The Royal and Ancient Golf Club of
St Andrews
andUnited States Golf Association

Technical Description of the Pendulum Test
Revised Version
Discussion of points raised during
Notice & Comment period(November 2003)
1

Contents
1 Introduction
1.1 Technical details of the science behind the test . . . . . . . . .
1.2 Technical details of the resolutions of questions raised concerning effects on the test . . . . . . . . . . . . . . . . . . . . . . .
1.3 Discussion of the test and submission protocol . . . . . . . . .

4
4
5
5

2 Technical Details of the Test
5
2.1 Simple checks on the acquired signals and data points . . . . . 7
2.2 The accuracy of the test . . . . . . . . . . . . . . . . . . . . . 8
2.3 The model used to predict the variation of characteristic timewith velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Setting of the limit and the correlation between COR and CT
9
2.5 Calibration procedure . . . . . . . . . . . . . . . . . . . . . . . 11
3 Technical details of the resolutions of questions raised concerning effects on the test
11
3.1 The effects of club loft, mass and shafting . . . . . . . . . . . 11
3.2 Clubs with coated faces . . . . . . . . . . . . . . . . . . . . . . 13
4 Discussion of the test and submission protocol
4.1 Submission process . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Description of the test protocol . . . . . . . . . . . . . . . . .
4.3 Details of the equipment . . . . . . . . . . . . . . . . . . . . .
4.4 Comments on the software and user interface . . . . . . . . . .

14
14
14
15
16

5 Conclusions

16

A Details of the mathematical methods behind the data analysis
18
A.1 Fourier series truncation . . . . . . . . . . . . . . . . . . . . . 18
A.2 Statistical analysis of the data . . . . . . . . . . . . . . . . . . 23
B Testing Hertz Contact Assumption of the Pendulum Test
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
23
23
24

C Velocity dependence of the Pendulum Test
24
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
C.2 Test Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2

C.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
C.4 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . 29
D Effect of club head mass on Characteristic Time
31
D.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
E Testing of coated clubs

33

List of Figures
1
2
3

4

5

6
7
8
9
10
11
12
13

14

Plot of correlation between characteristic time and coefficientof restitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
An image of a calibration club for use with the pendulum device. 12
A typical example of the voltage output from the scope. Thissignal has 6240 points and the resolution is determined by thefact that the scope returns a 12 bit signal. We also include
the corresponding filtered signal. . . . . . . . . . . . . . . . . . 18
Fourier spectrum for a typical signal shown for the first 100
modes. Here we see evidence of the noise (for n > 70 corresponding to 70KHz). . . . . . . . . . . . . . . . . . . . . . . . 20
Sample velocity profile for an impact between a club and aball. Notice that since we are using a longer signal collection
time and a corresponding longer pre-trigger this gives us alonger plateau region at the start and in fact also at the end. . 21
Results of experiment and numerical simulation for blocks ofaluminium and steel. . . . . . . . . . . . . . . . . . . . . . . . 25
Characteristic Time velocity dependence. . . . . . . . . . . . . 28
Linearised Characteristic Time velocity dependence. . . . . . . 29
Dependence of optimum exponent on slope at conformance limit. 30
Effect of constant non-linear exponent on intercept error for aclub with a slope of 80. . . . . . . . . . . . . . . . . . . . . . . 30
Distribution of club head mass. . . . . . . . . . . . . . . . . . 32
Theoretical effect of mass on characteristic time. . . . . . . . . 33
Effect of adding mass artificially to a club head, two examples.
Each figure shows the theoretical prediction and the actual
measurements from the pendulum. . . . . . . . . . . . . . . . 34
Variation of kcoating with coating thickness and slope. Dashedlines are used for thicknesses greater than 500µs. Note thatas the coating thickens the value of k returns to value of 0.33. 37

31Introduction

In this document we give details of the technical aspects of the pendulum.
Most of these are in response to questions which have been raised during the
Notice and Comment process. This report is intended to provide informationof a technical nature, but we shall start by summarising the salient points.
We note that the report is divided into sections: technical details of thescience behind the test; technical details of the resolutions of questions raised
concerning effects on the test and finally, discussion of the test and submission
protocol.
Before we start our description we mention that at points in this report
we shall highlight the fact that our intent is to determine the conformancestatus of the club and as such we concentrate on the range of clubs whose
performance is close to the limit. Although we note that the pendulum does
produce a good indication of the performance of clubs away from this limit we
do not intend to unduly compromise our ability to judge conformance by theconsideration of these clubs. In summary we intend to ensure that the smallest errors in the judgement of characteristic time occur in the neighbourhoodof the limit, although obviously we must ensure that correct judgements arealso made for clubs removed from the limit.

1.1

Technical details of the science behind the test

In this section we discuss the algorithm and the model used in the test,
including the statistical methods which are used to evaluate each individual
test. We also give details of a Gage R & R analysis which deals with therepeatability and reproducibility associated with two aspects of the test,
namely the marking of the centre location on the face and the variability
associated with machines and operators. In response to comments we haverevisited the choice of exponent in the model and have found that an exponentof k = 0.33 more accurately captures the physics of the impact between two
curved bodies where one is intrinsically flexible. This has necessitated achange in the associated limit to 239µs. We have further refined the data
analysis method to reduce the susceptibility of the technique to noise.
We have also modified the software to include the improved statisticalanalysis and to extend the time over which the test is run from 500 microseconds to 1000 microseconds. We shall also comment on the provision of acalibration object. We conclude this section with a discussion of the settingof the limit and the correlation between the coefficient of restitution and thecharacteristic time.

4

1.2

Technical details of the resolutions of questions
raised concerning effects on the test

In this section we comment on the issues raised concerning the effects of clubhead mass, loft, shafting and coatings on the face. These comments will bebased on our scientific findings. In brief the club head mass is found to havean effect, albeit secondary. In essence the characteristic time increases as
the club head mass increases. It has been found that the club’s loft will nothave an effect on the test provided the test is performed carefully in that it
must be assured that the correct location is hit on the face. We have shown
through our own tests, that the shafting of the club plays no rôle in the test;
although we add the slight caveat that the attachment must not add undue
mass to the head. In order to test a club with a ‘coated’ face it is necessary
to apply an additional test, namely a measurement of the thickness of thislayer. This extra measurement is only applied where the presence of thecoating may have influenced the conformance of a club. Note that no suchclubs have been submitted to the ruling bodies for conformance evaluation.

1.3

Discussion of the test and submission protocol

We shall include a discussion of the test protocol and the way in which a
conformance decision is reached. Details are also given of the way in whichclubs should be submitted for conformance measurements.

2

Technical Details of the Test

We shall start this section by describing the underlying method with which
the test is applied. This is to facilitate the forthcoming discussion.
• The centre of the club face is identified and marked with a small circle.
This actually circumscribes the intended impact location.
• Preferably, the club head shall be shafted. A removable shaft may be
used. This is then correctly mounted in the pendulum device; thisentails ensuring that the club face is perpendicular to the impact in
the heel-toe axis and that the centre of the face is in position (using
the laser pointer).
• The club is hit nine times (three times at each velocity setting). For
each hit the signal is analysed visually and the acquired characteristic
time is checked for gross errors.
5• The signal from the accelerometer is filtered and integrated to generate
the velocity as a function of time. The rising section of the curve isfitted using a cosine curve. The time taken to rise from 5% to 95% of
the maximum is recorded.
• The software analyses the data and checks that none of the points are
outliers: If they are, the offending point (or points) are identified andthese are retested (and the calculation repeated).
• A line of best fit is applied to the data (using a least squares approximation) of the form
T = A + BV −k ,where V is the velocity derived from the pendulum test and T is thetime for the velocity to rise from 5% to 95% of this value (as defined
above). The constants in the above expression: A the intercept and B
the slope are derived from the nine data points.
• A 99% confidence interval is calculated for the value of A (this is dubbed
the infinite velocity characteristic time herein) and this is used in theforthcoming conformance decision. If the confidence interval lies wholly
below the limit plus the tolerance then the club is deemed to conformto this test.
• If the upper limit of the confidence interval is above the limit then the
club is hit a further nine times and the above procedure is repeated,
except this time solely the calculated value of A is used to determine
the conformance of the club.
• We add the further point that if the slope, B, exceeds 20 units andcertain other conditions are met the club will be tested for a coatingon the face (this procedure will be detailed in due course).
We shall not dwell on the technical details of the acquisition of the data
and refer you instead to the previous technical report. Basically, the signal iscollected from the accelerometer via the signal conditioner and the electronic
oscilloscope ADC212/50 from Pico Tech (in fact we have now organized a
combined unit ADC212/50(Special Order S167) with a signal conditioner,
www.picotech.com). The signal is filtered using a truncated Fourier series(see Appendix A.1) and the filtered signal is integrated to find the velocity.
This represents the difference in velocity of the pendulum’s mass due tothe impact with the club face (within the moving frame of the mass). The

6

rising portion of this curve is fitted using a cosine curve which eliminates asensitivity to the presence of higher frequency modes within the signal.
Since the last release we have extended the time over which the test
is performed from 500µs to 1000µs. This was to accommodate clubs withhigh characteristic times and those with large values of the slope, such as
persimmon clubs. It is worth noting that it is necessary to acquire data
for a longer time than the actual characteristic time. It is also necessary
to increase the number of points acquired (in fact doubling it) so that theresolution of the data is retained. Since the truncation of the Fourier series
is based on the overall period of the data, we also need to retain twice thenumber of Fourier modes in the filtering process (again see Appendix A.1).
The fidelity of the signal can be checked visually. The raw acceleration
signal is shown, as is the filtered version, which should resemble the mean of
the unfiltered version. The associated velocity curve should appear to risefrom zero to a plateau, very much resembling a hyperbolic tangent function.
The acceleration signal should be close to zero at the start and end of thesignal. We note that the filtering requires this, else one could potentiallyencounter Gibbs’ phenomenon. For a properly acquired signal this should
not occur. If this does arise it is likely that the hit has not been properly
recorded.

2.1

Simple checks on the acquired signals and data
pointsAlthough there is some variation of velocity by machine and by club, it isinstructive to ensure that the velocities are similar to
Setting
HighestMiddleLowestApproximate velocity
1.3
1.0
0.65

This may vary between pendulums, but only slightly. This is another advantage of the use of the model, since the actual velocity is calculated andincorporated in the analysis. It is also worth considering the characteristic
times at each setting, each of the three measurements at a particular velocity
setting should be similar. For metallic clubs, that is those for which a small
slope is expected, one should expect quite similar measurements at all threevelocity settings; the characteristic time increasing for the lower settings ofthe pendulum. We add the caveat that for metallic clubs the ordering may
not be strict between the different settings. However, for a club with a higher
slope, namely a persimmon club (or one with a coated face), there should7be an observable change in the times associated with the different pendulum
settings (again the lower settings corresponding to higher times).
The size of the confidence interval provides an accurate reflection of thedegree to which the results can be used to determine conformance. The
confidence interval will generally be less than 8 microseconds wide, in fact
in many cases it will be substantially narrower than this. We note that theinclusion of the consideration of the confidence interval within the analysis
allows us confidence in our ability to extrapolate using the data.

2.2

The accuracy of the test

There have been many questions concerning the repeatability and reproducibility of the test. These comments fall into two main categories: variations associated the machine and its operation; and variations associatedwith the marking of the centre of the face (and its identification). It has
been found through a Gage R&R analysis (using 3 machines, 3 operatorsand 6 clubs) that the gage is 6 microseconds (giving 99% inclusion) andthis is associated entirely with operator variability (which actually includes
centre marking together with placing the club in the device alongside theoperation of the test) rather than inter-machine variation. A similar analysis
was run on the way in which the club’s centre is found and marked, in isolation. It was found that the gage in this case is 2.1mm, that is it is expectedthat 99% of all future measurements will be within a radius of 2.1mm of themean of the acquired measurements. We note that this variation is intrinsic
to the previous gage. The variation in terms of characteristic time associatedwith centre marking will of course vary from club to club, but it is very unlikely that this will lead to any substantial change in the characteristic timeas has been demonstrated by the overall gage.
We note that in the marking of the centre of the club face the positionof the intended impact could coincide with a face marking (punch marks,
grooves or decorative markings). In such instances, the position at which theimpact will occur is typically moved up the face, albeit slightly. For larger
markings, for instance decorative markings (which can be up to 0.375 inches(9.53mm) in length) it may be necessary to perform the measurement at
points around the marking and to use an alternative method for evaluating
the conformance of the club. In essence
If a marking prohibits the testing of the club at the centre of theface: if the marking is less than 4.27mm in vertical or horizontal
extent (ie groove or punch mark) then a point will be chosen
directly adjacent to the location of the intended impact as close8

as practical to the centre; if however the size of the centre marking
prohibits this (as for a larger decorative marking up to 9.53mm)
then we reserve the right to test the club in an appropriate mannerin the neighbourhood of this marking.

2.3

The model used to predict the variation of characteristic time with velocity

We have re-evaluated the model used to determine the variation of the characteristic time with velocity. We are now using an alternative exponent kof 0.33. Whilst it is accurate to use an exponent of k = 0.2 for the impact
between two curved solid bodies, this can be improved upon where one of
the bodies is flexible (as it is in this case). In fact the pendulum can be used
to accurately predict a true Hertzian contact and this analysis has been run
by the way of a check of the device, see Appendix B.
We appreciate that we are trying to measure the flexibility of the clubusing the test and we have evaluated the likely variation of the flexibility
associated with changes in the Young’s modulus. Over the range of Young’s
modulus associated with steel and titanium (as typically used in the manufacture of golf clubs) an exponent of around 0.33 is pertinent, especially for
those clubs whose flexibility corresponds to characteristic times close to thelimit. We reiterate the point that our intent is to design a machine which determines the conformance of a club rather than being a performance measure.
Further details can be found in Appendix C.

2.4

Setting of the limit and the correlation between
COR and CT

We have continued to test clubs on the pendulum and this has allowed us toextend our confidence in the ability of the test to provide information about
the performance of clubs, especially in the neighbourhood of the limit. To
date in excess of four hundred clubs have been tested. In this report we givedetails of the 60 clubs which have formed the core of our studies.
In Fig. 1 we show the correlation between characteristic time and coefficient of restitution. In order to set the limit we calculate a line of best fit
for the data. This shows that the limit of 0.822 corresponds to 239µs and
0.830 corresponds to 257µs. We should stress that this does not represent arelaxation of the limit proposed in a previous document at 232 and 250 respectively; it is merely a recalculation of the available data using the revised
exponent k = 0.33.
9

0.87

0.86

0.85

Coefficient of restitution

0.84

0.83

0.82

0.81

0.8

0.79

0.78
190

200

210

220

230

240
250
Char. time (µ s)

260

270

280

290

300

Figure 1: Plot of correlation between characteristic time and coefficient ofrestitution.

10

It has been shown in previous documents that the measurement of theinfinite velocity characteristic time is predominantly influenced by the flexibility of the club head. Therefore, it would be expected that the coefficientof restitution and the infinite velocity characteristic time may be highly correlated. To test this, a sample of 60 club heads was compiled and measured
via both procedures. Figure 1 shows the results of this comparison.
The best fit linear equation correlating the two quantities is:
e = 0.718 + 0.000436A,
where A is the infinite velocity characteristic time in microseconds.
To maintain consistency with the previous standard, the following limit
and tolerance is proposed:
Test Limit
Tolerance
Limit + Tolerance

Current Test
0.822
0.008
0.830

Proposed Test
239
18
257

We note that after January 1st January, 2004, conformance of a club tospring-like effect will be determined by its characteristic time.

2.5

Calibration procedure

Calibration objects are available from the ruling bodies. These objects have
two faces each with different characteristic times, see Fig. 2.

3

Technical details of the resolutions of questions raised concerning effects on the test

3.1

The effects of club loft, mass and shafting

We have run a variety of experiments to assess the likely variations associatedwith these aspects of club head characteristics. The club’s loft is of little
consequence when it comes to performing the pendulum test, it is however
essential that one ensures that the impact location is as it is intended to be.
It has been found that the loft of the club does not affect the characteristic
time as measured by the pendulum, although we appreciate that this may
not necessarily be so for the cannon test. In essence the pendulum is a truemeasure of the club head’s flexibility.

11

Figure 2: An image of a calibration club for use with the pendulum device.

12

The way in which the club head is shafted also does not play a rôle. The
most important aspect of the shaft or mounting of the club is its ability topresent the club in a consistent fashion for repeated hits. We have also foundthat some configurations are subject to low frequency oscillations and it isessential that these are stopped before the next impact. This is all so thatthe correct impact location is hit on the club face. The gripping of the club isalso not crucial, in fact it is possible to freely support the club head in frontof the pendulum and obtain the same results for the characteristic time. The
obvious addendum here is that the head must always be presented in thesame way. We add the final caveat that the shafting of the club head should
not add excessive amounts of weight to the head, as we are about to discussthat there is an effect associated with club head mass.
In Appendix D we give details of a study into the effect of changing theclub head mass. This comprises a theoretical and experimental analysis of
the issue. We have demonstrated that increasing the club head mass will
increase the characteristic time, although we add that this is a secondary
effect. We observe that for an increase of 1g we would expect to observe an
increase in characteristic time of 0.25µs. We should set this finding in context
and realize that the mass of most club heads submitted to the USGA in 2003
lie between 180g and 215g, with a mean of 197.4g.

3.2

Clubs with coated faces

Within our own analysis and at the suggestion of certain manufacturers wehave investigated the effect of coatings on the face. We should start by saying that most of our analysis has concerned coverings which are far thicker
than those currently being used, but nevertheless we want to ensure that thetechnique is robust. We shall start by giving an example of how the pendulum’s result for a club head might be affected by the addition of a covering.
In essence since the impact velocity of the pendulum is far lower than a real
impact the change in characteristic time can be affected by the addition ofa coating. As one might expect it has a disproportionate effect on the lower
velocity impacts. Applying a coating to the face will cause the characteristic
time to increase at each velocity (more so for the lower velocities). This has
the effect of artificially increasing the slope B (as currently calculated) andhence ultimately reducing the calculated intercept A. It is quite feasible thatthe application of a covering will change the characteristic times so that a
non-conforming club can be made to appear conforming. This is obviouslyof concern to us.
In essence the technique we are adopting uses the pendulum’s measurements and a further measurement to adjust the model so that the results
13

again correspond to the uncoated tests. This is accomplished by measuring the thickness of the coating and adjusting the value of the exponent k.
This adjustment is also based on the original value of the calculated slope
B with an exponent k = 0.33. We note that we have designed an algorithm
whereby this method is only applied where clubs might have changed theirconformance status by the presence of the coating. We discuss the details of
the procedure and analysis in Appendix E. We note that this technique has
been demonstrated as being very successful.

4

Discussion of the test and submission protocol

4.1

Submission process

We shall now discuss the process of club submission. Clubs will be tested
for their characteristic time by either the R&A or the USGA depending on
where the headquarters of the company resides. The only exception to thiswould be a club produced solely for use within the other governing body’s
territory.
Each governing body will have a pendulum, residing in St Andrews, Scotland and Far Hills, New Jersey, USA, respectively. These will be the pendulums which will determine the definitive measurement of conformance forclubs and once a club has been measured on one of these pendulums the acquired decision will be accepted by both organisations. Hence the club will
be deemed to exceed or not exceed the limit worldwide.

4.2

Description of the test protocol

A formal test protocol is being circulated, however it seems pertinent todescribe the basic method which will be employed. This is based on thealgorithm described on page 6.
• The club face’s centre will be marked and the club will be mounted
within the pendulum.
• The club will tested using nine hits (three at each velocity setting).
• If the calculated slope is less than 20:- If the upper limit of the confidence interval is below the limit plus
the tolerance (257µs) then the club will be deemed to conform to thependulum test.
14

- else the club needs to be tested with a further nine hits and therecalculated value of the intercept will be used to judge conformance(if A > 257µs then the club does not conform, if A 6 257µs then the
club will be deemed to conform under this test).
• If the calculated slope exceeds 20 the club should be tested with afurther 9 hits
- If the measured values of the characteristic times are all less than
257µs then the club is deemed to conform to this test.1- If the value of the intercept is greater than 257µs then the clubis deemed not to conform.2
Otherwise measure the coating thickness and calculate the value ofthe exponent kcoating and recalculate the value of the intercept, refer toAppendix E. If this value is strictly greater than 257µs then the club
will be deemed not to conform.
We should stress that this is purely to determine the results of the pendulum
test and the club will still need to be evaluated in other ways before its
conformance can be confirmed.

4.3

Details of the equipment

In order to use a pendulum a company needs to enter into a license agreement with either the R&A or the USGA. Copies of these can be obtainedby contacting the appropriate organisation . This includes access to thesoftware, drawings of the mechanical parts and an accompanying parts list
(including a description of the electronics). It is our intention to provide
calibration objects, described in Section 2.5, which will have been measured
on one of the two definitive pendulums. We are confident that the variations
between machines and operators are well within the allowed test tolerance of
18µs. The calibration object is purely to ensure that the machine is correctly
configured.
In addition to the equipment described above in order to completely test
the clubs in the methods described herein it may be necessary to acquirea device for measuring the thickness of coatings, see page 35. In order tomark the centre of the face we have been using a acetate sheet marked with1

In this case any recalculation of the intercept will still be below the limit, since theintercept is always lower than the measured values.
2
It is worth noting that any recalculation of the intercept is only likely to raise it
further.

15

a orthogonal set of axes and a circular hole at the origin. The axes aremarked at regular intervals so that the centre can be determined. Copies ofthese templates can be obtained from either governing body (electronicallyor physically). In order to test club heads (without shafts) we are using asimple reusable shaft which is designed not to add undue effective mass tothe club head.

4.4

Comments on the software and user interface

Within the comments from manufacturers, mention was made of the softwareand the general user interface. At the moment the software has been written
to be used by someone who is familiar with the test and the way in which
it is run. The new version of the software is, unsurprisingly, targeted at theversion of the test described herein. There are a variety of visual checks
which can be applied to individual signals (see page 7). To date these are
not included within the software. After the club has been hit nine times
the data is analysed and any outlying points are identified and the message
box suggests which points should be retested (this is done by clicking thecorresponding radio button and re-hitting the club - in order to rerun thecalculations it is necessary to press the recalculate button).
At the moment a simple version of datalogging is available which saves:the calculated conformance values in testlog.txt; the values of the measured times and velocities in biglog.txt; and each club’s individual data
in a filename which depends on the identification used for the club. There
are additional files which contain diagnostic information Tclubname.txt andFclubname.txt (Fourier spectra). The source code in Visual Basic is available on the signing of a license agreement, together with further details in
the user manual.

5

Conclusions

In this report we have given details of the answers to the questions raised bymanufacturers and we feel that the consideration of these points has allowed
us to develop a fuller understanding of the pendulum and the science behind
it. In order to answer the questions we have also undertaken studies of thelikely variation associated with the operation of the pendulum and we areconfident that this is well within the stated test tolerance of 18µs. We areconfident that the algorithm and use of the filtering scheme is robust andreliable.
The change to a more appropriate exponent in the model has resulted
16

in a modification of the limit to 239µs (with the additional test tolerance
giving 257µs as the value of the limit plus the test tolerance). We now havea fuller understanding of the effect of club head mass on the test and we areconfident that we are able to test clubs with coatings on the face.

17

A

Details of the mathematical methods behind the data analysis

A.1

Fourier series truncation

The mechanical device produces a signal (ti , Si ) which spans 1000µs. The
indices of the points run from i = 1 to i = N . The symbol S is used for thevoltage returning from the scope, which is representative of the acceleration.
The recommended scope is a 12 bit device sampling at 50MHz (ADC212/50
www.picotech.com, S167 with the inclusive signal conditioner). At this pointin this section we shall consider that the data has been extracted correctly.
For the sampling rate of the scope we are using this gives N = 6240, witha corresponding time step of 0.16µs. An example of a signal from the scopeis shown in figure 3. The accelerometer we are using has an intrinsic noise
10000

8000

Acceleration

6000

4000

2000

0

−2000

−4000

0

100

200

300

400

500
Time (µ s)

600

700

800

900

1000

Figure 3: A typical example of the voltage output from the scope. This signal
has 6240 points and the resolution is determined by the fact that the scope
returns a 12 bit signal. We also include the corresponding filtered signal.
associated with oscillations of the crystal at 65-75KHz (refer to Fig. 4). In
18

order to remove this and any other noise the Fourier spectrum of the data is
calculated. Using the period of 1000µs we have L = 500µs and the Fourier
coefficients are
1
a0 =
2L

ZL
S(t) dt
−L

ZL

1
an =
L

S(t) cosnπ(t − L)dt,
L

S(t) sinnπ(t − L)
dt.
L

−L

ZL

1
bn =
L

−L

Hence the reconstructed signal will take the form
SNm (t) = a0 +

Nm
X
n=1

an cosnπ(t − L)nπ(t − L)+ bn sin
L
L

where in the limit Nm → ∞ we will recover the original signal. However,using a finite value of Nm we are able to remove the noise. It has been foundsuitable to use a value of 50. Extensive testing has been performed to verify
that only small changes in the eventual answer are observed with variation inthis number. In order to determine the Fourier coefficients it has been foundto be sufficient to use a simple Trapezium method of numerical integration.
The Fourier spectrum associated with the typical signal shown in figure 3 isshown in figure 4. We note that the acceleration is close to zero at both endsof the signal and consequently we do not encounter Gibbs’ phenomenon.
We wish to construct the velocity and this is done by integrating thesignal S(t) from 0 to t, such that the initial velocity is zero. Hence we have
Nm
X
an L

nπ(t − L) bn L
vNm (t) = a0 t +
sin
−
nπ
L
nπ
n=1

µ

¶
nπ(t − L)n− (−1) , (1)cosLwhere the term (−1)n comes from cos nπ. We are now in a position to
manipulate this signal. Unsurprisingly, the resulting velocity is independentof whether the filtered signal is reconstructed and then integrated, or if it isconstructed directly from equation (1), as noted later. In figure 5 we show
a representative velocity plot (which again corresponds to the acceleration
signal shown in figure 3).
19

3000
2000

a

1000
0
−1000
−2000

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50
n

60

70

80

90

100

2500
2000
1500
b

1000
500
0
−500
−1000

Figure 4: Fourier spectrum for a typical signal shown for the first 100 modes.
Here we see evidence of the noise (for n > 70 corresponding to 70KHz).
In order to calculate a characteristic time for the club we determine themaximum velocity (simply by looking for the largest value of vN (t)). We
shall refer to this quantity as V (which gives a velocity representative of the
impact). We have discovered that the most important factor in determining
the flexibility of a club is the duration of the impact and our choice of protocol
is to eliminate factors which may unduly influence this calculation. We thencalculate the time for the velocity to rise from 5% of V to 95% of V , since theidentification of the start and end of the signal can be overly sensitive to smallchanges. In a similar manner the presence of localised vibrations in the clubhead can make the reliable calculation of the characteristic time troublesome;largely due to the presence of energy in higher modes of vibration. As suchit was found to be prudent to fit a cosine function to the rising portion of
the velocity profile and this serves to eliminate this phenomenon. For the
majority of clubs this makes no difference at all, it is only where the signal
contains energy in higher modes, for instance within clubs in which there issome degree of local irregularity near the centre of the face.
The method involves considering the function
g(t) = 1 − 2
20vN (t)
V

1.4

1.2

1

Velocity (ms−1)

0.8

0.6

0.4

0.2

0

−0.2

0

100

200

300

400

500
Time (µ s)

600

700

800

900

1000

Figure 5: Sample velocity profile for an impact between a club and a ball.
Notice that since we are using a longer signal collection time and a corresponding longer pre-trigger this gives us a longer plateau region at the startand in fact also at the end.

21

(which runs from 1 to −1 over the range of interest). We now seek a functioncos(αt + β) and rely on linear regression to minimise the quantity
E=

i=j2
X
¡

¢2
αti + β − cos−1 g(ti )i=j1

where j1 and j2 represent the start and end of the rising portion of the curve3 .
These values are further refined by minimising the function
Ecos =j2
X(cos(αti + β) − g(ti ))2

i=j1

by using a quasi-Newton method. Since we now have an analytic form forthe signal in the region of interest we can calculate the times corresponding
to vN (t) = 0.05V and vN (t) = 0.95V directly. When vN (t) = δV theng(t) = 1 − 2δ and hence tδ = (cos−1 (1 − 2δ) − β)/α. We now observe thattchar = t0.95 − t0.05 =¢1 ¡ −1cos (1 − 2(0.95)) − cos−1 (1 − 2(0.05)) .
α

We conclude this section by noting that the method has been designed
so that it is insensitive to changes in the algorithm, in the current method
we adopt the protocol:• Acquire signal, filter using a truncated Fourier series, directly integrate
to obtain the velocity from the Fourier series
However, one could also
• Acquire signal, integrate using a standard technique (Trapezium, Simpson’s 3/8th) and then filter using the truncated Fourier series (in thiscase one would need to use an even reflection of the signal to remove
Gibbs’ phenomenon, essentially using a cosine series)• Acquire signal, filter using a truncated Fourier series, invert the transform and then integrate using a standard method.
The results given by any of these methods differ by less than a microsecond.
We should say that the Fourier filter was adopted since we are aware of3

The method has been demonstrated to be insensitive to these choices, due to theminimisation of the quantity Ecos . We note that the minimisation of E and Ecos are
similar but subtly different. The advantage of the operation of minimising E is that thisis a direct calculation whereas Ecos has to be minimised using an iterative procedure.

22the frequency of the noise we wish to eradicate, although alternatives wereconsidered. It is essential that the filtering does not introduce a phase shift
into the data. The adoption of the 5-95% methodology was to remove thedifficulty in determining when the system had reached a maximum, which
would be far more sensitive to any remaining noise in the system.

A.2

Statistical analysis of the data

The data points are analysed using the method of deleted Studentised residuals for both the nine and eighteen point tests. The leverages are also calculated to ensure that no significantly different velocities have occurred. We
note that this is only likely to arise where the test has not been performed
correctly.
The calculation of the confidence intervals is performed using standard
analysis which can be found in most standard texts on data analysis (for instance ‘The Statistical Analysis of Experimental Data’ John Mandel, Dover,
ISBN: 0486646661).

B

Testing Hertz Contact Assumption of thePendulum Test

B.1

Introduction

One of the main central features of the pendulum test is the hypothesis thatthe total characteristic time may be separated into two contributions, that
due to the assumed linear stiffness of the club head and that due to thestiffness of the Hertzian contact between the steel pendulum mass and theface material of the club head.
To date, pendulum results have been provided for actual club heads
where, in general, the characteristic time is dominated by the linear stiffness of the club head. In order to test the Hertzian contact model, blocks ofsolid metal were tested.

B.2

MethodTwo blocks of metal, one 6061-T6 aluminium and the other a steel of unknown alloy were shafted using a removable shaft. The impact spot was
directed near the centre of gravity of the block. Before conducting the pendulum test, the pendulum mass was released three times into the block. It
was noticed that a small permanent indentation was left in the blocks.
23

The pendulum test was then conducted on the two blocks. Three impacts
were recorded at each of the three release heights starting with the highest
release point, followed by the middle release point and ending with the lowest
release point. This test was then conducted a second time for a total ofeighteen impacts.
Following the pendulum testing, the permanent indentations in the blocks
were measured using a contour tracing machine magnifying the indent by afactor of two hundred.

B.3

Results

It was found that the aluminium block had a permanent indentation having aradius of 62 mm and the steel block had a radius of 77 mm. This permanent
radius was then used, along with the material properties of the blocks andthe pendulum mass in the numerical simulation of the pendulum test.
The results of both the experiment (data points) and the simulation (solidline) are shown in Figure 6. It may be seen that there is good agreement
between the experiment and the simulation.
It should be noted that since the blocks are clearly rigid except for the
Hertzian contact, the non-linear exponent (k) is equal to 0.20 as opposed to
the value of 0.33 used for flexible club heads.
When a linear fit is applied to the experimental and numerical simulation
data given in Figure 6, it can be seen that(i) there again is very good agreement and(ii) the intercept is nearly zero.
It may be concluded therefore that the assumption that the pendulum test
isolates the club stiffness from the Hertzian contact is valid.

Numerical
Experiment

Aluminium
Intercept
-0.3
-5.4

Slope
62.5
64.9

Steel
Intercept
1.7
-3.2

Slope
46.3
52

C

Velocity dependence of the Pendulum Test

C.1

Introduction

The pendulum test has been successfully modelled with a one-dimensional
non-linear club model comprised of a linear spring representing the club
24

90

85

80
Aluminium
Char. Time (µ s)

75

70

65

60
Steel
55

50
1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

V−1/5

Figure 6: Results of experiment and numerical simulation for blocks of aluminium and steel.

25

head flexibility and a non-linear spring representing the Hertzian contact
between the club face and the pendulum mass. The model provides a basis
for isolating the local deformation flexibility from the overall flexibility of
the club head (which is the feature of interest) by testing at multiple impact
velocities.
As described in previous documents, the current procedure calls for linearizing the characteristic times as a function of the impact velocity using
the classical Hertzian relationship between contact time and impact velocity:
T = A + BV −1/5 .
However, the value of the exponent (-0.2) is strictly true only for the casewhere the club head is infinitely stiff. For realistic club flexibilities, the mostappropriate value for this exponent depends on both the club face material
and the underlying flexibility of the club head. It has been found that avalue of -0.33 adequately covers the range of face materials for clubs withflexibilities near the conformance limit.

C.2

Test Model

It has been found that a simple model of the club head is sufficient to predictthe predominant behaviour of the pendulum test. The pendulum mass isrepresented simply as a point mass having no elastic properties. Deformationof the pendulum mass is contained in an interface spring associated with theclub.
The club head is modelled as a single mass and two springs. The firstspring (Kc ) is linear and is intended to represent the overall flexibility of theclub head. This flexibility, along with the mass properties of the club head
contribute to the intercept value that is isolated by the pendulum test.
In addition to the overall flexibility of the club head, a second spring is
included to model the localized contact between the pendulum mass and theclub face. Hertzian contact definition and hence this spring is non-linear andtheir derivations are well described by several sources. For example, Johnson(1985)4 provides the following definition for the non-linear spring:1
1
4
kH = (R∗ ) 2 E ∗ (xH − xm ) 2
3
where (xH −xm ) includes the compression of both the club face and pendulum
mass at the localized area of contact. The model parameters are defined as:
1
1∗R∗ = 1
1 and E = 1−ν 22b+ Rc+ 1−νcRb
Eb

4

Johnson, K. L. (1985) Contact Mechanics, CUP, UK

26

Ec

where Rb and Rc are the radii of curvature of the pendulum mass and club
face respectively and ν and E are the Poisson’s ratio and modulus of elasticity.
Owing to the non-linear nature of the model, the equations of motion
must be solved numerically. In general terms, the equations of motion aresolved and the resulting velocity change of the pendulum mass is recorded.

C.3

Results

It may be shown that the characteristic time for a club head depends on
both the linear and non-linear springs of the model, and the initial velocityof the pendulum mass. However, owing to the non-linear nature of the Hertz
contact, as the impact velocity increases the characteristic times of various
face materials converge (for clubs with the same linear flexibility). It isthrough this feature that the pendulum test isolates the linear
flexibility of the club head and conformance is determined.
Using the model described above, various club stiffnesses and face materials were analyzed over a wide velocity range. Club stiffness values ranging
from 20 MN/m (highly conforming) to 3 MN/m (highly nonconforming) andface elastic modulus values ranging from 200 MPa (steel) to 3 MPa (polymer)
were considered. In all cases, the pendulum mass material was steel with aradius of curvature of one inch. The radius of curvature of the club head is
assumed to be ten inches.
For each combination of club stiffness and face material, the characteristic
time is plotted against the velocity change of the pendulum mass. A typicalplot is shown in Figure 7.
It has been found that the function:
T = A + BV −k(2)fits the data very well over a wide range of velocities and the asymptotic
nature of the function agrees with the physics of the equations of motion. It
has been found that the intercept (A) is a function primarily of the linear
club stiffness (with a small influence by the mass of the club head within therange of driving clubs). Therefore, by fitting experimental data to function(2), a relative measure of the club flexibility may be made. The intercept(A) is referred to as the infinite velocity characteristic time since the valueof the function at the intercept is the characteristic time for an infinitely
high impact velocity. The slope (B) indicates the magnitude of the velocity
dependence of characteristic time and is primarily a function of the club
face material and club stiffness. High values of the slope indicates soft facematerials.
27

280

275

Characteristic time (µ s)

270

265

260

255

250

245

240
−1
10

0

10

1

10

2

10

3

4

10
10
Impact Velocity m/s

5

10

6

10

7

10

Figure 7: Characteristic Time velocity dependence.
Figure 8 shows a plot of the data given in Figure 7 but with the velocity
raised to the power -0.329. The value of the exponent for this set of data
was generated by minimising the error between the model generated data
and function (2).
For the data plotted in Figures 7 and 8, the intercept value of 248.6 µs is
indicative of a marginally conforming club and the slope of 13.7 is indicativeof a metallic club face.
A similar procedure of finding the optimum value of the non-linear exponent for the range of club stiffness and face materials was used. The results
are given in Table 1.
Three options are now available for the practical application of data analysis:i. Fit a function over the range of data in Table 1. A preliminary functionthat fits the data well is:¡¢
k = exp −(1.07 + 87.6A−1.5 + 0.00099B) .
(3)
For a set of experimental data, one would first assume a value for k, findthe slope and intercept then use (3) to calculate a new k and iterate
until the solution converges.
28

280

275

Characteristic time (µ s)

270

265

260

255

T = 248.6 + 13.73 V−0.329

250

245

0

0.5

1

1.5

2

2.5

V−0.329

Figure 8: Linearised Characteristic Time velocity dependence.
ii. Since we are generally interested in conformance, concentrate on thevariability of k near the conformance limit intercept and make k afunction of slope only. The dependence of k on slope at the conformance
limit is shown in Figure 9. In this case, the procedure would be similar
to that in (i.) but with the exponent being adjusted based on slope
alone. i.e.,k = 0.3331 − 0.0003B.
iii. Select a constant value of the exponent that is most applicable andthat does not have an undue influence on any club. Table 1 shows thata value of -0.33 is applicable for metallic faced clubs and is suitable
for all others. The largest expected error using a value of -0.33 is 4
microseconds for a club with a very soft face having a slope of over 80,
refer to Figure 10.

C.4

Recommendation

A constant exponent value (-0.33) minimises the expected error on the intercept value for metallic clubs, does not have an undue influence on low
modulus faced clubs and does not complicate the data analysis procedure. It29

0.3350.33

k = 0.3331 − 0.0003 B

Optimum k

0.325

0.32

0.315

0.31

0.305

0.3

0

20

40

60
Slope

80

100

120

Figure 9: Dependence of optimum exponent on slope at conformance limit.
54Intercept Expected Error (µ s)

3

2

1

0

−1

−2

−3
0.305

0.31

0.315

0.32
Exponent

0.325

0.33

0.335

Figure 10: Effect of constant non-linear exponent on intercept error for aclub with a slope of 80.
30

γ
A

Kc
20
15
12
148.3 171.2 191.2

108209.5 234.2

7.1
248.6

5
296.24

4
331.2

3
382.5

200

k
B

0.325 0.326 0.328
15.50 14.10 13.10

0.329 0.328
12.35 11.53

0.329
11.06

0.331
9.85

0.331
9.14

0.334
8.28

120

k
B

0.324 0.325 0.327
18.62 17.05 15.84

0.328 0.329
14.44 13.89

0.329
13.38

0.330
11.91

0.331
11.07

0.331
10.08

69

k
B

0.320 0.323 0.325
23.93 21.86 20.34

0.326 0.327
19.17 17.89

0.328
17.18

0.329
15.33

0.329
14.26

0.330
12.96

20

k
B

0.310 0.314 0.317
46.55 42.77 39.95

0.319 0.321
37.78 35.24

0.323
33.90

0.325
30.36

0.327
28.26

0.327
25.80

10

k
B

0.301 0.306 0.310
69.80 64.42 60.36

0.312 0.315
57.31 53.61

0.317
51.65

0.321
46.31

0.323
43.18

0.325
39.40

5

k
B

0.289 0.295 0.300
104.9 97.57 91.84

0.303 0.308
87.44 81.89

0.310
79.14

0.315
71.32

0.318
66.62

0.321
60.94

3

k
B

0.279 0.286 0.291
141.1 131.9 124.8

0.295 0.300
119.1 112.1

0.302
108.5

0.309
98.14

0.312
92.00

0.316
84.37

Table 1: Variation of optimum exponent. The variables are: γ the elastic
modulus (GPa), A the intercept, B the slope, Kc the club linear stiffness(MN/m) and k the exponent.
is therefore recommended that the current proposed procedure by revised touse an exponent of -0.33 rather than -0.2 to linearize the experimental data.

D

Effect of club head mass on CharacteristicTime

The flexibility of the club head is the predominant parameter affecting thecharacteristic time for a club head. The club mass is a secondary effect. The
following quantifies this effect for clubs near the conformance limit using a31theoretical model described in previous documents. Experimental data isalso provided which shows good agreement with the theoretical data.
Figure 11 shows the distribution of club head mass for those driving clubs
submitted to the USGA in 2003. It may be seen that the vast majority ofclubs fall between 180 and 215 grams (99.5%). The average club head massis 197.4 g.
Figure 12 shows the infinite velocity characteristic time for a theoreticalclub head near the conformance limit (Kc = 8.0 MN/m) over the expected
range of club head mass (180 to 215 g).
It may be seen in Figure 12 that a change of 1 gram of club head mass
will result in an expected change of characteristic time by 0.25 µs (for clubs
near the conformance limit).
As a check on the model results, two club heads near the conformance
limit have been measured for characteristic time with mass added incrementally. Figure 13 shows the experimental values compared to the theoretical
expectation of the effect of mass. In each case, the model club flexibility was
adjusted in order that the infinite velocity characteristic times match before
mass was added using lead tape. It may be seen in Figure 13 that there isgood agreement between the experimental and the predicted mass effects.
60

Percentage of submissions

50

40

30

20

10

0
160

170

180

190
Club head mass (g)

200

210

Figure 11: Distribution of club head mass.
32

220

236

Infinite velocity char. time (µ s)

234

232

230

y = 0.253x + 180.89

228

R2 = 0.9988
226

224
170

175

180

185

190
195
200
Club head mass (g)

205

210

215

220

Figure 12: Theoretical effect of mass on characteristic time.

D.1

Conclusions

Over the expected range of driver club head mass, there is a direct but minor
effect of mass on the characteristic time. It is not believed that this effectwarrants the difficulty involved with obtaining the club head mass which
would almost certainly involve disassembling the club head.

E

Testing of coated clubs

We start by showing an example of how the application of a polycarbonate
coating could effect the pendulum test. We appreciate that the thickness
and type of coating described herein will not be durable within real impacts
but nevertheless we feel it acts as a good illustration of the phenomenon. We
shall give results for other coatings in due course
Polycarbonate Thickness(µm)
Fit
Uncoatedy = 248.72 + 7.08V −k248.9
y = 233.18 + 36.9V −k497.8
y = 235.72 + 44.02V −k749.3
y = 225.66 + 60.46V −k
33

255

254.5

Infinite velocity char. time (µ s)

254

253.5

253

252.5

252

251.5

251

250.5

250
184

186

188

190

192
194
Club head mass (g)

196

198

200

200

202

204

(a) Club A
247

246

Infinite velocity char. time (µ s)

245

244

243

242

241

240
188

190

192

194

196
198
Club head mass (g)(b) Club B

Figure 13: Effect of adding mass artificially to a club head, two examples.
Each figure shows the theoretical prediction and the actual measurementsfrom the pendulum.

34

As can be seen we have managed to reduce the value of the intercept from248.72µs to 225.66µs merely by adding a coating (this coating has been
demonstrated to have no effect on the cannon test).
We make the observation that most paints we have measured have beenless than 60µm thick. We also fully appreciate that a coating of the typeabove is likely to have other effects on the movement of the ball. We haveapplied a model to the solution of this problem and found that this adequately
captures the physics of the problem. The model includes a compliant layer
above a substrate and in essence our findings are that: for a thin coating theclub behaves as if it is uncoated, as the coating thickness increases the club’s
velocity dependence is changed by the presence of the coating and finally fora very thick coating the nature of the substrate becomes virtually irrelevant.
In these three régimes we find that k = 0.33 adequately models the velocity
dependence of the impact in the first and last cases, and for the second casethe value of k depends on the coatings thickness and its flexibility (reflectedin the value of the slope B).
At this point we add the comment that the application of a coating will
artificially increase the slope B and as such we can detect whether a coating
is present using the value of B. If the slope is below a certain threshold(namely 20) then we deem that the test using a value of the exponent of
0.33 adequately captures the physics of the impact. If the slope exceeds thisvalue then either the club is behaving as a plastic or persimmon club, or it iscoated. We seek to differentiate between these two cases using the thickness
probe.
The thickness probe automatically detects whether the club is ferrous or
non-ferrous (or more specifically whether the club is magnetic or not). In
the former case it uses magnetic inductance to measure the thickness of thecoating and in the latter case it uses an eddy. For plastic or persimmon clubs
the instrument fails in that there is effectively no substrate and in these cases
the results of the pendulum test stand. We note that this may also occur forclubs with excessive coating thicknesses on them and again the pendulum
result is an accurate reflection of how the club would behave in the cannon
test and thus in a real impact. We note that the thickness of the coating
may be measured in other ways if necessary.
The scope we are using is an Exacto/FN/e scope, for more informationsee
http://www.elektrophysik.de/e/p_exacto.html
We shall describe in detail the way in which the club is tested in thesection on the outline of the testing protocol. Here though we mention that35

the value of the exponent is given by the formula:kcoating = 0.33 + β(h) (ln(B) − ln(13.34)) ,where B is the slope calculated from the original data and β(h) will be given
in due course. The value of 13.34 corresponds to a slope of a typical clubnear the limit with a particular set of parameters chosen to characterise theclub. The value of the gradient β(h) can be found from the table:
Coating thickness (µm)
h
< 63.5
127
254
381
508
635
762
1016
1270
1524
2032
> 2032

Gradientβ(h)00.0752
0.09650.11570.1176
0.1091
0.09280.06430.04450.03270.01830

We note that the value of β(h) at intermediate values of h can be found bysuitable interpolation (either linear or cubic). We note the phenomenon thatas the thickness increases the value of k will ultimately return to 0.33, corresponding to a coating which is so thick that the behaviour has dominatedover that of the substrate. To set this within context five business cards
are approximately 1790µm thick. In figure 14 we show the values of kcoating
acquired in this way.
We now show results obtained using this method

36

0.65

0.6
381µ m508µ m

0.55635µ m254µ m762µ m127µ m

0.451016µ mkcoating0.51270µ m0.41524µ m2032µ m0.3563.5µ m
0.3
2.5

3

3.5

4

4.5

5

ln(Slope)

Figure 14: Variation of kcoating with coating thickness and slope. Dashedlines are used for thicknesses greater than 500µs. Note that as the coating
thickens the value of k returns to value of 0.33.

37

Coating
Uncoated
109.2µm Polycarbonate
248.9µm Polycarbonate
375.9µm Polycarbonate
497.8µm Polycarbonate
749.3µm Polycarbonate
396.2µm Delrin
500.4µm Delrin
645.2µm Delrin
838.3µm Delrin

Intercept withk = 0.33 (µs)
248.8
246.6
233.7
233.2
236.3
226.5
241.2
235.0
234.1
232.7

Error(µs)
0.0
2.2
15.1
15.6
12.5
22.3
7.6
13.8
14.7
16.1

Intercept with Revised Error
variable k (µs)
(µs)
248.8
0.0
246.6*
2.2*
245.3
3.5
249.4
-0.6
251.8
-3.0
246.0
2.8
248.7
0.1
246.3
2.5
246.1
2.7
245.1
3.7

In the case of the second row (marked with an asterisk) the slope was below
the threshold of 20 so the algorithm was not applied.

38