Experimental Mechanics

, Volume 59, Issue 1, pp 65–77 | Cite as

Solitary Waves to Assess the Internal Pressure and the Rubber Degradation of Tennis Balls

  • A. Nasrollahi
  • R. Lucht
  • P. RizzoEmail author


Rubber is a material present in many commodities, including tennis balls. The characteristics of tennis balls are specified by the International Tennis Federation and are evaluated using standard tests that are too cumbersome to be staged easily and quickly. In this article we present an experimental method based on the propagation of highly nonlinear solitary waves (HNSWs) to determine the internal pressure of tennis balls and to estimate rubber degradation. HNSWs are compact waves that can form and travel in a closely-packed assembly of systematically arranged particles that generally interact according to the Hertz contact law. In the study presented here, we developed a model that predicts the internal pressure of tennis balls and estimates rubber degradation by observing the waves propagating within a chain in contact with the ball to be estimated. The model was validated experimentally, by testing 18 identical balls played over a few weeks period. We found that the dynamic interaction between the waves and the rubber can successfully detect changes in internal pressure and bouncing characteristics and that these changes were barely detected using a conventional rebound test. In the future, the findings of this study may be expanded to characterize any rubber of any shape.


Highly nonlinear solitary waves Tennis balls Rubber Internal pressure Finite element methods 



This project was supported by the University of Pittsburgh CRDF seed funding. The second author conducted this research as part of a Pittsburgh Allderdice High School research class, where high school students work on projects much like the undergraduate research experience. The mentoring effort of Dr. Janet R. Waldeck, National Board Certified Teacher, is very much appreciated. Finally, we thank Drs. Vidic and Fazzari for using the test samples and providing fruitful comments and suggestions about the serviceability of tennis balls.


  1. 1.
    (ITF) ITF International Tennis Fedration (ITF) (2018) Accessed Feb 2018
  2. 2.
    Technical I. Accessed 2 March 2018
  3. 3.
    Steele C (2006) Tennis ball degradation. © Carolyn SteeleGoogle Scholar
  4. 4.
    International Tennis Fedration (ITF).
  5. 5.
    Cross R (1999) Dynamic properties of tennis balls. Sports Eng 2:23–34CrossRefGoogle Scholar
  6. 6.
    Haron A, Ismail K (2012) Coefficient of restitution of sports balls: a normal drop test. In: IOP conference series: materials science and engineering. vol 1. IOP Publishing, p 012038Google Scholar
  7. 7.
    Bridge NJ (1998) The way balls bounce. Phys Educ 33(3):174CrossRefGoogle Scholar
  8. 8.
    Hubbard M, Stronge W (2001) Bounce of hollow balls on flat surfaces. Sports Eng 4(2):49–61CrossRefGoogle Scholar
  9. 9.
    Andersen T (1999) Collisions in soccer kicking. Sports Eng 2(2):121–125CrossRefGoogle Scholar
  10. 10.
    Haake S, Carre M, Goodwill S (2003) The dynamic impact characteristics of tennis balls with tennis rackets. J Sports Sci 21(10):839–850CrossRefGoogle Scholar
  11. 11.
    Goodwill S, Haake S (2004) Modelling of tennis ball impacts on a rigid surface. Proc Inst Mech Eng C J Mech Eng Sci 218(10):1139–1153CrossRefGoogle Scholar
  12. 12.
    Nagurka M, Huang SA (2004) Mass-spring-damper model of a bouncing ball. In: American Control Conference, 2004. Proceedings of the 2004. IEEE, pp 499–504Google Scholar
  13. 13.
    Wadhwa A (2009) Measuring the coefficient of restitution using a digital oscilloscope. Phys Educ 44(5):517CrossRefGoogle Scholar
  14. 14.
    Wadhwa A (2012) Measuring the rebound resilience of a bouncing ball. Phys Educ 47(5):620CrossRefGoogle Scholar
  15. 15.
    Wadhwa A (2013) Study of the dynamic properties and effects of temperature using a spring model for the bouncing ball. Eur J Phys 34(3):703CrossRefGoogle Scholar
  16. 16.
    Bagheri A, Rizzo P (2017) Assessing the pressure of tennis balls using nonlinear solitary waves: a numerical study. Sports Eng 20(1):53–62CrossRefGoogle Scholar
  17. 17.
    Nasrollahi A, Rizzo P, Orak MS (2018) Numerical and experimental study on the dynamic interaction between highly nonlinear solitary waves and pressurized balls. J Appl Mech 85(3):031007–031011. CrossRefGoogle Scholar
  18. 18.
    Nasrollahi A, Orak MS, Kosinski K, James A, Weighardt L, Rizzo P (2018) An alternative noninvasive approach to characterize tennis balls. J Nondestr Eval Diagn Progn Eng Syst (in press)Google Scholar
  19. 19.
    Yang J, Silvestro C, Khatri D, De Nardo L, Daraio C (2011) Interaction of highly nonlinear solitary waves with linear elastic media. Phys Rev E 83(4):046606CrossRefGoogle Scholar
  20. 20.
    Bagheri A, La Malfa Ribolla E, Rizzo P, Al-Nazer L, Giambanco G (2015) On the use of l-shaped granular chains for the assessment of thermal stress in slender structures. Exp Mech 55(3):543–558CrossRefGoogle Scholar
  21. 21.
    Nesterenko V (1983) Propagation of nonlinear compression pulses in granular media. J Appl Mech Tech Phys 24(5):733–743CrossRefGoogle Scholar
  22. 22.
    Nesterenko V, Lazaridi A, Sibiryakov E (1995) The decay of soliton at the contact of two “acoustic vacuums”. J Appl Mech Tech Phys 36(2):166–168CrossRefGoogle Scholar
  23. 23.
    Nesterenko VF (2013) Dynamics of heterogeneous materials. Springer Science & Business Media, New YorkGoogle Scholar
  24. 24.
    Lazaridi A, Nesterenko V (1985) Observation of a new type of solitary waves in a one-dimensional granular medium. J Appl Mech Tech Phys 26(3):405–408CrossRefGoogle Scholar
  25. 25.
    Coste C, Falcon E, Fauve S (1997) Solitary waves in a chain of beads under hertz contact. Phys Rev E 56(5):6104CrossRefGoogle Scholar
  26. 26.
    Daraio C, Nesterenko V, Herbold E, Jin S (2005) Strongly nonlinear waves in a chain of Teflon beads. Phys Rev E 72(1):016603CrossRefGoogle Scholar
  27. 27.
    Daraio C, Nesterenko V, Herbold E, Jin S (2006) Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals. Phys Rev E 73(2):026610CrossRefGoogle Scholar
  28. 28.
    Job S, Melo F, Sokolow A, Sen S (2005) How Hertzian solitary waves interact with boundaries in a 1D granular medium. Phys Rev Lett 94(17):178002CrossRefGoogle Scholar
  29. 29.
    Job S, Melo F, Sokolow A, Sen S (2007) Solitary wave trains in granular chains: experiments, theory and simulations. Granul Matter 10(1):13–20CrossRefzbMATHGoogle Scholar
  30. 30.
    Carretero-González R, Khatri D, Porter MA, Kevrekidis P, Daraio C (2009) Dissipative solitary waves in granular crystals. Phys Rev Lett 102(2):024102CrossRefGoogle Scholar
  31. 31.
    Yang J, Daraio C (2013) Frequency-and amplitude-dependent transmission of stress waves in curved one-dimensional granular crystals composed of diatomic particles. Exp Mech 53(3):469–483CrossRefGoogle Scholar
  32. 32.
    Yang J, Gonzalez M, Kim E, Agbasi C, Sutton M (2014) Attenuation of solitary waves and localization of breathers in 1D granular crystals visualized via high speed photography. Exp Mech 54(6):1043–1057CrossRefGoogle Scholar
  33. 33.
    Li K, Rizzo P (2015) Energy harvesting using arrays of granular chains and solid rods. J Appl Phys 117(21):215101CrossRefGoogle Scholar
  34. 34.
    Li K, Rizzo P (2015) Energy harvesting using an array of granules. J Vib Acoust 137(4):041002CrossRefGoogle Scholar
  35. 35.
    Li K, Rizzo P, Bagheri A (2015) A parametric study on the optimization of a metamaterial-based energy harvester. Smart Mater Struct 24(11):115019CrossRefGoogle Scholar
  36. 36.
    Li K, Rizzo P (2017) Experimental parametric analysis of an energy harvester based on highly nonlinear solitary waves. J Intell Mater Syst Struct 28(6):772–781CrossRefGoogle Scholar
  37. 37.
    Daraio C, Nesterenko V, Herbold E, Jin S (2006) Energy trapping and shock disintegration in a composite granular medium. Phys Rev Lett 96(5):058002CrossRefGoogle Scholar
  38. 38.
    Deng W, Nasrollahi A, Rizzo P, Li K (2016) On the reliability of a solitary wave based transducer to determine the characteristics of some materials. Sensors 16:5. CrossRefGoogle Scholar
  39. 39.
    Nasrollahi A, Deng W, Rizzo P, Vuotto A, Vandenbossche JM (2017) Nondestructive testing of concrete using highly nonlinear solitary waves. Nondestr Testing Eval 32(4):381–399. CrossRefGoogle Scholar
  40. 40.
    Rizzo P, Nasrollahi A, Deng W, Vandenbossche J (2016) Detecting the presence of high water-to-cement ratio in concrete surfaces using highly nonlinear solitary waves. Appl Sci 6(4):104CrossRefGoogle Scholar
  41. 41.
    Schiffer A, Alkhaja A, Yang J, Esfahani E, Kim T-Y (2017) Interaction of highly nonlinear solitary waves with elastic solids containing a spherical void. Int J Solids StructGoogle Scholar
  42. 42.
    Ni X, Rizzo P (2012) Highly nonlinear solitary waves for the inspection of adhesive joints. Exp Mech 52(9):1493–1501CrossRefGoogle Scholar
  43. 43.
    Singhal T, Kim E, Kim T-Y, Yang J (2017) Weak bond detection in composites using highly nonlinear solitary waves. Smart Mater Struct 26(5):055011CrossRefGoogle Scholar
  44. 44.
    Schiffer A, Lee D, Kim E, Kim TY (2018) Interaction of highly nonlinear solitary waves with rigid polyurethane foams. Int J Solids Struct.
  45. 45.
    Yang J, Sangiorgio SN, Borkowski SL, Silvestro C, De Nardo L, Daraio C, Ebramzadeh E (2012) Site-specific quantification of bone quality using highly nonlinear solitary waves. J Biomech Eng 134(10):101001CrossRefGoogle Scholar
  46. 46.
    Bathe K-J (2006) Finite element procedures. Prentice Hall, Pearson Education, Inc. K-J Bathe, LondonGoogle Scholar
  47. 47.
    Belytschko T, Liu WK, Moran B, Elkhodary K (2013) Nonlinear finite elements for continua and structures. Wiley, West SussexzbMATHGoogle Scholar
  48. 48.
    Goodwill S, Kirk R, Haake S (2005) Experimental and finite element analysis of a tennis ball impact on a rigid surface. Sports Eng 8(3):145–158CrossRefGoogle Scholar
  49. 49.
    Frey PJ, George PL (2000) Mesh generation: application to finite elements. Hermes Science Europe, OxfordzbMATHGoogle Scholar
  50. 50.
    Goodwill S, Haake S (2004) Ball spin generation for oblique impacts with a tennis racket. Exp Mech 44(2):195–206CrossRefGoogle Scholar
  51. 51.
    Cross R (2014) Oblique bounce of a rubber ball. Exp Mech 54(9):1523–1536CrossRefGoogle Scholar
  52. 52.
    Herbold EB (2008) Optimization of the dynamic behavior of strongly nonlinear heterogeneous materials. University of California, San DiegoGoogle Scholar
  53. 53.
    Sissler L (2012) Advanced modelling and design of a tennis ball. © Lise Sissler, LoughboroughGoogle Scholar
  54. 54.
    Sissler L, Jones R, Leaney P, Harland A (2010) Viscoelastic modelling of tennis ball properties. In: IOP conference series: materials science and engineering. vol 1. IOP Publishing, p 012114Google Scholar

Copyright information

© Society for Experimental Mechanics 2018

Authors and Affiliations

  1. 1.Laboratory for Nondestructive Evaluation and Structural Health Monitoring Studies, Department of Civil and Environmental EngineeringUniversity of PittsburghPittsburghUSA
  2. 2.Allderdice High-SchoolPittsburghUSA

Personalised recommendations