Set Based Robust Design of Systems - Application to ...

Set Based Robust Design of Systems - Application to Flange Coupling (1,2)

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QURESHI Ahmed Jawad , DANTAN Jean-Yves , BRUYERE Jérôme , BIGOT Régis (1) LCFC, Arts et Métiers ParisTech Metz, 4 Rue Augustin Fresnel, 57078 METZ CEDEX 3, France. (2) Higher Education Commission Sector H-9, Islamabad, Pakistan. (3) LaMCoS UMR5259, Université de Lyon, CNRS, INSA-Lyon, F-69621 VILLEURBANNE, France

Abstract The paper discusses the mapping of the design space for existence of the robust solution for design of a mechanical system. Set based design is a popular contemporary design approach to designing systems with multiple parallel design solution possibilities and then narrowing through the possibilities towards the most optimal solution. In this paper we present an approach that provides means for the robust design space exploration of the system by integrating an algorithm providing the means for simultaneous exploration of the design and variation space. The design space is a set of design parameters in which each parameter is a set of discrete/continuous values of possible solution and the variation space is a set of variations with each variation being a set of associated uncertainties. In order to provide formal description of the problem the quantifier notion from the QCSP (Quantified Constraint Satisfaction Problem) has been utilised. The QCSP technique is an extension of the CSP (Constraint satisfaction problem), which helps integrate the robustness in the design of an assembly. This integration has been done with the help of an algorithm which brings together and integrates these notions using interval arithmetic. In order to demonstrate the approach, an example of design of rigid flange coupling with a variable number of bolts and a choice of bolts from ISO M standard has been resolved and demonstrated. Keywords: Set based design, Robust design, QCSP, Quantifiers, Tolerance integration. 1. INTRODUCTION One of the challenges that the design engineers deal with during the early design phase of a system is lack of detailed information. In order to create a conceptual design, the design engineers create initial concepts based on the client’s requirement. These concepts then compete with each other and are refined as more information becomes available during the product design cycle. In order to create the concepts the first task is to identify the functional requirements and define the main design parameters for the product. Once these parameters are defined the design process strives to find the appropriate values of these parameters such that an optimal balance between the desired functional requirements and constraints related to the product performance, quality and cost are achieved. Many design approaches exist which enable the design engineers to take the appropriate steps in order to reach this desired results. Traditionally the design process goes through a point based design approach where individual stake holders in the design process offer their specific design solution to other participating interfaces which then further develops the design based on the available information. Often this process results in the reworks due to the downstream incompatibilities that are reported via feedback loop and therefore result not only in rework but also in increasing time spent before the design can be finalised [1]. “Set-based concurrent engineering” (SCBE) or set based design is an approach utilised by Toyota to pursue its design process. SCBE begins by broadly considering sets of possible solutions and gradually narrowing the set of possibilities to converge on a final solution [1]. SBCE process consists of development and communication of sets of solutions in parallel and relatively independently instead of following a point based approach. These sets are then refined by elimination of set members that are not feasible, resulting in the narrowing of the possibilities. As the sets narrow, the depth and detail of information increases thereby refining the sets. Also the ability to take into account a number of sets concurrently instead of in a CIRP Design Conference 2009

point wise approach enables to achieve a robust solution space which is less susceptible to a rework or rejection by other design interfaces. In this paper we present an approach for robust design of mechanical systems that is based on the set-based design approach. The approach presented relates to the domain mapping stage of the set-based design and relying on the set-based design approach objectives it provides robust solution of mechanical systems. This is achieved by using the quantifier notion from QCSP and Interval arithmetic to perform design space exploration and separate the admissible design solution spaces containing the robust solutions from the departing design space. The approach permits taking into account the design parameters and uncertainty/variation parameters associated to the design of system in forms of sets, either discrete or continuous as well as integration of different design stake holders (design and manufacturing) simultaneously in the design solution space exploration. The integration of uncertainty/variation within the design space exploration results in the solutions being inherently robust. An algorithm has been developed which treats the example of a rigid flange coupling design with ISO standard screws and demonstrates the approach. 2. SET-BASED ROBUST DESIGN The main objective for robust design of a system is to design a system in such a way that its performance is not compromised beyond the minimum requirements in presence of variations and uncertainties in PLM as well as different environmental and other changing parameters. These uncertainties and variations can be classified into: Changing environmental and operating conditions; production tolerances and actuator imprecision; uncertainties in the system output and feasibility uncertainties. These uncertainties may be of deterministic type or probabilistic type or possibilistic type [2]. Robust design can be viewed as an optimisation approach which tries to account for uncertainties described above.

Traditionally the robust design is performed via sensitivity analysis of a given solution to the variations and then adjusting the design parameters such that the performance of the system is according to the requirements in presence of such variations. Different approaches exist for robust design such as Taguchi [4] [5] , robust optimization, axiomatic design [3] and variation risk management etc. which with help of deterministic or randomized approaches [2] allow the engineers to decide a robust parameter design of a system, Set based design is an approach to engineering design in which different design alternatives are evaluated by reasoning and comparing different design solutions based on possibilities offered by alternative possible configurations of “SETS” of design parameters. Set based design aims to delay commitments to a particular design in favour of gathering information about problem and to reduce imprecision to levels at which indeterminacy is resolved[6]. The domain mapping phase of a Set based design methodology consists of mapping or exploring the design space for feasible regions while keeping multiple alternatives and looking for intersections of feasible sets as proposed by different design/manufacturing interfaces [1] which satisfy the required minimum number of constraints while ensuring robustness. When viewed in terms of robust mechanical design, this translates into ensuring a choice of multiple design configurations, which may empower the design team to choose an appropriate solution while at the same time ensuring the robustness of the design with respect to the functional, as well as quality and cost constraints. Essentially it means that the different stake holders in the design process may be able to communicate their feasible sets at the start of the design process i.e. the design engineering may communicate the sets of design parameters where as manufacturing may communicate the sets of machining / process capability. The resulting design space consisting of the design parameter space and variation parameter space can then be explored for the feasible region. The intersection of the feasible design space can then lead to a possible solution and an intersection of design and variation space would lead to a solution which may be inherently robust. Using the principle described above, the approach presented in this paper provides a method to perform a simultaneous domain exploration of design and variation parameters, resulting in a robust solution. 2.1 Design as Constraint satisfaction problem The design parameters are the key variables which describe the product as well as its behaviour. The design variables may be of the geometric nature, engineering nature or manufacturing nature and may deal with shape, configuration, material, manufacturing process etc. Each product has some main performance criteria to fulfil. These criteria are the design constraints, i.e. the minimum performance requirements. These design constraints are generally expressions consisting of the design parameters, and constants. In an engineering model, the representation of constraints may be algebraic equations or predicates, sometimes with a few additional logical constraints [7]. Essentially it can be viewed as solving a system of simultaneous equation for homogenous solution. The first approach of solving the system is based on an iterative, numerical algorithm. It may be used when the constraints are expressed as algebraic equations in implicit form (f(x) = 0). The mathematical problem is to solve this set of non-linear equations simultaneously [1].

Different iterative methods for solving such systems exist which may be generalized in the category of Constraint Satisfaction Problems (CSP). 2.2 CSP adaptation to uncertainty A solution to a CSP generally seeks to identify the discrete point values or instances of research domains to evaluate a solution which satisfies the constraints. A set of these instances calculated in an iterative manner allow us to develop the final solution in terms of valid intervals which satisfy the constraints. This approach is feasible for the point based design but in terms of a set based design the need arises to have a system in place that can address the sets natively, i.e. a group of information. In addition, the need to separately quantify the variables existentially or universally is also required to integrate the notion of uncertainty/variation in the system. In such cases, the solution calculated by help of conventional CSP techniques might not suffice and may not perform according to the desired performance level. It is therefore useful to have a method which might integrate the incertitude in the system or in environment at an early design stage. In order to do so a methodology is being proposed which not only integrates the notion of uncertainty in the product but also enables quantification of the variables. As such the methodology being proposed allows the robust solution of a parametric design problem. 2.3 QUANTIFIED CONSTRAINT SATISFACTION PROBLEM INTEGRATION FOR ROBUST DESIGN [8] The quantified constraint satisfaction problem (QCSP) is a general extension of the constraint satisfaction problem (CSP) in which variables is totally ordered and quantified either existentially or universally. This generalization provides a better expressiveness for modelling problems. Model checking and planning under uncertainty are examples of problems that can be modelled with QCSP . The research in QCSP is recent. Bordeux and Montfroy [9] have extended the notion of Arc consistency of CSP to the QCSP. Mamoulis and Stergiou [10] have defined an algorithm for arc consistency for QCSP for binary constraints. In this paper, we demonstrate the effective application of the QCSP technique for integrating the notion of robustness in the Mechanical design process. 3. PROBLEM FORMALIZATION As mentioned earlier, an engineering design model may be expressed as a system of equations which may be generalized as a CSP. The general representation of the CSP may be done as a triple: {V , D,C }

(1)

Where V= Set of variables D= Domain of the variables C= Constraints governing the variables. In a general design problem, V represents all the design variables, D defines the total search space and C contains all the design / performance constraints. In CSP all the variables are existentially quantified, i.e. the interaction between the variables is existential and unique. This interaction is sufficient for modelling a parameter based problem but for the integration of the concept of uncertainty / noise, CSP modelling becomes complex. Modelling uncertainty and noise in a model stipulates that

the variables be able to quantify universally i.e. the ability to individually condition the interaction between the different variables of a model. QCSP offer us this flexibility. A General QCSP can be expressed as:

{QV , D,C}

(2)

Where Q is a quantifier ∃ or ∀, V is V= Set of variables D= Domain of the variables C= Constraints governing the variables. The QCSP allows the designer to explicitly condition the variables with help of quantifier to define its interaction in the system. 3.1 Integration of QCSP with Robust design In order to integrate the general design problem with help of QCSP given in eq. (2), we need to add the parameters of uncertainty to the design problem. There fore the new system that emerges can be described as:

{QV ,QN , D,C}

(3)

Where QV= Quantified design variables QN= Quantified noise or uncertainty variables. D= Domain of the variables C= Constraints governing the variables. The emerging system can now be formalized mathematically to a robust solution. A brief description of terms is given in order to illustrate the following example. • Design parameters “vi” are the parameters having an appreciable effect on the product performance and functional characteristic. These parameters maybe of a mechanical or geometric nature. • The noise parameters “nj” are the variables which model the noise. These parameters are the measure of the uncertainty of different factors, which might impact the product performance and therefore the conformity of the product performance to the desired design basis. Once these parameters have been designed, a model needs to be defined which embodies the relation between the product functional requirements, design basis and the above defined parameters. This is done with the help of the system of equations that model the mechanical behaviour of the system while taking into account the design as well as the noise parameters. The equation may be of the explicit or implicit form. In general they can be described in the following form: f(vi, nj) is a function which defines the relationship between the desired product performance, the design parameters and the noise parameters. 3.2 Conditions for Solution In order for a solution to be robust, it should respect the two global conditions concerning the design parameters as well as the noise parameters. Condition for existence of a solution The first condition for the existence of a robust solution is that; a solution must exist belonging to the domain of the design parameters such that the design constraints are satisfied. This can be defined as “At least one configuration of design parameters belonging to their respective domains must exist such that the functional requirements are fulfilled”. It can be translated mathematically as:

∃V ∈ DV :

(4)

Df (V ) ∈ Dsolution

Where V is the set of the design parameters belonging its respective domain DV. Condition for existence of a robust solution The second condition for the existence of a robust solution is that; there must exist a solution satisfying the constraints with for all the values of design parameters within their domains while keeping in account all possible values of noise variables within their domains. This can be defined as: “For all the values of design parameters and for all the values of noise, the constraints must be respected. This can be mathematically translated as: ∀ N ∈ DN , ∀V ∈ DV ;

(5)

Df (V ,N )i ∈ DSolution

These two conditions are verified for the successful existence of a solution. An algorithm has been developed which permits solving this problem with help of usage of exhaustive search and interval analysis techniques by using interval analysis to search for robust solution in the sub search spaces created by the algorithm. 4. ALGORITHM Having developed the integration of QCSP with robust design and the subsequent mathematical expressions establishing the basis for the robust solution in terms of the universal and existential quantifiers, in this section we will explain the basic concepts related to the transformation of the developed theory in an applicable algorithm which can then be programmed and run on computer to obtain the solution for a given problem. The main challenge in the transformation of the robust design with quantifier expression is the expression of quantifiers. For this purpose a number of approaches were explored out of which transformation into an interval analysis problem was adopted. In order to apply the developed approach with help of interval analysis we need to define some fundamental notions [11]. Definition 1: The design variables involved in the problem are expressed in forms of intervals except in case of design variables of discreet nature. Each interval is a set of connected reals with lowest and upper bounds as floating point intervals. The interval I for a design variable x defined as a real number would therefore be represented in form of an interval as follows:

{

= I x  x, x  ≡ x ∈ R | x ≤ x ≤ x

}

(6)

Definition 2: A Cartesian product of n intervals B =I1 × ..... × In is called a box; a domain D is either an interval I or a Union U of disjoint intervals. Definition 3: The set of the initial domains of all the involved variables is D-BOX. A D-Box BD with arity n is the Cartesian product of n intervals where n is the number of the design variables involved in the problem. It is denoted by I x1, I x 2 , I x 3 ,...I xn where each I is an interval. BD = {I x1, I x 2 , I x 3 ,...I xn }

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Definition 4: The set of the intervals of sub domains of the variables SD-BOX. A SD-Box BSD with airity n is the Cartesian product of n intervals where n is the number of the design variables involved in the problem. It is denoted

by

I x1, I x 2 , I x 3 ,...I xn

where each I is an interval. BSD

results when a D-BOX is split. BSD ⊆ BD  Isx1, Isx 2 , Isx 3 ,...Isxn | Isx1 ⊆ I x1, BSD =   Isx 2 ⊆ I x 2 ,...Isxn ⊆ I xn 

(8)

I x1, I x 2 , I x 3 ,...I xn ∈ Ι :

Definition 5 [11] An Interval extension of f : R n → R is a mapping F : Ι → Ι such that for all: n

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f ( x1,..., xn ) ∈ F (I x1,..., I xn ) An interval extension of a relation ρ ⊆ R n is a relation R ⊆ Ι n such that for all: I x1, I x 2 , I x 3 ,...I xn ∈ Ι : ∃x1 ∈ I x1,.....∃xn ∈ I xn

s.t .( x1,....xn ) ∈ ρ ⇒ ( I x1, I x 2 , I x 3 ,...I xn ) ∈ R

x1 ∈ I x1,.....xn ∈ I xn

(11)

Max (F (I x1,..., I xn )) ∨ Min(F (I x1,..., I xn )) ∈ Dsolution

I x1, I x 2 , I x 3 ,...I xn ∈ Ι : x1 ∈ I x1,.....xn ∈ I xn ⇒

Condition for box consistency of a solution: The equation 4 expresses the existence of solution in terms of the existential quantifier. Its transformation into the algorithm with help of the interval analysis stipulates that BSD should be consistent for the given constraints:

(10)

The function to be extended may be an inequality or equality. The software used to program the algorithm and later test it for the example mechanism is Mathematica. Mathematica contains the built in operators for the Universal quantifier and the Existential quantifiers. Initial applicability tests for the quantifier expression were undertaken while using these operators. The usage of these operators is however restricted to rudimentary verification only and soon becomes unfeasible for any applicability to even a simple mechanism. In order to address this, two alternative possibilities i.e Transformation to interval analysis and transformation into an optimization problem were studied. Among these two techniques, transformation of quantifiers with help of interval analysis was found to be fast and efficient therefore this technique was used. The algorithm is divided into three main steps: The first step takes the BD as an input and is responsible for dividing the BD in BSD and assigning the BSD to be evaluated to the next step for the evaluation of the existence of robust solution. Once this step is concluded the results are stored in the results module which then processes the results to present the domains related to the robust solution. A detailed flow chart describing the algorithm has been presented in Figure 2. The initial design domain specified by the design engineer is encapsulated in BD and is used as the starting search space for the algorithm. The algorithm then proceeds by the dividing BD in the number of BSD as specified by the design engineer. Each of these BSD is then successively passed on to the evaluation module for evaluation of box consistency existence of robust solution. The evaluation module is responsible for evaluation of the consistency of the BSD for finding the existence of robust solution. This module uses the two conditions expressed earlier to evaluate the robustness of the BSD under consideration. Using the quantifier conditions the relevant constraints arising from the transformation of the real functions f into the interval based functions F involving the interval variables obtained from the transformation of the real variables to interval variables are then used for evaluating the consistency of BSD. The functions maybe explicit or implicit expressed in terms of an equality or inequality. The quantifier conditions are translated by the following mathematical expressions:

For the BSD validated through the check performed by equation 4 robustness check is performed by the BSD consistency in presence of noise as stipulated by equation 5. This translates as following: Condition for box consistency of a robust solution: In presence of noise / uncertainty denoted by N={nx1,nx2,….,nxn}, where nxi is the noise/uncertainty related to the design variable xi , a solution is robust if the BSD is box consistent in presence of the noise parameters:

I x1, I x 2 , I x 3 ,...I xn ∈ Ι : nx1, nx 2 , nx 3 ,...nxn ∈ N : x1 ∈ I x1,.....xn ∈ I xn

(12)

Max (F (I x1,..., I xn , nx1, nx 2 ,...nxn )) ∧ Min(F (I x1,..., I xn , nx1, nx 2 ,...nxn )) ∈ Dsolution The transformation can be depicted by the following figure which shows the transformation of the components of the quantified Constraint satisfaction problem by interval analysis. Quantified Constraint Satisfaction Problem

Transformation to interval analysis

{QV , QN , D, C}

{V,N,D,C}

QV ={Qv1,Qv2,…Qvi}

V ={v1,v2,…vi}

vi ∈ R

Vi=[ vi , vi ]

QN ={Qn1,Qn2,…Qni}

N ={n1,n2,…ni}

ni ∈ R

ni=[ni , ni]

C ={c1,c2,…ci}

C={c1,c2,…ci} ci= f (v,n)

ci= f(v,n)

Figure 1 Quantifier transformation The quantified variables are replaced by the interval variables as shown in the diagram where each variable is assigned a upper and lower bound taken from the extremities of the interval. This operation is carried out for all the involved variables including the noise and design variables. Similarly the constraints are also transformed into interval constraints which are then able to take the interval variables. The constraints are then evaluated for the condition of existence of solution. If a BSD does not contain any solution, it is discarded and subsequently BD is reduced. Another BSD is then analyzed for the existence of solution. If an existence solution is found then this BSD is evaluated for global hull consistency of universal quantifier in presence of the uncertainty. In case of a successful evaluation the BSD saved as a robust design solution space. However if the space fails to evaluate for the consistency for robust solution, it is further decomposed into BSD until the robust solution space has been found. This process is repeated until the totality of BD has been explored for the robust solution. The final module stores the results as they are produced by the evaluation module. The results module serves two purposes. Firstly, it supplies the updated situation of the search during each iteration to the evaluation module and

secondly at the end of the simulation it presents the results in terms of the search space partition in terms of space without solution, space with robust solution and the space with a probable solution. The Figure 3 illustrates the algorithm with help of a simple example. The BD is [-6,6] which is then successively evaluated for the universal and existential quantifier in order to ascertain the consistency of the BD.

•

The Performance requirement is translated by the torque to be transmitted.

•

The safety and reliability requirement is translated by designing the coupling in a robust way to ensure the capacity of the coupling to transmit the torque while remaining within the zone of safe mechanical operations as given by the torque requirements and taking into account the uncertainty related to the design parameters.

5. APPLICATION

•

The cost versus quality factor to be evaluated by integrating the cost of material as well as the cost of the bolts.

In order to illustrate the approach an example has been tested. The example chosen is a design decision process regarding the dimensions of a Flange coupling as well as choice of bolts. The example is shown in Figure 4 and is a rigid flange coupling used to connect two shafts for torque transmission in varied applications. It may be used to connect a prime mover such as a small steam turbine or an electric prime mover such as a motor to the driven machinery such as a pump or a compressor etc. The prime design consideration is to transmit the power between the connected shafts in a reliable and safe manner with lowest possible loss of transmission as well as the optimum cost versus quality balance. The above mentioned factors being the prime considerations of the design, the approach presented earlier will be used t o integrate these requirements for a solution consistent with the reliability, performance and safety requirements while being economic at the same time.

x3≥0: x ∈[-6,6] x∈[-6,6]

x ∈[-6,-2]

∃

∃x: x3 ≥0? False

∀

x ∈[2,6]

∃x: x3 ≥0? True ∃x: x3 ≥ 0? True

∀x: x3 ≥0? False ∀x: x3 ≥0? True

Sub division

solution

x∈[-2,2]

Initial Search Space (D-Box) Split D-Box in BSDs

x ∈[-2,-0.6667] x ∈[-0.6667, 0.6667]

x ∈[0.6667,2]

Iteration 2

BSD box consistency for solution (Existential Quantifier)

NO

x ∈[-2,2]

Iteration 1

∃ ∀

Verification Success

∃x: x3 ≥0? False

∃x: x3 ≥0? True

∃x: x3 ≥ 0? True

∀x: x3 ≥0? False

∀x: x3 ≥0? True

YES Sub division

BSD consistency for Robustness (Universal Quantifier) Trim BD NO

Verification Success YES

Append BSD to Robust Solution Interval

Figure 3 Example with a single variable and single constraint Having defined the requirements and qualitative constraints, we can now proceed to develop the basic mathematical relations dictating the developed constraints. The description of the symbols and abbreviations used in the equation can be founding in appendix at the end of the article.

Append BSD « Solution Exists » Interval

t

Update The BD dn

YES

solution

D1

D1

d

D2

NO

D

Recursion count < N

Present the results in form of Robust, Soultion, Exist Solution and NO solution in terms of BD L

END

Figure 2 Algorithm flow chart 5.1 Design Constraints Once the requirements have been decided, the design constraints can then be laid down to ensure the adherence of the design process to the required criteria. Following main relation ships can be established.

Figure 4. Model assembly The most important design requirement being the torque transmission, the design constraints are oriented towards three main areas: •

Mechanical constraints related to the flange torque transmission capacity.

•

Mechanical constraints related to the bolt torque transmission capacity.

•

Dimensional and geometric constraints ensuring the assembly and insertion of the bolts and their tightening. The translation of the above three requirements results into 10 constraints which are: Mechanical constraints related to the flange torque transmission capacity D τ (π D2 )τ f 2 ,Thub ≥ T 2

Thub

= Tfriction i µ Fb rµ , Tfriction ≥ T

(13) (14)

T 16  D2  ≤ τf  , τ π  D24 − d 4  fcalculaτed (15) Mechanical Constraints related to the bolt torque transmission capacity

τf

calculaτed

Tbshear

 πd2  D i  n  τ b 1 , Tbshear ≥ T  4  2

Tbbearing

D i (d n t )σ b 1 , Tbbearing ≥ T 2

= i m 3, i ≥ i m

eq sst = +3 max max

= 16

(16)

(17)

2 bmax

C1 (p d ts3 )

d= d n − 0.938194 p ts

(19)

Fb At

Fb = a s F0min 0.9ss eqmax y ≥ Dimensional and geometric constraints ensuring the assembly and insertion of the bolts and their tightening. D1 ≥ D2 + 2bA /C + 2mb

(20)

D ≥ D1 + 2bA / C + 2mb

(21)

sc =2π

D1 − bA /C , sc ≥ πb i

(22)

Table 1. Shows the main design parameters used in the example with their starting sets and types. Out of the 14 design parameters selected above, 7 are continuous variables whereas 7 are discrete. The discrete variables may have additional defined attributes such as different material properties related to a specific bolt/flange material. In order to model the noise / uncertainty in the model, eight noise generating variables are defined related to the design variables. Table 2. Shows the noise variables, their sets and type. 5.2 Approach Once the set of constraints and the initial sets for the design parameters and the noise variables have been defined, the design problem is translated with help of the quantifiers for the evaluation of box consistency for existence of solution and consistency for a robust solution. The quantifier notion is then translated in terms

Description

Type

Domain

t

Thickness of flange

Real

[0.0015,0.02]

D

Outside diameter of flange

Real

[0.035,0.13]

D1

Bolt circle Diameter

Real

[0.03,0.11]

D2

Hub outside diameter

Real

[0.03,0.09]

µ

Coefficient of friction between flange surfaces

Real

[0.1,0.55]

f1

Bolt coefficient of friction

Real

[0.04,0.10]

f

Bolt Preload force

Real

i

Number of bolts

Discrete

dn

= C1 Fb (0.16 p + 0.583d 2f1 )

s max =

Variable

[3,4,5,6]

Bolt nominal diameter Discrete ISO M bolts

matb

Bolt material

Discrete Blot Classes

p

Thread pitch

Discrete ISO M bolts

d2

Pitch diameter

Discrete ISO M bolts

mb

Bolt edge clearance

Discrete ISO M bolts

Bolt tool clearance

Discrete Tool Charts

pb

(18) 2 bmax

tb

of computable form by help of interval mapping of each variable set to a corresponding interval and transforming the constraints into the interval form using the principles described earlier.

Table 1. Design Parameter sets and types Variable

Description

Type

Domain

∆t

Thickness of flange

Real

[-0.001,0.001]

∆D

Outside diameter of flange

Real

[-0.001,0.001]

∆D1

Bolt circle Diameter

Real

[-0.001,0.001]

∆D2

Hub outside diameter

Real

[-0.001,0.001]

∆µ

Coefficient of friction between flange surfaces

Real

±2.5%

∆f1

Bolt coefficient of friction

Real

±2.5%

∆f

Bolt Preload force

Real

±25%

∆matb

Bolt material

Discrete

±2.5%

Table 2 Uncertainty/variation variables sets and types For consistency check for existence of solution, the constraints defined in (13)-(22) are transformed into interval functions whereas in order to evaluate the robust solution consistency, the design constraints are modified by addition of the uncertainty/variation set to the corresponding design parameter in the constraint resulting in the interval based robust design constraints. Using the application of (4) and (5) via their interval transformations through (11) and (12), all the BSDs and hence BD is evaluated. If a BSD does not fulfil any box consistency expression, it is discarded as space without any solution. However if the solution box consistency expression (11) is validated, the BSD is regarded as having the possibility of solution and its evaluated for the box consistency of a robust solution via expression (12) which also integrates the noise parameters. If the latter expression is also validated then the space is regarded as a robust solution space else it is sub divided and the process is repeated. The results obtained for the given example are shown in form of three dimensional projections between three variables D, D1 and D2. In Figures 5 (a-d), the main box represents the total BD projected in terms of the three selected variables with the starting intervals along their respective axes.

Iteration 1

D1

D2

a

D

b

Iteration 2

c

c

st

st

Figure 5. (a)-BSDs for existence of solution (1 Iteration), (b)-BSDs for robust solution (1 iteration), (c)-BSDs for Existence nd nd of Solution (2 Iteration), (d)-BSDs for robust solution 2 iteration. In Figure 5. (a)-BSDs for existence of solution (1st robust and can accommodate the changes due to Iteration), (b)-BSDs for robust solution (1st iteration), different variations such as manufacturing variations (c)-BSDs for Existence of Solution (2nd Iteration), (d)which may be voluntary or involuntary, small variations in BSDs for robust solution 2nd iteration.Figure 5(a), light the material properties as well as variations that may grey boxes after the first iteration show the possible result to error such as bolt preload force in the previous search space (BSDs) marked by the algorithm for a example. Each robust solution in the approach presented consistency for existence of a solution. Figure 5 (b), is a set of possibilities which performs according to the shows the sets of robust solution within the search space given design requirements. The approach also gives the in form of dark grey boxes found after the first iteration designer a greater freedom over optimising the design consistent for a robust solution. In a similar fashion Figure solution with respect a given constraint as the solution is 5 (c) and (d) show results for consistency of solution and presented over an envelope of different values of design nd consistency of robust solution in 2 iteration. The choice parameters. of the discrete variables can also be shown in a similar The quantifier notion used to express the requirements way (Figure 6). allows us to explicitly define the design requirements on the individual variables involved in a product design phase. The approach proposed in this paper allows the 6. CONCLUSION design engineer to integrate the notion of uncertainty in The work presented in this paper proposes a new product design right from the early design phase and approach of concurrent engineering design with parallel helps him to find the sensitive as well as the robust design exploration of design parameters as well as uncertainty regions in the possible product design search space. and variation parameters of the mechanical systems with Different types of noise parameters can be treated by the help of the set based design and Quantifier notion. proposed approach. In the treated example, the types of Addressing the design parameters in terms of sets allows the uncertainties are of three types i.e. dimensional a greater freedom of design choices and their evaluation. uncertainties / variances, geometric uncertainties and When the robustness sets are also included in this material property uncertainties. Also, from a mathematical paradigm, the resulting design space / variation space point of view two different types of variables and intersection provides us with a solution that is inherently

uncertainties have been treated i.e. Discrete and continuous. The usage of interval analysis for conversion of the problem provides an appreciable gain in the computational time cost.

σb

max

= Max. tensile stress in bolt

µs = Coefficient of friction between flange surfaces τ f = Design shear stress in flange τ s = Design shear stress in shaft τ b = Max. torsional stress in the bolt max

8. REFERENCES [1] D1

D.K.Sobek,

A.C.Ward,

J.K.Liker,1999,

Toyota’s

Principles of Set-Based Concurrent Engineering, ISSN 0019-848X, Vol. 40, Nº. 2, pages. 67-83 [2] H.G. Beyer, B. Sendhoff, 2007,Robust optimization – A comprehensive survey, Comput. Methods Appl. Mech. Engrg. 196 Pages. 3190–3218

dn (mm)

i

[3] Park et al, 2006, Robust Design: An Overview, AIAA Journal (0001-1452) vol. 44 no. 1 pages 181191

Figure 6. Projection between real and discrete variables

[4] Taguchi, G.,1978, Off-line and On-Line quality control systems, Proceedings of International Conference

7. APPENDIX Description of abbreviations and symbols At = Tensile Stress Area bA /C = Bolt head length across corners C1 = Torsion moment in bolt due to preload D = Outside diameter of flange d = nominal diameter of the shaft/ hub internal diameter D1 = Bolt circle diameter D2 = Hub outside diameter d 2 = Pitch diameter of thread

on Quality Control, Tokyo Japan [5] Taguchi, G.,1987, System of experimental design, Edited by Don Clausing, American Supplier Institute, Dearborn, MI [6] R.J. Malak Jr., J.M. Aughenbaugh1, C.J.J. Paredis, 2009,Multi-attribute utility analysis in set-based conceptual design, Computer-Aided Design 41 Pages, 214-227

F0min = Minimum bolt tightening torque

[7] R Anderl, R Mendgen, 1995, Parametric design and its impact on solid modeling applications” Proceedings of the third ACM symposium on Solid modeling and applications, Salt Lake City, Utah, United States, Pages: 1 – 12

Fb = Tension load in each Bolt

[8] Verger Guillaume Verger, Bessiere Christian, 2006

d n = Bolt nominal diameter d ts = Diameter of stress area f1 = Friction coefficient between the bolt and the flange

i = Number of bolts mb = Minimum bolt center distance from edge

,BlockSolve: a Bottom-UP Approach for Solving

p = Pitch of thread

Quantifier

pb = tool clearance

Constraint Programming - CP, 978-3-540-46267-5

rm = Mean radius of surface Sp = Proof Strength of bolt T

= Torque to Thub Torque capacity based on shear of flange at

the outside hub diameter Tfriction = Torque transmission capacity due to friction

CSPs.

Principles

and

Practice

[9] Bordeaux, L., Montfroy, E., 2002, Beyond NP: Arcconsistency for quantified constraints, Proceedings CP’02, Ithaca NY 371-386. [10] Mamoulis, N., Stergiou, K.,2004, Algorithms for

Tbshear = Torque transmitted through bolts in shear

quantified

Tbbearing = Torque capacity based on bearing of boltsbe transmitted

,Proceedings CP’04, Toronto, Canada (752-756)

t = Thickness of flange

of

constraint

satisfaction

problems

[11] Benhamou, F. ,1999, Goulard F, Granvilliers L,

a s = Accuracy factor of tightening tool σ y = Bolt yield strength

Revising hull and box consistency, Proceedings of

σ b = Design stress in bolts σ eqmax = Von Mise stress

230-244, The MIT press

the 16

th

Intl. Conference on Logic programming, p.