Wiem Housseyni

Maryline Chetto
email: maryline.chetto@univ-nantes.frwiem.housseyni@univ-nantes.fr

ARTE: a Simulation Tool for Reconfigurable Energy Harvesting Real-time Embedded Systems

This report presents ARTE a new free Java based software, which we have developed for simulation and evaluation of reconfigurable energy harvesting real-time embedded systems. It is designed in the aim to generate, compare, and evaluate different real-time scheduling strategies on their schedulability performance as well as energy efficiency in the presence of external unpredictable reconfiguration scenarios. A reconfiguration scenario is defined as unpredictable events from the environment such as addition-remove of software tasks, and increase-decrease of the power rate. The occurrence of such events may evolve the system towards an overload state. For this purpose, ARTE provides a task sets generator, reconfiguration scenarios generator, and a simulator. It also offers to the designer the possibility to generate and run simulation for specific systems. In the actual version, ARTE can simulate the execution of periodic independent and (m,k)-firm constrained task sets on monoprocessor architecture. The energy consumption is considered as a scheduling parameter in the same manner as Worst Case Execution Time (WCET).

Introduction

Since decades, the literature has revealed a substantial interest in the real-time scheduling problem, where several approaches and algorithms have been ported over the years to optimize the scheduling of real-time tasks on single and multiple processor systems. The energy constraint has emerged as a major issue in the design of such systems, despite there are big research efforts addressing this problem. Research works have focused on either dynamic energy awareness algorithm in order to reduce system energy consumption or energy harvesting technology. In order to enhance the lifespan and to achieve energy autonomy in such system, there is a tremendous interest in the energy harvesting technologies recently.

In recent years, energy scavenging or harvesting technology from renewable sources such as photovoltaic cells, and piezoelectric vibrations emerges as new alternative to ensure sustainable autonomy and perpetual function of the system. By the same token, the literature has revealed a substantial interest in the scheduling research for energy aware and power management scheduling for real-time systems. Still, there is sufficient scope for research, although uni-processor real-time scheduling for energy harvesting based systems is well studied. On the other hand, scheduling techniques for the reconfigurable energy harvesting real-time systems are not mature enough to either be applicable or optimal as much as currently available uni-processor real-time scheduling techniques.

Nowadays, new criteria such as energy efficiency, high-performance and flexible computing arises the need of new generation of real-time systems. As a result, such systems need to solve a number of problems that so far have not been addressed by traditional real-time systems. Reconfigurable systems are solutions to providing both higher energy efficiency, and high-performance and flexibility. In recent years, a substantial number of research works from both academia and industry have been made to develop reconfigurable control systems [START_REF] Gharbi | Functional and operational solutions for safety reconfigurable embedded control systems[END_REF], [START_REF] Da Silva | Modeling of reconfigurable distributed manufacturing control systems[END_REF]. Reconfiguration is usually performed in response to both user requirements and dynamic changes in its environment such as unpredictable arrival of new tasks, and hardware or software failures. Some examples of such systems are multi-robot systems [START_REF] Chen | Combining re-allocating and rescheduling for dynamic multi-robot task allocation[END_REF], and wireless sensor networks [START_REF] Grichi | Rwin: New methodology for the development of reconfigurable wsn[END_REF]. From the literature we can drive different definitions of a reconfigurable system. The authors of [START_REF] Grichi | Rocl: New extensions to ocl for useful verification of flexible software systems[END_REF] define a reconfiguration of a distributed system as any addition/ removal/update of one/more software-hardware elements. In this work, we define a reconfiguration as a dynamic operation that offers to the system the capability to adjust and adapt its behavior i.e., scheduling policy, power consumption, according to environment and the fluctuating behavior of renewable source, or to modify the applicative functions i.e., add-remove-update software tasks.

Almost of embedded systems are real-time constrained. A real-time system involves a set of tasks where each task performs a computational activity according to deadline constraints. The main purpose of a real-time system is to produce not only the required results but also within strict time constraints. In order to check whether a set of tasks respects its temporal constraints when a scheduling algorithm is used or to evaluate the efficiency of a new approach, software simulation against other algorithms is considered as a valid comparison technique and it is commonly used in the evaluation of real-time systems. Over the last years, several simulation tools have been ported MAST [START_REF] Harbour | Mast: Modeling and analysis suite for real time applications[END_REF], CHEDDAR [START_REF] Singhoff | Cheddar: a flexible real time scheduling framework[END_REF], STORM [START_REF] Urunuela | Storm a simulation tool for real-time multiprocessor scheduling evaluation[END_REF], FORTAS [START_REF] Courbin | Fortas: Framework for real-time analysis and simulation[END_REF] and YARTISS [START_REF] Chandarli | YARTISS: A Generic, Modular and Energy-Aware Scheduling Simulator for Real-Time Multiprocessor Systems[END_REF]. However, none of these approaches provide support for reconfigurable based applications, yet. More recently, new simulation tools are proposed for reconfigurable computing systems Reconf-Pack [START_REF] Gammoudi | Reconf-pack: A simulator for reconfigurable battery-powered real-time systems[END_REF]. However, this attempt do not support energy harvesting requirements.

New challenges raised by reconfigurable real-time embedded energy harvesting systems in terms of reconfigurability, scheduling, and energy harvesting management. Indeed, in such a context, it is very difficult to evaluate and compare scheduling algorithms on their schedulability performance as well as energy efficiency. Unlike any prior work, we formulate and solve the challenging problem of scheduling real-time applications on reconfigurable energy-harvesting embedded system platforms. Based on these motivations, we investigate in this report the challenges and viability of designing an open and flexible simulation tool able to generate and simulate accurately the behavior of such systems. In this report, we introduce ARTE, a new simulation tool for reconfigurable energy harvesting real-time embedded systems, which provides various functions to simulate the scheduling process of real-time task sets and their temporal behavior when a scheduling policy is used. It provides the classical real-time scheduling policy EDF, the optimal scheduling algorithm for energy harvesting systems EDH, EDF scheduling for (m,k)-constrained task sets, and a new scheduling policy which is an extension of EDH for (m,k)-firm constrained task sets EDH-MK, and finally a new hierarchical approach Reconf-Algorithm. It implements also classical and new feasibility tests for both real-time and energy harvesting requirements. The main aim of this research work is to guarantee a feasible execution in the whole system in the presence of unpredictable events while satisfying a quality of service (QoS) measured first in term of the percentage of satisfied deadline, second the percentage of satisfied deadline while considering the degree of importance of tasks, and finally the overhead introduced by the proposed approach.

The remainder of this report is as follows: we review related works and examples of real-time simulators in Section 2. Section 3 presents a background about the EDF, EDH scheduling algorithm, and the (m,k)-firm model. In section 4 we detail the proposed new scheduling approach for reconfigurable energy harvesting real-time systems. Then in Section 5 we present the various functionalities of our simulation tool. A case study is given in Section 6. Finally, we discuss our future work in Section 7 and Section 8 brings a conclusion to this report.

Research Aims

New challenges raised by reconfigurable real-time embedded energy harvesting systems in terms of reconfigurability, scheduling, and energy harvesting management. Such systems work in dynamic environment where unpredictable events may occur arrival-remove of software tasks or increase-decrease of power rate. However, when unpredictable events from the environment occur, the system may evolve towards an unfeasible state (processor/energy overload) and the actual operation mode is no longer optimal. For this aim, the main focus of this research is on how to achieve system feasibility while satisfying a graceful QoS degradation. For this purpose we define three operating modes:

-Normal mode: where all the tasks in the system execute 100% of their instances while meeting all the deadlines -Degradation mode level 1: This is the case where the normal mode is not feasible. The K least important tasks execute in degraded mode according to the model (m,k)-firm and other tasks execute normally. The schedulability test is performed by considering iteratively the tasks according to their importance. -Degradation mode level 2. This is the case where the degradation mode level 1 is not feasible. Abandonable tasks are gradually eliminated by increasing importance.

When a system executing under an operating mode and an external event occurs it is imperative to verify the schedubility test. We identify three cases:

-The system may continue to execute in the actual operating mode, -the system may be executed under degraded mode, -the system may executed under normal mode.

The quality of service (QoS) is measured in term of:

-The percentage of satisfied deadline, -the percentage of satisfied deadline while considering the degree of importance of tasks, -the overhead introduced by the proposed approach.

Related Works

In this section we outline the existing works related to simulation of the scheduling of real-time tasks.

There are a lot of tools to test and visualize the temporal and execution behavior of real-time systems, and they are divided mainly into two categories: the execution analyzer frameworks and the simulation software. MAST [START_REF] Harbour | Mast: Modeling and analysis suite for real time applications[END_REF] is a modeling and analysis suite for real-time applications that is developed in 2000. MAST is an event-driven scheduling simulator that permits modeling of distributed real-time systems and offers a set of tools to e.g. test their feasibility or perform sensitive analysis. Another known simulator is Cheddar [START_REF] Hamdaoui | A dynamic priority assignment technique for streams with (m, k)-firm deadlines[END_REF][START_REF] Bernat | Combining (/sub m//sup n/)-hard deadlines and dual priority scheduling[END_REF] which is developed in 2004 and it handles the scheduling of real-time tasks on multiprocessor systems. It provides many implementations of scheduling, partitioning and analysis of algorithms, and it comes with a friendly Graphical User Interface (GUI). Unfortunately, no API documentation is available to help with the implementation of new algorithms and to facilitate its extensibility. Moreover, Cheddar is written in Ada programming language [START_REF] Mccormick | Building parallel, embedded, and real-time applications with Ada[END_REF] which is used mainly in embedded systems and it has strong features such as modularity mechanisms and parallel processing. Ada is often the language of choice for large systems that require real-time processing, but in general, it is not a common language among developers. We believe that the choice of Ada as the base of Cheddar reduces the potential contributions to the software from average developers and researchers. Finally, STORM [START_REF] Urunuela | Storm a simulation tool for real-time multiprocessor scheduling evaluation[END_REF] FORTAS [START_REF] Courbin | Fortas: Framework for real-time analysis and simulation[END_REF], and YARTISS [START_REF] Chandarli | YARTISS: A Generic, Modular and Energy-Aware Scheduling Simulator for Real-Time Multiprocessor Systems[END_REF] are tools which are written in Java. In 2009, STORM is released and it is described as a simulation tool for Real time multiprocessor scheduling. It has modular architectures (both software and hardware) which simulate the scheduling of task sets on multiprocessor systems based on the rules of a chosen scheduling policy. The specifications of the simulation parameters and the scheduling policies are modifiable using an XML file. However, the simulator tool lacks a detailed documentation and description of the available scheduling methods and the features of the software. On the other hand, FORTAS is a real-time simulation and analysis framework which targets uniprocessor and multiprocessor systems. It is developed mainly to facilitate the comparison process between the different scheduling algorithms, and it includes features such as task generators and computation of results of each tested and compared scheduling algorithm. FORTAS represents valuable contributions in the effort towards providing an open and modular tool. Unfortunately, it seems to suffer from the following issues: its development is not open to other developers for now, we can only download .class files, no documentation is yet provided and it seems that no new version has been released to public since its presentation in [START_REF] Courbin | Fortas: Framework for real-time analysis and simulation[END_REF]. More recently, YARTISS is proposed as a modular and extensible tool. It is a real-time multiprocessor scheduling simulator which provides various functions to simulate the scheduling process of real-time task sets and their temporal behavior when a scheduling policy is used. Functionalities of YARTISS: 1) simulate a task set on one or several processors while monitoring the system energy consumption, 2) concurrently simulate a large number of tasksets and present the results in a user friendly way that permits us to isolate interesting cases, and 3) randomly generate a large number of task sets. However, none of these simulation tools provide support for unpredictable reconfiguration scenarios yet. To date, only a few recent works target the real-time scheduling issue in reconfigurable systems. In [START_REF] Gammoudi | Reconf-pack: A simulator for reconfigurable battery-powered real-time systems[END_REF] a simulator tool is proposed for Reconfigurable Battery-Powered Real-Time Systems Reconf-Pack. Reconf-Pack is a simulation tool for analyzing a reconfiguration and applying the proposed strategy for real-time systems. It is based upon another tool Task-Generator which generates random tasks. According to the state of the system after a reconfiguration, Reconf-Pack calculates dynamically a deterministic solution. Moreover, it compares the pack-based solutions to related works. However, it seems to suffer from the following issues: its development is not open to other developers for now, and isn't available for download.

During the development of ARTE, we learned from those existing tools and we included some of their features in addition to others of our own. Our aim is to provide a simulation tool for reconfigurable energy harvesting real-time systems that is easily it can be used to generate, compare, simulate the scheduling of real-time tasks on reconfigurable energy harvesting real-time systems.

Background

This section gives a background about first the EDF scheduling algorithm, then EDH scheduling algorithm, and finally the (M,K)-model.

Earliest Deadline First scheduling EDF

EDF is probably the most famous dynamic priority scheduler. As a consequence of its optimality for preemptive uniprocessor scheduling of independent jobs, the runtime scheduling problem is perfectly solved if we assume there exists no additional constraints on the jobs. EDF is the scheduler of choice since any feasible set of jobs is guaranteed to have a valid EDF schedule [START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF]. Time-feasibility: A set ψ of n tasks

τ i = {C i , T i , E i } is time feasible if and only if n i=1 C i T i ≤ 1 (1)
However, when considering energy harvesting requirements EDF is no longer optimal and a necessary condition but non sufficient of schedulability is as follows: Time-feasibility: A set ψ of n tasks is time feasible if and only if

n i=1 C i T i ≤ 1 (2)
Energy-feasibility: A set ψ of n tasks is energy feasible if

n i=1 E i C + E P (0, t) ≤ 1 (3)
Let E p (0,t) be the amount of energy that will be produced by the source between 0 and t and C is the energy storage unit (supercapacitor or battery) capacity.

Earliest Deadline First-Energy Harvesting EDH

Dertouzos [START_REF] Dertouzos | Control robotics: The procedural control of physical processes[END_REF] shows that the Earliest Deadline First Algorithm (EDF) is optimal. EDF schedules at each instant of time t, that job ready for execution whose deadline is closest to t. But the problem with EDF is that it does not consider future jobs arrivals and their energy requirements. In [START_REF] Chetto | A note on edf schedulingfor real-time energy harvesting systems[END_REF], the authors prove that EDF is no longer optimal for RTEH systems. Jobs are processed as soon as possible thus consuming the available energy greedily.Although noncompetitive, EDF turns out to remain the best non-idling scheduler for uniprocessor RTEH platforms [START_REF] Chetto | A note on edf schedulingfor real-time energy harvesting systems[END_REF].

In [START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF], the authors describe the Earliest Deadline-Harvesting ED-H scheduling algorithm which is proved to be optimal for the scheduling problem of Energy-Harvesting Real-time systems. ED-H is an extension of the EDF algorithm that adds energy awareness capabilities by using the notion of slack-time and slack-energy. The idea behind ED-H is to order the jobs according to the EDF rule since the jobs issued from the periodic tasks have hard deadlines. Executing them in accordance with their relative urgency appears to be the best approach even if they are not systematically executed as soon as possible due to possible energy shortage. The difference between ED-H and classical EDF is to decide when to execute a job and when to let the processor idle. Before authorizing any job to execute, the energy level of the storage must be sufficient so that all future occurring jobs execute timely with no energy starvation, considering their timing and energy requirements and the replenishment rate of the storage unit. According to EDH a processor P j and battery B j , that performs tasks set ψ j should satisfy the schedulability tests described follows:

-Time-feasibility: [START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF] ψ j is time feasible if and only if

U Pj = n i=1 C i T i ≤ 1 (4)
-Energy-feasibility: [START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF] ψ j is energy feasible if and only if

U e j ≤ 1 (5)
U e j is the energy load of the set of tasks ψ j assigned to processor

P j U e j = sup 0≤t1≤t2≤H E cj (t 1 , t 2 ) C + E Pj (t 1 , t 2 )
Theorem 1 [START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF] The set of tasks ψ j assigned to processor P j is feasible if and only if

U Pj ≤ 1 and U e j ≤ 1.
Theorem 1 gives a necessary and sufficient condition of schedulability.

(M,K)-model

For its intuitiveness and capability of capturing not only statistical but also deterministic quality of service QoS requirements, the (m,k)-model has been widely studied, e.g., [START_REF] Bernat | Combining (/sub m//sup n/)-hard deadlines and dual priority scheduling[END_REF], [START_REF] Hamdaoui | A dynamic priority assignment technique for streams with (m, k)-firm deadlines[END_REF], [START_REF] Hua | Energy-efficient dual-voltage soft real-time system with (m, k)-firm deadline guarantee[END_REF], [START_REF] Quan | Enhanced fixed-priority scheduling with (m, k)-firm guarantee[END_REF], and [START_REF] Ramanathan | Overload management in real-time control applications using (m, k)firm guarantee[END_REF]. The (m,k)-model was originally proposed by Hamdaoui et al. [START_REF] Hamdaoui | A dynamic priority assignment technique for streams with (m, k)-firm deadlines[END_REF]. According to this model, a repetitive task of the system is associated with an (m,k) (0<m<k) constraint requiring that m out of any k consecutive job instances of the task meet their deadlines. A dynamic failure occurs, which implies that the temporal quality of service QoS constraint is violated and the scheduler is thus considered failed, if, within any k consecutive jobs, more than (k-m) job instances miss their deadlines. Based on this (m,k)-model, Ramanathan et al. [START_REF] Ramanathan | Overload management in real-time control applications using (m, k)firm guarantee[END_REF] proposed to partition the jobs into mandatory and optional jobs. So long as all of the mandatory jobs can meet their deadlines, the (m,k)-constraints can be ensured. The mandatory jobs are the jobs that must meet their deadlines in order to satisfy the -constraints, while the optional jobs can be executed to further improve the quality of the service or simply be dropped to save computing resources. Quan et al. [START_REF] Quan | Enhanced fixed-priority scheduling with (m, k)-firm guarantee[END_REF] formally proved that the problem of scheduling with the (m,k)-guarantee for an arbitrary value of mand k is NP-hard in the strong sense. They further proposed to improve the mandatory/optional partitioning by reducing the maximal interference between mandatory jobs.

Mandatory/Optional Job Partitioning With (M,K)-Pattern

The (m,k)-pattern of task τ i , denoted by Π i , is a binary string Π i = {π i0 , π i1 π i(ki-1 )} which satisfies the following: 1) π ij is a mandatory job if π ij = 1 and optional if π ij = 0 and 2)

ki-1 j=0 π ij = m i .
By repeating the (m,k)-pattern , we get a mandatory job pattern for τ i . It is not difficult to see that the (m,k)-constraint for τ i can be satisfied if the mandatory jobs of τ i are selected accordingly.

Evenly Distributed Pattern (Even Pattern) Even Pattern strategy was proposed by Ramanathan et al. [START_REF] Ramanathan | Overload management in real-time control applications using (m, k)firm guarantee[END_REF] as follows: the first release is always mandatory and subsequent distribution of mandatory and optional alternating. Mathematically,

Π i j =    1, if j = j × m i k i × k i m i , f orj = 0, 1, .., k i 0, otherwise (6)
In [START_REF] Niu | Energy minimization for real-time systems with (m, k)-guarantee[END_REF] a necessary and sufficient feasibility test for (m,k)-constrained tasks executing under EDF scheduling policy is proposed. Theorem 2: Let system T = {τ 0 , τ i , .., τ n-1 }, where τ i ={C i , T i , D i , m i , k i } and Ψ be the mandatory job set according to their E-patterns. Also, let L represent either the ending point of the first busy period when scheduling only the mandatory jobs or the LCM of T(i) , i = 0, ..., (n-1), whichever is smaller. Then, Ψ is schedulable with EDF if and only if (iff) all the mandatory jobs arriving within [0, L] can meet their deadlines, i.e., i

W i (0, t) = i ( m i k i × t -D i T i + ) × C i ≤ t (7)
5 New Scheduling Approach for Reconfigurable Energy Harvesting Real-Time Systems

Reconfiguration is usually performed in response to both user requirements and dynamic changes in its environment such as unpredictable activation of new tasks, removal of tasks, or increase-decrease of power supply of the system. Some examples of reconfigurable systems are multi-robot systems [START_REF] Chen | Combining re-allocating and rescheduling for dynamic multi-robot task allocation[END_REF] and wireless sensor networks [START_REF] Grichi | Rwin: New methodology for the development of reconfigurable wsn[END_REF]. At run-time, the occurrence of unpredictable task's activation makes the static schedule no longer optimal and may evolve the system towards an unfeasible state due to energy and processing overloads. Thereafter, some existing or added tasks may violate deadlines. The system has to dynamically adjust and adapt the task allocation and scheduling in order to cope with unpredictable reconfiguration scenarios. We identify mainly two function modes:

-Normal mode: where all the tasks in the system execute 100% of their instances while meeting all the deadlines -Degradation mode level 1: This is the case where the normal mode is not feasible. The K least important tasks execute in degraded mode according to the model (m,k)-firm and other tasks execute normally. The schedulability test is performed by considering iteratively the tasks according to their importance. -Degradation mode level 2. This is the case where the degradation mode level 1 is not feasible. Abandonable tasks are gradually eliminated by increasing importance.

At any instant, external unpredictable reconfigurations may occur to add or remove software tasks or to increase-decrease power supply on the system. The occurrence of such events provoke the execution of schedulability tests to identify in which mode the tasks should be executed.

Normal Mode

In the normal mode tasks are assumed to be executed under the optimal scheduler for real-time energy harvesting systems EDH algorithm. Then the tasks set should satisfy the following theorem:

Theorem 1 [START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF] The set of tasks ψ j assigned to processor P j is feasible if and only if U Pj ≤ 1 and U e j ≤ 1.

Degradation mode level 1: EDH-MK Algorithm

We propose in this work a new real-time scheduler for reconfigurale energy harvesting real-time systems EDH-MK. The proposed algorithm EDH-MK is an extension of the EDH algorithm with (m,k)-firm guarantee. When it is impossible to execute the tasks set in normal mode due to processor and/or energy overloads, we propose to execute the K least important tasks in degraded mode according to the model (m,k)-firm and other tasks execute normally. In this work we propose the necessary and sufficient schedulability conditions for the EDH-MK algorithm. Definition 1: The static slack time of a mandatory job set Ψ on the time interval

[t 1 , t 2 ) is SST Ψ (t 1 , t 2 ) = t 1 -t 2 - i W (t 1 , t 2 ) ( 8
)
SST Ψ (t 1 , t 2 ) gives the longest time that could be made available within [t 1 , t 2 ) after executing mandatory jobs of Ψ with release time at or after t 1 and deadline at or before t 2 . Definition 2: The total mandatory energy demand within interval [0,t] is

g(0, t) = i ( m i k i × t T i ) × En i (9)
Proof :

As shown [START_REF] Ramanathan | Overload management in real-time control applications using (m, k)firm guarantee[END_REF] that, if the mandatory jobs are determined according to (6), for the first p i jobs of τ i , there are l i (t) = mi ki × p i jobs that are mandatory. Therefore, the total mandatory energy load within interval [0,t] that has to be finished by time t, denoted by g(0,t), can be formulated as follows:

g(0, t) = i ( m i k i × t -D i T i + ) × En i (10) g(0, t) = i ( m i k i × t T i ) × En i (11)
Let E p (0,t) be the amount of energy that will be produced by the source between 0 and t and C is the energy storage unit (supercapacitor or battery) capacity. Definition 3: The total mandatory static slack energy on the time interval [0, t] is

SSE Ψ (0, t) = C + E p (0, t) -g(0, t) (12)
SSE Ψ (0,t) gives the largest energy that could be made available within [0,t] after executing mandatory jobs with release time at or after 0 and deadline at or before t.

U E Ψ ≤ 1 (15)
Proof : As proof of Lemma since

SSE Ψ (t 1 , t 2 ) ≤ 0 amounts to U E Ψ (t 1 , t 2 ) ≤ 1.
The necessary and sufficient schedulability conditions falls into two constraints which should be respected.

-Real-time constraints: For each processor P j the tasks set Ψ j assigned to P j should satisfy their deadlines. From equation ( 7) Time-feasibility: The set Ψ j is time feasible if and only if

i W i (0, t) = i ( m i k i × 1 T i ) × C i ≤ 1 (16)
-Energy constraints: Each processor P j must not, at any moment, lack energy to execute the tasks set assigned to processor P j . Energy feasibility: P j is energy feasible if and only if

U E Ψ ≤ 1 (17)
We give a necessary and sufficient condition for EDH-MK schedulability and feasibility. Theorem 3: Let Ψ be the mandatory job set according to their E-patterns. Also, let L represent either the ending point of the first busy period when scheduling only the mandatory jobs or the LCM of T(i ) , i = 0, ..., (n-1), whichever is smaller. Then, Ψ is schedulable with EDH-MK if and only if (iff) all the mandatory jobs arriving within [0, L] can meet their deadlines, i.e., i

W (0, t) ≤ 1andSSE Ψ ≤ 0 (18)
Proof : "If ": We suppose that constraint ( 17) is satisfied and Ψ is not schedulable by EDH-MK. Let us show a contradiction. First, we assume that Ψ is not schedulable by EDH-MK because of time starvation. of energy starvation. Lemma 2 states that there exists a time interval [t 0 , d 1 ) such that g(t 0 , d 1 ) > C + E p (t 0 , d 1 ) i.e., C + E p (t 0 , d 1 ) -g(t 0 , d 1 ) < 0. Thus, SSE Ψ < 0 and condition 17 in Theorem 2 is violated. "Only if ": Suppose that Ψ is feasible. Thus, Ψ is time-feasible and energy feasible.

From constraint (7) in theorem 2 and constraint [START_REF] Mccormick | Building parallel, embedded, and real-time applications with Ada[END_REF] in Lemma 2, it is the case that constraint (17) is satisfied.

Degradation mode level 2: Removal Algorithm

This is the case where the degradation mode level 1 is not feasible. Abandonable tasks are gradually eliminated by increasing importance. We sort all the abandonable tasks in an ascending order of degree of importance such that we can reject those with less importance one by one until establishing the system feasibility. Theorem 4: The set of tasks Ψ j assigned to processor P j is feasible under degradation mode level 2: Removal algorithm if and only if the set of non abandonable tasks set

Ψ na j U Ψ na j ≤ 1 and U e Ψ na j ≤ 1. (19)
Proof: Directly follows from the proof of the theorem 4 in [18]

Reconf-Algorithm

To adjust the framework to cope with any unpredictable external event such as new task arrivals, task removal, and increase-decrease power supply, we characterize a reconfiguration as any procedure that permits to reconfigure the system to be feasible, i.e., satisfying its real-time and energy constraints with the consideration of system performance optimization. We propose an approach with two successive adaptation strategies to reconfigure the system at run-time. The two adaptation strategies are performed in a hierarchical order as depicted in Fig. 1.

-Degradation mode level 1: EDH-MK Algorithm -Degradation mode level 2: Removal Algorithm

Functionalities

In this section we explain all functionalities of ARTE in details while showing their specifications and various characteristics regarding the problem of real-time scheduling in reconfigurable energy harvesting systems.

Task Set Generator

Task Model The current version proposes a task model according to the Liu and Layland task model with energy related parameters. All tasks are considered periodic and independent. Each task τ i is characterized by i) worst case execution time WCET C i , ii) worst case energy consumption WCEC E i , and iii) its period T i . It is considered that tasks have implicit deadlines, i.e., deadlines are equal to periods. In addition, each task is characterized by a degree of importance L i which define the functional and operational importance of the execution of the task vis-a-vis the application. Moreover, tasks are considered to be (m,k)-firm constrained deadlines. Tasks are classified into two categories: the first is firm task set with (1,1)-firm deadline constraints; the other is a set of soft tasks with (m,k)-soft deadline constraints. And a boolean A i to determine if a task is abandonnable or not. The used task sets can be loaded into the simulator either through the GUI by using a file browser or entering the parameters manually, or by using task set generator as depicted in Fig. 2 .

For the simulation results to be credible, the used task sets should be randomly generated and varied sufficiently. The current version includes by default a generator based on the UUniFast-Discard algorithm [START_REF] Bini | Measuring the performance of schedulability tests[END_REF] coupled with a hyper-period limitation technique [START_REF] Goossens | Limitation of the hyper-period in real-time periodic task set generation[END_REF] adapted to energy constraints. This algorithm generates task sets by dividing among them the CPU utilization (U = Ci Ti ) and the energy utilization (U e = Ei TiP r where Pr is the recharging function) chosen by the user. The idea behind the algorithm is to distribute the system's utilization on the tasks of the system. When we add the energy cost of tasks to the system, we end up with two parameters to vary and two conditions to satisfy. The algorithm in its current version distributes U and U e uniformly on the tasks then it finds the 2-tuple (C i , E i ) which satisfies all the conditions namely U i , U e and energy consumption constraints. The operation is repeated several times until the desired 2-tuple approaches the imposed conditions. Finally, the algorithm returns as a result a time and potentially energy feasible system. The (m,k)-firm parameters are randomly generated in the interval [START_REF] Gharbi | Functional and operational solutions for safety reconfigurable embedded control systems[END_REF][START_REF] Chandarli | YARTISS: A Generic, Modular and Energy-Aware Scheduling Simulator for Real-Time Multiprocessor Systems[END_REF]. We define three levels of importance then the degree of importance is randomly generated in the interval [START_REF] Gharbi | Functional and operational solutions for safety reconfigurable embedded control systems[END_REF][START_REF] Chen | Combining re-allocating and rescheduling for dynamic multi-robot task allocation[END_REF] where 1 is the higher importance level. The parameter A i is randomly generated in the interval [0, 1].

Reconfiguration Scenarios Generator

In order to represent as near as possible the real behavior of the physical reconfigurable energy harvesting real-time embedded systems we developed a reconfiguration scenarios generator tool. Through the GUI the user can use personalized reconfiguration scenarios by selecting the user personalization option, or using random reconfiguration scenarios by using the random task set generator as depicted in Fig. 3. The option user personalization offers to the user the possibility to generate reconfiguration scenarios that modify the applicative functions i.e., add-remove software tasks or increase-decrease the power supply of the system. For the simulation results to be credible, the used reconfiguration scenarios should be randomly generated and varied sufficiently. The current version includes by default a reconfiguration scenarios generator which offers the possibility to generate three kinds of reconfiguration scenarios: i) high dynamic system, ii) medium dynamic system, and iii) low dynamic system. The randomly reconfiguration scenarios algorithm calculates the number of jobs Njobs in the system upon one hyper-period.

-High dynamic system: the generator will add randomly n tasks where n is randomly in the interval [10%, 3%] from Njobs. -Medium dynamic system: the generator will add randomly where n is randomly in the interval [5%, 1%] from Njobs. -Low dynamic system: the generator will add randomly where n is randomly in the interval [ 1%, 0%] from Njobs.

Simulation Tool

The aim of this tool is to simulate the scheduling of a system according to the parameters and the assumptions of the user, mainly the task set, and the scheduling policy. But the purpose of ARTE is not only restricted to checking the feasibility of a given system but also to simulate and analyze the performance of the scheduling policies when unpredictable reconfiguration scenarios occur in the system. Through the main interface as depicted in Fig. 4 the user have the possibility to use task sets loaded into the simulator either through the GUI by using a file browser or entering the parameters manually, or by using task set generator. When the user creates a system the hyper period will be calculated automatically and displayed in the GUI.

For simulation the user choose the scheduling policy as well as the time interval for simulations. Two kinds of analyzes can be performed: scheduling simulation and test feasibility. The user can generate a set of reconfiguration scenarios to the system.

Case Study

This section presents a case study through which we can show the different features and functionalities implemented in ARTE as well as to explore the performance of the proposed EDH-MK algorithm and Reconf-Algorithm in order to keep feasible executions with graceful QoS after any external reconfiguration scenario that may evolve the system towards an unfeasible state. For this aim we create a new system where we generate randomly a task set with parameters in table 1. Initially, we have verified the task set feasibility using the simulator tool Fig. 5. Then, we have choose to generate a random high dynamic reconfiguration scenario using the reconfiguration scenario tool Fig. 6. Thereafter, the system evolves toward an unfeasible state Fig. 7.

In order to analyze the EDH scheduler performance we have run a simulation on 100 time units. The EDH scheduler provides 114 deadline miss Fig. 8. In order to analyze the EDHMK scheduler performance we have run a simulation on 100 time units. The EDHMK scheduler provides 16 deadline miss Fig. 9. 8 Future Works

The actual release offers many important features where the main purpose is to provide a simulator tool which provide the flexibility and the performance to deal with unpredictable reconfiguration scenarios. But, it has been made in a hurry and improvements are planned to address all features targeted by the proposed simulator tool.

The authors are now working on:

the development of an extension of the developed graphical user interface to facilitate the use of the simulator by a large number of users, and to provide our tool with a GUI to display the simulation results in an interactive and intuitive way. Three different views envisaged: a time chart, a processor view and an energy curve as well as a comparison results view permits to the user to see the simulation of selected scheduling policies, -implement other task models in the simulator, -implement other scheduling approaches in the simulator, such as the fixed priority approaches.

implement the multiprocessor platforms, and develop new scheduling techniques based on the migration of tasks between the different processors, -finally, we plan the use of a distributed system decentralizing the control, and more precisely the use of an multi-agent system MAS. We aim through the use of MAS to represent as near as possible the real behavior of the physical networked reconfigurable energy harvesting real-time embedded systems thanks to the developed simulator. Motivated by these considerations, we choose to deploy the intelligent agents to simulate the dynamic behavior of networked reconfigurable energy harvesting real-time embedded systems.

Conclusion

This report presents ARTE, a real-time scheduling simulator for reconfigurable energy harvesting real-time embedded systems. Presently, the ARTE simulator is able to simulate accurately the execution of task sets of a reconfigurable energy harvesting system. We briefly presented existing simulation tools. However, none of the aforementioned efforts respond to the new criteria of flexibility, agility and high-performance. Thus, there is a need to develop a new simulation tool that offers the ability to deal with the dynamic behavior of reconfigurable systems. We have detailed the different features provided by ARTE: 1) scheduling simulation, 2) feasibility tests, and 3) percentage of missed deadline measurement. We have described the three main tools

Fig. 1 .

 1 Fig. 1. The reconfiguration scenarios generator tool.

Fig. 2 .

 2 Fig. 2. The task set generator tool.

Fig. 3 .

 3 Fig. 3. The reconfiguration scenarios generator tool.

Fig. 4 .

 4 Fig. 4. The simulation tool.

Fig. 5 .

 5 Fig. 5. Random system generation.

Fig. 6 .

 6 Fig. 6. Random reconfiguration scenario generation.

Fig. 7 .

 7 Fig. 7. Test system feasibility.

Fig. 8 .

 8 Fig. 8. EDH simulation.

Fig. 9 .

 9 Fig. 9. EDH-MK simulation.

  At least one mandatory job with deadline after d 1 executes within [t 0 , d 1 ). Let t 2 be the latest time where a mandatory job, say τ 2 , with deadline after d 1 is executed. As d 1 is lower than d 2 and mandatory jobs are executed according to the earliest deadline rule in EDH-MK, we have r 2 < r 1 . At time t 2 , one of the following situations occurs. Case 2a: The processor is busy all the times in [t 0 , d 1 ). τ 2 is preempted by a higher priority job, say τ 3 , with d 3 ≤ d 1 . From rule 4.2 in[START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF], P SE Ψ (r 3 ) > 0 which implies that SE τ1 (r 3 ) > 0 and in consequence g(r 3 , d 1 ) < E(r 3 ) + E p (r 3 , d 1 ). All mandatory jobs that are executed within [r 3 , d 1 ) have release time at or after r 3 and deadline at or before d 1 . Consequently, the amount of energy they require is at most g(r 3 , d 1 ). That contradicts deadline violation and E(d 1 ) = 0. Case 2b: The processor is idle in [t 3 -1, t 3 ) with t 3 > t 2 and busy all the times in [t at t 3 because t 0 is the latest one. Furthermore, no mandatory job with deadline after d 1 is executed after t 2 and consequently after t 3 . In order not to waste energy, all the energy which arrives from the source is used to advance mandatory jobs with deadline after d 1 . The processor continuously commutes from active state to inactive state. The storage is maintained at maximum level until τ 1 releases. Consequently, we have E(r 1 ) = C. As τ 1 is feasible, g(r 3 , d 1 ) ≤ C + E p (r 1 , d 1 ). Thus, E(r 1 ) + E p (r 1 , d 1 ) ≥ g(r 1 , d 1 ). That contradicts deadline violation and E(d 1 ) = 0. Lemma 2: The set Ψ is energy-feasible if and only if Directly follows from Lemma 2. "Only If ": Since is energy-feasible, let us consider an energy-valid schedule produced within [0, d M ax ). The amount of energy demanded in each interval of time [t 1 , t 2 ), g(t 1 , t 2 ), is necessarily less than or equal to the actual energy available in [t 1 , t 2 ) given by E(t 1 ) + E p (t 1 , t 2 ). An upper bound on E(t 1 ) is the maximum storable energy at time t 1 , that is C. Consequently, g(t 1 , t 2 )is lower than or equal to C + E p (t 1 , t 2 ). This leads to ∀ (t 1 , t 2 ) ∈ [0, d M ax ), g(t 1 , t 2 ) ≤ C + E p (t 1 , t 2 ) i.e. SSE (t 1 , t 2 ) ≥ 0. Thus, SSE ≥ 0. Lemma 3: Ψ is energy-feasible if and only if

	SSE Ψ > 0	(14)
	Proof :
	"If ":

Definition 4: Let d be the deadline of the active job at current time t c . The preemption slack energy of a mandatory job set Ψ at t c is

P SE Ψ (t c ) = min tc≤ri≤di≤d SE τi (t c )

(13)

Lemma 1:

If

d 1 is missed in the EDH-MK schedule because of energy starvation there exists a time instant t such that g(t, d 1 ) > C + E p (t, d 1 ) and no schedule exists where d 1 and all earlier deadlines are met. Proof : Recall that we have to consider the energy starvation case where d 1 is missed with E(d 1 ) = 0. Let t 0 be the latest time before d 1 where a mandatory job with deadline after d 1 releases, no other mandatory job is ready just before t 0 and the energy storage unit is fully charged i.e. E(t 0 ) = C. The initialization time can be such time.

The processor is idle within [t 0 -1, t 0 ) since no mandatory jobs are ready. As no energy is wasted except when there are no ready jobs, the processor is busy at least from time t 0 to t 0 + 1. We consider two cases: Case 1: No mandatory job with deadline after d 1 executes within [t 0 , d 1 ). Consequently, all the mandatory jobs that execute within [t 0 , d 1 ). have release time at or after t 0 and deadline at or before d 1 . The amount of energy required by these mandatory jobs is g(t 0 , d 1 ). As is feasible, g(t 0 , d 1 ) is no more than the maximum storable energy plus all the incoming energy i.e., C + E p (t 0 , d 1 ). As E(t 0 ) = C, we conclude that all mandatory jobs ready within [t 0 , d 1 ) can be executed with no energy starvation which contradicts the deadline violation at d 1 with E(d 1 ) = 0. Case 2: 3 , d 1 ). The processor stops idle at time t 3 imperatively by rule 4.1

[START_REF] Chetto | Optimal scheduling for real-time jobs in energy harvesting computing systems[END_REF]

if E(t 3 ) = C. By hypothesis, there is no mandatory job waiting with deadline at or before d 1

Table 1 .

 1 Initial System Configuration.

	File Name	systemfile.txt
	Power Rate	10
	Processor Utilization 0.8
	Energy Utilization 0.8
	Number of tasks	10
	Emax	200
	Emin	2
	Battery Capacity	200
	Proc number	1

of ARTE: 1) random generator of task sets, 2) random generator of reconfiguration scenarios sets, and 3) a simulator tool. Finally, we presented some expanding features we will implement.


Son Quoc

Nguyen

Instability and friction Instabilité et frottement

Keywords: Friction, Unilateral contact, Instability, Stick-slip-separation waves Frottement, Contact unilatéral, Instabilité, Ondes adhérence-glissement-séparation

A review on the stability analysis of solids in unilateral and frictional contact is given. The presentation is focussed on the stability of an equilibrium position of an elastic solid in frictional contact with a fixed or moving obstacle. The problem of divergence instability and the obtention of a criterion of static stability are discussed first for the case of a fixed obstacle. The possibility of flutter instability is then considered for a steady sliding equilibrium with a moving obstacle. The steady sliding solution is generically unstable by flutter and leads to a dynamic response which can be chaotic or periodic. This dynamic response leads to the generation of stick-slip-separation waves on the contact surface in a similar way as Schallamach waves in statics. Illustrating examples and principal results recently obtained in the literature are reported. Some problems of frictioninduced vibration and noise emittence, such as brake squeal for example, can be interpreted in this spirit.

Résumé

On présente dans cette Note une synthèse des résultats de la littérature sur l'analyse de stabilité des solides sous contact unilatéral avec frottement de Coulomb. Le problème de contact frottant d'un solide élastique avec un obstacle fixe ou mobile est examiné. La possibilité d'instabilité par divergence et la recherche d'un critère de stabilité statique d'un équilibre sont examinées dans le cas d'un obstacle fixe. La possibilité d'instabilité par flottement est ensuite discutée pour un équilibre résultant d'un glissement stationnaire avec un obstacle mobile. La solution de glissement stationnaire est dynamiquement instable par flottement et conduit à une réponse dynamique qui peut être chaotique ou périodique. En particulier, la réponse dynamique génère des ondes d'adhérence-glissement-séparation sur la surface de contact d'une façon comparable aux ondes de Schallamach en statique. Des exemples et des résultats récents de la littérature sont rapportés. Quelques problèmes de vibration et d'émission acoustique induites par le frottement, comme le crissement des freins par exemple, peuvent être interprétés dans cet esprit.

Solids in frictional contact with an obstacle

Coulomb's law of dry friction

At a contact point of a solid with an obstacle, the relative velocity v = v sv o is by definition the difference between the velocities of the material points of the solid and of the obstacle in contact. The relative velocity v and reaction R can be decomposed into normal and tangential components

v = v T + v N n, R = T + Nn (1)
where n denotes the external normal vector to the obstacle at a contact point, v T is the sliding velocity vector, T the tangential reaction vector and N the normal reaction, N and v N are scalars. The unilateral condition of contact implies that the normal reaction must be non-negative N 0. When there is contact, Coulomb's law of dry friction states that the friction criterion must be satisfied and that the friction force must have the opposite direction to the sliding velocity,

φ = T -f N 0 and T = f N, T = -av T , a 0, if v T = 0 ( 2
)
where f denotes the friction coefficient. In particular, the dissipation by friction is -T • v T = f N v T . Coulomb's law has often been interpreted in the literature as a non-associated law since the velocity (v T , v N ) is not a normal to the domain of admissible forces. In particular, it has been discussed as a bi-potential law, cf. [START_REF] De Saxcé | A generalization of Fenchel's inequality and its application to constitutive laws[END_REF]. A more standard interpretation consists of saying that Coulomb's friction is a standard dissipative law with a state-dependent dissipation potential, cf. [START_REF] Moreau | On unilateral constraints, friction and plasticity[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF]. Indeed, normality law is satisfied by the flux v T and the force T since T and v T are related through a state-dependent dissipation potential D(v T , N)

T = -D, v T with D = f N v T ( 3
)
where D, v T is understood in the sense of sub-gradient. The set of admissible forces, which is a sphere of radius f N, depends on the present state through the present value of N .

Governing equations

The simple case of an elastic solid occupying a volume V in the undeformed position is considered. The solid is submitted to given forces and displacements r d = r d (λ(t)), u d = u d (λ(t)) respectively on the portions S r , S u of the boundary S, and λ(t) denotes a control parameter defining the loading history. On the complementary part S R , the solid may enter into contact with a moving obstacle h(m, t) < 0 and the non-penetration condition is

h x + u(x, t), t 0 ∀x ∈ S R (4)
The mechanical response of the solid are governed by unilateral contact conditions, Coulomb's law and classical equations of elastodynamics. In Lagrange description, the governing equations are

Div b = ρ ü, b = W, ∇u ∀x ∈ V b • n s = r d ∀x ∈ S r , b• n s = Nn + T ∀x ∈ S R , u= u d ∀x ∈ S u ( 5
)
where W (∇u) and b denote respectively the elastic energy per unit volume and the unsymmetric Piola-Lagrange's stress. In particular, the unknowns (u, N) must satisfy the local equations

u = u d ∀x ∈ S u , h 0, N 0, Nh = 0 ∀x ∈ S R (6)
and the variational inequality where U ca denotes the set of kinematically admissible rates

V (∇u * -∇ u) : W, ∇u (∇u) dV -S r r d • (u * -u) da + V ρ ü • (u * -u) dV -S R (v * N -v N )N da + S R f N( v * T -v T ) dS 0 ∀u * ∈ U ca (7)
U ca = u * | u * = ud on S u ( 8
)
and v * and v are the relative rates

v N = n • u -v oN , v T = u -(n • u)n -v oT , v * N = n • u * -v oN , v * T = u * -(n • u * )n -v oT , (9)
with n = h, m / h, m and v oN = -h, t / h, m . The system of solid in frictional contact under loads is associated with an energy potential and a dissipation potential:

E(u, λ, µ) = V W (∇u) dV -S r r d (λ) • u da -S R µh(x + u) da D(N, v T ) = S R f N v T da (10)
where µ denotes the Lagrange multipler associated with the constraints (4) and N = µ h, m . The variational inequality [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF] can be condensed as

(J + E, u ) • (u * -u) + D(N, v * T ) -D(N, v T ) 0 ∀u * ∈ U ca ( 11
)
where J denotes the inertial terms. Some regularizations of the frictional contact problem have been proposed in the literature:

-Non-local Coulomb's law, proposed by Duvaut [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF], in which the local normal reaction N is replaced by its mean value N on an elementary representative surface S re

N = 1 S re S re

N(x) ds

-Normal compliance law, discussed by Oden and Martins [START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF], Kikuchi and Oden [START_REF] Kikuchi | Contact Problem in Elasticity: A Study of Variational Inequalities and Finite Element Methods[END_REF], Andersson [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF][START_REF] Andersson | Existence and uniqueness for quasistatic contact problems with friction[END_REF], Klarbring et al. [START_REF] Klarbring | A global existence result for the quasistatic frictional contact problem with normal compliance[END_REF], . . .. It consists of replacing Signorini's relations of unilateral contact by a relationship giving the normal reaction as a function of the gap h(x + u(x)). For example, nonlinear springs of energy ϕ(h) per unit surface may be added to the system where ϕ(h) is a regular function permitting an approximation of Signorini's conditions. In particular, with ϕ(h) = k 2 h 2 -, then the normal compliance law consists of writing that N = k h, m h - General discussions on the existence of a solution of the quasi-static problem in small deformation have been given in the literature, when there is regularization by normal compliance cf. [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF][START_REF] Klarbring | A global existence result for the quasistatic frictional contact problem with normal compliance[END_REF] and by nonlocal Coulomb's law cf. [START_REF] Cocu | Analysis of an incremental formulation for frictional contact problems[END_REF]. In these cases, it has been proved that the existence of at least one solution is ensured when the friction coefficient is small enough.

Divergence instability of an equilibrium

There is flutter or divergence instability if, under disturbances, the system will leave the equilibrium position with or without growing oscillations. As usual, the possibility of divergence instability can be discussed in a purely static approach [START_REF] Hill | A general theory of uniqueness and stability in elastic/plastic solids[END_REF][START_REF] Petryk | A consistent approach to defining stability of plastic deformed processes[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF]. The quasi-static problem has been much discussed, cf. [START_REF] Kikuchi | Contact Problem in Elasticity: A Study of Variational Inequalities and Finite Element Methods[END_REF][START_REF] Cocu | Analysis of an incremental formulation for frictional contact problems[END_REF][START_REF] Klarbring | Derivation and analysis of rate boundary value problems of frictional contact[END_REF][START_REF] Andersson | Existence and uniqueness for quasistatic contact problems with friction[END_REF][START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF][START_REF] Klarbring | Stability and critical points in large displacement frictionless contact problems[END_REF]. For the sake of simplicity, the particular case of fixed obstacles is considered here. The obstacle is given by the time-independent domain h(m) 0. Normal and tangential relative velocities are v N = uN and v T = uT . The governing equations follow from ( 6), [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF] by deleting the inertia terms. If the rigid motion of the solid is not excluded by the implied displacements, it is well known that the existence of an equilibrium position under applied loads is not always ensured.

Rate problem in quasi-statics

The analysis of the static stability of an equilibrium follows from the consideration of the rate response as in the theory of elastic and plastic buckling of solids. With the following notation

V ad =    u * u * ∈ U ca and ∀x ∈ S Rc , u * N = 0 if N > 0 and u * N 0 if N = 0, u * T = 0 if φ < 0 and u * T = -bT , b 0 if φ = 0, N > 0    ( 12
)
it is clear that u ∈ V ad which is the set of admissible rates. The virtual work equation holds in rate form for all

u * ∈ V ad V ∇(u * -u) : L : ∇ u dV - S r ṙ • (u * -u) da - S R Ṙ • (u * -u) da = 0
where L = W, ∇u∇u denotes the elastic modulus. Since Ṫ • (u

* -u) = Ṫ • (u * T -uT ) + Ṫ • (u * N -uN
)n and Ṫ • n = 0 when T = 0, the last term can also be written as

-Ṙ •(u * -u) = -N(u * T -uT )•C • uT -Ṅ(v * N -v N )-Ṫ •(v * T -v T )
where C denotes the curvature tensor of the obstacle at contact point. Taking account of the fact that:

Ṫ (v T -v * T ) + f Ṅ v T -v * T 0 ( 13
)
which is a consequence of Coulomb's law, cf. [START_REF] Klarbring | Derivation and analysis of rate boundary value problems of frictional contact[END_REF][START_REF] Chateau | Buckling of elastic structures in unilateral contact with or without friction[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF], the following statement is obtained

The rate u ∈ V ad and satisfies the variational inequality

V (∇u * -∇ u) : L : ∇ u dV - S r ṙd • (u * -u) da - S Rc N(u * T -uT ) • C • uT da - S Rc (u * N -uN ) Ṅ( u) da + S Rc f Ṅ( u) u * T -uT da 0 ∀u * ∈ V ad ( 14
)
where

Ṅ( u) = d dt (n • b • n s ) = (n s ⊗ n) : L : ∇ u -R • C • u (15)
If a regularization of unilateral contact by normal compliances is introduced, there are no more unilateral constraints. For some particular cases of elastic structures in frictional contact with normal compliances, the framework of standard dissipative systems can be directly applied, cf. [START_REF] Biot | Mechanics of Incremental Deformation[END_REF][START_REF] Nguyen | Bifurcation and stability in dissipative media (plasticity, friction, fracture)[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF][START_REF] Mroz | Contact friction models and stability problems[END_REF]. Such a system is defined by an energy potential E(q, λ) and a dissipation potential D(q, q) which is convex and positively homogenous of degree 1 with respect to q. The quasi-static evolution is governed by

E, q +D, q = 0 ( 16
)
where D, q denotes the sub-gradient of D with respect to q. The associated rate equations are

E, qq • q + E, qλ λ + D, q q • q = 0 (17)
These equations can also be written in variational form as E, q • (q * -q) + D(q, q * ) -D(q, q) = 0 ∀q * (q * -q) • (E, qq • q + E, qλ λ) + q • D, q (q, q * ) -D, q (q, q) = 0 ∀q * ∈ V ad Rate equations ( 14) are written in the same spirit, with some additional difficulties due to the presence of constraints and to the state-dependent expressions of tangential and normal components in terms of q.

Static stability criterion

It is assumed that under the action of a given dead load, an equilibrium position of a solid in frictional contact with a fixed obstacle exists. The stability of this equilibrium is the subject of interest here.

The classical concept of static stability consists of defining the stability as the absence of additional displacement when the load does not vary λ = 0. The equilibrium is thus statically stable if the rate problem admits only the trivial solution u = 0. The following statement is then straightforward [START_REF] Chateau | Buckling of elastic structures in unilateral contact with or without friction[END_REF][START_REF] Nguyen | Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale[END_REF][START_REF] Nguyen | Bifurcation and stability in dissipative media (plasticity, friction, fracture)[END_REF][START_REF] Klarbring | Stability and critical points in large displacement frictionless contact problems[END_REF][START_REF] Martins | Some notes on friction and instabilities[END_REF][START_REF] Mroz | Contact friction models and stability problems[END_REF]:

The condition of positivity

I(u * ) = V ∇u * : L : ∇u * dV + S Rc Nu * T • C • u * T da + S Rc f Ṅ(u * ) u * T da > 0 for all u * = 0 ∈ V 0 ad ( 18
)
is a criterion of stability since it ensures the static stability of the considered equilibrium.

The significance of this criterion can be better understood from its energy interpretation in the same spirit as Hill's criterion of stability in plasticity, cf. [START_REF] Hill | A general theory of uniqueness and stability in elastic/plastic solids[END_REF][START_REF] Petryk | A consistent approach to defining stability of plastic deformed processes[END_REF][START_REF] Chateau | Buckling of elastic structures in unilateral contact with or without friction[END_REF][START_REF] Nguyen | Bifurcation and stability in dissipative media (plasticity, friction, fracture)[END_REF]. A perturbation of the equilibrium by perturbation forces is introduced for t 0. Let u(t) denotes the perturbed motion starting from the equilibrium u(0) = u eq . The energy balance at time t is

W per (t) = V W ∇u(t) -W (∇u eq ) dV - S r r d (λ) • u(t) -u eq da - t 0 S R R(τ ) • u da dt + C(t)
where W per (t) is the energy supplied by perturbation forces and C(t) is the kinetic energy of the solid at time t. At the begining of the perturbation, i.e., for small t, if the perturbed motion and the energy balance can be expanded as

u(t) = u eq + u 1 t + u 2 t 2 2 + • • • , W per (t) = W 0 + W 1 t + W 2 t 2 2 + • • • + C(t)
then, the following expressions are obtained W 0 = 0, W 1 = 0, W 2 = I (u 1 ), cf. [START_REF] Chateau | Buckling of elastic structures in unilateral contact with or without friction[END_REF][START_REF] Nguyen | Bifurcation and stability in dissipative media (plasticity, friction, fracture)[END_REF]. Thus, condition [START_REF] Mroz | Contact friction models and stability problems[END_REF] implies that the external world must supply energy at early times in order to perturbe the system from equilibrium. This criterion ensures a certain stability in the energy sense, also called directional stability [START_REF] Petryk | A consistent approach to defining stability of plastic deformed processes[END_REF]. Its violation in a direction of displacement leads to a divergence instability of the considered equilibrium since the system will leave this position with a growing kinetic energy.

Flutter instability of the steady sliding equilibrium

For the sake of clarity, the stability of the steady sliding equilibrium of an elastic solid in contact with a moving rigid half-space, in translation motion at a constant velocity w parallel to the free surface, is considered here in small deformation.

Steady sliding equilibrium

The steady sliding equilibrium u of the solid must satisfy

V ∇δu : L : ∇u dV - S r r d • δu da - S R (δu N N + f Nτ • δu T ) da = 0
This equation leads formally to a system of reduced equations of the form

N = k NN [u N ] + k NT [u T ] + N d , T = f Nτ = k T N [u N ] + k T T [u T ] + T d
The principal unknown u N must satisfy

u N = A[N] + B, N 0, u N 0, N • u N = 0 ( 19
)
with

A = k NN -k NT k -1 T T k T N -1 (I -f k NT h T N ) h T N [N] = k -1 T T [Nτ ], B = k NN -k NT k -1 T T k T N -1 N d -k NT k -1 T T T d ( 20
)
It is clear that the linear operator A is not symmetric if f = 0:

N * , A[N] = S R N * (x)A[N](x) dS = N, A[N * ] (21)
Thus, a linear complementary problem (LCP) must be considered. In particular, the existence and uniqueness of a steady sliding solution are ensured if A is positive-definite or P -positive [START_REF] Cottle | The Linear Complementarity Problem[END_REF][START_REF] Klarbring | A mathematical programming approach to three-dimensional contact problems with friction[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF].

Instability of the steady sliding equilibrium

The stability of the steady sliding position can be obtained from the study of small perturbed motions. However, the equations of motion cannot be linearized without the assumption of effective contact. Indeed, in the presence of a loose contact, a small perturbed motion is not necessarily governed by linear equations because of the possibility of separation. Under the assumption of an effective contact, if the sliding speed is never zero, the dynamic equations can be written as

V δu • ρ ü dV + V ∇δu : L : ∇u dV + S R Nδu N dS + S R f N uT -w uT -w • δu T dS = 0 ∀δu, δN (22)
The linearization is then possible for sliding motions. The nature of this particular problem can be better understood in the discretized form. After discretization, the equations of motion are

   U N = 0 M Y Y Ÿ + K Y Y Y = f Φ( Ẏ )N + F Y M NY Ÿ + K NY Y = N + F N ( 23
)
where u = (U N , Y ) and Φ( Ẏ ) is a matrix dependent on the direction of sliding. The linearized equations for sliding motions are

   U * N = 0 M Y Y Ÿ * + K Y Y Y * = f Φ N * + f Φ Ẏ Ẏ * M NY Ÿ * + K NY Y * = N * (24)
The general expression u * = e st U with U = (U N = 0, X) then leads to

s 2 M Y Y X + K Y Y X = f sΦ Ẏ X + f ΦN , s 2 M NY X + K NY X = N (25)
i.e., to the generalized eigenvalue problem

s 2 (M Y Y -f ΦM NY )X -sf Φ Ẏ X + (K Y Y -f ΦK NY )X = 0 (26)
Thus, the considered equilibrium is asymptotically stable (with respect to sliding motions) if (s) < 0 for all s and unstable if there exists at least one value s such that (s) > 0. This generalized eigenvalue problem can be written as (s 2 M + s C + K)X = 0 with non-symmetric matrices M, K and complex eigenvalues and eigenvectors. This analysis leads to the definition of a critical value f d 0 such that the considered equilibrium is unstable when f > f d .

Example on the sliding contact of two elastic layers

The simple example of the frictional contact of two elastic infinite layers is considered here as an illustrating example. This problem was discussed analytically by Adams [START_REF] Adams | Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction[END_REF], by Martins et al. [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF] and by Renardy [START_REF] Renardy | Ill-posedness at the boundary for elastic solids sliding under Coulomb friction[END_REF].

The contact in plane strain with friction of two infinite elastic layers, of thickness h and h * respectively as shown in Fig. 2, is considered. The lower face of the bottom layer is maintained fixed in the axes Oxyz. The upper face of the top layer, assumed to be in translation of velocity w in the direction Ox, is compressed to the bottom layer by an implied displacement δ < 0. At the interface y = 0, the contact is assumed to obey Coulomb's law of friction with a constant friction coefficient. The celerities of dilatation and shear waves are first introduced:

c 1 = λ + 2µ ρ , c 2 = µ ρ , τ = c 2 c 1
for the top layer and for the bottom layer (superscript *), to write the governing equations for the displacements of the top layer wte 1 + u(xwt, y, t) and of the bottom layer u * (x, y, t) under the form:

                   1 -τ 2 w c 2 2 u x,xx + τ 2 u x,yy + 1 -τ 2 u y,xy = τ 2 u x,tt -2 w c 2 u x,tx u y,yy + τ 2 1 - w c 2 2 u y,xx + 1 -τ 2 u x,xy = τ 2 u y,tt -2 w c 2 u y,tx u * x,xx + τ * 2 u * x,yy + 1 -τ * 2 u * y,xy = τ * 2 u * x,tt u * y,yy + τ * 2 u * y,xx + 1 -τ * 2 u * x,xy = τ * 2 u * y,tt (27)
Boundary and interface conditions are u(xwt, h, t) = 0, u * (x, -h * , t) = 0, u y (xwt, 0, t) = u * y (x, 0, t) σ yy (x, 0, t) = σ * yy (x, 0, t), σ xy (x, 0, t) = σ * xy (x, 0, t), f σ yy (x, 0, t) = -σ xy (x, 0, t) The stability of the steady sliding solution can be obtained by a linearization of the dynamic equation under the assumption of sliding perturbed motions near the steady sliding state. These motions are searched for under the form of slip waves of wave-length L = 1/k: u(xwt, y, t) = e 2πkct e 2ikπ(x-wt) X(y), u * (x, y, t) = e 2πkct e 2ikπx X * (y)

The condition of existence of non null displacement modes (X, X * ) requires that the pair c and k must be a root of the following equation:

F (c, k) = ρc 2 2 A(p, q, kh) iB(p * , q * , kh * ) + f C(p * , q * , kh * ) + ρ * c * 2 2 A(p * , q * , kh * ) iB(p, q, kh) -f C(p, q, kh) = 0 ( 28
)
where p, q, A, B, C are appropriate functions [START_REF] Moirot | Some problems of friction-induced vibrations and instabilities[END_REF]

                         p 2 = 1 + c -iw c 2 2 , q 2 = 1 + τ 2 c -iw c 2 2 A(p, q, kh) = -4pq 1 + p 2 + pq 4 + 1 + p 2 2 cosh(2πpkh) cosh(2πqkh) -1 + p 2 2 + 4p 2 q 2 sinh(2πpkh) sinh(2πqkh) B(p, q, kh) = q 1 -p 2 sinh(2πpkh) cosh(2πqkh) -pq cosh(2πpkh) sinh(2πqkh) C(p, q, kh) = pq 3 + p 2 -pq 3 + p 2 cosh(2πpkh) cosh(2πqkh) + 2p 2 q 2 + 1 + p 2 sinh (2πpkh) sinh (2πqkh)
The case of a rigid top layer is obtained when c 2 ⇒ +∞

F (c, k) = iB(p * q * , kh * ) + f C(p * , q * , kh * ) = 0 (29)
For an elastic half-plane compressed into a moving rigid half-plane, cf. Martins et al. [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF], the results are:

F (c) = iq * 1 -p * 2 + f 1 + p * 2 -2p * q * = 0 ( 30
)
In the case of two elastic half-planes, cf. Adams [START_REF] Adams | Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction[END_REF], this equation can be written as:

F (c) = ρc 2 2 1 + p 2 2 -4pq iq * 1 -p * 2 + f 1 + p * 2 -2p * q * + ρ * c * 2 2 1 + p * 2 2 -4p * q * iq 1 -p 2 -f 1 + p 2 -2pq = 0 ( 31
)
It has been established in each case that there exists a critical value f d 0 such that the steady sliding solution is unstable for f f d . For example, f d = 0 occurs for the system of two layers of finite depths while the possibility f d > 0 may happen in the sliding contact of half-spaces, cf. [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF][START_REF] Adams | Dynamic instabilities in the sliding of two layered elastic half-spaces[END_REF][START_REF] Martins | Some notes on friction and instabilities[END_REF][START_REF] Moirot | Etude de la stabilité d'un équilibre en présence du frottement de Coulomb[END_REF][START_REF] Ranjith | Slip dynamics at an interface between dissimilar materials[END_REF].

Stick-slip-separation waves

The fact that the steady sliding solution is unstable leads to the study of possible dynamic bifurcations of the sliding contact of solids. In the spirit of Hopf bifurcation [START_REF] Troger | Nonlinear Stability and Bifurcation Theory[END_REF], a periodic response can be expected as an alternative stable response. This possibility has been explored in the example of two coaxial cylinders [START_REF] Moirot | Etude de la stabilité d'un équilibre en présence du frottement de Coulomb[END_REF][START_REF] Moirot | An example of stick-slip waves[END_REF][START_REF] Oueslati | Transition vers une onde glissement-adhérence-décollement sous contact frottant de Coulomb[END_REF].

The mechanical response in plane strain of a brake-like system composed of an elastic tube, of internal radius R and external radius R * , in frictional contact on its inner surface with a rotating rigid cylinder of radius R + and of angular rotation Ω has been discussed, cf. Fig. 3. The mismatch 0 is a load parameter controlling the normal contact pressures. This model problem enables us to exhibit the existence of nontrivial periodic solutions in the form of stick-slip or stick-slip-separation waves propagating on the contact surface. The governing equations of the system follow from the kinetic relations, the fundamental law, the linear elastic constitutive equations, and the boundary unilateral contact conditions with Coulomb's friction:

                           ε = (∇ ū) s Div σ = γ ü σ = ν (1 + ν)(1 -2ν) Tr(ε)I + 1 1 + ν ε ūr (ξ, θ, t) = ūθ (ξ, θ, t) = 0 σrr (1, θ, t) = -p(θ, t), σrθ (1, θ, t) = -q(θ, t) ūr δ, p 0, p(ū r -δ) = 0 |q| fp, q(1 -uθ ) -fp|1 -uθ | = 0 (32)
where non-dimensional variables are introduced,

ū = u R , σ = σ E , r = r R , γ = ρR 2 Ω 2 E , ξ = R * R , δ = R , t = Ωt, u = d ū dt
The steady sliding solution is given by,

       ūer = δ 1 ξ 2 -1 ξ 2 r -r , ūeθ = δf 1 ξ 2 -1 ξ 2 r -r 1 + 1 ξ 2 (1 -2ν) p e = δ 1 ξ 2 -1 1 1 + ν ξ 2 + 1 1 -2ν > 0, q e = fp e (33)
Since closed form dynamical solutions cannot be generated, two complementary approaches has been followed. The first approach is semi-analytical after a reduction to a simpler system of equations. The second approach consists of a numerical simulation by the finite element method and appropriate time-integrations. An interesting simplification to the problem is obtained when the displacement is sought in the form

ūr = U(θ, t)F (r), ūθ = V (θ, t)F (r), F (r) = 1 ξ 2 -1 ξ 2 r -r (34)
In this approximation, the following local equations are obtained from the virtual work equation

         Ü -bU -dV + gU = P V -aV + dU + hV = Q P 0, U -δ 0, P(U -δ) = 0 |Q| f P, Q(1 -V ) -f P |1 -V | = 0 (35)
where denotes the derivative with respect to θ and a, b, g, h, d are material and geometry constants. All of them are positive except for the coupling coefficient d. Finally, only the non-dimensional displacements on the contact surface U(θ, t) and V (θ, t) and the non-dimensional reactions P (θ, t) and Q(θ, t) remain as unknowns in the reduced equations. The steady sliding solution, given by U e = δ, V e = δfg/ h, P = P e and Q e = f P e , is unstable for the reduced system. Indeed, under the assumption of sliding motions, a small perturbed motion is described by U = U e , V = V e + V * , P = P e + P * and

Q = Q e + Q * . It follows that V * -aV * + f dV * + hV * = 0 (36)
If a general solution is sought in the form V * = e s t e ikθ , then -s 2 = ak 2 + h + ikf d. When f = 0, it follows that s = ±iω k with ω 2 k = ak 2 + h. Thus two harmonic waves propagating in opposite senses of the form cos(kθ ± ω k t + ϕ) are obtained as in classical elasticity. When f > 0 and d > 0, then s = ±(s rk + is ik ), s rk > 0, s ik < 0, thus a general solution of the difference V * of the form V * = e ±s rk t cos(kθ ± s ik t + ϕ) is obtained and represents two waves propagating in opposite senses: an exploding wave in the sense of the implied rotation, and a damping wave propagating in the opposite direction. If f > 0 and d < 0, the exploding wave propagates in the opposite sense since the previous expression of s is still valid with s rk > 0 and s ik > 0.

It is expected that in some particular situations, there is a dynamic bifurcation of Poincaré-Andronov-Hopf's type. This means that the perturbed motion may evolve to a periodic response. This transition has been observed numerically in many examples of the literature, cf., for example, [START_REF] Oestreich | Bifurcation and stability analysis for a non-smooth friction oscillator[END_REF] or [START_REF] Vola | Friction and instability of steady sliding squeal of a glass/rubber contact[END_REF]. To explore this idea, a periodic solution has been sought in the form of a wave propagating at constant velocity:

U = U(φ), V = V (φ), φ = θ -ct (37)
where c is the non-dimensional wave velocity, U and V are periodic functions of period T = 2π/k. The physical velocity of the wave is thus c = | c|RΩ and the associated dynamic response is periodic of frequency | c|kΩ. The propagation occurs in the sense of the rotation when c > 0. According to the regime of contact, a slip wave, a stick-slip wave, a slip-separation wave or a stick-slip-separation wave can be discussed. The governing equations of such a wave follow from [START_REF] Oueslati | Transition vers une onde glissement-adhérence-décollement sous contact frottant de Coulomb[END_REF]:

         c2 -b U -dV + gU = P c2 -a V + dU + hV = Q P 0, U δ, P (U -δ) = 0 |Q| f P, Q(1 -V ) -f P |1 -V | = 0 (38)
The existence of stick-slip waves is obtained when the load is sufficiently strong or when the rotation is slow. For example, for ξ = 1.25 and f = 1, stick-positive slip solutions are obtained for 8 k 12. It is found that c must have the sign of d. These waves propagate in the sense of the previous exploding perturbed motions, thus opposite to the rotation of the cylinder when d < 0, with a frequency and a celerity independent of the rotation velocity Ω. The celerity is close to the celerities of dilatation and shear waves in the solid while the frequency is inversely proportional to the radius R. 4. For ξ = 1.15, for example, d is positive and the propagation goes in the rotation direction. The limiting case ξ ⇒ 1 can be interpreted as the sliding motion of a rigid plate on an elastic layer or of a rigid half-space on an elastic half-space [START_REF] Adams | Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction[END_REF][START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF]. The obtained solution is a wave with an oscillation about the steady sliding response. The amplitude of the wave is linearly proportional to the rotation Ω. It also increases with the friction coefficient f and decreases with the mismatch. Thus, for vanishing rotations, the steady sliding solution is recovered as the limit of the dynamic response. The stick-slip solution can no longer be available if the rotation is strong enough since the associated pressure may become negative. In the same spirit, for a small mismatch, the pressure may become negative under the assumption of a stick-slip regime everywhere. This means that the possibility of separation is not excluded when the mismatch is not strong enough or if the rotation or the friction coefficient is sufficiently high. A numerical simulation with an explicit scheme using Lagrange multipliers [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF][START_REF] Baillet | Tribologie de l'interface fibre/matrice. Approche théorique et expérimentale[END_REF] has been performed. The case ξ = 2 and f = 0.3 has been considered. Starting from a motionless initial state, the mismatch displacement is then increased linearly from 0 to its final value. A cyclic limit response is then obtained for large time. The numerical simulation leads to a stick-slip wave in mode 4 without forcing and the obtained response is close to the analytical solution of the reduced approach.

It was also checked that a stick-slip-separation wave is effectively obtained when the mismatch is small enough or when the friction is high enough. For example, when Ω = 50 rad/s, δ = 0.001 and f = 0.7, the limit cycle results as a stick-slip-separation wave. The result for radial displacements is shown in Fig. 5.

In fact, it is well known that a periodic response does not systematically result from the flutter instability of the steady sliding solution. It has been observed in various examples of discrete or continuous systems that the response may be quasi-periodic or non-periodic or chaotic [START_REF] Van Der Pol | On relaxation-oscillations[END_REF][START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF][START_REF] Popp | Stick-slip vibrations and chaos[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF]. Periodic responses prevail in the example of coaxial cylinders because of the special geometry of the system. Periodic solutions under the form of stick-slip waves have been also obtained by Adams in the sliding contact of two elastic half-spaces [START_REF] Adams | Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip[END_REF]. The generation of dynamic stick-slip-separation waves on the contact surface can be compared to Shallamach waves in the sliding contact of rubber in statics, cf. [START_REF] Zharii | Frictional contact between the surface wave and a rigid strip[END_REF][START_REF] Baillet | Tribologie de l'interface fibre/matrice. Approche théorique et expérimentale[END_REF][START_REF] Moirot | Etude de la stabilité d'un équilibre en présence du frottement de Coulomb[END_REF][START_REF] Martins | Dynamic stability of finite dimensional linear elastic system with unilateral contact and Coulomb's friction[END_REF][START_REF] Oueslati | Transition vers une onde glissement-adhérence-décollement sous contact frottant de Coulomb[END_REF][START_REF] Vola | Consistent time discretization for a dynamical frictional contact problem and complementary techniques[END_REF][START_REF] Renard | Modélisation des instabilités liées au frottement sec des solides, aspects théoriques et numériques[END_REF][START_REF] Adams | Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip[END_REF][START_REF] Vola | Friction and instability of steady sliding squeal of a glass/rubber contact[END_REF][START_REF] Martins | Some notes on friction and instabilities[END_REF][START_REF] Raous | Numerical characterization and computation of dynamic instabilities for frictional contact problems[END_REF].

Friction-induced vibrations and noises

It is well known that the presence of friction induces mechanical vibrations and noise emittences in the sliding contact of solids. For example, the creaking noise of a door, the unsteady motion with fits and starts of a windscreenwiper can be interpreted as stick-slip motions resulting from the instability of the steady sliding solution. In particular, the phenomena of squeal [START_REF] Crolla | Brake noise and vibration: state of art[END_REF] have been interpreted in the literature in this way. The squeals of band brakes in washing machines have been discussed in [START_REF] Nakai | Band brake squeal[END_REF]. The brake squeals of an automotive disk brake has been examined in [START_REF] Moirot | Etude de la stabilité d'un équilibre en présence du frottement de Coulomb[END_REF][START_REF] Moirot | Brake squeal: a problem of flutter instability of the steady sliding solution?[END_REF], cf. Fig. 6. The squeal of a system glass-rubber in finite deformation is considered in [START_REF] Vola | Friction and instability of steady sliding squeal of a glass/rubber contact[END_REF]. Fig. 6. Some unstable modes of the system pad-disk in an automotive disk brake [START_REF] Moirot | Etude de la stabilité d'un équilibre en présence du frottement de Coulomb[END_REF][START_REF] Moirot | Some problems of friction-induced vibrations and instabilities[END_REF]

Fig. 1 .

 1 Fig. 1. A solid in unilateral contact with an obstacle.

Fig. 2 .

 2 Fig. 2. Sliding contact of two elastic layers.

Fig. 3 .

 3 Fig. 3. The problem of coaxial cylinders in frictional contact.

  For example, for √ E/ρ = 1000 m/s, ξ = 1.25, f = 1, R = 1 m and Ω = 100 rad/s, the results obtained concerning the mode-8 wave are Ψ = 0.839, c = 1255 m/s and the associated frequency is 10045 Hz. If ξ = 2, f = 0.3, R = 0.5 m, Ω = 10 rad/s, a frequency 8240 Hz and a celerity 1030 m/s are obtained for k = 4 as shown in Fig.

Fig. 4 .

 4 Fig. 4. Semi-analytical approach: an example of stick-slip wave in mode 4 with ξ = 2, f = 0.3, Ω = 10 rad/s, R = 0.25 m and δ = 0.005. It is found that Ψ = 0.644, c = 1030 m/s. Phase diagram and variations of V /V e , P /P e and Q/Q e in [0, 2π/k].

Fig. 5 .

 5 Fig. 5. A mode-4 stick-slip-separation wave, obtained by numerical simulations for ξ = 2, Ω = 50 rad/s, f = 0.7, δ = 0.001. The isovalues of the radial displacement (mm): separation, slip and stick nodes on the contact surface are given respectively in red, green, blue.