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Tennis Racquet Finite Element Analysis
EMCH 461 – Finite Element Analysis
Samuel R. Wilton
Finite element analysis was done on a Wilson ncode six-one 95 tennis racquet frame. The model was
created using the combined power of solidworks and ANSYS to model a realistic interpretation of a tennis
racquet frame. Strings and ball impacts were not modeled physically in 3D, but rather modeled as
pressure distributions and point forces along the racquet frame. Results were found for the string-loaded
‘rest’ stresses and deflections as well as string loaded + impact force ‘active play’ stresses and
deflections. The results from both analyses were enlightening, and the model realistically behaved
within limits of the yield stress and ultimate tensile strength of the assumed tennis racquet frame
composite material.
Real world connections can be made from the FEA model to visible design
considerations in the shape of the racquet frame. Although finite element analysis data cannot be
compared directly with experimental data, the results seem to be logical within the scope of the problem.
I. Introduction
Finite element analysis of a tennis racquet was chosen for my topic because I have always been
interested in the forces, reactions, and material stress/deformation in a racquet frame since I started
playing in 2005. Initially, I hypothesized that the stress influence of a tennis ball impact on the racquet
frame would not be large enough to create a significant (near yield) stress contribution due to the small
mass of a tennis ball. However, when forces were estimated via impulse-momentum relations, it was
clear that further analysis was needed to understand the possible stresses and deflections caused by the
impact of a tennis ball.
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The racquet frame used to make a 3d solid model and perform finite element analysis
calculations is known as the “Wilson N-code Six-One 95” racquet frame. This particular racquet came
from a recent generation of tennis racquets that utilize nanoscopic silicon dioxide nanoparticles to fill in
voids between carbon fibers within the racquet frame, yielding a stronger and stiffer racquet frame. The
material composition of these tennis racquets are quite complex, and no single reliable source could be
found for the elastic modulus and tensile strength of this new material system that involves a myriad of
different materials to form the composite. According to various sources, the material system is most
likely made from a combination of graphite/carbon fibers, titanium, silicon dioxide nanoparticles, Kevlar,
and several other materials. I have contacted Wilson directly to ask if they could obtain data from their
engineering department about the modulus of elasticity, yield stress, and ultimate tensile strength of
this racket frame, but no reply has been received.
Due to the lack of availability of information about the material system, I resorted to using data
from a paper by Allen, Haake, and Goodwill published in 2009 in which the modulus of elasticity for a
high stiffness racquet frame was assumed to be 70GPa with a Poisson’s ratio of 0.3 [1]. The ultimate
tensile strength and yield stress for a tennis racquet were very difficult to find for such a complex
material, but research showed that a modern tennis racquet frame can be quasi-approximated by the
material properties of a material known as CFRP, or carbon fiber reinforced plastic. The modulus of
elasticity and ultimate tensile stress of CFRP were found to be 90GPa and 500MPa respectively [3].
Properties were considered from all sources and used to determine probable values for the modulus of
elasticity (70GPa), yield stress (250MPa), and ultimate stress (500MPa) of the tennis racquet. With
these material properties, finite element calculations could be performed. I decided to model the tennis
racquet using solidworks to obtain an accurate 3-dimensional representation of the volume and used
ANSYS finite element analysis software to evaluate the model and calculate the displacements and
stresses.
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II. Formulation & Approach
(a) Pre-Computer Calculations
Before any work was done using a computer, hand calculations were made to estimate the force of a
tennis ball impact on the strings during a fast paced game. To perform this calculation, Newton’s second
law is rearranged to solve for the impulse of the tennis ball on the strings. From the impulse
momentum relation, the force of impact can be determined. However, since the force F(t) is
complicated and graphical data could not be found, a simplified relation can be used to solve for the
average force of impact.
Newton’s Second Law
Impulse-Momentum Formulation
Simplified Impulse-Momentum
From knowledge of the game and several online sources, it was determined that professional players
can serve a tennis ball as fast as 150mph (67m/s) and rally the ball back and forth between 60mph
(35.8m/s) and 80mph (35.8m/s) [2][4]. In addition, the mass of a tennis ball was found to be 57 grams
(0.057kg) [5].
The last piece of missing data is the time contact of the tennis ball with the racquet strings. This
was another difficult piece of information to find, partly because the impact time is proportional to the
tension of the racquet strings. If the racquet strings have a higher tension, they rebound the energy
much more quickly than lower tension strings that are capable of flexing and holding the ball for a
longer, yet still imperceptible, period of time. However, data was found online, and the time of impact
of a tennis ball on racquet strings was estimated to be 5ms (0.005s) from approximate measurements
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taken from a high speed camera. A camera capable of capturing video at 1000 frames per second was
used to film one of the top tennis players in the world. Since the tennis ball was in contact with the
tennis racquet strings for approximately 5 frames during the video, it can be estimated that the time of
impact is five-thousandths of a second [6].
Using all of this information and substituting mass, velocity, and time values into the simplified
impulse-momentum relation, it was found that the average force of a tennis ball on impact is
approximately 800N during regular play and 750N during an extremely fast serve. These large values,
approaching the kilo-Newton range are quite surprising. The only large possible error in the analysis is
contributed from the time of impact since it is not known to a high degree of accuracy and only
measured approximately from one simple case. Therefore, verification was sought, and it was found
that the experimental values determined for the impact force of a tennis ball are quite a bit less than the
calculated values. The error is assumed to originate from slight error in the actual time of impact. The
experimentally determined values for the maximum impact force during a stroke were measured at 330
+/- 141N. Therefore, I decided to use a compromised value of 500N for all subsequent finite element
calculations.
The force of the string tension on the racquet frame was also taken into consideration for the
finite element analysis. Finding the value for string tension was easy because I already know the tension
that my (and most other) tennis racquets are strung at. My tennis racquet is strung at 60lbf (267N)
tension in both the main (vertical) string and 60lbf (267N) tension in the cross (horizontal) string. The
string weave is comprised of a 16 main x 18 cross pattern, so the total net force of the strings can be
estimated to be 267N * 68, or 18kN exerted toward the center of the ellipsoidal face. Therefore, it was
easy to model this in the finite element model as a pressure distribution. Having calculated the area of
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the inside of the ellipsoidal face to be .02m2, the total pressure exerted by the strings was determined to
be 1x106
.
(b) 3D Modeling
Once the material properties were estimated and hand calculations were made, the computer
model could be made with meaningful data. To model the virtual tennis racquet frame, I used
Solidworks to create an accurate three dimensional model of the Wilson ncode six-one 95 tennis racquet
by taking measurements from my own tennis racquet and using them to reconstruct the volume in
virtual space. It was important to accurately model the geometry so that the areas of high stress
contribution could be located and discussed. Inaccurate model geometry would result in relatively
meaningless data in determining the critical stress areas of the tennis racquet frame. The hand grip was
not modeled because it had virtually no contribution to the stress / displacement calculations.
Furthermore, the strings were not modeled due to the extreme complexity of the woven pattern.
Instead, forces of the strings were estimated by point forces acting on the racquet frame.
Figure 2.1: Comparison of real tennis racquet geometry to 3D model
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In figure 2.1, a comparison of the three dimensional model is made to a photograph of the actual tennis
racquet is shown via 50-50 overlay of the real photograph on the virtual screen-capture.
(c) Finite Element Analysis Formulation
The finite element analysis was done in ANSYS 12.1 by converting the solidworks part to a .iges
model compatible with ANSYS. Once the model was imported into ANSYS, an element type of SOLID
285 was used because the help file description listed that it was an appropriate element to use for CAD
created models. Material properties were defined for this tennis racquet composite frame as E=70GPa
and v=0.3. Then, a mesh was created using volume free-mesh with element size based on ANSYS’s preprogrammed mesh smart-size controls. Once an appropriate mesh refinement was found, a pressure
distribution of -1MPa along the inside edge of the ellipsoidal face was applied to simulate the racquet
strings as shown in figure 2.2(a). Then, results were obtained for the stress and deformation of the
string-stressed racquet frame with no ball impact force applied to the model. Next, 32 forces (32x
15.625N –z direction) were distributed over the head of the racquet frame to represent the stringdistributed forces on the racquet frame as shown in figure 2.2(b).
Fig. 2.2(a) – String Tension Approximation
Fig. 2.2(b) – Impact Force Approximation
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After the forces were modeled for both the string tension and the impact of the tennis ball, the system
was solved and the Von-Mises stress and deflections were measured to achieve the final solution.
III. Results & Discussion
(a) String Tension - Displacement & First Principal Stress
The results obtained from the finite element analysis of the racquet loaded by string tension only for the
high refinement mesh FE model are shown below.
Figure 3.1 – Displacement in X-Axis (String Tension Loading) [meters]
Figure 3.2 – Displacement in Y-Axis (String Tension Loading) [meters]
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Fig 3.3 - Racquet Frame Deformation
Fig. 3.4 - First Principal Stress (String Tension Loading) [Pascals]
Results for the string tension loaded racquet without impact forces are tremendously
interesting!The results obtained for the stress and deformation of the racquet frame under load from
string tension are very interesting for two specific reasons. First, the location of the stress concentration
at the head of the racquet frame coincides directly with the design of the racquet itself. Second, tennis
racquets I’ve seen in real life have shown small crack formations in the area near the stress
concentration shown at the throat of the racquet.
Both the maximum x-displacement and y-displacement of the racquet frame are on the order of
0.5mm, which makes sense intuitively because it’s very difficult (if not impossible) to detect a size
change of a tennis racquet before and after it has been strung. If the string-bed has a tension of 60lbf
(267N), and the strings are pulling the head inward from all around, the racquet should “squeeze” in the
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x direction and elongate in the y direction as shown in figure 3.3. This should happen because there are
more horizontal strings than there are vertical strings. Therefore, the net force pulling the racquet in on
itself in the x-direction is greater than the net force pulling the racquet in on itself in the y-direction.
The magnitude of the principle stresses also falls into the range of sensibility. With a maximum stress of
around 15-18MPa, these stresses are well below the possible range of yield stresses and ultimate
stresses for a tennis racquet frame, but still significant enough to appropriately warrant the 60lbf
maximum tension limit designated for this particular racquet frame.
The design of this particular racquet frame is such that, at the
two vertical edges of the ellipse of the racquet head (where the
high stress is indicated in orange in figure 3.4), the member is
thickened as shown in figure 3.5. This member thickening was
not accounted for in the modeling of the racquet frame. It is a
minor enough detail that I presumed it wouldn’t make a
difference to the end result. I’ve always wondered why this is
the case, and I’ve heard several different opinions from various
tennis coaches, but now I see that it’s possible Wilson foresaw
Figure 3.5 – Member thickening along
vertical edge of racquet head
excess stress in this particular area of the frame.
In addition, from my personal experience using this racquet frame, I have seen small crack formations
along the side of the throat (the upside-down triangular hole) shown as a high stress area in figure 3.4.
When I bought my first ncode six-one racquet, it was suggested to me to string the racquet over 60lbf if I
wanted the tension to rest at 60lbf due to tension loss accumulated by permanent strain in the strings.
By stringing the racquet between 62lbf and 65lbf, the recommended limit was breached. Therefore, it
makes sense that I could possibly see signs of fatigue in that particular area near the side of the throat.
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(b) String Tension + Ball Impact Force – Von Mises Stress & Displacements
The results obtained from the finite element analysis of the racquet loaded by string tension and ball
impact force for the high refinement mesh FE model are shown below.
Figure 3.6 – Vector Sum Displacement (String Stress + Impact Force) [meters]
Figure 3.7 – Von Mises Stress (String Stress + Impact Force) [meters]
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The displacements of the racquet and the Von Mises stress for the string tension + impact force
loading both turned out to fit surprisingly well within my expected results range. The maximum
displacement of the racquet is 0.02123m (2.123cm) and the maximum Von Mises stress is 131MPa. The
maximum displacement is quite large, but it fits precisely in my expected range due to my non-FE hand
calculations which I will explain later in section IV. However, it is important to note that the racquet is
loaded with the impact force for only a miniscule fraction of a second and some force is absorbed via
biomechanical interaction. Therefore, the racquet during real game situations will most likely not see
such drastic deflection. The racquet will see the Von-Mises stress during real game situations, and they
are actually quite high. Based on my estimates from section I, the Von Mises stress is more than half of
the yield stress for the racquet frame. If a professional tennis player was to hit the ball as hard as they
could and/or return a ball as hard as they could, it is definitely possible to see plastic deformation in a
racquet frame, or at least signs of fatigue.
Another interesting thing to note about the Von Mises stresses is the fact that the head of the
racquet sees stresses nearly two orders of magnitude less stress than the handle of the racquet. The
bending moment due to impact force at the grip is maximal and the bending moment at the tip of the
head is minimal, so it makes intuitive sense that the bending stress would be minimal at the head.
Another reason for several points of low stress along the head of the frame could be due to the prestressed condition from string tension loading causing a counter-balance to the forces exerted by a ball
on the strings at impact.
IV. The Finite Element Analysis
(a) Software & Hardware
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Finite element analysis was performed using ANSYS 12.1 software in 316 Hammond computer lab. The
computer I used had an Intel Core i5 660 quad-core processor @ 3.33Ghz and 4GB of ram.
(b) Element Type
The element type I used was the SOLID 285 element. The SOLID285 element is a three dimensional
tetrahedral structural solid that is a lower order element suitable for modeling irregular meshes (such
as, for geometry imported from CAD programs). Based on the description listed in the help file, I
presumed that it would be the most suitable element for the geometry I was evaluating.
(c) Time Required for Meaningful Simulation
The time needed for one meaningful simulation was very short, even when an extremely small element
size was used. When smart-size 10 (max) coarse elements were used, the solution took 7 seconds to
solve. When smart-size 5 (avg) elements were used, the solution took 32 to seconds to solve. No mesh
refinement beyond 5 was done because the mesh at 5 was already extremely refined, and results for
Von-Mises stress and displacement were consistent within a few percent across all mesh smart-sizes less
than 7.
(d) Material Properties
The material properties used for evaluation of the final ANSYS FEA model are:
E is the elastic modulus, v is Poisson’s ratio,
is the yield stress, and
(e) Input Code  Attached to electronic submission
is the tensile strength [1] [3].
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(f) Sensitivity Analysis
The mesh used to model the tennis racquet was generated using ANSYS’s smart-mesh system where a
general mesh refinement or coarseness is given by a single input number, 1 being the most refined mesh
with the smallest element sizes and 10 being the coarsest mesh with the largest element sizes. The
coarsest mesh possible (10) was still reasonably accurate compared to the more refined meshes and
took less than 7 seconds to solve, whereas the balanced refinement (5) took up to 32 seconds to solve.
The normalized % error in the maximum Von-Mises stress and the maximum displacement of the
coarsest mesh (10) and the refined mesh (5) were both coincidentally ~19%. I decided to work with a
final mesh smart-size of 0.5 because it gave the best balance between time needed for solution and
model updates, but it was also within 1% error of more refined mesh sizes that took 3 or 4 times as long
to evaluate. Side-by-side comparisons of the coarse-mesh (10) and refined-mesh (5) solutions are
pictured below in figure 4.1(a) & (b).
Fig 4.1(a) - Coarse Mesh Von-Mises Stress [Pascals]
Fig 4.1(b) - Refined Mesh Von-Mises Stress [Pascals]
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(g) Non FE Verification
For hand-calculated non finite element verification, it seems
appropriate to model a tennis racquet as a simple
intermediately loaded cantilever beam because of the way
the constraints and loads are placed on the model.
Not to mention the fact that the side view deflection of figure 3.6 looks very similar to the deflection
that one might see in an intermediately loaded cantilever beam. The equation for modeling deflection is
Where P is the applied load at point a, x is the x-coordinate of the cantilever beam that one wishes to
obtain the deflection for, E is the modulus of elasticity, and I is the moment of inertia of the beam.
Using this equation, the deflection at x=0 can be calculated assuming P=-500N, E=70GPa, a=0.5m, and I
is given by the equation
where b is the base of the beam, and h is the height of the beam. For simplicity, the tennis racquet is
being modeled as if the grip of the tennis racquet extended out to the tip of the head. For visualization
sake, this reduced model would look like a thin rectangular baseball bat. The base of the racquet grip is
3cm and the height of the racquet grip is 2cm, therefore, the moment of inertia is 2x10-8m4. Henceforth,
the deflection at x = L = 0.676m is calculated to be 0.02274m. From figure 3.6, it can be seen that the
finite element analysis calculated a deflection of 0.021274m, an error of only 6.9%!
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The bending stress can also be calculated for this simplified case by using the following equation.
where y is the distance between the neutral axis of the beam and the section of interest, and in this case
y=1cm. Solving the stress equation at x=0 yields the solution σB = 125MPa. The solution for the stress at
x=0 obtained in the finite element analysis model was 131MPa, another remarkably small error of 4.6%!
To verify my calculations, a website called engineeringcalculator.net was used to program a simple
intermediately loaded cantilever beam to compare solutions with my solutions to ensure validity. The
results from the mini-simulation are shown below [8].
δ(.676)=0.02274
Figure 4.2 – engineering calculator program [8]
Figure 4.3 – Graph of Deflection vs. Length [8]
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(h) Approximations & Assumptions & Limitations
The finite element model created for this project had numerous approximations, assumptions,
and limitations associated with the design. First and foremost, the tennis ball impact was approximated
by distributing point loads across the frame to simulate the distribution of force by the strings.
Moreover, the string bed was not modeled at all. In its place, a negative (pulling) pressure distribution
was applied in the inside ellipsoidal boundary of the frame head. A tennis racquet with accurately
modeled strings, tension, and ball impact would yield somewhat more informative results. A virtually
modeled impact would also be able to show time-dependent stresses and deflections in the racquet as
the energy is transferred from the ball to the strings and from the strings back to the ball.
In addition to modeling approximations, assumptions, and limitations, there were also
limitations with the calculations and material data gathered for the finite element analysis. The
modulus of elasticity, yield stress, and ultimate stress were estimated based on various values found for
materials associated with tennis racquet frames, but not measured directly from a tennis racquet frame.
(h) Suggestions for Future Work:
For future work on a tennis racquet stress analysis, I would suggest modeling an accurate
representation of the strings. The string bed is an extremely important factor in modeling stresses in a
tennis racquet because the force exerted by the strings on the racquet frame is exerted over a very
small area. Furthermore, holes that the strings are woven through have the potential to create stress
concentration factors that are unforeseen in my simplified model. However, modeling such complex
interwoven strings with the appropriate material properties would be a significant undertaking suitable
for a senior thesis type project. Moreover, a string analysis would be more beneficial than a frame
analysis because strings fatigue and fracture much more than the racquet frame itself. Most serious
tennis players break strings on a regular basis, ranging from days to months, depending on intensity and
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duration of play. However, most tennis players never break a racquet frame in their lives unless they get
angry and throw the racquet or there is an inherent defect in the frame.
In addition, a modal analysis of a tennis racquet frame might be an interesting topic to look into.
The “sweet spot” of a tennis racquet is dependent on hitting the ball at the correct place on the stringbed to ensure maximum energy transfer from the racquet to the ball and minimum racquet vibration.
Therefore, the full force of the tennis swing is utilized and the impact is very crisp. A modal analysis of
the natural frequencies of a tennis racquet frame, in addition to the string analysis might provide some
insight to the energy transfer and “sweet spots” of a the tennis racquet.
V. Conclusions
The tennis racquet finite element analysis model was a tremendous success. The finite element
analysis proved to be an effective way to analyze a tennis racquet frame even with large numbers of
assumptions and estimations. There were several surprising conclusions relating to the stresses in the
racquet due to the string tension, and the deflection & stress of the racquet when impacted with the
force of a high speed tennis ball. First, the high stress points in the string-tension loaded frame coincide
directly with design factors (thickening of members) and personal observation of fatigue and crack
formation. Secondly, although the deflections & stresses under impact force were very large, they were
within the material limits and coincided almost perfectly with the simplified model of treating the tennis
racquet as if it were an intermediately loaded cantilever beam. Overall, the results obtained from the
finite element analysis shed abundant light on the subject of stresses and deflections of a tennis racquet
during rest and during play.
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VI. References
[1] Allen, Tom, Steve Haake, and Simon Goodwill. "Comparison of a finite element model of a tennis
racket to experimental data." Sports Engineering 12 (2009): 87-98.
[2] "Who Are the Master Blasters?" BBC News - Home. Web. 12 Nov. 2010.
<http://news.bbc.co.uk/sportacademy/hi/sa/tennis/features/newsid_2181000/2181575.stm>.
[3] Davis, Claire, and Elizabeth Swinback. "Making a Racket: the Science of Tennis." Plus.maths.org.
11 Oct. 2010. Web. 12 Nov. 2010. <http://plus.maths.org/content/science-tennis>.
[4] "Forehand." Wikipedia.org. Wikimedia Foundation Inc. Web. 11 Nov. 2010.
<http://en.wikipedia.org/wiki/Forehand>.
[5] "Mass of a Tennis Ball." Hypertextbook.com. Ed. Glenn Elbert. Web. 12 Nov. 2010.
<http://hypertextbook.com/facts/2000/ShefiuAzeez.shtml>.
[6] "Tennis Raquet Physics." School of Physics - The University of Sydney. Web. 12 Nov. 2010.
<http://www.physics.usyd.edu.au/~cross/tennis.html>.
[7] Wu, S. K., M. T. Gross, W. E. Prentice, and B. Yu. "Comparison of Ball-and-racquet Impact Force
between Two Tennis Backhand Stroke Techniques." Journal of Orthopedics and Sports Physical
Therapy 31.12 (2001): 759-61.
[8] "Beam Calculator." Engineer's Calculator. Web. 13 Nov. 2010.
<http://www.engineeringcalculator.net/beam_calculator.html>.