TWI Knowledge Summary
Experimental design
by Ian Jones
Introduction
Welding processes, like most other manufacturing processes, involve setting of many machine parameters, and performance of the product is often affected by many factors. To optimise the procedures and to define which factors are most influential can involve many experiments.
Experimental procedures require a structured approach to achieve the most reliable results with minimal wastage of time and money. Experimental design, based on sound statistical principles, can be used to great effect to give an overall view of a manufacturing process using a limited number of experiments. The information gained can be used to optimise a process and define which parameters need to be placed under the most control to maintain the repeatability of a process. A mathematical model of the process is also produced that can help predict what results are expected when parameters are changed.
History
The basic concepts of the statistical design of experiments and data analysis were developed in the early part of the 20 th century as a cost effective research design tool to help improve yields in farming. Since then, many types of designed experiment and analysis techniques have been developed to meet the diverse needs of researchers and engineers.
Types of experimental design
The simplest form of experimental plan is the 2x2 factorial design. In this design four experiments are conducted consisting of all possible combinations of two process factors, each tested at two different levels.
Experimental designs for optimisation are often augmented by extending the basic 2x2 matrix, either by adding more factors, more levels for each factor, or by adding points around the design and at the centre (e.g. as in the central composite design). Usually some experiments are repeated to gain information about the errors existing in the experiment and to allow estimates of the stability of the process to be made.
A number of types of experimental design are available for different situations. Choice of design depends on the level of knowledge before the experiment, the resource available and the objectives, that is:
- Discovering important process factors
- - Plackett-Burman
- - Fractional factorial
- Estimating the effects and interactions of several factors
- - Factorial
- - Fractional factorial
- - Taguchi
- Optimisation
- - Central Composite
- - Box-Behnken
- - Simplex
- - D-optimal
- Others
- - Mixture designs for products that are mixed from several components
Example of the use of a factorial design
Take an adhesive bonding trial as an example. The aim was to discover which parameters have the strongest effects on the strength of the joint, and to allow prediction of joint strength for any combination of parameters.
The factors chosen for examination were adhesive thickness, heating time and temperature. A 3x2 factorial experiment was selected which normally has 8 experiments. In this design each experiment was repeated and a central design point was carried out three times, making a total of 19 experiments. A single-lap shear tensile test was used to assess the joint strength. The list of experiments and results is shown in Table 1.
Table 1: List of experiments in a factorial design to define strong effects for an adhesive bonding example.
| Standard order | Randomised run order | Adhesive thickness
µm |
Heating time
mins |
Temperature
°C |
Result,
Tensile MPa |
|---|---|---|---|---|---|
| 1 | 8 | 100.00 | 3.00 | 160.00 | 15.18 |
| 2 | 15 | 100.00 | 3.00 | 160.00 | 14.14 |
| 3 | 6 | 300.00 | 3.00 | 160.00 | 5.91 |
| 4 | 19 | 300.00 | 3.00 | 160.00 | 3.53 |
| 5 | 14 | 100.00 | 10.00 | 160.00 | 17.55 |
| 6 | 17 | 100.00 | 10.00 | 160.00 | 16.82 |
| 7 | 7 | 300.00 | 10.00 | 160.00 | 13.49 |
| 8 | 11 | 300.00 | 10.00 | 160.00 | 7.63 |
| 9 | 9 | 100.00 | 3.00 | 180.00 | 19.52 |
| 10 | 10 | 100.00 | 3.00 | 180.00 | 21.24 |
| 11 | 18 | 300.00 | 3.00 | 180.00 | 15.77 |
| 12 | 13 | 300.00 | 3.00 | 180.00 | 13.13 |
| 13 | 4 | 100.00 | 10.00 | 180.00 | 22.26 |
| 14 | 3 | 100.00 | 10.00 | 180.00 | 26.11 |
| 15 | 16 | 300.00 | 10.00 | 180.00 | 21.35 |
| 16 | 1 | 300.00 | 10.00 | 180.00 | 18.18 |
| 17 | 2 | 200.00 | 6.50 | 170.00 | 14.67 |
| 18 | 12 | 200.00 | 6.50 | 170.00 | 20.12 |
| 19 | 5 | 200.00 | 6.50 | 170.00 | 14.15 |
Analysis of the results is based on a statistical analysis of variance (usually called ANOVA). This assesses whether the effect of a given factor can be identified above the random variation of the measurements. It also allows a mathematical model of the process to be produced and assesses how valid the model is against the random variation of the measurements. The results are shown in Table 2 below.
The Model F-value of 30.99 implies the model is significant. There is only a 0.01% chance that a 'Model F-Value' this large could occur due to noise. Values of 'Prob > F' less than 0.0500 indicate model terms are significant. In this case adhesive thickness, heating time and temperature are significant model terms. The 'Curvature F-value' of 0.15 implies the curvature (as measured by difference between the average of the center points and the average of the factorial points) in the design space is not significant relative to the noise. There is a 70.16% chance that a 'Curvature F-value' this large could occur due to noise. The 'Lack of Fit F-value' of 0.68 implies the Lack of Fit is not significant relative to the pure error. There is a 61.91% chance that a 'Lack of Fit F-value' this large could occur due to noise. Non-significant lack of fit is good -- we want the model to fit.
Table 2. Analysis of variance table (ANOVA) for the factorial model
| Source | Sum of squares | DF | Mean square | F value | Prob > F | |
|---|---|---|---|---|---|---|
| Model | 508.05 | 3 | 169.35 | 30.99 | < 0.0001 | significant |
| Thickness | 181.10 | 1 | 181.10 | 33.14 | < 0.0001 | |
| Time | 76.43 | 1 | 76.43 | 13.98 | 0.0022 | |
| Temperature | 250.51 | 1 | 250.51 | 45.84 | < 0.0001 | |
| Curvature | 0.84 | 1 | 0.84 | 0.15 | 0.7016 | not significant |
| Residual | 76.51 | 14 | 5.47 | |||
| Lack of Fit | 16.43 | 4 | 4.11 | 0.68 | 0.6191 | not significant |
| Pure Error | 60.08 | 10 | 6.01 | |||
| Model | Tensile =
-48.85902 -0.033644 * Adhesive Thickness +0.62446 * Time +0.39569 * Temperature |
|||||
The model can then be used to display and predict the results. A typical graph from the above experiment is shown in Fig.1.
Concluding remarks
A structured approach should be taken for any series of experiments. Statistical experimental design provides a powerful tool for analysing the effects of processing parameters for any manufacturing process.
A certain degree of care must be used in preparing the experiment design. It is important to select suitable limits for the process factors to ensure that the trends are identified satisfactorily. It is also useful to have some background knowledge of the process to be examined, so that appropriate factors and levels for those factors can be chosen initially. After any experiment it is always advisable to carry out a confirmation study to ensure that the results are robust.
Fig.1. Effect of adhesive thickness and heating time on the tensile strength of the adhesive bond predicted from the model of the experiment results.
Further information
You can use the Weldasearch literature database to supplement what you find in JoinIT.
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