Thárcylla Clemente

Adiel Almeida-Filho
email: atalmeidafilho@yahoo.com.br

Marcelo Alencar
email: marcelohazin@gmail.com

Cristiano Cavalcante
email: cristianogesm@gmail.com

A Decision Support System Based on RCM Approach to Define Maintenance Strategies

Keywords: Decision Support System, Reliability Centered Maintenance, Maintenance Strategies

This article presents a Decision Support System which focuses on an important organizational context: maintenance decisions. Usually, a specific management system is required to deal with data on failure, reliability, maintainability and availability regarding equipment. The maintenance context is surrounded by uncertainty and does not provide a clear profit from its activities, and thus requires a tool which fits in with organizational needs. The tool presented in this paper considers the requirements and approaches of an industrial production system, using concepts and methodologies from Reliability Centered Maintenance (RCM) to establish maintenance strategies, such as inspection and preventive maintenance planning, and also takes into consideration sophisticated MCDA and FMECA models for strategic decision making.

Introduction

The globalized context requires organizations to formulate a strategic plan to ensure their permanency in the competitive market. Despite the heavy investments that many organizations have been making in communications, operations and negotiations, competitiveness requires organizations to maintain a continuous and updated flow of information within the environment of their organization in order to ensure their survival and to position themselves vis-à-vis constant change [START_REF] Campos | New patents on Business Process Management Information Systems and Decision Support[END_REF]. Thus, companies are constantly seeking to improve their performance [START_REF] Dekker | Applications of Maintenance Optimization Models: A Review and Analysis[END_REF].

Improving performance is linked to both the production function and the business function. Both are associated with the strategic objective of the organization. The activities performed in an organization all contribute to maintenance management, since all activities are related to maintaining the reliability of the operational system at a high level by replacing equipment or parts or maintaining operational conditions [START_REF] Pintelon | Maintenance management decision making[END_REF].

To fulfill its strategic objectives, an organization needs to adopt a management plan that is able to integrate its resources, and its personnel and operational policies. It must also acquire engineering capabilities and technologies that enable it to maximize its performance. In this scenario, maintenance actions are part of the production process and can influence the company's position in relation to its competitors as well as to its business strategy. If this is done, maintenance management may be a production activity that adds to profits rather than only an unavoidable business cost [START_REF] Verdecho | Prioritization and management of inter-enterprise collaborative performance[END_REF].

The main objective of maintenance management is to plan the operational system and to manage the product (benefit and/or service). Thus, it is premised on its being deployed to supervise and control the occurrence of failures in a production plant. This task can become complicated and the use of methodologies can facilitate understanding how to analyze the context.

One of the methodologies used for the effective management of failure is preventive maintenance. This methodology specifies maintenance activities to ensure the plant functions for a given period of time, and to avoid production stopping unexpectedly. One of the approaches that adopt this methodology is Reliability Centered Maintenance (RCM).

RCM is understood as a procedure that is able to identify requirements for preventive maintenance in complex systems in order to preserve or maintain materials, people and equipment on the production line in acceptable conditions [START_REF] Moubray | Reliability-centered Maintenance: RCM II[END_REF]. When RCM is combined with other approaches, it provides a complete understanding of the operational context, and provides management and financial information throughout the organization.

From this perspective, the use of information technology resources can offer some advantages to organizations worldwide, since they enable complex data to be managed and operations to be planned [START_REF] Deshpande | Application of RCM for safety considerations in a steel plant[END_REF].

The main contribution of this paper is to put forward a Decision Support System for maintenance management. To this end, it will present the methodologies and framework concepts used to model the system as well as a brief study of the system requirements for the maintenance area.

Reliability Centered Maintenance

RCM is an approach that considers the economic prioritization and the security function of the production system; thus, its main contribution is to identify opportunities to do preventive maintenance in order to ensure that any physical component continues to fulfill its operational functions [START_REF] Moubray | Reliability-centered Maintenance: RCM II[END_REF]. This is why the flow of preventive maintenance activities must be consistently modeled.

In addition, RCM is used to develop and select maintenance alternatives based on the operational, economic and safety criteria of a production system. This method is shown to be a systematic process for managing functional and operational failures [START_REF] Jones | Reliability-Centered Maintenance (RCM) for Command, Control, Communications, Computer, Intelligence, Surveillance, and Reconnaissance (C4ISR) Facilities[END_REF].

First of all, RCM identifies the appropriate maintenance tasks for maintaining the current reliability level of resources and people in the production environment, which will reduce costs in the maintenance operation [8].

Essentially, RCM presents a methodology based on subjective basic questions that may be extended to any applications [START_REF] Jia | Case Experience Comparing the RCM Approach to Plant Maintenance with a Modeling Approach[END_REF]:  Is the cause of failure obvious to the operator?  What is the consequence of failure?  What tasks can be selected to prevent this failure?  What is the reason for selecting a specific task?  What is the suggested scope of the task?

These simple questions enable extensive knowledge about the operational context of organizations to be gained. While the answers are being given, the system is mapped and relevant information about the occurrence of failures is highlighted. This mapping may be presented in the form of a logical diagram in which the steps correspond to those of the RCM.

Main steps of the RCM

Different sequences for implementing the RCM [START_REF] Deshpande | Application of RCM for safety considerations in a steel plant[END_REF][START_REF] Rausand | Reliability centered maintenance[END_REF][START_REF] Matteson | Airline experience with RCM[END_REF][START_REF] Nowlan | Reliability-centred maintenance[END_REF] are identified, but the main benefit offered by this methodology, is knowledge about the entire maintenance process, which depends on the detail level given in the process [START_REF] Smith | RCM: Gateway to World Class Maintenance[END_REF].

In the first instance, the RCM process follows recognizing and identifying the functional requirements of the items in the production context and it establishes the desired reliability levels. For each item, it identifies specified functions and the functional failures associated with them. Then, an analysis is carried out on the failure mode and its effects [START_REF] Alencar | An MCDM model for potential failure causes ranking from FMECA[END_REF]. This concept is very important for the maintenance context because it analyzes the existing risk in the operational production [START_REF] Braglia | MAFMA: multi-attribute failure mode analysis[END_REF][START_REF] Dong | Failure mode and effects analysis based on fuzzy utility cost estimation[END_REF]. Then, the process defines the consequence type observed because of the occurrence of the given failure. These consequences are present in four categories: hidden, safety or environmental, operational, and non-operational [START_REF] Nowlan | Reliability-centred maintenance[END_REF]. Thus, the process identifies the maintenance actions to be put in place in the production system. In general, two profiles of maintenance actions are applied: preventive maintenance and corrective maintenance. Figure 1 presents the simple process of RCM. The RCM process offers advantages and opportunities to improve the performance and safety of equipment throughout the production process by minimizing waste and machinery downtime [START_REF] Moubray | Reliability-centered Maintenance: RCM II[END_REF]. The effort to maintain a consistent assessment process enables accurate and timely information to be obtained, which are important features for maintenance management [START_REF] Abdul-Nour | A reliability based maintenance policy: a case study[END_REF].

In the maintenance area some data are considered strategic, especially those regarding the failure rate, downtime, time to repair and all historic data related to these variables [START_REF] Christer | An Operation Research Approach to Breakdown Maintenance: Problem Recognition[END_REF]. These data are obtained in general from the performance records of a corrective maintenance service and from experts' knowledge.

The logical structure of the RCM process presents consistency for the specification of an information system that is able to process and provide information for the maintenance area. In this treatment, the focus is placed on the context of industrial production, and aims to extract information from the strategic level for maintenance actions [START_REF] Kans | An approach for determining the requirements of computerised maintenance management systems[END_REF].

Strategic level information is characterized by the non-structured data that are presented and therefore the introduction of a Decision Support System (DSS) is timely. The consideration of requirements to shape the system model is important before introducing a DSS.

Maintenance Management System Requirements

Using Information Technology resources presents data processing advantages in the business context. Data treatment offers gains in speed and certainty in the decision making process. Thus, the use of information systems adds value to the operational functions of an organization [START_REF] Melville | Information technology and organizational performance: an integrative model of IT business value[END_REF].

Adding value in the production context is crucial for acquiring a competitive advantage in the context of current business, since the trend for consumers to require product differentiation is increasing. In light of these arguments, the acquisition of an appropriate information system is essential. An appropriate information system is one that not only meets the needs of management in the organization as it provides relevant information for decision making but is also an information system that helps the organization achieve its objectives at different levels: operational, tactical and strategic [START_REF] Delone | Information systems success: the quest for the dependent variable[END_REF].

In general, an information system should meet the development requirements imposed by the user of the service. The user's role is to provide information about his/ her preferences and needs with regard to using the information system. Among the requirements are those that refer directly to the characteristics of the information system and the reliability and quality associated with the system. These groups of requirements are defined by the context in which the information system is inserted. In general, an information system must present values for six crucial aspects [START_REF] Powell | Information technology as competitive advantage: the role of human, business, and technology resources[END_REF], which are described in Table 1.

Table 1. Description of the requirements of an information system

Requirement of Information System

Description System Quality

The information system should meet the client´s needs satisfactorily Information Quality

The information system should provide relevant and timely information System Use

The information system should deal with cognition requirements and usability User Satisfaction

The information system should be easy to operate and/or keep in operation Individual Impacts

The information system should enable the user to learn, thus encouraging individual development Organizational Impacts

The information system should have wellinformed data on operational performance For each operation in the organization there is one information system type that is the most appropriate. In the exercise to define maintenance strategies, the appropriate information system is the one that, in addition to providing operational information, supports the decision-making process. A Decision Support System is a type of information system that has specific functions that enable simulations to be conducted and scenarios to be modeled using a structure with a database of the data on thwe requirements/criteria and a database of the model obtained from dialogic interaction with the user [START_REF] Watson | Case-based reasoning is a methodology, not a technology[END_REF].

For the context of maintenance management, data processing is essential. It is present at all organizational levels. The demand for maintenance management illustrates different perspectives on implementing an information system. These perspectives take into consideration the goal, purpose, use and user of each organizational level.

At the strategic level, the information system for the maintenance area has the following features: its goal is to ensure the cost-effectiveness of maintenance; its purpose, decision support; its main use is for data output (report generating facility, integrating information technology); and its user is the maintenance manager [START_REF] Melville | Information technology and organizational performance: an integrative model of IT business value[END_REF].

The process of identifying the demands on the information system in the maintenance area follows different methodologies. It is important that the steps of the process are outlined so as to meet the real needs of the context.

Implementing a Decision Support System for the maintenance area contributes to structuring the subjectivity of the context. Therefore, the occurrence of failures and interruptions to production are registered consistently and this enables the performance of the maintenance to be maximized with regard to preventing or correcting failures. Thus, it comes to be seen as a management tool that adds value.

The use of specific concepts and specific methodologies of the maintenance area assists in processing knowledge and identifying the several factors and elements involved, in context. The RCM process enables interactions of these factors to be identified. Thus, the logical structure of RCM was used to facilitate the modeling of the DSS presented for maintenance management.

A DSS for Maintenance Management

The main goal of this section is to describe and justify a decision support system for the maintenance area. In this perspective, the RCM process was used to outline the application context and the features of the system as per management needs. Thus, the DSS enables the maintenance actions that best suit the scenarios analyzed to be visualized and identified.

System Architecture

Maintenance actions can be applied to different items belonging to the production system. To this end, what must be known are the several interactions between the components of the system and the equipment in operation. This is a requirement addressed by the DSS proposed, which enables complete knowledge of the production system to be obtained.

To address this consideration, the DSS presents an architecture comprising modules that enable the complete identification of components and their operations in the production system. DSS has two main modules: (i) a physical module and (ii) a functional module. These are described in Table 2 The cognitive aspects considered include the usability of the components and constant access to information. This requirement is critical to the maintenance area, in view of the constant monitoring of changes. Both modules provide a complete view of the context studied in the maintenance area and allow a direct flow to the users of the system. However, the Functional Module provides consistent features to define the maintenance strategies desired by organization.

Main Flow of the Functional Module

The main flow of the Functional Module of DSS is to insert user data, to validate data by means of the several interactions and to analyze data and, finally, to provide an analysis the results for decision making. All functional features of the module correspond to steps developed by the RCM process. Thus, there are occurrences of failures analysis and criticality analysis of failure modes that form the foundation to define strategic actions to maintain the items in the analysis of the production system. The process undergone by the Functional Module may be represented in five steps which are illustrated in Figure 2. The steps of the Functional Module of DSS enable a sequence designed for a simple navigation in the system to be viewed. It is important that the requirements of the system are obtained so as to construct this sequence. A hierarchy of the functional components was created in order to design a productive context for the DSS.

Thus, consider a productive process. A system is characterized by a set of items that has a higher functionality. This system can consist of subsystems, and those for other subsystems. Each system or subsystem has a function. This function may comprise several potential functional failures that are described by the conditions of operation of the system or subsystem. Each functional failure can occur in an item or a set of items in the production system. For each occurrence of failure on an item there is a failure mode associated with it. This failure mode informs the manner and context that led to the occurrence of a functional failure. Each failure mode presents essential characteristics to determine its criticality as it is important to identify how critical the failure mode for the production system is.

In DSS this sequence is shown in a hierarchy, which facilitates visualizing and compiling elements in the production context. Figure 3 represents the interface of the functional module of the DSS, with emphasis on the evaluation stage of the critical failure mode.

An analysis of the criticality of the failure mode is a very important resource used to define maintenance actions. This issue is closely related to the levels of risk the organization is willing to face. Many studies have been developed to provide a consistent approach so as to obtain the criticality level of the failure modes [START_REF] Alencar | An MCDM model for potential failure causes ranking from FMECA[END_REF][START_REF] Dong | Failure mode and effects analysis based on fuzzy utility cost estimation[END_REF].

In the DSS, the criticality analysis is structured by a mathematical model that classifies the severity and risk associated with a failure mode. The information required for this analysis suggests that the user should set qualifying levels of detectability and frequency of failure, as well as levels of severity in line with o the damage: human, environmental, operational and financial, caused by the occurrence of the failure mode in specific equipment the production line, as illustrated in Figure 3. Fig. 3. This is the interface for defining the criticality of failure modes. The hierarchy of the items of the production system is shown on the left. The definition of criticality of failure modes is calculated from the data on detectability, frequency and severity.

After acquiring this information, what follows is to define strategic maintenance actions for the production system, especially for the operations of each piece of equipment/item.

Definition of maintenance actions

The information extracted from the DSS depends on the frequency of the occurrence of the failure modes of the production system under study. The information contributes to a summarized analysis of the occurrence of failure, a critical step in defining strategies for maintenance actions.

Classifying the severity and the risks is the basis for defining the critical failure mode. This definition is based on a correlation matrix that identifies how critical the occurrence of a failure is for the production system. In turn, the criticality encourages the mechanism to define the maintenance action.

The process of defining the maintenance action considers several aspects, including cost, timing of maintenance and operational safety in the production system. Thus, to establish the most appropriate strategy for the context under study, the DSS follows a specific logical diagram which is guided by the RCM methodology.

Based on the basic issues of the RCM process, questions are designed to outline the context, given the objects observed (system, subsystem, function failure, and failure mode group) that were selected. The questions are provided in a logical sequence while the maintenance action recommended by the system is being defined. Figure 4 represents the interface to define the maintenance action. Fig. 4. This is the interface for defining the maintenance strategy. The user must answer the questions by following a sequence diagram. This process will recommend the most appropriate maintenance action for the situation.

It is simple to obtain the process for a maintenance strategy. The questions listed are direct ones and the process enables answers to be categorized. This feature adds value to the process, and aims at the usability of the information system.

Among the alternatives offered by the DSS are: scheduled restoration, scheduled discards, on condition, redesign, failure-finding, not scheduled, combination of tasks and capacity and training of staff. Table 3 gives a description of each maintenance action recommended by the DSS.

The type of maintenance action recommended will depend on the context analyzed. However, it can be helpful to list the aspects considered relevant to define the strategy process: cost, quality and production performance. From this consideration, it can be concluded that there is a relationship between business strategy and maintenance strategy [START_REF] Pinjala | An empirical investigation on the relationship between business and maintenance strategies[END_REF]. The benefits of DSS are moreover based on the type of maintenance action recommended. For these actions that enable the time interval to be analyzed, modeling to estimate the maintenance period based on the inspection and maintenance based on replacing pieces of equipment/items are suggested.

This resource is central to the maintenance strategy, since after learning what to do, it is essential to know when to do it. The results of time analysis are based on mathematical modeling of the specific items of the maintenance area and these are illustrated graphically to the user. Thus, the reports of maintenance activities are consistent since all that needs to be considered doing in the same time period is shown.

Another contribution to define maintenance strategies is the use of multicriteria methods to criticality analysis. The use of multicriteria methods is often in maintenance area in special when complex problems are highlighted in maintenance management systems [START_REF] Almeida | Multicriteria decision making on maintenance: spares and contracts planning[END_REF][START_REF] Cavalcante | A preventive maintenance decision model based on multicriteria method PROMETHEE II integrated with Bayesian approach[END_REF]. Some examples these problems in maintenance area include risk evaluation [START_REF] Brito | Multi-criteria decision model for selecting repair contracts by applying utility theory and variable interdependent parameters[END_REF][START_REF] Brito | Multi-attribute risk assessment for risk ranking of natural gas pipelines[END_REF], outsourcing and logistics [START_REF] Almeida | Multicriteria modeling of repair contract based on utility and Electre I method with dependability and service quality criteria[END_REF], project management [START_REF] Alencar | Assigning priorities to actions in a pipeline transporting hydrogen based on a multicriteria decision model[END_REF][START_REF] Mota | A multicriteria decision model for assigning priority classes to activities in project management[END_REF] and many others.

Conclusion

Investment in information technology in the maintenance area presents is an important resource to strategic position in the competitive context [START_REF] Almeida | Multicriteria decision making on maintenance: spares and contracts planning[END_REF]. Since this investment enables value to be added to the decision-making process, it ensures the operational conditions of the organization are satisfactory.

DSS provides a logical sequence for the particular definition of maintenance strategies and provides features that allow a wide understanding of the scenario studied. The application of the DSS satisfies the needs of the maintenance area in the industrial context and is able to deliver results and benefits in any organizational context. Thus, it becomes a tool to support operations management and on which recommends strategic activities.

Thus, this prompts the development of research into the interaction between the concepts involved in the maintenance area with the concepts of decision theory. This article also shows that the multicriteria decision-making process can make a contribution to constructing a new perspective on the occurrence and definition of maintenance actions based on analysis of the criticality failure mode.

Fig. 1 .

 1 Fig. 1. Simplified representation of the RCM process [17].

Fig. 2 .

 2 Fig. 2. The structure of the Functional Module consists of five main steps, which are defined by the features of the system.

Table 2 .

 2 Description of the modules of a DSS.

	Module	Activity	Data
	Physical	Registry	Item of production system
			Occurrence of failures
			Failure detection levels
	Functional	Registry	Data on the system, subsystem,
			function, functional failure, equipment
			and components, failure modes
		Analysis	Criticality of failure modes
		Definition	Maintenance Strategy

Table 3 .

 3 Description of maintenance strategies recommended by the system.

	Strategy Maintenance	Description
	Scheduled restoration	Restoration of the initial production capacity of an
		item or component
	Scheduled discard	Discard of an item or component before the age
		limit specified
	On condition	Is the evaluation of operational conditions and
		perceptions of potential failures
	Redesign	This identifies the need for change in design
		specifications of the item or component
	Failure-Finding	To find hidden failures in regular time intervals
	Not scheduled	This is adopted when failure is hidden but does not
		affect the performance of the production system
	Combination of tasks	This is the effective combination of other
		maintenance activities to solve a problem
	Capacity and training of staff	This consists of the organization's ability to invest in
		training staff to operate the machinery available


Farid Baleh
email: farid.baleh@gmail.com

Collatz conjecture

that becomes periodic converges)

Introduction

The Collatz conjecture (or Syracuse conjecture, Ulam conjecture or 3x+1 problem) claims that the following sequence of natural numbers reaches the value 1 after a certain rank (in this article, it will be specified that the sequence converges): s 0 ≥ 1, and for all natural number n :

s n+1 = sn 2 if s n is even, 3s n + 1 if s n is odd.
3p+1 being even if the natural number p is odd, the compressed sequence (c n ) of the sequence (s n ) is defined as follows: c 0 ≥ 1, and for all natural number n :

c n+1 = cn 2 if c n is even, 3cn+1 2
if c n is odd.

An uncompressed (or compressed) sequence that converges continues, after a certain rank, with the trivial cycle 1-4-2 (or 1-2) that is infinitely repeated.

2 General expression of an odd element of a Collatz sequence

Considering the extracted sequence (u n ) composed of the odd elements of the sequence (s n ), two successive elements have the following relationship:

u n = 3u n-1 + 1 2 k ′ n (1)
Where k ′ n is the number of divisions by 2 of the first even element of (s n ) (following u n-1 ) before reaching the first successive odd element of (s n ), i.e. u n . By developing the previous expression:

u n = 3 n u 0 2 n j=1 k ′ j + 3 n-1 2 n j=1 k ′ j + 3 n-2 2 n j=2 k ′ j + ... + 3 0 2 k ′ n
Then:

u n = 3 n u 0 2 n j=1 k ′ j + n i=1 3 n-i 2 n j=i k ′ j
After factorization of the u 0 multiplier and simplification of the second term:

u n = 3 n 2 n j=1 k ′ j u 0 + n i=1 3 -i * 2 i-1 j=1 k ′ j Moreover:
∀j ≥ 1 : k ′ j = 1 + k j Because the successor of an odd element of (s n ) is always an even element. k j is the number of divisions by 2 of the first even element of (c j ) (following u j-1 ) before reaching the first successive odd element of (c j ), i.e. u j . Therefore:

n j=1 k ′ j = n + n j=1 k j and i-1 j=1 k ′ j = (i -1) + i-1 j=1 k j n j=1
k j is the number of the even elements, at the rank n, of the compressed sequence (c n ) that follow u 0 . By introducing k j , the expression of u n becomes:

u n = 3 2 n 1 2 n j=1 kj u 0 + 1 3 n i=1 2 3 i-1 * 2 i-1 j=1 kj
After a shift on the index i, the expression of u n , depending on a given odd element u 0 , of n and of the n first elements of the sequence (k n ), we have that:

u n = 3 2 n 1 2 n j=1 kj u 0 + 1 3 n-1 i=0 2 3 i * 2 i j=1 kj
Consequently:

u n = 1 a n u 0 + x n 3 (2)
x n and a n being defined by the following expressions:

x n = n-1 i=0 2 3 i * 2 i j=1 kj a n = 2 3 n * 2 n j=1 kj (3)
For all n, x n and a n are strictly positive. Therefore the sequence (x n ) is strictly increasing. Moreover:

a n = x n+1 -x n and x n = n-1 i=0 a i (4)
Notice that: x 0 = 0. The Excel file used to verify the formula of the equation ( 2) tends to show that, for sufficient large n, a n → +∞ and xn 3an → 1 (which corresponds to u n = 1). It is the case if the Collatz sequence becomes periodic (see the demonstration below).

Convergence of a periodic sequence after a certain rank

We hypothesize that the sequence (s n ) is periodic after a certain rank n 0 . Therefore it is the case of the sequence (u n ).

If T is its period:

∀n ≥ n 0 , u n+T = u n . In particular: u n0+T = u n0
In the rest of this paragraph, the sequence (u n ) is considered after u n0 : then the index n begins to 1 (for exemple, u 1 is noted u n0+1 , successor of u n0 ). Consequently, according to the equation ( 2), with u n0 as the first reference element:

1 a T u n0 + x T 3 = u n0
Then: (a T -1)u n0 = x T 3 Therefore, like u n0 and x T are strictly positive:

a T > 1 (5)
3.1 a n → +∞ when n → +∞ For all n, the Euclidean division of n by T implies that: n = q n T + r n , with: 0 ≤ r n < T .

According to the equation (3), by replacing n:

a n = 2 3 qnT +rn * 2 qn T +rn j=1 kj
Then:

a n = 2 3 qnT * 2 qn T j=1 kj * 2 3 rn * 2 qn T +rn j=qn T +1 kj
The sequence (u n ) being periodic, the sequence (k ′ n ) is also periodic by reference to equation (1); therefore, it is also the case of the sequence (k n which implies the following two equalities:

qnT j=1 k j = q n * T j=1 k j qnT +rn j=qnT +1 k j = rn j=1 k j
And consequently:

a n = (a T ) qn * a rn (6)
The expressions of a T and a rn being as follows:

a T = 2 3 T * 2 T j=1 kj a rn = 2 3 rn * 2 rn j=1 kj
Moreover, for all n :

a rn ≥ I, with:

I = inf(a i ) i∈[0,T [
r n belonging to the interval [0, T [. I is a stricly positive number because it is the case of a i , for all i.

Then, for all n: a n ≥ (a T ) qn * I If n → +∞, then q n → +∞ and a T is strictly greater then 1. We can conclude that: lim n→+∞ a n = +∞ And that: lim n→+∞ 1 a n = 0

xn an converges when n → +∞

According to the equation ( 4) :

x n a n = 1 a n n-1 i=0 a i = n-1 i=0 a i a n
Like for the index n previously, the Euclidian division of i by T is expressed by: i = q i T + r i , with : 0 ≤ r i < T . According to the equation ( 6):

For all i: a i = (a T ) qi * a ri (7)

The table below gives the values of q i , r i and a i related to the values of the index i (from 0 to n = q n T + r n ).

i q i r i a i 0 0 0 a 0 1 0 1 a 1 2 0 2 a 2 - - - - T-1 0 T-1 a T -1 T 1 0 a T * a 0 T+1 1 1 a T * a 1 T+2 1 2 a T * a 2 - - - - 2T-1 1 T-1 a T * a T -1 2T 2 0 a 2 T * a 0 2T+1 2 1 a 2 T * a 1 2T+2 2 2 a 2 T * a 2 - - - - 3T-1 2 T-1 a 2 T * a T -1 - - - - q n-1 T q n-1 0 a qn-1 T * a 0 q n-1 T +1 q n-1 1 a qn-1 T * a 1 q n-1 T +2 q n-1 2 a qn-1 T * a 2 - - - - q n T -1 q n-1 T-1 a qn-1 T * a T -1 q n T q n 0 a qn T * a 0 q n T +1 q n 1 a qn T * a 1 q n T +2 q n 2 a qn T * a 2 - - - - q n T + r n -1 q n r n -1 a qn T * a rn-1 q n T + r n q n r n a qn T * a rn
Therefore, by reference to this table and to the equation ( 4):

x n = a 0 T (a 0 +a 1 +...+a T -1 )+a 1 T (a 0 +a 1 +...+a T -1 )+...+a qn-1 T (a 0 +a 1 +...+a T -1 )+a qn T (a 0 +a 1 +...+a rn-1 )

Then:

x n = qn-1 i=0 a T i T -1 i=0 a i + a qn T rn-1 i=0 a i
We have:

x T = T -1 i=0 a i and qn-1 i=0 a T i = 1 -a qn T 1 -a T (because q n-1 = q n -1; see table) and x rn = rn-1 i=0 a i
Therefore:

x n = 1 -a qn T 1 -a T x T + a qn T x rn
Consequently, according to the previous formula and to the equation ( 6):

x n a n = 1 (a T ) qn * a rn 1 -a qn T 1 -a T x T + a qn T x rn
Then:

x n a n = 1 a rn 1 -( 1 aT ) qn a T -1 x T + x rn
When n → +∞, then q n → +∞ and 1 aT is strictly lower than 1, according to (5). Consequently, ( 1 aT ) qn tends to 0, which implies that the sequence ( xn an ) is convergent because it tends to the stricly positive following limit:

1 a rn 1 a T -1
x T +x rn = 1 a rn 3u n0 +x rn = 3u rn (see equality before (5) and equation ( 2))

Remark that the sequence ( xn 3an ) tends to the limit u rn , this natural number belonging to the following set of T elements: u n0 , u n0+1 , ..., u n0+T -1 . As this limit is necessarily unique, that fact tends to show that there is only one element in this set, and then T = 1 and u rn = u n0 . At this point of that demonstration, let simply precise that ( xn 3an ) converges.

u

n → 1 if n → +∞
According to the equation (2) (replacing u 0 by u n0 ), u n is the sum of two convergent sequences:

un 0 an and 1 3 ( xn an ). Therefore the sequence (u n ) converges when n → +∞. If l is its limit (l is greater or equal to 1 because it is the case of u n , for all n). Note that we also have: l = u rn (see end of section 3.2). After a certain rank, u n is equal to the number l (due to the fact this is a sequence of natural numbers). According to the equation (1), and for sufficient large n:

u n+1 = 3u n + 1 2 1+kn+1
Therefore:

l = 3l + 1 2 1+kn+1 Which implies the following equality: l(2 1+kn+1 -3) = 1
The product of these two natural numbers is equal to 1. Consequently, each of theses two numbers is equal to 1. Effectively, if pq = 1 (p and q being integers), q divides 1 (because p = 1 q ), which implies that q = 1 (1 is divisible only by itself), and then p = 1. Therefore: l = 1. Moreover k n+1 = 1, due to the fact that 2 1+kn+1 -3 = 1. The sequence (k n ) tends towards 1. Therefore, u n = 1 after a certain rank, and the Collatz sequence (s n ) converges.

Note that this last demonstration proves also that if two successive elements of the sequence (u n ) are equal, then this sequence converges and tends towards 1.

Moreover, we can remark that, at the end of section 3.2, we found that the limit of the sequence ( xn 3an ) is equal to u rn ; as T = 1 and l = 1, we have r n = 0, and then u rn = u n0 = 1, which is consistent.

Conclusion

A Collatz sequence that is periodic after a certain rank converges, i.e. it reaches the value 1. Therefore, the Collatz trivial cycle 1-4-2 (or 1-2 for a compressed sequence) is unique, which solves a half of the Collatz conjecture. Note that a consequence of this demonstration is the following: a Collatz sequence that is upper bounded is convergent. Effectively, such a sequence becomes periodic after a certain rank, with a maximum period that is equal to the number of odd elements of the sequence (u n ) lower than its maximum. To prove the Collatz conjecture, it is for example sufficient to demonstrate that each Collatz sequence has an upper bound.


Xavier Beudaert
email: xbeudaert@ideko.es

Iker Mancisidor

Luis Miguel Ruiz

Asier Barrios

Kaan Erkorkmaz

Jokin Munoa

Analysis of the feed drives control parameters on structural chatter vibrations

Keywords: Chatter, Machine tool drive, Control

Chatter vibrations are a major limitation for machining operations and lead to productivity reduction, low tool life and poor surface finish. When structural chatter occurs, the whole machine is vibrating and the feed drive behaviour cannot be neglected. Indeed, the proportional gain of the velocity loop is a key parameter influencing the damping provided by the drives and it can be adjusted to increase the stability limit. The objective of this article is to analyse the effect of the feed drive control parameters on the structural chatter vibrations. The process stability is studied with stability lobe diagrams and with root locus techniques applying the Padé formulas to approximate the regenerative effect delay by a rational function. This way, a single degree of freedom turning process is analysed from different point of views and the impact of the feed drive parameters is presented.

INTRODUCTION

The structural chatter vibrations can limit the productivity of large machine tools when the dynamic stiffness of the machine is lower than the one required by the cutting process [START_REF] Munoa | Chatter suppression techniques in metal cutting[END_REF]. At the design stage, the machine structures can be optimized to increase the static stiffness and the natural frequencies but it is difficult to estimate the damping which is one of the most important dynamic parameters to predict the stability. As result, some structural modes can be poorly damped and cause chatter vibrations. In order to increase the dynamic stiffness and hence, the stability, some authors proposed the introduction of external devices to add damping to the machine's structure. Passive absorbers increase the damping of the critical mode and are characterized by their relatively low cost and simplicity [START_REF] Den | Mechanical Vibrations[END_REF] [START_REF] Sims | Vibration absorbers for chatter suppression: A new analytical tuning methodology[END_REF]. However, they have problems in situations in which the dynamic parameters change during the process. The employment of self-tuned passive dampers is a recently reported alternative [START_REF] Munoa | Design of self-tuneable mass damper for modular fixturing systems[END_REF]. Nevertheless, the use of active dampers overcomes all these problems since they actuate over a range of frequencies using a feedback control algorithm [START_REF] Cowley | Active dampers for machine tools[END_REF] [6] [START_REF] Munoa | Chatter suppression in ram type travelling column milling machines using a biaxial inertial actuator[END_REF]. Another interesting solution was presented by the authors [START_REF] Munoa | Active suppression of structural chatter vibrations using machine drives and accelerometers[END_REF], where the damping is increased by an additional feedback loop in the machine's feed drives. To obtain an efficient result, an accelerometer was located close to the cutting point and the active force was introduced by means of the feed drives, avoiding the necessity of an external device. Hence, the ability of the feed drives to suppress structural chatter vibrations has been demonstrated, but the introduction of an additional control loop with the external accelerometer can induce instability issues [START_REF] Beudaert | Limiting factors for the active suppression of structural chatter vibrations using machine's drives[END_REF]. Structural chatter vibrations are characterized by low frequency modes ) with mode shapes inducing vibrations that can be measured by the drives encoders. Various studies analysed the control tuning of the feed drives from different point of view such as drive static accuracy, bandwidth, tracking accuracy, disturbance rejection, and provided general feed drive commissioning rules [START_REF] Groß | Electrical feed drives in automation[END_REF] [START_REF] Zirn | Machine Tool Analysis -Modelling, Simulation and control of machine tool manipulators[END_REF]. A review of the work related to active damping of drive vibrations caused by the machine movements has been presented in [START_REF] Altintas | Machine Tool Feed Drives[END_REF]. These active damping techniques are focused on the reduction of tool tip vibrations which are caused by inertial loads but the chatter vibrations are not considered. New feed drive control strategies are presented in the literature to suppress vibrations but the classical cascade control structure used on the feed drive can also have a significant impact on the dynamic stiffness of the machines. Significant variations of the structural chatter stability limit have been observed experimentally when the feed drives tuning parameters are changed. Initial investigations were carried out by Alter and Tsao [START_REF] Alter | Stability of turning processes with actively controlled linear motor feed drives[END_REF] regarding the interaction between the control system of a feed linear motor and the stability of the turning process. More recently, Albertelli et al. [START_REF] Albertelli | The effects of dynamic interaction between machine tool subsystems on cutting process stability[END_REF] suggested a control tuning criteria that maximize cutting process stability. In this work, the capacity of adding damping by means of the usual control loops of feed drives is analysed. For that purpose, the effect of each loop gain on the machine dynamics is analysed and after that, their impact on the stability is studied.

MODELLING

To analyse the effect of the control gains on the feed drives dynamics and on the process stability, a simple single degree of freedom model that allows understanding the effect of each parameter is studied.

Feed drive modelling

Figure 1 presents schematically the single degree of freedom system with a vibration mode on a turning tool realizing an orthogonal cutting operation. A closed loop control of the velocity and position of the mass 𝑚 1 is simulated to represent the behavior of a linear motor. 𝑚 2 represents the vibrating structure.

Figure 1: Schematic model of the system.

The system is represented by the following equations: 𝑋 2 (𝑚 2 𝑠 2 + 𝐶𝑠 + 𝐾) = 𝑋 1 (𝐶𝑠 + 𝐾) -𝐹 𝐶

(1)

𝑋 1 (𝑚 1 𝑠 2 + 𝐶𝑠 + 𝐾) = 𝑋 2 (𝐶𝑠 + 𝐾) + 𝐹 𝐷 (2)

𝐹 𝐷 = -𝑋 1 (𝑠 + 𝐾 𝑣 )𝐾 𝑝 (1 + 1 𝑇 𝑖 𝑠 ) (3)
where 𝑚 1 and 𝑚 2 are the first and second mass linked by a rigidity 𝐾 and a damping 𝐶. The controller of the linear motor is composed of a velocity loop with a proportional gain 𝐾 𝑝 and an integral time 𝑇 𝑖 . The position loop is composed of another proportional gain 𝐾 𝑣 . The drive is generating a force 𝐹 𝐷 on the first mass while the cutting force 𝐹 𝐶 is applied to the second mass.

The proportional gain of the velocity loop 𝐾 𝑝 is the main parameter to define the bandwidth of the drive 𝑓 𝑏𝑤 in Hz. 𝑓 𝑏𝑤 should be adjusted according to the first natural frequency 𝑓 0 of the machine and as a rule of thumb it can be chosen as 𝑓 𝑏𝑤 = 0.8 * 𝑓 0 . Using 80% of the natural frequency, the optimum damping is approximately achieved, for the usual mass ratio (𝑚 2 < 𝑚 1 ) and damping of the machine tools (𝜉 = 2 -7%). Once this parameter is selected the velocity proportional gain 𝐾 𝑝 can be calculated with Eq. 4, considering a rigid system of mass 𝑚 1 + 𝑚 2 . Eq. 5 provides a damping of 70% to the integral pole and Eq. 6 ensures not to have overshoot on the position loop. This tuning will be referred as optimum damping for the rest of the article.

𝐾 𝑝 = 2𝜋(𝑚 1 + 𝑚 2 )𝑓 𝑏𝑤 (𝑁/(𝑚/𝑠) (4)
𝑇 𝑖 = 1 𝜋 𝑓 𝑏𝑤 (𝑠) (5)
𝐾 𝑣 = 𝜋 𝑓 𝑏𝑤 2 . 60 1000 ((𝑚/𝑚𝑖𝑛)/𝑚𝑚) (6)
The integral term is important for disturbance rejection, whereas the position gain 𝐾 𝑣 increases the tracking accuracy. However, when the velocity feedforward is used, the position loop has a secondary importance. In general, the given formulas provide a satisfactory starting point for the commissioning of feed drives.

In this study, two different objectives should be fulfilled by the control parameters. On the one hand, the axis response should be satisfying in terms of settling time, overshoot and disturbance rejection [11] [12]. On the other hand, the process stability should be as high as possible.

Process model

An orthogonal turning process is modelled as a control system with feedback loop as introduced by Merrit [START_REF] Merrit | Theory of self-excited machine-tool chatter. Contribution to machine-tool chatter. Research 1[END_REF] (Fig. 2). The static chip thickness ℎ 𝑠 is modulated by the current vibration 𝑋 2 and by the vibration occurred in the previous revolution 𝑋 2 𝑒 -𝜏𝑠 with 𝜏 the time needed to have a complete revolution of the workpiece. The process is characterized by a specific cutting coefficient 𝐾 𝑓 and a width of cut 𝑏. The dynamics of the structure including the control loops is represented by the transfer function Φ(𝑠) relating the cutting force 𝐹 𝐶 to the tool displacement 𝑋 2 .

ℎ(𝑠) = ℎ 𝑠 (𝑠) + 𝑋 2 (1 -𝑒 -𝜏𝑠 ) (7)
𝐹 𝐶 = ℎ(𝑠) 𝐾 𝑓 𝑏 (8) ℎ(𝑠) ℎ 𝑠 (𝑠) = 1 1+𝐾 𝑓 𝑏(1-𝑒 -𝜏𝑠 )𝛷(𝑠) (9)
Figure 2: Regenerative chatter model [START_REF] Merrit | Theory of self-excited machine-tool chatter. Contribution to machine-tool chatter. Research 1[END_REF].

The analysis of the characteristic equation 9 allows the evaluation of the process stability using the classical chatter theory [17] [18]. Considering the Laplace variable as 𝑠 = 𝜎 + 𝑗𝑤 𝑐 , the absolute minimum of the stability diagram can be obtained analytically with the following formula:

𝑏 𝑎𝑏𝑠 = -1
2𝐾 𝑓 𝐺(𝑤 𝑐 ) [START_REF] Iglesias | Optimisation of face milling operations with structural chatter using a stability model based process planning methodology[END_REF] with 𝐺 being the real part of the transfer function Φ = G + iH and 𝑤 𝑐 the chatter frequency.

Structure dynamics with control parameters

Previous section shows that the stability is related to the transfer function Φ(𝑠) of the machine structure between the cutting force 𝐹 𝐶 and the tool displacement 𝑋 2 . The literature offers several studies in which the structure is modeled as an ideal mass-spring-damper system which is subject to an oscillatory cutting force. This model results in a second order transfer function. Such approximation dismisses the effect of the feed drives on the transfer function, since they are considered as fixed. However, the feed drives control parameters can modify considerably the overall response of the tool. Thus, the present work adds the drive velocity and position control loops to the ideal mass-spring-damper model which represents the vibrating structure (Fig. 1). Therefore the expression of the transfer function Φ(𝑠) when taking into account the control loops of the linear drive is given in Eq. 11.

For the rest of the article, the following numerical value will be used: 𝑚 1 = 5522.6 𝑘𝑔, 𝑚 2 = 788 𝑘𝑔, 𝐾 = 28 10 6 𝑁/𝑚, 𝐶 = 14854 𝑁/(𝑚/𝑠), 𝐾 𝑓 = 1500 10 6 𝑁/𝑚². If the linear motor is stopped (𝑚 1 is fixed), these numerical values lead to a vibration mode at 𝑓 0 = 30Hz with 5% damping.

ANALYSIS OF THE EFFECT OF THE CONTROL PARAMETERS ON THE PROCESS STABILITY

The present section is focused on the study of the effect of the feed drive control parameters on the cutting capability of the machine. The absolute stability limit will be evaluated for different values of the feed drives control parameters.

As well, the overall shape of the stability lobes and the process root locus will be analysed for specific parameter sets.

Effects of Kp, Ti, Kv on the absolute stability limit

To evaluate the effect of the feed drive control parameters on the stability limits, the minimum width of cut is calculated with the equations 10 and 11.

𝛷(𝑠) =

(𝑠+𝐾 𝑣 )𝐾 𝑝 (1+ Figure 3 shows the evolution of the minimum stability limit as a function of the integral time and the position proportional gain, for two velocity loop gain cases. When the velocity proportional gain is adequately tuned with the commissioning rule given in section 2.1, it can be seen that the absolute width of cut is increased compared to the fixed drive configuration except when the integral time is too low. In this configuration, the pole related to the integral term becomes poorly damped and is at the origin of the chatter vibration. The root locus of the velocity loop presented in Fig. 6 shows that effect when the locus of the mechanical and integral term poles are interacting. Following the standard commissioning rules of section 2.1 for the tuning of the control parameters, this situation should not appear. With an adequate 𝐾 𝑝 gain, a large range of 𝑇 𝑖 and 𝐾 𝑣 combination gives high value of the absolute limit of stability. However, when the proportional gain of the velocity loop is too high, for example 𝐾 𝑝 = 3 10 6 𝑁/(𝑚/𝑠), 𝑏 𝑎𝑏𝑠 is significantly lower because the feed drive control does not provide any additional damping to the mechanical pole.

1 𝑇 𝑖 𝑠 )+𝑚 1 𝑠 2 +𝐶𝑠+𝐾 (𝑚 2 𝑠 2 +𝐶𝑠+𝐾)(𝑠+𝐾 𝑣 )𝐾 𝑝 (1+ 1 𝑇 𝑖 𝑠 )+𝑠 2 (𝑚 1 𝑚 2 𝑠 2 +(𝑚 1 +𝑚 2 )𝐶𝑠+(𝑚 1 +𝑚 2 )𝐾) (11)
Figure 4 shows the evolution of the absolute stability limit with conservative values of the integral time and position proportional gain (𝐾 𝑣 = 2 (𝑚/min )/𝑚𝑚 and 𝑇 𝑖 = 0.010 𝑠). In this configuration, the optimum value of 𝐾 𝑝 for process stability is close to 𝐾 𝑝 = 10 6 𝑁/(𝑚/𝑠) given by Eq. 4.

Stability lobes

The stability lobes diagram is a widely employed tool for designing proper cutting process parameters [START_REF] Dombovari | General milling stability model for cylindrical tools[END_REF]. These stability lobes can be obtained using the frequency domain method presented by Altintas in [START_REF] Altintas | Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design[END_REF]. In this work, the lobes diagram will be used to show the stability prediction variations employing different control parameters sets. Apart from the optimum damping tuning proposed in the section 2.1, it is possible to set the gains to maximize the minimum of the real part 𝐺 to increase 𝑏 𝑎𝑏𝑠 , similarly to the tuning proposed by Sims [START_REF] Sims | Vibration absorbers for chatter suppression: A new analytical tuning methodology[END_REF] to adjust passive dampers.

In the present work, the four parameters sets presented in Fig. 5a are compared: 1-The feed drive is rigidly blocked so 𝑚 1 is fixed 𝑥 1 = 𝑥1 = 𝑥1 = 0 (mechanical brake is activated). 2-Control parameters are set with high gains. 3-Optimum damping tuning is used. 4-The control parameters are optimized to maximize the minimum stability limit. The colour code defined in Fig. 5a will be used for the rest of the figures. Figure 5b shows that the feed drive can have a significant impact on the Frequency Response Function. It can be seen that when the control gains are set too high, the feed drive behaves more as a rigid element and is not able to provide extra damping to the system. However, when they are properly tuned, the flexibility of the critical mode can be considerably reduced (curve 2 in green). This improvement on the frequency response function involves an increase of the stability of the machine. Analysing figure 5c, it can be observed that the minimum stability limit is higher when parameters are set to maximize the negative real part of the frequency response function (curve 4 in red). Nevertheless, although this parameter set provides interesting chatter stability properties, this kind of optimization leads to poor performance in terms of overshoot and vibrations during drive movements as presented in the section 4.

Root locus of the turning process

In section 3.2, the process stability is studied with stability lobe diagrams in the frequency domain. A parallel analysis can be carried out using root locus techniques. This analysis allows to have a different view of the process stability problem and to know which pole of the system is causing the instability. From Eq. 9, the characteristic equation that determines the stability of the turning process is:

1 + 𝐾 𝑓 𝑏(1 -𝑒 -𝜏𝑠 )𝛷(𝑠) = 0 ( 12
)
As presented by Olgac & Hosek [START_REF] Olgac | A new perspective and analysis for regenerative machine tool chatter[END_REF] and by Ganguli [START_REF] Ganguli | Chatter reduction through active vibration damping[END_REF], the stability limit of the process can be obtained using a root locus. The term (1 -𝑒 -𝜏𝑠 ) has an infinite number of roots which can be approximated as a rational function by the Padé approximation. The accuracy of the approximation is linked to the order of the polynomials. For the chatter stability analysis, the Padé order should be chosen based on the expected chatter frequency and on the minimum spindle speed that should be studied. Indeed, higher is the lobe order, lower is the accuracy of the approximation. In this work, a Padé approximation of order 20 has been employed.

Increasing 𝜏 or decreasing the spindle speed vertically compresses the root locus as the zeros are moving along the imaginary axis, Fig. 6. The zeros located at a lower frequency than the mechanical pole corresponds to the stability lobes which cannot be made unstable at this spindle speed. For 20, 25 and 30 rev/s, the lower branch of the root locus is never crossing the imaginary axis. Similarly, in Fig. 5c, the lobe 0 is not limiting the stability for those spindle speeds.

As shown in Fig. 7, the infinite number of roots associated with the delay term (1 -𝑒 -𝜏𝑠 ) correspond to the infinite number of lobes which can be crossed at a given spindle speed. The next figures, shows two different zones in the stability lobes diagram. At 50 rev/s, the locus of the mechanical pole is at the origin of the process instability. Hence, the higher damping provided by an adequate tuning of the control parameters can increase significantly the stability limit. However, in the sweet spot zone of the lobe, at 28.3 rev/s (Fig. 8), the large improvement of the mechanical pole damping has no effect because the stability limit is given by the delay terms which are almost not affected. Figure 8: Root locus of turning process at 28.3 rev/s for fixed drive and adequately tuned drive. Thanks to this analysis, it can be concluded that a different strategy should be used when the stability limit is not governed by the locus of the mechanical mode. For this area of the lobes, the control feed drive gains have little effect on the stability. In the literature, other control strategies such as the delayed resonator [START_REF] Olgac | A new perspective and analysis for regenerative machine tool chatter[END_REF] and delayed position feedback [START_REF] Bilbao-Guillerna | Control laws for chatter suppression in milling using an inertial actuator[END_REF] [START_REF] Mancisidor | Optimal control laws for chatter suppression using inertial actuator in milling processes[END_REF] have been presented for cases in which the instability is not related to the mechanical pole and the increase of damping does not improve significantly the stability limit.

FEED DRIVE BEHAVIOUR

The control tuning should fulfil different objectives at the same time: the process stability and the motion performance. The root locus of the velocity loop shows the evolution of the system poles as a function of the proportional gain of the velocity loop 𝐾 𝑝 (Fig. 9). The colour code corresponds to the integral time of the parameter sets of Fig. 5a. For the given 𝐾 𝑝 of the parameter sets, the location of the system poles is marked by the squares. For this analysis, the cutting process and the position loop are not considered. When the proportional gain of the velocity loop is too high, the feed drive does not provide significant damping improvement to the mechanical pole. Indeed, the system pole corresponding to the mechanical mode gets close to the mechanical zero which has 6% damping. With the optimum gain 𝐾 𝑝 in green, the drive can give a damping of 10.5% to the mechanical pole. When the integral time is too low, the shape of the root locus changes drastically: the circular locus related to the integral term interacts with the locus of the mechanical pole and the damping of the pole related to the integral time is significantly reduced (red curve). This effect corresponds to the important reduction of the absolute stability limit in the top plot of Figure 3. The dark green root locus corresponds to the adequate axis tuning which aims at providing the optimum damping for the mechanical pole and important damping to the integral pole. To analyse the axis response with the different tuning, Fig. 10 shows the motor and tool tip velocity and position bode plots. General commissioning rules allow a +3dB overshot on the motor side knowing that the vibration will be amplified on the tool tip. However, the position loop is closed on top of the velocity loop and the position bode plot should not exceed the 0dB line. Comparing the 3 parameters sets, the adequate axis tuning gives lower gain magnification while preserving a high bandwidth for the velocity loop (Fig. 10a &10b). On the position loop, the low value of 𝐾 𝑣 for the red curve leads to a really low bandwidth. However, if the gain 𝐾 𝑣 is increased the absolute stability limit decreases quickly. Hence, the parameter sets 4 (in red) for the optimization of the absolute width of cut cannot fulfil both stability and motion performances. Regarding the parameter set 2 (in blue), the high tuning of the gain is detrimental both for the stability limit and for the motion response of the drive. Practical tuning on the machines are realized looking only at the motor response because no measuring system is available to get the response of the tool tip. Hence, this kind of high gain tuning can often be encountered on the machine tools because the bode plots on the motor side (Fig. 10a &10c) does not reflect large amplification of the vibration on the tool tip (Fig. 10b &10d).

With the proposed model, the key parameter influencing the absolute stability limit is the proportional gain of the velocity loop. Using the formulas (Eq. 4, 5 & 6) to obtain the 𝐾 𝑝 , 𝑇 𝑖 and 𝐾 𝑣 parameters a good balance is achieved regarding chatter stability limit and the motion drive characteristics.

CONCLUSION

This article studied the effect of the feed drive control parameters on the process stability limit. The model used is simple but it allows evaluating the effect of the key parameters. First, it can be seen that the control tuning of the feed drive can have an impact on the process stability limit. The proportional gain of the position loop 𝐾 𝑣 has a secondary effect compared to the velocity loop controller (𝐾 𝑝 and 𝑇 𝑖 ). For process stability purpose, the integral time should not be too low and the proportional gain should be adjusted to provide the highest damping. When 𝐾 𝑝 is too high, the control does not provide additional damping which is detrimental both for chatter and for control performance. This situation can occur on the machine tools because the bode plots on the motor side does not put in evidence the problem related to the vibration on the tool side. The optimum damping tuning rules provided in section 2.1 allow to obtain parameters for the velocity and position loops which are interesting both for the chatter stability and for the motion performances. To go further, a mechatronic model of the machine tool combining Finite Element model of the structure and feed drive simulation could be used to extend this analysis to complex structures.

Figure 3 :

 3 Figure 3: Minimum stability limit with varying control parameters.

Figure 4 :

 4 Figure 4: Evolution of the minimum stability limit as a function of Kp with (𝐾 𝑣 = 2 (𝑚/min )/𝑚𝑚 & 𝑇 𝑖 = 0.010 𝑠).

Figure 5 :

 5 Figure 5: a) Parameter sets, b) Frequency Response Functions and c) stability lobes.

Figure 6 :

 6 Figure 6: Root locus of the stability process for the fixed drive configuration.

Figure 7 :

 7 Figure 7: Root locus of turning process at 50 rev/s for fixed drive and adequately tuned drive.

Figure 9 :

 9 Figure 9: Root locus of the velocity loop.

Figure 10 :

 10 Figure 10: Bode plots of the position and velocity loop at the motor and tool tip points.

ACKNOWLEDGMENT

This work was partially supported by the European Community under the MC-Suite project (H2020-FoF-2015-680478).