ABSTRACT

All engineering problems are dependent on multiple design variables and the solutions for these needs to satisfy different responses. Compromises in deciding values for these design variables need to be made so that all the functional requirements are met with in the budgeted cost and time. To arrive at the values for these design variables, there is a need to understand the main and interaction effects of all these input parameters on the response variables and hence, necessitates conducting scores to hundreds of experiments. Performing these numbers of experiments may not be feasible every time given the budget and cost constraints.

Methods like DoE based on statistical basis are found useful in such scenario where the effects of all the input parameters are modeled with a minimum number of experiments. Application of mathematical tools to engineering problems poses several challenges in arriving feasible solutions and DoE is no different. This article describes the challenges that are faced in applying DoE to automotive engineering problems. A six-sigma based methodology thus arrived based on how these challenges were overcome with two case studies is presented in this article.

Key Words: Design of Experiments, Response Variable, Automotive Seating System

1. Introduction

The crucial factors that govern engineering and design of automotive systems are safety and cost. Several specifications, Legal and Customer specified, that guide the design of the systems from package and strength points of view need to be satisfied in designing these systems. Automotive Seating Systems also play a major role in providing safety to the passengers apart from satisfying other important requirements like comfort. Crashworthiness requirements can be met only through proper balancing and/or optimization of several design variables like material and sectional properties of different components. Thus, the influence of many variables and the need for satisfying many requirements makes the optimization process more difficult. Trial and error methods are very much in vogue in such design problems and these methods seldom hit targeted responses.

Of late, methods of designing the experiments based on statistical techniques are developed and found to be useful in undestanding the effects of different variables on the output. Design of Experiments is one such tool through which one can correlate between the design parameters and the responses with a minimum number of experiments done in a structured order. Tools like Regression and Optimization aid the engineer in arriving at the correlations and deciding the design parameters. These techniques can reduce thus the Product Design Cycle time and also enhance the Product quality due to the scientific basis of the tools.

Although, the mathematical tools help in designing, the challenge lies in their application to engineering problems. A systematic methodology to be formulated for every typical engineering problem in transforming this real life problem to a mathematical one without affecting the underlying physics and also in a form where the other tools can be applied. Such a methodology, which outlines the different phases, involved in transformation of automotive engineering problems to mathematical problems where six sigma tools like DoE, Regression and Optimization can be applied is described here.

2. Methodology

A four-phased approach to solving the Automotive Engineering problems using different Six Sigma Tools has been proposed as in Fig. 1. The four phases are 1) Problem Definition 2) Problem Formulation 3) Six Sigma and 4) Engineering Judgement and Conclusion.

In the Problem Definition phase, the Engineering Problem to be solved should be defined elaborately with the outcome expected and the constraints. Also, to be identified at this stage are the variables that influence the outcome as understood at that stage and feasibility of their modification. The real task involved in the whole process is in the Problem Formulation stage where the identified design variables need to be normalized or combined so as to give meaningful and practical regression equations formulated in the Six-Sigma stage. Further to this, the design parameters, which do not have a real value i.e. those, belong to attribute type need to be properly translated into quantifiable parameters. Similar to these, all the responses, both quantitative and qualitative need to be quantified using proper combination of the design variables.

Finally, all the variable definitions including the input and response variables, the ranges for the input variables and optimization criteria for all the response variables should be clearly identified in this phase. The Six-Sigma phase involves in deciding the right set of experiments to be carried out, fitting regression equations and solving them for the optimization. This phase also involves in giving a lot of insight in finding the main effects of the design variables on the response variables and also the interaction effects among different design variables through different plots. Different charts depicting the degree of influence of each of the design variables on the response variables can be deduced through these tools. Any ambiguity that occurred during the Problem definition phase in deciding the design variables can be exploded in this stage.

Hence, it is best in the initial phase to include as many design variables as possible so that none of the important design variables can be ignored and at the same time those which have negligible influence on the response variables can be eliminated in this phase. Feasibility of using the values arrived for the design variables in the third phase are evaluated in the final phase called as the Engineering Judgement and Conclusion phase. Wherever, unrealistic values are arrived, closer realistic values are chosen and these are fed back into the regression equations to obtain the real responses. This whole exercise is concluded once the responses are found satisfactory and a final experimental run is conducted to validate the choice of the design variables.

To reinforce the basis for the methodology, two instances of successful application of six sigma tools to automotive engineering problems are described in ensuing sections. The tool used for deciding in the set of experiments, fitting the regression equations and optimizing is MINITAB [1].

3. Case studies

3.1 Optimization of Lift Mechanism

3.1.1 Problem Definition

Figures 2a, 2b and 2c show three configurations a typical front seat needs to provide for an occupant. The three configurations shown are needed to provide different sized drivers of a vehicle the required visibility and the comfort. This requirement is normally stated as the required “H-point” travel of the seat in Z and X co-ordinate axes as shown in Fig. 3. H-Point is a well-accepted reference point in automotive design and can be defined as the center of the joint line between the upper and lower torso of an occupant of the seat. In this particular problem, the H-Point travel in Z-axis is targeted as 30mm up and down at the same time the travel in X-axes is as 30mm forward and rearward respectively.

Figure 2(a), 2(b) & 2(c). The Design, the Low and the High Configurations of the seat respectively

3.1.2 Problem Formulation

The mechanism to be designed is considered as a wire-frame model with constraints shown (Fig. 4) and simulated using I-DEAS [2] software of EDS-PLM solutions. These constraints are arrived from the requirements of the problem like the H-point in the Design configuration should stay in the same position for all possible link lengths. To achieve the exact H-point travel in Z-axes, it was decided to vary the other parameters of the mechanism over a practical range. Even though in the present problem, the length of the ground link, L4 is fixed, it has been varied along with the A, the angle of the front link in Design configuration for getting the required travel. Thus, four response variables are defined that are to be calculated from the simulation experimentation viz. Link Length Ratio ?, the ratio of the length of the Ground Link to that of the Intermediate Link, H-X, the H-point travel in X-direction when the mechanism is in High configuration, H-L, the H-point travel in X-direction when the mechanism is in Low configuration and angle A. From simple reasoning it can be deduced that for an efficient design angle A should be approximately equal to 45 and ? should be approximately equal to unity. Based on these conditions, the problem is stated mathematically as given in Table. 1.

Figure 3. Travel of H-Point in Design, Low and High Configurations

Figure 4. Constrained Wire-frame used in the study

Table 1. Variable Definitions for Lift Mechanism Problem

Input Variables

Front Link Length, L1

Rear Link Length, L3

Inter-mediate Link Length, L2

Range to be included

Response Variables

H-Point travel along negative X-axis, H-X

H-Point travel along positive X-axis, H+X

Traverse angle, A

Link Length Ration, ?

NA

Optimization Criteria

1. Maximize H-X

2. Maximize H+X

3. Target A ~ 45ï¿½

4. Target ? ?1

NA

3.1.3 Six-Sigma

Three six-sigma tools – Design of Experiments, Regression and Optimization have been used in the present problem. The inputs with their possible ranges in levels of 3 has been considered and been fed into the six sigma tool so that a set of full factorial experiments to be conducted are designed. For these combinations of the input variables, the responses H-X, H+X and A are recorded and fed back into the tool to understand the effects of different variables on the responses and also among themselves. A close inspection of the main effects plots as in Fig 5 show that the relationships between different input variables and responses are linear. This suggests that the use of linear regression fits between the extremes will not induce any inaccuracies.

Figure 5. Main Effects Plots of the Input Variables and Responses

Using these recorded outputs, regression equations are formed as below for the three response variables.

H+X = 51.5 + 0.1*L1 + 0.4*L2 – 0.4*L3

H-X = -1.1 + 0.004*L1 – 0.2*L2 + 0.08*L3

A = -492.9 + 1.1*L1 -0.2*L2 +1.2*L3

? = 3.4 + 0.004*L1 + 0.003*L2 – 0.005*L3

These equations are thus subjected to Optimization techniques to arrive at proper link lengths. The lengths of the different links are calculated as in Table 2 below.

Table 2. Link Lengths arrived for the Lift Mechanism Problem

Design Parameter

Value

Front Link Length, L1

92 mm

Rear Link Length, L3

79 mm

Inter-mediate Link Length, L2

356 mm

3.1.4 Engineering Judgement and Conclusion

The link lengths thus arrived are verified for engineering correctness by simulating the mechanism using the wire-frame. No locking of the mechanism is observed and the mechanism is found to be in favorable conditions for effort through out the activation.

Thus, the above case study suggests that using a structured methodology, better designs which can accurately hit the targets in automotive design is possible.

3.2 Optimization of Seat Mounting Feet

3.2.1 Problem Definition

A typical Front seat is mounted on to the floor of the vehicle through four mounting feet as shown in Fig. 6. The mounting feet secure the seat to the vehicle and at the same time contribute to the over all seat deflection at the tip of the Seat back. This Seat back deflection is one of the essential parameters useful in evaluating the performance of the seat in a rearward impact scenario. The current problem involves in identifying a suitable material and thickness for the three of the four mounting feet viz. Front Inboard, Front Outboard and the Rear Outboard, which optimises the weight of the feet with out affecting the rearaward impact performance of the seat. Table 3 gives the properties of the three available materials from which the required material is to be identified. A value of 425 mm is assumed to be safe for the seat back deflection under a rearward impact in the present study.

Table 3. Properties of the three Materials with Attribute Value

Sl. No.

Material

Attribute

Properties

Yield Stress (N/mm2)

Ultimate Stress (N/mm2)

% Elongation

1

Material-1

2.00

140

280

38

2

Material-2

1.83

170

280

23

3

Material-3

1.00

400

460

20

3.2.2 Problem Formulation

FE simulation using LS-Dyna [3] software is considered for evaluating the performance of the seat under rearward impact scenario. A 95-percentile dummy is used for evaluating the seat performance. As the plastic strain limits for each material varies, a normalized plastic strain ratio is defined for evaluating the failure criteria. The plastic strain ratio is defined as the ratio of the maximum of the plastic strains observed on different feet to the plastic strain limit defined for the material. To have least weight for the feet assembly it was decided to vary the thickness and material for the feet so as to maximise the plastic strain ratio and the seat back deflection until they reach unity and 425 mm respectively. Different input parameters and their values used in the present study are given in Table 4.

Table 4. Variable Definitions for Feet Optimization Problem

Input Variables

Thickness of the Front Inboard and Outboard Feet, TF

1.5 – 2.5 mm

Thickness of the Rear Outboard Feet, TR

2 -2.5 mm

Material for the Feet

Material-1, Material -2, Material-3

Response Variables

Plastic Strain Ratio, ?R

Seat Back Deflection at the Top, X

Optimization Criteria

1. Maximize Plastic Strain Ratio till unity

2. Maximize Deflection till 425 mm

?R ? 1

X ? 425 mm

3.2.3 Six-Sigma

Again the Design of Experiments, Regression and Optimization tools are used in this problem to arrive at the set of experiments to be performed and finding the optimum values for the Design parameters. Reduced factorial experimentation is resorted to in this problem as each experiment consumes large computational time and hence is costly. A total of 4 experiments are designed as in Table 5 below.

Table 5. Set of Experiments Designed using Reduced Factorial Approach

Sl. No

TF

TR

Material

1

1.5 mm

2 mm

Material-1

2

2.5 mm

2 mm

Material-2

3

1.5 mm

2.5 mm

Material-2

4

2.5 mm

2.5 mm

Material-1

The results from the Simulation runs are then used in formulating the regression equations and solved for the optimisation criteria mentioned earlier in Table 4. The values for each of the Design variables thus obtained are as follows.

Thickness of the Front Inboard and Outboard Feet, TF = 1.5 mm

Thickness of the Rear Outboard Feet, TR = 2.0 mm

Material Attribute = 1.16

3.2.4 Engineering Judgement and Conclusion

As the material with the attribute vale 1.16 is not available, the material with value 1 is selected. To avoid any deterioration of the performance of the seat because of the selection of the lower grade material, the thickness of the Front Feet, which shows high plastic strain ratio, is taken as 2.0 mm, the next higher value available. Using these inputs, the plastic strains and the seat back deflection are calculated using the regression equations formed earlier. These values are found to be well with in the design constraints and a verification simulation has been carried out using LS-Dyna. The results that are obtained from the simulation run show a close agreement between the values obtained from the Regression output confirming the validity of the exercise (Refer Table. 6).

Table 6. Comparison between the Results obtained by solving Regression equations and from Simulation Run

Sl. No.

Output Parameter

Results obtained by solving Regression Equations

Results obtained from Simulation run

1

Percentage Plastic Strain in Front Inboard Foot

28

25

2

Percentage Plastic Strain in Front Outboard Foot

34

35

3

Percentage Plastic Strain in Rear Inboard Foot

23

25

4

Deflection of Seat back in X

405.3

410

5

Deflection of Seat back in Z

– 341.5

-360

4. Discussion and Conclusion

It has been evident from the two case studies that Six- Sigma tools have a definite applicability in the automotive engineering problems when proper transformations in converting the practical problem into a mathematical one for optimized solution. Also, the importance of these tools can be found in hitting the targets more precisely with in the ambit of the posed constraints. As there is no definite possibility in case of the problem described in the first case study, that one will hit the target with trial and error methods, application of such tools becomes inevitable. Moreover, tools like Design of Experiments help in achieving higher productivity, which is an important requirement for present day’s corporations to maintain the competitive edge.

5. Acknowledgements

The authors gratefully acknowledge the review efforts spent by Dr. M.S.S. Prabhu, Mr. Padmanabhan Venkataraman, Mr. M.R. Ravishankar, Mr. Hursh Kumar Donde and Mr. Ramanath, K.S. in finalizing the methodology proposed in the present article and in validating the case studies.

References

[1] MINITAB, MINITAB Incorporation.

[2] I-DEAS software, Electronics Data Systems.

[3] LS-Dyna Software, Livermore Software Technology Corporation.