NOVEL DESIGN OF SPORTS EQUIPMENT THROUGH
IMPACT VIBRATION ANALYSIS
S. Knowles, Ft. Brooks, J. S. B. Mather
Depariment of Mechanical Engineering,
Universly of Nottingham, University Park
Nottingham, NG7 ZRD, UK.
ABSTRACT. Modern sports equipment, for games such as
cricket. baseball and goii is the subject of increasing
amounts of research. Vibrational analysis has proved a
particularly useful tool with most work to date concentrating
on the modal response of the body to a given input. During
impact, however, the vibrational response also has an effect
upon the force of contact which is a critical parameter in
design.
%
w
0
i
Subscripts
1
bat
2
ball
I
representing the ih mode
representing the n’” time interval
:F
representing a simplefiree
beam
CF
representing a clamped/free beam
Acomputer model has been developed to simulate the case
of a ball impacting upon cricket and baseball bats. The use
of composite materials increases the scope of the designer
and makes the impact and vibrational analysis all the more
important. By altering the design and material properties
the vibrational response of the bat can be ‘tuned’to achieve
increased performance.
INTRODUCTION
The increasing demand among consumers for the latest
high-performance sporl equipment is fuelling scientlic and
engineering research by sports equipment manufacturers.
Such research, together with the use of stiff, light-weight
composite materials has spawned many novel features and
designs, particularly for golf clubs and baseball bats. One
sport which has seen no such development is cricket. For
this reason cricket bats have been chosen for our test and
analysis but many of the techniques and results could be
applied to other types of sports, particularly baseball bats.
Impact tests performed on two different cricket bats are
used to verify the computer predictions. These two sources
of data can be used as an effective tool in the design and
analysis of a high performance composite bat.
NOMENCLATURE
A
a
E
P
I
k,
L
h
Ill
P
0
:
T’
AT
7
initial velocity
deflection
undamped natural frequency
distance along length of the beam
mode shape function of the beam
beam crowx.ectional area
approach of bat and ball ( = w, - w,)
Young’s modulus
coefficient of restitution (= -v&J
contact force
second moment of area
Hertz contact constant
beam length
eigenvalue where ?+4 = pA&El
mass
beam density
factor for evaluating y for beams with nonclassical
boundary conditions
time
torsional
stiffness
TWEI
time interval used in numerical calculations
variable of integration
In order to understand the factors affecting bat performance
an understanding of the mechanics of impact are required.
In the simplified case a cricket bat and ball can be modelled
as a sphere impacting upon a beam. This analysis uses a
combination of contact and vibration theory [l], however the
problem cannot be accurately defined using classical
boundary conditions.
The batsman’s hands can be regarded as a clamp and the
bat can be modelled
as a beam attached to a torsion spring
representing a relatively flexible handle opposite a free end
[a
Modifying the impact theory [l] to take account of the nonclassical boundary conditions [Z] gives a model for the
390
impact of acricket ball on a bat which can be programmed
into a computer. This model can then be used to assess
the performance of various bat designs.
the following equation [2].
X(xj = (sinhx
+ sinJ.4
(3)
In order to validate the model, experimental tests have been
performed on two different bat designs and deflection
measurements have been taken at various impact points.
* cc&h Ax)
Modal analysis as used on baseball bats to assess the size
and position of the “sweet spot” [3], can also be applied to
a cricket bat. This paper also uses modal analysis to
assess the accuracy of the impact response theory and to
validate the experimental results.
The natural frequency oi and eigen values & are calculated
from the equation
T’ = A L
(SinhL coshhL - sinALcoskL)
1 + coshLcosh~L
THEORY
and lie between the corresponding values for a simple/ free
beam and a clampedflree
beam depending on the torsional
stiffness (T = 0 w = osF; T = - o I 61~~). This natural
frequency can thus be expressed as;
By combining the general equation for forced vibration of a
continuous beam [4] with the Hellz contact theory, a
general&d equation can be derived. This links the forcing
function to the beams vibrational response [l] for impact at
some point x = c.
where C! varies with the torsional stiffness T’. Tables of Q
values for a range of values of T* [2] greatly simpliiy the
evaluation of wi and 4.
These values can then be used in equation (2) to evaluate
how the ball and beam deflections, contact force and ball
velocity vary with time.
The above equation, containing the force function on both
sides, does not have an analytical solution. A numerical
solution, however may be found by using the small
increment method, in which the contact force is regarded as
constant over any time increment AT.
TESTS
Cricket bat behaviour
different tests:
1.
2.
3.
For the nh time interval t = EAT, equation (1) can be written
as
has been investigated using three
Modal analysis
Theoretical computer model
Cricket ball impact tests
Two different bats have been studied; a traditional style
lightweight willow cricket bat and a rather more heavy glass
reinforced plastic (GRP) composite test bat. A photograph
of the two bats is shown in Figure 1. Table 1 shows the
key properties of the two bats.
w h e r e t h e t e r m (AT)’ i On.,.,
1-t
F, is the numerical
evaluation of j d i F df
0
0
Table 1 - Physical Properties of the Two Test Bats
The mode shape function X, and the natural frequencies o,
are dependent on the beam properties and the boundary
conditions. For a beam with a torsion spring at one end
and free at the other the mode shape function is given by
The values of El for the traditional bat are based on an
average cross-section along the length of the blade. For
the GRP bat with a constant cross-section of several
391
component materials, the overall flexural rigidity is
calculated from (El),, = (El), + (El), + The torsional
stiffnesses were measured in static deflection tests.
Modal Analysis
For the modal analysis of the two bats a standard impact
The accelerometer was
excitation test was used.
positioned at the bat tip and five hammer impacts were
made at each of twenty-three impact points along the blade.
For both bats the test was performed with a clamped
handle boundary conditions in order that the results could
be compared with the other methods.
Table 2 - Resonant Frequencies
of the Traditional Bat
Theoretical Computer Model
It can be seen that the theoretical model is not very
accurate at predicting the frequencies of the higher modes.
This is due to the non-uniform cross-section along the
length of the willow blade.
A computer program has been written to solve equation (2)
numerically. Results have been obtained for various impact
points along the length of the blade at various impact
velocities. For each case the resuits were output in
graphical form showing impact point deflection versus time.
The summation included the first ten modes although
modes four and above make a negligible contribution.
The composite GRP bat does have a constant cross-section
and therefore the match between the modal analysis data,
the theoretical computer model and the impact test (Figures
6, 7 and 8 respectively) is much closer as can be seen in
Table 3.
Cricket Ball Impact Test
This test is used to measure the deflections caused by
impact of a cricket ball upon the bat. The experimental set
up is shown in Figure 2. An accelerometer is attached to
the underside of the bat at the impact point for each of the
nine positions along the length of the bat. The signal from
the accelerometer is fed into a PC via a charge amp and an
ADC. The data acquisition is triggered using an infrared
beam and detector. As the ball breaks the beam the data
sampling commences. The time between the first reading
and the impact is used to determine the impact velocity.
The accelerationRime
data is imported into a Microsoft
Excel spreadsheet. A simple small interval numerical
integration can then be performed to give first the velocity
and then a deflection I time curve. showing the magnitude
and frequency of the bat oscillations. For each impact point
several sets of impact data are acquired. The peak bat
deflection is also found for each test and an average for
each impact point is calculated.
Table 3 Resonant Frequencies
of the GRP Composite Bat
It should, however, be pointed out that for the third mode
the frequency found by the theoretical model differs quite
significantly from the experimental value.
RESULTS
From the experimental results (Figures 5 8 8) and the
theoretical computer model (Figures 4 & 7) for each of the
bats the peak deflection at the impact point can be
determined. Data was acquired at nine evenly spaced
points along the length of the bat (at 65 mm intervals, from
25 mm from the handle to 545 mm). The values of peak
deflection are shown in Figures 9 & 10.
Frequency
Figure 3 shows the deflected mode shapes and the
resonant frequencies for the traditional willow cricket bat.
Frequency data was also obtained from the computer
model, Figure 4. and the impact test, Figure 5. A
comparison of the results from these three methods is given
Table 2.
It should be noted that these graphs do not represent mode
shape as they encompass the contributions of all modes
and represent the response of each point to an impact at
that particular position.
392
model can predict, with reasonable accuracy, the first
vibrational mode of a bat. It is this first mode of vibration
which is dominant in the evaluation of the bat deflection.
DISCUSSION
The results of the frequency analysis ie. the comparison
between the modal analysis. the theoretical model and the
impact tests show very good agreement for the first mode
of both bats. It is this mode which dominates the response
of the bat, particularly for impact in the end two-thirds of the
blade which is the main hitting area.
For the uniform blade structure, the frequency of the semnd
mode is also identified. The shape of the impact point
deflection curve is also very well predicted with the factor
between the theory and experiment being near constant.
This allows the theory, although not accurate enough to be
used to predict exact bat deflections. to be used in a
qualitative manner to compare two or more alternative bat
designs.
The second mode of the composite GRP bat is also
predicted by the theory with reasonable accuracy with
significant errors occurring only at the third mode. The
higher modes of the traditional bat are not very accurately
predicted due to the non-uniform cross-section of the blade.
The theoretical model assumes constant properties along
the length of the blade and consequently for other than the
first mode where the bending of the handle is dominant, the
experimentally recorded frequencies vary significantly from
those predicted by the theory.
The theory, as well as calculating the bat deflection with
time, also evaluates the ball’s displacement and can
consequently be used to predict the output velocity of the
ball for a given set of impact conditions. This is particularly
useful when assessing the performance of a particular bat
design. The following section uses the model to assess the
relative merits of some different bat designs.
From Figure 10 it is evident that the theoretical model is
consistently over estimating the bat deflection by a factor of
around 1.6 although the shape of the two cutves matching
very closely. A similar over estimate (1.65) exists for the
impact at the tip of the traditional bat. For intermediate
points (0.155 to 0.35), however, the theoretical model and
the test results are in good agreement. Again this variation
is likely to be caused by the assumption of a uniform crosssection beam in the theory.
DESIGN ANALYSIS
In order to assess the contribution of various design
parameters on the overall bat performance, the model of
the GRP composite bat was used with changes in the
following parameters:
flexural rigidity El
handle stiffness T
mass per unit length (of the blade) pA
The fact that the theoretical model over estimates the
impact point deflection implies that in practice some of the
energy which the theory predicts as going into the
transverse vibration of the bat, is being lost. Possible
causes of this energy loss are:
1.
2.
For the analysis of each of these factors the values of the
other properties are those originally used, as given in Table
1. In each case an impact of 5 m/s, 6cm from the free end
was used.
inelastic collision between the bat and ball which
the theory assumes is perfectly elastic.
excitation of other vibrational modes not measured
by the accelerometer eg. torsional and lateral
vibrations.
The flexural rigidity was varied over a range of El = 10006000 Nm’ in stages of 1000 Nm’ (original value El-4000
Nm’). Figure 11 shows the effect of these changes on the
coefficient of restitution e ( = v,,,&,). The shape of this
curve indicates that the stiffer the bat (the higher El) the
greater the rebound velocity of the ball. Large increases in
the value of El, however, only give a small increase in e.
The theory based around equation (2) assumes all the
kinetic energy initially in the ball is transferred into the bat
vibrations and the outward velocity of the ball. In practice
some of this energy is lost as heat and noise during the
impact resulting in less energy being transferred to the bat
and ball after impact. Equally, when torsional and lateral
modes of vibration are excited less energy goes into the
transverse mode being measured. The experimental impact
test does not measure the accelerations resulting from
these modes because they act in planes normal to that of
the measured transverse deflection.
A range of handle stiffnesses from T = 500-3000 Nmlrad in
500 Nm/rad steps were studied (original value T=lZOO
Nmlrad). Figure 12 illustrates how the coefficient of
restitution is affected by these changes. As can be seen
there is very little effect.
Figure 13 shows the effect of changing values of pA from
0.6 kg/m to 3.0 kg/m in 0.2 kg/m steps (Original value
pA=2.24kg/m).
Changing the value of pA which, for a bat of
a given length, is equivalent to increasing its mass. has a
dramatic effect on the hitting power of the bat. A very
lightweight bat gives a very low coefficient of restitution. As
the mass of the bat is increased so does e. For
heavyweight bats, increasing the mass still further has less
Applications of the Model
From the comparison with the modal analysis and the
cricket ball impact tests, the theoretical model has been
shown to be a useful tool in analysing the behaviour of
cricket bat structures during impact with a cricket ball. The
393
of an effect than for lighter bats.
(41
Timoshenko, S.P.
Vibration problems in Engineering,
2nd Edition. Van Nostrand Company Inc. [19X’).
The obvious conclusion from a purely analyiical viewpoint
is that a very stift. very heavy bat would be capable of
hitting the ball the furthest. In practice the performance of
a batsman also depends on his ability to swing the bat
effectively. For this reason these results can be sssn as
the first step in the analysis and prediction of criiket bat
performance.
Figure 1 -The Two Test Bate
Lefi GRP Composite Bat, Right Traditional Bat
CONCLUSIONS & FURTHER WORK
As identified in the discussion the torsion spring#ree beam
model does not take any variations in cross-section into
account. Modification to the mods shape function (2) and
the frequency equation can be made to accommodate this.
It is also envisaged that the other vibrational modes of the
bat will be investigated in order to assess their effect on the
measured transverse deflections and the inclusion of the
effect of an inelastic collisions should improve the overall
accuracy of the model.
The hands of a batsman cannot provide a rigid clamp
around the bat handle. An anaiysis of various boundary
conditions and how closely they can simulate hand-held bat
will enable a better understanding of the factors affecting
bat performance.
Notwithstanding these desirable modifications, it has been
shown that the model, in its present form, provides a good
basis for gaining a greater insight into the performance of
cricket bats. Furthermore, this model has been validated
for the dominant first mode of vibration using two
independent experimental techniques. As a design tool, the
model is already giving a good indication of the parameters
controlling performance and thus will allow novel design
solutions invohring corr~osite materials to be investigated
with confidence.
ACKNOWLEDGEMENTS
Figure 2 - Impact Test Layout
The authors wish to acknowledge the financial support and
encouragement of CadCam Technology Ltd. and the
EPSRC (UK) for the award of a CASE studentship.
III
PC
REFERENCES
PI
Golcfsmfth.
W.
Impact, the Theory and Physical Behaviour
Colliding So/ids,
Charge
Amplifier
of
Ed. Arnold Ltd. London, pp. 106-l 11 (1960).
PI
Gorman. D.J.
Clamped
Free Vibration Analysis of Beams and Shafts,
John Wiley 8 Sons, pp 7-16, (1975).
PI
Tognarelli, D and Dunbar, E.
How Sweet ir is!! - Can your Basebll Baf Measure
UP?,
Accelerometer
Sound &Vibration, January 94 pp 7-14, (1994).
394
\
56800 :
Deflection Im
Deflection Im
Figure 9 - Peak impact
Point Deflection: Traditional Bat
.c 0.025 r -cricket
Figure12 - Effect of Varying
The Torsional Stiffness T of the Handle
on the Coefficient of Restitution e
0.475
Ball Impact Tad -7
0.473
0.471 ’
OO.469
0.467
0.465
Distance from Clamped End Im
Figure 10 - Peak Impact
Point Deflection: GRP Composite Bat
E 0.025
:E
Figure 13 - Effect of Varying
The Mass per Unit Length pA
on the Coefficient of Restitution e
-,
0.02
0.6
0.5
0.4
0.3
0.2
0.1
0
5
5 0.015
‘ij
g 0.01
%
n 0.005
Y
%
0
P
Mass per Unit Length pA /kg/m
Distance from Clamped End Im
Figure 11 - Effect of Varying
The Flexural Rigidity El
on the Coefficient of Restitution e
396
+