi), \tag{66} \\
\bar{z}_B - Q \cos \varphi_B &= \bar{z}_0 - Q \cos \varphi \notag
\end{align}
$$

From the relation:

$$
\begin{align}
z'_{B} &= -Q\omega \sin \varphi_{B} - (z'_{0} + Q\omega \sin \varphi), \tag{67} \\
z'_{2B} &= z'_{20} = 0 \notag
\end{align}
$$

we obtain $\sin \varphi_B = -\sin \varphi$, hence

$$
\begin{align*}
\varphi_B &\equiv \omega\tau + \varphi = -\varphi, \quad \varphi = -\omega\tau/2 \quad \text{or} \\
\varphi_B &\equiv \omega\tau + \varphi = \pi + \varphi, \quad \text{hence } \tau = \pi/\omega = \tau_1
\end{align*}
$$

In the first case, ($\varphi_B \equiv -\varphi$, $\varphi = -\omega\tau/2$), from (67)
$z'_B = -z'_0$ and from (66):

$$
z_B = z_0 - Q \cos \varphi + Q \cos \varphi_B = z_0
$$

which is impossible because for $0 < t < \tau$, $z'_2 < 0$,
hence: $z_{2B} < z_{20}$.

In the second case, ($\tau = \pi/\omega = \tau_1$), hence $H_i =$
$h_i$, $i = 1, 2, 3$

$$
z_B + Q \cos \varphi = z_0 - Q \cos \varphi, \quad \text{i.e.,}
$$

$$
x_B + Q \cos \varphi = x_0 - Q \cos \varphi.
$$

From (56), we get

$$
\begin{align}
z_B + \bar{z}_B + 2Q \cos \varphi \nonumber \\
&= H_1(z_0 + \bar{z}_0 - 2Q \cos \varphi) \quad \text{i.e.} \tag{68} \\
x_B + Q \cos \varphi \nonumber \\
&= H_1(x_0 - Q \cos \varphi) = x_0 - Q \cos \varphi \nonumber
\end{align}
$$

Hence, if $\det(H_1 - I) \neq 0$, we obtain

$$
x_0 = Q \cos \varphi, \quad x_B = -Q \cos \varphi = -x_0 \quad (69)
$$

2. $\det(P) = 0$

From $P = H_2^{-1}(H_1 + I) + h_2^{-1}(h_1 + I)$: we deduce

$$
\begin{align}
P &= \Lambda \tilde{P} \Lambda^{-1}, \quad \tilde{P} = \begin{pmatrix} \rho_1 & 0 \\ 0 & \rho_2 \end{pmatrix}, \notag \\
\Lambda &= \begin{pmatrix} 1 & 1 \\ \lambda_1 & \lambda_2 \end{pmatrix} \tag{70} \\
&= \frac{\sin(\alpha_i + \beta_i)}{(\sin \alpha_i)(\sin \beta_i)}, \quad (i=1,2) \notag \\
&\qquad \alpha_i = \omega_i \tau/2, \quad \beta_i = \omega_i \tau_1/2 \notag
\end{align}
$$

It results $\det(P) = 0$ if $\sin(\alpha_i + \beta_i) \equiv \sin(\omega_i \pi / \omega) = 0$, i.e., $\omega_i = k\omega$, ($k = 1, 2, \dots$) (resonance).
Except this particular case of resonance with the
natural frequencies of the system, only symmetrical
periodic solutions with a phase of slipping motion
and a phase of overshooting motion exist.

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