# On convex combinations of grazing linear modes for 2-degree-of-freedom vibro-impact systems with 1:2 resonance

Huong Le Thi, Stéphane Junca

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Huong Le Thi, Stéphane Junca. On convex combinations of grazing linear modes for 2-degree-of-freedom vibro-impact systems with 1:2 resonance. 2026. hal-05504162

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# On convex combinations of grazing linear modes for 2-degree-of-freedom vibro-impact systems with 1:2 resonance

Huong Le Thi*¹ and Stephane Junca²

¹VNU University of Engineering and Technology, Hanoi, Vietnam,
huonglt.uet@vnu.edu.vn

²Université Côte d'Azur, CNRS, LJAD Mathematics, Nice, France,
Stephane.Junca@univ-cotedazur.fr

## Abstract

Grazing periodic orbits in vibro-impact mechanical systems are known to generate nonlinear normal modes. In the absence of internal resonance, many of the essential dynamical features already appear in two-degree-of-freedom (2-dof) mass-spring chains. In this short note, we focus on such a system under the simplest internal resonance condition. Two grazing linear modes, denoted $GLOM_1$ and $GLOM_2$, associated with the first and second linear modes, play a key role. We show that the grazing linear orbit generating one-impact-per-period (1IPP) solutions is neither $GLOM_1$ nor $GLOM_2$, but rather a convex combination of the two. This leads to the introduction of the set $GLO_c = [GLOM_1, GLOM_2]$ of grazing linear orbits. For all $GLO_c$ except the endpoint modes, the free-flight time to impact admits a particularly simple structure, providing a foundation to study the nonlinear instability expected of such Grazing linear orbits.

**Key words:** Unilateral contact, Grazing Linear Mode, free flight time, Internal Resonances

## Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Model of 2-dof chain with a unilateral stop</td><td>2</td></tr><tr><td>3 Main results</td><td>5</td></tr><tr><td>4 The set of GLO<sub>c</sub></td><td>5</td></tr><tr><td>5 Continuity of 1IPP to a GLO<sub>c</sub></td><td>6</td></tr></table>

<footnote>*Corresponding author</footnote>
<table><tr><td>6</td><td>Zero or an infinite number of contacts and FFT ≤ T<sub>2</sub></td><td>8</td></tr><tr><td>7</td><td>Simplified dynamics near a GLO<sub>c</sub></td><td>8</td></tr><tr><td>8</td><td>Conclusion</td><td>10</td></tr><tr><td>9</td><td>Acknowledgement</td><td>10</td></tr></table>

# 1 Introduction

Grazing orbits are known to introduce new and often delicate features in the dynamics of vibro-impact mechanical systems [1–3]. In particular, the occurrence of grazing may lead to square-root singularities in the return map [4], grazing bifurcations [2], and the start of chaotic dynamics [5, 6]. These phenomena reflect the intrinsic non-smoothness induced by unilateral contact and play a central role in the qualitative behavior of impact oscillators.

For linear mass-spring systems without internal resonance, grazing periodic orbits have been extensively studied. In such systems, linear grazing orbits are relatively rare [7, 8], yet they organize the principal branches of nonlinear normal modes [9, 10]. Moreover, many results obtained for mass-spring systems with $N$ degrees of freedom can already be observed and verified in the two-degree-of-freedom setting, in particular concerning the structure of the nonsmooth modal spectrum [11, 12]. For this reason, two-degree-of-freedom mass-spring chains provide a natural and minimal framework for understanding the essential mechanisms of vibro-impact dynamics.

In contrast to the non-resonant case, the presence of internal resonance substantially modifies the structure of the linear dynamics. While resonance simplifies the free (contactless) motion by reducing the number of independent frequencies [13], it also considerably enlarges the set of linear grazing orbits. This enrichment motivates a focused study of grazing phenomena under internal resonance, where classical results from the non-resonant case no longer directly apply [1, 3].

In this work, we therefore consider a particular class of grazing periodic linear orbits, denoted by $GLO_c$, defined as the set of all convex combinations of the grazing linear modes previously identified in the non-resonant setting [7, 8]. We show that, in the presence of internal resonance, a specific element of $GLO_c$ generates a family of non-smooth modes, including 1IPP solutions, which were first identified in [9] and are closely related to modal structures observed in impact problems on elastic bars [14].

Furthermore, we prove that the dynamics in a neighborhood of a $GLO_c$ exhibit a remarkably simple structure: no sticking phase occurs and the free-flight time remains almost constant near such orbits. This behavior sharply contrasts with the non-resonant case, where the free-flight time is unbounded and discontinuous near grazing [7]. Finally, we state a general result describing the dynamics of all orbits in this setting, which provides a unified framework for understanding grazing phenomena in vibro-impact systems with internal resonance.

# 2 Model of 2-dof chain with a unilateral stop

We consider a 2-dof mass-spring chain in which the second mass is subject to a unilateral constraint (rigid stop) located at a distance $d > 0$, Fig. 1.
<figure><img src="image_3.png" /><figcaption>Figure 1: The spring-mass chain system with unilateral contact condition</figcaption></figure>

The governing equations are as follows.

$$ M\ddot{u} + Ku = r \quad (1a) $$

$$ u(0) = u_0, \quad \dot{u}(0) = \dot{u}_0 \quad (1b) $$

$$ u_2(t) \le d, \quad R(t) \le 0, \quad (u_2(t) - d)R(t) = 0 \quad (1c) $$

$$ \dot{u}^+(t)^\top M \dot{u}^+(t) + u^\top(t) Ku(t) = E(u(t), \dot{u}^+(t)) = E(u(0), \dot{u}(0)) \quad (1d) $$

with

$$ M = \operatorname{diag}(m_j)_{j=1}^2; \quad K = (k_{ij})_{i,j=1}^2; \quad u(t) = (u_j)_{j=1}^2; \quad r(t) = (0, R(t)) $$

where $\dot{u}_j$ and $\ddot{u}_j$ represent the velocity and acceleration of mass $j$, $j = 1, 2$, respectively. The model in question is a chain, for which $k_i$ is the stiffness of the spring number $i$ and the stiffness matrix is

$$ K = \begin{pmatrix} k_1 + k_2 & -k_2 \\ -k_2 & k_2 \end{pmatrix}. $$

Condition (1c) says that mass 2 is constrained on the right side by a rigid obstacle at a distance $d > 0$ from its equilibrium. There is only one constraint on mass 2. The other masses are not constrained in any way. The quantity $R(t)$ is the reaction force induced by the obstacle on mass $N$ at the time of gap closure. Generally, $R(t)$ is a measure. However, it is a Lipschitz function which is as regular as $\ddot{u}_2$ for solutions with sticking phases [15]. The present work targets non-dissipative dynamics and condition (1d) is enforced: the total energy of the system is preserved during the motion.

Matrices $M$ and $K$ are symmetric positive definite. Hence, there exists a matrix $P$ of M-orthogonal eigenmodes which diagonalizes both $M$ and $K$, that is $P^\top MP = I$ and $P^\top KP = \Omega^2 = \operatorname{diag}(\omega_j^2)|_{j=1,2}$ where $I$ is the $2 \times 2$ identity matrix; $\omega_j^2$ are the eigenfrequencies and $T_j$, the linear periods with $\omega_j T_j = 2\pi$, $j = 1, 2$.

**Internal resonance.** The system is said to be in *internal resonance* if the frequency ratio is rational, i.e. $\omega_2/\omega_1 \in \mathbb{Q}$. Equivalently, there exist coprime integers $p, q \in \mathbb{N}$ such that $q\omega_2 = p\omega_1 \Leftrightarrow T_2 = \frac{p}{q}T_1$. Through out this paper, we focus on the special case $p = 2, q = 1$, which gives the 1:2 internal resonance condition

$$ T_2 = 2T_1 \quad (2) $$

**Contact hyperplane as a section and free flight time.** Let $\mathcal{H} = \{W = (u, \dot{u}^+) \in \mathbb{R}^4: u_2 = d\}$ be the contact hyperplane. For $W \in \mathcal{H}$, let $u(t; W)$ be the corresponding hybrid trajectory. Following a possible initial sticking episode, characterized by $\dot{u}_2 = 0$ over a finite time interval, the system evolves according to the free-flight dynamics until the next closing contact with the stop. The free flight time (FFT) to the contact hyperplane is,

$$ \text{FFT}(W) := \min\{t > 0: u_2(t; W) = d\}, \quad (3) $$
where $t = 0$ is the departure time from $\mathcal{H}$. In other words, FFT denotes the return duration to $\mathcal{H}$. The notion of FFT used in this work should not be confused with the first return time (FRT) appearing in related studies [7, 8]. In contrast to FRT, which is defined with respect to a Poincaré section, FFT measures only the time interval over which the system evolves under free-flight dynamics, from release to the next closing contact. In the non resonance case, FRT can be very large and may even fail to exist [7, 8]. Here, with the 1 : 2 resonance, we will show that FFT always exists, moreover FFT $\le T_2$.

We now define a class of special orbits, namely, grazing linear orbits on $\mathcal{H}$ with linear dynamics. These orbits are well known to trigger grazing bifurcations [3, 5]. Due to internal resonance, they have a period $T_2$ and are more numerous than in the absence of internal resonance.

**Definition 1** (Grazing linear orbit (GLO)). A *grazing linear orbit* (GLO) is a *periodic* orbit of the free-flight linear system that reaches the contact hyperplane $\mathcal{H}$ only at a grazing event.

After a time shift we often assume that a grazing instant is at $t = 0$, so that

$$u_2(0) = d, \quad \dot{u}_2(0) = 0.$$

In particular, two pure-mode grazing linear orbits are fundamental to our analysis.

**Definition 2** (Grazing linear orbit modes (GLOM$_1$, GLOM$_2$)). For $i \in \{1, 2\}$, the *grazing linear orbit mode* GLOM$_i$ is the pure-mode periodic solution

$$\mathbf{u}^i(t) = A_i \mathbf{P} \mathbf{e}_i \cos(\omega_i t), \quad A_i := \frac{d}{P_{2i}},$$

so that $u_2^i(0) = d$ and $\dot{u}_2^i(0) = 0$, and $u_2^i(t) < d$ for all $0 < t < T_i$, where $\mathbf{P}$ is the matrix that diagonalizes both $\mathbf{M}$ and $\mathbf{K}$.

We now extend the notion of GLOMs to include their convex combinations.

**Definition 3** (GLO$_c$ - Convex combinations of GLOMs). Denote by $W_0^1$ and $W_0^2$ the data on the contact hyperplane $\mathcal{H}$ of GLOM$_1$ and GLOM$_2$. The set of all convex combinations $W^\lambda$ of $W_0^1$ and $W_0^2$ corresponding to a unique grazing linear orbit represents all GLO$_c$,

$$W_0^\lambda = (1 - \lambda)W_0^1 + \lambda W_0^2 = \left( (1 - \lambda) \frac{P_{11}}{P_{21}} + \lambda \frac{P_{12}}{P_{22}}, d, 0, 0 \right), \quad \lambda \in [0, 1]. \quad (4)$$

We also write $\{\text{GLO}_c\} := [\text{GLOM}_1, \text{GLOM}_2]$ for this family.

**Remark 1.** In the 2-dof case, the grazing-start constraints $u_2(0) = d$ and $\dot{u}_2(0) = 0$ define a two-dimensional affine plane in $\Sigma$, which may be parameterized by $(u_1(0), v_1(0))$. The set of GLO$_c$ is a one-dimensional subset of this plane.

Finally, an important family of nonsmooth periodic orbits, previously investigated in [9], is closely related to grazing bifurcations and the family GLO$_c$, as will be shown in this paper.

**Definition 4** (One-impact-per-period (1IPP) orbit). A *one-impact-per-period* (1IPP) orbit is a $T$-periodic hybrid trajectory whose *minimal* period is $T$ and that undergoes exactly one closing impact with the unilateral constraint during each period [9].
## 3 Main results

We consider the 2-dof unilaterally constrained chain governed by (1a)-(1d) and the internal resonance condition (2). First, the central role of GLO$_c$ is highlighted since one of them generates an invariant manifold of non-smooth modes as for a bar [14, 16].

**Theorem 1** (Nonlinear modes with one impact-per-period are generated by a GLO$_c$). Let $W^T$ be the data on $\mathcal{H}$ of a 1IPP nonlinear mode with period $T$, for $T \notin T_1\mathbb{N} \cup T_2\mathbb{N}$. Then, there exists a unique GLO$_c$, that is $\lambda \in (0, 1]$, such that

$$W^T \rightarrow W^\lambda \quad \text{as} \quad T \rightarrow T_2. \qquad (5)$$

Second, the dynamics in a neighborhood of a GLO$_c$ is shown to be particularly simple.

**Theorem 2** (Near a GLO$_c$). The FFT near a GLO$_c \neq GLOM_1$ is close to and less than $T_2$.

The above result follows from the simplicity of the underlying linear dynamics. Owing to internal resonance, linear solutions are $T_2$-periodic, and $T_2$ is the minimal period for all solutions except those associated with mode 1. In the same spirit, the following theorem about zero or infinitely many contacts is specific to the resonant case, which is different from the corresponding result in the non-internal resonance setting [8], where trajectories may exhibit zero, one, or infinitely many contacts.

**Theorem 3** (Zero or infinitely many contacts). There are only two kinds of orbits:

(i) No contact: $u_2(t) < d$ for all $t \in \mathbb{R}$.

(ii) Infinitely many contacts: the set of all contact times $CT = \{t \in \mathbb{R}, u_2(t) = d\}$ is infinite, $\inf CT = -\infty$, $\sup CT = +\infty$ and between two consecutive contacts FFT $\le T_2$.

Moreover, FFT = $T_2$ if and only if the solution is a GLO different from GLOM$_1$.

In particular, for any nonlinear solution, the free flight time is strictly less than $T_2$.

## 4 The set of GLO$_c$

Taking a convex combination of GLOMs yields a grazing linear orbit (GLO). This fact is straightforward, and we briefly explain it here. Let

$$u(t) = (1 - \lambda)u^1(t) + \lambda u^2(t), \quad 0 < \lambda < 1,$$

be a convex combination of the two GLOMs. Since both $u^1$ and $u^2$ are solutions of the linear system with period $T_2$, their convex combination $u$ is also a $T_2$-periodic solution of the linear system. Moreover, for each $i \in \{1, 2\}$, the displacement of the second mass along the $i$th GLOM is given by $u_2^i(t) = d \cos(\omega_i t)$. Consequently,

$$u_2(t) = (1 - \lambda)u_2^1(t) + \lambda u_2^2(t) \le d \quad \text{for all } t.$$

Thus, $u$ satisfies the unilateral constraint $u_2 \le d$ at all times and is therefore a solution of the vibro-impact system. Furthermore, this solution grazes the contact hyperplane $\mathcal{H}$, since $u_2(0) = d$ and $\dot{u}_2(0) = 0$.
The internal resonance case exhibits a qualitative difference compared with the non-resonant setting. In particular, linear combinations of GLOMs that reach the wall simultaneously give rise to GLOs, leading to a strictly larger set of GLOs than the set of all $GLO_c$. For $1 < \lambda \le 4/3$, the corresponding solutions form a nontrivial family of GLO that do not belong to {$GLO_c$}.

**Lemma 1.** $\max((1 - \lambda)\cos(2t) + \lambda\cos(t)) = 1$ if and only if $\lambda \in [0, 4/3]$.

*Proof.* Let $f(t) = \cos(2t) = 2\cos^2 t - 1$ and $g(t) = \cos t$. Both functions are $2\pi$-periodic and attain their maximum value 1 at $t = 0$. Consider the convex combination $h(t) = (1 - \lambda)f(t) + \lambda g(t)$ and write it as a function of $X = g(t) \in [-1, 1]: h(X) = (1 - \lambda)(2X^2 - 1) + \lambda X$.

* If $0 \le \lambda \le 1$, then $h(X) \le (1 - \lambda) + \lambda = 1$, with equality at $X = 1$, hence $\max h = 1$.

* If $\lambda < 0$, taking $X = -1$ yields $h(-1) = 1 + 2|\lambda| > 1$.

* If $1 < \lambda \le 4/3$, writing $\lambda = 1 + \mu$ with $0 < \mu \le 1/3$ gives $h(X) = \mu + (1 + \mu)X - 2\mu X^2$, which is strictly increasing on $[-1, 1]$, so $h(X) \le h(1) = 1$.

* If $\lambda > 4/3$, then $h(0) = 1$, $h'(0) = 0$, $h''(0) = 3\lambda - 4 > 0$, so $h(t) > 1$ for $t \ne 0$ small enough.

Therefore, $\max h = 1$ if and only if $\lambda \in [0, 4/3]$. $\square$

# 5 Continuity of 1IPP to a $GLO_c$

In this section, we will show that there is a branch of the 1IPP orbits emerges from a GLO, that is Theorem 1. First, let us recall the definition of the nonlinear solution with 1-IPP [9] and define the initial data for the set of grazing linear orbits which are proven to be the limit of the 1-IPPs.

**Initial data for 1-IPP.** Denote the post-impact state as $W = (u, \dot{u}^+) \in \mathbb{R}^4$ with $u_2 = d$ on $\mathcal{H}$. For $T \notin T_1\mathbb{N} \cup T_2\mathbb{N}$, the unique 1-IPP orbit of period $T$ (Definition 4, cf. [9]) is characterized by the post-impact initial data

$$W^T := (u(0; T), \dot{u}^+(0; T)), \quad u(0; T) = \dot{u}_2^+(0; T) w(T), \qquad (6)$$

$$\dot{u}^+(0; T) = \dot{u}_2^+(0; T) e_2, \quad \dot{u}_2^+(0; T) = \frac{d}{w_2(T)}, \qquad (7)$$

$$w_k(T) = \sum_{j=1}^{2} a_{kj} \Phi_j(T), \quad a_{kj} = P_{kj} P_{j2}^{-1}, \quad \Phi_j(t) = \frac{\sin(\omega_j t)}{\omega_j(1 - \cos(\omega_j t))}. \qquad (8)$$

**Grazing limit and resonant GLO data.** Assume the internal resonance condition (2). The associated GLO is the $T_2$-periodic solution whose data on $\mathcal{H}$ is

$$W_0 := (u(0; T_2), 0) \in \mathcal{H}, \quad u_k(0; T_2) = d \frac{\sum_{j \in I} a_{kj} \omega_j^{-2}}{\sum_{j \in I} a_j \omega_j^{-2}}, \quad k = 1, 2, \qquad (9)$$
where $I \subset \{1, 2\}$ is the set of resonant modes satisfying $T_2 \in T_j\mathbb{N}$ and $a_j := a_{2j} = P_{2j}P_{j2}^{-1}$. Theorem 1 can now be proven as follows.

*Proof.* Fix $T \notin T_1\mathbb{N} \cup T_2\mathbb{N}$ sufficiently close to $T_2$. By [9], the corresponding 1IPP orbit of period $T$ is uniquely determined by the post-impact initial data $W^T = (\mathbf{u}(0; T), \dot{\mathbf{u}}^+(0; T)) \in \mathcal{H}$ given by (6)–(7).

Let $I \subset \{1, 2\}$ be the set of resonant indices, i.e. those $j$ such that $T_2 \in T_j\mathbb{N}$ (equivalently, $\omega_j T_2 \in 2\pi\mathbb{Z}$). For $j \in I$, a Taylor expansion about $T_2$ yields

$$ \sin(\omega_j T) = \omega_j(T - T_2) + O((T - T_2)^3), \quad 1 - \cos(\omega_j T) = \frac{\omega_j^2}{2}(T - T_2)^2 + O((T - T_2)^4), $$

and hence

$$ \Phi_j(T) = \frac{\sin(\omega_j T)}{\omega_j(1 - \cos(\omega_j T))} = \frac{2}{\omega_j^2(T - T_2)} + O(1), \quad T \to T_2. \quad (10) $$

For $j \notin I$, the denominator $1 - \cos(\omega_j T)$ stays bounded away from 0 near $T_2$, so $\Phi_j(T) = O(1)$ as $T \to T_2$.

Using (10) and the boundedness of the non-resonant terms, we obtain for $k = 1, 2$,

$$ w_k(T) = \sum_{j=1}^{2} a_{kj} \Phi_j(T) = \frac{2}{T - T_2} \sum_{j \in I} a_{kj} \omega_j^{-2} + O(1), \quad T \to T_2. \quad (11) $$

In particular,

$$ w_2(T) = \frac{2}{T - T_2} \sum_{j \in I} a_{2j} \omega_j^{-2} + O(1). $$

Since $\sum_{j \in I} a_{2j} \omega_j^{-2} \neq 0$ [9], it follows that $|w_2(T)| \to \infty$ as $T \to T_2$ and therefore

$$ \dot{u}_2^+(0; T) = \frac{d}{w_2(T)} \to 0, \quad \text{as } T \to T_2, $$

which proves $\dot{\mathbf{u}}^+(0; T) \to \mathbf{0}$.

Finally, since $\mathbf{u}(0; T) = \dot{u}_2^+(0; T)\mathbf{w}(T)$, we may write componentwise

$$ u_k(0; T) = \dot{u}_2^+(0; T) w_k(T) = d \frac{w_k(T)}{w_2(T)}, \quad k = 1, 2. $$

Combining this with (11) gives

$$ u_k(0; T) \to d \frac{\sum_{j \in I} a_{kj} \omega_j^{-2}}{\sum_{j \in I} a_{2j} \omega_j^{-2}} =: u_k(0; T_2), \quad k = 1, 2. $$

Therefore $W^T \to W_0 := (\mathbf{u}(0; T_2), \mathbf{0})$ as $T \to T_2$. This proves (5). $\square$

The last step is to show that $W_0$ corresponds to a GLO$_c$.

**Lemma 2** (Explicit convex combinations). Let $W_0$ be given by (9). Then, there exists $\lambda \in (0, 1]$ such that $W_0 = (1 - \lambda)W_0^1 + \lambda W_0^2$.
Proof. With $I = \{1, 2\}$, formula (9) yields

$$u_1(0; T_0) = d \frac{a_{11}\omega_1^{-2} + a_{12}\omega_2^{-2}}{a_{21}\omega_1^{-2} + a_{22}\omega_2^{-2}}, \quad u_2(0; T_0) = d, \quad \dot{u}(0; T_0) = 0,$$

where $a_{kj} = P_{kj}P_{j2}^{-1}$. Since $\frac{a_{1j}}{a_{2j}} = \frac{P_{1j}P_{j2}^{-1}}{P_{2j}P_{j2}^{-1}} = \frac{P_{1j}}{P_{2j}}$, $j = 1, 2$, we may rewrite

$$u_1(0; T_0) = (1 - \lambda) d \frac{P_{11}}{P_{21}} + \lambda d \frac{P_{12}}{P_{22}}, \quad \lambda := \frac{a_{22}\omega_2^{-2}}{a_{21}\omega_1^{-2} + a_{22}\omega_2^{-2}}.$$

Because $\omega_j^{-2} > 0$ and $a_{2j} > 0$ for $j = 1, 2$, we have $\lambda \in [0, 1]$. The identity for $W_0$ follows since $u_2(0; T_0) = d$ and $\dot{u}(0; T_0) = 0$ for $W_0^1$ and $W_0^2$, then for $W_0$. $\square$

# 6 Zero or an infinite number of contacts and FFT $\le T_2$

This section completes the proof of Theorem 3. Theorem 3 will be used to prove Theorem 2. We show that trajectories either experience no contact or infinitely many contacts, and that whenever a contact occurs the associated FFT satisfies $FFT \le T_2$. If no contact occurs, the motion is governed entirely by the linear dynamics and is therefore $T_2$-periodic. We henceforth assume that a contact occurs. By invariance under time translation, we can assume that a contact occurs at $t = 0$. Denote by $u^\#$ the solution of the unconstrained linear system whose position and velocity coincide with those of the constrained trajectory $u$ at $t = 0^+$, immediately after detachment. Then $u(t) = u^\#(t)$ throughout the free-flight time. If a sticking phase occurs, it has finite duration for the chain model under consideration [15]; in that case, we redefine $t = 0$ as the end of the sticking phase so that the motion again starts with free flight. The linear trajectory $u^\#$ is $T_2$-periodic, although $u$ need not be periodic. Since $u_2^\#(T_2) = d$, the constrained trajectory must experience a contact no later than time $T_2$, which implies

$$FFT \le T_2.$$

If the contact occurs with nonzero velocity, i.e. $u_2(0^-) > 0$, then $u_2^\#(0) = -u_2(0^-) < 0$, and hence $u_2^\#(-\tau) > d$ for $\tau > 0$ sufficiently small. By $T_2$-periodicity of $u^\#$, this yields $u_2^\#(T_2 - \tau) > d$, implying that a new contact occurs strictly before $T_2$. Thus, for a true impact,

$$FFT < T_2. \qquad (12)$$

For a grazing contact, $FFT = T_2$ is possible but not mandatory. Repeating the above argument after each detachment shows that once a contact occurs, the trajectory necessarily undergoes infinitely many contacts. Moreover, the number of contacts (when entering or exiting $H$, a sticking counts as two, an impact and a grazing count as one) is finite on any compact sets [17]. This completes the proof of Theorem 3.

# 7 Simplified dynamics near a $GLO_c$

In this section, we now prove Theorem 2. We show that, in a neighborhood of a $GLO_c$ distinct from $GLOM_1$, the dynamics simplifies and the associated free flight time is close
to, and strictly less than, $T_2$. A key preliminary step is to exclude the occurrence of micro- contacts near GLO$_c$. Here, a micro-contact refers to an additional contact occurring arbitrarily close to the contact at $t = 0$ for trajectories with initial data on the contact hyperplane $\mathcal{H}$.

**Proposition 1.** There is no micro-contact in the vicinity of a GLO$_c$ which is not a GLOM.

The proof follows the non-resonance case, Section 4.2 in [7].

*Proof.* Let $W_0 = W_0(\lambda)$ be a GLO$_c$ with $\lambda \in (0, 1)$. Then, no micro-contact means that there exist $\varepsilon > 0$, $\eta > 0$, and $\delta > 0$ such that

$$u_2(t; W_0) \le d - \eta, \quad \forall t \in [\varepsilon, T_2 - \varepsilon], \qquad (13)$$

and, for every $W \in \mathcal{H} \cap B_\delta(W_0)$,

$$u_2(t; W) < d, \quad \forall t \in [\varepsilon, T_2 - \varepsilon]. \qquad (14)$$

Consequently, any contact time after $t = 0$ (FFT($W$)) belongs to $(T_2 - \varepsilon, T_2]$. In partic- ular, there is no micro-contact on $(0, T_2 - \varepsilon]$ for all such nearby initial data. Since $W_0$ is associated to a GLO$_c$ and the free-flight dynamics is linear, the second component along the free-flight trajectory issued from $W_0(\lambda)$ can be written as

$$u_2(t; W_0) = d((1 - \lambda)\cos(\omega_1 t) + \lambda\cos(\omega_2 t)), \quad \omega_1 = 2\omega_2.$$

Hence $u_2(t; W_0) \le d$ for all $t$, and equality $u_2(t; W_0) = d$ can occur only if $\cos(\omega_1 t) = \cos(\omega_2 t) = 1$. Under $\omega_1 = 2\omega_2$ this implies $t \in T_2\mathbb{Z}$. Therefore,

$$u_2(t; W_0) < d \quad \forall t \in (0, T_2).$$

Fix any $\varepsilon \in (0, T_2/2)$. By continuity of $t \mapsto u_2(t; W_0)$ and compactness of $[\varepsilon, T_2 - \varepsilon]$, the maximum

$$m_{\varepsilon} := \max_{t \in [\varepsilon, T_2 - \varepsilon]} u_2(t; W_0)$$

is attained and satisfies $m_{\varepsilon} < d$. Define $\eta := d - m_{\varepsilon} > 0$. This proves (13). Next, since the free-flight solution depends continuously (indeed, linearly) on the initial data, there exists a constant $C > 0$ such that

$$\sup_{t \in [\varepsilon, T_2 - \varepsilon]} |u_2(t; W) - u_2(t; W_0)| \le C \|W - W_0\| \quad \text{for all } W \in \mathbb{R}^4.$$

Choose $\delta := \eta/(2C)$. Then, for any $W \in B_\delta(W_0)$ and any $t \in [\varepsilon, T_2 - \varepsilon]$,

$$u_2(t; W) \le u_2(t; W_0) + \frac{\eta}{2} \le (d - \eta) + \frac{\eta}{2} = d - \frac{\eta}{2} < d,$$

which is (14). Finally, if a contact occurs at some time $\tau \in (0, T_2]$, then necessarily $u_2(\tau; W) = d$. By (14), such a $\tau$ cannot lie in $[\varepsilon, T_2 - \varepsilon]$. Thus any contact time after $t = 0$ must belong to $(T_2 - \varepsilon, T_2]$, which yields the claimed bound on the free flight time and excludes micro-contacts on $(0, T_2 - \varepsilon]$. $\square$

The same proof is valid for $GLO_2$. For $GLO_1$ the situation is slightly more complicated since FFT can be also near $T_1$. We can now prove theorem 2.
*Proof.* By Theorem 3, we have $FFT \le T_2$. Consider initial data $W$ in a neighborhood of $W^\lambda$ such that $GLO_c \ne GLOM_1$. For such initial data, the associated linear dynamics (ignoring the unilateral contact) is exactly $T_2$-periodic. We restrict to a neighborhood of $W^\lambda$ that does not intersect the first linear mode.

Taking the unilateral contact into account and using the neutral stability of the free dynamics, the trajectory issued from $W$ closely follows the corresponding linear orbit. By Proposition 1, no micro-contact occurs along this evolution. Consequently, the trajectory completes almost one full loop before the next impact, and therefore the first return time satisfies $FFT \approx T_2$. □

# 8 Conclusion

We have presented a concise analysis of a 2-dof vibro-impact oscillator with internal resonance, focusing on a mass-spring chain and a distinguished subset of grazing linear orbits ($GLO_c$). We have shown that the 1IPP family is generated by a special $GLO_c$, and established the continuity of FFT, in a neighborhood of this orbit. These results clarify the key mechanisms governing the grazing limit in the presence of internal resonance. A complete stability analysis of the $GLO_c$ lies beyond the scope of the present short note and will be addressed in forthcoming work in a more general setting.

# 9 Acknowledgement

The authors acknowledge Mathias Legrand for insightful discussions that contributed to this work.

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