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Chủ đề chính: Grip, Racquet
Tóm tắt nội dung (trích từ tài liệu gốc): Vibration absorption in the tennis grip and the effects on racquet dynamics Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy By Nicholas James Savage BSc. Supervisor Professor Aleksandar Subic School of Aerospace, Mechanical and Manufacturing Engineering RMIT University Submitted August 2006 Declaration I certify the work presented in the thesis is that of the candidate alone, except where due acknowledgement is given, and has not previously been submitted, in whole or in part, to qualify for any other academic awards. The content of the thesis is the result o
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Nội Dung Gốc (Tiếng Anh)¶
Vibration absorption in the tennis grip and the effects on
racquet dynamics
Submitted in fulfilment of the requirements for the degree of Doctor of
Philosophy
By
Nicholas James Savage BSc.
Supervisor
Professor Aleksandar Subic
School of Aerospace, Mechanical and Manufacturing Engineering
RMIT University
Submitted August 2006
Declaration
I certify the work presented in the thesis is that of the candidate alone, except where due
acknowledgement is given, and has not previously been submitted, in whole or in part, to
qualify for any other academic awards. The content of the thesis is the result of the work
carried out since the official commencement date of the approved research program. Any
editorial work, paid or unpaid, carried out by a third party contribution is acknowledged.
Signed
Nicholas James Savage
Date
i
Acknowledgements
I would like to thank everyone who has contributed to my research. This thesis has been a
long and difficult adventure that has taught me much more than I expected. The work
presented in thesis would not have been possible without the help and advice of many people.
I would firstly like to thank my family for supporting me throughout my PhD. My father,
Neil Savage, has been instrumental in helping me complete my thesis, for which I am
eternally grateful. His knowledge and support have helped me conquer the challenges I have
faced during my candidature. My mother has also been incredibly supportive during my PhD
and this has helped me keep my feet on the ground and achieve my best. Grandparents are
great people and should be treated as such. My grandparents (Edith, Joan and George) have
shared their knowledge, experience and support with me throughout my research, and it has
proven invaluable for helping me complete my work. To my family and friends in England
and Australia, I thank you all.
My supervisor, Prof. Aleksandar Subic, proved to be one of the best mentors a researcher
could ask for. His technical knowledge and determination were an invaluable help to me in
achieving my goals throughout my PhD. The support provided by Aleks throughout my
research drove the project and allowed me to achieve my objectives. I would like to take this
opportunity to thank him for his patience and belief in me.
ii
"If I have seen further it is by standing on the shoulders of giants."
-Sir Isaac Newton
iii
Table of contents
Declaration ............................................................................................................................................. i
Acknowledgements .............................................................................................................................. ii
Table of contents ................................................................................................................................. iv
List of figures ..................................................................................................................................... viii
List of tables........................................................................................................................................ xii
Nomenclature..................................................................................................................................... xiii
Publications ....................................................................................................................................... xiv
Summary ............................................................................................................................................... 1
1 Introduction .................................................................................................................................. 4
1.1 Rationale.............................................................................................................................................. 7
1.2 Literature review ................................................................................................................................. 8
1.2.1 Overview of lateral epicondylitis......................................................................................................... 9
1.2.2 Dynamic behaviour of tennis racquets .............................................................................................. 12
1.2.3 Mechanics of the tennis grip.............................................................................................................. 20
1.2.4 Active damping technology................................................................................................................ 28
1.2.5 Summary of relevant body of knowledge ........................................................................................... 29
1.3 Research objectives and scope .......................................................................................................... 33
1.3.1 General objectives ............................................................................................................................. 34
1.3.2 Specific objectives ............................................................................................................................. 34
1.4 Thesis overview................................................................................................................................. 35
1.4.1 Chapter 2........................................................................................................................................... 36
1.4.2 Chapter 3........................................................................................................................................... 36
iv
1.4.3 Chapter 4........................................................................................................................................... 37
1.4.4 Chapter 5........................................................................................................................................... 37
1.4.5 Chapter 6........................................................................................................................................... 38
2 Comparing the structural dynamic properties of two tennis racquets ................................ 39
2.1 Methodology ..................................................................................................................................... 43
2.1.1 Experimental set-up........................................................................................................................... 44
2.1.2 Test racquets...................................................................................................................................... 48
2.2 Experimental results .......................................................................................................................... 50
2.2.1 Racquet A .......................................................................................................................................... 51
2.2.2 Racquet B .......................................................................................................................................... 55
2.3 Discussion of results.......................................................................................................................... 58
2.3.1 Effect of racquet strings..................................................................................................................... 61
2.3.2 Vibration excitation ........................................................................................................................... 63
2.4 Conclusions and significance ............................................................................................................ 64
3 Characterisation of tennis grip pressure distributions ......................................................... 67
3.1 Identification of locations in the tennis grip with the greatest contact pressure ................................ 71
3.1.1 Instrumentation ................................................................................................................................. 72
3.1.2 Testing protocol................................................................................................................................. 73
3.1.3 Results and discussion....................................................................................................................... 75
3.2 Use of strain gauge cantilever system grip characterisation.............................................................. 80
3.2.1 Experimental set-up........................................................................................................................... 83
3.2.2 Testing protocol................................................................................................................................. 83
3.2.3 Results and discussion....................................................................................................................... 87
3.3 Real time analysis of tennis grip pressure distribution characteristics .............................................. 96
3.3.1 Experimental set-up........................................................................................................................... 98
3.3.2 Testing protocol............................................................................................................................... 102
3.3.3 Results and discussion..................................................................................................................... 104
v
3.4 Conclusions and significance .......................................................................................................... 127
4 Experimental investigation of damping in tennis racquets................................................. 130
4.1 Experimental set-up......................................................................................................................... 133
4.2 Results and discussion..................................................................................................................... 137
4.2.1 Time based damping estimation ...................................................................................................... 149
4.2.2 Signal processing ............................................................................................................................ 153
4.2.3 Quantifying the effectiveness of the piezoelectric damping system ................................................. 160
4.3 Conclusions and significance .......................................................................................................... 163
5 The effect of grip pressure distribution on racquet frame vibrations damping................ 167
5.1 Establishing correlations between grip pressure and racquet vibration damping ............................ 169
5.1.1 Data Exclusion ................................................................................................................................ 176
5.1.2 Defining an appropriate grip pressure............................................................................................ 178
5.2 Grip damping results ....................................................................................................................... 181
5.2.1 Grip damping with respect to mode shapes..................................................................................... 182
5.3 Grip damping model........................................................................................................................ 188
5.4 Discussion of findings ..................................................................................................................... 191
6 Conclusions and recommendations...................................................................................... 194
6.1 Conclusions ..................................................................................................................................... 195
6.1.1 General outcomes............................................................................................................................ 196
6.1.2 Specific outcomes ............................................................................................................................ 199
6.2 Recommendations ........................................................................................................................... 202
7 References................................................................................................................................ 205
8 Appendices............................................................................................................................... 214
vi
8.1 Appendix 1 ...................................................................................................................................... 214
8.1.1 Development of a strain gauge cantilever system for the measurement of tennis gripping forces .. 214
8.2 Appendix 2 ...................................................................................................................................... 223
8.2.1 Locations of hydrocell pressure sensor attachments on the racquet handle ................................... 223
8.3 Appendix 3 ...................................................................................................................................... 226
8.3.1 Half power bandwidth damping calculation ................................................................................... 226
8.4 Appendix 4 ...................................................................................................................................... 230
8.4.1 Subjective Gripping data................................................................................................................. 230
vii
List of figures
Figure 1. Anatomical diagram of lateral epicondylitis (Source: med.umich.edu)...................10
Figure 2. Sweet spot and other important locations on the tennis racquet (Source: Brody et al.
2002; Kotze et al. 2000)...................................................................................................14
Figure 3. Mode shapes for clamped and freely suspended racquet condition (Source: Kotze et
al. 2000) ...........................................................................................................................15
Figure 4. Example of node lines and locations of a tennis racquet (Source: Cross 2001).......16
Figure 5. Wave propagation from the centre of percussion along the racquet strings shown at
different time intervals (Source: Brannigan and Adali 1981)..........................................19
Figure 6. Example of tennis gripping force traces for the forehand stroke (Source: Knudson
and White 1989)...............................................................................................................22
Figure 7. Forces in the tennis grip resulting from the tennis ball impact (Brody et al. 2002).23
Figure 8. Schematic of tennis racquet modal test set-up .........................................................45
Figure 9. Racquet geometry showing excitation points (response measured at point 31).......46
Figure 10. Racquet dimensions................................................................................................48
Figure 11. Frequency response measurements for racquet A (with strings)............................51
Figure 12. Frequency response measurements for racquet B (with strings)............................55
Figure 13. Node location associated with the first bending mode for racquets A and B.........58
Figure 14. Comparison of average FRF's for racquets A and B .............................................60
Figure 15. Comparison of average FRF's for racquet B with and without strings ..................62
Figure 16. Pressure film layout................................................................................................72
Figure 17. Continental gripping technique ..............................................................................73
Figure 18. Developed pressure film attached to racquet handle ..............................................75
viii
Figure 19. Pressure film results for a forehand stroke for the metacarpals and thumb ...........77
Figure 20. Pressure film results for a forehand stroke for the MP joint and phalanges...........78
Figure 21. Strain gradient: a) relationship between strain and distance from load applied to
the beam; b) measurement locations on the cantilever beam ..........................................82
Figure 22. Location of cantilever beams with respect to the gripping hand for a)distal
phalanx; b) proximal phalanx; c) MP joints and distal metacarpals; d) metacarpals ......85
Figure 23. Diagram of drop test set-up ....................................................................................86
Figure 24. Threshold calculation .............................................................................................89
Figure 25. Sample of measured strain gauge for a visual test at: a) distal phalanx; b) proximal
phalanx; c) MP joints and distal metacarpals; d) metacarpals.........................................94
Figure 26. Schematic of hydrocell sensor data collection set-up...........................................101
Figure 27. Accelerometer locations .......................................................................................102
Figure 28. Experimental set-up on a tennis court ..................................................................103
Figure 29. Racquet handle: a) upper and b) lower gripping sections ....................................105
Figure 30. Sample of right-handed forehand stroke handle section pressure variation
measurements during impact .........................................................................................108
Figure 31. Pre-impact pressure distribution in a forehand stroke for: a) upper handle and b)
lower handle...................................................................................................................110
Figure 32. Post-impact pressure distribution in a forehand stroke for: a) upper handle and b)
lower handle...................................................................................................................112
Figure 33. Sample of service stroke handle section pressure variation measurements during
impact.............................................................................................................................113
ix
Figure 34. Pre-impact pressure distribution for the service stroke at: a) upper handle and b)
lower handle...................................................................................................................115
Figure 35. Post-impact pressure distribution for the service stroke at: a) upper handle and b)
lower handle...................................................................................................................116
Figure 36. Sample of backhand slice stroke handle section pressure variation measurements
during impact .................................................................................................................118
Figure 37. Pre-impact pressure distribution for the backhand slice stroke at: a) upper handle
and b) lower handle........................................................................................................120
Figure 38. Post-impact pressure distribution for the backhand slice stroke at: a) upper handle
and b) lower handle........................................................................................................121
Figure 39. Free suspension racquet-ball impact experiment: a) front view; b) side view .....134
Figure 40. Schematic diagram of hand-held ball drop test ....................................................136
Figure 41. Average frequency response of freely suspended tennis racquet (B) for tennis and
golf ball impacts.............................................................................................................138
Figure 42. Average frequency response of hand-held tennis racquet (B) for tennis and golf
ball impacts ....................................................................................................................139
Figure 43. Comparison of racquet frequency responses for different gripping conditions ...140
Figure 44. Damping and gripping correlations for racquet A................................................147
Figure 45. Damping and gripping correlations for racquet B................................................147
Figure 46. Decaying vibration ...............................................................................................150
Figure 47. Sample of raw acceleration data...........................................................................154
Figure 48. Sample of acceleration data after frequency filtering...........................................155
Figure 49. Sample of acceleration data after smoothing (Savitzky and Golay, 1964) ..........156
x
Figure 50. Damping ratio box-plots for freely suspended and hand held racquets ...............159
Figure 51. Definition of vibration ( a ) and pressure ( p ) peak parameters ...........................172
Figure 52. Frequency response of a tennis racquet with nodal and non-nodal impacts ........177
Figure 53. Mode shape of tennis racquets 1st bending mode .................................................179
Figure 54. Damping correlations using the total pressure applied to the racquet handle ......181
Figure 55. Damping correlation using the pressure applied to the racquet handle in the z
direction .........................................................................................................................184
Figure 56. Damping correlations using the grip pressure applied to the upper handle section
........................................................................................................................................187
Figure 57. Linear damping correlations using the grip pressure applied to the lower handle
section ............................................................................................................................187
Figure 58. Non-linear damping correlation using the grip pressure applied to the lower handle
section ............................................................................................................................189
Figure 59. Schematic of the cantilever beam and circuit diagram.........................................215
Figure 60. Full Wheatstone bridge orientation on a cantilever beam ....................................216
Figure 61. Wheatstone bridge circuit diagram.......................................................................217
Figure 62. Test racquet handle butt (end view dimensions) ..................................................219
Figure 63. Test racquet handle butt (side view dimensions) .................................................220
Figure 64. Hand grip cantilever test system...........................................................................221
Figure 65. Calibration chart for beams A: D .........................................................................222
Figure 66. Racquet handle side configuration .......................................................................223
Figure 67. Half power damping parameter identification......................................................227
Figure 68. Example of half power damping parameter identification ...................................228
xi
List of tables
Table 1. Summary of existing knowledge relevant to this investigation.................................33
Table 2. Racquet mass and centre of mass location.................................................................49
Table 3. Modal analysis results for racquet A ........................................................................53
Table 4. Modal analysis results for racquet B..........................................................................56
Table 5. Outline of experimental investigation of tennis gripping pressure............................71
Table 6. "Visual" drop test gripping time results ....................................................................90
Table 7. "Blind" drop test gripping time results ......................................................................90
Table 8. Evaluation of response parameters of racquet A for tennis ball impacts................145
Table 9. Evaluation of response parameters of racquet B for tennis ball impacts.................145
Table 10. Logarithmic decrement calculations for a hand held racquet ................................157
Table 11. Logarithmic decrement calculations for a freely suspended racquet.....................158
Table 12. Logarithmic decrement and damping ratio of racquet A .......................................161
Table 13. Logarithmic decrement and damping ratio of racquet B .......................................161
Table 14. Vibration and pressure measurements used for grip damping correlations...........173
Table 15. Hydrocell attachment locations for the continental forehand grip ........................224
Table 16. Hydrocell attachment locations for the service and backhand slice grips .............225
Table 17. Response data of racquet A for tennis ball impacts...............................................230
Table 18. Response data of racquet B for tennis ball impacts ...............................................231
xii
Nomenclature
F - Force
a - Acceleration
k - Stiffness
m - Mass
t - Time period
n - Natural frequency
d - Damped natural frequency
- Period of oscillation
d - Damped period of oscillation
- Logarithmic decrement
- Damping ratio
xiii
Publications
Savage, N. & Subic, A. (2006). Relating grip characteristics to the dynamic response of
tennis racquets. In: Moritz, E.F. & Haake, S.J. ed. 6th International Conference on the
Engineering of Sport, 2006, Munich, pp.155-160.
xiv
Summary
The modern game of tennis has changed in recent years as a result of lightweight, stiffer
racquets. The evolution of the tennis racquet, with respect to both design and materials,
has increased the speed of the game but also the levels of stress placed on the player's
bodies. Larger racquet heads generate greater top spin on the ball, allowing the player to
strike the ball harder and still be able to place the ball in court. However, by striking the
ball harder the strains on the player's upper extremities caused by the transmission of
ball-racquet impact energy are increased. Injuries such as lateral epicondylitis (tennis
elbow) are thought to be both instigated and aggravated by the transfer of racquet shock
and vibration. Therefore, it is important to manage the levels of shock and vibration
transmission to the player, in order to reduce the associated performance inhibiting
effects.
Racquet energy that causes upper extremity injuries is transferred to the tennis player via
the tennis grip in the form of shock and vibration. Parameters defining the degree of
shock and vibration transmission are the inherent properties of the racquet and the
mechanics of the tennis grip. This thesis presents an experimental investigation into the
transmission of racquet vibration to the player's hand and forearm. Experimental
techniques have been used to quantify the main parameters defining the transmission of
vibration via the tennis grip.
The mechanics of grip damping show precisely how the transfer of racquet vibration to
the player occurs. The tennis grip has been experimentally quantified using various
sensing equipment. Gripping devices used in previous research have been modified,
1
manufactured and used in conjunction with pressure sensitive film and hydrocell sensors.
Each of the experimental techniques used in this research has been designed to examine
different aspects of the tennis grip. Manufactured strain gauge cantilever systems have
been utilised for a real-time analysis of the grip tightness variations during impact. The
cantilever technique enabled estimations of anticipation times, allowing for a description
of the tennis grip regarding the time of maximum grip force and the initial increase in
grip force with respect to the time of impact. Specialised pressure sensitive film has also
been utilised to identify important contact locations within the tennis grip where the
magnitudes of pressure are greatest. These two primary laboratory tests provided
information for further experiments, allowing for the analysis of grip pressure distribution
during different stroke types using real-time data acquisition.
Variations in the distribution of grip pressure during impact for three stroke types have
been measured by attaching hydrocell pressure sensors to the racquet handle at multiple
contact locations. Calculated pressure distributions show the magnitudes of gripping
pressure at multiple contact locations in the tennis grip. These pressure distribution
characteristics have been used to analyse the applied gripping pressure of the player's
hand together with the reactions force imparted on the player's hand, generated by
racquet rotation during impact.
Correlations between racquet vibrations and grip pressure distribution could only be
made if the degree to which the vibrations are dampened could be quantified. The half-
power bandwidth method (Quality factor) has been applied to estimate the magnitude of
racquet damping in the frequency domain. Racquet damping estimations have been
2
correlated with the grip pressure characteristics to show the mechanics of the grip
damping phenomena. Estimates of logarithmic decrement have been utilised to relate
variations in grip pressure distribution to the damping of racquet vibrations. Using the
modal properties of the racquet (also established in this thesis) the mechanics by which
the tennis grip absorbs racquet vibrations, have been described.
Previous research has shown the hand to have a profound effect on the dynamic response
of the tennis racquet in terms of frame vibration damping. It has been shown that the
tighter a tennis grip, the greater the level of vibrations transferred to the player's hand and
forearm. This research has investigated the grip damping phenomena and built upon the
current body of knowledge by interpreting the mechanics of grip damping, showing
precisely how the tennis grip dampens tennis racquet frame vibrations, and how they are
absorbed by the player at contact locations on the hand. Future racquet designs can now
incorporate the findings of the present research to optimise the vibration attenuation
systems (whether they are passive or active) to aid in the management of upper extremity
injuries such as lateral epicondylitis.
3
Chapter 1
1 Introduction
4
Over recent years the dramatic evolution of tennis racquet design has lead to an increase
in game speed that has resulted in increased physical forces being imparted on the player.
These increased forces are thought to have given rise to increases in the development and
aggravation of injuries. The most common injuries are those that involve the player's
upper extremities, and are believed to come primarily as a result of the forces transmitted
to the player during racquet - ball impact. The most common injury resulting from such
impact forces is lateral epicondylitis (tennis elbow). Tennis elbow is not the only upper
extremity injury encountered by players, but with tennis elbow affecting 40-50% of
recreational players (Roberts et al. 1995; Nirschl 1986), the injury inhibits player
performance on a large scale. Recent surveys have shown that 55.6% of recreational
players occasionally suffer from symptoms of tennis elbow, and 42.2% of those injured
said that tennis elbow reduced the amount of tennis they played (Sports Marketing
Surveys 2003). The management of tennis elbow is therefore in high demand.
Although tennis elbow represents an acute problem for many players across the world,
additional upper extremity injuries, such as wrist and shoulder strains, also affect the
players. Upper extremity injuries are thought stem from the transfer of large impact
forces in a repetitive manner, to the player via the racquet-hand interface known as the
tennis grip. Injuries, such as tennis elbow, can be better managed only if the causes can
be better understood. This in turn means that the transmission of racquet forces to the
player needs to be better understood.
The impact forces transmitted to the player are in the form of impact shock and post-
impact racquet vibrations. However, the degree of racquet shock and vibration
5
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[Cuối tài liệu]
8.3 Appendix 3
8.3.1 Half power bandwidth damping calculation
The half power damping calculation method (often referred to as the Quality Factor (Q))
is an estimation of the damping associated with the modes of oscillation of a structure.
The estimation of damping is based in the frequency domain and can be used with any
expression of magnitude (i.e. acceleration (m/s2), velocity (m/s), displacement (m)). The
method can be used with either frequency response function or frequency response
measurements.
The half power estimation uses the points at either side of the resonant frequency for an
identified mode. The half power points are calculated by finding the frequencies at
magnitudes equal to that of the resonance x 0.707 (or 3dB below the peak magnitude)
(Taylor 1994; Thompson 1993; David and Cheeke 2002). The half power points are
chosen because they identify the frequency range that needs to be excited to produce the
resonance/mode peak. This frequency range is known as the mode's amplification factor.
A mode is referred to as a single frequency, but in-order for this mode to be excited there
must be adequate excitation of frequencies both greater and smaller than the peak
frequency. A large frequency bandwidth means excitation of a greater number of
frequencies is required to excite the mode of the system. The further apart the half power
points are from the resonant frequency (i.e. the wider the resonant peak) the greater the
associated damping of the vibrations at the resonant frequency. A sharp resonance peak
means that the excitation of fewer frequencies is required for the excitation of the system
mode and therefore there is less associated damping.
226
Magnitude (m/s2; m/s; m)Figure 68 shows an example of a resonance peak obtained from a frequency response
analysis. A peak magnitude is identified along with the half power magnitude (peak
magnitude x 0.707). The peak frequency is identified as n and the corresponding half
power frequency are identified as 1 and2 .
Peak magnitude
Peak magnitude x 0.707
Frequency
1 n 2
Figure 68. Half power damping parameter identification
Once the parameters of the half power damping estimation have been identified they are
applied to the equation (1.27).
227
Q = 2 - 1 = 1 (1.27)
n 2
Equation (1.27) shows the expression of Q using the half power points. The equation also
shows the relationship between the half power damping estimation and the damping ratio
of the mode. Figure 69 shows an example frequency response and the identification of
the half power damping estimation parameters.
Log Magnitude (m/s2) 0.3
0.2101 m/s2
0.25
0.2
0.15
0.2101 x 0.1
0.707 =
0.1485 m/s2
0.05
0
100 120 140 160 180 200 220 240
Frequency (Hz)
1 =131.25Hz n =142.5 Hz 2 =153.75 Hz
Figure 69. Example of half power damping parameter identification
The half power damping parameters identified in figure 69 have been used in equation
(1.28) to estimate the Quality factor using the half power bandwidth technique.
228
Q = 2 - 1 (1.28)
n
= 153.75 -131.25
142.5
= 22.5
142.5
Q = 0.15789
Equation (1.29) shows the expression of the modes damping ratio by using the Quality
factor estimation in equation (1.28).
= 1 (1.29)
2Q
=1
2(0.15789)
=1
0.31578
= 3.1668
The half power bandwidth damping estimation (Quality factor) is based in the frequency
domain. As a result of the damping estimation being based in the frequency domain,
correlations with time based variables are not possible. The estimation is an expression of
the damping present in the system assuming that the parameters defining the dynamic
response of the system remain constant throughout the data collection period. Variable
defining parameters may result in variation in the rate of decay of vibration and this is not
considered when using the half power bandwidth damping estimation. If time based
variables are included in the investigation, then time based damping estimation should be
utilised.
229
8.4 Appendix 4
8.4.1 Subjective Gripping data
The data presented in this appendix shows the response of the two racquets A and B
during ball impacts. A summarised for of the results is presented in chapter 4 together
with a discussion of the findings. The variation of modes 2-4 is very little so the data
presented here is focused on the first mode ( 1 ). Table 17 shows the response data for
racquet A during ball impacts while table 18 shows the data for racquet B. Both tables
show the frequency resonant frequency for the first mode of oscillation under different
gripping conditions together with the associated half-power damping estimation. The
tables also include the standard deviations and averages of the two response parameters.
Grip 1 Standard Average (Q) Standard
Deviation Deviation
Condition (Hz) 0.020408163
Trial 0.020408163 0 Average
0.020408163
Free 1 183.75 0.020408163 0.027067 0.020408
0.020408163 0.124698
Free 2 183.75 0.095588235 0.007364 0.127322
0.111111111 0.149539
Free 3 183.75 0 183.75 0.133333333 0.020001
0.166666667
Free 4 183.75 0.116788321
0.138461538
Free 5 183.75 0.126865672
0.129770992
Light 1 170 0.120300752
0.121212121
Light 2 168.75 0.146153846
Light 0.157894737
3 168.75 1.629801 170.25 0.132743363
0.157894737
Light 4 172.5 0.155172414
Light 5 171.25
Medium 1 162.5
Medium 2 167.5
Medium 3
163.75 1.976424 165
Medium 4 166.25
Medium 5 165
Tight 1 162.5
Tight
Tight 2 142.5
Tight
3 141.25 1.629801 162.25
4 142.5
Tight 5 145
Table 17. Response data of racquet A for tennis ball impacts
230
Grip 1 Standard Average (Q) Standard Average
Deviation Deviation
Condition (Hz) 0.023076923 0.023077
Trial 0.023076923 0 0.120865
0.023076923 0.129927
Free 1 162.5 0.023076923 0.013336 0.156139
0.023076923
Free 2 162.5 0.117647059 0.007161
Free 3 162.5 0 162.5 0.1 0.015726
0.127118644
Free 4 162.5 0.13559322
0.123966942
Free 5 162.5 0.11965812
0.139130435
Light 1 148.75 0.133333333
0.128205128
Light 2 150 0.129310345
0.17699115
Light 3 147.5 1.629801 149 0.157894737
0.132743363
Light 4 147.5 0.157894737
0.155172414
Light 5 151.25
Medium 1 146.25
Medium 2 143.75 2.338536 146.25
Medium 3 150
Medium 4 146.25
Medium 5 145
Tight 1 141.25
Tight 2 142.5
Tight
3 141.25 1.530931 142.5
Tight 4 142.5
Tight 5 145
Table 18. Response data of racquet B for tennis ball impacts
231