A systematic framework for the design and material selection of composite for tennis racket upon impact

Abstract

The present study explores the comprehensive composite material selection methodology of a tennis racket involving coupled Multi-Criteria Decision-Making (MCDM) and numerical analysis. Initially all the possible material alternatives from the composite family were shortlisted using CES Selector 2016. A design space was developed incorporating MCDM technique naming VICKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje), PROMETHEE (Preference Ranking Organisation Method for Enrichment Evaluation) and TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) to rank the most suitable composite material based on critical mechanical properties to identify the most suitable composite materials for applications requiring optimal combinations of rigidity, strength, and structural performance. Using finite element analysis, the structural integrity of each of the materials was evaluated on static and dynamic stress analysis modules of ANSYS. Out of all material alternatives, T300 carbon fiber epoxy was considered as the most suitable for tennis rackets. Further, a comparison between the selected material was also made with conventional materials and were found in good agreement with selected material.

Introduction

The utilization of composite material has increased significantly in various products, which includes robotic components, medical equipment, construction industry, surgical implants, helicopter rotor/turbine blades, engine parts, and satellites [1]. This is attributed to their distinguishing characteristics, which include tailored properties achieved through design flexibility, specific strength, and weight reduction [2, 3]. One of the most prevalent applications of composites is in the fabrication of lightweight sporting equipment. Despite the extensive range of applications for composites, researchers suggested that a critical aspect in maximizing their capabilities is by comprehending the methodology employed in their design and material selection [4, 5]. In contrast to the isotropic behavior observed in conventional metals, the design process for composites is comparatively complicated due to the inclusion of mechanically distinct components naming, fibers and matrix. These constituents collectively contribute to the highly orthotropic behavior of composites. Research investigations have demonstrated that composites, when prudently selected and designed, can offer advantages including reduced weight through customized mechanical properties. Researchers have used this technique to study various aspects of tennis racket design utilizing finite element analysis. Glynn et al. [6] performed a finite element simulation analysis to investigate the impact of string tension, head size, weight, length, and balancing point's impact on racket performance. Goodwill et al. [7] simulated angled impacts between a tennis ball and rigid surface, the finite element simulation was validated with an experiment using a high-speed camera to quantify the velocity, angle, and spin of the ball upon impact [8]. Developed finite element models to examine racket stiffness, balance, and mass effects on performance. Banwell et al. [9] employed finite element analysis to investigate a tennis racket’s natural frequency and related mode shapes. Yao-dong et al. [10] performed a further finite element study on the tennis racket. The vibration properties of two different string tensions were compared to identify the mode shape of the racket. The study revealed 70% reduction in vibration frequency when the handle was clamped with higher tension. Lang et al. [10, 11] studied string tension's influence on vibration and coefficient of restitution (COR). The study demonstrated that lower string tension improved both vibration and COR. Furthermore, minimizing racket vibration may be accomplished by hitting the ball at certain node positions in the stringed area. Allen et al. [12] developed a finite element model of a tennis racket's string-bed with an interwoven pattern to replicate real-world scenarios. The research work utilized the simulation to examine how the consequent rebound velocity and spin affect a simulated groundstroke. Allen et al. [13] examined several racket models with various design features like as stiffness, balance point, and mass to investigate the influence of string-bed area on racket performance. Results identified that shifting the racket's balance point towards the tip might improve the ball's rebound speed and topspin.

Selection of the right material requires managing data on material qualities and user needs. Ranking algorithms help to further narrow down the options once the initial screening is done. Multi-criteria decision-making (MCDM) [14–18] and optimization methods [19–21] have been used in the selection process. MCDM includes two primary categories: Multiple Objective Decision Making (MODM) and Multiple Attribute Decision Making (MADM). Each of these were further categorized according to the limits and uses that were imposed on them. MADM subdomains as VICKORS, AHP, TOPSIS, and so on. Materials are evaluated based on various aspects, not just one criterion. MADM helps to select the suitable material in engineering product and process design considering user-required features. Decision-making involves quantitative and qualitative approaches. Some traits can be measured, while others are binary. Binary traits like abrasion and corrosion resistance are categorized as: not good, sufficient, or exceptional. Analytical Hierarchy Process (AHP), a strong MCDM method, recognizes the variable i importance of each parameter by considering stakeholder preferences and expert viewpoints for a comprehensive investigation.

A detail insight of the existing research related to the design and analysis of tennis racket revealed that considerable efforts have been devoted by researchers to enhance the knowledge spectrum in this field where the vast amount of work is done related to vibrational stability and structural integrity of the tennis racket. However, limited studies talked about the comprehensive design methodology and development of design space for tennis racket. It is pertinent to mention that the existing methodology of composite tennis racket is different from that conventional tennis racket and was not identified in the literature. In a similar context, a comprehensive design and material selection methodology for tennis racket has been proposed and employed on the tennis racket which is the novel contribution in the literature. For this purpose, CES selector [22] has been used to identify all the possible alternative material for the design of tennis rackets. Similarly, MCDM was used to rank the most suitable candidate for the FE analysis. Furthermore, static FE analysis was initially conducted to find the best alternative with the highest strength to weight ratio. Similarly, the material qualifying the static analysis was further analyzed in transient analysis to assess the integrity at varying loading conditions. Further, a comparison between the T300 carbon fiber/epoxy and conventional material was made to evaluate the performance of structure. The methodology presented in this research work provides a systemic framework for the development of sport products using composite material. The design procedure of composite is different from conventional materials. The study has offered an insight to integration of multi criteria decision making and finite element analysis in early design phase in computational thrusts. The presented approach applies to tennis rackets and can be used for other sport equipment including boats, bats, skateboards etc. The complete methodology of the work is also depicted in Fig. 1.

Fig. 1 [Images not available. See PDF.]

Methodology of the current research work

The remainder of the paper follows the following schema. After the introduction in Sects. 1, 2 deals with the material selection methodology and development of design space. The methodology of the FEA analysis has been presented in Sect. 3 whereas Sect. 4 deals with results and discussion. The paper has been culminated by proposing the conclusions in Sect. 5.

Material selection

The selection of the material from scrap is a significant process for the material selection. It was seen that the most conventional tennis rackets were composed of Aluminum and wood, so the selected material must exhibit the properties similar or better than the initially selected materials. For this purpose, CES Selector 2016 (Design, 2018) was used to initially identify the material that can be used as an alternative to this. Strength to weight ratio, impact resistance and fatigue strength were the comparison attributes of the target material. The first iteration yielded 3943 materials, however, after applying multiple filters the number was reduced to 15. It is pertinent to mention that all material selected are from composite family and are shown in Table 1.

Table 1. Total material candidate yielded from CES Selector

Sr. No#	Candidate material
1	BMS/HS C.F. QI Lamiate
2	Epoxy/aramid fiber, UD
3	Epoxy/HS carbon fiber, UD
4	Epoxy/S-glass fiber, UD
5	Cyanate Ester/HMI Carbon fiber UD Laminate 0°
6	Cyanate Ester/HMI Carbon fiber UD Laminate 90°
7	Epoxy/HS Carbon Fiber (fabric), biaxial laminate
8	Epoxy/HS Carbon Fiber (fabric), QI laminate
9	Fluoro elastomer (FEPM/ Aflas, 15–30% carbon black)
10	Fluoro elastomer (FKM/ Aflas, 20–35% carbon black)
11	PAI (graphite and PTFE high PV)
12	Perfluoro elastomer (FFKM, carbon black)
13	Polyimide/HS carbon fiber (fabirc), QI laminate
14	Polyimide/HS carbon fiber (fabirc), biaxial laminate
15	Polyvinylidene Fluoride PVDF (30% CF)

It is pertinent to mention that analyzing each of the selected material separately using FEA is difficult and computationally expensive. To further narrow down the list to list to five most suitable candidate materials, Multi-Criteria Decision Making (MCDM) has been utilized. Weights were assigned to each on the selection attributes and three MCDM techniques were applied on the set of the data which includes VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) [23]. TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) [15] and PROMETHEE (Preference Ranking Organisation Method for Enrichment Evaluation) [24, 25] to rank the materials. The top 5 high ranked materials were selected and are presented in Table 2 along with their industrial names and mechanical properties. Figure 2 presents a flowchart illustrating the operational workflow of the codes implemented for the TOPSIS, PROMETHEE, and VIKOR techniques used in this study.

Table 2. Comparison assigned value

Assigned value	Denotation	Explanation
1	Slightly more important	The first criterion is slightly more important than the second criterion
2	Moderately more important	The first criterion is moderately more important than the second criterion
3	Significantly more important	The first criterion is significantly more important than the second criterion

Fig. 2 [Images not available. See PDF.]

Flowchart outlining ranking techniques

Multi criteria decision making

Using a multi-criteria decision-making (MCDM) technique, composite materials were chosen. With this method, a database of composite materials' built-in criteria was used to shortlist fifteen options from the CES Selector 2016.This strategy involved applying the TOPSIS, PROMETHEE, and VIKOR techniques to the materials that were shortlisted. The parameters that were considered were density, Young's modulus, tensile strength, and shear modulus. Each criterion was given a weight based on how important it was to determine the material's overall performance in a given application, particularly in structural applications where stiffness, strength, and durability are crucial.

Weight assignment and justification using the MDL method

The Modified Digital Logic (MDL) method, a systematic approach to assigning relative importance to diverse criteria in decision-making processes, was used to establish the weights for each criterion [26]. As stated in Table 2, the MDL method compares pairs of criteria and assigns values according to their relative importance.

Four important parameters have to be weighed in the material selection process: density (0.14), shear modulus (0.18), tensile strength (0.36), and Young's modulus (0.32). Density was devalued because high-performance industries like automotive and aerospace place a higher value on mechanical attributes like strength and rigidity than on weight. Young's Modulus had a higher weight because it demonstrated how important stiffness and structural integrity are. The greatest weight was given to tensile strength, highlighting its vital function in withstanding tensile stresses. Although significant, shear modulus was given less weight to ensure that it wouldn't dominate the more important characteristics of stiffness and tensile strength. The following factors were compared in this study as reported in Table 3.

Table 3. MDL weights

		AB	AC	AD	BC	BD	CD	Sum	Weight
A	Density	1	1	1				3	0.14
B	Young’s modulus	2			3	2		7	0.32
C	Compressive strength		3		3		2	8	0.36
D	Shear strength			1		1	2	4	0.18
Total								22	1

Application of MCDM methods

Fifteen short listed composite materials using CES selector 2016 were ranked using the computed weights in the PROMETHEE, TOPSIS, and VIKOR methodologies as mentioned in Table 4. These techniques provide a thorough assessment of the items, taking into account how well they performed in each category. The PROMETHEE technique is useful for managing both qualitative and quantitative criteria since it use pairwise comparisons and preference functions to create an easy-to-understand ranking of alternatives [27]. When dealing with conflicting requirements, VIKOR provides a strong method for balancing the ideal and anti-ideal solutions in order to reach a compromise solution [28]. With its ability to rank options according to how close they are to the perfect answer, TOPSIS excels in simplicity and readability, making it simple to compare how well each option achieves the intended results [29]. Each method brings unique strengths to the evaluation process, ensuring a comprehensive and well-rounded assessment of options. The results of each method were analyzed and compared to identify the materials that consistently ranked highly across all methods as listed in Table 5. Properties of Aluminum and Wood are listed in Table 6 for reference.

Table 4. Ranking of 15 CES shortlisted materials

Composite	Density (Kg/m3)	Youngs modulus (GPa)	Tensile strength (MPa)	Shear modulus (GPa)	TOPSIS	Ranks	PROMETHEE	Ranks	VICKOR	Ranks
BMS/HS C.F. QI Lamiate	1610	55.40	430	16.90	0.30	7	0.51	6	0.00	8
Epoxy/aramid fiber, UD	1440	8.30	1979	3.90	0.39	5	0.27	4	0.22	5
Epoxy/HS carbon fiber, UD	1540	20.90	1979	15.00	0.44	3	0.26	3	0.43	3
Epoxy/S-glass fiber, UD	2000	50.00	1700	4.70	0.42	4	0.16	8	0.49	4
Cyanate Ester/HMI Carbon fiber UD Laminate 0°	1620	15.00	1890	4.69	0.39	6	0.11	5	0.51	6
Cyanate Ester/HMI Carbon fiber UD Laminate 90°	1620	5.60	33	4.69	0.07	11	0.06	12	0.53	12
Epoxy/HS Carbon Fiber (fabric), biaxial laminate	1600	101.00	1800	7.17	0.56	2	0.05	2	0.61	2
Epoxy/HS Carbon Fiber (fabric), QI laminate	1750	202.00	2690	5.25	0.85	1	0.03	1	0.66	1
Fluoro elastomer (FEPM/ Aflas, 15 -30% carbon black)	1580	0.01	13	0.00	0.03	13	0.00	13	0.66	13
Fluoro elastomer (FKM/ Aflas, 20 -35% carbon black)	1800	0.00	13	0.00	0.01	14	-0.17	14	0.88	14
PAI (graphite and PTFE high PV)	1580	9.46	115	3.25	0.07	12	-0.18	11	0.88	10
Perfluoro elastomer (FFKM, carbon black)	1920	0.00	9	0.00	0.01	15	-0.20	15	0.91	15
Polyimide/HS carbon fiber (fabirc), QI laminate	1590	42.20	465	16.10	0.28	9	-0.24	7	0.94	9
Polyimide/HS carbon fiber (fabirc), biaxial laminate	1590	58.10	738	4.50	0.28	8	-0.31	9	0.98	7
Polyvinylidene Fluoride PVDF (30% CF)	1720	18.80	92	7.08	0.12	10	-0.34	10	1.00	11

Table 5. Finalized material after MCDM technique along with mechanical properties

Parameters	Epoxy/HS carbon fiber, UD	Epoxy/aramid fiber, UD	Epoxy/S-glass fiber, UD	Epoxy/HS carbon fiber, QI	Epoxy/HS Carbon fiber, QI
Industrial names commonly used	Carbon HM CFRP Epoxy	Aramid Fiber Epoxy	Torayca S-Glass Epoxy	T300 Carbon Fiber Epoxy	M35J Carbon Fiber Epoxy
Density	1540 kg/m3	1440 kg/m3	2000 kg/m3	1600 kg/m3	1750 kg/m3
Young’s Modulus in X	20,900 MPa	8300 MPa	50,000 MPa	101,000 MPa	202,000 MPa
Young’s Modulus in Y	9450 MPa	7000 MPa	8000 MPa	9800 MPa	1000 MPa
Young’s Modulus in Z	9450 MPa	7000 MPa	8000 MPa	9800 MPa	1000 MPa
Poisson’s Ratio in XY	0.27	0.41	0.3	0.28	0.3
Poisson’s Ratio in YZ	0.4	0.41	0.4	0.28	0.3
Poisson’s Ratio in ZX	0.27	0.4	0.3	0.28	0.3
Shear Modulus in XY	5500 MPa	2100 MPa	4700 MPa	7170 MPa	5250 MPa
Shear Modulus in YZ	3900 MPa	1886 MPa	3500 MPa	3440 MPa	3616.2 MPa
Shear Modulus in ZX	5500 MPa	2100 MPa	4700 MPa	7170 MPa	5250 MPa
Tensile Strength in X	1979 MPa	1377 MPa	1700 MPa	1800 MPa	2690 MPa
Tensile Strength in Y	500 MPa	18 MPa	35 MPa	80 MPa	68 MPa
Tensile Strength in Z	500 MPa	18 MPa	35 MPa	80 MPa	68 MPa
Compressive Strength in X	− 893 MPa	− 235 MPa	− 1000 MPa	− 1560 MPa	− 1400 MPa
Compressive Strength in Y	− 450 MPa	− 53 MPa	− 120 MPa	− 246 MPa	− 200 MPa
Compressive Strength in Z	− 450 MPa	− 53 MPa	− 120 MPa	− 246 MPa	− 200 MPa
Shear Strength in XY	34 MPa	34 MPa	60 MPa	98 MPa	131 MPa
Shear Strength in YZ	34 MPa	34 MPa	35 MPa	62 MPa	83.6 MPa
Shear Strength in ZX	20 MPa	20 MPa	60 MPa	98 MPa	131 MPa

Table 6. Mechanical properties of aluminum [30] and wood [31]

Component	Density (kg/m3)	Elastic Modulus (GPa)	Poison ratio
Aluminum	2770	71.0	0.33
Wood	60	13.8	0.38

Numerical procedures

CAD model and composite lay up

Computer Aided Design (CAD) is considered as one of the most crucial steps for the evaluation of the mechanical response using FE analysis. The accuracy of the FEA results is majorly dependent on the corrected CAD modelling and problem setup. For this purpose, a tennis racket was designed using SOLIDWORKS®. Details of the 3D CAD model are depicted in Fig. 3. The CAD model was exported into STEP file before importing into ANSYS Workbench. CAD Cleaning was also performed on the tennis racket to perform the composite layup on the model. ANSYS Space Claim was used to convert the solid model to shell model, and all the small surfaces were merged to single surface. Face repair and face merge features were used to ensure to reduce the occurrence of undesired meshing errors.

Fig. 3 [Images not available. See PDF.]

Solid CAD model developed on SOLIDWORKS. (All dimensions are in mm)

After the modeling of the racket, composite layup is another crucial step as the structural integrity is dependent on the accuracy of the modeling of composite. For this purpose, the racket has been divided into different sections head, throat and frame handle [32]. The segmentation of the racket will help us to apply distinct lay-up patterns on each section. The details of layering scheme for all three segments of racket are presented in Fig. 4. Figure 4 shows the layering strategy in detail for each of the racket's three components. In the figure (a) denotes the ply’s angle (degrees), and (t) represents the thickness of each layer that makes up the laminate (mm). For this study, each layer is configured with a 0-degree ply angle and a thickness of 1mm to construct the composite layup. All candidate materials have the same layup pattern, allowing for a more accurate evaluation of the strength to weight ratio.

Fig. 4 [Images not available. See PDF.]

Overall composite layup used for racket FEA for all materials, a (ply’s angle in degree) and t (thickness of one layer in mm)

Mesh convergence study

Prior to analysis, a mesh sensitivity/convergence analysis was carried out to determine the ideal mesh size for producing effective analysis results. To achieve this, a mesh convergence study for T300 Carbon Fiber Epoxy as the material was carried out for six distinct mesh sizes, findings of study are listed in Table 7. A mesh size of 1 mm was selected for analysis purposes due to an error of less than 5% for both the deformation and stress measurements. The meshed body with a mesh size of 1 mm showed 85,399 nodes and 85,214 elements. The structure of the tennis racket was examined utilizing quad form Shell 181 with quadratic element type, which consists of 6 DOF (3 translation and 3 rotation). A stress probe was positioned in proximity high stress regions to evaluate the convergence of the results. The mesh contour of the tennis racket has been depicted in Fig. 5.

Table 7. Mesh convergence study for tennis racket

Sr No.	Mesh size(mm)	Number of nodes	Number of elements	Deformation (mm)	Equivalent stress (MPa)
1	5	18,937	19,022	25.945	393.55
2	4	19,556	19,649	26.582	399.85
3	3	21,275	21,320	26.981	405.45
4	2	28,577	28,535	27.122	408.76
5	1	85,399	85,214	27.376	410.86
6	0.5	335,499	335,179	27.378	411.32

Fig. 5 [Images not available. See PDF.]

Mesh contour of tennis racket

Structural analysis

Static stress analysis

After complete composite modeling, the static stress analysis was conducted to ANSYS. The purpose of the stress analysis is to evaluate all five candidate materials and characterize the most suitable material based on the strength to weight ratio. To simulate real-world conditions, the racket strings will be subjected to 20 stresses, each with a magnitude of 25 N in the z-direction, uniformly distributed across the racket frame. This setup replicates the effect of an impact force of 500 N, as the total force is distributed evenly among the 20 points (500 N ÷ 20 = 25 N per point) [33]. In a similar context, the string of the racket has been removed and the equivalent load is applied on the frame of racket where the strings are attached. Moreover, additional inward pressure is also applied to the inner side of frame. The pressure is applied to model the tension of string attached to the frame. Figure 6 below depicts the boundary condition of the tennis racket.

Fig. 6 [Images not available. See PDF.]

Boundary conditions of the tennis racket at static loading

A shell 2D surface model with thickness of 2 mm was utilized in analysis for modeling of racket made of conventional material (Aluminum and wood). The racket for conventional material was analyzed for same loading and boundary conditions under which composite racket was analyzed for best comparison of stress and FoS behavior.

Modal analysis

The modal analysis is used to evaluate the mode frequencies of the structure and mode shape associated with each frequency. The forced modal analysis was performed using the previous static results coupled with modal analysis module of ANSYS Workbench. The total of four mode frequencies were evaluated and the critical failure mode was identified. The force excitation used for analyzing the modes in modal analysis. The model analysis is also considered as the basis of explicit analysis and is used to calculate the minimum mesh size and time step of the problem and is also considered as the pre-requisite of explicit analysis using LS Dyna.

Explicit analysis

The explicit analysis was also conducted to simulate the response of the structure under the actual conditions using LS Dyna solver. For this purpose, the string and ball are modeled separately by using CAD package and then assembled with the frame of racket by using assembly module. The string and ball has been modelled using nylon and hyper elastic rubber material that will help in transferring impact load to the frame [34]. The material properties used for nylon and rubber ball are presented in Table 8. The mesh size for nylon strings was set to 0.5 mm with total 26,777 number of nodes. The strings were connected to perforation holes in the racket frame during the transient phase of the simulation, using Bonded face to face contact option available in LS Dyna. While, the strings were free to move in vertical direction at impact of ball and this contact between ball and stringer was modeled using stiffness adjustment body interaction option of LS Dyna, to ensure that the ball deforms realistically and does not penetrate too much into the racket strings upon impact.

Table 8. Material properties of ball and nylon used in LS Dyna analysis [31]

Component	Material type	Density (kg/m3)	Elastic modulus (GPa)	Poison ratio
Nylon	Elastic	0.000107	1.95	0.23
Ball	Hyper elastic	0.0001	2.98	0.25

It also assesses the stress distribution and dynamic response after the collision between the ball and racket. The explicit analysis was performed only on one composite material that was finalized after the static stress analysis. The speed of the ball is maintained at 40 m/s, as described in the literature [32]. The racket has been fixed from the handle and the ball is allowed to strike on the strings. Boundary conditions of the analysis are shown in Fig. 7.

Fig. 7 [Images not available. See PDF.]

Analysis description of tennis racket during LS Dyna analysis

Results and discussion

This section deals with the results of numerical analysis performed to provide a suitable composite material for tennis rackets. This section is divided into two parts. The first section deals with the result of stress analysis at static state while transient analysis results are discussed in the other section. Further, a comparison has been made with conventional material in the later part of this section. For the stress analysis, only deformation and the factor of safety have been reported.

Static analysis results

Computational analysis was performed on the tennis racket using all the seven materials. It was observed that the area prone to failure is in the vicinity of fixed support. All the material exhibits the same behavior except the value of factor of safety and deformation differ in each case. Table 9 below depicts the comparison of the mechanical attributes with all five materials. It is pertinent to mention that T300 Carbon Fiber Epoxy has passed the static analysis requirement of the tennis racket and has the maximum strength to weight ratio of 1.92 as depicted in Table 9. Moreover, Tsai-Wu failure criteria was utilized for evaluation of FOS because it consider the interaction of shear stress in its evaluation as compared to other failure criteria [35, 36]. Figure 8 depicts Comparison of mechanical parameters of all the candidate materials and Comparison of FOS/W ratio of all the candidate materials is shown in Fig. 9.

Table 9. Comparison of results for FOS, weight, strength to weight ratio and deformation

Sr. No#	Material CATEGORY	FOS	Deformation (mm)	Stress (MPa)	Weight (kg)	FOS/W
1	Carbon HM CFRP Epoxy	1.24	37.571	337.98	1.17	1.059
2	Aramid Fiber Epoxy	1.22	51.150	463.11	0.846	1.442
3	Torayca S-Glass Epoxy	1.34	75.732	375.09	1.31	1.022
4	T300 Carbon Fiber Epoxy	1.38	27.376	410.86	0.72	1.917
5	M35J Carbon Fiber Epoxy	1.06	24.739	534.81	0.79	1.342
6	Wood	0.695	77	247.17	1.28	0.545
7	Aluminum	1.10	29.783	122.19	1.45	0.759

Fig. 8 [Images not available. See PDF.]

Comparison of mechanical parameters of all the candidate materials

Fig. 9 [Images not available. See PDF.]

Comparison of FOS/W ratio of all the candidate materials

The contours of factors of safety, deformation and stress of the racket composed of composite T300 Carbon Fiber Epoxy materials are depicted in Fig. 10. The static finite element modeling of the tennis racket indicated a deformation of 27.37 mm in the edges of the rim which was not constraint during the analysis and minimum Factor of Safety of 1.38 assuring the carbon HM fiber/epoxy resistance to failure. A comparison between the conventionally used materials was also provided along with T300 to ensure that the tennis racket has better performance and is lighter in weight as compared to others. The results were found in good agreement with selected material.

Fig. 10 [Images not available. See PDF.]

a Deformation contours, b stress plots and c factor of safety computed for the tennis racket composed of T300 Carbon fiber/epoxy

Dynamic analysis results

Similarly, based on the static analysis, T300 carbon fiber epoxy has passed the static analysis requirement of the tennis racket and has the maximum strength to weight ratio of 2. The modal analysis of the tennis racket was also conducted to determine the failure modes and frequencies associated with each failure mode. Table 10 depicts the frequencies of T 300 Carbon fiber epoxy. According to the analysis, mode I, or out-of-plane bending mode, will predominately fail under the current stress conditions and occur at a frequency of 43.72 Hz. Figure 11 shows the common forms of tennis rackets and the modes types that correlate with them. This mode is more prone to failure since, in a real-world situation, the ball tends to bend the racket's strings in the direction that the ball is traveling in. The bend direction in question aligns with dynamic analysis mode I.

Table 10. Mode shape frequency acquired from modal analysis

Sr. No#	Material category	Mode I	Mode II	Mode III	Mode IV
1	Carbon HM CFRP Epoxy	34.952	38.906	66.533	163.5
2	Aramid Fiber Epoxy	30.216	35.531	50.82	116.51
3	Torayca S-Glass Epoxy	25.702	27.684	50.40	110.81
4	T300 Carbon Fiber Epoxy	43.72	57.04	82.84	173.05
5	M35J Carbon Fiber Epoxy	43.27	53.23	74.032	156.74
6	Aluminum	46.82	51.02	108.07	261.34
7	Wood	46.47	50.67	106.82	258.86

Fig. 11 [Images not available. See PDF.]

Modal analysis of T300 carbon fiber/epoxy a Bending Mode 1 (Out of Plane), b Torsional Mode, c Bending Mode (In Plane), d Bending Mode 2 (Out of Plane)

To ascertain the failure modes and ensure the structural integrity at final state, explicit structural analysis has been conducted by using LS Dyna and it has been found that the maximum deformations occur on the frame. Figure 12 shows the deformations and maximum stresses incurred on the racket upon impact. The stiffness and string tension of a tennis racket are two important features that affect how the ball returns after a hit. To be more precise, the ball tends to rebound faster off a stiffer racket because of this. Higher string tension, on the other hand, typically causes a slower rebound because the strings take up more energy from the ball. Comprehending these variables is essential as they have a direct impact on the racket's performance during play, affecting the ball's speed and control.

Fig. 12 [Images not available. See PDF.]

a Deformation contours, b stress plots of the tennis racket composed of T300 fiber epoxy

Conclusions

This research article outlines a comprehensive procedure for selecting composites in the design of engineering products. The study is divided into two parts. The first part focuses on shortlisting ideal candidate materials for a tennis racket, utilizing CES selector and MCDM techniques to identify the top five materials based on their mechanical properties, thus optimizing computational efficiency for the subsequent finite element analysis (FEA). FEA was then performed on these five shortlisted materials to determine the most suitable option for the tennis racket. Static analysis revealed that T300 carbon fiber epoxy exhibited the highest strength-to-weight ratio of 1.917 among all candidates. The structure composed of T300 fiber epoxy experienced a total deformation of 27.37 mm with a Factor of Safety (FoS) of 1.38. Additionally, modal analysis identified the critical failure mode as out-of-plane bending, occurring at 43.72 Hz. To further validate the racket’s performance, explicit analysis was conducted on both conventional aluminum and T300 carbon fiber epoxy. The results showed that the T300 carbon fiber epoxy outperformed the conventional aluminum racket by 39% in terms of strength-to-weight ratio, confirming its superior performance. Additionally, the T300 carbon fiber epoxy demonstrated a lower deformation (27.37 mm vs. 29.78 mm) and higher FoS (1.38 vs. 1.10) compared to the aluminum structure. This methodology can be extended to other composite sports equipment for design and material selection during the early phases of development.

Author contributions

M.M developed methodology, data analysis, manuscript write-up and FEA; A.K.Z formal analysis, review; A.A revision, material selection methodology, review; MSUR, review and revisions;

Data availability

Data will be made available on request.

Declarations

Competing interests

The authors declare no competing interests.

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