G Morel

Control, C E Pereira

Historical Survey and Emerging Challenges of Manufacturing Automation Modeling and Control: a Systems Architecting perspective

Keywords: Holistic Control, Logistics, Knowledge Orchestration, Systems Engineering, System of Systems Control

 highlights the impact of digitalization both on isolated machines and devices, and on the architecting of large-scale manufacturing and logistics systems. To distribute the holistic control over related components, two main paradigms have been followed for the engineering of those System of Systems (SoS): (1) technology-enhanced, and (2) bio-inspired. In both, the cognitive orchestration of the knowledge and skills involved in a system project remains challenging for the interdisciplinary togetherness and harmony required beyond the disciplinary boundaries. Finally, lessons learned, as well as acquired knowledge through this innovative history are addressed with several open questions and emerging challenges related to the cyber-physical systems evolution.

Introduction and Context

A historical survey of the "Factory of Future [1]" paradigm over time points out a recurrent search of "system in the loop" manufacturing enterprise control aligned with the increasing "hardware-software in the loop" capabilities of automation technology. This systemic perspective is characterizing the integration-in-manufacturing and the intelligence-in-manufacturing paradigms which intricately have been sharing this common thread over the past forty years under industry-led initiatives [START_REF] Boland | Roadmap of the Network of Excellence in Intelligent Control and Integrated Manufacturing[END_REF]. Two different architectural patterns of control can be identified in meeting the systemic purpose of product-service expected symbiosis:

• Centralized/hierarchical approaches, resulting in the well-established technology-enhanced Computer Integrated

Manufacturing for Enterprise-Control System Integration issues; • Distributed/heterarchical approaches, resulting in certain promising biology-inspired generation of Intelligent Manufacturing Systems for Enterprise-Control System Openness issues.

Large-scale manufacturing and logistics systems exhibit properties (Figure 1, left) characterizing System of Systems (SoS) behaviors interoperating both as parts and wholes for a period of time, while sharing a common purpose. The compelling challenges (Figure 2, [START_REF] Monostori | Cooperative control in production and logistics[END_REF] raise issues on advantages and disadvantages relating to the balance of these characteristics (Figure 1, right) --between automation and Human in the control loop (Figure 5, chapter 10, Millot et al., Boy, 2011), and between system architecting alternatives:

• Beyond closed change, suggesting control and Integration (manufacturing system and its multiple related typology);

• To meet open-ended change, suggesting openness and distribution (extended production system);

• Under contained change, suggesting engineering management of both (adapted from [START_REF] Brier | Problem frames for socio-technical systems: predictability and change[END_REF].

The long-time search of manufacturing systems automation and modeling at large scale has been recently termed "Design for Resilience" by Reyes Levalle and Nof (2017), as well as "Design for the Unexpected" by [START_REF] Valckenaers | Design for the Unexpected: From Holonic Manufacturing Systems towards a Human Mechatronics Society[END_REF].

The objective is to better prepare effective responses ahead of time (by design) to future uncertainty, which too often is revealed a posteriori in real operation by systemic failures [START_REF] Boardman | Systemic thinking: building maps for worlds of systems[END_REF].

Architectural patterns are discussed in more detail in sections 2 and 3. The reality/knowledge illustration aims to suggest some partition between the technology-based deductive architecting process in reality, and the model-based inductive architecting process in knowledge [START_REF] Maier | The art of systems architecting[END_REF] before synthesis with regard to a system perspective. Therefore, section 4 explores their interdisciplinary necessary coupling → ← as a complete knowledge-based architecting process to encode/decode (Figure 5, Rosen, 2012) a responding control system (Figure 3.1, Lawson, 2015) to a situation with certain technology.

Finally, from this synthesis of lessons learned, and of the acquired knowledge from these return-of-experiences, section 5 concludes with a possible regeneration of the scientific corpus in order to explore what is at stake with respect to the cyber-physical and sustainability challenging paradigms.

Figure 1: Manufacturing Enterprise System of Systems

Integrated Manufacturing Systems and Engineering

With the emergence of computer communications and personal computers, in the 1980s, the industrial manufacturing automation community initiated the development of Computer-Integrated Manufacturing (CIM) systems. Integration in Manufacturing has been a first systemic paradigm, aiming to organize Humans and Advanced Manufacturing Technologies as a whole system. The integration was meant not only at the field level, but also at the management and corporate levels, resulting in a centralized/hierarchical architecture [START_REF] Mesarovic | Theory of Hierarchical, Multilevel, Systems[END_REF]) around an information system backbone (Figure 2, right).

Component-based integrated-system architecting

A host of data-intensive components are achieving to network, step-by-step, the entire integration of Manufacturing Plant Control within the overall control of an Extended Enterprise System. Eventually, such networking has been accomplished virtually.

Progress in real-time communications (Neumann, Morel et al., 2007), web-enabled and wireless technologies have strengthened the two-dimensional integration of automation in order to manage the customized manufacturing of both goods and services, as desired by the Internet society for agile Business-to-Manufacturing (B2M) and Business-to-Business (B2B) purposes:

• Synchronic, in time, integration of shop-floor process controls into plant-wide information management systems,

• Diachronic, through time, integration of Product Life-cycle Management over the manufacturing chain.

Many competing technologies for Factory Automation (Figure 2, left) have inferred a complication of the operational system architecture, both in terms of heterogeneous material means (dedicated computers, communications networks, supply chain operations, etc.) and in terms of the software functions (scheduling, control, supervisory control, monitoring, diagnosis, reconfiguration, etc.) it embeds. To address these increasing technological and functional interoperability issues, many efforts are being deployed to design architectures that are more flexible, by federating intra-and inter-zones of hardware/software facilities which share a common purpose (Business, Security, Logistics, etc.)

Enterprise modeling frameworks have been proposed [START_REF] Zachman | A framework for information systems architecture[END_REF] in order to functionally cope with the complexity to engineer these complicated Enterprise systems by defining modeling rules and generic infrastructure to be instantiated. The ISO 19439 Enterprise integration Framework for enterprise modeling resulted from the 1990s of world-wide joint works of IFAC and IFIP as well as international R&D programs leading to CIMOSA [Computer Integrated Manufacturing Open System Architecture] [START_REF] Vernadat | CIMOSA: Open System Architecture for CIM[END_REF] and GERAM [Generalised Enterprise Reference Architecture and Methodology] [START_REF] Nell | An Overview of GERAM[END_REF]. The Purdue Enterprise Architecture Framework [START_REF] Williams | The Purdue enterprise reference architecture[END_REF] has been first a standardized reference for Business-to-Manufacturing plantcontrol functional integration issues relating to Batch, and Continuous as well as Discrete Processes. This framework (

Service-oriented integrated-system architecting

Service orientation architecting (SOA) has intensified in the 21 st century. It was motivated by the scale-up trend in manufacturing that cannot just add more ICT (Information Communication Technology) agents, virtual and augmented reality devices and processes (Figure 3, left). There is the evident need to integrate their use for collaborative services to support new, distributed and complex architectures that comply with the convergence to sustainable production modes. Sustainable production, manufacturing, and logistics require interactions and mutual support among multiple virtual and physical facilities, enabling sustainable, efficient and cost-effective operations based on rich computational and cyber-collaborative autonomous points of view [START_REF] Moghaddam | The Collaborative Factory of the Future[END_REF][START_REF] Vernadat | Information systems and knowledge management in industrial engineering: recent advances and new perspectives[END_REF]. The emerging SOA concept (Figure 3, right) is a change in distributed and collaborative architectures, towards a distributed set of cyber-physical entities with service orientation: enhanced cells, startup organizations, and other production and supply sources, all with defined identity and purpose, and capability to collaboratively combine different processes as "services-components" [START_REF] Moghaddam | Collaborative service-component integration in Cloud Manufacturing[END_REF]. This service orientation is envisaged as a complete symbiosis between services and products [START_REF] Silva | Manufacturing Service: From e-work and Service-Oriented Approach towards a Product-Service[END_REF][START_REF] Dutra | Product-Service Architecture (PSA): towards a Service Engineering perspective in Industry 4.0[END_REF]Moghaddam et al., 2016;[START_REF] Moghaddam | Collaborative service-component integration in Cloud Manufacturing[END_REF][START_REF] Postigo | Formal Requirements Specification for Microgrid based on Architecture IEC 61.850[END_REF].The emergence of manufacturing service orientation in the architecting perspective (Figure 4, left) has advanced, following beneficial trends in the digital and cyber-physical convergence [START_REF] Schmenner | Manufacturing, service, and their integration: some history and theory[END_REF][START_REF] Tao | Cloud manufacturing: a computing and service oriented manufacturing model[END_REF][START_REF] Nof | Research Advances in Manufacturing with Service-Oriented e-Work and Production[END_REF][START_REF] Devadasan | Collaborative intelligence in knowledge based service planning[END_REF][START_REF] Nof | Perspectives on Manufacturing Automation under the Digital and Cyber Convergence[END_REF][4]). A recent review of smart manufacturing architectures [START_REF] Moghaddam | Collaborative service-component integration in Cloud Manufacturing[END_REF] has pointed out the convergence of manufacturing and industrial reference architectures and international standards towards service orientation, based on its advantages (IBM, 2008;[START_REF] Li | Cloud manufacturing: a new service-oriented manufacturing model[END_REF][START_REF] Gao | Service-oriented manufacturing: a new product pattern and manufacturing paradigm[END_REF][START_REF] Jiang | Towards a cyber-physical-social-connected and service-oriented manufacturing paradigm: Social Manufacturing[END_REF][5]).

Figure 3: Multi-dimensional service-oriented architecting

Synthesis

Both for CIM and SOA, formal ontologies (Figure 4, left) under standardization aim to provide guidelines to solutions by addressing the structural perspective of an architecting process. Therefore, Process Modeling Techniques [START_REF]European Commission Enterprise Architecture Framework[END_REF] must complete the dynamics of the overall Enterprise system architecting process (Figure 4, right) in order to functionally depict workflows in form of sequences of business activities before their mapping onto organizational units and operational components. All these modeling techniques [START_REF] Panetto | Towards a Classification Framework for Interoperability of Enterprise Applications[END_REF] reflect basically an integrative approach, despite some systemic traits, such as the enterprise system life-cycle.

These works and other complementary ones have not fully taken up the challenge of engineering an Enterprise System as a whole, while contributing, among others, to world-wide technology such as the well-known ERP. Even though the roots of this approach can be traced to manufacturing relative to Taylorism, the scientific basis of these process-based frameworks and techniques came from another community; those aware of relational data management, and information systems modeling, rather than automation engineering. The main drawback was that information-based process modeling, or workflow modeling, may entail more feedbacks than those constrained by the physics of the operational processes to be controlled in real-time. The resulting enterprise system architecting gap is that the information bridge between the operational and management enterprise levels is not fully integrated, despite the data bridge of the Manufacturing Execution System technology. The expected Enterprise control system is a large puzzle of networked automatons based on legacy or de-facto standardized, as well as proprietary hardware and software parts. The overall behavior of which is more or less well-managed, yet failing to be wholly contained. The mature "pyramid" remained the prevailing 2-dimension integrative approach, even with more or less digitalized technology. Information Control Problems in Manufacturing are still reflecting some lack of systems engineering knowledge by differentiating component-based bottom-up from system-based top-down integration to efficiently cope with the increasing overlapping of hardware, Software, and Humans, interacting in the control loop. Note that the evolution from Enterprise-control synchronic-integration to diachronic-integration is impacting the system togetherness and harmony which is expected to be ensured by a web of multiple, interacting data conduits. In return, Enterprise systems are accumulating so much "big data" relative to product variety (Giovannini., 2015) that key milestones in the evolution of dependability in manufacturing plant control pointed out the threat that software growth may limit further automation progress for the entire enterprise system (Figure 4, Johnson, Morel, et al.., 2007), with an ecological cost related to increasing use of data servers.

Figure 4: Two systemic perspectives of enterprise architecting frameworks

Intelligent Manufacturing Systems and Engineering

As argued by [START_REF] Monostori | Cooperative control in production and logistics[END_REF], integration-in-manufacturing often resulted in rigid, centralized or hierarchical control architectures, even if computerized, which could not cope with a changing and uncertain manufacturing environment. Thus, distribution relates to cooperative control architectures with more or less autonomous agents, ensuring local control and wholly interoperation-ability are necessary to ensure certain collective, holistic goals. For instance, design for better responsiveness to open-ended changes, which is required by the mechatronic society. Transitioning from integrated to distributed automation for product life-cycle issues has been strengthened by the considerable amount of interest -occasionally converging or controversial relating to integrating/distributing a form of technical intelligence into control architectures. Two complementary threads have been explored relating to the shifting intelligence-in-manufacturing paradigm:

Computational agents' architectures --embedding more computational techniques into automation agents from the upper to the field level, to achieve numerical control while enabling interactivity among participating entities, including interactions with Human agents. Early techniques for intelligent control have been associated to Artificial Intelligence, for example, to develop a unifying Cell Management Language for Factory Automation (Wright, 1988) and for diagnostic expert systems. They are in general applied in manufacturing and process automation in order to deal with well-defined or imperfect knowledge depending on domain local control goals (Figure 1, [START_REF] Zaremba | Integration and Control of Intelligence in Distributed Manufacturing[END_REF]. They are currently associated to computational decision-making [START_REF] Dolgui | Supply chain engineering: useful methods and techniques[END_REF][START_REF] Filip | Computer-Supported Collaborative Decision-Making[END_REF] in order to distribute some parts of the overall control on networked units for a wide-range of applications, such as model-based predictive control (Figure 12, [START_REF] Monostori | Cooperative control in production and logistics[END_REF]. The system architecting process remains limited to the reducible "divide-and-conquer" principle, even if applied to more cooperative control.

Bio-inspired architectures --On the other hand, embedding certain level of technical intelligence beyond computational techniques led the IMS community, envisions new distributed automation paradigms by designing bio-inspired architectures. The related form of technical intelligence advances beyond simple data through information to knowledge to be also partly embedded not only into intelligent components, but into the products themselves. Thus, products and devices can interact together as an intelligent system by means of remote communication, e.g., machine communication protocols, and Industrial Internet of Things. Continuing roadmaps have aimed at defining the desired structure of the factory of the twenty-first century since the world-wide industry-led IMS initiative arguing that manufacturing technology as a whole has not been systematized in any demonstrable or explainable form [START_REF] Yoshikawa | Manufacturing and the 21 century -Intelligent Manufacturing Systems and the Renaissance of the Manufacturing Industry[END_REF]. Two main bio-inspired paradigms are MAS (Multi-Agent System) and HMS (Holonic Manufacturing System). Their purpose is to contribute to the IMS systemic search to reflect "on the fly' the surrounding world-of-interest. With these two paradigms, any so-called system must service by exhibiting behavior in actual reality, and by generating meaning in knowledge without which the participating individual components cannot achieve the collaborative systems goals.

Agent-based intelligent-system architecting

The product-driven control paradigm is a major contribution to the full observability of flowing agents by embedding automaticidentification technology [AUTO-ID]. Discrete-Event Control deals currently with theoretical modeling techniques applied to formally define, even to automatically synthesize, the (unknown) control rules from both the (known) dynamics and (known) goal that an automation system must achieve in a specified time according to: {Process Dynamics Ù Control rules É Goal} [START_REF] Fusaoka | A description and reasoning of plant controllers in temporal logic[END_REF]). These dynamics are resulting from sensing processes triggered by a flowing discrete object through actuation processes in order to close the control loop for the shape, space and time operations to which a product is subject during its life cycle.

Embedding AUTO-ID led to formally revisiting the full observability of the (discrete-event) state space for direct product sensing, rather than inferring indirectly from process dynamics as it is usually done (Figure 5, right) according to: {{Product Ù Process}} Dynamics Ù Control rules É Goal}. Continuing efforts [6] towards the Internet of Things pointed out the pivotal role of the flowing interactive products to trigger the holistic dynamics of a responsive automation system in a wide range of industrial applications related to the concept of "intelligent products" (Figures 1 and2, [START_REF] Leitao | Intelligent products: The grace experience[END_REF]. They have contributed to anticipate the symbiosis between the tangible flow of goods, and the non-tangible flow of the related services; by breaking off the control hierarchy, they enable an "intelligent product" [START_REF] Baina | Model -Driven Interoperability: product-oriented approach to enterprise systems interoperability[END_REF] to actively participate as control agent of the manufacturing enterprise resources (Figure 5, left). The PHM paradigm [Prognostics and Health Management] results also from a multidisciplinary, integrative approach of embedding accurate algorithms into control units. By applying such algorithms at any enterprise-system level, it is possible to improve the precision of customized information in assessing systems in complex situations. The Watchdog agents TM infotronicsbased approach [START_REF] Djurdjanovic | Watchdog Agent-an infotronics-based prognostics approach for product performance degradation assessment and prediction[END_REF] is becoming technologically mature [7] to move from traditional failures, and predict and prevent by industrial maintenance practices for cost-oriented automation [START_REF] Erbe | Technologies for Cost Effective Automation in Manufacturing (Low Cost Automation)[END_REF]. Beyond technology, the deployment of such intelligent maintenance systems [START_REF] Iung | Conceptual framework for e-maintenance: Illustration by e-maintenance technologies and platform[END_REF] relates also to IT BPM best practices (Léger and Morel, 2001). The objective is to architect together a set of techniques and tools for enabling the prognosis when performance is becoming unacceptable, the diagnostic of why the performance is degrading, and the decision as to what maintenance action to perform. It also includes the performance benchmarking resulting from cumulative learning (machine learning) based on previous, similar operating processes. Data-driven and model-based prognostics can be combined with Knowledge-based Management technique in order to enable the maintainability of a targeted system-of-interest in operational conditions [START_REF] Chen | Conflict and Error Prevention and Detection in Complex Networks[END_REF][START_REF] Regal | Building an ontology for intelligent maintenance systems and spare parts supply chain integration[END_REF]. Relevant properties-of-interest to be observed in-situ can be predicted a priori from relating instantiated ontology, encapsulating the knowledge gained from previous experiments (Figure 6, right). Relevant data-of-interest are thus embedded into a computer platform in order to be checked for remote maintenance operation reflecting in real-time the related situation system (Figure 6, left). Note the instantiation process (Figure 6, middle) which enables the coupling relationship between reality and generic knowledge, as detailed in section 4.3. The bio-inspired MAS paradigm advances further to implement agent-based intelligent systems by bringing a computing technology for dynamics control issues as a set of interacting software agent exhibiting properties of autonomy, social ability, reactivity, proactiveness [START_REF] Wooldridge | An Introduction to MultiAgent Systems[END_REF]. Designing a MAS is both concerned by the agent design, such as the Beliefs-Desires-Intentions model [START_REF] Rao | BDI Agents: From Theory to Practice[END_REF] and about the society design as for Manufacturing Systems applications. Stimergy [START_REF] Grassé | Concept of Stimergy[END_REF], mimicking that an insect does not direct its work, but it is guided by its work, is one of the mechanisms among many others, which has been explored in the field of swarm intelligence to indirectly coordinate, through the environment, the trace left by an operating agent to cooperate with others. The principle is that the traces left in the environment by actions stimulate related close actions by the same or a different agent. In that way, subsequent actions tend to reinforce performance and build on each other, leading to the spontaneous emergence of coherent, apparently systematic activity. The resulting network of autonomous, yet interacting reasoning elements, enables to dynamically and adaptively architect complex distributed systems without localized decision-making processes. Nevertheless, the overall system performance at the right scale of manufacturing applications can be only assessed by simulation as evidenced by the multitude of experiments (Figure 1, [START_REF] Leitao | Past, present and future of industrial agent applications[END_REF]. The lack of a consensual MAS engineering framework [START_REF] Marik | Industrial Adoption of Agent-Based Technologies[END_REF][START_REF] Vrba | Rockwell Automation's Holonic and Multiagent Control Systems Compendium[END_REF] is another limitation to meeting system readiness level [START_REF] Adcock | The Trustees of the Stevens Institute of Technology[END_REF] in reality. This limitation is added on top of industry's uncertainty whether emerging behaviors (Figure 3, [START_REF] Monostori | Cooperative control in production and logistics[END_REF] can be contained to holistically target control goals, while being grounded to the plant physical level.

Holon-based intelligent-system architecting

The focus of component-based approaches, modeled by cooperative agents that operate as a whole artefact, may not be sufficient to satisfy a contextualized mission that can overcome open-ended changes in future, unexpected situations.

Pioneering works on science and engineering of intelligent systems led to the basic structure of the Real-Time Control System Architecture (RCS). RCS aims to integrate various intelligent techniques in order to bridge the gap between deliberative and reactive controllers. The fundamental structure of a control loop (Figure 1, Albus 1991) of the overall hierarchical architecture combines real-time sensory perception (SP) with prior information. The purpose is to reflect a model of the world-of-interest (WM) for behavioral (BG) knowledge-valued (VJ) decision-making in automation contexts. In particular, it aims to handle uncertain and unstructured operating environments with the 4D/RCS extension adding time as another dimension. Among many application domains, the ISAM (Intelligent System Architecture for Manufacturing) engineering framework (Figure 14, in [9]) is an RCS application in the Manufacturing domain, which consists of three hierarchically layered sets: behavioural, computational and organizational processing nodes, organized as a nested series of control loops. Despite possible computer performance issues to reflect WM in real-time with regard to situation system, the three fundamental reflections on what is matter-energy; what is life; and what is mind should remain relevant for any generation of IMS. They have inspired the progress discussed next on distributed intelligent automation.

The works on the HMS paradigm are combining a vision related both to intelligent beings and intelligent agents. They are designed to tackle expected heterarchical control issues. The holonic concept originated from the work of [START_REF] Koestler | The ghost in the machine[END_REF] who termed Holon the hybrid nature of organization units in biological and social systems. These units behave partly {h} as dependent parts and wholly {H} as self-contained wholes according to the way you consider them. The relating open-ended hierarchy was termed holarchy, since the architecture is comprised of holons and is not bounded in either downward or upward directions. Thus, autonomy of communicating agents cooperating in a decentralized manner to meet holistic objectives must be under coordination to contain the so-called emergence constrained in real-time practice by physical relationships. The HMS computational features equip MAS through time as a suitable technology for implementation [START_REF] Babiceanu | Development and applications of holonic manufacturing systems: a survey[END_REF]. A key difference is the part-whole aggregation of entities, encapsulating both software and hardware building blocks (Figure 7, right) to cope with the large-scale two-dimensional changes that are dynamically affecting an entire B2M2B (Business to Manufacturing to Business) chain. The holonic control theoretical framework has inspired the community of Manufacturing and Logistics Systems since its inception. The related PROSA (Figure 7, left), standing for {Product-Resource-Order-Staff Architecture}, is a prominent experiment from the early 1990's [START_REF] Van Brussel | Reference architecture for holonic manufacturing systems: PROSA[END_REF], via PROSA-DMAS, to ARTI [START_REF] Valckenaers | ARTI Reference Architecture -PROSA revisited[END_REF]. The reference holarchy is composed of recursive agents to be instantiated for specific manufacturing applications. A key system property is to reflect "on the fly" the behavior of a related real-world {Resource Holon} by processing relevant data with other Holons, in order to maintain its local control objective requested by {Order Holon} when executing the operational tasks of a {Product Holon}. Note the separation of concerns of each of the three basic (above) Holons. Each of which can aggregate supervisor holons according to the specific underlying domain; and an optional {Staff Holon} with advisory role, which can be adapted when facing unexpected situations.

To cope with scale up and edge effects of combinatorial data explosion, and for short-term forecasting and predicting unexpected events, the associated DMAS, ({Delegate Multi Agent System}) has been designed. It refers to a coordination mechanism of the dynamics of the HMS, which is needed to support Holons to fulfil their task responsibilities with a swarm of ants-like for feasibility, exploration, and intention. The approach has been recently revisited [START_REF] Valckenaers | Design for the Unexpected: From Holonic Manufacturing Systems towards a Human Mechatronics Society[END_REF] to consider architectural design guidelines based on Laws of Artificial [START_REF] Simon | The Sciences of the Artificial[END_REF] and on relevant insights about complex-adaptive systems [START_REF] Waldrop | Complexity: The Emerging Science at the Edge of Order and Chaos[END_REF]. Despite this effort, a wide range of laboratory experiments focused mainly on MAS technology to deploy Intelligent Agents based platforms. These experiments have focused on how the system knowledge corpus can handle dynamic changes, rather than on how to mirror the reality of a targeted world-of-interest. PROSA-related efforts for industry adoption (e.g. in the automated warehousing domain) at the higher TRL levels [START_REF] Adcock | The Trustees of the Stevens Institute of Technology[END_REF] are not using agent technology but Erlang/OTP, a technology from the telecom industry whose scalability is suitable for embedded systems. That approach may be a prerequisite property for any service oriented architecture and standards. The objective would be to plug-in, on the fly, executable models reflecting open changes of a situation system, such as IOT (Internet of Things) based changes. However, addressing the coupling between reality and a related IT architecture requires to reverse some of the current industrial practices. Thus, ARTI ({Activity-Resource-Type-Instance}), (Figure 7, middle), revisits PROSA to cover application domains beyond manufacturing. The approach emphasizes a generic nature, aimed at building an IT platform based on Intelligent Beings as activity performers accessing reality through (and not from) their digital twins as Intelligent Agents [START_REF] Valckenaers | ARTI Reference Architecture -PROSA revisited[END_REF].

Synthesis

An archetypical shift for distributed automation has been initiated by continuous process industry-led R&D [10] [START_REF]ESPRIT III-6188 PRIAM 'Pre-normative Requirements for Intelligent Actuation and Measurement[END_REF] in the late 1980s from the technological opportunity to unify a continuum of data communication around a real-time fieldbus [START_REF] Thomesse | A review of Field-Bus[END_REF]. An important architectural pattern has then resulted from a wide-large European R&D program as an integrated control, maintenance, and technical management system [CMMS] allocating intelligent actuation and measurement functions [IAMS] to distributed intelligent actuators and sensors (Figure 6, [START_REF] Pétin | Distributed intelligent actuation and measurement (IAM) system within an integrated shopfloor organisation[END_REF]. Digital technology can now encapsulate the matterenergy flow through a data-driven channel, not only between technical objects, but also with human agents whose interacting capabilities are augmented. The related overall digitalized interaction is normalized by a service channel to enable human operators access to information anywhere, at any time (Figure 15, Dobre, 2010a) as addressed by on-going R&D [START_REF]Innovative architecture for command and control platforms adapted to nuclear power plants www[END_REF] for nuclear power industry instrumentation and control (Figure 11). Note that these works have explored over time some aspects of the matter-energy, physico-physiological and data-information interactions between CMMS/IAMS agents [START_REF] Bouffaron | Integrative construct for model-based human-system integration: a case study[END_REF].

Compared with the discrete process domain, the product-driven paradigm contributed to ensure the diachronic flow of both services and goods [START_REF] Mcfarlane | Product Intelligence in Industrial Control: Theory and Practice[END_REF] in the manner of matter-energy in the continuous process domain, by means of a digitallyactive flowing object. Thus, it raised a question on the form-space-time partition between planning and control in manufacturing and logistics systems. In addition, it sought to envision with an energy perspective (Dembele et al., 1994) how to ensure togetherness and harmony of the interactions in the related, digitalized system of systems [START_REF] Koubeissi | Methodology for unified modeling of system of systems engineering[END_REF].

In both area, the meaning of "intelligence" must be interpreted as based on the designer's knowledge, embedded into digital agents.

Embedding is designed by means of computational technology [START_REF] Dobre | Contribution to industrial-process interactive-aided control modeling[END_REF], to reflect the reality of a given world-of-interest, in which the targeted system-of-interest has to behave, operate, and achieve its purposes.

The relative immaturity of IMS experiments for the next generation of manufacturing systems can be overcome by the relatively powerful holistic approach of the holonic control paradigm for facing resilience issues of large-scale SoS. It can be applied towards a broader range of applications, such as recently addressed by (Figure 4, Le [START_REF] Mortellec | Embedded holonic fault diagnosis of complex transportation systems[END_REF] for Holonic Diagnosis of a train system. That makes a "holon" in particular a knowledge building block of the system's DNA (Boardman, et al., 2009a) to recursively define a SoS at multi-scale as a holarchy {H, h}. Note that this holonification should be built from a real situation awareness by a sentient being {h} (Wilber, Mella 2009) requiring certain wholeness to bound a limited part of reality it is perceiving complex at that time [START_REF] Kuras | A Multi Scale Definition of a System[END_REF]. Additionally, next generation manufacturing systems design should be explored more extensively at another engineering time for designating a situation system [START_REF] Mayer | Exploring the notion of situation for responsive manufacturing systems specification issues[END_REF], and overcoming errors and conflicts in complex interactions among agents and holons, by the design principles of Collaborative Control Theory, CCT [START_REF] Nof | Collaborative control theory for e-work, e-production, and e-service[END_REF][START_REF] Nof | Revolutionizing Collaboration through e-Work, e-Business and e-Service[END_REF].

Interdisciplinary Systems Engineering

"System", as conceptualization unit of reality of a situation perceived as complex [START_REF] Pénalva | Modeling by systems in complex situation[END_REF], is suggesting a required wholeness to cope with some hidden aspects occurring between related interacting parts in reality and in knowledge. The respondent architected whole is a result of a system engineering process. In this process, the related body of interdisciplinary system knowledge is a prerequisite to cope with the hybrid nature of the physico-physiologico-artefactual interactions to be controlled in operation. Thus, harmonizing by orchestrating together the individual disciplinary knowledge and best practices also relates to the cognitive nature of the collaborative system thinking and acting process. Together, they can reconcile the easy-touse approach of the digital technology with the architecting approach required for manufacturing and logistic system resilience concerns.

Non-linear Systems Thinking and Acting

The Human role in the control loop is to be aware of a situation and to face the 'unanticipated' in real operation [START_REF] Boy | The handbook of human-machine interaction: a human-centered design approach[END_REF]. Furthermore, there is the question of what can be automated, or not, or both (Millot et al., Boy 2011;Patton, chapter 18, Nof, 2009). Even if Humans have limitations leading to recurrent errors (not always failures) at work, requiring enabling artefactual control [START_REF] Galara | Roadmap to master the complexity of process operation to help operators improve safety, productivity and reduce environmental impact[END_REF], that behavior points out the unique human capability to build by detour [START_REF] Berthoz | Simplexity: Simplifying Principles for a Complex World[END_REF] a cognitive control based on certain embedded knowledge blocks, previously acquired. This property, called simplexity, may be understood as a constitutive process to make the most likely decision by cobbling alternative solutions to complex problems in a given situation.

A sensory prerequisite (Lieber, 2013a) is that the artefact affords a right quantity of physical assets for enabling a Human to perceive correctly, before acting accordingly (photon, as quantum of energy, for orange sensory, in Figure 8, left). Paradoxically, this essential property, and human factors in general, are not adequately addressed beyond technology-based enhancements by the manufacturing operation control community, even when targeting resilience issues.

In the engineering domain, a set of artificial modeling principles has been drawn by biologically inspired conceptualizations on 'what is a system' since the early times of Systemics [START_REF] Bertalanffy | General System Theory: Foundations, Development, Applications[END_REF]Pouvreau, 2013). Its relationship to cybernetics [START_REF] Wiener | Cybernetics »[END_REF][START_REF] Forrester | Industrial Dynamics[END_REF]) remains essential, as addressed in recent works of the worldwide community of Systems Engineering (BKCASE, 2017). It is, therefore, possible to agree that the "whole-part" decoding-encoding relationship of a reality [START_REF] Rosen | Life Itself[END_REF] is not as trivial as currently addressed in disciplinary engineering. It is necessary to inquire about its effectiveness for integrative foundation of any system-centered interdisciplinary engineering. Coping with the related non-formal part of a reality involves system thinker's attitudes (Figure 8, top right). For example, to reassess the uncertainty of what can be patterned in an un-ordered domain, from what is known in the ordered (formal) domain [START_REF] Kurtz | The new dynamics of strategy: Sense-making in a complex and complicated world[END_REF], such as interactions taken implicitly (and mistakenly) as contained by the surrounding environment of a system, when they may actually not be contained by it. For example, what may or may not damage the togetherness and harmony [START_REF] Boardman | Systems thinking: Coping with 21st century problems[END_REF] of a whole structure and behavior (Figure 8, bottom right) is of importance for situational assessment to correctly designate the reinforcing and balancing causal loops of a situation system, before the design of a relevant control system. That realization should be explored particularly for indirect connectivity links of IOT-based infrastructures, which must maintain the togetherness and harmony among internetworked human, physical, and digital entities (Figure 9, left). Togetherness and harmony are essential for integrating tangible and intangible, whole cyber-physical systems, while mirroring, if possible 'on the fly', the dynamics of a related world-of-interest. Thus, the many alternative pathways (Figure 8, right) are challenging the cognitive, collaborative orchestration of the interdisciplinary knowledge togetherness and harmony between the many engineering actors involved along the entire life-cycle stages of a system-of-interest, from its conceptualization to its retirement.

Process-Guided versus Model-Based Systems Engineering

A variety of techniques compiled by the Guide to the Systems Engineering Body of Knowledge focus on certain generic principles of a system approach to perform standardized engineering processes [START_REF]S standard: Systems and software engineering -System life cycle processes[END_REF] in a recursive, iterative, and concurrent manner, according to project templates. The Model-Based Systems Engineering (MBSE) approach aims to contrast with the traditional document-centric approach by replacing the basic project artefact of process to a model for expected system modeling added value. Nevertheless, the tautological MBSE (any engineering process is model-based) does not necessarily consider that "system", as primary informal model of the entire architecting, enables an interdisciplinary reduction of the combinatorial complexity of the related multidisciplinary models of parts.

System thinking knowledge elements can be diagrammed in a form of systemigrams, causal-loop diagrams. The objective is to display the dynamics of a situation under designation, specially reinforcing and balancing loops to explore "what if" questions. The holistic completeness of this informal process for system-of-interest definition can be then systematically guided by the "conceptagon" framework ordering twenty-one system thinking concepts into seven triads [START_REF] Boardman | The conceptagon: A framework for systems thinking and systems practice[END_REF]. This pragmatic architectural framework highlights that {communication command control} forms together one triad, surrounded by six other ones. This approach results in the alignment of any related disciplinary knowledge, such as automation, with the interdisciplinary knowledge reflecting the required respondent system under modeling. Transitioning this architecting system knowledge to traditional engineering knowledge requires unifying system-oriented modeling languages [14] [START_REF] Cloutier | Transitioning Systems Thinking to Model-Based Systems Engineering: Systemigrams to SysML Models[END_REF], or domain-oriented [15] [START_REF] Retho | Model of intention: A concept to support models building in a complex system design project[END_REF]. Despite ontological representations efforts for knowledge interoperability (Giovannini et al., 2015a), both process-guided and model-based approaches focus mainly on the technical result of the overall engineering process; while possibly being limited in the ability to draw on human capabilities for heuristic processing [START_REF] Nugent | The Hidden Perils of Addressing Complexity with Formal Process -A Philosophical and Empirical Analysis[END_REF].

Technology is emerging which dismantles the silos of traditional systems multidisciplinary engineering. Its purpose is to meet the whole digitalization addressed by the "digital twin" paradigm shift. A unique and extensible development platform (Figure 9, right) can fully integrate the cross-discipline modeling, simulation, verification, and business process. Thus, it can significantly enable the exchange and collaborative execution of a set of models, built in their own environments as required by a common system under engineering design. Functional mock-up interface aims to ensure a standardized openness interface to be used in computer simulations to develop complex cyber-physical systems based on off-the-shelf models, even 'on the fly' for some responsive user-driven control related to IOT-based applications (Figure 9, left). At the same time, however, the increasing range of digital collaboration technologies, albeit useful, are only means to facilitate the cognitive role of a product, project, or team chief in managing the multidisciplinary exchanges of documents, as well as of models.

Figure 9: Technology Enhancing Model Based Systems Engineering (courtesy of 3DEXPERIENCE Dassault Systèmes)

Interdisciplinary Knowledge Orchestrating

The coupling relationship → ← points out the feedback that a respondent system has to bind into wholeness relative to a requesting situation system, as recently addressed by [START_REF] Lawson | A journey through the systems landscape[END_REF]. It focuses on "why humans make systems?" to cope with a problem or an opportunity occurring in real life (Figure 10). [START_REF] Lawson | A journey through the systems landscape[END_REF] giving meaning to Interdisciplinary Systems Engineering relative to a given situation system [START_REF] Dupont | System-Centered Specification of physico-physiological interactions of sensory perception[END_REF] It also highlights the need for systems to be designed to provide effective and respective services for each other if the whole system of systems is designed to accomplish intelligently its designated collaborative interactions and goals. One of the instantiated system assets is a control element, directing the respondent system mission outputs according to the situation system inputs resulting from operating elements. The interactions of systems and of elements comprise the phenomenological source of requirements' measurable properties. Note that the proposed coupling relationships shown in Figure 10 may reason and interpret by interactive exchanges the meaning and implications to dynamically enhance the togetherness and harmony of the respondent interdisciplinary knowledge, relative to the requesting situation system to be controlled. Also note that these mechanisms can be designed by the HUB-CI model of CCT, a virtual hub for collaborative intelligence [START_REF] Nof | Revolutionizing Collaboration through e-Work, e-Business and e-Service[END_REF]. It enables multiple sub-systems to interact, exchange, and learn to improve their interactions by collaborative intelligence and machine learning.

A cognitive interpretation of the overall process consists of orchestrating the appropriate interdisciplinary knowledge into a coherent whole to be allocated {↔} from the multidisciplinary knowledge assets available in a given project system. The cognitive nature of these coupling relationships has been recently explored by [START_REF] Bouffaron | Executable Model Based System co-Specification: application to critical process interactive control[END_REF] for interdisciplinary MBSE issues on the basis of the problem frames approach previously addressed for software engineering [START_REF] Jackson | The meaning of requirements[END_REF][START_REF] Jin | Revisiting the meaning of requirements[END_REF]. The proposed heuristics differentiate for each source-sink interaction between a requesting knowledge and the related responding one in order to preserve the individual ability to build by detour [START_REF] Berthoz | Simplexity: Biologically Inspired Principles for Complex Systems[END_REF] a model as part of a whole-system model according to the set of predicates:

-Descriptive specification from a domain problem-space according to:

The twin-peaks model-based orchestration process [START_REF] Termitologia | Relating software requirements and architectures using problem frames[END_REF] results from iterative refinements all along a system project life cycle, between interacting knowledge. Knowledge are, respectively, source of problem-oriented requirements (Equation 1), and sink of solution-oriented models (Equation 2), both being verified [START_REF] Zaytoon | Synthesis and implementation of logic controllers: A review[END_REF] before contractual validation (Equation 3) in silico ⊩ .

Technology-enhanced modeling platforms (Figure 9, right) may enable designers to meet TRL proof of the "right job right" [START_REF] Fanmuy | Requirements Verification in the Industry[END_REF] of the system response to a given situation system, by execution of the multidisciplinary models together.

To cope with the difficulty of translating any human intention source into a model for a related human understanding sink, the focus may be to exchange only the essential parts of the interdisciplinary knowledge to be shared. For example, it can be modeled with the de facto-standardized SysML for event-driven orchestration of the multidisciplinary models running in their own working boundaries [START_REF] Bouffaron | Executable Model Based System co-Specification: application to critical process interactive control[END_REF]. Note that the validation in situ of the specified situation system control with regard to the perceived reality of a world-of-interest is a non-formal entailment relationship {├ instead of ⊧ in equation 3} between Humans involved respectively in the requesting tangible (operational) domain, and the responding non-tangible (engineering) domain.

Synthesis

A main advantage of the digital technology is that it can be designed by various suppliers as individual parts, or assets to be cobbled together with relative simplicity, to meet system wholeness. This "hardware-software in the loop" approach may present a possible drawback. It can challenge the "model-in-the-loop" interdisciplinary systems engineering, which aspires to enable collaborating individuals working together [START_REF] Retho | Collaborative methodology based on virtual product aided building for the design of electrically propelled aircraft[END_REF], even though increasingly better trained with simple-do-it-yourself integrative approaches. At the same time, it is possibly desirable to not collectively frame a "system-in-the-loop" in certain design situations. The orchestration metaphor highlights the "human-in-the-loop" approach by intelligent detour to a synthesis containing the whole system. It is also contextualized in its external environment, and enables analysis of its related internal architecting by investigating relevant component partial solutions [START_REF] Edson | Systems Thinking[END_REF]. From this analysis, artefacts are only considered as means to encompass the system togetherness and harmony, as required by a given situation. Failing a priori in system thinking for acting, often leads to complicating the design of an essential respondent system by patching a posteriori in-silo digital parts.

Notwithstanding its appropriate use, embedding more digital technology in the architecture may ensure a just-in-time remote continuum of data, information and knowledge flows between plants operated activities, and model-based systems engineering activities. The objective is to enhance complex situation awareness and control "on the fly", as addressed by on-going R&D [START_REF]Innovative architecture for command and control platforms adapted to nuclear power plants www[END_REF] in process industry (Figure 11). More generally, the "Digital Twin" paradigm (Kusiak, 2018) should not mask that it is the tangible relationship between reality and virtuality which maintains any system in its wholeness with certain energy level. Otherwise, digital artefacts may cause user's cognitive overload in real life when facing unexpected system situations.

Conclusion and perspectives

The Discrete Process Automation and Modeling community members have been creative and innovative to face the challenging architecting of hardware, software and information in the control loop of large-scale manufacturing and logistics system. It brought together with the computing community a body of knowledge anticipating to certain extent the digital revolution to industry 4.0, [START_REF] Zuehlke | SmartFactory -Towards a factory-of-things[END_REF][START_REF] Nogueira | Digital Twin Data Modeling with Automation ML and a Communication Methodology for Data Exchange[END_REF], and the digital and cyber conversion [START_REF] Pereira | Survey and Emerging Challenges of Manufacturing Automation Modeling and Control --A Global Perspective[END_REF][START_REF] Nof | Perspectives on Manufacturing Automation under the Digital and Cyber Convergence[END_REF]. Its systemic technology-enhanced trait has been to deal over time with the architecting of integrated and intelligent systems related to a wide range of applications and their challenges, such as: shop-floor control; process and planning plant and logistics control; factory, supply chains, and supply networks information and communication networking and integration, SoS interoperability and resilience, and service-oriented issues. The resulting cross-fertilization has addressed through time improvement and management methods as well as heterarchical, interactive and collaborative control design principles. The related innovative traits led, among many, to the well-known ERP technology, as well as to several efficient paradigms such as the Product-driven and PHM ones, Collaborative Control workflow protocols and algorithms, as well as the emerging HMS.

From a historical perspective, despite an impressive production of concepts and experiments, the effectiveness [START_REF] Nof | Design of effective e-Work: review of models, tools, and emerging challenges[END_REF] of many architecting paradigms of these technological SoS [START_REF] Nof | From plant and logistics control to multi-enterprise collaboration[END_REF] is not yet proven in real work and the Human capabilities have manifestly been insufficiently explored in operation, as well as in interdisciplinary engineering. Several systemic failures and weaknesses are repeatedly reminding us that this corpus of knowledge should be collectively improved, by rethinking beyond technology-enhanced in-silo approaches. To some extent, a complementary trait may be necessary: A symbiotic approach, as suggested by multiple aspects presented the above sections 2 and 3. It may also be possible to consider that the building of an interdisciplinary knowledge, as described in section 4 is one relevant form of symbiosis. The objective is to yield significantly more than just the addition of specialist knowledge which is limited in its ability to reflect the togetherness and harmony of real situations which require full systemic integrative reference.

That realization challenges us to consider better approaches that take into account, in the control loop, the "simplex" domains of human and nature beyond the "causal" nature of man-made artefacts to be able to efficiently cope with cyber-physical [START_REF] Zhong | The dynamic lines of collaboration model: Collaborative disruption response in cyber-physical systems[END_REF] as well as resilience and sustainability [START_REF] Moldavska | Development of Manufacturing Sustainability Assessment Using Systems Thinking[END_REF][START_REF] Reyes Levalle | Resilience by teaming in supply chains and networks[END_REF] issues. Understanding and responding to the meaning of "why humans make systems?" may require an earlier [START_REF] Ponto | Sustainable Tomorrow: A Teachers' Guidebook for Applying Systems Thinking to Environmental Education Curricula[END_REF] and faster fundamental shift [16] in education, engineering and service, as people need to be as early as they are primary, secondary and university students; and throughout their career

  Figure 2.1, Williams, Nof 2009) remains the standardized backbone of the set of modeling techniques (Figure 3, Brandl, 2016) simplifying the federation of components systems for factory automation [2].

Figure 2 :

 2 Figure 2: Multi-dimensional component-based architecting

Figure 5 :

 5 Figure 5: AUTO-ID based product-driven control paradigm

Figure 6 :

 6 Figure 6: Coupling between remote PHM-IT infrastructure and IMS Engineering (courtesy of www.Predict.fr )

Figure 7 :

 7 Figure 7: Holonic Manufacturing System-of-Systems Architecting

Figure 8 :

 8 Figure 8: Simplexity giving meaning to Systems Thinking and Acting in a given situation

Figure 10 :

 10 Figure 10: Cognitive interpretation of the System Coupling Diagram[START_REF] Lawson | A journey through the systems landscape[END_REF] giving meaning to Interdisciplinary Systems Engineering relative to a given situation system[START_REF] Dupont | System-Centered Specification of physico-physiological interactions of sensory perception[END_REF]

Figure 11 :

 11 Figure 11: Towards Operation and Engineering knowledge-driven harmony[START_REF] Galara | Connexion cluster: concepts and frameworks[END_REF]

Acknowledgement

The authors wish to express thanks and gratitude to all our colleagues and students in IFAC, IFIP, ICPR, and beyond for their contributions to the knowledge summarized in this article, as well as to the valuable reviewer's reports.


D Lavazec

G Cumunel

D Duhamel
email: denis.duhamel@enpc.fr

C Soize

Experimental evaluation and model of a nonlinear absorber for vibration attenuation

Keywords: nonlinear absorption, vibrations control, noise control, homogenization, metamaterials

Because of their large wavelength, the noise and the vibrations at low frequencies cannot easily be reduced in the structures by using dissipative materials contrarily to the waves at middle and high frequencies. A possible technique for obtaining an attenuation is to randomly distribute absorbers in a matrix to attenuate acoustic waves and vibrations in such a low-frequency band. If linear absorbers are used, then it is difficult to obtain an attenuation on a broad low-frequency band with a reasonable number of absorbers. On the other hand, it is known that nonlinear absorbers allow the absorption bandwidth to be increased. However, such technique requires to properly design such a nonlinear absorber and to perform its experimental validation. The objective of the paper is to present the design of such an absorber with nonlinear geometrical effects and its experimental evaluation. It is experimentally proved that this nonlinearity type induces an attenuation on a broad frequency band around its first resonance. The novelty of this work is to present the design of an inclusion that has the potentiality to be reproduced at any scale, in particular for microstructured materials. For such an inclusion, we present its mechanical model for the simulation, its design, its manufacturing that is realized with a 3D-printing system, the experiments, and the experimental identification of the model. The results show that a significant attenuation is effectively obtained over a broad low-frequency

Introduction

At middle and high frequencies, the reduction of acoustic waves and vibration in structural elements can be obtained by using adapted poroelastic dissipative materials. However, for low frequencies, the wavelength are larger than the pores, and dissipative materials are less efficient. To get around this problem at low frequencies, oscillators-based absorbers have been designed in order to pump the vibratory energy. This type of technologies has extensively been studied in the litterature. Hereinafter, we present a non-exhaustive short bibliographical summary. Among the first papers devoted to the enery pumping by simple oscillators, the work by Frahm [1] in 1911 can be cited, in which the author proposes to add an auxiliary body to a structure to damp or avoid the vibration due to periodic impacts. The resonance vibrations of the main body are annulled by the secondary resonance vibrations of the smaller auxiliary body. In nearer years, tuned-mass dampers have been studied. The principle is to add an oscillator (generally a mass-spring-damper system) to a main structure that exibits an annoying resonance. The eigenfrequency of the damped oscillator is then adjusted for attenuating the resonance amplitude. With that, the peak is splited into two peaks of low amplitudes. A review made by Gutierrez Soto et al [START_REF] Soto | Tuned mass dampers[END_REF] presents a representative study on tuned-mass dampers. Metamaterials have also been used for the reduction of noise and vibration. A metamaterial is generally a composite for which these mechanical properties do not exist at the natural state. For instance, there are materials with both negative permittivity and negative permeability such as in [START_REF] Veselago | The electrodynamics of substances with simultaneously negative values of and ν[END_REF]. In the framework of the absorption of vibration and noise, numerous papers have been published with metamaterials, see for instance [START_REF]Handbook of Micromechanics and Nanomechanics[END_REF][START_REF] Del | Dynamic problems for metamaterials: Review of existing models and ideas for further research[END_REF] for review papers. For instance metamaterials exhibiting a negative refracting index in the field of optics has been presented by [START_REF] Smith | Negative refractive index in left-handed materials[END_REF] and equivalent negative mass in the field of acoustic material by [START_REF] Ding | Metamaterial with simultaneously negative bulk modulus and mass density[END_REF][START_REF] Sheng | Dynamic mass density and acoustic metamaterials[END_REF][START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF][START_REF] Lee | Acoustic metamaterial with negative density[END_REF][START_REF] Liu | An elastic metamaterial with simultaneously negative mass density and bulk modulus[END_REF][START_REF] Wang | Dynamic behaviour of a metamaterial system with negative mass and modulus[END_REF]. Another works in the field of acoustic metamaterial can be found in [START_REF] Chen | Acoustic cloaking in three dimensions using acoustic metamaterials[END_REF] for acoustic cloaking, in the field of vibration attenuation with linear multiple discrete oscillators [START_REF] Zhou | Analytic model of elastic metamaterials with local resonances[END_REF][START_REF] Hussein | Metadamping: an emergent phenomenon in dissipative metamaterials[END_REF][START_REF] Zhu | A chiral elastic metamaterial beam for broadband vibration suppression[END_REF] or multiple oscillators induced by continuous systems [START_REF] Yang | Acoustic metamaterial panels for sound attenuation in the 50-1000 Hz regime[END_REF][START_REF] Auriault | Long wavelength inner-resonance cut-off frequencies in elastic composite materials[END_REF][START_REF] Xiao | Sound transmission loss of metamaterialbased thin plates with multiple subwavelength arrays of attached resonators[END_REF][START_REF] Varanasi | The low frequency performance of metamaterial barriers based on cellular structures[END_REF][START_REF] Wang | Membrane-constrained acoustic metamaterials for low frequency sound insulation[END_REF] have presented metamaterials with linear behaviors.

In 1952, Roberson [START_REF] Roberson | Synthesis of a nonlinear dynamic vibration absorber, Portions of a dissertation submitted to the Department of Applied Mechanics[END_REF] has presented the equations of a nonlinear dynamic vibration absorber and highlighted the fact that an auxiliary body with nonlinear dynamical behavior offers significant advantages over a linear absorber. Then, some works have been performed for analyzing vibration of nonlinear oscillators in a stochastic framework, such as in [START_REF] Bouc | The power-spectral density of response for a strongly nonlinear random oscillator[END_REF][START_REF] Soize | Stochastic linearization method with random parameters for SDOF nonlinear dynamical systems: Prediction and identification procedures[END_REF][START_REF] Bellizzi | Analysis of multi-degree of freedom strongly nonlinear systems with random input[END_REF][START_REF] Spanos | On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: An alternative perspective[END_REF] in which it is demonstrated the role plays by the nonlinearities in the transfer of the vibratory energy over a broad frequency band. Another works concerning the energy pumping by discrete nonlinear mechanical oscillators with linear damping without experimental results have been published such as [START_REF] Soize | Vibration damping in low-frequency range due to structural complexity. A model based on the theory of fuzzy structures and model parameters estimation[END_REF][START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators: part I-dynamics of the underlying hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part II-resonance capture[END_REF][START_REF] Vakakis | Shock isolation through the use of nonlinear energy sinks[END_REF][START_REF] Carrella | Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic[END_REF][START_REF] Alexander | Exploring the performance of a nonlinear tuned mass damper[END_REF][START_REF] Ji | Suppression of super-harmonic resonance response using a linear vibration absorber[END_REF][START_REF] Ji | Design of a nonlinear vibration absorber using three-to-one internal resonances[END_REF][START_REF] Viet | On a nonlinear single-mass two-frequency pendulum tuned mass damper to reduce horizontal vibration[END_REF], and with nonlinear damping [START_REF] Cveticanin | Vibration of a two-mass system with non-integer order nonlinear connection[END_REF].

The work presented in this paper has been developed in the framework of the reduction of vibration and induced noise in structures at macro-scale for low frequencies for which the first structural modes are excited. The final objective of this work is to reduce vibrations and induced noise on a broad low-frequency band by using a microstructured material by inclusions that are randomly arranged in the material matrix. The novelty of this work is to present the design of an inclusion that has the potentiality to be reproduced at any scale, in particular for microstructured materials for which both the modelling approach and the parametric experiments performed for several structures manufactured with a 3D-printer, see [START_REF] Lavazec | Attenuation of mechanical vibrations at low frequencies by a nonlinear dynamical absorber[END_REF] for a first example of this approach. The first step of this work is to design and to analyze the efficiency of an inclusion, which is made up of a cantilever beam with a mass at its end. This inclusion behaves as a nonlinear oscillator due to nonlinear geometrical effects, which is designed so that the energy pumping is effective on a broad frequency band around its resonance instead of a narrow frequency band as for an equivalent linear oscillator. The objectives of the paper are (1) to design the nonlinear oscillator in terms of form, dimension, and materials, in order that the vibratory energy is pumped on a broad frequency band around its first resonance, (2) to experimentally manufacture this system with a 3D-printing system, (3) to perform parametric experiments allowing to understand the bandwidth absorption versus the nonlinear amplitude levels, (4) to develop stochastic mechanical model to predict the responses, and (5) to perform the identification of the mechanical model with the experiments. The results that will be obtained will exhibit the physical attenuation over a broad low-frequency band, which is expected. Such a validated mechanical model is useful for designing a structural element including a random distribution of such a nonlinear oscillator. Compared to the communication [START_REF] Lavazec | Attenuation of mechanical vibrations at low frequencies by a nonlinear dynamical absorber[END_REF], published in a French congress, this article develops the bibliography and the design of the inclusion. Above all more experimental results are presented in this article and are discussed in much more details. New and more detailed experimental results are the true novelty of this article which can be interesting for people working in this field.

In Section 2, the mechanical model of the inclusion and the related stochastic solver are introduced. In Section 3 the experimental design is performed in order to obtain an effective attenuation in the frequency band of analysis. The experimental measurements and the identification of the mechanical model are presented in Section 4. The last section is devoted to the conclusions on the results obtained and to the perspectives of the future work.

2. Construction of a nonlinear model for representing the absorber and stochastic solver

Model of the nonlinear oscillator

As explained in Section 1, a nonlinear oscillator with one-DOF is proposed to model the nonlinear dynamical behavior of the inclusion. The one-DOF nonlinear model is composed of a mass-spring-damper system with a nonlinear spring and a nonlinear damper, subjected to an excitation of its base (see the scheme displayed in Figure 1). We introduce a parameterized family of one-DOF nonlinear oscillators, for which the proposed algebraic model is inspired/coherent with the linear viscoelasticity theory in finite displacements (nonlinear model) without memory [START_REF] Coleman | Foundations of Linear Viscoelasticity[END_REF][START_REF] Desceliers | Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization[END_REF]. The motivation of this choice is the following: the nonlinear absorber that will be designed will not have material nonlinearities but only nonlinear geometrical effects and will have a viscoleastic type behavior related to the choice of the material. This is the reason why we have constructed a model coherent with the finite viscoelasticity theory.

In addition, we want that the stochastic response of the one-DOF nonlinear oscillator submitted to a centered stochastic excitation be a centered stochastic process. This constraint will be taken into account in the construction of the nonlinear damping force and of the nonlinear stiffness force deduced from the viscoelasticity theory in finite displacements. Let X exp imp (t) be the displacement imposed at the support in the absolute frame and let X s (t) be the relative displacement of the point mass with respect to the support. Let { Ẍexp imp (t), t ∈ R} be the acceleration imposed to the support, which is assumed to be a Gaussian stationary second-order centered stochastic process, defined on the probability space (Θ, T , P), for which the power spectral m X s (t)

X exp imp (t) F exp s (t) f K (X s (t)) f D (X s (t), Ẋs (t))
Figure

(t)) + f D (X s (t), Ẋs (t)) + f K (X s (t)) = 0 ,
which is rewritten as

m Ẍs (t) + f D (X s (t), Ẋs (t)) + f K (X s (t)) = F exp s (t) , (1)
in which the applied stochastic force F exp s (t) is written as

F exp s (t) = -m Ẍexp imp (t) , (2)
and where the nonlinear damping force f D and the nonlinear stiffness force f K are written as

f D (X s (t), Ẋs (t)) = c 1 + c 2 |X s (t)| Ẋs (t) , (3)
f K (X s (t)) = k 1 X s (t) + k 3 (X s (t)) 3 . (4)
The absolute displacement X a of the mass is such that

X a (t) = X s (t) + X exp imp (t) . (5)
In these equations, m is the participant mass of the inclusion, associated with the one-DOF oscillator, c 1 and c 2 are the two parameters describing the family of nonlinear damping force, and k 1 and k 3 are the two parameters related to the nonlinear stiffness force. It should be noted that the algebraic expression of the nonlinear terms that comes from the viscoelasticity theory without memory in finite displacements, should induce, for the nonlinear stiffness force, linear, quadratic, and cubic terms, corresponding to the nonlinear geometrical effect, and for the nonlinear damping force, a linear function of the velocity with a nonlinear coefficient of the displacement. Due to the constraint that we have introduced (centered response for a centered excitation), the quadratic term in the stiffness force must vanish and the nonlinear coefficient of the displacement in the damping force must be an even function of the displacement.

As we are interested in quantifying the vibratory energy absorbed by the oscillator, we introduced the mean input power Π in and the mean power dissipated Π diss , which are such that

Π in = E{F exp s (t) Ẋs (t)} , Π diss = E{f D (X s (t), Ẋs (t)) Ẋs (t)} , (6)
in which E is the mathematical expectation. Since the stochastic processes F exp s and Ẋs are stationary, these two quantities are independent of time and can be rewritten

Π in = R π in (ω) dω , Π diss = R π diss (ω) dω ,
in which the density π in (ω) and π diss (ω) are such that

π in (ω) = S F exp s Ẋs (ω) , π diss (ω) = S f D Ẋs (ω) . (7)
In Eq. ( 7), S F exp s Ẋs is the cross-spectral density function of the stationary stochastic processes {F exp s (t), t ∈ R} and { Ẋs (t), t ∈ R}, and S f D Ẋs (ω) is the cross-spectral density function of the stationary stochastic processes {f D (X s (t), Ẋs (t)), t ∈ R} and { Ẋs (t), t ∈ R}. The frequency-dependent energy pumping is therefore characterized by π diss (ω), which is equal to π in (ω) due to the power balance equation of the stochastic system. In order to qualify the efficiency of the energy pumping as a function of the intensity of the nonlinearity that will be characterized by the value of the dimensionless response amplitude defined in Section 4.2, we introduce the normalized quantity,

π in,norm (ω) = π in (ω) S F exp s (ω) . (8)
In order to experimentally identify the vector-valued parameter of the model, denoted by w = (c 1 , c 2 , k 1 , k 3 ), which belongs to an admissible set C ad , we introduce the frequency dependent function FRF(ω; w) 2 , defined on B o by

FRF(ω; w) 2 = |S ẊsF exp s (ω)| 2 |S F exp s (ω)| 2 = |π in,norm (ω)| 2 , (9)
in which the w-dependence in the right-hand side of Eq. ( 9) has been removed.

It should be noted that if f D (x, ẋ) = c 1 ẋ and f K (x) = k 1 x, the model corresponds to the one of a linear oscillator and FRF 2 represents the square of the modulus of the frequency response function of the associated linear filter for which F exp s is the input and Ẋs is the output.

Stochastic solver and signal processing

Stochastic solver. For constructing the stationary stochastic solution of the nonlinear differential equation defined by Eq. (1), the Monte Carlo method (see [START_REF] Rubinstein | Simulation and the Monte Carlo Method[END_REF]) is used. Let {F exp s (t; θ ), t ∈ R} be a realization of the stochastic process F exp s for θ in Θ. We consider L independent realizations θ = 1, ..., L in Θ constructed with the generator described in a next paragraph of this section. For each realization θ , we would have then to solve the deterministic nonlinear differential equation associated with stochastic Eq. (1): calculate X(t; θ ) for all t in R such that

m Ẍ(t; θ ) + f D (X(t; θ ), Ẋ(t; θ )) + f K (X(t; θ )) = F exp s (t; θ ) . (10)
This deterministic Eq. ( 10) cannot numerically be solved because t belongs to an infinite interval. Consequently, this problem is replaced by the following approximate problem with initial conditions over a finite interval of length T : calculate X(t; θ ) for t belonging to the finite interval ]0, T ], solution of Eq. ( 10) with the initial conditions at time t = 0,

X(0, θ ) = 0 , Ẋ(0, θ ) = 0 . (11)
With such initial conditions, the transient part of the response would be 0 for a linear second-order dynamical system and it is assumed to be 0 or negligeable for the family of the nonlinear second-order dynamical systems that are considered (we have effectively checked the validity of these hypotheses for the numerical application presented in the paper). The part of the trajectory on [0, T ] corresponds to the stationary response, X s (t; θ ) X(t; θ ) for t in [0, T ]. Time T that is related to the frequency resolution is defined in a next paragraph. The deterministic problem defined by Eq. (10) for t in ]0, T ] with the initial conditions defined by Eq. ( 11) for t = 0 will be solved by using the Störmer-Verlet scheme presented in a next paragraph.

Time and frequency sampling. For solving the nonlinear deterministic equation with initial conditions associated with a given realization θ , we need to generate realizations of the Gaussian stationary stochastic process F exp s on the time interval [0, T ]. Consequently, the time and the frequency samplings are constructed with respect to this time interval. The signal processing requires a time sampling with a constant time step ∆ t that is performed using the Shannon theorem for the stationary stochastic processes [START_REF] Papoulis | Probability, Random Variables and Stochastic Processes[END_REF]. The sampling frequency is f e = 2 f max where f max is the maximum frequency that exists in the frequency band of analysis and the time step is ∆ t = 1/f e . The corresponding time sampling is t α = α ∆ t with α = 0, 1, ..., N -1 in which the integer N is chosen in order that the time duration T = 8 s, with T = N ∆ t yielding N = 16, 384. The frequency resolution ∆ f = 1/T = 0.125 Hz. The corresponding sampling points in the frequency domain are

f β = -f max + (β + 1/2)∆ f for β = 0, 1, ..., N -1.
Generation of independent realizations of Gaussian stationary stochastic process F exp s . The usual second-order spectral representation of the stationary stochastic processes is used [START_REF] Guikhman | The Theory of Stochastic Processes[END_REF][START_REF] Priestley | Spectral Analysis and Time Series[END_REF]. The power spectral density function S F exp s (ω) of the Gaussian stationary second-order centered stochastic process second-order stochastic process Ẍexp imp is based on the usual spectral representation (see [START_REF] Shinozuka | Simulation of multivariate and multidimensional random processes[END_REF][START_REF] Poirion | Numerical methods and mathematical aspects for simulation of homogeneous and non homogeneous Gaussian vector fields[END_REF]). Let ψ 0 , . . . , ψ N -1 be N mutually independent uniform random variables on [0, 1] and let φ 0 , . . . , φ N -1 be N mutually independent uniform random variables on [0, 2π], which are independent of ψ 0 , . . . , ψ N -1 . The spectral representation used is written, for all t, as

F exp s is such that S F exp s (ω) = m 2 S Ẍexp
Ẍexp imp (t; θ ) 2∆ ω Re N -1 β=0 S Ẍexp imp (ω β ) Z β e -iω β t e -iφ β , (12)
with ∆ ω = 2π ∆ f , where Z β = -log(ψ β ), where ω β = 2π f β , and where Re denote the real part of complex numbers. From Eq. ( 12), it can be deduced that, for all t, the realization Ẍexp imp (t; θ ) is written as

Ẍexp imp (t; θ ) 2∆ ω Re N -1 β=0 g β, e -iω β t , (13)
in which θ ) . In practice, for each , Eq. ( 13)

g β, = S Ẍexp imp (ω β ) Z β (θ ) e -iφ β (
is computed for t = t α with α = 0, . . . , N -1. Consequently, the numerical calculation is done by using the FFT { g 0, , . . . , g N -1, } of {g 0, , . . . , g N -1, }, which is such that g α, = N -1 β=0 g β, exp -2i π αβ/N and which yields, for α = 0, 1, . . . , N -1,

Ẍexp imp (t α ; θ ) = 2∆ ω Re exp -iπα 1 -N N g α, . (14)
Störmer-Verlet integration scheme. The Störmer-Verlet integration scheme is well suited for the resolution of dynamical Hamiltonian systems [START_REF] Verlet | Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules[END_REF][START_REF] Hairer | Geometric numerical integration illustrated by the Störmer/Verlet method[END_REF] and stays very efficient for dissipative Hamiltonian systems as proposed in [START_REF] Soize | Time-domain formulation in computational dynamics for linear viscoelastic media with model uncertainties and stochastic excitation[END_REF]. Such a scheme preserves the mechanical energy during the numerical integration. Equation [START_REF] Lee | Acoustic metamaterial with negative density[END_REF] with initial conditions defined by Eq. ( 11) is then written as the following first order differential equation, for t in ]0, T ],

         Ẋ(t; θ ) = 1 m Y (t; θ ) , Ẏ (t; θ ) = -f D (X(t; θ ), Y (t; θ ) m ) -f K (X(t; θ )) + F exp s (t; θ ) . (15)
We use the notations x α = X(t α ; θ ) and y α = Y (t α ; θ ). The Störmer-Verlet integration scheme for Eq. ( 15) is then written, for α = 0, 1, ..., N -1, as

                             x α+1/2 = x α + ∆ t 2m y α , y α+1 = y α + ∆ t - 1 2 (f D (x α+1/2 , y α /m) + f D (x α+1/2 , y α+1 /m)) -f K (x α+1/2 ) + F exp s (t α+1 ; θ ) , x α+1 = x α+1/2 + ∆ t 2m y α+1 , (16)
in which F exp s (t α+1 ; θ ) = -m Ẍexp imp (t α+1 ; θ ) and where x 0 = y 0 = 0.

Signal processing. For estimating, the power spectral density functions and the cross-spectral density functions defined in Eqs. ( 7) and ( 9), the periodogram method [START_REF] Papoulis | Probability, Random Variables and Stochastic Processes[END_REF] is used for which the rectangular time-window on ]0, T ] is used.

Experimental design

In Section 2, we have presented the nonlinear dynamical model of the inclusion and the related stochastic solver. In this section, we present the design of an inclusion whose response can be simulated with this nonlinear dynamical model.

Defining the constraints for the design of the inclusion and its consequences on the design. The following constraints have been taken into account for defining the design:

• The dimension of the inclusion has been setted at about two centimeters in order to be representative of the scheduled technology for which such inclusions would be incorporate in a composite material such as a concrete panel.

• The inclusion must have the capability to exhibit a strong nonlinear geometrical effect with reasonable finite displacements to remain in the bulk whose geometry has previously been defined.

• The inclusion must exhibit a relatively small damping in order to optimize the transmitted mechanical power to the nonlinear oscillator in order to optimize the absorption.

The first system that we have worked was made up of a point mass centered in a spherical steel shell with an internal elastomer layer around the mass. The main advantage of this system was to have an isotropic absorption property no matter the direction of excitation. The elastomer had a hyperelastic behavior and after experimental tests, we have rejected this system because of the important damping due to the elastomer. After that, we studied a system inspired by the nonlinear energy sinks (NES) composed of a point mass linked to two prestressed traction springs clamped at their ends. In this configuration, the springs had effectively a nonlinear behavior when the point mass is excited in the transverse direction. However, large displacements are needed for obtaining a significant geometrical nonlinear effect what was not possible. Then, we focused on two systems composed of a point mass attached to a beam. For the first one, the beam was biclamped and the point mass was located at its center. The second one was a cantilever beam with a point mass located at its free end. In these two cases, the excitation is transverse to the beam and the nonlinear behavior is obtained by the geometric nonlinearities of the beam submitted to finite displacements.

For limiting the developments in this paper, only the inclusion that has been designed with a cantilever beam and its mass at its end is presented. The point mass m of the inclusion is equal to the sum of the mass of the cube, with the mass of the accelerometer attached to it for the measurements and with the mass of the accelerometer cable.

Preliminary experiments have been performed with a monopoint velocimeter Polytech PDV-100 without accelerometer for testing possible damping and stiffness effects of the accelerometer cable. No significant effect has been found with respect to the material damping and structural stiffness by comparison with the same results obtained with the accelerometer.

Design of the inclusion. To determine the length of the beam, its bending stiffness EI, and the point mass m of the inclusion,we have used the equation of the resonance frequency of a cantilever beam with a point mass at its end:

f p = 1 2π 3EI m 3 , (17)
for which it is assumed that the mass of the beam is negligeable with respect of the point mass. The magnitudes of these parameters are shown in Table 1, in which f p , , E, d max have been defined taking into account the experimental constraints for the manufacturing of the inclusion and where I has arbitrarily be fixed for limiting the number of design parameters. Material of the inclusion. The first idea for the material of the inclusion was to use metallic parts (steel, aluminum) for the beam and the point mass.

Parameters

However, it was difficult to correctly embed the point mass to the beam and the beam to its support. That is why we have chosen to use a 3D-printing system and to print the beam, its support, and the point mass in a single piece, using a polymer material.

Final design of the inclusion with its support. The inclusion is attached to a rigid frame for which the CAO of the final design is shown in Figure 8(a).

The point mass of the inclusion is constituted of a cube, embedded at the free end of the beam. The other end of the beam is continuously connected to the frame. The beam length is 0.026 m and the width of its square section is around 0.001 m. The width of the cube is about 0.008 m. The point mass m of the inclusion is equal to the sum of the mass of the accelerometer attached to it for the measurements (0.4 g), with the mass of the accelerometer cable (0.2 g), and with the mass of the cube (0.55 g), that is to say a total point mass of 1.2 g (the mass of the beam is neglected as we have assumed). The mass density of ABS is 1, 780 kg/m 3 . Some experimental traction tests have been carried out to identify the mechanical properties of ABS. The experiments give 2.2×10 9 P a for the Young modulus and 0.35 for the Poisson coefficient. This inclusion has been designed so that the first eigenfrequency of the frame is 1, 200 Hz and the first eigenfrequency of the inclusion around 24 Hz, which shows that the frame can effectively be considered as rigid for analyzing the dynamical response of the inclusion in the observed frequency band B o that is centered around the first eigenfrequency.

Experimental measurements and identification of the model

Manufacturing of the test inclusion

The inclusion is manufactured using a 3D-printing system. The inclusion and the rigid frame on which the inclusion is attached, are printed in one piece in ABS (Acrylonitrile Butadiene Styrend) that is a material commonly used for 3D printing with the fused deposition modeling technique (see Figure 2(b)).

Description of the experimental procedure

We are interested in analyzing the stationary random response of the inclusion in the frequency band of analysis B a = [0, f max ] with f max = 1, 024 Hz, induced by the stationary random excitation generated by an imposed acceleration of the embedded end of the beam via the rigid frame (see Figure 3). This acceleration is equal to the acceleration that is imposed to the frame (that can be considered as rigid in the observed frequency band), on which a stationary random external force is applied. The observed frequency band is B o = [START_REF] Wang | Membrane-constrained acoustic metamaterials for low frequency sound insulation[END_REF][START_REF] Spanos | On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: An alternative perspective[END_REF] Hz ⊂ B a , which contains the resonance frequency for the all amplitudes that are considered for the excitation. A Gaussian stationary random excitation in the frequency bandwidth [5 -800]Hz is applied, which is delivered by a shaker (Brüel & Kjaër 4808) with a power amplifier (Brüel & Kjaër 2719). The excitation signal, Ẍexp imp , is measured by an accelerometer (Brüel & Kjaër 4507-B004) attached to the frame and the response Ẍexp of the inclusion point mass by an accelerometer Endevco 2250A-10 attached to the point mass (see Figure 3). We have chosen to attach an accelerometer to the point mass for measuring its response, in order to avoid some problems that would be induced by the use of different instrumentations adapted to different excitation levels. The sensors data are acquired with National Instrument Hardware. A CompactDAQ module 9234 is used for the measurements of the two IEPE accelerometers and a module 9263 is used for the signal generation of the excitation. The modules are used with a 4-slot USB chassis 9174. The software used to process the measurements is SO Analyzer from m+p international. The useful bandwidth is set to 1, 024 Hz that leads us to a time sample rate of 2, 048 Hz. The block size is 16, 384 points, which corresponds to an acquisition time of 8 s for a measurement block and so the frequency step is equal to 0.125 Hz. The overlap factor is chosen equal to 66%. The number of blocks that are processed is 101, which corresponds to a total acquisition time of 280 s for each excitation level. The H 1 estimator with Hanning window is used to estimate the FRF functions. The experimental configuration can be viewed in Figure 3. Experimental responses have been measured for seven excitation levels. These cases are identified by the dimensionless response amplitude d/h in which h is the thickness of the beam (h = 0.001 m) and d is defined as the statistical mean of the " peak-to-peak" deflection amplitudes relative to the frame obtained for each measurement block, such that, using Eq. ( 5),

d = 1 2 E{max t (X exp s (t)) -min t (X exp s (t))} ( 18
)
where X exp s (t) = X exp a (t) -X exp imp (t). For instance, if d/h = 1, the displacement of the inclusion is of the order of magnitude of the thickness of the beam. Seven values of d/h are considered for the experiments and are, in percent, 1.9, 11, 52.3, 82.4, 96.7, 131, and 134.

Experimental results

The experimental power spectral density (PSD) functions S exp Fx of the excitation are displayed in Figure 4 for the seven considered values of d/h. Figure 5 depicts the measured transmissibility functions obtained from the post-processing of the time data of the two accelerometers for each values of d/h. For each block, the displacement signals are obtained by numerical double integration of the acceleration signals. A high-pass filter is applied before each numerical integration. The square of the modulus FRF exp (2πf ) 2 of the experimental frequency response function is displayed in Figure 6 and the experimental normalized input power density π exp in,norm (2πf ) is displayed in Figure 7 (which is the square root of FRF exp (2πf ) 2 as noted in the Eq. ( 9)).

Experimental identification of the model

As explained in Section 2.1, for all the considered values of d/h, the experimental identification of the damping and stiffness parameters is performed by minimizing over the observed frequency band B o , the distance between the function FRF(ω, w) 2 (defined by Eq. ( 9)) computed with the model and its experimental counterpart FRF exp (ω) 2 that is to say, by solving the following optimization problem,

w opt = arg min w∈C ad J(w) , (19)
in which the objective function J(w) is written as

J(w) = Bo |FRF(ω, w) 2 -FRF exp (ω) 2 |dω . (20)
Note that the right-hand side member of the above equation is the L 1norm of the function ω → FRF(ω, w) 2 -FRF exp (ω) 2 . The optimal value

w opt = (c opt 1 , c opt 2 , k opt 1 , k opt 3 ) obtained is        c opt 1 = 0.0038 N s/m , c opt 2 = 10 N s/m 2 , k opt 1 = 26.8 N/m , k opt 3 = -4 × 10 6 N/m 3 . (21)
During the identification process, we have noted that the damping effect is effectively nonlinear, that is coherent with the nonlinear dynamical model introduced in Section 2. This damping nonlinearity can also be explained by the fact that the beam of the 3D-printing inclusion exhibits a stratified structure of layers with pores (miss of matter) generated by the 3D-printing process (Figure 8). For high response amplitudes, the shear deformation could induce small slides at the interface of the layers. agreement between the experiments and the identified model. The prevision could certainly be improved by using a model with more than one degree of freedom. Figure 10 Furthermore, the results presented in these figures confirm a strong effect of the nonlinearity that allows the pumping energy phenomenon to be efficient over a broader frequency band around the resonance frequency than for the linear case.

Conclusions

In this paper, we have presented the results related to the first stage of a work devoted to the design and the analysis of a nonlinear microstructured material to reduce noise and vibration at low frequencies. We have designed an inclusion that is a nonlinear absorber at macroscale, which has been manufactured with a 3D printing system. The dimension of this inclusion could easily be reduced with the same technology, in order to be inserted in a macro-structural element. A nonlinear dynamical model of the inclusion has been developed and its parameters have been identified with experiments. Both the predictions given by the nonlinear dynamical model and the experiments confirm that the pumping energy phenomenon is more efficient over a broad frequency band around the resonance frequency than for a linear dynamical system. Note that the parametric experiments, which have been presented, allow for demonstrating the role played by the nonlinear geometrical effect absorption bandwidth, which is the expected phenomenon. The work in progress is the use of such inclusions for analyzing the attenuation of acoustics waves and vibrations in structural elements.

  imp (ω), in which S Ẍexp imp (ω) = ω 4 S X exp imp (ω). The autocorrelation function τ → R Ẍexp imp (τ ) of stationary stochastic process Ẍexp imp is such that R Ẍexp imp (τ ) = E{ Ẍexp imp (t + τ ) Ẍexp imp (t)} and R Ẍexp imp (τ ) = R e iωτ S Ẍexp imp (ω) dω. The generator of realizations of the Gaussian stationary

Figure 2 :

 2 Figure 2: CAO and 3D printing of the test structure.

Figure 3 :

 3 Figure 3: The experimental configuration.

  Measured transmissibility for the considered values of d/h. Zoom around the first eigenfrequency of the measured transmissibility for the considered values of d/h.

Figure 5 :

 5 Figure 5: Transmissibility functions for the different excitation levels of the inclusion base (the frame).

For

  each one of the seven values of d/h, Figure 9 displays the experimental function ω → FRF exp (ω) 2 (upper figure) and the optimized function ω → FRF(ω, w opt ) 2 (lower figure) corresponding to the experimental identification of the model. The comparison of these two figures shows a reasonable

  displays the experimental function ω → π exp in,norm (ω) (upper figure) and optimized model function ω → π in,norm (ω, w opt ) (lower figure), for the seven values of d/h. A reasonable agreement can also be seen between the prediction and the experiments.

  (a) Top view of the inclusion beam. (b) Side view of the inclusion beam. (c) View of a cross section of the inclusion beam.

Figure 8 :

 8 Figure 8: Several views of the 3D-printing beam of the inclusion.

4 )Figure 9 :

 49 Figure 9: For the seven values of d/h, experimental function ω → FRF exp (ω) 2 (upper figure) and optimized model function ω → FRF(ω, w opt ) 2 (lower figure).

2 )Figure 10 :

 210 Figure 10: For the seven values of d/h, experimental function ω → π exp in,norm (ω) (upper figure) and optimized model function ω → π in,norm (ω, w opt ) (lower figure).

Table 1 :

 1 Magnitudes of parameters for the design of the inclusion.

		Magnitudes
	f p	≤ 300 Hz
		≤ 0.03 m
	I	8.3 × 10 -14 m 4
	E	{10 9 , 250 × 10 9 } P a
	m	≥ 4.4033 × 10 -6 kg

  Square of the modulus FRF exp (2πf ) 2 of the experimental frequency response function for the seven values of d/h.

							9%
							d/h=11%
							d/h=52.3%
							d/h=82.4%
							d/h=96.7%
							d/h=131%
							d/h=134%
	10 2 Figure 6: 21 π in,norm (2π f) (W 2 /N 2 ) 10 1 Experimental	22	23	24	25	26	d/h=1.9% d/h=11% d/h=52.3% d/h=82.4% d/h=96.7% d/h=131% d/h=134%
			f (Hz)
	Figure 7: Experimental normalized input power density π exp in,norm (2πf ) for the seven values
	of d/h.

Acknowledgements

This work has benefited from a French government grant managed by ANR within the frame of the national program investments for the Future ANR-11-LABX-002-01.

Bibliography

[1] H. Frahm, Device for damping vibrations of bodies, United states patent office (1911) 1-9.