Dynamic Analysis of the Biomechanical Model of Head Load Impact Using Differential Transform Method

Document Type : Research Paper


1 Department of Systems Engineering, University of Lagos, Akoka, Lagos, Nigeria

2 University of Lagos, Department of Mechanical Engineering, Akoka, Lagos, Nigeria

3 University of Lagos, Department of Systems Engineering, Akoka, Lagos, Nigeria


The dynamic analysis of the biomechanical model of the head load impact using the Differential Transform Method is presented in this paper. In many parts of the world, the problem of traumatic brain injuries (TBI) has led to neurodegenerative dementing disorders and diseases as a result of head load impact from sporting activities, accidents involving the head, etc. have serious effects on humanity. The head load impact and its control have been modeled as a rigid linkage head-neck manipulator. This rigid link manipulator is governed by a system of nonlinear ordinary differential matrix equations with three degrees of freedom which requires special techniques for its solution. The system of equations was solved using Differential Transform Method (DTM) and the results were compared with results obtained in earlier studies and validated with the fourth-order Runge-Kutta numerical method (RK4). Good agreements are reached in all these results. From the model, the effects of head loads, head mass, neck mass, upper and lower linkage lengths, head and neck moments of inertia were investigated. As the head loads increased, there were increases in both axial and angular displacement of the head motion and the neck region. The study provides a theoretical basis for the design and understanding of the effects of head load carriage on vital organs that are susceptible to pains, damages, and even failure.


Main Subjects

1. Guskiewicz, K.M., Marshall, S.W., Bailes, J., McCrea, M., Cantu, R.C., Randolph, C., Jordan, B.D., Association between Recurrent Concussion and Late-Life Cognitive Impairment in Retired Professional Football Players, Neurosurgery, 57(4), 2005, 719–726.
2. Graves, A.B., White, E., Koepsell, T.D., Reifler, B.V., van Belle, G., Larson, E.B., Raskind, M., The association between head trauma and Alzheimer’s disease, Am. J. Epidemiol., 131, 1990, 491–501.
3. Holsinger, T, Steffens, D.C., Phillips, C., Helms, M.J., Havlik, R.J., Breitner, J.C., Guralnik, J.M., Plassman, B.L., Head injury in early adulthood and the lifetime risk of depression, Arch. Gen. Psychiatry, 59, 2002, 17–22.
4. Khatib, O., Demircan, E., De Sapio, V., Sentis, L., Besier, T., Delp, S., Robotics-based synthesis of human motion, J. Physiol. Paris, 103(3–5), 2009, 211–219.
5. Demircan, E., Murai, A., Khatib, O., Nakamura, Y., Muscular effort for the characterization of human postural behaviors, Springer Tracts Adv. Robot., 109, 2016, 685–696.
6. May, P.R.A., Fuster, J.M., Haber, J., Hirschman, A., Woodpecker drilling behavior: an endorsement of the rotational theory of impact brain injury, Arch. Neurol., 36(6), 1979, 370–373.
7. Drake, A., Haut Donahue, T.L., Stansloski, M., Fox, K., Wheatley, B.B., Donahue, S.W., Horn and horn core trabecular bone of bighorn sheep rams absorbs impact energy and reduces brain cavity accelerations during high impact ramming of the skull, Acta Biomater., 44, 2016, 41–50.
8. Gennarelli, T.A., Thibault, L.E., Graham, D.I., Diffuse Axonal Injury: An Important Form of Traumatic Brain Damage, Neurosci., 4(3), 1998, 202–215.
9. Kleiven, S., Predictors for traumatic brain injuries evaluated through accident reconstructions, Stapp Car Crash J., 51, 2007, 81–114.
10. Hrysomallis, C., Neck Muscular Strength, Training, Performance and Sport Injury Risk: A Review, Sport. Med., 46(8), 2016, 1–14.
11. Simoneau, M., Denninger, M., Hain, T.C., Role of loading on head stability and effective neck stiffness and viscosity, J. Biomech., 41(10), 2008, 2097–2103.
12. Anderson, G.B.J., Winters, J.M., Role of muscle in postural tasks: spinal loading and postural stability. In: Winters, J.M., Woo, S.L.-Y. (Eds.), Multiple Muscles Systems: Biomechanics and Movement Organization. Springer, New York, 1990.
13. McClure, P., Siegler, S., Nobilini, R., Three-dimensional flexibility characteristics of the human cervical spine in vivo, Spine, 23, 1998, 216–223.
14. Eckner, J.T., Oh, Y.K., Joshi, M.S., Richardson, J.K., Ashton-Miller, J.A., Effect of neck muscle strength and anticipatory cervical muscle activation on the kinematic response of the head to impulsive loads, Am. J. Sports Med., 42(3), 2014, 566–576.
15. Gutierrez, G.M., Conte, C., Lightbourne, K., The relationship between impact force, neck strength, and neurocognitive performance in soccer heading in adolescent females, Pediatr. Exerc. Sci., 26(1), 2014, 33–40.
16. Mansell, J., Tierney, R.T., Sitler, M.R., Swanik, K.A., Stearne, D., Resistance training and head-neck segment dynamic stabilization in male and female collegiate soccer players, J. Athl. Train., 40(4), 2005, 310–9.
17. Lisman, P., Signorile, J.F., Rossi, G.D., Asfour, S., Eltoukhy, M., Stambolian, D., Kevin Allen Jacobs, K.A., Investigation of the Effects of Cervical Strength Training on Neck Strength, EMG, and Head Kinematics during a Football Tackle, Int. J. Sport. Sci. Eng., 6(3), 2012, 131–140.
18. Mihalik, J.P., Guskiewicz, K.M., Marshall, S.W., Greenwald, R.M., Blackburn, J.T., Cantu, R.C., Does cervical muscle strength in youth ice hockey players affect head impact biomechanics?, Clin. J. Sport Med., 21(5), 2011, 416–21.
19. Jin, X., Feng, Z., Mika, V.H., Li, H., Viano, D., Yang, K.H., The Role of Neck Muscle Activities on the Risk of Mild Traumatic Brain Injury in American Football, J. Biomech. Eng., 139(10), 2017, 101002.
20. Eckersley, C.P., Nightingale, R.W., Luck, J.F., Bass, C.R., Effect of Neck Musculature on Head Kinematic Response Following Blunt Impact Christopher, IRCOBI Conf., 2017, 674–676.
21. Alvarez, V.S., Halldin, P., Kleiven, S., The Influence of Neck Muscle Tonus and Posture on Brain Tissue Strain in Pedestrian Head Impacts, Stapp Car Crash J., 58, 2016, 63–101.
22. Lincoln, A.E., Caswell, S.V., Almquist, J.L., Video incident analysis of concussions in boys’ high school lacrosse, Am. J. Sports Med., 41, 2013, 756–61.
23. Mihalik, J.P., Blackburn, J.T., Greenwald, R.M., Cantu, R.C., Marshall, S.W., Guskiewicz, K.M., Collision type and player anticipation affect head impact severity among youth ice hockey players, Pediatrics, 125(6), 2010, 1394–1401.
24. Heck, J.F., Clarke, K.S., Peterson, T.R., Torg, J.S., Weis, M.P., National Athletic Trainers’ Association position statement: head-down contact and spearing in tackle football, J. Athl. Train., 39(1), 2004, 101.
25. Viano, D.C., Pellman, E.J., Concussion in professional football: Biomechanics of the striking player—Part 8, Neurosurgery, 56(2), 2005, 266–280.
26. Mukherjee, S., Goswami, D., Roy, B., Solution Of Higher-Order Abel Equations By Differential Transform Method, Int. J. Modern Phys. C, 23(9), 2012, 1250056.
27. Chen, X., Lou, J.X., Dai. Y., Differential Transform Method for the Brooks-Corey Model, in Applied Mechanics and Materials. Trans Tech Publ., 2014.
28. Garg, M., Manohar, P., Kalla, S.L., Generalized differential transform method to space-time fractional telegraph equation, Int. J. Diff. Equ., 2011, 2011, 548982.
29. Elzaki, T.M., Hilal, E.M.A., Solution of linear and nonlinear partial differential equations using mixture of Elzaki transform and the projected differential transform method, Math. Theo. & Model., 2, 2012, 50-59.
30. Agboola, O.O., Opanuga, A.A., Gbadeyan, J.A., Solution of Third Order Ordinary Differential Equations Using Differential Transform Method, Glob. J. Pure Appl. Math., 11(4), 2015, 2511-2517.
31. Kuo, C., Fanton, M., Wu, L., Camarillo, D., Spinal Constraint Modulates Head Instantaneous Center of Rotation and Dictates Head Angular Motion, J. Biomech., 76, 2018, 220-228.
32. Michael Fanton, Calvin Kuo, Jake Sganga, Fidel Hernandez, David B. Camarillo, Dependency of Head Impact Rotation on Head-Neck Positioning and Soft Tissue Forces, IEEE Trans. Biomed. Eng., 66(4), 2019, 988-999.
33. Elkin, B.S., Gabler, L.F., Panzer, M.B., Siegmund, G.P., Brain tissue strains vary with head impact location: A possible explanation for increased concussion risk in struck versus striking football players, Clin. Biomech., 64, 2019, 49–57.