![354 21stInternationalConferenceon Computersand IndustrialEngineering
above lines. Figure 1 shows the spring at various load conditions. To formulate this problem, following
additional notations, expressions and data are assumed based on [1, 2]:
fo •L! -L.
e
/ I I Installed
,, ,, L, ~- -----rrop,,,t,, ~ T
I I I I I I length, Sollid
i, ,i ,i L. 17,th.
ip w n
adjacentcoila)
Figure 1" Compression Spring at Various Load Conditions ( Adapted from Mott, R.L.)
Number of inactive coils, (Q) = 2; Pitch (p) = (Lf- 2d)/N; Shear modulus (G) = (1.15E+07) lb/in2;
Weight density of spring material ('1,) = 0.285 lb/in3 ; Gravitational constant (g) = 386 in/sec 2
Mass density of material (p = T/g); and p = (7.383 42E-04) lb-sec2/in4
Frequency of surge waves ta - 2riD 2N
Pitch Angle ~.= tan-I [__~__] (2)
Coil Clearance cc = (Lo-Ls)/N (3)
Spring Index C = D/d (4)
Load deflection equation P = K~ (5)
d4G
Spring Constant K = - - (6)
8DSN
8kPD
Shear stress z = ~ (7)
nd 3
Wahl stress concentration factor k= (4 D - d) + 0.615d (8)
(4D - 4d) D
Mass of the spring M=I / 4(N + Q)n 2Dd~p (9)
Mathematical Model for Optimization
For a spring under tension or compression, the wire experiences twisting which causes the shear stress.
This is one of the most critical factors in spring design. The cost of the spring is another important
factor and may be considered as directly related to the mass of the spring. Objective of the design
problem is to minimize mass and shear stress while satisfying other constraints. A mathematical model
for spring design optimization problem will be as follows:
Minimize Z:
Z=WI(1/4(N+Q)~2Dd2p}+W2( --~5-~8PD.(-~ --4"d )(4D - d) ÷ 0.615dD ) ]" (10)
Subject to:
8PDSN _ ~ (11)
d4G
d "12~ > (12)
2~D2N - 6J oI - r
D + (ll/10)d <- D (13)
tan-,[n__.~._]< ~0 (14)](https://image.slidesharecdn.com/amodelingapproachforintegratingdurabilityengineeringandrobustnessinproductdesign-150527162448-lva1-app6892/85/A-modeling-approach-for-integrating-durability-engineering-and-robustness-in-product-design-2-320.jpg)
![356 21st International Conference on Computers and Industrial Engineering
In previous formulations we have put an equality constraint on the deflection of spring to ensure that
final solution provides us target deflection at the mean value of decision parameters. However, in reality
our objective is to ensure functionality and not the target value. Taguchi has recommended the concept
of loss function. According to this concept we have no loss if outcome is at the target value. But as it
deviates from the target, the loss is depicted by a quadratic function [3]. We are recommending a loss
function represented by a step function instead of a continuos function. Advantage of step function is
that calculation of loss as in our case of spring deflection becomes very simple. Further, this will be
more representative of real life situations in many cases as there is always a limit on sensitivity of the
customers or of the system for which our output is an input. Spring design problem was solved by
applying the concept of step loss function. We assumed that the loss for 0.45 to 0.60 inch of
deflection as 0 units; for 0.40 to 0.45 and 0.60 and 0.70 inch of deflection a loss of 1 units and for
less than 0.40 and greater than 0.70 inch of deflection a loss of 2 units. Objective function for this
model will be:
Z = Wl{Quality Loss} + W2{ 1/TD}+W3{1/Td} (26)
The solution # 2 in Table 1 is for this model and it provides a good robust design with minimum quality
loss and best possible combination of tolerances selected. However, this solution fails to consider the
aspect of durability which depends upon the degradation. As product becomes older, its performance
degrades. To represent this real life scenario we should integrate the information on degradation of
design parameters with the changing customer expectations to calculate loss in quality. We calculated
loss in quality at the end of warranty period (3 Yrs) and also at the end of designed life (10 Yrs) of the
spring assuming degradation in spring wire and mean coil diameter. With usage and environmental
effects spring wire diameter will reduce at certain critical sections, and mean coil diameter will increase.
We are assuming an empirical relationship( C = A tB ) for degradation of D and d.. Here C is the
amount of degradation in t time units while A and B are empirically determined parameters [4]. The
specific relationship we have used are: d(t) = d - 0.0009t°65and D(t) = D + 0.001t°'65
Solution # 3, in Table 1 provides the results for this model. This solution considers degradation, and
interesting point to note is the value of deflection, which is towards lower side as we understand that
spring will provide more deflection with usage. It will be better to have little less deflection in the
beginning though it may mean more rejection at the manufacturing stage as tolerance is tight towards
lower side, but over the complete life of spring it will be a better choice.
Table 1: Summar of All the Solutions
Solution D d N TD Td Mass Deflec Stress Reliab
# inch inch # millie millie lb. inch Kpsi. %
1 0.513 0.069 9 6 2 0.050 0.500 62.47 95.00
2 0.516 0.076 13 5 1 0.081 0.523 49.21 99.90
3 0.513 0.078 13 4 1 0.085 0.463 45.57 95.00
Discussion
By comparing the models developed and solved in this example, many useful and interesting facts
comes out. First, model brings out the advantage of considering manufacturing options in terms of
design parameter tolerances. Second, model shows how concept of loss function should be integrated
in the design optimization model. Finally, model provides a Decision Support System for designer to
integrate the aspects of robust, and durable product design. Results of these models will change if we
select different weight factors. However, these models provide a decision support to the designer who
may try different combinations before making final choice.
References
1. Mott, R.
2.
3.
4.
L., 'Machine Elements in Mechanical Design', 2~d ed. Maxwell Macmillan
International,1992
Arora J. S, Introduction to Optimum Design., New York: McGraw-Hill, 1989
Taguchi, G. Introduction to Quality Engineering, Asian Productivity Organization, 1990
Albrecht P. , Naeemi A. H., Performance of Weathering Steel in Bridges, National Cooperative
Highway Research Program report 272, p. 65, July 1984](https://image.slidesharecdn.com/amodelingapproachforintegratingdurabilityengineeringandrobustnessinproductdesign-150527162448-lva1-app6892/85/A-modeling-approach-for-integrating-durability-engineering-and-robustness-in-product-design-4-320.jpg)
A modeling approach for integrating durability engineering and robustness in product design
- 1. Pergamon Computers ind. Engng Vol. 33, Nos 1-2, pp. 353-356, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-8352197 $17.00 + 0.00 PII: S0360-8352(97)00110-1 A Modeling Approach for Integrating Durability Engineering and Robustness in Product Design Parveen S. Goel and Nanua Singh Department of Industrial and Manufacturing Engineering Wayne State University, Detroit, MI 48202 Abstract This paper presents a set of multi-objective non-linear programming models to integrate durability and robustness in product design. A spring design problem is used for illustration. © 1997 Elsevier Science Ltd Key Words Design, Robustness, Durability and Optimization Introduction. After unit cost, quality and lead time, durability is the fourth and perhaps the last frontier in the battle for increased market share and profitability of organizations. Designers need some aids to understand the behavior of factors effecting robustness and durability at early stage of design. We present a set of mathematical models to integrate robustness and durability engineering at the design stage in this paper. The models can be used as a Decision Support System (DSS) by design engineers to make more informed decisions engineer when selecting a particular design or making a choice among manufacturing alternatives.. The basic strength of the models lies in mathematically capturing design and manufacturing options, degradation, variability and target of changing functional requirements simultaneously. The approach is demonstrated by an example of compression spring design. Integration of Robustness and Durability Integration of robustness and durability aspects is one of the most desired activity in product design. Many attempts have been made to increase the robustness of products, but in most of the cases, the issue of durability is left out for verification and testing stage of product development. Information on degradation pattern of many functions and product is available up front or otherwise it may be generated with the help of accelerated experiments. The knowledge of degradation pattern adds a very important dimension to product design. For example, if we have nominal the best type of quality characteristic and based on degradation pattern we know that its value with usage is going to increase or decrease, we may mathematicallydecide what the best initial target should be for this. This initial target may be quite different than the nominal value of the characteristic, but, overall, it will provide more robust and durable product. Example of Sprin2 Desien To demonstrate this approach, we will design a compression spring of round music wire with square and ground ends. The three design variables for the problem are: wire diameter(d), mean coil diameter(D) and number of active coils(N). It is planned to use this spring in an application where it should have a free length (Lf) of 1.75 inches with a deflection (5) of 0.50 inches under the operating load (P) of 14.0 lb. Design stress (~d)should be less than 130 Kpsi, and maximum allowable stress (xa)at the solid length of spring should be less than 150 Kpsi. Spring will be installed in a hole of 0.60 in. diameter (D) and we require the frequency of surge waves (a)) for the spring should be at least 100 Hertz(t~). It is also desired that proper coil clearance and pitch angle ~ is provided. Spring should be designed such that it has minimum mass and shear stress at operating load while satisfying all other requirement stated in 353
- 2. 354 21stInternationalConferenceon Computersand IndustrialEngineering above lines. Figure 1 shows the spring at various load conditions. To formulate this problem, following additional notations, expressions and data are assumed based on [1, 2]: fo •L! -L. e / I I Installed ,, ,, L, ~- -----rrop,,,t,, ~ T I I I I I I length, Sollid i, ,i ,i L. 17,th. ip w n adjacentcoila) Figure 1" Compression Spring at Various Load Conditions ( Adapted from Mott, R.L.) Number of inactive coils, (Q) = 2; Pitch (p) = (Lf- 2d)/N; Shear modulus (G) = (1.15E+07) lb/in2; Weight density of spring material ('1,) = 0.285 lb/in3 ; Gravitational constant (g) = 386 in/sec 2 Mass density of material (p = T/g); and p = (7.383 42E-04) lb-sec2/in4 Frequency of surge waves ta - 2riD 2N Pitch Angle ~.= tan-I [__~__] (2) Coil Clearance cc = (Lo-Ls)/N (3) Spring Index C = D/d (4) Load deflection equation P = K~ (5) d4G Spring Constant K = - - (6) 8DSN 8kPD Shear stress z = ~ (7) nd 3 Wahl stress concentration factor k= (4 D - d) + 0.615d (8) (4D - 4d) D Mass of the spring M=I / 4(N + Q)n 2Dd~p (9) Mathematical Model for Optimization For a spring under tension or compression, the wire experiences twisting which causes the shear stress. This is one of the most critical factors in spring design. The cost of the spring is another important factor and may be considered as directly related to the mass of the spring. Objective of the design problem is to minimize mass and shear stress while satisfying other constraints. A mathematical model for spring design optimization problem will be as follows: Minimize Z: Z=WI(1/4(N+Q)~2Dd2p}+W2( --~5-~8PD.(-~ --4"d )(4D - d) ÷ 0.615dD ) ]" (10) Subject to: 8PDSN _ ~ (11) d4G d "12~ > (12) 2~D2N - 6J oI - r D + (ll/10)d <- D (13) tan-,[n__.~._]< ~0 (14)
- 3. 21st International Conference on Computers and Industrial Engineering (Lo-Ls)/N > d/10 Did > C 8PD((4D-d) + 0.615d )< '~d /td 3 (4D - 4d) D dG(L/ - dN) ( (4D - d) 0.615d r~D2N (4D - 4d) + --D~ 1/4(N +Q)g2Dd2p <.085 where: N is an integer and d, D and N > 0 (15) (16) (17) ) -< Xa (18) (19) 355 Integration of Robustness & Durability In design of any component it is necessary to assign tolerances to all the dimensions and consider variability of all the inputs and outputs. The combination of all the tolerances with variability of inputs should guarantee, at least from a statistical basis, that the system will perform as expected. Usually there is a conflict in design and manufacturing engineers' interest for tolerances. This conflict of interest can be resolved if design engineer consider tolerances as a decision parameter in overall design optimization problem. We need to understand how these tolerances effect the output of the system and tolerances which has significant impact may be kept tighter while other ones may be kept loose. In our example of spring design we will consider tolerances of mean coil diameter (D), spring wire diameter (d), and the variability of applied load as uncontrollable or noise. Desired outcome of the spring is deflection at the given load condition. We are considering; tolerances of mean coil diameter (D) = .004, .005 and .006 in.; tolerances of spring wire diameter (d) = .001, .002 and .003 in.; variability in load (uncontrollable) = 14.00 + 0.25 and variability in deflection (acceptable range) = 0.5_o~÷o5 We know the relationship of deflection with mean coil diameter (D), spring wire diameter (d) and applied load (P), and will use Taylor's series expansion to fred out variability of deflection. (G ~ ), as function of D, d and P and their variances. Using this relationship we can fred out the probability of getting deflection with in acceptable range for a given value of D, d and their tolerances. We are assuming that tolerances are equal to the standard deviation. Therefore standard deviations of D, d and P (Go,G d and Ge), can be estimated using the respective tolerance. To calculate the out of specification product or probability of getting acceptable deflection, we need to fred the value of standard normal variate Z for upper and lower limit of deflection using the following relationship: Z = x - ~t (20); R = q~(Zt: ) --q~(ZL) (21) (Y Here tp(Z) represent the area under standard normal probability distribution up to the point Z. In our case kt = 0.5 in. and x= 0.4 and 0.7 in., for ZL and Zu respectively. We need to add terms for the cost associated with tolerances in the objective function also extra constraints will be added to ensure minimum acceptable probability of getting deflection with in acceptable range. New optimization problem will have five decision variables D, d, P, To (Tolerance for D) and Td ( Tolerance for d). It is assumed that, (1000*TD) and (1000*Td) are integer. Equation (22) represent the objective function for this optimization problem. Equations (23) & (24) ensures proper selection of tolerances and equation (25) ensures that more than 95% of springs provide deflection within specified limits. 8PD ( (4D-d)Z=W1{I/4(N +Q)x2Od:p }+W2{ nd 3 x + 7-ff--47) 6 > (1000*TD) >4 3 _>(1000*Td) >1 R > 0.95 0.615d ) }+Wa{1/TD}+W4{1/Td} (22) D (23) (24) (25) New objective function represented by equation (22) and constraints are reflected in equations (11) to (19) and (23) to (25). As we do not have any information about preferences we have selected weights for objective function such that each term is of the same magnitude. The solution # 1 in Table 1,is for this model and it provides satisfactory mass and stress condition while ensuring that 95% of springs provide deflection with in acceptable range.
- 4. 356 21st International Conference on Computers and Industrial Engineering In previous formulations we have put an equality constraint on the deflection of spring to ensure that final solution provides us target deflection at the mean value of decision parameters. However, in reality our objective is to ensure functionality and not the target value. Taguchi has recommended the concept of loss function. According to this concept we have no loss if outcome is at the target value. But as it deviates from the target, the loss is depicted by a quadratic function [3]. We are recommending a loss function represented by a step function instead of a continuos function. Advantage of step function is that calculation of loss as in our case of spring deflection becomes very simple. Further, this will be more representative of real life situations in many cases as there is always a limit on sensitivity of the customers or of the system for which our output is an input. Spring design problem was solved by applying the concept of step loss function. We assumed that the loss for 0.45 to 0.60 inch of deflection as 0 units; for 0.40 to 0.45 and 0.60 and 0.70 inch of deflection a loss of 1 units and for less than 0.40 and greater than 0.70 inch of deflection a loss of 2 units. Objective function for this model will be: Z = Wl{Quality Loss} + W2{ 1/TD}+W3{1/Td} (26) The solution # 2 in Table 1 is for this model and it provides a good robust design with minimum quality loss and best possible combination of tolerances selected. However, this solution fails to consider the aspect of durability which depends upon the degradation. As product becomes older, its performance degrades. To represent this real life scenario we should integrate the information on degradation of design parameters with the changing customer expectations to calculate loss in quality. We calculated loss in quality at the end of warranty period (3 Yrs) and also at the end of designed life (10 Yrs) of the spring assuming degradation in spring wire and mean coil diameter. With usage and environmental effects spring wire diameter will reduce at certain critical sections, and mean coil diameter will increase. We are assuming an empirical relationship( C = A tB ) for degradation of D and d.. Here C is the amount of degradation in t time units while A and B are empirically determined parameters [4]. The specific relationship we have used are: d(t) = d - 0.0009t°65and D(t) = D + 0.001t°'65 Solution # 3, in Table 1 provides the results for this model. This solution considers degradation, and interesting point to note is the value of deflection, which is towards lower side as we understand that spring will provide more deflection with usage. It will be better to have little less deflection in the beginning though it may mean more rejection at the manufacturing stage as tolerance is tight towards lower side, but over the complete life of spring it will be a better choice. Table 1: Summar of All the Solutions Solution D d N TD Td Mass Deflec Stress Reliab # inch inch # millie millie lb. inch Kpsi. % 1 0.513 0.069 9 6 2 0.050 0.500 62.47 95.00 2 0.516 0.076 13 5 1 0.081 0.523 49.21 99.90 3 0.513 0.078 13 4 1 0.085 0.463 45.57 95.00 Discussion By comparing the models developed and solved in this example, many useful and interesting facts comes out. First, model brings out the advantage of considering manufacturing options in terms of design parameter tolerances. Second, model shows how concept of loss function should be integrated in the design optimization model. Finally, model provides a Decision Support System for designer to integrate the aspects of robust, and durable product design. Results of these models will change if we select different weight factors. However, these models provide a decision support to the designer who may try different combinations before making final choice. References 1. Mott, R. 2. 3. 4. L., 'Machine Elements in Mechanical Design', 2~d ed. Maxwell Macmillan International,1992 Arora J. S, Introduction to Optimum Design., New York: McGraw-Hill, 1989 Taguchi, G. Introduction to Quality Engineering, Asian Productivity Organization, 1990 Albrecht P. , Naeemi A. H., Performance of Weathering Steel in Bridges, National Cooperative Highway Research Program report 272, p. 65, July 1984