A Barrot

M Sartor

M Paredes

Investigation of torsional teeth stiffness and second moment of area calculations for an analytical model of spline coupling behaviour

Keywords: spline coupling, teeth stiffness, second moment of area, experimental design

Torque distribution from within spline coupling is an important matter for design engineers wishing to study the behaviour of spline couplings. This distribution can be estimated thanks to an analytical equation, based on material, torsional teeth stiffness, and second moments of area of the shaft and sleeve. These parameters have to be determined precisely. The material properties can be readily determined but calculation of the other parameters requires finite element calculations or complex algorithms. The aim is to propose simplified equations so as rapidly to obtain values for both teeth stiffness and second moments of area. Experimental designs were implemented to determine the most appropriate equations. Torsional stiffness is the first parameter studied in the present paper. This is the result of rotations due to various phenomena involving distortions such as bending, shear, compression, rotation of the foundation of the teeth and the teeth sliding. Two simplified equations are finally expressed to define torsional stiffness. The first takes into account the four first phenomena and the second one the teeth sliding. The second topic in the present paper covers the influence on torque distribution of various formulae for calculating second moments of area. A solution, which takes into account shear of the teeth, is highlighted.

Involute spline couplings are used as an easy dismountable link between two rotating parts as shown in Fig. 1. In high power transmission, the sizing of spline coupling can be complex and often requests the development of three dimensional finite element models. From the economic point of view, in most everyday industrial applications, the need to dimension systems quickly remains a priority. Several studies [START_REF] Adey | Development of analysis tools for spline couplings[END_REF][START_REF] Barrot | An assistance tool for spline coupling design[END_REF][START_REF] Tatur | Stress Sources and Critical Stress Combinations for Splined Shaft[END_REF][START_REF] Tjernberg | Load distribution in the axial direction in a spline coupling[END_REF][START_REF] Volfson | Stress sources and critical stress combinations for splined shaft[END_REF] show that the axial load distribution on the teeth is non-uniform and therefore contradicts standardisation assumptions related to spline coupling sizing [START_REF]American standardization[END_REF][START_REF]Spline coupling (in inch)[END_REF][START_REF]Cannelures cylindriques droites à flancs en développante et centrage sur flancs -Angles de pression 30[END_REF]. This can have harmful consequences. For example, errors in dimensioning can lead to failure despite correct dimensioning in accordance with standards. It is therefore advantageous to be able to develop fast analytical methods allowing for direct analysis of the most significant parameters describing the behaviour of spline coupling, especially avoiding Finite Elements model development.

According to Tatur and Vygonnyi [START_REF] Tatur | Stress Sources and Critical Stress Combinations for Splined Shaft[END_REF], axial torque distribution can be calculated thanks to the equation (1), where the running torque t(z) is the torque transmitted from the external spline to the internal spline along the axial direction z. t(z) is directly linked to the mean pressure acting on section z:

= = [ - ] e i e dT (z) t(z) c (z) (z) dz ϕ ϕ ϕ , (1)
where T e (z) is the shaft torque (for every z, the sum of T e (z) and T i (z), the sleeve torque, being equal to the external torque T o ), c ϕ is the torsional stiffness of the joint, considered as constant whatever the value of z, and ϕ i (z) and ϕ e (z) are the twisting angles for the internal spline and external spline: where G i and G e are the shear modulus for the internal spline and external spline, I i and I e are the second moment of area for the internal spline and external spline.

( ) ( ) i i i i d z T z dz G I ϕ = and
Equation (1) can be developed to obtain the distribution torque along the shaft or sleeve. Using the expression of the twisting angles, (1) gives:

        - = i i i e e e e I G z T I G z T c dx z T d ) ( ) ( ) ( 2 2 ϕ , A new equation is obtain by replacing T i (z) = T o -T e (z):         - - = i i e o e e e e I G z T T I G z T c dx z T d ) ( ) ( ) ( 2 2 ϕ
After simplification and resolution of this differential equation, the shaft distribution torque is obtained. It expression is gives at the equation (2).

(

) ( )

( )= 1 1 e k z k z A B T z e e k k - -- -, (2)
where:

1 1 = i i e e k c G I G I ϕ   +     , ( )
1 = k L i i o k L k L c k e k G I A T e e ϕ - -   + -     -       , = + o i i c T B A k G I ϕ , L the spline coupling length.
Equation (2) is governed by three parameters: torsional stiffness, shaft, and sleeve second moment of area. The accuracy of the results is highly dependent on the values given to these parameters. It is therefore important to assign the appropriate values to c ϕ , I e and I i .

A number of studies have been devoted to the evaluation of c ϕ . Analytical methods were developed by Hayashi I. and Hayashi T. [START_REF] Hayashi | Miniaturisation of involute splined couplings. Discussion from torsional stiffness and yield torque[END_REF] or Marmol et al. [START_REF] Marmol | Spline coupling induced nonsynchronous rotor vibrations[END_REF]. These methods are analysed in a recent work by Barrot et al. [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF]. However, it appears that they fail to take all phenomena that can be present in such a joint into account. In the study by Barrot et al., an algorithm was proposed to determine the torsional stiffness of involute spline couplings. This algorithm takes into account the pressure distribution on the teeth flanks, geometric properties and teeth distortion. In the preliminary study in the first stage of the design process, this method turned out to be difficult to apply.

Few articles address the determination of I e and I i for spline coupling. Tjenberg [START_REF] Tjernberg | Load distribution in the axial direction in a spline coupling[END_REF] and

Isakower [START_REF] Isakower | Don't guess on non-circular shafts[END_REF] propose two methods but do not evaluate the consequence of their choice.

Second moments of area remain difficult to determine, due to the behaviour of the spline coupling teeth.

The present paper firstly presents how the algorithm developed in Barrot et al. [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF] is used to construct a simplified method to calculate c ϕ .. Secondly, a process to calculate the shaft and sleeve second moment of area is detailed.

Torsional stiffness estimation, c ϕ ϕ ϕ ϕ

As with teeth gear stiffness determination, studied by example by Cornell [START_REF] Cornell | Compliance and stress sensitivity of spur gear teeth[END_REF], the spline geometry requires the shaft and sleeve teeth to be divided up into segments. Moreover, an analytical method, dedicated to determining torsional stiffness of standard involute spline couplings, was developed in a previous work [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF]. This method seeks to consider the interdependence between teeth distortions and pressure distribution in its calculations.

In the present paragraph, a simplified and quicker method is proposed. This can be used for involute spline couplings. Torsional teeth stiffness estimation commences by analysing its expression as follows:

' = ext g T c ϕ θ , (3)
where, T' ext is the moment applied on the contact surface per unit width and θ g is the global rotation. Thanks to the algorithm developed [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF]

θ θ θ θ θ θ + ,
where θ b is the rotation due to bending of the teeth of both sleeve and shaft, θ s the rotation due to shear of the teeth of both sleeve and shaft, θ c the rotation due to compression at the contact, θ f the rotation due to teeth foundation rotation of the sleeve and the shaft. These rotations depend on a number of factors: contact pressure, teeth geometry, and material. θ sl is the rotation due to sliding between the internal and external spline teeth. This rotation depends on the same factors as the others but to be determined also requires knowledge of the internal and external diameters of the spline coupling. All these rotations are illustrated in Fig. 2. The dependence between rotations and the cited factors precludes describing global rotation using a single equation.

One solution is to consider global rotation as a sum of two entities. The first entity takes into account rotations that do not depend on the inside and outside diameters, while the second takes into account rotation due to sliding. This method can be used if certain basic assumptions are respected, with all sections being straight before and during distortion and only long spline couplings being used.

Rotations not dependent on inside and outside diameters

According to Barrot et al. [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF], rotation equations due to bending, shear, compression, and foundation rotation do not depend on the outside and inside diameter of the spline coupling. A simplified analytical equation to determine the sum of these four rotations, known as θ ws , has then to be determined:

= + + + ws b s c f θ θ θ θ θ
To describe a phenomenon, an equation can be used accounting for a set of parameters. Not all such parameters have the same weight in the final equation and may lead to unnecessary complexity. Identifying the most appropriate parameters is therefore essential in formulating the equation that fits.

-Search for relevant parameters An attempt will now be made to reduce the number of parameters.

The equations describing these rotations depend on the circular pitch, P, the number of spline coupling teeth, N and finally the material parameters G i and G e . To facilitate calculation, a modulus m, equal to P/π is applied.

The modulus and number of teeth parameters are linked by the pitch diameter, D. One way of reducing the number of parameters could be to replace m and N by D alone. This would be acceptable if two different involute spline couplings having the same pitch diameter also have identical values for θ ws rotations. A test is conducted for a first spline coupling where N=18, m=2.5mm and a further test where N=27, m=1.667mm leading to a common value for D = 45mm. The gap between the results is 34 per cent, precluding the D alone being considered in the equations.

The material parameters G i and G e can also be studied. A solution to reduce the number of parameters could involve introducing the ratio between these two parameters.

Thus, a test is conducted with the same spline coupling where the material couple is

G i /G e = 1
with steel-steel and aluminium-aluminium couples. A Gap of θ ws is obtained close to 200 per cent. This result attests to the impossibility of considering only the G i /G e ratio to describe the material parameters. According to American standardisation [START_REF]American standardization[END_REF][START_REF]Spline coupling (in inch)[END_REF] and the most conventional uses of involute spline couplings, this study proposes to establish the law of torsional stiffness for two material couples: steel-steel and brass-steel (brass for the sleeve). The work presented is available for the commonest modulus and number of teeth.

-

Search for influential parameters

Various parameters to calculate θ ws were identified. The next step was to highlight the most influential ones. To this purpose, an approach using experimental design was developed.

This method is commonly used and is explained through for example in the work by Goupy [START_REF] Goupy | Introduction aux plans d'expériences. Technique et ingénierie[END_REF] and by Brisset et al. study [START_REF] Brisset | Optimization with experimental design. An approach using Taguchi's Methodology and Finite Element simulations[END_REF].

This process comprises inputs known as factors and outputs known as responses. The factor values can be continuous or discrete. The plan for a two-level design uses +1 and -1 notation to denote respectively the "high level" and the "low level" for each factor. In searching for the θ ws equation, the proposed factors are m and N. The levels are defined as:

For the modulus, -1 = 1.25mm and +1 = 2.5mm.

For the number of teeth, -1 = 12 and 1 = 22.

Remark: this study is available for a steel-steel spline coupling.

This process requires four trials to be conducted with each factor set to high or low values as described in the matrix of Tab. 1 in the three first columns. The fourth column, m N, corresponds to the interaction between factors on the response, obtained by multiplying m level with N level for each trial.

Finally, the model for the experiment is:

E = Y X ,
where Y is the vector consisting of the four trial responses, E (E 0 , E 1 , E 2 , E 3 ), the unknown coefficients called factor effects, and X the vector consisting of the factors and interaction. E 0 correspond to the average of responses, while E 1 , E 2 and E 3 correspond respectively to the effect of the factors m and N and the interaction m N. For the E i effect of the i factor, the equation is:

4 i j j j 1 1 . 4 E x y = = ∑
with y j the response of the i factor of j trial, and x j the level of this factor for the trial.

The experimental design graph is shown in Fig. 3. Each effect of the factors or interaction is represented by a straight line, which can be expressed by the generic equation i 0

y E x E = + ,
with y the response, and x the level for the factor.

It appears that all factors and interactions give different response values for low and high levels. Thus all parameters can be considered to be influential in the equation for determination of rotation without sliding.

To find the simplified equation for θ ws , its variations according to m and N must be known. Using analytical results from Barrot's algorithm, the change in θ ws can be determined. The form of the analytical result of the rotation evolution suggests a polynomial form of the solution. Then θ ws can be:

θ ws = f(N 3 , N 2 , N, m 2 , m, m N).
Optimisation using the least squares method to reduce the gap between the analytical values and the sought equation gives the result shown in Fig. 4. A comparison between the analytical and optimised results highlights the difference between them. θ ws must be known with greater accuracy. It is therefore impossible to describe it using the first suggested equation. Another proposition is to establish an equation for each material couple and for each modulus. θ ws form is yet:

θ ws = f(N 3 , N 2 , N).
A new comparison between the θ ws simplified equation and the analytical result is shown in Fig. 5. This represents a case for m = 1.25mm and the steel-steel material couple.

This figure reveals good correlation between the results, and suggests that generalisation to the other modulus can be performed. Finally, equations for the most popular modulus and for two commonest material couples are summarised in Tab.2. The N interval is between 12 and 30. The last column represents the correlation between the analytical result and the proposed equation. Correlation is measured using the Pearson Product Moment Correlation. A correlation of +1 means that there is a perfect positive relationship between results. The good correlation results show that θ ws can be easily calculated by using the equation of Tab. 2.

Rotation depending on inside and outside diameters

The second part of global rotation is rotation due to sliding [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF]. Equations describing sliding depend on inside and outside diameters. The principal equations are recalled below:

( ) π +2 (tan( )-) 2 = + tan + 4 e i sl sl sl N v v P N φ φ π θ φ          
, where:

φ is the contact angle at the pitch radius in radians. A

( )

2 2 e 2 2 = 1-υ + 1+ υ E - e e r re sl re int re int p R v R R R R , - ( ) ( ) ( )
( )

2 2 i 2 2 = 1-υ + 1+ υ E - i i r ri sl ri ext ext ri p R v R R R R , -R ext
  = - - +    
, R b is the base radius and F the force applied on the contact surface per unit width.

All parameters dependent on geometry or material except the force F, which depends on the tilted angle, α, shown in Fig. 6. To evaluate exactly F, a method, applied in Barrot [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF], takes into account the equilibrium law applied to the external spline on the shaft section centre. This equation establishes a link between F, T' ext , R b , N and the teeth distortion represented by the tilted angle α, as shows equation (4). c represents the half contact length of tooth and χ is Kolosov's constant and µ the modulus of rigidity. Calculation of the tilted angle requires the pressure distribution on teeth flank to be known, imposing an iterative calculus.

The difficulty in determining the tilted angle requires a simplified form of F to be established.

The suggested equation, where k F is a corrective coefficient that optimises the rotation due to sliding, is equation (5).

  

     + = K b b ext A R c R N T F 2 2 '
πα where µ χ

4 1 A K + = (4) b ext F R N T k F ' = (5)
Estimation of the coefficient k F requires determination of the validity domain. As in the previous paragraph, an experimental design is developed to highlight the different influential factors. The graphic representation is shown in Fig. 7.

The factors taken into account are the ratios for the outside diameter, D ext by the pitch diameter D, the inside diameter, D int by the pitch diameter, the modulus and the number of teeth. Experimental design is conducted for a steel-steel material couple. The levels are:

For D ext /D, -1 = 1.25 and +1 = 2.

For D int /D, -1 = 0 and +1 = 1.2.

For N, -1 = 16 and +1 = 28.

For m, -1 = 1.25mm and +1 = 2.5mm. Finally, a value of k F can be found for the various material couples. For the steel-steel couple, k F = 0.9879 and for the brass-steel couple, k F = 0.9898. These coefficients are available for range of N between 12 to 30 for a modulus from 1.25mm to 5mm. This factor increases the accuracy of the value of the θ sl of less than 2 per cent. It can be thus considered that for the presented cases k F can be removed.

Torsional stiffness validation

For each modulus, the simplified equations to estimate c ϕ are checked. A selection of comparisons between the torsional stiffness obtained by the analytical method from Barrot et al. [START_REF] Barrot | Determining both radial pressure distribution and torsional teeth stiffness of involute spline coupling[END_REF] and the simplified equations is shown in Tab. 3. It appeared to be of interest also to compare the simplified equation results with torsional stiffness calculated by the FE model.

The gap between the analytical method and simplified equations for the different tests is less than 5 per cent, being enough to consider that torsional stiffness is known with a good estimation. The developed equations are available for long spline couplings, where the sleeve material can be steel or brass and the shaft material is steel. This model is valid for the considered domain, where N can vary between 12 and 30 and m between 1.25mm and 5mm.

Shaft and sleeve second moment of area determination I e , I i

According to the literature, various solutions exist to determine second moments of area of internal and external spline. During this study, various calculations of I e and I i are proposed. To determine the most appropriate one, torque distribution along the spline coupling was analysed. This distribution is obtained thanks to the Tatur and Vygonnyi differential equation (2) solution and Finite Element model results. The value for torsional stiffness used was obtained by applying the simplified equations presented in the first section of the present article.

Different calculation possibilities

Four determinations were checked as illustrated in Fig. 8.

Second moment of area calculated with the part bodies (Fig. 8a)

The ends of the two parts are the outside limits D ext and D int and the teeth are ignored.

The other limits are the teeth foundations, D re and D ri . Finally, shaft and sleeve second moments of area equations are respectively:

( ) 32

ext ri i b D D I π - = . ( 6
)
This Tjenberg [START_REF] Tjernberg | Load distribution in the axial direction in a spline coupling[END_REF] method is used to apply equation (1). In Isakower [START_REF] Isakower | Don't guess on non-circular shafts[END_REF], the final second moments of area are obtained with an increase in 10 per cent in this value.

Second moments of area calculated with the teeth end diameters (Fig. 8b)

As previously, external limits are identical, and the other diameters are the major diameters. Shaft and sleeve second moment of area equations are respectively:

( ) 32

ext i i t D D I π - = . ( 7
)
Second moments of area are calculated with the pitch diameter (Fig. 8c)

When influence of the teeth is taken into account with the pitch diameter the equations for shaft and sleeve second moments of area are respectively:

( )

Second moments of area calculated with influence of teeth shearing (Fig. 8d)

Second moments of area can be divided into two parts: the body and the teeth. The second moment of area of the two parts is the sum of the individual second moments of area.

For the external spline, the second moment of area of the body is calculated at the centre of the spline section:

( ) ( )

4 4 2 32 4 i ext i i s D D S N D I π - = + , ( 9
)
where S i is the sleeve shear surface in a section plane.

Method for determination of spline coupling second moments of area

In order to select the best method of second moment of area evaluation, a reference has to be chosen. The 3D FE method is used for this. Indeed, various research works as with Three dimensional finite element models using ABAQUS have been developed for different spline couplings. The meshing used to model the two parts making up the coupling, the shaft and sleeve are shown in Fig. 9 and is composed by 26000 solid linear brick elements and 30000 nodes. The boundary conditions are defined as follows. The contact is specified on the teeth flanks (with no friction). This condition imposes a non linear resolution. The nodes of a sleeve's extremity are locked but radial expansion is allowed, as the figure illustrates with the small arrows. Nodes of the opposite shaft end can't move in the axial direction and a torque is applied to the same shaft surface. The loading is introduced by forces on nodes of the shaft surface, in order to introduce a linear shear stress on the section. Cyclic symmetry conditions are used on the two lateral surfaces to simulate behaviour of the complete spline coupling.

Comparison is made between the axial torque distribution according to FE models and the analytical model with different second moments of area calculation. The test is performed for four different involute spline couplings as described in Tab. 4 and the results are shown in Fig. 10. The torque transfer between the internal and external spline is located in particular at the ends of spline coupling. Indeed, it has been established for the load configuration shown in Fig. 9 that the pressure peaks are at the extremities [START_REF] Barrot | An assistance tool for spline coupling design[END_REF][START_REF] Tjernberg | Load distribution in the axial direction in a spline coupling[END_REF][START_REF] Volfson | Stress sources and critical stress combinations for splined shaft[END_REF]. If the gap between torque distribution result of FE and analytic models decreases, accuracy of knowledge on maximum pressure will be enhanced.

In accordance with the results shown in Fig. 10, the best calculation is that which takes shear teeth influence into account.

In calculation of the second moments of area, the error computed can be due to the influence of the teeth. This influence is less significant for the sleeve second moment of area than for the shaft. This explains why it is necessary to determine whether an optimisation of the shaft second moment of area can increase accuracy of the torque distribution result. In the next study, the new second moments of area considered are I i = I i s and I e = β I e s .

Moreover, thanks to this optimisation, and if the influence of the external diameter of the spline coupling is taken into account, the errors in the basic hypotheses of equation (1) can also diminish the end-contact effect, which can be neglected.

An experimental design is developed to estimate the influence of N, m and D ext /D on a population of spline couplings. For every spline couplings, the ratio D ext /D allows the same skin thickness above the sleeve teeth to be checked. An experimental design with 3 factors and 2 levels, corresponding to 9 tests, is conducted. The different levels are as follows:

For N -1 = 14 and +1 = 30

For m -1 = 1.25mm and +1 = 5mm

For D ext /D -1=1.25 and +1=2

The responses β 0 are obtained with minimisation of the gap between analytical and FE torque distribution using the least squares method.

Fig. 11 stresses the influence of factors and interactions taken into account in the experimental design. This graph shows that only the interaction of a high order appears not to be influential, i.e. having a value that has little effect on the response.

To determine the equation for the corrective coefficient β, a linear form is proposed. This is a function of the previous factors and interactions. To find the effect values of these parameters, the least squares method is used. It is important to note that to increase the The following equation is found:

D m N D D D m N m N D D β - - - - - - = × + × + × - × + × - × (10)
To estimate the accuracy of (10), a comparison between β and β 0 is highlighted in Tab. 5, which presents the gap, Gap β in per cent, corresponding to:

0 0 100 Gap β β β β - =
The average result is about 6%.

To evaluate the influence of the corrective coefficient, a calculation of the σ ratio between the model before and after correction is made. σ values are shown in the last column. This ratio is the division of the sum of the least squares fit between the analytical equation and The ratio represents the gain due to β. Tab. 5 reveals that β allows the accuracy of almost all torque distributions to be improved by a factor of two on average.

Second moment of area calculation validation

The utility of the corrective coefficient on the shaft second moment of area determination was highlighted in the previous paragraph. It is important to validate equation (10) on other spline couplings that are not used to determine this equation. Tests are conducted for four involute spline couplings, described in Tab. 6. It is possible to calculate the gap between the FE results and the analytical results with and without β. Taking the FE as reference, Fig. 12 illustrates these gaps along z-axis. According to Tab. 6 and Fig. 12, introducing β to calculate I e increases torque distribution accuracy on all the studied spline couplings.

Conclusion

Thanks to Tatur, there is an equation to estimate torque distribution along the spline coupling. This equation requires knowledge of torsional stiffness and shaft and sleeve second moments of area. In a previous study, a complex algorithm to determine the torsional stiffness of involute spline couplings was performed. In this paper simplified equations are proposed that are easily and rapidly used.

To determine the most appropriate equations, experimental designs are analysed.

Torsional stiffness is the result of rotations due to various distortion phenomena such as bending, shear, compression, rotation of teeth foundation and sliding of the teeth. These rotations are divided into two parts. The first takes into account the four first phenomena and the second addresses the issue of sliding. Two simplified equations are finally expressed to define stiffness. In this study two couples of materials are considered: steel -steel and brasssteel. The influential factors are highlighted through various experimental designs, and the equations are developed using the least squares method. A comparison with the simplified method and the complex algorithm underlines the good estimation of torsional stiffness.

This paper analyses several analytical solutions to describe these parameters, and reveals that the second moments of area have considerable influence on distribution. One particular solution that takes the teeth shear into account, is proposed as being the most effective. To increase accuracy, a β coefficient was added and its analytical detailed. The said coefficient is capable of enhancing accuracy of torque distribution analysis three-fold.

Fpstβ

  the second moment of area of the shaft which minimizes the gap between analytical and FE torque distribution β Corrective coefficient for the second moment of area of the shaft θ b Rotation due to bending of the sleeve and shaft in radians θ c Rotation due to compression of the sleeve and shaft in radians θ f Rotation due to the foundation rotation of the sleeve and shaft in radians θ g Sum of all rotations: θ ws , θ sl in radians θ s Rotation due to shear of the sleeve and shaft in radians θ sl Rotation due to sliding of the sleeve and shaft in radians θ ws Sum of four rotations: θ b , θ s , θ f , θ c in radians ϕ i , ϕ e Twisting angles for the internal spline and external spline in radians φ Contact angle at the pitch radius in radians φ p Contact load angularity at the pitch radius in radians c ϕ Torsional teeth stiffness per unit width in N/rad D Pitch diameter in metres D ext , R ext External diameter and radius of the sleeve in metres D o , D i Shaft and sleeve major diameter in metres D int , R int Diameter and radius corresponding to the hole of the shaft in metres D re , R re Diameter and radius of the teeth foundation of the shaft in metres D ri , R ri Diameter and radius of the teeth foundation of the sleeve in metres E e , E i Young's modulus of internal spline and external splines in Pa E 0 Average of responses E 1, E 2 E 3 Effects respectively of factors m, N and m N interaction Force applied on the contact surface per unit width in N/m G i , G e Shear modulus of the internal and external spline in Pa I e , I i Shaft and sleeve tooth second moment of area in m 4 Second moments of area of the shaft and sleeve calculated with the pitch diameter in m 4 Second moments of area of the shaft and sleeve calculated with the influence of the teeth shear in m 4 Second moments of area of the shaft and sleeve calculated with the teeth extremities diameters in m 4 k F Corrective coefficient, optimising rotation due to sliding L Spline coupling length in metres m Shaft and sleeve major radius in metres R b Base radius of the shaft in metres S e , S i Teeth shear surface of the shaft and sleeve in m² t(z) Running torque in Nm T e Shaft torque of the analytical equation with optimised I e in Nm T e , T i Shaft torque at section z in Nm T e EF Shaft torque of the FE model in Nm T' ext External torque per unit width in N T o External torque applied to the spline coupling in Nm v sl e , v sl i Shaft and sleeve radial displacement due to sliding in metres 1. Introduction

Fig. 4

 4 Fig. 4 represents this change for a fixed modulus and a fixed number of teeth respectively.

12 π

 12 (30°) value is commonly used for involute spline coupling deisgn, -

  is the external radius of the sleeve, R int is the hole radius of the shaft -R re , R ri are the foundation radii of the external and internal spline -The radial component of the contact pressure is uniformly distributed over the outside diameter for the shaft and inside diameter for the sleeve. For the shaft, is the contact load angularity at the pitch radius. The expression of φ p

Fig. 7

 7 Fig. 7 reveals that all factors and interactions have a negligible influence on the

  tooth second moment of area, at the tooth's centre of gravity is negligible. This moment, at the centre of the shaft section can be determined from the Huygens theorem: S e is the shaft shear surface in a section plane. Finally, for a spline coupling the equations are:

  precision of the β equation, 27 tests are performed. In fact a new level is added: N = 22, m = 2.5mm, D ext /D = 1.625.

FE

  model results and sum of the least squares fit between the analytical equation and FE model results.

  Where, T e is the shaft torque value of the analytical equation, e FE T is the torque value of the FE model and e T β is the torque value of the analytical equation, with the optimised shaft second moment of area.

FiguresFig. 1

 1 FiguresFig. 1 Spline coupling descriptions

Spline coupling descriptionsFig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 3D

 23456789 Fig. 1 Spline coupling descriptions

Fig. 10 .Fig. 11

 1011 Fig. 10. Axial torque distribution for four spline couplings with various second moment of

Fig. 12

 12 Fig. 12 Comparison of gap between FE torque distribution and analytical torque distributions

  , T' ext can be used to determine θ g . Then, for different T' ext values, a unique c ϕ .value is obtained. Thus in the present study, to determine torsional stiffness the considered value of T' ext is the unit (1N). θ g represents the sum of various movements due to the superposition of five phenomena identified byMarmol et al.

	and Barrot et al.: bending of the teeth, shear of the teeth, compression, tooth foundation
	rotation and sliding.
	g	= + + + b s c	f	sl

  Barrot et al., Leen et al., Limmer et al., Sum et al., [2, 16-19] have shown that FE simulation and experimental studies give similar results.

  Gap between the optimised and calculated corrective coefficient and the influence of this coefficient on torque distribution Second moment of area calculation validation for four different spline couplings

	Tables Designations	Modulus	Number of teeth	Outside diameter	Inside diameter
	Number of the experiment Trial 1 Trial 2 Trial 3 Trial 4 Tab. 1 Analysis matrix for the rotation without sliding rotation m N m N θ ws [rad] -1 -1 +1 2.03×10 -5 +1 -1 -1 5.08×10 -6 -1 +1 -1 5.80×10 -6 +1 +1 +1 1.46×10 -6 Modulus Material Sleeve Material Shaft Equations 1.25 Steel Steel -9 3 -7 2 -5 ws = -5.963 10 N +4.468 10 N -1.229 10 N +1.101 10 θ × × × × Brass Steel -9 3 -7 2 -5 -4 ws =-8.424 10 N + 6.621 10 N -1.743 10 N + 1.566 10 θ × × × × 1.667 Steel Steel -9 3 -7 2 -6 -5 -4 ws = -4.095 10 N + 3.070 10 N -7.723 10 N + 6.657 10 θ × × × × Brass Steel -9 3 -7 2 -5 -5 ws = -5.441 10 N + 4.093 10 N -1.039 10 N + 9.085 10 θ × × × × 2.5 Steel Steel -9 3 -7 2 -6 -4 ws = -1.825 10 N + 1.362 10 N -3.417 10 N + 2.944 10 θ × × × × Brass Steel -9 3 -7 2 -6 -5 ws = -2.265 10 N + 1.738 10 N -4.482 10 N + 3.966 10 θ × × × × 5 Steel Steel -10 3 -8 2 -7 -6 ws = -2.314 10 N + 1.862 10 N -5.105 10 N + 4.875 10 θ × × × × Brass Steel -10 3 -8 2 -7 -6 ws = -3.318 10 N + 2.674 10 N -7.343 10 N + 7.025 10 θ × × × × Tab. 2 Equations of rotation without sliding for each couple (modulus, materials) R² 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 Modulus [mm] 1.25 1.667 2.5 5 Number of teeth 20 25 24 30 Outside diameter [mm] 37.5 60 90 250 Materiel Couple Steel-steel Brass-steel Steel-steel Brass-steel Torsional stiffness obtained by the analytical method from Barrot[9] ×10 6 [N/rad] 90.1 150 561 2590 Torsional stiffness obtained by the FE method ×10 6 [N/rad] 87.6 152 552 2555 Torsional stiffness obtained by the simplified equations ×10 6 [N/rad] 94.0 153 542 2550 Gap between FE model and simplified equations [%] +7.3 +0.65 -1.81 -0.19 Gap between analytical method and simplified equations [%] +4.30 1.85 -3.43 -1.49 m N D ext D int A 1.25 22 34.4 10 B 1.66 27 55 30 C 2.5 18 70 15 D 5 30 188 20 Tab. 4 Spline coupling designation for second moment of area calculus research m [mm] N D ext /D β 0 β Gap β [%] σ 1.25 14 1.25 0.89 0.76 14.33 1.95 1.25 14 1.625 0.82 0.86 4.60 9.91 1.25 14 2 0.88 0.96 8.70 2.84 1.25 22 1.625 0.98 1.00 1.88 1.01 1.25 22 2 1.11 1.08 2.02 1.99 1.25 22 1.25 1.03 0.91 11.39 1.36 1.25 30 2 1.15 1.21 4.78 1.80 1.25 30 1.25 0.97 1.06 8.82 1.13 1.25 30 1.625 1.01 1.13 11.59 1.01 2.5 14 1.25 0.91 0.82 9.31 1.99 2.5 14 1.625 0.85 0.92 9.11 5.69 2.5 14 2 0.92 1.03 11.88 1.77 2.5 22 2 1.09 1.12 2.41 1.10 2.5 22 1.25 0.96 0.94 1.99 1.40 2.5 22 1.625 0.97 1.03 6.16 1.42 2.5 30 1.25 0.99 1.05 6.70 1.05 2.5 30 1.625 1.10 1.13 3.35 1.38 2.5 30 2 1.33 1.21 9.16 2.35 5 14 1.25 0.94 0.94 0.86 4.60 5 14 1.625 0.99 1.05 5.75 1.00 5 14 2 1.18 1.16 1.98 2.52 5 22 1.25 0.96 1.00 3.44 1.26 5 22 1.625 1.07 1.09 2.49 1.19 5 22 2 1.27 1.19 6.20 3.27 5 30 1.25 0.99 1.05 5.88 1.02 5 30 1.625 1.06 1.13 6.78 1.18 5 30 2 1.26 1.22 3.42 3.00 coupling m [mm] N D ext /D I e s [mm 4 ] β I e [mm 4 ] σ A 1.25 18 1.56 2.40×10 4 0.923 2.14×10 4 3.57 B 1.667 20 1.49 1.19×10 5 0.927 1.11×10 5 3.45 C 1.667 27 1.22 3.20×10 5 0.998 3.19×10 5 2.17 D 5 18 1.66 63.3×10 5 1.08 68.3×10 5 1.12 Tab. 5 Spline Tab. 6
	Tab. 3 Comparison of torsional stiffness for different spline couplings between the complete
			and simplified methods

Author's name: Barrot


Manuel Paredes

Marc Sartor

Cédric Masclet

Finding the Optimal Stock Spring from Optimal Spring Design Characteristics

This paper presents two methods for selecting the best stock spring from within a database. A way of calculating the optimal custom spring design which takes into account the database properties is described and two methods that explore the neighbourhood of the optimal continuous solution are detailed. Two methods for exploring the neighbourhood of the optimal continuous solution are discussed. The first, called the "expanding" method explores the database in successive layers, and stops once a solution has been found which cannot be improved upon. The second method also uses the "expanding" technique but stops once a spring which matches the specifications has been found. The algorithm then performs a "sliding" operator, whereby the immediate neighbourhood of the existing solution is explored. When no better spring can be found, this process stops. Two examples are shown. They demonstrate the effectiveness for our approach in selecting a stock spring without exploring the whole database. The most time-consuming stage is the calculation of the optimum custom-made spring design. Both these methods are particularly useful when applied to large database.

INTRODUCTION

The creation of mechanical objects is often the end result of a long design process where standard component selection is perhaps the simplest, but nonetheless frequent class of design decision problems. As catalogues become increasingly common and voluminous, this process can be time-consuming. Text-only assistance is not an efficient means for designers to find a stock spring which respects not only certain geometrical properties stored in a database but also several operating characteristics. Thus, optimal operating points have to be calculated for each potential stock spring depending on the requirements. Finding the best stock spring and calculating its associated operating parameters is a Mixed Discrete Optimization problem. The most common mathematical solution is first to approximate an optimum by treating all variables as continuous and then to use algorithms to find the best feasible discrete point in the region of the continuous variable optimum. Optimization procedures dealing with continuous variables are numerous and well known. Technical literature provides mathematical methods which calculate design parameters corresponding to an optimal custom spring design [START_REF] Sandgren | Nonlinear Integer and Discrete Programming in Mechanical Design Optimisation[END_REF], [START_REF] Kannan | An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimisation and its Applications to Mechanical Design[END_REF]Kramer (1994), Deb and[START_REF] References Deb | A Flexible Optimisation Procedure for Mechanical Component Design Based on Genetic Adaptive Search[END_REF]). To help designers select stock springs, [START_REF] Yuyi | Application of Expert System for Spring Design and Procurement[END_REF] and [START_REF] Motz | An Integrated Approach to Knowledge-Aided Design and Optimisation of Mechanical Springs[END_REF]] have implemented methods that propose some stock springs close to the optimal custom spring design. But, in every case, the designer has to calculate the operating parameters for each proposed spring by hand, in order to select the one that best fit his specifications. Moreover, as the best stock spring is not necessarily close to the custom-designed optimum, the designer is never sure he has selected the best spring. To find the discrete optimum, many algorithms have been developed such as the optimal discrete search by [START_REF] Pappas | Mathematical Programming Procedures for Mixed-Continuous Design Problems[END_REF], the sequential linearization approach by Han [START_REF] Tong | Computational Implementation and Test of a Sequential Linearization Algorithm for Mixed-discrete Nonlinear Design Optimization[END_REF] or the Boolean logic method by [START_REF] Peng | A Boolean Local Improvement Method for General Discrete Optimization Problems[END_REF]. These methods explore the discrete neighbourhood of the continuous optimum but are difficult to implement. When there is a high number of discrete variables, they can however help to find the discrete optimum without enumerating the total neighbourhood of the continuous point. Considering that spring catalogues give a maximum of four discrete variables in the optimization problem, our objective is to find methods that would extract the most suitable solution by just testing a few springs. Thus two methods have been developed. They use a particular optimal spring as a starting point to explore the database.

STARTING FROM A PARTICULAR OPTIMAL SPRING

Usually, optimal spring design is calculated by solving a conventional optimisation problem: Minimise or Maximise F(x) where x is a vector of continuous variables and F(x) is either the mass, or the fatigue life, or the operating load P2, or the operating length L2... In addition to the objective function, a large set of constraints is considered, so as to express design specifications (operating length limits, operating load limits…), the standards (fatigue life, buckling length…) and the technical capability limits of the spring manufacturer (maximum outside diameter, minimum outside diameter, maximum wire diameter…). But catalogues often provide a much smaller exploration domain than the hyper cube generally considered in the continuous optimisation process. The global optimum can therefore be a long way from the discrete approach. To match catalogue properties and decrease the gap between the two optima, constraints related to the database limits can be added to the initial optimisation problem. The augmented formulation will lead to a particular optimal spring which can be retained as an appropriate starting point for a catalogue search. The first task is thus to model the catalogue data in order to obtain the boundaries of the exploration domain.

Modelling the database

For this study, the catalogue of the «Ressort Vanel» company, comprising 5050 springs, has been used. For each spring, the characteristics detailed in the catalogue are stored in a table as shown in Figure 1. A spring reference can be for instance the index of the associated line in the table. Table 1

. Link between parameters

Within this catalogue, springs are classified using three parameters : Do (outside diameter), d (wire diameter) and L0 (free length). Each spring can be represented by a point in the [Do, d, L0] space. Our suggestion is to add a matrix M[X, Y, Z] to this discrete space. The table 1 show the link between Do and X, d and Y, L0 and Z. Matrix dimensions are thus : M [44,33,19]. This matrix will contain only the spring references. Note that M[] is not fulfilled by the database references. Some points of the discrete space have associated spring references, but not the others.

Analysing the database limits

Figure 2 shows rectangles that surround the catalogue springs. The objective is to find the database limits in the three following planes : d/L0, Do/d and Do/L0. In each of these planes, a curve is sought that minimises the distance from the database (in number of cells within M[] ) while staying outside the database limits. Equations of the curves detailed in Figure 3 are: dmax = 0.246 Do 0.9108 , dmin = 0.07066 Do 0.9949 L0max = 23.1 Do 0.6677 , L0min = 1.4177 Do 0.8301 L0max = 175.56 d 1.098 , L0min = 0.07066 d 0.9049 These equations are added to the initial optimisation problem in order to find an optimal custom spring design within the catalogue limits. In this study, x is defined with Do (outside diameter), d (wire diameter), L0 (free length), R (spring rate), L1 (maximum operating length), L2 (minimum operating length). From the result obtained, the values of Do, d, L0 are retained as starting point coordinates. Then, two exploration methods to select the most suitable spring from the database are presented. 1 -Read the characteristics stored in the spring table 2 -Calculate the optimum operating points according to the specifications, the objective function and the standards (fatigue life, buckling length...) as described by [START_REF] Paredes | Stock Spring Selection Tool[END_REF] 3 -Calculate the whole range of characteristics 4 -Calculate the objective function value 5 -Evaluate how the spring satisfies the requirements.

First exploration method : «Expanding»

The proposed method tests points that surround the starting point in successive layers, beginning with the points closest to the starting point (first layer), the others being gradually tested as described in Figure 4.

: starting point (optimal custom design)

: points of the first layer : points of the second layer In a n dimension space, a continuous point is surrounded by 2 n discrete points. In our case n=3, the first layer contains 8 points. The Nth layer contains the points located on the faces of a cube with sides 2N long. Thus, it contains [6 (2N-2) 2 + 24N -16] points.

The progression is stopped when the exploration of a layer does not improve the solution obtained on the preceding layer. The stop criterion selected is thus as follows : the algorithm stops when, at the end of the exploration of the Nth layer, the optimal stock spring still belongs to the (N-1)th layer.

Second exploration method : "Expanding, then sliding a window"

When the dimensions of the database spatial model are large (n>2), the exploration of numerous layers results in the testing of a large number of points. An alternative approach can then be used : "Expanding, then sliding a window". The exploration starts as defined previously but when at least one spring matching the specifications has been found on a layer, the expansion is stopped and the process continues by sliding a window. At this stage, the algorithm only tests the points from within a window that surrounds the optimal point previously found as shown in Figure 5 (discrete neighbourhood). For an n dimension space and a R window radius, a discrete point is surrounded by (1+2 R ) n -1 points. In our case, the unit-neighbourhood is tested inducing n = 3 and R = 1 : 26 points are thus evaluated. If a better point is found, the window is slid onto that new point. The algorithm is stopped when the stage has not improved the preceding solution.

EXAMPLES

Two examples are given below to illustrate and compare the efficiency of the two methods. Note that, all the springs in the database have been simultaneously evaluated in order to control the results. Both these examples deal with, steel springs with closed and ground ends.

Minimum mass

In this example, the spring with the smallest mass value and a spring travel of 10 mm, a minimum inside diameter (Di) of 4 mm, F1 equal to 15 N and a maximum L1 of 35 mm is sought. The optimal custom spring design within the database limits induces Do = 6.42 mm, d = 0.75 mm, L0 = 39.69 mm. Thus, according to table 1, the first layer cube coordinates are [13,14,11], [14,15,12] (points of the diagonal).

Results obtained with both methods are shown in Table 2. The algorithm proposes in every case [17,15,13] as a solution, i.e. Do = 8.0 mm, d = 0.8 mm, L0 = 50 mm, R = 0.68 N/mm, working between L1 = 27.94 mm and L2 = 17.94 mm. Checking procedure: the evaluation of the entire database (5050 springs) leads to the same stock spring (298 springs match the specifications). In this example, the best method is "Expanding, then sliding a window". In this case, the optimal stock spring can be found after evaluating only 142 springs.

Minimum operating length

Prototype design specifications are : Do ≤ 38mm, Di ≥ 27mm, L1 ≤ 50mm, spring travel equal to 11mm. 5N ≤ P1 ≤ 15N and 50N ≤ P2 ≤ 100N (spring loads). The spring with the smallest operating length is sought. Using the optimal design in continuous variables obtained without adding the database constraints induces first layer coordinates of : [40,22,10], [41,23,11]. When these constraints are added to the optimization problem, the first layer coordinates become [40,23,9], [41,24,10].

The results obtained are detailed in Table 3. In every case, [41,24,10] is proposed as a solution, i.e. Do = 32 mm, d = 2.2 mm, L0 = 25 mm, R = 5.78 N/mm, working between L1 = 22.4 mm and L2 = 11.4 mm. Checking procedure : the evaluation of the entire database leads to the same stock spring from the 12 springs that match the specifications. Once again, it shows the efficiency of the "Expanding, then sliding a window" method. It also shows the importance of adding the database constraints to the initial optimization problem in order to reduce the number of springs evaluated.

CONCLUSION

This paper presents two methods for selecting stock springs from within a catalogue. These methods are less time consuming than that which tests all the springs, especially for large catalogues. The first method consists in testing the springs that surround the optimal custom spring design. The algorithm called "Expanding" tests springs by successive layers, moving away gradually until the exploration of a layer no longer improves the solution selected on the preceding layer. The second method starts with a similar expansion process and stops as soon as a spring that satisfies the specifications is found. Then the springs from within a window that surrounds that spring are tested. If a better spring is found, the window is slid onto this point and a new step is carried out until no better solution remains. The results obtained are satisfactory and have a high "result" VS "number of tested springs" ratio. This ratio can be increased by adding the database limit constraints to the initial optimization problem of finding the optimal custom spring design. As it reduces calculation time, this method is particularly useful for large data-bases. It can also help in selecting a temporary stock spring to replace a broken custom-made spring, as here the starting characteristics are already known. In addition, it is particularly important to reduce operating time in integrated design procedures.

AFFILIATION

PAREDES Manuel, SARTOR Marc, MASCLET Cédric

LGMT -INSA, 135 avenue de Rangueil 31077 Toulouse (France) Tel : 33 5 61 55 97 18 Fax : 33 5 61 55 97 00 email: manuel.paredes@insa-tlse.fr web: http://www.meca.insa-tlse.fr/lgmt

Figure

  Figure 1. Spring table

  For example M[10,11,11] = 2371 (see Fig 1) but M[1,33,19] = 0, as no spring exists with the dimensions Do = 1.6 mm, d = 6 mm, L0 = 200 mm.

Figure 2 .

 2 Figure 2. Do, d, L0 space

Figure

  Figure 4. Expanding

:

  Figure 5. Sliding a window

Table 2 .

 2 Minimum mass

	Method	Expanding	Exp. then sliding
	Nb of layers	5	2
	Nb of windows	-	3
	Nb of springs	656	142
	Result coordinates	17, 15, 13	17, 15, 13

Table 3 .

 3 Minimum operating length

	Starting point	without adding constraints	when adding constraints
	Method	Expanding	Exp. Then sliding	Expanding	Exp. then sliding
	Nb of layers	3	2	2	1
	Nb of windows	-	1	-	1
	Nb of springs	43	20	15	10
	Result coordinates	41, 24, 10	41, 24, 10	41, 24, 10	41, 24, 10


Manuel Paredes
email: manuel.paredes@insa-toulouse.fr

Marc Sartor

Jean-Christophe Wahl

OPTIMISATION TECHNIQUES FOR INDUSTRIAL SPRING DESIGN

The main industrial software available to a designer during a spring definition work use exhaustive calculations from standards to make a full check of the proposed spring's compatibility with the given specifications. As far as we are aware, they don't really exploit optimisation process capabilities. This paper presents tools that exploit several optimisation techniques for custom spring design by linking both industrial and mathematical knowledge.

We have defined tools that manage not only design parameters but also operating parameters including designer specifications, calculation from standards (buckling, fatigue life, operating limits) and the capability limits of the spring manufacturer.

Each tool uses a specification sheet where data is set with interval values. Setting data in this way provides a powerful and efficient means of expression for designers during the early stages of the design process. The tools we have developed for custom spring design are specific for each type of springs.

The compression spring optimisation implies solving an optimisation problem with 6 continuous variables (4 design variables and 2 operating variables) while satisfying 43 constraints. The resolution process is based on a mathematical programming process. An algorithm to automatically intialise the variables has been developed.

The extension spring optimisation tool is based on the same process. In that case, when a specific angle between end loops is required, the optimisation problem is defined by 5 continuous variables, one integer variable (the number of coils) and 43 constraints. The algorithm uses the previous resolution method for continuous variables and adds a branch and bound process in order to find the mixed variables optimum.

The torsion spring optimisation has required the use of a hybrid resolution algorithm. The optimisation problem is defined by 7 continuous variables (3 design variables and 4 operating variables) and 45 constraints. In that case the constraints can define a non convex allowable space for the variables. For that kind of problems, deterministic mathematical programming fails in jumping over the several intervals. For that reason, we have implemented a process that first uses an evolutionary strategy. The best solution found is then improved by a mathematical programming process.

To deal with multi-objective optimisation, we have chosen to use an interactive dialog with the designer : each possible objective function can be set as a constraint in the specification sheet. That way, the designer can try several configurations and find the best Pareto optimal solution for his application.

All the presented tools have been implemented on visual-basic for Excel. They have been successfully tested by a spring manufacturer on industrial problems. All this study shows the efficiency and the benefits that can be obtained on industrial applications using design tools that exploit both deterministic and stochastic optimisation processes.

Introduction

The design of machines imposes the dimensioning of numbers of common mechanical components (gears, cams, shafts, springs). These components have their own dimensioning rules and require specific manufacturing knowledge. The problem of designing a mechanical component is often solved by using tables and charts for certain pre-selected specifications and certain pre-selected objectives. These calculations can be carried out manually, but without computer assistance, designers are often obliged to oversimplify the procedures, e.g. by assigning a value to certain parameters in order to reduce the number of problem variables to only 2 or 3 [START_REF] Kothari | Optimum design of helical springs[END_REF][START_REF] Metwalli | CAD and Optimization of helical torsion springs[END_REF]. So, they are unable to exploit all the specification possibilities and consequently to optimize design.

Progress in computer assisted design should lead to successful solutions to this kind of problem. In theory, it should be possible, for a given component, to define the problem globally and find the appropriate resolution method that would provide the optimal solution whatever the specifications. This paper focuses on the optimum design of helical springs with circular wire [START_REF] Wahl A M | Mechanical Springs[END_REF] which are the most common springs in use today.

Spring design has been often used to illustrate optimisation algorithms. Yokota, Taguchi and Gen [START_REF] Yokota | A Solution Method for optimal Weight Design Problem of Herical Spring Using Genetic Algorithm[END_REF] present an optimum spring design using genetic algorithms [START_REF] Davis | Handbook of genetic algorithms[END_REF]. Deb and Goyal [START_REF] Goyal M | A Flexible Optimization Procedure for Mechanical Component Design Based on Genetic Adaptive Search[END_REF], Kannan and Kramer [START_REF] Kannan B K | An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design[END_REF] and Sandgren [START_REF] Sandgren E | Nonlinear Integer and Discrete Programming in Mechanical Design Optimization[END_REF] compare the efficiency of their optimisation algorithms using the same example dealing of spring design. Unfortunately, all the examples are very simplified and can not be directly used by designers in industry.

From another point of view, the main industrial software available to a designer during the spring definition work (from IST [9], from Hexagon [10] or from SMI [11]) use standard calculations only to make a full check of the proposed spring's compatibility with the given specifications. "Spring Design Software" from SMI can carry out minimum mass optimisation to adjust one design parameter but three other design parameters have to be known beforehand. Industrial software dedicated for spring design can be considered as robust validation software. They are thus mainly used at the very end of the design cycle, when the detailed design is considered.

There is a need for tools that could be used at any steps of the design cycle, from the very beginning where most of the data are not known or uncertain, to the detailed design where data are quite well defined. This paper proposes an approach which intends to satisfy this need.

Requirements for the tools

It is difficult to propose optimisation tools to industry. Most of designers did not learn optimisation at school and thus do not like using optimisation techniques. For that reson, we have chosen to developp our tools using Excel and visual-basic. These well known software are commonly used in industry. This approach can be a good way to introduce optimisation knowledge to designers.

Considering the expected wide range of applications for the tools, we have decided that each specification sheet should allow the designer to give information upon each design parameters and each operating parameters. In a specification sheet, each parameter can be defined by a lower limit and/or an upper limt as shown in figure 1. The idea is to let the designer fullfiled the knowned cells without taking care of the empty ones. In order to be able to defined the best design, an objective function as to be selected in the specification sheet.

Automatically build the optimisation problem

Once the requirements are defined, the results have to be automatically calculated. The first step consists on building the problem associated to the specifications.

The problem can be expressed as an optimisation problem : Minimise or Maximise F(X)

Subjected to constraints

The objective function F is selected by the designer in the specification sheet. It can be any parameter that is considered in the specification sheet. The vector of variables X is chosen in order to define the spring (design variables) and the associated use of the spring (operating variables). The constraints are defined not only by the limits defined on the specification sheet but also by the manufacturing constraints and the requirements from standards. Depending on the considered type of spring, on the selected objective function, on the fullfiled cells and on the associated values, very differents optimisation problems can be obtained. In any case, the resolution process has to automatically find a solution without any assistance from the designer. Considering this, we have defined resolution methods dedicated for each type of spring.

Automatically solve the problem for compression springs

Figure 2 presents the main parameters defining helical compression springs. All the parameters that can be used by a designer to define and use a spring appear in the specification sheet presented in figure 3. The objective function selected in the specification sheet, the requirements defined by the designer, the requirements from standards and from the manufacturers define an optimisation problem with 6 continuous variables (4 design variables and 2 operating variables) while satisfying 43 constraints.

Considering that we have to manage continuous variables and convex spaces, we have sought that we could take benefits of mathematical programming [START_REF] Vanderplaats | Numerical optimization Techniques for Engineering Design[END_REF] algorithms. We have selected a direct method (GRG) as it is able to provide an acceptable solution, if non optimal, even if the resolution process is stopped before end (for example if the resolution process is too long). The GRG method is already implemented in the Excel solver. Direct methods need the variables to be well initialised i.e., that the calculation starting point is an acceptable solution. For that reason, we have developed another algorithm to initialise the variables. It uses interval arithmetic [13] to perform a sequential meshing of the space of the design variables in order to build a virtual catalogue related to the specifications. The best spring form this virtual catalogue is used to initialise the variables for the mathematical programming process.

To build the virtual catalogue, limits from standards, manufacturer constraints and the most usual data (Do, D, Di, d, L0, R, F1, F2, L1, L2, Sh) from the specification sheet are exploited using interval arithmetic in order to define springs which respect the majority of the problem constraints.

Interval arithmetic is thus exploited to reduce the variables limits depending on the specifications. To build the virtual catalogue, a basic method could consist on first determining all variable limits depending on the specifications and on secondly regularly meshing the variable space (each node representing a spring of the virtual catalogue). In fact, sequential meshing is a more efficient method. The idea is to recalculate variable limits before meshing a one variable space. Once a first regular meshing (on one variable is performed), another meshing can be performed on another variable for each fixed value of the preceding variable depending on those values. Using the basic meshing method, the allowable variation space for the two variables are first calculated and then the obtained space is regularly meshed. Using the sequential meshing, the algorithm starts meshing the available variation space for the first variable d. Secondly, for each obtained value of d, the algorithm recalculates the available limits for the second variable D and meshes the associated spaces. In this example, a virtual catalogue containing 16 springs is built. Using a basic meshing, only 6 springs fit the specifications whereas using a sequential meshing, all the springs respect the specifications. That is the reason why we use a sequential meshing of the variables space to build the virtual catalogue.

Once the virtual catalogue is built, the algorithm developed by Paredes et al. [START_REF] Paredes | Stock spring selection tool[END_REF] is used to select the best stock spring from the virtual catalogue and determine the associated operating lengths.

We have called the proposed resolution process SM + MP (Sequential Meshing and Mathematical Programming).

In the example presented in figure 3, the SM process has built a virtual catalogue containing 1256 springs. From within this catalogue, 262 springs fit the specifications. The best spring has been selected to initialise the variables for the MP process. The evolution of the objective function value during the MP process is presented on figure 5. The GRG algorithm has quickly converged towards the optimal solution. The overall process is no longer than 6 seconds on a 300Mhz computer.

Figure 5. Evolution of the objective function during the MP process

These results illustrate the efficiency of the proposed SM + MP process.

Automatically solve the problem for extension springs

The main parameters that can be used to defined extension springs are presented on figure 6. The extension spring optimisation implies solving an optimisation problem with 6 variables (4 design variables and 2 operating variables) while satisfying 43 constraints.

Most of the time, the variables can be considered as continuous variables thus the SM + MP process presented before is used.

When a designer needs a fixed angle between end loops, the number of coils n becomes a discrete variable. To solve this mixed variable problem, a branch and bound process is added to the first resolution process. The branch and bound process is already implemented the Excel solver.

Automatically solve the problem for torsion springs

Torsion springs differ from both compression and extension springs in both load application and mode of operation. Helical torsion springs are often used in non highly stressed applications. This is mainly due to the fact that there are few fatigue life data available as friction problems appear between the coils and at the end of the legs where the load is applied. The shape of legs is usually determined by the required mode of operation of the component to which it is attached. However, designers should always try to use the simplest legs i.e., torsion springs with tangential legs. Design and operating parameters of torsion springs with tangential legs are presented in figure 7. The optimisation problem can be defined by 7 continuous variables (3 design variables and 4 operating variables) and 45 constraints. In some cases, constraints can define a non convex allowable space for the variables.

An example is presented on figure 8. Specifications on the angle between ends such as 90° < γ < 180° implies a number of coils defined by intervals : N + 0.25 < n < N + 0.5 where N is an integer value. For that kind of problems, deterministic mathematical programming fails in jumping over the several intervals. For that reason, we have implemented a process that first uses an evolutionary strategy (ES). The idea is that ES is able to find the area where the optimum solution is located. The best solution found by ES is then improved by the MP process previously described.

Constraint : 90

≤ γ ≤ 180 γ = f(n) 360° 270° 180° 90° 0° : valid areas n n = N n = N+1 n = N+2 n = N+3 Figure 8.
The number of coils can be defined on a non convex space

Multi-objective optimisation

There are many way to deal with multi-objective optimisation [START_REF] Deb | Multi-Objective Evolutionary Optimization : Past, Present, and Future[END_REF]. We have chosen to use an interactive dialog with the designer : a constraint can be set in the specification sheet on each possible objective function. That way, the designer can try several configurations and find the best Pareto optimal solution for his application as described in figure 9.

Conclusion

We have shown that defining the best custom spring for an application could be set as an optimisation problem. Tools for compression, extension and torsion spring design have been defined. Several resolution methods depending on the considered type of springs have been implemented. They use interval arithmetic, a direct optimisation method, branch and bound process and an evolutionary strategy.

The proposed tools dedicated to optimal spring design have been implemented on visual-basic and Excel to be easily used in industry. They have been successfully tested by a spring manufacturer on industrial problems. The short resolution and preparing time of the proposed tools allows designers to tests several configurations. They thus can use them as exploration tools by adding data, refining data or changing the objective function from a to another. This type of tool increases design process flexibility as the component design is made easier and more efficient.

All this study shows the efficiency and the benefits that can be obtained on industrial applications using design tools that exploit both deterministic and stochastic optimisation processes.

Figure 1 .

 1 Figure 1. Proposed specification sheet

Figure 2 .Figure 3 .

 23 Figure 2. Compression spring parameters

Figure 4 .

 4 Figure 4. Comparison on two meshing methods An example of the difference between a basic and a sequential meshing is illustrated on figure 4. This example uses a simplified problem which involves two variables (d and D) and only 6 constraints representing d, D and w (w=D/d ) limits.Using the basic meshing method, the allowable variation space for the two variables are first calculated and then the obtained space is regularly meshed. Using the sequential meshing, the algorithm starts meshing the available variation space for the first variable d. Secondly, for each obtained value of d, the algorithm recalculates the available limits for the second variable D and meshes the associated spaces. In this example, a virtual catalogue containing 16 springs is

Figure 6 .

 6 Figure 6. Extension spring parameters

Figure 7 .

 7 Figure 7. Torsion springs parameters Even considering static applications, custom torsion spring design is more complicated that the design of compression or extension springs. The torsion spring optimisation has required the use of a hybrid resolution algorithm.

Figure 9 .

 9 Figure 9. Find the best solution on the Pareto front

Acknowledgements

The financial and technical support of the spring manufacturer « Ressorts VANEL » is gratefully acknowledged.


Manuel Paredes
email: manuel.paredes@insa-tlse.fr

Cédric Marc Sartor

Masclet Lgmt

ADVANTAGES IN USING A STOCK SPRING SELECTION TOOL THAT MANAGES THE UNCERTAINTY OF THE DESIGNER REQUIREMENTS

This paper analyses the advantages of using a stock spring selection tool that manages the uncertainty of designer requirements. Firstly, the manual search and its main drawbacks are described. Then a computer assisted stock spring selection tool is presented which performs all necessary calculations to extract the most suitable spring from within a database. The algorithm analyses data set with interval values using both multi-criteria analysis and fuzzy logic. Two examples, comparing manual and assisted search, are presented. They show not only that the results are significantly better using the assisted search but it helps designers to detail easily and precisely their specifications and thus increase design process flexibility.

Introduction

The creation of mechanical objects is often the end result of a long design process. Standard component selection is perhaps the simplest, but nonetheless an important, class of design decision problems as catalogues are becoming increasingly common and voluminous.

Let us analyze the method commonly used by design engineers to select stock springs in order to highlight the difficulties they encounter, the help they can find today and what could be added to improve it.

The usual method for selecting stock springs can be divided in three steps: • Step 1 : evaluate, from the requirements, certain spring design parameters among those classified in the catalogue of the chosen spring manufacturer. • Step 2 : find springs that are within parameter limits.

• Step 3 : calculate the operating parameters for each spring short-listed, so as to select one that satisfies the specifications. Designers are confronted here with the following problem. On the one hand, when the specification is vague, it is difficult to choose the best spring from the large range available. When the specification is precise, on the other hand, choice of an appropriate spring becomes limited.

As paper-based methods are tedious and time-consuming, Yuyi [START_REF] Yuyi | Application of Expert System for Spring Design and Procurement[END_REF] has implemented the first two steps of the search in an expert system but where step 3 has to be calculated. Technical literature provides mathematical methods to calculate the design parameters corresponding to the optimal design (Sandgren [START_REF] Sandgren E | Nonlinear Integer and Discrete Programming in Mechanical Design Optimization[END_REF], Kannan and Kramer [START_REF] Kannan | An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design[END_REF], Deb and Goyal [START_REF] Goyal M | A Flexible Optimization Procedure for Mechanical Component Design Based on Genetic Adaptive Search[END_REF]). These methods can be used in step 1 but problems have been simplified and the practical existence of the spring is never envisaged.

Text-only systems such as "SPEC" [START_REF] Innomech | SPEC Select, Spring Selection[END_REF] have been developed which assist the designer during step 2. As with many computer-based methods, if the requirements are not specified precisely enough or lie outside the catalogue range, "all or nothing" search results can be obtained [START_REF] Harmer | Design-led component selection[END_REF].

Finally, industrial software available for a designer during the spring definition work such as "Compression spring Software" from IST  [7], can be used in step 3.

The first drawback of the common component selection method is that it may take a long time when the requirements are advanced. The process time remains significant even when dealing with the simplest applications. The second disadvantage is the approximate nature of the procedure. Usually, a spring is selected without being sure of the pertinence of the choice. Generally, the designer cannot affirm that a spring best matching his specifications does not exist.

But the main drawback is probably the decrease in the design process flexibility as illustrated in Fig. 1. Once a stock spring has been selected, the designer is inclined to keep it unchanged for as long as possible to avoid the timeconsuming work of making a new search. The choice is usually kept until the end of the design process. So, there is a need for software tools dedicated to reliable and less time-consuming catalogue search.

Step N

Step N+1

Stock Spring Selection

Yes

Yes, if the spring can be kept

No, when possible

Fig. 1 Effects on the design process

This paper presents a tool armed at meeting designer expectations which also takes advantage of all specifications possibilities. The following capabilities, that are not usually presented, are proposed :

• taking uncertain parameters into consideration (data set with interval values),

• automatically performing all the necessary calculations (buckling, fatigue life...)

to check that a spring satisfies the specifications, • excluding "all or nothing" search results,

• introducing an objective function in order to propose the most suitable component. This tool may be used even in the early design stages. To fit perfectly with the designer's incomplete knowledge, the method determines springs from a specification sheet where data can be uncertain. The associated algorithms select the best spring by calculating the operating parameters for a given objective. Using this kind of tool, the designer can express his specifications in a very formal and practical way. He can obtain search results instantaneously (number of springs available and the one selected). The present study deals only with helical compression springs with closed ends and with closed and ground ends.

A Stock Spring Selection Tool Working from Toleranced Specifications

First, spring characteristics are detailed in order to illustrate whether the tool can accept over-definite requirements. Then the resolution algorithm is described. Finally, the two different methods available to compare competing springs are presented.

Specifications for the Assisted Search

The parameters which define the spring geometry are: Do, D, Di, d, R, L0, Ls, n, z, p. Fig. 2 illustrates these parameters which characterize the intrinsic properties of the spring. A spring works traditionally between two configurations, one corresponding to the least compressed state W1, the other corresponding to the most compressed state W2. Thus the operating parameters which define the use of a spring are: P1, P2, L1, L2 and sh (see Four independent design parameters have to be known in order to calculate the six others. When the design parameters are known, only two independent operating parameters (to be taken among P1, P2, L1, L2 and sh) are necessary to determine the two operating points W1 and W2. To express all the calculations inside the tool which is presented below, springs of the database are defined by Do, d, L0 and R and the chosen operating parameters are L1 and L2. Nevertheless, each particular parameter illustrated in Fig. 2 or Fig. 3 can be used by the designer to express his specifications.

The designer can decide on design and operating parameters by giving their bounds (lower and/or upper limits: L S L 0 , L S U 0 ,... P S L 1 , P S U 1 ...) in the specification sheet (see Fig. 4). Each fixed parameter simply involves the specification of lower limit equal to upper limit. Moreover, the designer can define a number of other characteristics with interval values:

• Natural Frequency of surge waves • Spring mass • Overall space taken up when uncompressed (L=L0)

• Overall space taken up when compressed (L=L2).

• Internal energy during the operating travel Designers can provide additional data to calculate other characteristics:

• The number of cycles (Ncycles) to calculate the fatigue life factor (to check that it is higher than unity) • The end fixation factor (ν) to calculate the buckling length and check that it is less than L2. [START_REF] Wahl | Mechanical Springs[END_REF] The designer can also specify the material and the spring ends required. Finally, to be able to select the most suitable spring, the objective function (maximize fatigue life, minimize mass, minimize L2...) has to be given.

Fig. 4 Specification sheet

Any data not defined in the user specification sheet is set to a default value : 0 for a lower limit and 10 7 for an upper limit. Then the proposed resolution algorithm is performed.

Proposed Resolution Algorithm

General methods dedicated to component selection problems, as the one proposed by Bradley and Agogino [START_REF] Bradley | An Intelligent Real Time Methodology for Component Selection: An Approach to Managing Uncertainty[END_REF], could be applied to the stock spring selection problem. Significant reduction in development costs and processing times can be obtained using a more direct method which is able to take advantage of the spring problem characteristics such as the COSAC system [START_REF] Vogwell | A strategy for selecting engineering components[END_REF] developed at Bath University.

The method chosen here can be considered as the most reliable, since all springs are successively tested. The first spring of the catalogue is evaluated and set as the potential optimum. Then the second spring is evaluated and compared to it. If it is better, it becomes the new potential optimal spring. All the springs of the catalogue are thus evaluated and compared to the last potential optimal spring. When the end of the catalogue is reached, the spring that best matches the requirements is the potential optimal spring. This method provides an acceptable processing time (less than ten seconds) with the catalogue used in this paper which contains about five thousands references.

To evaluate a spring, its four associated design parameters are read from the database and the two operating parameters are automatically calculated in order to optimize the objective value (maximize P2, minimize P2, maximize L2 …) [START_REF] Pardes | Stock Spring Selection Tool[END_REF]. All previously detailed design and operating parameters are then calculated. To fit with real-life industrial problems, other properties are added such as fatigue life, price, mass, buckling length or solid length. 23 criterions are thus calculated. When all the spring criteria have been calculated, it remains to know how they fit the specifications. To manage the various needs of designers, two different analysis are proposed.

Comparing Springs Using Multi-criteria Analysis

In the first steps of the design cycle, when most part and shapes have not been chosen, specifications are often imprecise and constraint violations can be admitted. Multi-criteria analysis has been chosen to solve this problem. For each spring, the following equation is used to evaluate the constraint violations.

Comparing Springs Using the Fuzzy Logic Analysis

The previous analysis often ends in proposing a spring that is close to certain limit values of the specifications. In the first steps of the design cycle, a spring near the centre of the solution domain can be the best choice, even if its objective function value is less interesting. To solve this kind of problem, fuzzy logic analysis is proposed here.

As the goal is to find springs within the limit values of the specifications, a basic comparison is first carried out on the number of constraints violated (ncv). A spring that has the lowest ncv value is definitely considered has better than the others. When both springs have the same ncv value, the comparison with fuzzy logic is performed. The first step is to evaluate how the tested spring matches the specifications. The comparison between this evaluation and that obtained for the potential optimal spring is made at the second step. In the third step, the comparison between the objective function values of these two springs is done. Then, a final comparison is made to select the new potential optimal spring.

Step 1 : How a Spring Matches the Specifications

First, an evaluation is made of each criterion. To evaluate how a criterion matches the specifications, the following method is used. FuzzyMarkc (Lc, Uc, Vc) = [VB, B, M, G, VG] c is calculated as follows.

Step 2 : Comparison of Specifications

The comparison of the previous fuzzy values is performed using the Mamdani [START_REF] Mamdani | Application of fuzzy algorithms for control of simple dynamic plant[END_REF] definition for the AND connector as described in table 1. For the case presented in the previous tables, as S value is superior to I value, the "old" potential optimal spring is superior to the tested spring. Thus, it is kept as the potential optimal one (otherwise the tested spring would have replaced it) and the next spring is tested.

Examples and Comparison Between Manual and Assisted Search

In order to compare the results between the manual and assisted search, the two examples have been performed using the same catalogue with both methods. Each assisted search has been carried out with the two different analysis. The average time of the first analysis is 3 seconds increasing to 6 seconds for the analysis using fuzzy logic.

A Spring for a Clamping Pin

. Assisted Search

During a reengineering procedure, it was decided to reselect a spring in order to reduce the main drawback of the clamping pin : its high axial length. Now, the proposed search tool can be used and the specifications can be detailed precisely. Specifications : the maximum outside diameter of the spring is 38 mm, the minimum inside diameter is 27 mm, spring travel must be 11 mm, the maximum value of L1 is 50 mm, the maximum spring rate is 5.5 N/mm, the load P1 must be between 5 and 15N and the load P2 between 50 and 100N. The goal is to obtain the spring with the smallest value of L2.

According to the chosen objective, the algorithm calculates the operating parameters of each spring in order to have the minimum operating length L2 while satisfying the specifications.

Results : there are 7 springs that fit the given specifications. The first analysis proposes a spring that is close to specification requirements (in terms of R) : As the properties of the two proposed springs are automatically calculated, the designer can easily choose the one that best matches his specifications. In fact, the spring found with the first method fits the geometrical parameters and has a law Violation value (= 0.051), whereas its L2 value is much lower than the one selected by the fuzzy logic method. This spring is chosen for the new design described in Fig. 8. The assisted search proposes springs significantly better than those found with the paper search, leading to useful changes in the clamping pin design.

A Spring for an Axial Displacement Sensor

Manual Search

In this example, the manual search in the catalogue led to the following spring : Do = 12.5 mm, d = 1.25 mm, L0 = 100 mm, R = 0.8 N/mm, L1 = 93.75 mm, L2 = 33.75 mm (P2 = 53N , steel with closed and ground ends).

Assisted Search

Once again, the assisted search allows to the following requirements to be expressed :

Specifications : the maximum outside diameter of the spring is 13 mm, the minimum inside diameter is 5 mm, spring travel must be 60 mm, the minimum P1 value is 3 N and the length L2 must be between 30 and 45 mm. The goal is to obtain the spring with the smallest value of P2.

According to the chosen objective, the algorithm calculates the operating parameters of each spring in order to have the minimum operating load P2 while satisfying the specifications.

Results : there are 14 springs that match the given specifications. Using the first analysis, a spring that fits the specifications (Violation = 0) is selected : Do = 11.0 mm, d = 0.9 mm, L0 = 100 mm, R = 0.3 N/mm, L1 = 90 mm, L2 = 30 mm (P2 = 21N , steel with closed and ground ends). Fuzzy logic analysis proposes another spring that fit the specifications : Do = 11.0 mm, d = 1 mm, L0 = 100 mm, R = 0.374 N/mm, L1 = 92 mm, L2 = 32 mm (P2 = 25.4N , stainless steel with closed and ground ends).

In this case, the spring found by fuzzy logic analysis is within the geometrical constraints. In order to obtain a reliable design, this spring is included in the mechanism. Once again, the assisted search results in the choice of a much better spring than the one obtained by the paper based method.

Conclusion

Paper-based methods for selecting stock springs are tedious, time consuming and decrease design process flexibility. A stock spring selection tool managing uncertain parameters and including all the necessary calculations in order to suggest the most suitable spring is proposed. It has been developed and tested for one year in collaboration with a spring manufacturer. It has shown that this kind of tool changes the designer's approach during the catalogue search. With the assistance of the proposed tool, the designer can specify his needs and quickly choose the spring that best matches his requirements. Finally, this type of tool increases design process flexibility as the component choice is made easier and more efficient.

Fig. 3

 3 Fig.3).

IF 6 Fig. 5

 65 Fig. 5 FuzzyMarkc evaluationThen, the final values of VB (very bad), B (bad), M (medium), G (good), and VG (very good) are calculated using the formula :

Table 1 : 3 .Fig. 6 [

 136 Fig. 6 [VI, I, E, S, VS]Obj value 2.4.4. Step 4 : Final Comparison Springs are finally compared using both [VI, I, E, S, VS]Obj and [VI, I, E, S, VS]Spec with the same rules as in step 2 in order to calculate I, E, S values. Table 3 : Final comparison [VI, I, E, S, VS]Spec [VI, I, E, S, VS]Obj VI (0.3) I (0.50) E (0.50) S (0) VS (0) VI (0) I (0) I (0) I (0) I (0) E (0) I (0) I (0) I (0) I (0) E (0) S (0) E (0) I (0) I (0) E (0) S (0) S (0) S (0.57) I (0.30) E (0.50) S (0.50) S (0) S (0) VS (0.43) E (0.30) S (0.43) VI (0) S (0) S (0)

3. 1

 1 .1. Manual Search A clamping pin has been custom-designed for an industrial manipulating robot. The time-consuming manual procedure resulted in the selection of the spring Do = 36.0 mm, d = 2.5 mm, L0 = 50 mm, R = 3.54 N/mm, L1 = 47 mm, L2 = 36 mm. (steel with closed and ground ends) from the paper catalogue. The result is shown in Fig. 7 and the clamping pin was added to the robot.

Fig. 7

 7 Fig. 7 Old clamping pin design

  Do = 32.0 mm, d = 2.2 mm, L0 = 25 mm, R = 5.78 N/mm, L1 = 22.4 mm, L2 = 11.4 mm. (steel with closed and ground ends). Using fuzzy logic the following spring is selected : Do = 32.0 mm, d = 2.2 mm, L0 = 32 mm, R = 4.34 N/mm, L1 = 28.54 mm, L2 = 17.54 mm. (steel with closed and ground ends).

Fig. 8

 8 Fig. 8 New clamping pin design

Table 4 :

 4 Value of I, E, S

	I	E	S
	0.30	0.50	0.43

Acknowledgements

The financial and technical support of the spring manufacturer « Ressorts VANEL » is gratefully acknowledged.