Investigating automotive polyurethane foam hardness testing methodologies. (Cover Story).

A key component in industrial quality control, as well as applied scientific research, is the gathering of reliable data by means of a measurement procedure. The importance of this data leads to the development of new measurement methods that will be superior in terms of reduced cost and time, easier to administer, more accurate and/or more precise (ref. 1). In many instances, there are multitudes of methods that may be used to measure a common property, such as polyurethane foam (PUF) hardness. Choosing such a method is dictated by many economic and scientific factors. Thus, the choice between alternative methods for measuring foam hardness is both a technical and economic consideration.

In automotive seating, PUF components (cushions, backs) significantly influence seat functionality. Thus, all OEMs require the measurement of part hardness. However, many OEMs have their particular methods for measuring hardness. As a result, there does not exist any one universal method for the evaluation of automotive PUF hardness. Given the multitude of methods, does any one method show technical and economic characteristics that are superior to the others? This question is increasingly important within the framework of procedural standardization and ultimately in the reduction of hardness measurement variation globally. Consequently, there exists a strong need to evaluate hardness testing methods. As such, the goal of this investigation is to illustrate a statistical method that seeks to effectively evaluate hardness testing methods both technically and economically. Our ultimate intention is to recommend a global hardness testing procedure in the near future. This contribution illustrates the statistical framework used to evaluate two hardness testing procedures and suggests a method that is statistically valid and economically attractive.

Experimental

PUF sample preparation

This study examined five different PUF samples formulated for specific hardnesses that represent a range of cushion hardness used globally (ref. 2). All PUF sample hardness measurements were conducted in our testing laboratory in Woodbridge, Ontario, Canada under controlled conditioning at 23 [+ or -] 2 [degrees] C and 50 [+ or -] 5% relative humidity.

Hardness measuring apparatus

The hardness measurements were conducted using a Zwick model 050 universal testing machine, a 5KN load cell, and a standard 203 mm diameter indentor foot.

Hardness measurement specification

The two hardness testing procedures investigated were the ASTM D3574 50% indentation force deflection test and a modified test based on this same procedure. The modified test differed only in the amount of recovery time between the preflex and test cycle. Instead of the specified recovery time of six minutes, three seconds were used. The testing parameters are shown in table 1.

Derivation -- hardness measurement methods comparison (ref. 3)

The only constant among all OEM firmness testing methods is that they all measure the same property, that being hardness. When measuring the same property using different measurement procedures, it follows that the various methods must be functionally related since each is functionally related to the property being measured. Therefore, the first among many steps in such an investigation requires that we verify that such a functional relationship between the methods exists.

The process of comparing measurement methods may be derived from an understanding of the functional relationship that exists between the method of measurement and the property being measured. In this investigation, the chemical composition of various PUF samples was changed to obtain a measurable change in hardness. Such a change is illustrated in figure 1. The relationship between hardness, H, and the chemical composition, C, is illustrated. The relationship that exists between H and C may be expressed as H = f(C).

[FIGURE 1 OMITTED]

This expression simply states that H is a function of C. In figure 1, H is some linear function of C and may be expressed as H = K * C. In this expression, K is the rate of change in H per unit change in C.

Using the law of propagation of errors, the variation in H may be expressed as:

[[sigma].sub.H.sup.2] = [K.sup.2] * [[sigma].sub.C.sup.2]

Of interest is the standard deviation in the chemical composition, C. Rearranging this expression produces the following:

[[sigma].sub.C] = 1/K * [[sigma].sub.H]

This expression provides for deeper insight into the comparison of testing methods. In an industrial setting, having the ability to discriminate changes in the chemical composition, C, is critical. As a result, the precision with which we can measure the chemical composition in question becomes increasingly important. Therefore, that method that produces a higher level of precision in the chemical composition being measured is technically superior. This leads to the comparison in the precision estimates of property, C. To illustrate, say two different methods of measurement were employed. Suppose the standard deviation in property C for each of the two methods, expressed as a ratio, is equal to unity. In such a case, the methods in question would not be considered statistically different. This leads to the following expression when comparing two methods for measuring a common property, C.

[[sigma].sub.c.sub.1]/[[sigma].sub.c.sub.2] = [[sigma].sub.H.sub.1]/[K.sub.1]/ [[sigma].sub.H.sub.2]/[K.sub.2]

In the expression above, K is the change in hardness per unit change in C. Therefore, K may be expressed as follows:

[K.sub.2]/[K.sub.1] = [DELTA] [H.sub.2]/[DELTA] C/[DELTA] [H.sub.1]/[DELTA] C = [DELTA] [H.sub.2]/[DELTA] [H.sub.2] = [beta]

This expression illustrates how two methods of measurement may be compared without any knowledge in the property being measured. Consequently, our final expression for the comparison of methods of measurement is as follows:

[[sigma].sub.c.sub.1]/[[sigma].sub.c.sub.2] = [beta] * [[sigma].sub.H.sub.1]/[[sigma].sub.H.sub.2]

This expression states that the ratio of standard deviations in property C is equal to the product of the change in hardness and the ratio of standard errors for each of the hardness testing methods. This comparison can be made without any knowledge in the nature of the chemical changes that were made to affect the hardness, H.

Linear relationship when both variables are subject to error

Using standard regression to derive the linear relationship between testing methods is not appropriate. Such an approach assumes that the regressor is known and without error. An alternative fitted line is required that describes the relationship between two measuring methods. Below is an equation that describes the relationship between two variables subject to error.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Economic considerations

Suppose two methods of measurement have different property precision estimates with [[sigma].sub.CA] < [[sigma].sub.CB]. Using the central limit theorem, the number of samples required of method B to achieve the same level of precision as method A may be determined. An expression for a sample size is shown.

n = [[sigma].sup.2.sub.CB]/[[sigma].sup.2.sub.CA]

Using the property precision estimates derived from methods A and B, this expression says an average of n samples are required of method B to achieve the same precision as that of method A. Since the cost of a single test is known or can be easily determined, the economic costs associated with each of the methods may be derived. If the same testing apparatus is employed, the cost of a test becomes a function of the time required to administer such a test. Consequently, the previous expression may be modified accordingly. This new expression takes into account the number of samples, n, and amount of time, t, required for method B to achieve the same economic merit as method A. Using this expression, the economic value of a test may be judged.

t = n * [t.sub.B]/[t.sub.A]

Results and discussion

In table 2, the hardness summary statistics are shown. Listed are the averages and standard deviations for each of the five samples used in this investigation. An overall pooled standard deviation was used for each of the two hardness testing methods shown in table 2. This was computed by finding the square root of the mean variance for each hardness testing method. Figure 2 shows this is an appropriate assumption.

[FIGURE 2 OMITTED]

In table 2, the modified ASTM procedure showed a 5% decrease in the average hardness result, but a 14% improvement was noted in the pooled standard deviation. In figure 2, the standard deviation patterns are consistent, as expected. Again, the ASTM procedure exhibits a slightly higher average standard deviation when compared with the modified ASTM hardness testing method.

Since two different hardness testing procedures were used to measure a common property, it follows that both methods must be functionally related. In figure 3, we show the functional relationship between the hardness testing methods.

[FIGURE 3 OMITTED]

The numerical value that represents the linear relationship shown in figure 3 is calculated based on the fact that both hardness testing methods are subject to error. The numerical calculation is illustrated. Since both hardness testing methods are expressed in Newtons, a 0.884 N change in the modified ASTM method corresponds to a 1 N change in the ASTM method.

S[S.sub.H1] = N [summation over (i)] [([x.sub.i]-[bar]x).sup.2] = 91,907.6

S[S.sub.H2] = N [summation over (i)] [([y.sub.i]-[bar]y).sup.2] = 71,853.7

S[S.sub.H1H2] = N [summation over (i)] ([x.sub.i]-[bar]x)([y.sub.i]-[bar]y) = 81,051.3

[beta] = 0.884

This result is expected given the modified ASTM hardness results were lower when compared with the standard ASTM method. Removing the six-minute recovery time between the preflex and testing cycle results in a lower hardness outcome.

Comparing both hardness testing methods technically, we find the result shown below is approximately unity. This leads to the conclusion that both hardness testing methods are technically similar.

[sigma] [c.sub.1]/[sigma] [c.sub.2] = [beta] * [sigma] [H.sub.1]/[sigma] [H.sub.2] = 1.03

The time required to complete an ASTM test cycle is ten minutes. In comparison, four minutes are required to complete a modified ASTM test cycle. Comparing both hardness testing methods economically, the result shown below is 2.6. Essentially, the modified ASTM method is economically superior compared with the standard ASTM method.

t = n * [t.sub.B]/[t.sub.A] = 1.03 * 2.5 = 2.6

An average 1.03 samples are required for the standard ASTM method to achieve the same precision as that of the modified ASTM procedure. Essentially, they are sample size equivalent. The standard ASTM methods take 2.5 times longer to perform compared to the modified ASTM procedure. For equal levels of precision, the standard ASTM procedure will take 2.6 times longer to administer compared to the modified ASTM method.

Summary and conclusion

A statistical procedure is described that may be used to evaluate hardness testing methods. Illustrated are two hardness testing methods, and the linear relationship that exists between these methods. We show that both methods are technically similar, and have shown that the modified ASTM method is economically superior when compared to the standard ASTM procedure.

Table 1 -- hardness measurement parameters

Testing parameters     ASTM               Modified ASTM

Preload speed          50 mm/min.         50 mm/min.
Preload force          4.5 N              4.5 N
Preflex speed          240 mm/min.        240 mm/min.
Preflex set point      75% deflection     75% deflection
Dwell @ set point      1 sec.             1 sec.
No. of preflexes       2                  2
Recovery time          6 min.             3 sec.
Test speed             50 mm/min.         50 mm/min.
Test set point         50% deflection     50% deflection
Dwell @ test point     60 sec.            60 sec.
Table 2 -- hardness summary statistics

Sample number        ASTM D3574 (N)      Modified ASTM (N)
                       ([H.sup.1])          ([H.sup.2])
                   Average   Std. dev.  Average   Std. dev.

1                   314.27      4.82     298.97       3.58
2                   345.77      2.15     334.30       3.40
3                   386.60      3.96     371.07       3.46
4                   432.17      6.65     411.67       6.01
5                   484.53      8.07     449.77       6.38
Average             392.67               373.16
Pooled std. dev.               5.5296                4.7576

References

(1.) Tan, Y.T., Iglewicz, B. August 1999. "Measurement-methods comparison and linear statistical relationship," Technometrics, Volume 41, p. 192.

(2.) Blair, G.R., Milivojevich, A., Pageau, J-G. and Van Heumen, J. 1999. "Automotive seating comfort; defining comfort properties in polyurethane foam," SAE Technical Paper 1999-01-0587.

(3.) Mandel, J. "The statistical analysis of experimental data," Dover Publications, New York, Chapter 14, pp. 374-375, 381-383.