Reliability enhancement of manufacturing systems through functions

Abstract

A reliable system means being able to perform its intended functions. Therefore, ensuring performing of its required functions will help to enhance its reliability. For a manufacturing system (e.g. computer numerical control machines), there are a large number of functions, which complicate and make analysis difficult. In this article, a logical and systems approach of graph theory, which is effective to eliminate such difficulties, is employed. The graph theoretic models do consider the system structure explicitly and are applied to model functions at various hierarchical levels of a manufacturing system. These function digraph models are analysed using matrix approach to examine the cause and effect, which helps to evaluate importance of the function and hence provide direction for system reliability enhancement. A step-by-step methodology is presented, which is illustrated by an example of manufacturing system: computer numerical control drilling machine.

Introduction

In the present manufacturing scenario, the challenging issues such as cost, quality, agility ¹ and mass customized goods make enormous demand on reliability and availability of the systems and their elements in the manufacturing plants. System functions play a crucial role in this. Manufacturing systems such as machining centres (MCs), flexible manufacturing systems (FMSs) and computer numerical control (CNC) turning centres, and drilling and grinding machines, are the basic elements of a large manufacturing plant. Ensuring performing of required functions of these will help to improve their reliability, and hence overall plant availability. Each of these systems comprises various functional paths, which are constitutes of various structural elements that contribute to the system functions. ² The functional paths of a manufacturing system, in general, are designed to perform the required actions of the system (e.g. turning, grinding operations) and other related actions (e.g. changing spindle speed), which are not essential in the required action or function, but are necessary to select the correct machining condition. Generally, a manufacturing system, for example, CNC drilling machines, may have two or more functional paths: linear, rotary axis drives, which are designed to serve the main purpose, and pneumatic and hydraulic actuations, which are used to perform actions such as component clamping. Each of the functional paths is an electromechanical system, which has various elements integrated into it. The functions of elements, be it mechanical assembly or hydraulic or electrical subsystem, interact with each other through structural interconnections, which may be of mechanical joints or hydraulic piping or electrical cabling. If an element in the functional paths fails to perform its intended function, it fails to satisfy its function and it will directly or indirectly affect the function of other connected subsystems, assemblies or components, and the performance as a whole. This will lead to complete or partial failure of the system and will result in poor or lower system reliability. Therefore, consideration of functions will help to minimize system failures attributed to these and will hence enhance the system reliability.

Reliability enhancement in this article means system reliability increase or improvement. The reliability enhancement can be carried out for a manufacturing system by analysing its functions during design and development stage or for an operating manufacturing system experiencing failures and reliability problem. The term reliability growth, which is practiced in the field of reliability engineering, is the improvement in the reliability of system over a period of time due to the changes in system design and the manufacturing process. It includes activities such as quantification and assessment of parameters relating to the system reliability growth over time and planning and management of system reliability growth as a function of time and resources. The reliability growth process is a broader area and the reliability enhancement considered in this article is a subset of this.

Failure-related studies were initiated in manufacturing industries in early 1990s to improve reliability and maintainability (R&M) of manufacturing machinery and equipment. Blache and Shrivastava ³ defined and classified failures and suggested a comprehensive failure reporting, analysis and corrective-action program. A considerable effort was made to improve the R&M of CNC machine tools, especially in the production of automobiles. ⁴ A few researchers^5,6 addressed failure issues through implementing R&M program for automotive manufacturing plants. Some research works have been directed towards developing field failure database and failure models for CNC machines.^7–11 Although the model has been used successfully in solving some practical issues, the operational failure in the manufacturing systems still remains a challenging issue. A hierarchical fault diagnosis model ¹² for diagnosing operational faults of an FMS was developed based on fault tree analysis (FTA). The FTA permits systematic identification of failure causes logically, that is, starting from its top failure event to its failure causes. However, it may miss crucial failure events, as it does not consider the system structure explicitly. ¹³ Some works have been undertaken to develop a method of diagnosing operational faults in CNC machines based on switching function. ¹⁴ Soft computing approaches^15,16 have also been attempted for carrying out failure mode and effects analysis (FMEA) and criticality analysis on CNC machines. Rao and Gandhi ¹⁷ carried out failure cause analysis of machine tools at design stage by considering structure. The concept of reliability-centred maintenance (RCM) has also been applied to manufacturing systems to help choosing best maintenance tasks based on the failures. ¹⁸

There have been numerous research works on failure analysis and reliability improvement based on structure and functional models in conceptual stage of products. ¹⁹ Milne ²⁰ has used topological representation of components at various hierarchies for obtaining complete knowledge of the system for strategic diagnosis. Chittaro et al. ²¹ developed functional models for fault diagnosis of electrical and electronic systems using structural and functional aspects. The essential features of knowledge-based approach to FMEA were described by Wirth et al. ²² Various models, based on functional reasoning, are described by Umeda and Tomiyama. ²³ Hawkins and Woollons ²⁴ extended functional models to represent an electrically driven gear pump. An automatic generation of FMEA model based on functional reasoning techniques has been proposed by Teoh and Case. ²⁵ A few studies have dealt in linking functionality to failure modes of aerospace systems.^26,27 Wani and Gandhi ²⁸ suggested a methodology for failure analysis of mechanical systems using function-cum-structure approach. Kurtoglu and Tumer ²⁹ developed functional-failure identification and propagation framework for assessing functional-failure risk of physical systems during conceptual design. Kurtoglu et al. ³⁰ introduced a new methodology for reasoning the functional failures during early design of aerospace electrical power system. Noh et al. ³¹ proposed a functional model for FMEA of a car air purifier. Their study was able to identify the failures that depended not only on the system hierarchical structure but also on the module interactions. Functional and structural models based on bond graph representations ³² have been employed in detection and isolation of faulty elements of a three-tank hydraulic system.

From the above, it is clear that the researchers have attempted structural and functional models in assessing failures and enhancing reliability, including application of FTA and FMEA that have been mainly applied to hydraulic, mechanical, electronic and electrical systems. However, these models are applied only at subsystem level and have not really been applied to manufacturing systems. Moreover, these models do not explicitly represent functions and their interactions at various hierarchical levels. For manufacturing systems, the models that ignore the complex functional interactions at various hierarchical levels are insufficient. This is because the malfunctioning of the system is attributed to improper functional interaction between its subsystems, assemblies and components. Therefore, the above methods do not provide realistic solution to the failure of manufacturing systems, and if these are applied, there will be a serious implication for the overall system reliability.

For consideration of reliability enhancement of a system, it is desirable to consider the system functions and their interactions from structure point of view. The system structure means the system elements and their interconnections. A system approach is needed to express the functions and their interactions, in conjunction with the system structure, for example, as a digraph model or network. This can be conveniently represented in a matrix form, which can aid in analysis of functions in the context of reliability enhancement. Graph theoretic models have been extensively applied in numerous engineering applications.^30,33–37 Digraph models provide a convenient method to represent the functions and their interactions explicitly through nodes and edges. This will be simplified further using graph theory techniques to evaluate the importance of the functions. Fukuda ³⁸ proposed a digraph-based approach for importance evaluation of failure events in a complicated welded structure. This concept is extended in this article for functional analysis of a manufacturing system. The main objective of the article is to establish a methodology for function consideration for reliability enhancement of manufacturing systems based on digraph models and matrix method. This article is aimed at carrying out the function importance evaluation and its analysis for the reliability enhancement of a CNC drilling machine using digraph models and matrix method.

The article is organized, with the next section on understanding of the system through generic structure and function decomposition of a manufacturing system. Subsequently, function digraphs are defined and developed for the system hierarchical levels. This is followed by developing matrix representation. The step-by-step methodology is described in the subsequent section, which is followed by a case study. Finally, the conclusion is drawn.

Understanding the system – structure and functions

Manufacturing systems are complex, and therefore, it is required to decompose these to obtain multiple levels of abstractions of their structure and functions that will help in better understanding. A generic abstraction of the manufacturing systems is presented below.

System structure

Let a manufacturing system consists of subsystems, assemblies and components. However, depending on the system complexity, additional levels can be defined. The structural decomposition of the system is carried out as below.

Structure decomposition

The structure decomposition provides an insight of the levels of hierarchical abstraction that is based on top–down approach. This helps to gain the knowledge of structural elements and their interconnections. For better understanding and ease of handling vast number of its elements, each element of the system is designated in the hierarchical order. It is already mentioned that the system (S) consists of subsystems, assemblies and components. A subsystem is designated as S_i, where i = 1, 2,…, N; where the mth assembly of ith subsystem is $a_m^i$ , m = 1, 2,…, M, and the kth component of mth assembly of ith subsystem is $c_k^{\mathrm{im}}$ , k = 1, 2,…, L. Based on this, a four-level generic hierarchical structure tree that represents the structural decomposition of a manufacturing system is developed, which is shown in Figure 1.

Figure 1.

Generic hierarchical structure tree of a manufacturing system.

After the structural elements are identified, the system functions are considered at various hierarchical levels. This is described in the following section.

System functions

The system functions involve decomposition and interactions among functions.

Function decomposition

Functional decomposition means identifying functions of subsystems, assemblies and components. For this, it is required to follow standard function taxonomies and basis.^39,40 The functions of a system are classified into three groups: overall, primary function and secondary function. The overall function means the desired output of the system or subsystems, while primary function performs the basic and important action that is the main purpose or existence of an element, be it a component or assembly. The secondary function on the other hand is the useful function performed by component or assembly. ²⁸ To represent hierarchy of system functions, the functions of the each element are designated as follows.

Let ‘F’ be the overall system function (set of functions of the system), and let ${\mathrm{F}}_{\mathrm{i}}$ (i = 1, 2,…, N) be the overall function of ith subsystem. At assembly and component levels, there will be a primary function and one or more secondary functions. ²⁸ These are designated as follows: ${\mathrm{F}}_{\mathrm{mp}}^{\mathrm{i}}$ (m = 1, 2,…, M; p = 1) and ${\mathrm{F}}_{\mathrm{ms}}^{\mathrm{i}}$ (m = 1, 2,…, M; s = 2,3,…, P) be the pth primary function and sth secondary function of mth assembly in ith subsystem, and ${\mathrm{F}}_{\mathrm{kp}}^{\mathrm{mi}}$ (k = 1, 2,…, L; p = 1) and ${\mathrm{F}}_{\mathrm{ks}}^{\mathrm{mi}}$ (k = 1, 2,…, L; s = 2,3,…,Q) be the pth primary function and sth secondary function of kth component of mth assembly in ith subsystem. Based on this, a generic hierarchical function tree that shows the decomposition of the overall system function into smaller, fundamental subfunctions, primary and secondary functions of elements, is developed for a manufacturing system as shown in Figure 2.

Figure 2.

Generic hierarchical function tree of a manufacturing system.

It is clear from the above that the function decomposition helps to understand the functions at various system levels. Functional interactions among functions are dealt in the next section, which is explained by means of an example.

Interactions among functions

Each element of the system is designed and developed as per its specifications, which ensures that it performs its intended function. For this, one needs to examine design and operational parameters related to the function, which help to identify functional relationships. ²⁸ These can be identified from the function equation. Some functions interact with the other functions, while others do not interact with any other functions. Let us consider timing belt drive assembly (Figure 3) of a CNC machine.

Figure 3.

Timing belt drive assembly.

The timing belt drive assembly transmits torque from driving to driven shaft. The driving and driven pulleys are connected to their shafts through key way connection. The belt has internal teeth, which mesh with the external teeth of the pulley. A primary function of shaft is to ‘transmit power’, which is represented by

$${\mathrm{P}}_{\mathrm{shaft}}=\left[\frac{2\pi {\mathrm{nd}}_{\mathrm{s}}\mathrm{f}}{120}\right]\mathrm{W}$$

(1)

where ‘P_shaft’ is the power transmitted by the shaft; ‘n’ and ‘d_s’ are the speed and diameter of the shaft, respectively; and ‘f’ is the tangential force acting on the shaft.

On the other hand, the primary function of pulley is to ‘transmit torque’, which is presented as

$${\mathrm{T}}_{\mathrm{pulley}}=\left[\frac{{\mathrm{d}}_{\mathrm{p}}\mathrm{f}}{2}\right]\mathrm{N\; m}$$

(2)

where ‘T_pulley’ is the torque transmitted by the pulley, ‘d_p’ is the diameter of the pulley and ‘f’ is the tangential force acting on the pulley. Equations (1) and (2) ⁴¹ help to identify the functional interaction between driving shaft and pulley.

The parameters of transmit power and transmit torque are tangential force, diameter and speed, and tangential force and diameter, respectively. It means that the functions, transmit power and transmit torque, are related through the parameter, tangential force.

In a similar way, relationships for the remaining functions of the timing belt drive assembly are identified and presented in Table 1. It may be noted that the functions, support load ( ${\mathrm{F}}_{12}^{11}$ ), avoid slippage ( ${\mathrm{F}}_{32}^{11}$ ) and support load ( ${\mathrm{F}}_{52}^{11}$ ), do not interact with any other functions.

Table 1.

Timing belt drive assembly functions and their interactions.

Component	Functions	Parameters	Interacting functions
Driving shaft ( ${\mathrm{c}}_1^{11}$ )	Primary function – transmit power ( ${\mathrm{F}}_{11}^{11}$ )	Force, diameter and speed	${\mathrm{F}}_{21}^{11}$
	Secondary function – support load ( ${\mathrm{F}}_{12}^{11}$ )	Force	–
Driving pulley ( ${\mathrm{c}}_2^{11}$ )	Primary function – transmit torque ( ${\mathrm{F}}_{21}^{11}$ )	Force and diameter	${\mathrm{F}}_{11}^{11}$ , ${\mathrm{F}}_{31}^{11}$
	Secondary function – develop rotation ( ${\mathrm{F}}_{22}^{11}$ )	Speed	${\mathrm{F}}_{33}^{11}$
Timing belt ( ${\mathrm{c}}_3^{11}$ )	Primary function – transmit torque ( ${\mathrm{F}}_{31}^{11}$ )	Force and diameter	${\mathrm{F}}_{21}^{11}$ , ${\mathrm{F}}_{41}^{11}$
	Secondary functions
	Avoid slippage ( ${\mathrm{F}}_{32}^{11}$ )	Frictional force, coefficient of friction	–
	Develop rotation ( ${\mathrm{F}}_{33}^{11}$ )	Speed	${\mathrm{F}}_{22}^{11}$ , ${\mathrm{F}}_{42}^{11}$
Driven pulley ( ${\mathrm{c}}_4^{11}$ )	Primary function –transmit torque ( ${\mathrm{F}}_{41}^{11}$ )	Force and diameter	${\mathrm{F}}_{51}^{11}$ , ${\mathrm{F}}_{31}^{11}$
	Secondary function – develop rotation ( ${\mathrm{F}}_{42}^{11}$ )	Speed	${\mathrm{F}}_{33}^{11}$
Driven shaft ( ${\mathrm{c}}_5^{11}$ )	Primary function – transmit power ( ${\mathrm{F}}_{51}^{11}$ )	Force, diameter and speed	${\mathrm{F}}_{41}^{11}$
	Secondary function – support load ( ${\mathrm{F}}_{52}^{11}$ )	Force	–

From the above, it seems that the function interactions are causal interactions that mean the functional deviation (i.e. failure or its degradation), that is, an element can cause functional deviation to the other connected elements. So there can be cause function(s) for each failed function. The element reliability decreases when its function begins to deteriorate, which will cause a functional deviation of its connected element(s). The reliability can, therefore, be improved either by modifying the cause function or by increasing the capacity of the failed element. This can be analysed by expressing the functions and their interactions in graph model as discussed in the following section.

Function digraphs

A system has a number of functions to perform at its hierarchical levels but there does exist causal interactions among these functions. The causal interactions may be of direct or through one or more intermediate functions. If the causality of functional interaction is direct, it may be easy to carry out the functional analysis. But if it is through other intermediate functions, there will be a danger of failing to note some of these for large systems. It is, therefore, necessary to represent it based on a systems approach to carry out its analysis. Graph or digraph representations ⁴² are effective, as these consider mutual interactions among functions. Here, it is used to represent functions and their causal interactions.

The following digraph models are defined for a manufacturing system:

• System function digraph (SFD);

• Subsystem function digraph (SSFD);

• Assembly function digraph (AFD).

SFD

SFD represents subsystem functions, ‘F_i’, through its nodes, while the directed edges, ‘F_ij’, represent the interaction between functions of subsystem. The SFD provides a visual aid to understand the relationships between functions of subsystems. This does help to carry out functional analysis at system level. However, the subsystem-, assembly- and component-level function digraphs will help in detailed analysis of the functions.

SSFD

SSFD is defined on lines of the SFD to represent subsystem function in terms of its assembly functions, that is, ‘ ${\mathrm{F}}_{\mathrm{mp}}^{\mathrm{i}}$ ’ and/or ‘ ${\mathrm{F}}_{\mathrm{ms}}^{\mathrm{i}}$ ’ through its nodes, while the directed edges, ‘ ${\mathrm{F}}_{\mathrm{mpnp}}^{\mathrm{i}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{mpns}}^{\mathrm{i}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{msnp}}^{\mathrm{i}}$ ’, and ‘ ${\mathrm{F}}_{\mathrm{msns}}^{\mathrm{i}}$ ’, correspond to the interaction between assembly functions; where ‘ ${\mathrm{F}}_{\mathrm{mp}}^{\mathrm{i}}$ ’ and ‘ ${\mathrm{F}}_{\mathrm{ms}}^{\mathrm{i}}$ ’ represent primary function ‘p’ and secondary function ‘s’ of mth assembly of ith subsystem and ‘ ${\mathrm{F}}_{\mathrm{mpnp}}^{\mathrm{i}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{mpns}}^{\mathrm{i}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{msnp}}^{\mathrm{i}}$ ’ and ‘ ${\mathrm{F}}_{\mathrm{msns}}^{\mathrm{i}}$ ’ represent interaction between primary and secondary functions of assemblies ‘m’ and ‘n’ in ith subsystem.

AFD

AFD is also defined on lines of the SFD and SSFD to represent assembly function in terms of its component functions ‘ ${\mathrm{F}}_{\mathrm{kp}}^{\mathrm{mi}}$ ’ and/or ‘ ${\mathrm{F}}_{\mathrm{ks}}^{\mathrm{mi}}$ ’ through its nodes, while the directed edges ‘ ${\mathrm{F}}_{\mathrm{kplp}}^{\mathrm{mi}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{kpls}}^{\mathrm{mi}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{kslp}}^{\mathrm{mi}}$ ’ and ‘ ${\mathrm{F}}_{\mathrm{ksls}}^{\mathrm{mi}}$ ’ correspond to the interaction between component functions; where ‘ ${\mathrm{F}}_{\mathrm{kp}}^{\mathrm{mi}}$ ’ and ‘ ${\mathrm{F}}_{\mathrm{ks}}^{\mathrm{mi}}$ ’ represent primary function ‘p’ and secondary function ‘s’ of kth component of mth assembly in ith subsystem and ‘ ${\mathrm{F}}_{\mathrm{kplp}}^{\mathrm{mi}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{kpls}}^{\mathrm{mi}}$ ’, ‘ ${\mathrm{F}}_{\mathrm{kslp}}^{\mathrm{mi}}$ ’ and ‘ ${\mathrm{F}}_{\mathrm{ksls}}^{\mathrm{mi}}$ ’ represent the interaction between primary and secondary functions of components ‘k’ and ‘l’ of mth assembly in ith subsystem.

The above digraph models will be useful to analyse the causal relationship among functions. These will aid in identifying the cause functions for failed functions, and hence provide direction to enhance the system reliability. The causality of failure due to functional interactions among elements can be understood by representing functions. This is described below.

Let us reconsider the example that is discussed under section ‘Interactions among functions’, that is, timing belt drive assembly, and its function digraph is to be developed, that is, AFD. Table 1 helps to identify its functional interactions among its components, and hence its AFD. The digraph representation of functions for the timing belt drive assembly is developed, which is shown in Figure 4. It may be noted that the functions, which are not interacting with the other functions, are shown as an isolated node. The digraph developed is simplified, as not all the functions have been included. It does, however, include the basic functions and their interactions.

Figure 4.

Digraph representation of functions for timing belt drive assembly.

The graph models are convenient for visual analysis and explicitly describe the functions and their connectional/causal relationships. However, these tend to become complex and difficult to visualize, as the number of functions and interaction increases. Therefore, function digraph models are expressed using matrix method. This is explained in the following section.

Matrix representation

In this section, functional analysis is carried out by defining an equivalent matrix for the digraph model at each hierarchical system level, and the corresponding matrix is called node–node function connection matrix. The matrix representation helps not only to clarify the causality of functional interaction between elements that are connected directly but also to carry out the importance evaluation of functions based on their number of stages of connection. This will provide a direction for reliability enhancement.

System node–node function connection matrix

A node–node function connection matrix for the SFD is defined, with both the rows and columns representing nodes (i.e. subsystem functions). This matrix is called ‘system node–node function connection matrix’ (SNNFCM), which represents direct relationships, among subsystem functions (i.e. nodes). The (i, j)th entity of the matrix is assigned a value ‘1’, if there is a directed edge from ith node to jth node; otherwise, ‘0’. Although it visually shows the presence of causality of interactions among functions, it does not indicate whether these are direct or through different stages of interactions. It is, therefore, desirable to identify these functional interactions, which will help in prioritizing their involvement in reliability enhancement. Graph theory, the systems approach, is rich in terms of procedures and algorithms that can be exploited to take care of this, and even beyond. ⁴² In line with this, the approach suggested by Fukuda ³⁸ is extended to evaluate the number of stages between functions. This is described below.

System function importance evaluation matrix

For evaluating the number of stages between functions, the SNNFCM is processed further by applying Fukuda method to obtain a matrix, called ‘system function importance evaluation matrix’ (SFIEM). This is defined, with both the rows and columns representing nodes (i.e. subsystem functions). The (i, j)th entity of this matrix represents a value n, if there are n directed edges from ith node to jth node; otherwise, 0. This shows the number of stages of connections between the function of a row and a function of a column. The SFIEM represents not only the direct relationships among subsystem functions (i.e. nodes) but also the multi-stage relations between the function of a row and the function of a column. This is included in Step VI in section ‘Step-by-step methodology’. On the similar lines, the function importance evaluation matrix for SSFDs and AFDs is defined.

Step-by-step methodology

The step-by-step proposed methodology for reliability enhancement for manufacturing systems through functions is as below:

• Step I: select a manufacturing system and understand and study its description.

• Step II: define the structure of the chosen system (Step I) on the lines of section ‘System structure’, including Figure 1, that is, identifying its subsystems, assemblies, components and their interconnection.

• Step III: identify the system functions on the lines of section ‘System functions’, including Figure 2, that is, resolving its overall function into various elemental functions and identifying the interactions through design and operational parameters obtained.

• Step IV: develop digraph models at various hierarchical levels, that is, SFD, SSFDs and AFDs. Refer section ‘Function digraphs’ for details.

• Step V: obtain node–node function connection matrix for the digraphs developed in Step IV to evaluate their function importance evaluation matrices. Refer section ‘Matrix representation’ for details.

• Step VI: evaluate importance of functions using function importance evaluation matrices from the node–node function connection matrix obtained for each hierarchical levels. Its steps are given below.

• Multiply ‘M’ by ‘M’, one after another. A non-zero or unity element, which appears for the first time after n multiplication, implies that there is an n-stage relation between the function of a row and the function of the column.

• Continue the above step, that is, Step 2 until no more new matrix appears which helps to obtain all connectional relations between subsystem functions.

• Show the value of n at location of a non-zero or unity element in the matrix, which appears for the first time in the matrix. This matrix is called ‘SFIEM’.

Step VII: identify actions for enhancing system reliability based on the above step, that is, Step VI.

Step VIII: using Steps I–VI, carry out reliability enhancement at assembly and subsystem levels of the selected manufacturing system.

Case study

To demonstrate the proposed methodology, example of a manufacturing system, CNC drilling machine, is selected. This is an automated manufacturing system that is operated by structured commands, stored in a computer memory unit. Its failure rates and reliability characteristics are difficult to predict, with its complex structure involving several elements that undergo static and dynamic loads. This demands well-defined functional requirements, which pose a challenge for designers and operating personnel to sustain or improve the system reliability. The step-by-step case study is illustrated below.

Step I

A CNC drilling machine, turret-based and three-axis, used in an automotive manufacturing plant, is selected. This is shown in Figure 5. The machine is equipped with several drilling heads mounted on a turret. Each turret head can hold drill bits of different sizes. The turret allows the needed drill bit to be quickly indexed into position. The component to be machined is clamped using hydraulically operated fixtures. The X-axis (not shown) traverses the work table in the longitudinal direction, while the Y-axis moves the saddle back and forth. The saddle supports the complete X-axis drive unit. The Z-axis moves the drill head unit up and down. All three axes are powered by servo motor. The rotary motion of servo motor shaft is converted into the linear motion of saddle and table by ball screw assembly. The position and speed of servo motors are controlled by numerical control (NC) programming, which is embedded in the CNC subsystem. The axis drives receive control signals (shown as long dashed line) from CNC subsystem, which controls three-phase supply current. The rotary encoder, which is a speed-sensing device, provides feedback signal through the control loop (shown as square dotted line). This sends speed and position signals from servo motor to axis drives. Other subsystems such as coolant, hydraulic and lubrication are also controlled by CNC subsystem. The mechanical subsystem is indirectly controlled by CNC subsystem through servo motor of electrical subsystem.

Figure 5.

CNC drilling machine.

Step II

The structure of the CNC drilling machine is developed by decomposing into its various hierarchical elements. The interconnections among these are of moving, sliding or stationary, through mechanical shaft or piping or electrical cabling. The structure of the CNC machine is composed of six subsystems: hydraulic power (HP), shown in tan colour; power supply (PS), including electrical and motor drives, shown in yellow; CNC, shown in green; mechanical (ME), including power transmission, slides and machine bed, shown in blue; central lubrication (CLUB), shown in purple, and coolant (CL), shown in red. A four-level structure tree is developed to represent the CNC drilling machine, which is shown in Figure 6.

Figure 6.

Structure tree – CNC drilling machine.

Step III

To identify various system functions, the desired output, ‘drill hole of specified size’, which is the overall function (F) of the CNC drilling machine, is decomposed into elemental functions. Function decomposition of CNC drilling machine is developed and is shown in Figure 7.

Figure 7.

Functions decomposition – CNC drilling machine.

The structure tree (Figure 6) and the function tree (Figure 7) do capture mapping of structure and function of the CNC drilling machine, for example, the function ‘dissipate hydraulic power to actuators’ (F₁) in Figure 7 is mapped to the HP subsystem (S₁) in Figure 6. In a similar manner, each function in the function tree is mapped to the corresponding hierarchical elements in the structure tree. The mapping ensures that no function is excluded from the analysis.

Refer section ‘Interactions among functions’. The functional interactions are identified by determining the associated parameters. Let us identify the functional interactions at the component level for the CNC drilling machine by considering an example of its assembly; ball screw drive assembly is shown in Figure 8.

Figure 8.

Ball screw drive assembly.

The primary function of ball screw drive is to ‘convert rotary motion to linear motion’ ( ${\mathrm{F}}_{11}^4$ ), which is further decomposed into component-level functions (Figure 7). The primary function of ball screw shaft transmit power ( ${\mathrm{F}}_{71}^{41}$ ) is represented by ⁴³

$${\mathrm{P}}_{\mathrm{s}}=\left[\frac{\mathrm{TN}}{9550}\right]\mathrm{W}$$

(3)

For recirculating balls, the primary function, transmit power ( ${\mathrm{F}}_{81}^{41}$ ), is represented by ⁴³

$${\mathrm{P}}_{\mathrm{s}}=0.9\left[\frac{\mathrm{TN}}{9550}\right]\mathrm{W}$$

(4)

where ‘P_s’ is the power transmitted by the ball screw shaft; ‘T’ and ‘N’ are the torque and speed of the shaft, respectively. From equations (3) and (4), it is clear that the power transmitted by the ball is 0.9 times the power transmitted by the ball screw shaft, which means the ball can transmit power up to 90% of the total power supplied by the ball screw shaft. It is also evident from these equations that the above functions interact with each other through the parameters, torque and speed. Similarly, other functions and their interactions are identified for ball screw drive assembly and presented in Table 2. On the same lines, the functional interactions among other assemblies and subsystems are identified from their design specifications for the CNC drilling machine.

Table 2.

Ball screw drive assembly functions and their interactions.

Components	Functions		Parameters	Interacting functions
	Primary function	Secondary function
Duplex bearing (front) ( ${\mathrm{c}}_1^{41}$ )	Support radial load ( ${\mathrm{F}}_{11}^{41}$ )		Force	${\mathrm{F}}_{71}^{41}$ , ${\mathrm{F}}_{132}^{41}$
		Support thrust load ( ${\mathrm{F}}_{12}^{41}$ )	Force	$F_{72}^{41}$
Duplex bearing (rear) ( ${\mathrm{c}}_2^{41}$ )	Support radial load ( ${\mathrm{F}}_{21}^{41}$ )		Force	${\mathrm{F}}_{71}^{41}$ , ${\mathrm{F}}_{142}^{41}$
		Support thrust load ( ${\mathrm{F}}_{22}^{41}$ )	Force	${\mathrm{F}}_{72}^{41}$
Front duplex bearing seal (front) ( ${\mathrm{c}}_3^{41}$ )	Prevent leakage ( ${\mathrm{F}}_{31}^{41}$ )		Flow rate	–
		Transmit force ( ${\mathrm{F}}_{32}^{41}$ )	Force	${\mathrm{F}}_{71}^{41}$
Front duplex bearing seal (rear) ( ${\mathrm{c}}_4^{41}$ )	Prevent leakage ( ${\mathrm{F}}_{41}^{41}$ )		Flow rate	–
		Transmit force ( ${\mathrm{F}}_{42}^{41}$ )	Force	${\mathrm{F}}_{71}^{41}$
Labyrinth seal (front) ( ${\mathrm{c}}_5^{41}$ )	Prevent leakage ( ${\mathrm{F}}_{51}^{41}$ )		Flow rate	–
		Transmit force ( ${\mathrm{F}}_{52}^{41}$ )	Force	${\mathrm{F}}_{71}^{41}$ , ${\mathrm{F}}_{92}^{41}$
Labyrinth seal (rear) ( ${\mathrm{c}}_6^{41}$ )	Prevent leakage ( ${\mathrm{F}}_{61}^{41}$ )		Flow rate	–
		Transmit force ( ${\mathrm{F}}_{62}^{41}$ )	Force	${\mathrm{F}}_{71}^{41}$ , ${\mathrm{F}}_{92}^{41}$
Ball screw ( ${\mathrm{c}}_7^{41}$ )	Transmit power ( ${\mathrm{F}}_{71}^{41}$ )		Torque and speed	${\mathrm{F}}_{52}^{41}$ , ${\mathrm{F}}_{62}^{41}$ , ${\mathrm{F}}_{81}^{41}$ , ${\mathrm{F}}_{11}^{41}$ , ${\mathrm{F}}_{21}^{41}$ , ${\mathrm{F}}_{151}^{41}$
		Support load ( ${\mathrm{F}}_{72}^{41}$ )	Force	${\mathrm{F}}_{12}^{41}$ , ${\mathrm{F}}_{22}^{41}$
Recirculating balls ( ${\mathrm{c}}_8^{41}$ )	Transmit power ( ${\mathrm{F}}_{81}^{41}$ )		Torque, speed and coefficient of friction	${\mathrm{F}}_{71}^{41}$ , ${\mathrm{F}}_{91}^{4^4}$
		Support radial load ( ${\mathrm{F}}_{82}^{41}$ )	Force	${\mathrm{F}}_{72}^{41}$ , ${\mathrm{F}}_{92}^{41}$
Ball nut ( ${\mathrm{c}}_9^{41}$ )	Transmit thrust force ( ${\mathrm{F}}_{91}^{41}$ )		Force	${\mathrm{F}}_{81}^{41}$ , ${\mathrm{F}}_{101}^{41}$
		Support load ( ${\mathrm{F}}_{92}^{41}$ )	Force	${\mathrm{F}}_{52}^{41}$ , ${\mathrm{F}}_{62}^{41}$ , ${\mathrm{F}}_{82}^{41}$ , ${\mathrm{F}}_{102}^{41}$
Ball nut housing ( ${\mathrm{c}}_{10}^{41}$ )	Transmit thrust force ( ${\mathrm{F}}_{101}^{41}$ )		Force	${\mathrm{F}}_{91}^{41}$
		Transmit load ( ${\mathrm{F}}_{102}^{41}$ )	Force	${\mathrm{F}}_{92}^{41}$
		Isolate ball nut ( ${\mathrm{F}}_{103}^{41}$ )	–	–
Rear duplex bearing seal (front) ( ${\mathrm{c}}_{11}^{41}$ )	Prevent leakage ( ${\mathrm{F}}_{111}^{41}$ )		Flow rate	–
		Transmit force ( ${\mathrm{F}}_{112}^{41}$ )	Force	${\mathrm{F}}_{71}^{41}$
Rear duplex bearing seal (rear) ( ${\mathrm{c}}_{12}^{41}$ )	Prevent leakage ( ${\mathrm{F}}_{121}^{41}$ )		Flow rate	–
		Transmit force ( ${\mathrm{F}}_{122}^{41}$ )	Force	${\mathrm{F}}_{71}^{41}$
Front duplex bearing housing ( ${\mathrm{c}}_{13}^{41}$ )	Isolate bearing ( ${\mathrm{F}}_{131}^{41}$ )		–	–
		Transmit load ( ${\mathrm{F}}_{132}^{41}$ )	Force	${\mathrm{F}}_{11}^{41}$
Rear duplex bearing housing ( ${\mathrm{c}}_{14}^{41}$ )	Isolate bearing ( ${\mathrm{F}}_{141}^{41}$ )		–	–
		Transmit load ( ${\mathrm{F}}_{142}^{41}$ )	Force	${\mathrm{F}}_{21}^{41}$
Lock nut ( ${\mathrm{c}}_{15}^{41}$ )	Transmit clamping force ( ${\mathrm{F}}_{151}^{41}$ )		Force	${\mathrm{F}}_{12}^{41}$

Step IV

This step concerns function digraph models at system, subsystem and assembly levels. However, only the function digraph model for an assembly, that is, ball screw drive assembly, (Figure 8) is demonstrated.

For simplicity of the digraph model, some components, ${\mathrm{c}}_3^{41}$ , ${\mathrm{c}}_4^{41}$ , ${\mathrm{c}}_{11}^{41}$ ${\mathrm{c}}_{12}^{41}$ , and hence their functions will not be included. It may be mentioned that high accuracy and mechanical efficiency are the prime requirements of the assemblies and components of CNC machines. Each function plays a vital role in meeting these requirements. In their work, Wani and Gandhi ²⁸ considered the functions responsible for geometric motion for the functional analysis. However, other functions that do not contribute to geometric motion but help in isolation and enclosing of elements also influence the reliability. For instance, function(s), say, ‘isolate bearing’ or ‘transmit load’ of a bearing housing, is necessary to achieve mounting accuracy. Although a ball screw shaft has better transmission accuracy, it will not give the benefit of the accuracy to a machine when it is not mounted accurately. Improper mounting of bearing housing will lead to component loosening as the vibration increases. It is, therefore, appropriate to consider all functions for the functional analysis of elements that are designed for high accuracy and mechanical efficiency in the context of reliability consideration. Based on the above, Table 2 is developed for functions and their interaction for ball screw drive assembly.

AFD for the ball screw drive assembly is developed by considering the details given in Table 2. This is shown in Figure 9. The AFD represents functions and their interactions at assembly level. The nodes of the digraph represent the component functions and the directed edges represent the functional interactions among components.

Figure 9.

Assembly function digraph (AFD) – ball screw drive assembly.

The AFD is helpful to carry out the cause-and-effect function analysis visually, but in a limited way. Let us consider the functional interaction between transmit power ( ${\mathrm{F}}_{71}^{41}$ ) and ‘transmit clamping force’ ( ${\mathrm{F}}_{151}^{41}$ ). If the function ‘ ${\mathrm{F}}_{71}^{41}$ ’ deviates from its nominal value (e.g. ball screw shaft power fluctuates due to the change in drive motor current), the ball screw will vibrate and subsequently it will affect the function ‘ ${\mathrm{F}}_{151}^{41}$ ’ (e.g. clamping force of the lock nut decreases). This will result in loss of function of lock nut, that is, nut loosening. Here, the failed function is ‘clamping force’ and the cause function is the ‘transmit power’. The direct functional interactions can be analysed in this way using the AFD.

It is observed from the AFD that the functions represented by isolated nodes, ${\mathrm{F}}_{51}^{41}$ (prevent leakage), ${\mathrm{F}}_{61}^{41}$ (prevent leakage), ${\mathrm{F}}_{103}^{41}$ (isolate ball nut), ${\mathrm{F}}_{131}^{41}$ (isolate bearing) and ${\mathrm{F}}_{141}^{41}$ (isolate bearing), do not interact with any of the other functions. However, loss of these functions may be caused by their improper interaction with the other functions that help in mounting and enclosing of the assembly in the housing.

In this way, the functional analysis can be carried out visually for other elements by developing their digraph models. However, for detailed analysis of functions, the AFD will be converted into equivalent matrix, which is carried out in the next step.

Step V

The node–node function connection matrix is obtained at various system levels based on the discussion under section ‘Matrix representation’. The value 1 in the matrix means that there is a direct interaction between two functions and 0 indicates that there is no direct interaction. To illustrate, let us consider functional interaction between ${\mathrm{F}}_{71}^{41}$ and ${\mathrm{F}}_{11}^{41}$ , with ${\mathrm{F}}_{71}^{41}$ (transmit power) interacting with ${\mathrm{F}}_{11}^{41}$ (support redial load) through direct connection; hence, the matrix entity in the function connection matrix is assigned the value 1, while other elements are 0. In this way, assembly node–node function connection matrix (ANNFCM; F) is written for the AFD (Figure 9), which is written as matrix, A (equation (5))

$$\begin{array}{c}\mathrm{A}=\left[\begin{array}{cccccccccccccccccccccc}\hfill F_{11}^{41}\hfill & \hfill F_{12}^{41}\hfill & \hfill F_{21}^{41}\hfill & \hfill F_{22}^{41}\hfill & \hfill F_{51}^{41}\hfill & \hfill F_{52}^{41}\hfill & \hfill F_{61}^{41}\hfill & \hfill F_{62}^{41}\hfill & \hfill F_{71}^{41}\hfill & \hfill F_{72}^{41}\hfill & \hfill F_{81}^{41}\hfill & \hfill F_{82}^{41}\hfill & \hfill F_{91}^{41}\hfill & \hfill F_{92}^{41}\hfill & \hfill F_{101}^{41}\hfill & \hfill F_{102}^{41}\hfill & \hfill F_{103}^{41}\hfill & \hfill F_{131}^{41}\hfill & \hfill F_{132}^{41}\hfill & \hfill F_{141}^{41}\hfill & \hfill F_{142}^{41}\hfill & \hfill F_{151}^{41}\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\begin{array}{c}\multicolumn{1}{c}{F_{11}^{41}}\\ \multicolumn{1}{c}{F_{12}^{41}}\\ \multicolumn{1}{c}{F_{21}^{41}}\\ \multicolumn{1}{c}{F_{22}^{41}}\\ \multicolumn{1}{c}{F_{51}^{41}}\\ \multicolumn{1}{c}{F_{52}^{41}}\\ \multicolumn{1}{c}{F_{61}^{41}}\\ \multicolumn{1}{c}{F_{62}^{41}}\\ \multicolumn{1}{c}{F_{71}^{41}}\\ \multicolumn{1}{c}{F_{72}^{41}}\\ \multicolumn{1}{c}{F_{81}^{41}}\\ \multicolumn{1}{c}{F_{82}^{41}}\\ \multicolumn{1}{c}{F_{91}^{41}}\\ \multicolumn{1}{c}{F_{92}^{41}}\\ \multicolumn{1}{c}{F_{101}^{41}}\\ \multicolumn{1}{c}{F_{102}^{41}}\\ \multicolumn{1}{c}{F_{103}^{41}}\\ \multicolumn{1}{c}{F_{131}^{41}}\\ \multicolumn{1}{c}{F_{132}^{41}}\\ \multicolumn{1}{c}{F_{141}^{41}}\\ \multicolumn{1}{c}{F_{142}^{41}}\\ \multicolumn{1}{c}{F_{151}^{41}}\end{array}\hfill \end{array}$$

(5)

The ANNFCM (equation (5)) expresses the connectional relation among the functions of ball screw drive assembly. This represents direct functional relations, that is, single-stage functional connections between the functions of elements of ball screw drive assembly. However, multi-stages of connections between functions will help to evaluate functions’ importance.

Step VI

For evaluation of function importance of ball screw assembly, the ANNFCM (equation (5)) is processed as per the discussion under section ‘System function importance evaluation matrix’. The following are its steps.

Step VI(1)

Assign non-zero non-diagonal elements 1 and diagonal element 0 in the ANNFCM obtained in Step V.

Steps VI((2) and (3))

Multiply ‘A’ by A, one after another. Let a new matrix be ‘B’ (equation (6)), which is obtained by multiplication of A by A after n multiplication. Here, n represents the multiplication, which is equal to ‘2’ after carrying out the multiplication, that is, A × A. The value of n will change as 3, 4,… and so on for subsequent multiplications. A non-zero or unity element, which appears for the first time in matrix B (equation (6)) after ‘2’ multiplication, that is, n = 2, shows that there is a two-stage connection between the function of a row and the function of the column.

$$\begin{array}{c}\mathrm{B}=\left[\begin{array}{cccccccccccccccccccccc}\hfill F_{11}^{41}\hfill & \hfill F_{12}^{41}\hfill & \hfill F_{21}^{41}\hfill & \hfill F_{22}^{41}\hfill & \hfill F_{51}^{41}\hfill & \hfill F_{52}^{41}\hfill & \hfill F_{61}^{41}\hfill & \hfill F_{62}^{41}\hfill & \hfill F_{71}^{41}\hfill & \hfill F_{72}^{41}\hfill & \hfill F_{81}^{41}\hfill & \hfill F_{82}^{41}\hfill & \hfill F_{91}^{41}\hfill & \hfill F_{92}^{41}\hfill & \hfill F_{101}^{41}\hfill & \hfill F_{102}^{41}\hfill & \hfill F_{103}^{41}\hfill & \hfill F_{131}^{41}\hfill & \hfill F_{132}^{41}\hfill & \hfill F_{141}^{41}\hfill & \hfill F_{142}^{41}\hfill & \hfill F_{151}^{41}\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\begin{array}{c}\multicolumn{1}{c}{F_{11}^{41}}\\ \multicolumn{1}{c}{F_{12}^{41}}\\ \multicolumn{1}{c}{F_{21}^{41}}\\ \multicolumn{1}{c}{F_{22}^{41}}\\ \multicolumn{1}{c}{F_{51}^{41}}\\ \multicolumn{1}{c}{F_{52}^{41}}\\ \multicolumn{1}{c}{F_{61}^{41}}\\ \multicolumn{1}{c}{F_{62}^{41}}\\ \multicolumn{1}{c}{F_{71}^{41}}\\ \multicolumn{1}{c}{F_{72}^{41}}\\ \multicolumn{1}{c}{F_{81}^{41}}\\ \multicolumn{1}{c}{F_{82}^{41}}\\ \multicolumn{1}{c}{F_{91}^{41}}\\ \multicolumn{1}{c}{F_{92}^{41}}\\ \multicolumn{1}{c}{F_{101}^{41}}\\ \multicolumn{1}{c}{F_{102}^{41}}\\ \multicolumn{1}{c}{F_{103}^{41}}\\ \multicolumn{1}{c}{F_{131}^{41}}\\ \multicolumn{1}{c}{F_{132}^{41}}\\ \multicolumn{1}{c}{F_{141}^{41}}\\ \multicolumn{1}{c}{F_{142}^{41}}\\ \multicolumn{1}{c}{F_{151}^{41}}\end{array}\hfill \end{array}$$

(6)

For instance, the value of elements, 3, 6, 8, 11 and 22 in the first row of matrix B (equation (6)), is ‘1’, which is a unity element that appeared for the first time in matrix B after ‘2’ multiplication. This shows that there is a two-stage connection between functions, ${\mathrm{F}}_{11}^{41}-{\mathrm{F}}_{21}^{41}$ , ${\mathrm{F}}_{11}^{41}-{\mathrm{F}}_{52}^{41}$ , ${\mathrm{F}}_{11}^{41}-{\mathrm{F}}_{62}^{41}$ , ${\mathrm{F}}_{11}^{41}-{\mathrm{F}}_{81}^{41}$ and ${\mathrm{F}}_{11}^{41}-{\mathrm{F}}_{151}^{41}$ . The next multiplication will be B by A, which will produce a new matrix C and the value of n will change to ‘3’. Based on this, the three-stage connection between functions will be obtained from matrix, C. This procedure is continued until no more new matrix. The matrix, which is obtained after 15 repetitions or continuation of the procedure, provides the value of n for all functional connections of ball screw drive assembly. MATLAB ⁴⁴ program is used to evaluate this.

Step VI(4)

The value of n at location of all non-zero or unity elements in the matrix obtained in Step VI(3) is shown. This is called ‘assembly function importance evaluation matrix’ (AFIEM) of ball screw drive assembly, which is obtained as

$$\begin{array}{c}\mathrm{Z}=\left[\begin{array}{cccccccccccccccccccccc}\hfill F_{11}^{41}\hfill & \hfill F_{12}^{41}\hfill & \hfill F_{21}^{41}\hfill & \hfill F_{22}^{41}\hfill & \hfill F_{51}^{41}\hfill & \hfill F_{52}^{41}\hfill & \hfill F_{61}^{41}\hfill & \hfill F_{62}^{41}\hfill & \hfill F_{71}^{41}\hfill & \hfill F_{72}^{41}\hfill & \hfill F_{81}^{41}\hfill & \hfill F_{82}^{41}\hfill & \hfill F_{91}^{41}\hfill & \hfill F_{92}^{41}\hfill & \hfill F_{101}^{41}\hfill & \hfill F_{102}^{41}\hfill & \hfill F_{103}^{41}\hfill & \hfill F_{131}^{41}\hfill & \hfill F_{132}^{41}\hfill & \hfill F_{141}^{41}\hfill & \hfill F_{142}^{41}\hfill & \hfill F_{151}^{41}\hfill \\ \multicolumn{1}{c}{0}& \hfill 3\hfill & \hfill 2\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill \\ \multicolumn{1}{c}{6}& \hfill 0\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 7\hfill & \hfill 3\hfill & \hfill 8\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 7\hfill & \hfill 0\hfill & \hfill 7\hfill & \hfill 6\hfill \\ \multicolumn{1}{c}{2}& \hfill 3\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \multicolumn{1}{c}{6}& \hfill 2\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 7\hfill & \hfill 3\hfill & \hfill 8\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 7\hfill & \hfill 0\hfill & \hfill 7\hfill & \hfill 6\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{2}& \hfill 3\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{2}& \hfill 3\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill \\ \multicolumn{1}{c}{1}& \hfill 2\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{5}& \hfill 1\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 7\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 5\hfill \\ \multicolumn{1}{c}{2}& \hfill 3\hfill & \hfill 2\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill \\ \multicolumn{1}{c}{4}& \hfill 2\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 4\hfill \\ \multicolumn{1}{c}{3}& \hfill 4\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 3\hfill \\ \multicolumn{1}{c}{3}& \hfill 3\hfill & \hfill 3\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 3\hfill \\ \multicolumn{1}{c}{4}& \hfill 5\hfill & \hfill 4\hfill & \hfill 7\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 6\hfill & \hfill 1\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 4\hfill \\ \multicolumn{1}{c}{4}& \hfill 4\hfill & \hfill 4\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 4\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 4\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 5\hfill & \hfill 3\hfill & \hfill 5\hfill & \hfill 4\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 3\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{3}& \hfill 4\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 5\hfill & \hfill 3\hfill & \hfill 5\hfill & \hfill 4\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \multicolumn{1}{c}{7}& \hfill 1\hfill & \hfill 7\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 7\hfill & \hfill 3\hfill & \hfill 8\hfill & \hfill 4\hfill & \hfill 9\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8\hfill & \hfill 0\hfill & \hfill 8\hfill & \hfill 0\hfill \end{array}\right]\begin{array}{c}\multicolumn{1}{c}{F_{11}^{41}}\\ \multicolumn{1}{c}{F_{12}^{41}}\\ \multicolumn{1}{c}{F_{21}^{41}}\\ \multicolumn{1}{c}{F_{22}^{41}}\\ \multicolumn{1}{c}{F_{51}^{41}}\\ \multicolumn{1}{c}{F_{52}^{41}}\\ \multicolumn{1}{c}{F_{61}^{41}}\\ \multicolumn{1}{c}{F_{62}^{41}}\\ \multicolumn{1}{c}{F_{71}^{41}}\\ \multicolumn{1}{c}{F_{72}^{41}}\\ \multicolumn{1}{c}{F_{81}^{41}}\\ \multicolumn{1}{c}{F_{82}^{41}}\\ \multicolumn{1}{c}{F_{91}^{41}}\\ \multicolumn{1}{c}{F_{92}^{41}}\\ \multicolumn{1}{c}{F_{101}^{41}}\\ \multicolumn{1}{c}{F_{102}^{41}}\\ \multicolumn{1}{c}{F_{103}^{41}}\\ \multicolumn{1}{c}{F_{131}^{41}}\\ \multicolumn{1}{c}{F_{132}^{41}}\\ \multicolumn{1}{c}{F_{141}^{41}}\\ \multicolumn{1}{c}{F_{142}^{41}}\\ \multicolumn{1}{c}{F_{151}^{41}}\end{array}\hfill \end{array}$$

(7)

In matrix Z (equation (7)), matrix entity or element 0 means that there is no any directed edge from a functional node to another functional node of AFD of ball screw drive assembly. This implies that the function is isolated, that is, it does not interact with other function. The elements other than 0 give the multi-stage relationship between the function of a row and the function of a column, that is, 1 means direct relationship, 2 means two-stage relationship and so on. This can be used to identify which function is relatively important by carrying out cause-and-effect functions analysis in the context of reliability enhancement for all elements of the assembly, which is discussed in the following lines.

Let us reconsider AFIEM, that is, matrix Z (equation (7)), for example, column of the function, ${\mathrm{F}}_{12}^{41}$ (support thrust load) for evaluation of the importance of functions for the ball screw drive assembly. There are two paths along which the functional variations occur: ${\mathrm{F}}_{72}^{41}$ – ${\mathrm{F}}_{12}^{41}$ and ${\mathrm{F}}_{72}^{41}$ – ${\mathrm{F}}_{151}^{41}$ . It is observed that the function ${\mathrm{F}}_{151}^{41}$ (transmit clamping force) is in one-stage relationship with ${\mathrm{F}}_{12}^{41}$ . It is clear that the function, transmit clamping force (cause function), of lock nut is quite influential on function, supports thrust load (effect or failed function) of duplex support bearing. This implies that no matter how well the bearing performs its function, that is, supports the thrust load, the problem of improper load distribution in the bearing occurs unless the lock nut performs its function properly, that is, imparting clamping force consistently. The function, transmit clamping force, helps to ensure proper load distribution in the bearing. This, in turn, provides long service life of the bearing and hence helps to enhance the reliability of the ball screw assembly. This emphasizes the importance of right amount of clamping force, that is, function, transmit clamping force, in the context of reliability enhancement. The similar argument is applicable for the function, ${\mathrm{F}}_{72}^{41}$ (support load, cause function), which also has one-stage relationship with the bearing function, ${\mathrm{F}}_{12}^{41}$ . If the ball screw drive assembly is able to perform its function, support load, the bearing function will be performed properly.

This emphasizes the importance of capacity of ball screw shaft to support the load, that is, function, support load. The action to either modify the cause function or improve the effect/failed function will be followed subsequently. The cause functions mentioned above are equally important and these, however, may be prioritized based on the type of function according to their purpose of application, be it static or dynamic supporting and assembling function. As a thumb rule, the supporting functions are relatively more important than the assembling functions. Therefore, the function, support load (supporting function), is important than transmit clamping force (assembling function).

Now, let us consider an example for two-stage functional connection. The function, ${\mathrm{F}}_{82}^{41}$ (support radial load), of the recirculating balls interacts with the function, ${\mathrm{F}}_{12}^{41}$ (support thrust load), of front duplex support bearing in two stages through the intermediate function, ${\mathrm{F}}_{72}^{41}$ (support load), of ball screw shaft, that is, through the path, ${\mathrm{F}}_{82}^{41}$ – ${\mathrm{F}}_{72}^{41}$ – ${\mathrm{F}}_{12}^{41}$ . This corresponds to such case that the function ${\mathrm{F}}_{12}^{41}$ deteriorates as a result of deviation of function ${\mathrm{F}}_{72}^{41}$ due to the degradation of function ${\mathrm{F}}_{82}^{41}$ . In the first stage, the recirculating balls, when failed to support radial load, will cause loss of ball screw function, support load, that is, the ball screw will be unable to withstand the additional load caused by the failure of balls. In the second stage, this will cause the front duplex support bearing to loss its function, support thrust load. The function ${\mathrm{F}}_{82}^{41}$ is the cause function and its variation propagates to the effect/failed function ${\mathrm{F}}_{12}^{41}$ through the intermediate function ${\mathrm{F}}_{72}^{41}$ . An appropriate action will have to be taken to improve the effect–cause function. It may also be mentioned here that the function ${\mathrm{F}}_{72}^{41}$ (support load) of ball screw has more immediate influence on ${\mathrm{F}}_{12}^{41}$ than ${\mathrm{F}}_{82}^{41}$ . This implies that the function ${\mathrm{F}}_{12}^{41}$ deviates unless the ball screw shaft supports load properly no matter how proper the recirculating balls supports radial load. This points out the importance of the function ${\mathrm{F}}_{72}^{41}$ (support load) of ball screw. Similarly, the function ${\mathrm{F}}_{22}^{41}$ (support thrust load) is also in two-stage relationship with the function ${\mathrm{F}}_{12}^{41}$ , through the path ${\mathrm{F}}_{22}^{41}$ – ${\mathrm{F}}_{72}^{41}$ – ${\mathrm{F}}_{12}^{41}$ . The function ${\mathrm{F}}_{72}^{41}$ (support load) of ball screw has immediate influence on ${\mathrm{F}}_{12}^{41}$ than ${\mathrm{F}}_{22}^{41}$ .

From the above, it is evident that the ball screw shaft should support load properly to ensure proper functioning of duplex bearing as these are in direct connection. This emphasizes the importance of one-stage functional connection over two-stage functional connection. This lies in the fact the functional variation can quickly propagate through direct connection, which will result in severe causality of functional loss, whereas the propagation of functional variation through multi-stage connection will cause less severe causalities.

In this way, matrix Z is utilized for enhancing overall reliability of ball screw assembly through function importance evaluation. The function important evaluation for the function ${\mathrm{F}}_{12}^{41}$ (support thrust load) as discussed above is given in Table 3, which helps to provide directions for reliability enhancement. In a similar manner, the cause-and-effect functions analysis is carried out to evaluate function importance up to n-stages of functional connections to help in prioritizing the functions in the context of reliability enhancement. The priority number of a cause function, for example, is 1-1, which means that the cause function is in direct relationship, that is, one-stage functional connection, with the effect function and hence, is to be taken care of on first priority for any action.

Table 3.

Function importance evaluation of function ${\mathrm{F}}_{12}^{41}$ (support thrust load).

Effect/failed function	Cause functions	Path along which functional variation occurs	Effect	Action to improve the effect–cause function	Type of function according to the purpose of application	Priority number of the cause function
${\mathrm{F}}_{12}^{41}$ (support thrust load of front duplex support bearing)	${\mathrm{F}}_{72}^{41}$ (support load of ball screw shaft)	${\mathrm{F}}_{72}^{41}$ – ${\mathrm{F}}_{12}^{41}$	Increases thrust load on the bearing	Ensure better load capacity of ball screw or increase thrust load-carrying capacity of bearing to provide long service life of bearing	Supporting function – static	1-1
	${\mathrm{F}}_{151}^{41}$ (transmit clamping force of lock nut)	${\mathrm{F}}_{151}^{41}$ – ${\mathrm{F}}_{12}^{41}$	Improper load distribution on the bearing	Ensure optimum level of clamping force or increase thrust load-carrying capacity of bearing to provide long service life of bearing	Assembling function – static	1-2
	${\mathrm{F}}_{82}^{41}$ (support radial load of recirculating balls)	${\mathrm{F}}_{82}^{41}$ – ${\mathrm{F}}_{72}^{41}$ – ${\mathrm{F}}_{12}^{41}$	Increases additional load on ball screw shaft and thrust load on the bearing	Ensure better radial load-carrying capacity of recirculating balls or increase thrust load-carrying capacity of bearing to provide long service life of bearing	Supporting function – dynamic	2-1
	${\mathrm{F}}_{22}^{41}$ (support thrust load of rear duplex support bearing)	${\mathrm{F}}_{22}^{41}$ – ${\mathrm{F}}_{72}^{41}$ – ${\mathrm{F}}_{12}^{41}$	Increases additional load on ball screw shaft and thrust load on the bearing	Increase thrust load-capacity of rear duplex support bearing or of front support bearing	Supporting function – static	2-2

Step VII

Based on the above, the appropriate actions are identified to enhance the overall reliability of a ball screw assembly which are discussed below.

Creating database

Based on the details presented in Steps I–VI, not only assembly level but also at higher and lower levels database can be created to help to provide directions for reliability enhancement through functions. A basic type of ball screw assembly of a CNC machine involves as many as 22 functions (refer Table 2). Besides providing detailed evaluation of function importance, the database may help to retrieve other basic information such as the list of element functions including primary and secondary, parameters through which the functions interact, path along which the functional variations occur and list of cause functions and failed functions. More qualitative functional descriptions including types of function (e.g. supporting or assembling function) can also be added to the database to further determine which function has to be given priority when two or more functions are in direct connections. Such a database will also help to identify appropriate data, which can help to eventually more accurate quantitative evaluation of functions (e.g. assigning probability of functional loss) to enhance the assembly reliability. This helps to identify not only the functions of ball screw assembly that are more vulnerable to partial or complete loss but also the mode of functional loss. Using the database, it is also possible to plan redundancies (e.g. redundant ball nuts for ball screw assembly) early at the design stage. During system operation, maintenance tasks assist to improve reliability. This database may provide useful information not only to choose appropriate maintenance tasks (e.g. Condition Based Maintenance (CBM) for monitoring shaft and bearing functions) for each element of the assembly but also to provide guidelines in performing machine alignments. The database can then be extended to the remaining assemblies, subsystems and system.

Identifying critical functions

Functions are prioritized based on the importance evaluated using function importance evaluation matrix. One can also determine critical functions in the context of reliability enhancement by obtaining causalities due to improper functional interactions. A function will be more critical when the severity of causality due to its interaction with another function is more.

Redesign/modification

The functions, which are closely connected, can be improved by modification/or redesign of their corresponding structural elements, taking into account the optimum level of design, operational and environmental parameters. This will require detailed design and manufacturing process knowledge of various elements of ball screw assembly.

Step VIII

On the lines of the above, reliability enhancement is carried out at subsystem and system levels of CNC drilling machine. Based on the approach and its illustration for the case study, it is evident that methodology is effective in analysing functions of manufacturing system by considering structure. The approach takes care of system complexity by identifying each element including its functions and their interdependencies/interactions. One can improve the reliability at any level of system hierarchy by identifying its functional interdependencies/interactions. The proposed method can, therefore, be potentially applied to all hierarchical levels of a manufacturing system including its subsystems. The method helps not only the design engineers in enhancing system reliability in design stage but also for improving its operational reliability during operation by helping the plant engineers in redesign/modification of the system elements.

Conclusion

A methodology for reliability enhancement of a manufacturing system has been proposed through functions. This has been carried out by identifying functions and their interactions at various system levels. A systems approach, digraph model, has been applied to develop function digraphs. The analysis is carried out using matrix method to evaluate function importance at each system level to examine cause and effect. This provides direction in reliability enhancement. The proposed methodology is illustrated for a CNC drilling machine.