# Analytical investigation of the dynamics of a nonlinear structure with two degrees of freedom

Madeleine Pascal

► To cite this version:

Madeleine Pascal. Analytical investigation of the dynamics of a nonlinear structure with two degrees of freedom. ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Sep 2005, Long Beach, CA, United States. pp.1959-1967, 10.1115/DETC2005-84306. hal-00342877

HAL Id: hal-00342877

https://hal.science/hal-00342877v1

Submitted on 26 Jul 2016

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Analytical investigation of the dynamics of a nonlinear structure with two degree of freedom

MADELEINE PASCAL

Laboratoire Systèmes Complexes, Université d'Evry
Val d'Essonne et CNRS FRE 2494, 40 rue du
Pelvoux, 91020 Evry cedex, France.
mpascal@aramis.iup.univ-evry.fr

## ABSTRACT

A two degree of freedom oscillator with a colliding component is considered. The aim of the study is to investigate the dynamic behavior of the system when the stiffness obstacle changes from a finite value to an infinite value. Several cases are considered. First, in the case of rigid impact and without external excitation, a family of periodic solutions are found in analytical form. In case of soft impact, with a finite time duration of the shock, and no external excitation, the existence of periodic solutions, with an arbitrary value of the period, is proved. Periodic motions are also obtained when the system is submitted to harmonic excitation, in both cases of rigid or soft impact. The stability of these periodic motions is investigated for these four cases.

Keywords: Nonlinear vibrations, two degree of freedom oscillator, rigid and soft impact, periodic motion, stability, forced and unforced system.

## 1. INTRODUCTION

Vibrating systems with clearance between the moving parts are frequently encountered in technical applications. These systems with impacts are strongly nonlinear; they are usually modeled as a spring-mass system with amplitude constraint. Such systems have been the subject of several investigations, mainly in the simplest case of a one degree of freedom system [1-4] and more seldom for multi degree of freedom systems [5-9]. The system behavior during the contact between the moving parts can be described as rigid impact, usually associated with a restitution coefficient, or modeled as soft impact, with a finite time duration of the shock. Several other parameters such as damping, external excitation, influence the behavior of the system. The work is the continuation of a previous paper [10], in which a two degree of freedom oscillator is considered. The nonlinearity in this case comes from the

nonlinearity in this case comes from the presence of two fixed stops limiting the motion of one mass. Assuming no damping and no external excitation, the behavior of the system is investigated when the obstacles stiffness changes from a finite value to an infinite one. In both cases, a family of symmetrical periodic solutions, with two impacts per period, is obtained in analytical form.

In the present paper, a two degree of freedom system in presence of one fixed obstacle is considered. Assuming that no damping occurs, we investigate four cases: unforced system with rigid impact, unforced system with soft impact, forced system (with harmonic excitation) with rigid impact and, at last, forced system with soft impact. In all cases, periodic solutions are found and stability results of these particular motions are obtained.

## 2. PROBLEM FORMULATION

The system under consideration (Fig.1) is a generalization of the double oscillator investigated in the paper [10]. It consists of two masses $m_1$ and $m_2$ connected by linear springs of stiffness $k_1$ and $k_2$. The displacement $z_1$ of the mass $m_1$ is limited by the presence of a fixed stop. When $z_1$ is greater than the clearance, a contact of the first mass with the stop occurs; this contact gives rise to a restoring force associated to a spring stiffness $k_3$.

<figure><img src="image_2.png" /><figcaption>Figure 1: Double oscillator</figcaption></figure>
The mathematical model of the system is given by:

$$ 
\begin{aligned}
M\ddot{z} + Kz &= F + P \cos(\omega t + \varphi) \quad z = (z_1, z_2)' \\
F &= \begin{pmatrix} f(z_1) \\ 0 \end{pmatrix}, \quad f(z_1) = \begin{cases} -k_3(z_1 - 1) & z_1 \ge 1 \\ 0 & z_1 \le 1 \end{cases}
\end{aligned} 
\quad (1) $$

$$ M = \begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix}, \quad K = \begin{pmatrix} k_1 & -k_1 \\ -k_1 & k_1 + k_2 \end{pmatrix}, $$

$P = \begin{pmatrix} P_1 \\ P_2 \end{pmatrix}$ and $\varphi$ are the amplitude and the phase angle of the harmonic excitation.

### 3. UNFORCED SYSTEM

Let us consider the system without external excitation ($P = 0$).

#### 3.1 RIGID IMPACT

When the stiffness of the obstacle $k_3$ tends to infinity, a rigid impact of the first mass against the stop occurs. Starting from initial positions $z_0 = \begin{pmatrix} 1 \\ y \end{pmatrix}$ and initial velocities $\dot{z}_0 = \begin{pmatrix} u \\ w \end{pmatrix}$ corresponding to a contact of the first mass against the stop, assuming a perfect elastic impact, the new positions $z_c$ and the new velocities $\dot{z}_c$ after the shock are obtained by:

$$ \begin{pmatrix} z_c \\ \dot{z}_c \end{pmatrix} = \begin{pmatrix} I & 0 \\ 0 & E \end{pmatrix} \begin{pmatrix} z_0 \\ \dot{z}_0 \end{pmatrix}, \quad I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad E = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \quad (2) $$

After the impact, the system performs a free motion defined by:

$$ \begin{pmatrix} z \\ \dot{z} \end{pmatrix} = C(t) \begin{pmatrix} z_c \\ \dot{z}_c \end{pmatrix}, \quad C(t) = \begin{pmatrix} \Gamma_1(t) & \Gamma_2(t) \\ \Gamma_3(t) & \Gamma_1(t) \end{pmatrix}, \quad (3) $$

$$ \Gamma_i = \Lambda B_i(t) \Lambda^{-1} \quad (i = 1, 2, 3) $$

$$ \Lambda = \begin{pmatrix} 1 & 1 \\ \lambda_1 & \lambda_2 \end{pmatrix}, \quad B_1(t) = \begin{pmatrix} C_1(t) & 0 \\ 0 & C_2(t) \end{pmatrix}, \quad B_2(t) = \begin{pmatrix} \frac{S_1(t)}{\omega_1} & 0 \\ 0 & \frac{S_2(t)}{\omega_2} \end{pmatrix} \quad (4) $$

$$ B_3(t) = \dot{B}_1(t), \quad C_i(t) = \cos \omega_i t, \quad S_i(t) = \sin \omega_i t \quad (i = 1, 2) $$

In these formulas, $(\omega_1, \omega_2)$ are the roots of the characteristic equation:

$$ \Delta(\omega^2) = \det(K - M\omega^2) = 0 \quad \text{while} \quad \Psi_i = \begin{pmatrix} 1 \\ \lambda_i \end{pmatrix} \quad \text{are defined by} $$

$$ (K - M\omega_i^2)\Psi_i = 0 \quad (i=1,2). $$

The following properties for the $\Gamma_i$ matrices hold:

$$ \begin{aligned} \Gamma_1^2(t) - \Gamma_2(t)\Gamma_3(t) &= I, \\ \Gamma_i(t)\Gamma_j(t) &= \Gamma_j(t)\Gamma_i(t) \quad \text{for } i,j=1,2,3 \end{aligned} \quad (5) $$

Moreover, the coefficients $C_{ij}(t)$ of the 4 by 4 matrix $C(t)$ satisfy the property:

$$ C_{ij}(t) = C_{i-2,j-2}(t), \quad (i, j = 3, 4) \quad (6) $$

Let us investigate if for a set of initial conditions $Z_0 = \begin{pmatrix} z_0 \\ \dot{z}_0 \end{pmatrix}$ related to contact of the first mass against the stop, it is possible to obtain a periodic solution of period T, with one impact per period.

The free motion performed by the system after the rigid impact finishes at time $t=T$ when $z_1(T)=1$ and $\dot{z}_1(T)>0$. Let us denote by $Z_f = \begin{pmatrix} z_f \\ \dot{z}_f \end{pmatrix}$ the positions and the velocities reached by the system at $t=T$.

The condition to obtain such a periodic motion is given by:

$$ Z_f \equiv C(T)Z_c = Z_0 \quad (7) $$

$$ Z_c = H_0 Z_0, \quad H_0 = \begin{pmatrix} I & 0 \\ 0 & E \end{pmatrix} $$

Let us introduce the position $Z_s$ reached by the system from the initial position $Z_0$ after a backward motion of duration T:

$$ Z_s = C(-T)Z_0. \quad \text{The condition (7) is equivalent to: } Z_s = Z_c. $$

It results for the determination of the four scalar parameters $(y, u, w, T)$ the four scalar equations:

$$ (-H_0 + C(-T))Z_0 = 0, \quad Z_0 = (1, y, u, w)' \quad (8) $$

or equivalently:

$$ \begin{aligned} (\Gamma_1 - I)z_0 - \Gamma_2 \dot{z}_0 &= 0 \\ -\Gamma_3 z_0 + (\Gamma_1 - E) \dot{z}_0 &= 0 \end{aligned} \quad \Gamma_i = \Gamma_i(T) \quad (i=1,2,3) $$

Taking into account the properties (5) of the $\Gamma_i$ matrices, the system (8) leads to:

$$ z_0 = (\Gamma_1 - I)^{-1} \Gamma_2 \dot{z}_0 \quad (E + I) \dot{z}_0 = 0 \quad (9) $$

The last equation of (9) reduces to $w=0$. From the first one, $y$ and $u$ are obtained in terms of the period:

$$ y = \frac{(\omega_2 t_2 - \omega_1 t_1) \lambda_1 \lambda_2}{\lambda_2 \omega_2 t_2 - \lambda_1 \omega_1 t_1}, \quad u = \frac{(\lambda_1 - \lambda_2) \omega_1 \omega_2 t_1 t_2}{\lambda_2 \omega_2 t_2 - \lambda_1 \omega_1 t_1}, \quad (10) $$

$$ t_i = \tan\left(\frac{\omega_i T}{2}\right) \quad (i = 1, 2) $$

In case of rigid impact, a family of periodic solutions is obtained for which the initial conditions are defined in terms of the period and the initial velocity $w$ of the non impacting mass is zero. For these particular motions, the conditions giving the positions and the velocities after the shock can be formulated by

$$ z_c = z_0, \quad \dot{z}_c = -\dot{z}_0 \quad (11) $$

These results are similar to the results obtained in [10]. The system considered in this previous paper is a symmetrical system with respect to the position $z_1$ of the first mass and it can be expected that the obtained results are due to this
property. But it is not the real explanation because the system investigated now is not symmetrical.

Remark

In more general cases, the impact is described by a restitution coefficient $r$ ($0<r<1$). In this case, the new positions and the new velocities after the shock are given by:

$$Z_c = \begin{pmatrix} I & 0 \\ 0 & E_r \end{pmatrix} Z_0, \quad E_r = \begin{pmatrix} -r & 0 \\ 0 & 1 \end{pmatrix} \qquad (12)$$

The initial conditions corresponding to a periodic orbit of period $T$ are obtained from the relation:

$$z_0 = -(\Gamma_1 - I)^{-1} \Gamma_2 \dot{z}_0, \quad (E_r + I) \dot{z}_0 = 0 \qquad (13)$$

leading to the solution $z_0 = \dot{z}_0 = 0$. In conclusion, if the restitution coefficient $r$ is not equal to 1, no periodic solution with one impact per period exists. This fact, of course, is due to the non conservation of the total energy in this case: it is not possible for the system to perform a periodic orbit if no external excitation occurs.

## 3.2 SOFT IMPACT

Let us assume that the stiffness obstacle is bounded. The mathematical model of the system is given by:

$$\begin{array}{l} \text{For } z_1 \le 1 \\ M\ddot{z} + Kz = 0 \end{array} \qquad (14)$$

$$\text{For } z_1 \ge 1 \qquad M\ddot{z} + K_1 z = K_3, \quad K_1 = \begin{pmatrix} k_1+k_3 & -k_1 \\ -k_1 & k_1+k_2 \end{pmatrix}, \quad K_3 = \begin{pmatrix} k_3 \\ 0 \end{pmatrix} \qquad (15)$$

Let us assume that the initial conditions are given by: $Z_0 = (1, y, u, w)'$, $u > 0$.

A periodic solution is defined in two steps:

- For $0 \le t \le \tau$, $z_1 > 1$, the system is defined by the motion equations (15). The time duration $\tau$ of this constraint motion is defined by the condition:

$$z_1(\tau) = 1 \qquad (16)$$

Let us denotes by $Z_c = Z(\tau) = (1, y_c, u_c, w_c)'$ the value of the parameters at the end of shock, with the condition $u_c < 0$.

- For $\tau \le t \le \tau + \bar{T}$, a free motion obtained from equations (14) and initial conditions $Z_c$ occurs. This motion finishes when $z_1(\tau + \bar{T}) = 1$.

Let us denotes by $Z_f = Z(\tau + \bar{T}) = (1, y_f, u_f, w_f)'$ the value of the parameters at the end of free motion ($u_f > 0$).

The condition to obtain a periodic orbit of period $\tau + \bar{T}$ is given by the condition:

$$Z_f = Z_0 \qquad (17)$$

The piecewise linear systems (14) and (15) give the two parts of the motion in analytical form.

- For $0 \le t \le \tau$, the constraint motion is deduced from a modal analysis of system (15):

$$Z(t) = H(t)(Z_0 - d) + d, \quad d = (d_1, d_2, 0, 0)' \\ d_1 = \frac{k_3(k_1 + k_2)}{k_1 k_2 + k_3(k_1 + k_3)}, \quad d_2 = \frac{k_3 k_1}{k_1 k_2 + k_3(k_1 + k_2)} \qquad (18)$$

$$H(t) = \begin{pmatrix} H_1(t) & H_2(t) \\ H_3(t) & H_1(t) \end{pmatrix}, \qquad (19) \\ H_1(t) = \Sigma G_i(t) \Sigma^{-1} (i=1,2,3), \Sigma = \begin{pmatrix} 1 & \mu_2 \\ \mu_1 & 1 \end{pmatrix}$$

$$G_1(t) = \begin{pmatrix} c_1(t) & 0 \\ 0 & c_2(t) \end{pmatrix}, \quad G_2 = \begin{pmatrix} \frac{s_1(t)}{\sigma_1} & 0 \\ 0 & \frac{s_2(t)}{\sigma_2} \end{pmatrix}, \quad G_3 = \dot{G}_1 \qquad (20) \\ c_i(t) = \cos(\sigma_i t), \quad s_i(t) = \sin(\sigma_i t), \quad (i=1,2)$$

In these formulas, $\sigma_1, \sigma_2, \Phi_1 = \begin{pmatrix} 1 \\ \mu_1 \end{pmatrix}, \Phi_2 = \begin{pmatrix} \mu_2 \\ 1 \end{pmatrix}$ define the characteristic frequencies and the eigenvectors of the constraint system (15). For the $H_i$ matrices, the properties (5) obtained for the $\Gamma_i$ matrices hold, together with the property $H_{ij}(t) = H_{i-2,j-2}(t), (i, j = 3,4)$ for the coefficients $H_{ij}(t)$ of the matrix $H(t)$.

- For $\tau \le t \le \tau + \bar{T}$, the free motion is obtained from $Z(t) = C(t-\tau)Z_c$ where the matrix $C$ is defined by formulas (3) and (4) and $Z_c = H(\tau)(Z_0 - d) + d$.

Let us introduce the positions and the velocities $Z_s = (z_{1s}, z_{2s}, \dot{z}_{1s}, \dot{z}_{2s})'$ of the system after a backward motion of duration $\bar{T}$ from the initial position $Z_0$. The condition (17) of periodicity is reformulated as

$$Z_s \equiv C(-\bar{T})Z_0 = Z_c \qquad (21)$$

$$Z_c = \begin{pmatrix} \dot{z}_c \\ \dot{z}_s \end{pmatrix} = \begin{pmatrix} H_1(z_0 - d_0) + H_2\dot{z}_0 + d_0 \\ H_3(z_0 - d_0) + H_1\dot{z}_0 \end{pmatrix}, \quad d_0 = (d_1, d_2)'$$

$$Z_s = \begin{pmatrix} \dot{z}_s \\ \dot{z}_x \end{pmatrix} = \begin{pmatrix} \Gamma_1 z_0 - \Gamma_2 \dot{z}_0 \\ -\Gamma_3 z_0 + \Gamma_1 \dot{z}_0 \end{pmatrix}, \qquad (22) \\ H_i = H_i(\tau), \Gamma_i = \Gamma_i(\bar{T}) \quad (i=1,2,3)$$

The condition (21) is equivalent to

$$X_1 = X_2, \quad Y_1 = Y_2 \qquad (23)$$

$$\begin{align} X_1 &\equiv z_c - z_0 = (H_1 - I)(z_0 - d_0) + H_2\dot{z}_0, \notag \\ X_2 &\equiv z_x - z_0 = (\Gamma_1 - I)z_0 - \Gamma_2\dot{z}_0 \tag{24} \end{align}$$

$$\begin{align} Y_1 &\equiv \dot{z}_c + \dot{z}_0 = H_3(z_0 - d_0) + (H_1 + I)\dot{z}_0, \notag \\ Y_2 &\equiv \dot{z}_s + \dot{z}_0 = -\Gamma_3 z_0 + (\Gamma_1 + I)\dot{z}_0 \end{align}$$

From the properties of $\Gamma_i$ and $H_i$, we deduce:
$$Y_i = P_i X_i \quad (i=1,2)$$
$$P_i = H_2^{-1}(H_1+I), \quad P_2 = -\Gamma_2^{-1}(\Gamma_1+I) \tag{25}$$

The condition (22) leads to

$$X_1 = X_2, \quad P_1 X_1 = P_2 X_2 \tag{26}$$

Two possible cases of periodic solutions can be deduced from (26), namely:

$$X_1 = X_2 = 0 \text{ or } X_1 = X_2, \det(P_1 - P_2) = 0 \tag{27}$$

## 3.2 EXISTENCE OF PERIODIC MOTIONS (SOFT IMPACT)

Let us discuss the first conditions (27). In this case, from (23), we deduce:

$$z_c = z_0, \quad \dot{z}_c = -\dot{z}_0 \tag{28}$$

The condition (16) is fulfilled and the initial conditions are obtained from the equations:

$$(H_1 - I)(z_0 - d_0) + H_2 \dot{z}_0 = 0 \tag{29}$$
$$[\Gamma_1 - I + \Gamma_2 H_2^{-1}(H_1 - I)]z_0 = \Gamma_2 H_2^{-1}(H_1 - I)d_0$$

This system provides four scalar equations for the determination of the five parameters (y, u, w, $\tau$, $\bar{T}$). It results that, as in the case of rigid impact, $\bar{T}$ and hence the period can be chosen arbitrarily. Moreover, the conditions (28) and (11) obtained at the end of the shock are the same for both rigid and soft impacts.

From (29), we deduce:

$$\begin{gathered} \dot{z}_0 = -H_2^{-1}(H_1 - I)(z_0 - d_0) \tag{30} \\ [\Gamma_1 - I + \Gamma_2 H_2^{-1}(H_1 - I)]z_0 = \Gamma_2 H_2^{-1}(H_1 - I)d_0 \end{gathered}$$

The last equation (30), after the elimination of y, provides a relation $F(\tau, \bar{T})=0$ between the time duration $\tau$ of the shock and the time duration $\bar{T}$ of the free motion. The other case $X_1 = X_2, \det(P_1 - P_2) = 0$ leads to no solution [10].

In both cases (soft or rigid impact), a family of periodic motions is obtained, with an arbitrary value of the period. Moreover, the conditions (28) obtained at the end of the shock for soft impact are consistent with Newton rules of rigid impact (11) with a restitution coefficient equal to one, i.e. with assumption of ideal elastic impact. This rather remarkable result have been already obtained for the symmetrical system of [10].

## 4. FORCED SYSTEM

Let us assume that the two masses are subjected to harmonic external excitations of period $2\pi/\omega$, constant amplitudes $P_1, P_2$ and constant phase angle $\phi$. From the results obtained in the previous paragraph, where a family of periodic orbits is found with an arbitrary value of the period, it can be expected that for

the forced system, periodic solutions of period $2\pi/\omega$ exist.

### 4.1 RIGID IMPACT

Let us investigate the case of rigid impact, with a restitution coefficient $r=1$. Starting from the initial conditions $Z_0 = (1, y, u, w)'$ ($u>0$), the conditions $Z_c = (1, y_c, u_c, w_c)'$ after the shock are obtained from (2) and the free motion performed by the system is given by:

$$\begin{aligned} z &= \Gamma_1(t)(z_0 - R \cos \varphi) + \Gamma_2(t)(\dot{z}_c + R \omega \sin \varphi) + R \cos(\omega t + \varphi) \\ \dot{z} &= \Gamma_3(t)(z_0 - R \cos \varphi) + \Gamma_1(t)(\dot{z}_c + R \omega \sin \varphi) - R \omega \sin(\omega t + \varphi) \end{aligned} \tag{31}$$

where $R = (R_1, R_2)'$ is the amplitude of the response:

$$\begin{aligned} R_1 &= A_1 + A_2, \quad R_2 = \lambda_1 A_1 + \lambda_2 A_2 \\ A_i &= \frac{P_i + \lambda_i P_2}{(\omega_i^2 - \omega^2)(m_1 + \lambda_i^2 m_2)}, \quad (i=1,2) \end{aligned} \tag{32}$$

The free motion finishes at time $t=T$ when $z_1(T) = 1$, $\dot{z}_1(T) > 0$. Let us denote by $Z_f = (z_f, \dot{z}_f)'$ the positions and the velocities reached by the system at this time. The condition to obtain a periodic motion of period T is:

$$Z_f = Z_0 \tag{33}$$

or

$$\begin{aligned} z_0 &= \Gamma_1(T)(z_0 - R \cos \varphi) + \Gamma_2(T)(\dot{z}_c + R \omega \sin \varphi) + R \cos(\omega T + \varphi) \\ \dot{z}_0 &= \Gamma_3(T)(z_0 - R \cos \varphi) + \Gamma_1(T)(\dot{z}_c + R \omega \sin \varphi) - R \omega \sin(\omega T + \varphi) \\ \dot{z}_c &= E \dot{z}_0 \end{aligned}$$

Let us assume that $T = 2\pi/\omega$, $\varphi = 0$, $\dot{z}_c = -\dot{z}_0$. A periodic motion of period $2\pi/\omega$ is obtained if the initial conditions $Z_0 = (1, y, u, 0)'$ are defined by the system:

$$\begin{aligned} (\Gamma_1 - I)(z_0 - R) - \Gamma_2 \dot{z}_0 &= 0 \\ \Gamma_3 (z_0 - R) - (\Gamma_1 + I) \dot{z}_0 &= 0 \\ \Gamma_i &= \Gamma_i (2\pi/\omega), \quad (i=1,2,3) \end{aligned} \tag{34}$$

Taking into account the properties (5) of the $\Gamma_i$ matrices, this system reduces to

$$\dot{z}_0 = \Gamma_2^{-1}(\Gamma_1 - I)(z_0 - R) \tag{35}$$

and the corresponding values of y and u are obtained:

$$\begin{aligned} y &= R_2 + \frac{(\omega_2 t_2 - \omega_1 t_1) \lambda_1 \lambda_2}{\lambda_2 \omega_2 t_2 - \lambda_1 \omega_1 t_1} (1 - R_1), \\ u &= (1 - R_1) \frac{(\lambda_1 - \lambda_2) \omega_1 \omega_2 t_1 t_2}{\lambda_2 \omega_2 t_2 - \lambda_1 \omega_1 t_1}, \quad t_i = \tan(\frac{\omega_i \pi}{\omega}) \quad (i=1,2) \end{aligned} \tag{36}$$

**Remark**

In more general cases, the impact is described by a restitution coefficient $r$ ($0<r<1$). The initial conditions and the phase angle related to a periodic solution of period $2\pi/\omega$ can also be obtained in analytical form. A similar solution has been studied in paper [6].
## 4.2 SOFT IMPACT

When the stiffness obstacle is bounded, the motion equations of the system are given by:

$$ 
\begin{aligned}
M\ddot{z} + K_1 z &= K_3 + P \cos(\omega t + \varphi) & z_1 \ge 1 \\
M\ddot{z} + K z &= P \cos(\omega t + \varphi) & z_1 \le 1
\end{aligned}
\quad (37) $$

From the initial condition $Z_0 = (1, y, u, w)'$ ($u > 0$), the solution is defined in two parts:

- For $0 < t < \tau$, $z_1 \ge 1$, the solution is given by:

$$ 
\begin{aligned}
z &= H_1(t)(z_0 - d_0 - Q \cos \varphi) + H_2(t)(\dot{z}_0 + Q \omega \sin \varphi) \\
  &\quad + d_0 + Q \cos(\omega t + \varphi) \\
\dot{z} &= H_3(t)(z_0 - d_0 - Q \cos \varphi) + H_1(t)(\dot{z}_0 + Q \omega \sin \varphi) \\
  &\quad - Q \omega \sin(\omega t + \varphi)
\end{aligned}
\quad (38) $$

$Q = (Q_1, Q_2)'$ is the response amplitude defined by:

$$ 
\begin{aligned}
Q_1 &= \frac{P_1(1 + \mu_2^2) + P_2(\mu_1 + \mu_2)}{(\sigma_1^2 - \omega^2)(m_1 + m_2\mu_1^2)} \\
Q_2 &= \frac{P_1(\mu_1 + \mu_2) + P_2(1 + \mu_1^2)}{(\sigma_2^2 - \omega^2)(m_2 + m_1\mu_2^2)}
\end{aligned}
\quad (39) $$

The time duration $\tau$ of this motion is obtained from the condition $z_1(\tau) = 1$. Let us denote $Z_c = (1, y_c, u_c, w_c)'$ the value of the parameters at $t = \tau$ ($u_c < 0$).

- For $\tau < t < \tau + \bar{T}$, the motion of the system is defined by:

$$ 
\begin{aligned}
z &= \Gamma_1(t - \tau)(z_c - R \cos \psi) + \\
  &\quad \Gamma_2(t - \tau)(\dot{z}_c + R \omega \sin \psi) + R \cos(\omega t + \varphi) \\
\dot{z} &= \Gamma_3(t - \tau)(z_c - R \cos \psi) + \\
  &\quad \Gamma_1(t - \tau)(\dot{z}_c + R \omega \sin \psi) - R \omega \sin(\omega t + \varphi)
\end{aligned}
\quad (40) $$

where $R = (R_1, R_2)'$ is defined by (32) and $\psi = \omega\tau + \varphi$.

This motion finishes at time $t = \tau + \bar{T}$ when $z_1(\tau + \bar{T}) = 1$, $\dot{z}_1(\tau + \bar{T}) > 0$. If $Z_f = (z_f, \dot{z}_f)'$ denote the positions and the velocities reached by the system at this time, the condition to obtain a periodic motion of period $\tau + \bar{T}$ is $Z_f = Z_0$. Let us assume that $\tau + \bar{T} = 2\pi/\omega$, $\varphi = -\omega\tau/2$. At the end of the first part of the motion ($t = \tau$), the positions and the velocities are given by:

$$ 
\begin{aligned}
z_c &= H_1(z_0 - d_0 - Q \cos \varphi_0) + H_2(\dot{z}_0 - Q \omega \sin \varphi_0) + d_0 + Q \cos \varphi_0 \\
\dot{z}_c &= H_3(z_0 - d_0 - Q \cos \varphi_0) + H_1(\dot{z}_0 - Q \omega \sin \varphi_0) - Q \omega \sin \varphi_0 \\
\varphi_0 &= \omega\tau/2, \quad H_i = H_i(\tau) \quad (i = 1, 2, 3)
\end{aligned}
\quad (41) $$

At the end of the second part of the motion ($t = 2\pi/\omega$), we obtain:

$$ 
\begin{aligned}
z_f &= \Gamma_1(z_c - R \cos \varphi_0) + \Gamma_2(\dot{z}_c + R \omega \sin \varphi_0) + R \cos \varphi_0 \\
\dot{z}_f &= \Gamma_3(z_c - R \cos \varphi_0) + \Gamma_1(\dot{z}_c + R \omega \sin \varphi_0) + R \omega \sin \varphi_0 \\
\Gamma_i &= \Gamma_i(2\pi/\omega - \tau) \quad (i = 1, 2, 3)
\end{aligned}
\quad (42) $$

The conditions of periodicity are reformulated as:

$$ 
\begin{aligned}
\bar{X}_1 &= \bar{X}_2, & \bar{Y}_1 &= \bar{Y}_2 \\
\bar{X}_1 &\equiv z_c - z_0 = (H_1 - I)(z_0 - d_0 - Q \cos \varphi_0) + H_2(\dot{z}_0 - Q \omega \sin \varphi_0) \\
\bar{Y}_1 &\equiv \dot{z}_c + \dot{z}_0 = H_3(z_0 - d_0 - Q \cos \varphi_0) + (H_1 + I)(\dot{z}_0 - Q \omega \sin \varphi_0) \\
\bar{X}_2 &\equiv z_c - z_0 = (\Gamma_1 - I)(z_0 - R \cos \varphi_0) - \Gamma_2(\dot{z}_0 - R \omega \sin \varphi_0) \\
\bar{Y}_2 &\equiv \dot{z}_c + \dot{z}_0 = -\Gamma_3(z_0 - R \cos \varphi_0) + (\Gamma_1 + I)(\dot{z}_0 - R \omega \sin \varphi_0)
\end{aligned}
\quad (43) $$

As in the case of unforced system taking into account the properties of the $H_1$ and $\Gamma_1$ matrices, the solution of system (43) is given by:

$$ \bar{X}_1 = \bar{X}_2 = 0, \quad \bar{Y}_1 = \bar{Y}_2 = 0 \quad (44) $$

We deduce for the forced system, the existence of a periodic motion of period $2\pi/\omega$ for which the conditions at the end of the shock are $z_c = z_0$, $\dot{z}_c = -\dot{z}_0$. The time duration $\tau$ of the shock and the initial conditions (y, u, w) are obtained from the first part of system (44):

$$ 
\begin{aligned}
\dot{z}_0 &= Q \omega \sin \varphi_0 - H_2^{-1}(H_1 - I)(z_0 - d_0 - Q \cos \varphi_0) \\
&[\Gamma_1^{-1}(\Gamma_1 - I) + H_2^{-1}(H_1 - I)]z_0 = (Q - R) \omega \sin \varphi_0 + \\
&\Gamma_2^{-1}(\Gamma_1 - I)R \cos \varphi_0 + H_2^{-1}(H_1 - I)(d_0 + Q \cos \varphi_0)
\end{aligned}
\quad (45) $$

These formulas give the relations (30) when $Q = R = 0$ (unforced system).

# 5. STABILITY OF PERIODIC MOTIONS (RIGID IMPACT)

## 5.1 UNFORCED SYSTEM

Let us consider a periodic motion of period $T$ related to initial conditions $z_{00} = \begin{pmatrix} 1 \\ y_0 \end{pmatrix}$, $\dot{z}_{00} = \begin{pmatrix} u_0 \\ 0 \end{pmatrix}$, where $z_{00}$, $\dot{z}_{00}$ are defined by:

$$ 
\begin{aligned}
(\Gamma_1 - I)z_{00} - \Gamma_2 \dot{z}_{00} &= 0 & \Gamma_i &= \Gamma_i(T) \\
-\Gamma_3 z_{00} + (\Gamma_1 - E) \dot{z}_{00} &= 0
\end{aligned}
\quad (46) $$

Let us consider the perturbed motion defined by a set of new initial conditions:

$$ z_0 = z_{00} + dz_0 \quad \text{where} \quad dz_0 = \begin{pmatrix} 0 \\ y \end{pmatrix}, \quad d\dot{z}_0 = \begin{pmatrix} u \\ w \end{pmatrix} \quad (47) $$

This motion is defined for $t>0$,

$$ 
\begin{aligned}
z &= \Gamma_1(t)z_0 + \Gamma_2(t)E\dot{z}_0 \\
\dot{z} &= \Gamma_3(t)z_0 + \Gamma_1(t)E\dot{z}_0
\end{aligned}
$$

This motion ends at $t = T + dT$, when $z_1(T + dT) = 1$ and $\dot{z}_1(T + dT) > 0$. Let us denote by $z_f, \dot{z}_f$ the positions and the velocities of the system at this time.

$$ 
\begin{aligned}
z_f &= \Gamma_1(T + dT)z_{00} + \Gamma_2(T + dT)E\dot{z}_{00} \\
\dot{z}_f &= \Gamma_3(T + dT)z_{00} + \Gamma_1(T + dT)E\dot{z}_{00}
\end{aligned}
$$

Assuming small perturbations $dz_0, d\dot{z}_0$ of the initial conditions,
$$ 
\begin{aligned}
dz_f &= z_f - z_0 = \Gamma_1 dz_0 + \Gamma_2 Ed\dot{z}_0 + L_1dT \\
d\dot{z}_f &= \dot{z}_f - \dot{z}_0 = \Gamma_3 dz_0 + \Gamma_1 Ed\dot{z}_0 + L_2dT
\end{aligned}
\quad (48) 
$$

$$ L_1 = \dot{\Gamma}_1 z_{00} + \dot{\Gamma}_2 E \dot{z}_{00} $$

$$ L_2 = \dot{\Gamma}_3 z_{00} + \dot{\Gamma}_1 E \dot{z}_{00} $$

$$ \dot{\Gamma}_i = \dot{\Gamma}_i(T) $$

From the relations

$$ \dot{\Gamma}_1 = \Gamma_3, \dot{\Gamma}_2 = \Gamma_1, \dot{\Gamma}_3 = \Gamma_2^{-1}\Gamma_1\Gamma_3, E\dot{z}_{00} = -\dot{z}_{00}, \text{ we deduce:} $$

$$ L_1 = \dot{z}_{00}, L_2 = \Gamma_2^{-1}(\Gamma_1 + I)\dot{z}_{00} $$

By elimination of dT, (48) gives:

$$ (\Gamma_1 + I)dz_f - \Gamma_3 d\dot{z}_f = (\Gamma_1 + I)dz_0 + \Gamma_2 Ed\dot{z}_0 \quad (49) $$

From (49) and the second equation related to $dz_f$ in (48), we deduce:

$$ 
\begin{aligned}
y_f &= C_{22}y - C_{23}u + C_{24}w \\
C_{12}y_f - C_{13}u_f - C_{14}w_f &= C_{12}y - C_{13}u + C_{14}w
\end{aligned}
\quad (50) 
$$

$$ (C_{22} + 1)y_f - C_{23}u_f - C_{24}w_f = (C_{22} + 1)y - C_{23}u + C_{24}w $$

or $AX_f = BX$, $X_f = (y_f, u_f, w_f)$, $X = (y, u, w)$

$$ A = \begin{pmatrix} 1 & 0 & 0 \\ C_{22} - C_{23} & C_{24} \\ C_{12} - C_{13} & -C_{14} \end{pmatrix}, \quad B = \begin{pmatrix} C_{22} - C_{23} & C_{24} \\ 1 & 0 & 0 \\ C_{12} - C_{13} & C_{14} \end{pmatrix} $$

The stability of the periodic impact solution is determined by the eigenvalues of the matrix $A^{-1}B$. If all the eigenvalues are inside the unit circle, the periodic solution is stable. If one of them is outside the unit circle, the solution is unstable. Critical cases occur if some eigenvalues lie on the unit circle, the other ones being strictly inside this circle.

Let us introduce the characteristic polynomial $P(\lambda)$ of the matrix $A^{-1}B$:

$$ P(\lambda) = \det(A^{-1}B - \lambda I) = -(\lambda^3 + a_3\lambda^2 + a_2\lambda + a_1) $$

$$ P(\lambda) = 0 \text{ is equivalent to } D(\lambda) = 0, \quad D(\lambda) = \det(B - \lambda A) $$

From the properties: $D(1) = \det(B - A) = 0$, $\det(A) = \det(B)$, we deduce that one eigenvalue of $A^{-1}B$ is $\lambda_1 = 1$ and $a_1 = -\det(A^{-1}B) = -1$. The characteristic polynomial of $A^{-1}B$ takes the following form:

$$ P(\lambda) = (1 - \lambda)[\lambda^2 + (1 + a_3)\lambda + 1] $$

It results that when $\delta = (a_3 - 1)(a_3 + 3)$ is positive, the two other eigenvalues $\lambda_2, \lambda_3$ of $A^{-1}B$ are real and $\lambda_2\lambda_3 = 1$ leads to the instability of the periodic solution. For $\delta < 0$, $A^{-1}B$ has a complex conjugate pair of eigenvalues on the unit circle.

## 5.2 FORCED SYSTEM WITH IDEAL ELASTIC IMPACT

A periodic solution of period $2\pi/\omega$ is obtained for initial conditions $z_{00} = \begin{pmatrix} 1 \\ y_0 \end{pmatrix}$, $\dot{z}_{00} = \begin{pmatrix} u_0 \\ 0 \end{pmatrix}$, and phase angle $\tilde{\phi}_0 = 0$, $(y_0, u_0)$ defined by (34).

Let us consider a perturbed motion related to initial conditions (47) and phase angle $\varphi = \tilde{\phi}_0 + d\varphi$. The corresponding free motion performed by the system for $t>0$, is obtained from (29), with $\dot{z}_c = E\dot{z}_0$. This motion ends at $t= \frac{2\pi}{\omega} + dT$, when

$z_1(\frac{2\pi}{\omega} + dT) = 1$ and $\dot{z}_1(\frac{2\pi}{\omega} + dT) > 0$. Let us denote by $z_f, \dot{z}_f$ the positions and the velocities of the system at this time.

Assuming small perturbations of the initial conditions and of the phase angle:

$$ 
\begin{aligned}
dz_f &= \Gamma_1 dz_0 + \Gamma_2 (E_r d\dot{z}_0 + R\omega d\varphi) + n_1 dT \\
d\dot{z}_f &= \Gamma_3 dz_0 + \Gamma_1 (E_r d\dot{z}_0 + R\omega d\varphi) - R\omega d\varphi' + n_2 dT
\end{aligned}
\quad (51) 
$$

$$ n_1 = \dot{z}_{00} = \begin{pmatrix} u_0 \\ 0 \end{pmatrix}, \quad n_2 = \Gamma_2^{-1}(\Gamma_1 + I)n_1 $$

From (52), we deduce:

$$ Y_f = N_0 Y, \quad N_0 = \tilde{A}^{-1} \tilde{B} \quad (52) $$

$$ 
\tilde{A} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ C_{12} - C_{13} & -C_{14} & -\omega l_1 & 0 \\ C_{22} - C_{23} & -C_{24} & -\omega l_2 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \quad \tilde{B} = \begin{pmatrix} C_{22} & -C_{23} & C_{24} & \omega l_2 \\ C_{12} & -C_{13} & C_{14} & \omega l_1 \\ 1 & 0 & 0 & 0 \\ -\tilde{\chi} C_{13} & \tilde{\chi} C_{13} & -\tilde{\chi} C_{14} & 1 - \omega \tilde{\chi} l_1 \end{pmatrix} \\
l_1 = C_{13}R_1 + C_{14}R_2, \quad l_2 = C_{23}R_1 + C_{24}R_2, \quad \tilde{\chi} = \frac{\omega}{u_0}
$$

From the property $\det(\tilde{A}) = \det(\tilde{B}) = \det(\Gamma_2)$, we deduce

$$ a_0 = P(0) = \det(\tilde{A}^{-1}\tilde{B}) = 1 \quad (53) $$

In this case, it is impossible that all the eigenvalues of the matrix $N_0$ lie strictly inside the unit circle. The periodic solution is unstable except if all these eigenvalues lie on the unit circle.

# 6. STABILITY OF PERIODIC MOTIONS (SOFT IMPACT).

## 6.1 UNFORCED SYSTEM

When the stiffness of the obstacle is bounded and when there is no external excitation, the mathematical model of the system is given by (14) for the free motion and (15) for the constraint motion.

Let us consider a periodic motion of period $\tau_0 + T_0 = 2\pi/\omega$, where $\omega$ is an arbitrary positive value in this case. This periodic solution is related to the initial conditions

$z_{00} = \begin{pmatrix} 1 \\ y_0 \end{pmatrix}$, $\dot{z}_{00} = \begin{pmatrix} u_0 \\ w_0 \end{pmatrix}$, where $(y_0, u_0, w_0, \tau_0, T_0)$ are defined in terms of $\omega$ by
$$
\begin{gather}
\dot{z}_{00} = -H_2^{-1}(H_1 - I)(z_{00} - d_0) \tag{54} \\
[\Gamma_1 - I + \Gamma_2 H_2^{-1}(H_1 - I)]z_{00} = \Gamma_2 H_2^{-1}(H_1 - I)d_0
\end{gather}
$$

$$
H_i = H_i(\tau_0), \quad \Gamma_i = \Gamma_i(T_0), \quad (i = 1, 2)
$$

and the condition $\tau_0 + T_0 = 2\pi / \omega$.

Let us consider the perturbed motion defined by a set of new
initial conditions (47).

This motion is defined in two steps:

-For $0 \le t \le \tau = \tau_0 + d\tau$ :

$$
z = H_1(t)(z_0 - d_0) + H_2(t)\dot{z}_0 + d_0
$$

$$
\dot{z} = H_3(t)(z_0 - d_0) + H_1(t)\dot{z}_0
$$

This motion ends when $z_1(\tau) = 1$ and $\dot{z}_1(\tau) < 0$. Let us denotes
by $z_c = z_{00} + dz_c$, $\dot{z}_c = -\dot{z}_{00} + d\dot{z}_c$, $dz_c = \begin{pmatrix} 0 \\ y_c \end{pmatrix}$, $d\dot{z}_c = \begin{pmatrix} u_c \\ w_c \end{pmatrix}$ the
positions and the velocities reached by the system at this final
time.

-For $\tau \le t \le \frac{2\pi}{\omega} + d\theta$, the motion is defined by:

$$
z = \Gamma_1(t - \tau)z_c + \Gamma_2(t - \tau)\dot{z}_c
$$

$$
\dot{z} = \Gamma_3(t - \tau)z_c + \Gamma_1(t - \tau)\dot{z}_c
$$

This motion ends when $z_1(\frac{2\pi}{\omega} + d\theta) = 1$, $\dot{z}_1(\frac{2\pi}{\omega} + d\theta) > 0$.

Let us denotes by $z_f = z_{00} + dz_f$, $\dot{z}_f = \dot{z}_{00} + d\dot{z}_f$, the
positions and the velocities reached by the system at this time.
Assuming small perturbations $dz_0, d\dot{z}_0$ of the initial conditions,

$$
\begin{align}
dz_c &= H_1 dz_0 + H_2 d\dot{z}_0 + p_1 d\tau \tag{H_i = H_i(\tau_0), i = 1,2,3} \\
d\dot{z}_c &= H_3 dz_0 + H_1 d\dot{z}_0 + p_2 d\tau \nonumber
\end{align}
$$

$$
d\dot{z}_c = H_3 dz_0 + H_1 d\dot{z}_0 + p_2 d\tau
$$

$$
\begin{align*}
p_1 &= \dot{H}_1(z_{00} - d_0) + \dot{H}_2 \dot{z}_{00} \\
p_2 &= \dot{H}_3(z_{00} - d_0) + \dot{H}_1 \dot{z}_{00}
\end{align*}
\qquad (\dot{H}_i = \dot{H}_i(\tau_0), i = 1, 2, 3)
$$

$$
p_2 = \dot{H}_3(z_{00} - d_0) + \dot{H}_1 \dot{z}_{00}
$$

From the properties of the $H_i$ matrices and the results obtained
in paragraph 3.2, we deduce:

$$
p_1 = -\dot{z}_{00}, \quad p_2 = -H_2^{-1}(H_1 + I)\dot{z}_{00} \tag{56}
$$

In a same way

$$
dz_f = \Gamma_1 dz_c + \Gamma_2 d\dot{z}_c + p_3 d\theta
$$

$$
d\dot{z}_f = \Gamma_3 dz_c + \Gamma_1 d\dot{z}_c + p_4 d\theta
$$

$$
p_3 = \dot{\Gamma}_1 z_{00} - \dot{\Gamma}_2 \dot{z}_{00} = \dot{z}_{00}
$$

$$
p_4 = \dot{\Gamma}_3 z_{00} - \dot{\Gamma}_1 \dot{z}_{00} = \Gamma_2^{-1} (\Gamma_1 + I) \dot{z}_{00}
$$

From the system (55), after the elimination of $d\tau$, we deduce:

$$
\begin{pmatrix} y_c \\ u_c \\ w_c \end{pmatrix} = A_1 \begin{pmatrix} y \\ u \\ w \end{pmatrix}, \quad A_1 = \tilde{H} + M_2 M_1, \tag{58}
$$

$$
\tilde{H} = \begin{pmatrix} H_{22} & H_{23} & H_{24} \\ H_{32} & H_{11} & H_{12} \\ H_{42} & H_{21} & H_{22} \end{pmatrix}, M_2 = \begin{pmatrix} -w_0 \\ p_2 \end{pmatrix}, M_1 = \frac{1}{u_0} (H_{12}H_{13}H_{14})
$$

From the system (57), after the elimination of $d\theta$, we obtain:

$$
\begin{pmatrix} y_l \\ u_f \\ w_f \end{pmatrix} = A_2 \begin{pmatrix} y_c \\ u_c \\ w_c \end{pmatrix}, \quad A_2 = \tilde{C} + N_2 N_1 \tag{59}
$$

$$
\tilde{C} = \begin{pmatrix} C_{22} & C_{23} & C_{24} \\ C_{32} & C_{11} & C_{12} \\ C_{42} & C_{21} & C_{22} \end{pmatrix}, N_2 = \begin{pmatrix} w_0 \\ p_4 \end{pmatrix}, N_1 = -\frac{1}{u_0} (C_{12}C_{13}C_{14})
$$

- The stability of the periodic solution is determined by the
eigenvalues of the matrix $A_3$ giving the linear correspondence
between the initial perturbations and the final ones :

$$
\begin{pmatrix} y_f \\ u_f \\ w_f \end{pmatrix} = A_3 \begin{pmatrix} y \\ u \\ w \end{pmatrix}, \quad A_3 = A_2 A_1 \tag{60}
$$

The periodic motion is stable if all the eigenvalues of $A_3$ lie
inside the unit circle.

6.2 FORCED SYSTEM

Let us consider a periodic solution of system (37), of period
$\tau_0 + T_0 = 2\pi / \omega$, with $\varphi = -\varphi_0 = -\omega\tau_0 / 2$, related to the initial
conditions $z_{00} = \begin{pmatrix} 1 \\ y_0 \end{pmatrix}$, $\dot{z}_{00} = \begin{pmatrix} u_0 \\ w_0 \end{pmatrix}$ where $(\tau_0, y_0, u_0, w_0)$ are
deduced from the system (45).

The stability of this periodic solution is investigated by
considering the motion related to the new initial
conditions $z_0 = z_{00} + dz_0$, $\dot{z}_0 = \dot{z}_{00} + d\dot{z}_0$ and new phase angle

$$
\varphi = -\frac{\omega \tau_0}{2} + d\varphi.
$$

This motion is defined in two steps:

-For $0 \le t \le \tau = \tau_0 + d\tau$: $z, \dot{z}$ are defined by (38).This motion ends when $z_1(\tau) = 1$ and $\dot{z}_1(\tau) < 0$. Let us denotes by $z_c = z_{00} + dz_c$, $\dot{z}_c = -\dot{z}_{00} + d\dot{z}_c$ the positions and the velocities reached by the system at this final time.

-For $\tau \le t \le \frac{2\pi}{\omega} + d\theta$, ($d\theta = d\tau + dT$), the motion is defined by
(40). This motion ends when $z_1(\frac{2\pi}{\omega} + d\theta) = 1$. Let us denote by $z_f = z_{00} + dz_f$, $\dot{z}_f = \dot{z}_{00} + d\dot{z}_f$, the positions and the velocities reached by the system at this time. Assuming small perturbations $dz_0, d\dot{z}_0$ of the initial conditions,

$$ 
\begin{aligned}
dz_c &= H_1 dz_0 + H_2 d\dot{z}_0 + \tilde{p}_1 d\tau + q_1 d\varphi \\
d\dot{z}_c &= H_3 dz_0 + H_1 d\dot{z}_0 + \tilde{p}_2 d\tau + q_2 d\varphi \\
(H_i &= H_i(\tau_0), i = 1, 2, 3)
\end{aligned}
\quad (61) $$

$$ 
\begin{aligned}
\tilde{p}_1 &= -\dot{z}_{00}, \quad \tilde{p}_2 = -H_2^{-1}(H_1 + I)(\dot{z}_{00} - Q\omega \sin \varphi_0) \\
&\quad -Q\omega^2 \cos \varphi_0
\end{aligned}
$$

$$ 
\begin{aligned}
q_1 &= -(H_1 + I)Q \sin \varphi_0 + H_2 Q \omega \cos \varphi_0 \quad (q_1 = (q_{11}, q_{12})') \\
q_2 &= H_2^{-1}(H_1 - I)q_1
\end{aligned}
\quad (62) $$

In a same way

$$ 
\begin{aligned}
dz_f &= \Gamma_1 dz_c + \Gamma_2 d\dot{z}_c + \tilde{p}_3 dT + q_3 d\varphi' \\
d\dot{z}_f &= \Gamma_3 dz_c + \Gamma_1 d\dot{z}_c + \tilde{p}_4 dT + q_4 d\varphi'
\end{aligned}
\quad (63) $$

$$ 
\begin{aligned}
d\varphi' &= \omega d\tau + d\varphi \\
\tilde{p}_3 &= \dot{z}_{00}, \quad \tilde{p}_4 = \Gamma_2^{-1}(\Gamma_1 + I)(\dot{z}_{00} - R\omega \sin \varphi_0) - R\omega^2 \cos \varphi_0
\end{aligned}
$$

$$ 
\begin{aligned}
q_3 &= (\Gamma_1 + I)R \sin \varphi_0 + \Gamma_2 R \omega \cos \varphi_0 \quad (q_3 = (q_{31}, q_{32})') \\
q_4 &= \Gamma_2^{-1}(\Gamma_1 - I)q_3
\end{aligned}
$$

From systems (61) and (63), after the elimination of $d\tau$ and $dT$, we deduce the correspondence between the initial perturbations and the final ones :

$$ 
\begin{gathered}
\begin{pmatrix} y_f \\ u_f \\ w_f \\ d\varphi' \end{pmatrix} = \tilde{A}_3 \begin{pmatrix} y \\ u \\ w \\ d\varphi \end{pmatrix} \\
\tilde{A}_3 = \begin{pmatrix} \tilde{A}_2 \tilde{A}_1 + \omega \tilde{A}_3 M_1 & \tilde{A}_2 (1 + \omega \frac{q_{11}}{u_0}) + \tilde{A}_2 \tilde{A}_1 \\ \omega M_1 & 1 + \omega \frac{q_{11}}{u_0} \end{pmatrix}
\end{gathered}
\quad (64) $$

$$ 
\begin{aligned}
\tilde{A}_1 &= \tilde{H} + \tilde{M}_2 M_1, \quad \tilde{A}_1 = \begin{pmatrix} q_{12} \\ q_2 \end{pmatrix} + \frac{q_{11}}{u_0} \tilde{M}_2, \quad \tilde{M}_2 = \begin{pmatrix} -w_0 \\ \tilde{p}_2 \end{pmatrix} \\
\tilde{A}_2 &= \tilde{C} + \tilde{N}_2 N_1, \quad \tilde{A}_2 = \begin{pmatrix} q_{32} \\ q_4 \end{pmatrix} - \frac{q_{31}}{u_0} \tilde{N}_2, \quad \tilde{N}_2 = \begin{pmatrix} w_0 \\ \tilde{p}_4 \end{pmatrix}
\end{aligned}
$$

The stability of the periodic motion is determined by the eigenvalues of the matrix $\tilde{A}_3$.

## 7. NUMERICAL RESULTS

Some numerical investigations are performed for the following values of the parameters [6]:

$$ k_1 = k_3 = 1, k_2 = 5, m_1 = 1, m_2 = 2, P_1 = 2/3, P_2 = 0 $$

The corresponding eigenvalues of the free system are: $\omega_1 = 1.7958$, $\omega_2 = 0.8805$ while the eigenvalues of the constraint system are: $\sigma_1 = 1.8347$, $\sigma_2 = 1.2783$.

In Figures 2 and 3, the behavior of the periodic solutions is compared for unforced (ZiF) and forced system(ZiNF), in the rigid impact case. Figures 4 and 5 are related to soft impact: the bold part of the curves shows the free motion, the other part (ZicF or ZicNF) the constraint motion. Other results about the stability conditions when $\omega$ varies, will be presented at the Conference.

## 8. CONCLUSIONS

The main objective of this paper is to compare the behavior of the system when the stiffness of the obstacle changes from a finite value (soft impact) to an infinite value (rigid impact). The results obtained for unforced systems about the existence of periodic solutions show that there is a smooth transition between the two cases. For both cases, the period T can be chosen arbitrarily, the initial conditions related to the periodic motion are obtained in terms of T. Moreover, in case of rigid impact, the initial velocity of the non impacting mass being zero, we can formulate the conditions giving the positions and the velocities after the shock by formula (11). In case of soft impact, we showed analytically that the conditions obtained at the end of the constraint motion (when the contact of the first mass with the stop finishes), that the positions and the velocities at this time are obtained by the same rule. The case of forced systems is a more standard one. The investigations performed in this paper show how the results obtained in the case of unforced systems can lead very simply to obtain the periodic solutions in case of forced systems. The stability of periodic motions is based on mapping. In all cases, (rigid or soft impact, unforced or forced systems), the matrix of the linear correspondence between the initial perturbations and the final ones, is obtained in close form. In several cases, some interesting properties of the corresponding eigenvalues are also obtained.

## REFERENCES

1. Shaw, S.W. and Holmes, P.J., "A periodically forced piecewise linear oscillator", *Journal of Sound and Vibration*, **90** (1), 1983, 129-155.

2. Shaw, S.W. and Holmes, P.J., "A periodically forced impact oscillator with large dissipation", *Journal of Applied Mechanics*, **50**, 1983, 849-857.

3. Hindmarsh M. B. and Jeffries D. J., "On the motions of the impact oscillator", *Journal of Physics A*, **17**, 1984, 1791-1803.
4. Peterka, F., "Dynamics of oscillator with soft impacts", in *Proceedings of the ASME 2001 Design Engineering Technical Conferences, (CDROM)*, Pittsbourg (USA), September 9-12, 2001.

5. Aidanpaa, J. O and Gupta, R. D., "Periodic and chaotic behavior of a threshold-limited two degree of freedom system", *Journal of Sound and Vibrations*, **165** (2), 1993,305-327.

6. Luo, G. W. and Xie, J. H, "Hopf bifurcation of a two degree of freedom vibro-impact system", *Journal of Sound and Vibrations*, **213** (3), 1998,391-408.

7. Valente, A.X., McClamroch, N.H., and Mezie, I. "Hybrid impact of two coupled oscillators that can impact a fixed stop", *International Journal of Non-Linear Mechanics*, **38**, 2003,677-689.

8. Natsiavas, S., "Dynamics of multiple-degree-of-freedom oscillators with colliding components", *Journal of Sound and Vibrations*, **165** (3), 1993, 439-453.

9. Pun, D., Lua, S.L., Law, S.S. and Cao, D.Q., "Forced vibration of a multidegree impact oscillator.", *Journal of Sound and Vibrations*, **213** (3), 1998, 447-466.

10. Pascal, M., Stepanov, S. and Hassan, S., "An analytical investigation of the periodic motions of a two degree of freedom oscillator with elastic obstacles", to appear in *Journal of Computational Methods in Sciences and Engineering*.

<figure><img src="image_3.png" /><figcaption>Figure 2: Motion of the impacting mass for forced and unforced cases (rigid impact)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Motion of the non impacting mass (forced and unforced cases, rigid impact)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Motion of the impacting mass for forced and unforced cases (soft impact)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Motion of the non impacting mass (forced and unforced cases, soft impact)</figcaption></figure>