Interval computation and constraint propagation for the optimal design of a compression spring for a linear vehicle suspension system

Mechanism and Machine Theory 84 (2015) 67–89

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Interval computation and constraint propagation for the optimal design of a compression spring for a linear vehicle suspension system Hassen Trabelsi a,b, Pierre-Alain Yvars a,⁎, Jamel Louati b, Mohamed Haddar b a b

LISMMA, Institut Supérieur de Mécanique de Paris (SupMeca), 3 rue Fernand Hainaut, 93407 Saint Ouen Cedex, France U2MP, Ecole Nationale d' Ingénieurs de Sfax Université de Sfax (ENIS), BP N 1173-3038, Sfax, Tunisia

a r t i c l e

i n f o

Article history: Received 16 May 2014 Received in revised form 22 September 2014 Accepted 26 September 2014 Available online xxxx Keywords: Interval computation Constraint satisfaction problem Compression spring Optimization Preliminary design Linear suspension system

a b s t r a c t In this paper an optimization design method based on existing intervals and constraint satisfaction problem “CSP” computer tools is proposed. The method was used in the preliminary design to size a compression spring implemented in a linear vehicle suspension system. Compared to conventional design methods, our design method avoids the passing through two stages of sizing (static and dynamic). Using the numeric CSP approach, static and dynamic requirements can be coupled in the same step of sizing. It also avoids the falling on the loop “design–simulate–back to the initial step in case of failure”, as the design parameter values of the compression spring generated by the numeric CSP satisfy all imposed requirements, and the simulation results of the system behavior are always successful and respect all posted constraints. This is due to the fact, that in the CSP, all analytical relation types static and/or dynamic defining the product and its behavior are implemented and integrated from the beginning. So the production of a qualifying system can be achieved from the first time without any need for resizing the system. The general idea of the proposed design method consists of expressing the design variables by intervals; integrate all imposed constraints of different types before the simulation step and solve the problem using the CSP. The generated intervals represent the domains of possible values for the design variables of the product. The obtained result which can be a solution or set of solutions, affirms that the suggested method is valid and potentially useful to size dynamic systems easily and effectively. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction The design activity of mechanical systems is now part and partial of the context of integrated and collaborative design. It requires supporting tools and methodologies adapted throughout the design process [1–4]. We are interested here in problems of optimal design [5–7] of components and we propose an improvement to these tools and methodologies using constraint satisfaction techniques [6–17]. Our approach is applied to an optimal sizing case of a compression spring [18–27]. The springs are structural elements designed to maintain and store the energy and mechanical work based on the principle of the flexible deformation of the material. They are among the components of the most heavily loaded machines and are usually used as: - Energy absorbing and for commands and reversing devices, - Interceptors of static and dynamic forces, - Elements for the creation of strength joints, ⁎ Corresponding author. Tel.: +33 149452925; fax: +33 149452929. E-mail address: [email protected] (P.-A. Yvars).

http://dx.doi.org/10.1016/j.mechmachtheory.2014.09.013 0094-114X/© 2014 Elsevier Ltd. All rights reserved.

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- Dampers in the anti-vibration protection, - Devices for control and measuring force. Several software tools are available for sizing springs, particularly compression springs one. In most cases it is either a software validation of a given size, or tools allowing very low variability specifications (predefined choices of design variables). We find in [18–22] design optimization tools for springs according to one or more values of the performance variables (weight, cost …). Generally the algorithm used in those software tools is based on the design process described in Fig. 1. In this approach we generally find three main steps: The first one consists of fixing the design variable values which requires high expertise to provide the initial dimensions. Before moving to the dynamic study, the designer carries out a static test and sets the safety factors according to the imposed requirements. The next step is to achieve the dynamic modeling then to make the dynamic test. So the designer is before two situations. In the first case, if the resulting behavior of the system fulfills the imposed constraints in the specifications document subsequently, the design parameters used in the simulation will be taken as a solution. In the second case, if the system response does not satisfy the imposed constraints, the designer has to change the parameters taking into account the previous simulation. The same sizing steps must be repeated until obtaining the optimal solution (in our case we look to optimize the weight of the compression spring). For most of these works, the designers use mathematical evolutionary algorithm or simulated annealing to minimize the cost function. The optimization phase is mainly based on stochastic methods. These methods do not provide the optimum global, they are just approximation methods. The optimization is global only when the function to be optimized is differentiable and convex.

Fig. 1. Conventional method for a system dimensioning.

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This technique has the disadvantage of being very sensitive to the initial conditions otherwise it remains trapped in a local optimum and it depends on high expertise. In addition there is no connection between the static and dynamic designs. The passage through two steps of sizing (static and dynamic) leads to the oversizing of the spring especially when the designer takes high safety coefficient values. This step which is carried out after the static modeling may lead to dynamic problems of the entire system. The designer has to run several simulations to determine an optimal solution without being sure if the obtained solution is the optimal global one within the solution space. It comes from the fact that the number of simulations that can be done is limited by the time and cost constraints which leads also to the oversize of the system. To overcome these limits [20] and optimize the design process we propose to couple the static and dynamic sizing in one step aiming optimize the choice of the safety factors first, and then to use intervals instead of fixed values in order to minimize the number of simulations and to obtain a set of solutions instead of a single one. Therefore to achieve these goals a global method based on interval arithmetics [9,10] and constraint propagation and resolution is used to identify better the area of research and to have more accurate results. The CSP approach is integrated in a new design process described in Fig. 2. This approach suffers from a low genericity but thanks to our work we can overcome this limitation. Fig. 2 shows a different sizing approach based on intervals and CSP. This sizing process consists of three main stages: the first stage is to express design variables by intervals. Here the choice of the design variables values does not require expertise. We can use a blind choice and long intervals, but we cannot deny that expertise may reduce the calculation time. In the next step, the designer identifies the requirements that must be satisfied and express them as constraints on the design variables. These constraints provide declarative descriptions of important requirements related to engineering objectives. Then all the types of constraints are implemented in the CSP code. Finally the designer spreads these constraints on the intervals of the design variables to define the parameter areas defining the system design. Here comes the role of the CSP approach. In fact the CSP eliminates all the values of the design variables that do not respect the imposed requirements. The sets of values that remain represent the entire generated solution. They can be placed in the case when there is no solution. In this case the problem must be reviewed and for example, some constraints can be eliminated. With the CSP it is possible to couple static and dynamic constraints. This avoids going through two steps of sizing (static and dynamic) as in the case of a conventional sizing. Thanks to this methodology based on interval computation it is certain that the chosen design parameters would satisfy the imposed constraints and the success of the dynamic and static test is sure. Furthermore, the CSP approach can be used for optimization through imposing such optimization criteria as constraints like weight and cost. This calculation was intended and applied for the design of the resistance and dimensioning of cylindrical compression springs made by circular section rods, submitted to the action of static and/or dynamic forces. The main purpose of this study is to resize a spring with no need to make many calculations taking into account all types of constraints together in one sizing step.

Fig. 2. New method for a system dimensioning.

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The advantages of the proposed design method are numerous; a summary table of these advantages is presented in the following. Table 1 presents a short comparison between the CSP based design method and conventional design methods. The remaining of this paper was organized as follows. First, the technique used to optimize the sizing process was defined. Then, in the third and forth parts a brief example of sizing of a compression spring integrated in the linear vehicle suspension system was exhaustively discussed. Section 5 was devoted to present the application of the CSP approach and to detail the obtained results. Finally some conclusions are drawn based on the obtained results. 2. Constraint propagation and interval computation 2.1. Constraint-satisfaction problem The constraint satisfaction problem “CSP” [6–17] is a programming paradigm that emerged in the 1980s to solve combinatorial problems of large sizes such as problems with planning and scheduling. It was intended to solve the mathematical problem that looks for states or objects satisfying a number of constraints. This technique is extensively useful to treat problems manipulating intervals. A CSP [6,7] is defined by a 3-tuple (X, D, C) in such a way that: - X = {x1, x2, x3, xn} is a finite set of variables which we call constraint variables with n being the integer number of variables in the problem to be solved. - D = {d1, d2, d3, dn} is a finite set of variable value domains of X such that: ∀i∈f1; …; ng; xi ∈di :

ð1Þ

A domain should be a real interval or a set of integer values. - C = {c1, c2, c3, cp} is a finite set of constraints, p being any integer number representing the number of constraints of the problem. ∀i∈f1; …; pg; ∃X i ⊆ X=ci ðX i Þ

ð2Þ

A constraint is a relationship between one or more variables that can take their limit values by the constraint. It can be any type of mathematical relationship (linear, quadratic, nonlinear, Boolean, …) covering the values of a set of variables. The variable domain can be: - Discrete: in the form of sets of possible values. - Continuous: in the form of intervals on real numbers. A solution is an assessment that satisfies all the constraints. Solving a CSP [8–14] boils down to instantiating each of the variable of X while meeting the set of constraints C, and at the same time satisfying the set of constraints C.

Table 1 CSP based design method vs conventional design methods. Conventional design methods

CSP based design method

- The design variables are expressed as fixed values - Single assessment of the system - Obtaining a one solution only after the simulation step - The passage through two steps of sizing (static and dynamic) leads to the oversizing of the system - The requirements are implemented progressively according to the level of design - Possibility of failure and the success of the design for each proposed solution

- The design variables are expressed by intervals - An envelope of system performance - Obtaining a set of solution after the simulation step - Possibility of coupling between the static and dynamic study

- Redesigning the system in case of failure - Local optimization problem → over-sizing of the system - Several simulation implying an important computation time - High cost of design process

- The requirements are included in the initial stages before the solving process - All the solutions generated by this method satisfy the constraints imposed since all requirements are integrated from the beginning - The design loops as (design–simulation–back at the initial stage in case of failure) are avoided - Global optimization - Optimization of computation time since the simulation by interval allows to generate an envelope of performance - Cost optimization of the design process

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2.2. Interval arithmetic Specific interval arithmetics was developed by Moore [7,9,10] in the context of modeling imprecision and uncertainty, and gives arithmetics rules between interval variable types. It provides basic support for resolution of digital CSPs. The evaluation of a function applied in interval variables with the use of the interval computation method can provide lower and upper bounds. Here are the four basic mathematical operations between intervals used in interval arithmetics: h i h i ½x; x þ y; y ¼ x þ y; x þ y

ð3Þ

h i h i ½x; x− y; y ¼ x−y; x−y

ð4Þ

 3 2 i min x  y; x  y; x  y; x  y  5 ½x; x  y; y ¼ 4 ; max x  y; x  y; x  y; x  y h

½x; x h i¼ y; y

" min

! x ;x x x ; ; ; max y y y y

!# h i x x x x ; ; ; ; 0∉ y; y : y y y y

ð5Þ

ð6Þ

Interval arithmetics has been extended to include other operators (quadratic, trigonometric, logarithmic, …), which has made the development of the CSP easier. So it is worth mentioning that the CSP approach is still being developed and has been extended to continuous field so far. As an example: considering the function f(x) = tan(x) + exp(x), the interval evaluation of f for x ∈ [−1; 1] can be computed as follows: f ð½−1; 1Þ ¼ tanð½−1; 1Þ þ expð½−1; 1Þ ¼ ½−1:5575; 1:5575 þ ½0:3678; 2:7183 ¼ ½−1:1896; 4:2757:

ð7Þ

2.3. Consistency Several consistency [13,14] techniques exist in the literature [7]. Let us present the two main categories: - Hull consistency: Let (X, D, C) a constraint satisfaction problem involving a vector X of n variables and let [x] be the domain of x (X, D, C). is said to be hull consistent if for every constraint c in C and for all i|(1 ≤ i ≤ n|), there exists two points in [x] which satisfy c and whose ith coordinates are xi and xi respectively. The key property of hull consistency lies in the combination of local reasoning and interval representation of domains. This concept brought a decisive improvement to the traditional Newton-based numerical solvers that were basically only able to contract domains globally. - Box consistency: Box consistency is a relaxation of hull consistency [15]. The principle is to replace the constraint satisfaction test over the real domain with a refutation procedure over the interval domain. More precisely, the definition of arc consistency is examined. A value υk ∈ Dk is inconsistent, if ∀ v1 ∈ D1, …, vk ∈ Dk, …, vn ∈ Dn, c(v1, …, vn) ≠ 0. An equivalent statement is that the range of c over the domain D1 × D2 × … × Dk − 1 × vk × Dk + 1 × … × Dn is non-zero. The key idea is to compute a superset of this range using any interval extension of c. In so doing, a value can be eliminated if any superset of the range is non-zero. Then, the definition of box consistency is the following: If we have xk is a CSP variable and Dk the domain of xk and ∀ vk, 0 ∈ c(D1, …, Dk − 1, D1, Dk − 1, …, Dn) then the domain of xk is box consistent. Otherwise, the goal is to find the extreme values in Dk that are consistent. The standard implementation uses a dichotomous search procedure, which exploits the monotonicity property of interval evaluation. If a numerical CSP is Hull or Box consistency, the main property is that if an interval solution of a numerical CSP exists, it is inside the intervals returned by the hull or box consistency algorithm. In practice, this condition is necessary but not sufficient and we need complementary algorithm to find the true interval solutions.

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2.4. Branch and prune algorithm and strategies Branch and prune algorithms [7] are used for finding the interval solutions of a numerical CSP as in Algorithm 1. The goal is to obtain for each variable an upper value and a lower value as close as possible to each solution of the numerical CSP (X, D, C). To achieve this goal, a branch and prune algorithm applies recursively a pruning operator (i.e. a hull or box consistency operator).

Algorithm 1. Branch and Prune algorithm

The process starts with a given CSP (X, D, C) and the list of solutions L is empty (i.e. {}). At each step, a variable is chosen (via the Choose Variable function), its interval D is bisected in two subintervals D1 and D2. The pruning operator is applied to each of the two subintervals on all the constraints of C. It reduces the intervals of the other variables of the CSP. If a good precision is obtained on D (i.e. one solution is reached), the resulting set of intervals is added to the set of interval solutions, L. If not, the splitting process continues until one interval becomes empty. The good precision test is done by the OKPrecise function. At the end of the process, the algorithm returns L which is the list of the interval solutions of the given numerical CSP (X, D, C). Search algorithms such as branch and prune start the process by selecting a variable to bisect. The order in which this choice is done is referred the variable ordering. A correct ordering decision can be crucial to perform an effective solving process in case of real-life problems. Several heuristics exist for selecting the variable ordering. After selecting the variable to bisect, the algorithms have to select a subinterval form of the variable's domain. This selection is called the value ordering. It can also have an important impact on the duration of the solving process. 2.5. Optimization and constraint satisfaction problem The optimization principle adopted to minimize the value of a CSP [7] real variable f is described in Algorithm 2. In practice, f should be a variable equal to a constraint expression representing the criteria to be minimized.

Algorithm 2. Minimizing a function value using CSP

The key point is to solve by dichotomy a sequence of CSP where the set of constraints increases from one CSP to the next. Each CSP is solved by a branch and prune procedure. At each step, we add a constraint expressing that the next CSP has to be better than the current one according to the minimization of the f variable. The process stops on the CSP which minimizes the f variable value when the required precision ε is reached. Let us notice that the solution found by this algorithm is a global optimum in the algorithmic point of view. The method presented in this paper is not purely a theoretical calculation one but rather a practical method already applied for the design of a compression spring. The general purpose of this study is to design the compression spring whatever the environment

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where it is placed. The case of the design of a compression spring integrated into an automotive suspension was chosen. There are too many factors that influence the calculation of the spring and its life, so it would be preferable to stay in a simplified form in order to achieve the spring sizing as quickly as possible. 3. Problem of optimal design of an elastic link — case of a compression spring In this work we are interested in optimally sizing an elastic link type spring. This design problem has been extensively studied in many excellent scientific works [18–22] and treated by several techniques that are mainly based on stochastic methods to ensure the optimization of the design, but in these studies the static sizing was studied separately from the dynamic sizing. So we propose in this study to combine the consideration of static and dynamic requirements in the same preliminary design phase which represents our own scientific contribution. This idea is applied in the case of a compression spring integrated in a linear vehicle suspension system. A spring is a system which function is to deform under the action of a force or a torque and to restore the energy stored by resuming of its original form [18–20]. The art of the spring-maker is to reduce the overall weight while keeping the qualities described. This study focuses on the most used type of spring namely the compression coil spring with round wire, axial pitch and outer diameter constant (Fig. 3). 3.1. Design variables for a compression spring Design variables (see Appendix A) for a compression spring [21,22] can be classified into three types. - The variables characterizing the material are related to the used material. Knowing the material kind, the variables (G, E, ρ) are well known. The other variables depend on the values of the geometric variables of the spring (Table 2). - The geometric variables (De, Di, R, Lo, Lc, d, n, z and m) are used to define the geometry of a compression spring. The spring is a component whose geometry varies significantly during its use. Traditionally a compression spring works between two configurations: one corresponding to the less compressed state, the second corresponding to the most compressed state. - The operating variables (F1, F2, L1, L2 and Sh) define the spring usage. 3.1.1. Technological relations between design variables The design variables of the compression spring are interrelated through a set of equations [22,26] which are detailed as follows: D ¼ De −d

ð8Þ

Di ¼ D−d

ð9Þ

F 1 ¼ RðL0 −L1 Þ

ð10Þ

F 2 ¼ RðL0 −L2 Þ

ð11Þ

Fig. 3. Parameterizing of a compression spring.

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Table 2 Limits materials for compression springs. Material

Steel DH

Stainless steel 302

Limits of the manufacturer (mm) G E Rm Maximum permissible stress τzul(% of Rm) ρ

0.3 ≤ d ≤ 12 81,500 206,000 2230 − 355.94Ln(d) 50 7.85

0.15 ≤ d ≤ 15 70,000 192,000 1919 − 255.86Ln(d) 48 7.91

Fc

th

¼ RðL0 −Lc Þ

ð12Þ

F n ¼ RðL0 −Ln Þ

3560d fe ¼ nD2

k¼

ð13Þ

sffiffiffiffi G ρ

ð14Þ

w þ 0:5 w−0:75

ð15Þ

L0 ¼ m n þ ðni þ nm Þd

ð16Þ

Lc ¼ dðn þ ni þ nm Þ

ð17Þ

 Ld ¼ πD 2 þ nm þ

n  cos z

Ln ¼ dðn þ ni þ nm Þ þ Sa

ð18Þ

ð19Þ

3

Lr ¼ L0 −

πd τ zul 8DRk

−3

M¼

Ld πd 4

2

ð21Þ

L0 −dðni þ nm Þ n

ð22Þ

Gd4 8RD3

ð23Þ

m¼

n¼

ρ10

ð20Þ

nt ¼ n þ nm þ 2

ð24Þ

4

R¼

Gd 8nD3

sh ¼ L1 −L2

ð25Þ

ð26Þ

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2

V 0l0 ¼

πDe L0 4000

V 0l2 ¼

πDe L2 4000

ð27Þ

2

W ¼ 0:5ð F 1 þ F 2 ÞðL1 −L2 Þ

w¼

D d

tan z ¼

ð28Þ

ð29Þ

ð30Þ

m πD

ð31Þ

  3 τk2 ¼ 8DRðL0 −L2 Þk= πd

ð32Þ

  3 τkc th ¼ 8DRðL0 −Lc Þk= πd :

ð33Þ

Additional relationships such as inequalities, compatibility tables, and conditional relations are also taken into account in the design of a compression spring. 3.1.2. Choice of compressing spring material The type of material selected imposes values on certain parameters and a restriction on the bounds for other parameters (Table 2). Moreover, DIN [27] sets the scope of these formulas for helical compression springs: d≤17

ð34Þ

D≤200

ð35Þ

L0 ≤630

ð36Þ

n≤2

ð37Þ

4≤w ≤20:

ð38Þ

3.1.3. Choice of extremities The choice of extremities determines the value of the ni variable which can be chosen from the set {1, 1.5, 2, 3}. In practice, extremities with simple cut (ni = 1) or just grinded (ni = 1) are rarely used because they cause a force dispersion. It is preferable to use closer extremities (ni = 3) or even close and round (ni = 1.5 or ni = 2). 3.1.4. Choice of the number of dead spiral turns To increase the length of a spring, it is possible to add spiral turns named “dead spirals” (nm) without changing its stiffness. 3.1.5. Winding ratio The winding ratio w (also called index of the spring) is the ratio between the average diameter of the spring and the wire diameter. DIN [27] indicates that w may vary between 20 maximum and 4 minimum: 4≤w ≤20:

ð39Þ

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3.1.6. Minimum operational length The minimum length Ln is the minimum operational length based on geometrical considerations. DIN [27] imposed to respect a minimum distance between the spiral turns named Sa as: ! D2 þ 0:1d : Sa ¼ n 0:0015 2

ð40Þ

Moreover, when the number of failed cycles is around N N 104 then Sa is multiplied by coefficient 1.5. 3.1.7. Buckling of the spring Compression springs are subject to buckling solicitation [28,29]. In a conventional way, the length of the compression spring decreases under an axial load. Under a critical length called Lk, some springs flex laterally instead of continuing to decrease in length. To avoid the buckling phenomenon, the lengths L1 and L2 of the spring should be longer than Lk. L1 NLk

ð41Þ

L2 NLk

ð42Þ

In addition, according to DIN norm [27], we have: μ þ1¼

E : 2G

ð43Þ

And if L ν 0bπ D

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μ þ 1 : μ þ2

ð44Þ

Then LK ¼ 0:

ð45Þ

Else 0

0 11 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ð μ þ 1 Þ 2μ þ 1 μD @1− 1− AA: LK ¼ L0 @1− 2μ þ 1 μ þ 2 νL0

ð46Þ

3.1.8. Maximum static stress To ensure that the spring resists in static, the stress shear τK2 must not exceed the maximal eligible stress of the spring material: τK2 bτzul :

ð47Þ

3.1.9. Resistance to fatigue There are many curves to define the fatigue life of a material [30–32]. We cite the Wohler curve, Goodman diagram and Haigh diagram. The safety fatigue factor β (Tables 3 and 4) is calculated based on the Haigh diagram. τm ¼

4DRð2L0 −L1 −L2 Þk πd3

ð48Þ

Table 3 Safety factor according to the type of material. Material

Shot peened steel

No shot peened steel

Stainless steel

β τd

1.6

2

3

350:8 d0:1769

293:5 d0:1786

303:0 d0:268

τzul

0.5(2230 − 355.94ln d)

0.5(2230 − 355.94ln d)

0.48(1918 − 255.86ln d)

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Table 4 Values of τd(N) according to the cycle numbers before failure N. τd(N)

N 7

N ≥ 10 104 b N b 107 N ≤ 104

τa ¼

τd [(τd − τzul)log N + 7τzul − 4τd]/3 τzul

4DRðL1 −L2 Þk πd3

ð49Þ

βτ zul −τdðNÞ   α F ¼ τdðNÞ   τa βτ zul −τdðNÞ þ ðβ−1Þτm τdðNÞ

ð50Þ

The choice of τd(N) is according to Table 4. 3.1.10. Constraint of contiguously spiral turn Constraint of contiguously spiral turn imposes an upper bound on the value of the theoretical stress shear adjusted for Lc which s s . The value of τkc variable according to the type of relaxation is given in Table 5. must be smaller than τkc s

τkc th b τkc

ð51Þ

Almost all the constraints that define the static physical behavior of the compression spring were included. External constraint related directly to the environment of the compression spring where it is placed is studied in the next section. The vehicle suspension system was chosen as an environment example for the compression spring. Our next objective is to determine the dynamic requirements imposed on the compression spring. The static and dynamic constraints would be coupled in order to size the compression spring in an optimal way, according to the proposed design approach based on intervals and CSP as shown in Fig. 2. However, as shown in Fig. 4 the proposed design approach provides not only a single evaluation of the system behavior but also a set of solutions which satisfies all the requirements imposed from the beginning; this is due to the fact that the design variables are expressed in intervals and verified by CSP. 4. Vehicle suspension The vehicle suspension, also called “ground contact” is the set of elements designed to absorb shocks and ensure permanent adhesion of the wheels on the ground. The spring is one of its masterpieces: it is the one that sets the frequency of oscillation of the sprung mass and the amplitude of the vertical movements. It is also an element of comfort perceived by passengers. There are various types of automotive suspension, with specific characteristics, strengths and limitations, briefly described in [25]. The steel coil spring is selected among the most common suspension configurations: MacPherson, double wishbones, and axle rigid (Fig. 5). The springs are wound in a helix diameter varying between 120 and 200 mm and a height approximately between 150 and 300 mm, the basic turns are smaller, even off center relative to the body of the spring based on the shock absorber retainers. The steel bar diameter used for the production of a compression spring can vary between 9 and 16 mm. The number of active turns (turns that compress during the use of the vehicle) of springs is between 3 and 8. The dynamic study was done to obtain the dynamic requirements of the suspension system to avoid its destruction in case of resonance. The goal is to determine certain conditions so that the behavior of the system responds well and resists in case of disturbance and excitation phenomena. Suspension systems (Fig. 5), are placed at the front and rear of motor vehicles, and are of top quality. The forms of the spring ends are calculated to control the lateral forces of the moving vehicle. The goal is to offer great comfort for passengers and good handling in all circumstances. Knowledge of the level of damping in the vehicle suspension system is important in the utilization. It avoids the production of damaging motions when the system is subjected to a disturbance phenomenon. In the design of the suspension system, knowledge of damping in constituent devices, components and support structures is important. Such knowledge serves to make design modifications in case the system failed the acceptance test. It is also useful in imposing dynamic environmental limitations to get the correct functioning. The nature and the level of component damping are studied relying on the dynamic model of the suspension system Table 5 s according to the type of relaxation chosen. Values of τkc Type of relaxation (aff)

s τkc steel × Rm

τskc stainless steel × Rm

Without relaxation (aff = 0) Slight relaxation (aff = 1) Permanent relaxation (aff = 2)

0.5 0.56 0.75

0.48 0.56 0.70

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Fig. 4. Conventional method for a system dimensioning.

model presented in Fig. 6. The dynamic analysis was made to determine the dynamic constraints [33,34] imposed on the compression spring. The dynamic and static constraints are integrated in the preliminary design of the compression spring in order to have the best functioning of the vehicle suspension. Fig. 6 presents a simple linear suspension model of a quarter vehicles in vertical mode with one degree of freedom. Indeed, the linear suspension model is made up of the chassis (m2) which is connected to the wheel (m1) by a linear spring of stiffness (k2), and in parallel with a linear viscous damper provided with a damper coefficient (η). The wheel–ground contact is modeled by a linear spring of stiffness (k1) (which represents the stiffness of the tire and the rim). Two studies were made first by imposing a snap displacement (transient) and second by imposing a profile displacement (the vibration regime) which corresponds to two modes of operation of the suspension. By applying the fundamental principle of the dynamics, the isolation of each mass gives the following equation of motion:   m2 :z˙ ˙2 þ η: z2 −z1 þ k2 :ðz2 −z1 Þ ¼ 0

ð52Þ

  m1 :z˙ ˙1 þ η: z1 −z2 þ k2 :ðz1 −z2 Þ þ k1 :ðz1 −z0 Þ ¼ 0:

ð53Þ









The Laplace transform is applied: 2

m2 :p : z2 þ η:pðz2 −z1 Þ þ k2 :ðz2 −z1 Þ ¼ 0   2 ⇔ m2 :p þ η:p þ k2 z2 −ðη:p þ k2 Þz1 ¼ 0

ð54Þ

Fig. 5. MacPherson suspension system.

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79

Fig. 6. Linear model of a quarter vehicle suspension system.

2

m1:p :z1 þ η:pðz1 −z2 Þ þ k2 :ðz1 −z2 Þ þ k1 :ðz1 −z0 Þ ¼ 0 : 2 ⇔ m1 :p þ η:p þ k1 þ k2 z1 −ðη:p þ k2 Þz2 −k1 :z0 ¼ 0

ð55Þ

We obtain the following transfer functions: z1 ¼

m2 :p2 þ η:p þ k2 z2 η:p þ k2

z2 ¼ 

ð56Þ

k1 ðη:p þ k2 Þ z :   2 0 m1 :p2 þ η:p þ k1 þ k2 m2 :p2 þ η:p þ k2 −ðη:p þ k2 Þ

ð57Þ

The global transfer function is: F t ðpÞ ¼

z2 ðη:p þ k2 Þk1 ¼ : z0 m2 :m1 :p4 þ η:ðm2 þ m1 Þ:p3 þ ½m2 ðk2 þ k1 Þ þ m1 :k2 :p2 þ η:k1 :p þ k2 :k1

ð58Þ

The harmonic transfer function is obtained simply by substituting “p” with “jω”. F t ð jωÞ ¼

k1 :k2 þ j:ω:k1 :η m2 :m1 :ω4 −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω2 þ k2 :k1 −j η:ðm2 þ m1 Þ:ω3 −η:k1 :ω

ð59Þ

The study of the stability of the suspension system is carried out to determine the roots of the denominator (poles of the transfer function). 4

2

m2 :m h 1 :ω −½m2 ðk2 þ k1 Þ þ mi1 :k2 :ω þ k2 :k1 3 − j η:ðm2 þ m1 Þ:ω −η:k1 :ω ¼ 0 4

2

⇔m2 :m1 :ω −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω þ k2 :k1 ¼ 0 3 and η:ðm2 þ m1 Þ:ω −η:k1 :ω ¼ 0

ð60Þ

So the roots of this equation are: For the real part

ω1;2;3;4

1 ¼  pffiffiffi 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi m2 :k2 þ m2 :k1 þ m1 :k2 ∓ Δ : m1 m2

ð61Þ

With 2

2

Δ ¼ m2 :ðk2 þ k1 Þ þ m1 :k2 ½m1 :k2 þ 2m2 :ðk2 −k1 Þ:

ð62Þ

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For the imaginary part pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm2 þ m1 Þ:k1 : m1 þ m2

ω5;6;7 ¼ 0; 

ð63Þ

Magnitude calculation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 u ðk1 :k2 Þ þ ðω:k1 :ηÞ j F t ð jωÞj ¼ u h i2 : u u m2 :m1 :ω4 −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω2 þ k2 :k1 u i2 t h 3 þ η:ðm2 þ m1 Þ:ω −η:k1 :ω

ð64Þ

Phase angle calculation: arg ð F t ð jωÞÞ ¼ argðk1 :k2 þ j:ω:k1 :ηÞ 4

−arg

2

m2 :m1 :ω − h ½m2 ðk2 þ k1 Þ þ m1 :k2 :ωi 3 þk2 :k1 −j η:ðm2 þ m1 Þ:ω −η:k1 :ω

0

!

ð65Þ

1

k1 :k2 B C ffiA arg ð F t ð jωÞÞ ¼ arccos @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðk1 :k2 Þ þ ðω:k1 :ηÞ 0

1

ð66Þ

m2 :m1 :ω4 −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω2 þ k2 :k1 B C −arccos @q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 A:

2 m2 :m1 :ω4 ½m2 ðk2 þ k1 Þ þ m1 :k2 :ω2 þ k2 :k1 þ η:ðm2 þ m1 Þ:ω3 −η:k1 :ω In the following part the Bode diagram is represented for the case where the variables (m2, m1, k2, k1, η) are replaced by the numerical values shown in Fig. 6. The poles of the transfer function are [−40.3013 +83.0930i, −40.3013 −83.0930i, −1.6987 +5.8851i, −1.6987 −5.8851i]. The magnitude is expressed in decibels (dB). It is calculated by multiplying the log10(magnitude) by a factor of 20. We notice in Fig. 7 that the magnitude is maximum at the angular frequency wr = 6.19 rad/s but the system is stable. The shapes of amplitude and phase curves appear to be like the response of a second order system.

Fig. 7. Bode diagram of the transfer function of the linear vehicle suspension.

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81

To sum up the dynamic constraints: assuming that xmax and xmin maximum and minimum displacements imposed in the suspension system requirement (which represents the maximum and the minimum elongation of the compression spring) and F(t) = F0sin Ωt is the Ω is the excitation frequency. Then for the proper functioning of the suspension system and to avoid sinusoidal excitation force with F e ¼ 2π that the system oscillates until destruction, it has to obey the following constraints:

xmin

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 F0 u ðk1 :k2 Þ þ ðω:k1 :ηÞ uh bxðt Þ ¼ i2 b xmax : 4 k1 u u m2 :m1 :ω −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω2 þ k2 :k1 þ u i2 th 3 η:ðm2 þ m1 Þ:ω −η:k1 :ω

ð67Þ

Also the excitation frequency of the system should be different from its natural frequency. This constraint is expressed with the following mathematical relation: h i 4 2 3 m2 :m1 :ω −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω þ k2 :k1 −j η:ðm2 þ m1 Þ:ω −η:k1 :ω ≠0:

ð68Þ

4.1. Step response The dynamic behavior of the suspension system when excited by a displacement of 20 mm on the wheel is presented in Fig. 8. The displacement z2 converges to zero, which is natural due to the role of a suspension which is to mitigate the vibrations caused by the road in a minimum time. 4.2. Harmonic response Fig. 9 shows the vehicle suspension system response when excited by a sinusoidal force (F(t) = 0.02k1. sin(10t)). We observe that the displacement z2 of the chassis is sinusoid (the motion is harmonic) and follows the trajectory road. Thus it is clear that the response of the suspension system is π/2 phase lead compared to the excitation force due to the viscous damper. This is because the velocity has a π/2 phase lead compared to displacement. We notice also that the magnitude was decreased after the first sinusoid. Fortunately this is a part of the suspension to delay and reduce the abrupt changes of the road. In this study, to determine the dynamic constraints, we used a linear dynamic model. But in the case when we deal with non-linear model, it is necessary to linearize it. However, the system is put in the form of state equation then is linearized around an equilibrium point with linearization techniques (i.e. the Jacobian matrix). 5. Constraints formalization and resolution by interval 5.1. Definition of variables in CSP Each design variable present in the nomenclature of a compression spring is considered as a variable in our CSP which involves an initial range of values. They can be found in the column “initial value” in Table 11. Some variables like Ld and F1 are expressed using long intervals with infinity as upper bound since there is no requirement for an upper bound in the imposed

Fig. 8. Linear vehicle suspension system response when excited with snap-displacement.

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Fig. 9. Linear vehicle suspension system response when excited by a sinusoidal force.

constraints in conventional norms. In order to reduce the size of the intervals and eventually the number of simulations, designers relied on experience and expertise which is not the case in our design method. In fact, even using conventional materials in the design of a spring, with the CSP approach we don't resort to expertise in designing the compression spring. In spite of using long intervals, we found about the same dimensions as in industry. Other variables like D and Vol0 are described by intervals with well defined bounds based on norm. It is useful to remind that the use of small intervals minimizes the computation time. 5.2. Implementation of technological relationships between variables by CSP A constraint is a relationship between entities that should be satisfied from certain engineering technology. Thus the technological relations of the compression spring are easily modeled as a set of constraints on the design variables. It is possible to dynamically post new constraints during the process of CSP resolution. Thus, the variable domains are progressively contracted each time a constraint is added. 5.3. Implementation of tables Table 6 represents valid combinations of material parameter study in 1 and 2. It is modeled as a table constraint which is called global constraint representing the possible combinations of values for a set of variable constraints. The term global constraint, means a constraint that should be propagated on complex data structures. Each row of a table is considered as a tuple constrained of consistent values. For example, in Table 6, if the value of G must be less than 81,500, lines 1 and 2 are automatically removed from the table due to the constraint propagation. Only line number 3 remains in the table constrained. The effect of such a constraint is propagation of an event across the table. Likewise, Table 7 is a reformulation of Table 5. It represents all possible combinations of s parameters. material parameters at the level of type of relaxation and τkc The possibility of implementing a constraint table shows the high efficiency of the design method based on interval calculation and CSP approach. Indeed, to determine the appropriate values of some design parameters (as in the case of structural design), designers use abacuses which comes from experience instead of equations and mathematical relationship. In this case thanks to our proposed design method we can implement directly these abacuses as a constraint table with no need to encode it as an algorithm to represent the possible combinations like in the case of conventional design methodologies. Also the efficiency of the proposed design can be seen when the designer looks to optimize a function of a design parameter; the use of stochastic methods requires that this function must be differentiable contrary to our design method which allows optimization of a non-differentiable function as its different values can be implemented as a table of constraints. In addition, the optimization with the use of stochastic methods is global only if the objective function is convex, which is not the case for the design method

Table 6 Constraints Table 1 — transformation Tables 2 and 3. Material

E

Coeff τzul

ρ

dmin

dmax

β

G

1 2 3

206,000 206,000 192,000

0.5 0.5 0.48

7.85 7.85 7.91

0.3 0.3 0.15

12 12 15

2 1.6 3

81,500 81,500 70,000

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83

Table 7 Constraints Table 2 — transformation of Table 5. s Type τkc

aff

Material

s Coeff τkc

1 2 3 4 5 6 7 8 9

0 1 2 0 1 2 0 1 2

1 1 1 2 2 2 3 3 3

0.5 0.56 0.75 0.5 0.56 0.75 0.48 0.56 0.70

by intervals. In fact the interval computation allows determining in a guaranteed manner the global optimum regardless of the form of the function to be optimized.

5.4. Static constraints and dynamic requirements The static and dynamic models of the compression spring described previously are implemented together in the same CSP model (Fig. 10). Fig. 10 shows an extract from the model of the dimensioning problem of a compression spring integrated in a linear vehicle suspension system. All static and dynamic constraints are written in a text file which will be inserted into the CSP software resolution “Constraint Explorer” used in this study. Compared to the previous work in [6,35], in this paper, the CSP model takes into account the buckling and the fatigue studies of the spring and the constraint of contiguously spiral turn. Also the CSP model besides the consideration of natural frequency, it considers the dynamic behavior of the suspension so the values of the design parameter of the compression spring generated by “Constraint Explorer” satisfy the condition that the movement of the suspension is between the two limit values x min and xmax. Constraint Explorer [36] is a software tool which is the major result of the CO2 project, a research project granted by the French Ministry of Research in 2002 and carried out by Dassault Aviation project leaders. It has been specified for modeling, sizing and decision making in Design. It addresses design problems that can be expressed by a set of relations on integer or real variables. These relations are either equations or inequalities; they can be linear or non-linear. Constraint Explorer supplies numerical algorithms based on solving methods supplied by the numerical constraint programming community. The solving techniques implemented in “CE” rely on a usual branch-and-prune algorithm. Pruning is provided by recent interval constraint consistency techniques (namely Hull and Box consistency [15]). Branching allows one to explore the search space by bisecting the variable domains. Generally all design variables of the compression spring are expressed by intervals as shown in Table 11. Parameters and constants are expressed with fixed values and presented in Table 8. The next step of our design method is to propagate the constraints in the intervals of the design variables to determine all the possible values that satisfied the requirements.

5.5. Numerical results of the compression spring design 5.5.1. Static aspects To show the interest of the proposed design method based on interval computation and CSP approach, we present in Table 10 a comparison between results obtained by the usual method1 (Conjugate Gradient) [26] and results generated by CSP based method for the example of compression spring. In this case only the static requirements are considered. The calculation is made with the same assumptions and initial values (Table 9) used by M. Paredes [26] in his thesis. Table 10 presents a comparison for some main design parameters of the compression spring. According to these results we notice that the values obtained after the step of constraint propagation are identical to those determined by the method of Conjugate Gradient except for the mass M and the factor of fatigue life αf of the spring. So even without applying the optimization the CSP Solver converges approximately to the results determined by the Conjugate Gradient method. In column 4 of Table 10 we find the values of the spring parameters obtained in the case where its mass is optimized. Thus, compared to the results of M. Paredes, we have better results for most of the spring design parameters and especially for its mass, which confirms the efficiency of CSP based design method in optimization. In the next section, the total design model of the spring with all the static and dynamic constraints is implemented in the Constraint Explorer solver.

1

Results recovered from the thesis of M. Paredes.

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Fig. 10. Extract from CSP model of the static and dynamic sizing problem of a compression spring of a linear vehicle suspension.

5.5.2. Static and dynamic aspects In this part all static and dynamic requirements are considered in the CSP model. Indeed, the dynamic requirements studied in Section 4 and defined by Eqs. (67) and (68) are incorporated into the CSP resolution model as the following algebraic equations: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 F0 u ðk1 :k2 Þ þ ðω:k1 :ηÞ uh xðt Þ ¼ i2 bxmax k1 u u m2 :m1 :ω4 −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω2 þ k2 :k1 þ u i2 th 3 η:ðm2 þ m1 Þ:ω −η:k1 :ω

ð69Þ

and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 F0 u ðk1 :k2 Þ þ ðω:k1 :ηÞ uh xðt Þ ¼ i2 Nxmin : 4 k1 u u m2 :m1 :ω −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω2 þ k2 :k1 þ u i2 th 3 η:ðm2 þ m1 Þ:ω −η:k1 :ω

ð70Þ

The constraint “different” (≠) in Eq. (68) cannot be integrated into a CSP solver because it means nothing. Indeed two disjoint intervals can always be different and in two intervals with a common intersection nothing can spread. Thus, Eq. (68) is rewritten as follows:     4 2 m2 :m1 :ω −½m2 ðk2 þ k1 Þ þ m1 :k2 :ω þ k2 :k1  Nε

ð71Þ

Table 8 Constants. ε m2 η Ωe

0.01 500 7000 10

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85

Table 9 Initial values of design parameters of compression spring. Variables

Initial values

Material G Rm D De d n ni nm m L1 L2 N ν aff

1 (No shot peened steel) 81,500 1839 30 30 3 10 3 0 9.1 90 80 105 1 0

and     3 η:ðm2 þ m1 Þ:ω −η:k1 :ωNε

ð72Þ

with ε = 0.0000001 very small. The calculation process converges to results presented in Table 11 (column 3) from the first evaluation. Computation time takes 2 min and 30 s on a regular computing machine (IntelCore i5-3317U [email protected]). Table 11 (column 3) represents the generated results with CSP due to constraint propagation. According to these results we notice s a drastic reduction in certain intervals. The solution for some parameters is intervals (example: τkc , Fn, ω, …) and for other parameters it is just one value (example: Rm, D, m, …). The general dimensions of the compression spring obtained by the CSP are: The mean diameter D is 50 mm and accordingly the wire diameter is about 12 mm then the external diameter of the spring is about 62 mm. The total length of the spring calculated by the CSP approach is 200 mm which satisfies the condition that the spring must be longer than the critical length of buckling Lk which is between 160.957 mm. These measures confirm the harmonization between imposed constraints. The total number of the spiral turns is equal to 12. The extremities of the spring end are grinded as ni = 2. The axial pitch of the spring m is 17.60 mm and the winding angle is about 0.111≗. The mass of the spring is about 1.678 kg. According to the suspension springs which are marketed, such a weight of the spring is acceptable. The spring stiffness is 168,000 N/m. Compared to the value obtained in the static dimensioning (Table 10) we remark that this value is increased from 3060 N/m to 168,000 N/m. This increase is due to the coupling of consideration between static and dynamic requirements in a global manner. Indeed, in the process of resolution, the constraints of resonance, displacement limitation (xmax and xmin) and the resistance to fatigue are involved with the static constraints in the dimensioning of the spring. Hence the need for a higher value for more rigidity of the suspension system. This shows the interest of the proposed design method which allows generating the dimensions of the system to design from the first simulation and which satisfy at the same time the static and dynamic requirements. The natural frequency of the spring is 173,856.88 Hz and since τk2 = 11.381 N/mm2 which is by far lower compared to τzul = 673.024 N/mm2 we conclude that the compression spring has good resistance in static. In fatigue, the number of cycles before ruptures equal 100,000. We also notice that the DH Steel material is the only one which can satisfy the imposed requirement.

Table 10 Comparison between optimization with Conjugate Gradient method and CSP based design method. Variables 2

E (N/mm ) d (mm) Ln(mm) R(N/m) Lk(mm) M(g) αf

Conjugate Gradient

CSP

CSP optimization

206,000 3 50.25 3060 69.63 62.76 4.68

206,000 3 50.25 3060 69.63 63 5.02

206,000 2.86 48.6 2540 69 57.4 5.43

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H. Trabelsi et al. / Mechanism and Machine Theory 84 (2015) 67–89 Table 11 Numerical results of the compression spring dimensions obtained by CSP. The material parameters Variables

Initial values

Results

Material G E Rm ρ Coeff τzul τzul μ

{1, 2, 3} {70,000, 81,500} {192,000, 206,000} [0; 2230] {7.85, 7.91} {0.48, 0.5} [0, 10,000] {0.27, 0.30, 0.31}

1 81,500 206,000 1346.049 7.85 0.5 673.024 0.27

Principal constructive parameters of the spring d dmin dmax D De Di L0 Lc R n ni nm m z

[0.15; 15] {0.15, 0.3} {12, 15} [0; 200] [0; 217] [0; 200] [0; 630] [0; 630] [0; 200,000] [2; 2e9] {1, 2, 3, 4} [0; 2e9] [0, 315] [−π/2, π /2]

11.982 0.3 12 50 61.982 38.017 200 143.786 168,000 10 2 0 17.603 0.111

Secondary constructive parameters of the spring Ld Vol0 nt w fe M

[0; +∞] [0; 24,000] [4; 2e9] [4; 20] [0; +∞] [0; +∞]

1894.788 603.468 12 4.172 173,856.88 1.677243

[0; +∞] [0; +∞] [0; 630] [0; 630] [0; 630] [0; +∞] [0; 24,000]

[3.971; 3.975] [5632.342; 5632.347] 199.976 166.474 33.502 [94,414.468; 94,414.474] 502.309

[0; 630] [0; +∞] [1; 2] [0; +∞] {0, 1, 2} [0; 2230] [0; +∞] {1, 2, 3, 4, 5, 6, 7, 8, 9} {0.48, 0.5, 0.56, 0.70, 0.75}

160.351 [5704.612; 5704.619] 1.365 954.196 0 [954.206; 2230] 11.381 1 0.5

[0; 2e9] {1.6, 2.0, 3.0} [0; +∞] [0; 630] [0; +∞] [0; +∞] [0; +∞] [0; +∞] [0; +∞] [1; +∞]

100,000 2 22.667 166.454 [5635.618; 5635.623] 188.356 511.468 [288.349; 288.350] 284.343 1.114

Functional parameters of the spring F1 F2 L1 L2 Sh W Vol2 Performance parameters (static rupture) Lr Fcth k τkcth aff s τkc τk2 s Type τkc s Coeffτkc Performance parameters (fatigue rupture) N β Sa Ln Fn τd τd(N) τm τa αF

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87

Table 11 (continued) The material parameters Variables

Initial values

Results

Performance parameters (buckling) ν Lk

{0.5, 0.7, 1, 2} [0; 630]

1 160.957

Dynamic study ω F0 x0

[0; +∞] [0; +∞] [0; +∞]

[7.039; 93.857] 4000 0.029

This material has a density ρ equal to 7.85 kg/dm3, a Young's modulus E equal to 206,000 N/mm2 and a shear modulus G equal to 81,500 N/mm2. Dynamically and based on the design parameter values of the suspension system, the angular frequency of excitation must be different from its natural angular frequency ω = [7.039; 93.857] Hz. The interval limits of the natural angular frequency are obtained according to the values of the design parameters which some of them are expressed by intervals (m1 = 25, m2 = [200; 500], k1 = 200,000, k2 = [19,000; 21,000] and η = [0; 2000]). According to the type of the imposed excitation force, the displacement stroke of the compression spring is about 0.029 m. With the numeric CSP the calculation is made by intervals which explain the accuracy of some obtained values. Moreover, with the CSP we can couple more than one analysis to design a system which provides an exact decision. Let's consider a mechanical system and the dynamic analysis as an example. This analysis takes into account the kinematic constraint relationships and fundamental principle of dynamics to formulate the motion of the mechanical components of the vehicle suspension system. Hence this analysis can determine whether the dynamic behavior of the system is satisfactory relative to some dynamic performance requirements. However, the dynamic analysis does not evaluate stress concentration factors, deformations, and the fatigue life of the specific mechanical component, because the quantities cannot be determined from the dynamic analysis. Therefore thanks to CSP we can combine more than one analysis in one global model to the dynamic analysis, stress, deformation, and fatigue life. The coupling between these studies allows the right choice of static dimensions and safety coefficients. The advantages of the proposed design method are many but it's still applicable to the case of linear or nonlinear-linearized systems defined by algebraic equations. This is due to the weakness of the CSP tool in solving differential equations. In addition to convergence to an acceptable solution that satisfies all requirements imposed, the proposed design method requires a study at solving strategies that allow ordering the variables in the resolution.

6. Conclusion In this paper a study of the numeric CSP application in the case of a compression spring was made. The benefits of a CSP over conventional approaches for the optimal design of mechanical components and systems are pointed as follows: - The approach is declarative and does not require the simplification of analytical models; - It allows the verification of the validity of an existing design as well as the generation of one or more feasible solutions and optimal ones in the sense of a given criterion; - It is non-causal and thus allows a wide variety of specifications for a model; - The ability to integrate compatible tables of values and alternative design; - The ability to treat a wide variety of numerical constraints (arithmetic, quadratic, trigonometric, …); - The principle of contraction ensures that all feasible solutions of a problem are necessarily inside the block results; - It is possible to operate in an approach based on a set-based design by adding and/or removing constraints under study; - When an optimum is found, it is a global one, whatever the function to be optimized; - The computation time is satisfactory: in this study the optimization results are few minutes on a computer type standard PC. The objectives and the advantages of the proposed design compared to the conventional design are verified. The method was applied for the design of one component of a linear system. Prospects in this work are numerous. Indeed, we believe that many components and complex systems can be modeled using numerical CSPs and we are developing this idea. Future research should be focused on the validation of the capability of the proposed design based on CSP and intervals to size a full non linear system (structure + all components) and the integration of other types of dynamic requirements (controllability, stability, observability …) defined on the basis of differential equations.

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Appendix A

Variables

Units

Nomenclature

D De Di d m L0 L1 L2 Lc Ld Lk Ln Lr Sa Sh Vol0 Vol2 F1 F2 Fcth Fn E G Rm τk2 τkcth τzul τm τa s τkc R W fe M ρ z N n ni nm nt z w αf μ ν β

mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm cm3 cm3 N N N N N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/m N.mm Hz g kg/dm3 rad – – – – – – – – – – –

Mean diameter of spiral turns External diameter of spiral turns Internal diameter of spiral turns Wire diameter Axial pitch of the spring Free length Spring length in charge for force F1 Spring length in charge for force F2 Block length Developed length Critical length of buckling Shorter length of eligible work (geometrically) Shorter length of eligible work (maximum stress) Sum of minimum space between the active spiral turns Course Volume envelope for L0 Volume envelope for L2 Spring force for the length L1 Spring force for the length L2 Theoretical force of the spring for the length Lc Spring force for the length Ln Modulus of the material elasticity Shear modulus Minimum value of the tensile strength Stress shear adjusted for L2 Theoretical stress shear adjusted Lc Maximal eligible stress Mean stress Alternating stress s Maximal value authorized by τkc th Spring stiffness Work of the spring Natural frequency of the spring Mass of the spring Density Winding angle Number of cycles before rupture Number of active spiral turns Number of spiral turns for the extremities Number of dead spiral turns Total number of spiral turns Winding angle Winding ratio Factor of fatigue life Poisson coefficient End fixation constant Safety factor in fatigue

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