Medical and biological engineering

, Volume 14, Issue 3, pp 263–273 | Cite as

Mathematical model for investigating combined seatback—head restraint performance during rear-end impact

  • J. C. Fox
  • J. F. Williams


A mathematical model of the seated driver subjected to a rear-end impact is developed using a lumped parameter model which,inter alia, allows for the investigation of the effect of an elastic—perfectly plastic head restraint device on the overall motion of the head and torso. The most satisfactory seatback-head restraint combination is found to be a seatback having a high rotational stiffness with a viscous damping coefficient of near ‘critical’ value combined with an energy absorbing head restraint having e plastic collapse load of approximately 890 N. The model confirms recent results which indicate that tensing of neck musculature prior to impact reduces injury potential.


Whiplash Head restraints Rear end collisions Impact Mathematical model 



constant collapse load of head restraint


viscous damping coefficient of seatback


initial head offset


moment of inertia of torso/seatback unit


moment of inertia of head/neck unit


distance between pivot point and centre of gravity of torso/seatback


distance between pivot point and C7


distance between C7 and centre of gravity of head


mass of torso/seatback


mass of head/neck


generalised forces


stiffness of torso/seatback


stiffness of neck


stiffness of head restraint


stiffness of chin stop


kinetic energy


potential energy



\(\dot x\)


\(\ddot x\)



unit vectors


radius of gyration—head/neck

k t

radius of gyration—torso/seatback


duration of impact acceleration pulse


generalised co-ordinates


angular displacement of head relative to vertical


angular displacement of seatback relative to vertical


Un modèle mathématique du conducteur assis subissant un choc à l'arrière est développé, en utilisant un modèle à paramètres localisés qui, entre autres, permet l'étude de l'effet d'un dispositif de maintien de la tête élastique et parfaitement plastique sur le mouvement global de la tête et du buste. La combinaison dossier/maintien-tête s'avère être un dossier ayant une grande rigidité rotationnelle avec un coefficient d'amortissement visqueux d'une valuer quasi-critique conugué avec un maintien-tête absorbant d'énergie dont l'effort d'effondrement plastique est environ 890 N. Le modèle confirme des résultats récents indiquant que le raidissement de la musculature du cou avant l'impact limite la blessure éventuelle.


Unter Verwendung eines punktförmig verteilten Parametermodells wird ein mathematisches Modell des sitzenden Fahrers entwickelt, der einen Heckaufprall erleidet. Bei diesem Modell ist u.a. die Untersuchung der Wirkung einer elastischen, perfekt plastischen Kopfstütze auf die Gesamtbewegung von Kopf und Körper möglich. Es wurde festgestellt, daß die beste Kombination von Rückenlehne und Kopfstütze aus einer Rückelehne mit hoher Verdrehungssteifigkeit besteht, deren viskoser Dämpfkoeffizient fast den ‘kritischen’ Wert erreicht, verbunden mit einer kraftabsorbierenden Kopfstütze mit einer plastischen Knickbelastung von etwa 890 N. Das Modell bestätigt kürzlich erzielte Ergebnisse, die darauf schließen lassen, daß sich durch eine Spannung der Nackenmuskulatur vor dem Aufprall die Möglichkeit einer Verletzung mindert.


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Copyright information

© International Federation for Medical & Bological Engineering 1976

Authors and Affiliations

  • J. C. Fox
    • 1
  • J. F. Williams
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of MelbourneMelbourneAustralia

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