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Analytical viscoelastic modelling of whiplash using lumped-parameter approach

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Abstract

A two-dimensional lumped-parameter simulation of the head and neck has been carried out to capture the dynamic whiplash response during rear-end collisions in the sagittal plane. Joint locations are positioned at the instantaneous axis of rotation locations to capture the realistic intervertebral motions of the cervical spine for the 50th percentile male. Rotational viscoelastic joints are defined at each joint using linear stiffness and damping parameters in a Voigt configuration. Lagrangian mechanics is used to derive the whiplash response for a given collision acceleration. An inverse analysis is used to determine the viscoelastic lumped-parameter values based on cadaver whiplash test results. The results of four models are presented, demonstrating the response and influence of stiffness and damping joint parameters for the following models: (1) uniform joint stiffness with no damping, (2) joint stiffness with no damping, (3) joint damping with no stiffness, and (4) joint stiffness and joint damping. The rigid-linkage viscoelastic joint stiffness and joint damping lumped-parameter model is seen to provide quick solution results and good overall results for the head and intervertebral response (peak head x-displacement, y-displacement and rotation within 10, 20 mm and 7.5°) compared with cadaver test results. This model can be easily scaled to accommodate anthropometric differences in occupant height, weight, gender, and posture for the desired collision dynamics. These properties make this rigid-linkage model well-suited for injury assessment and use in head-restraint optimization to minimize injury and improve neck biofidelity in anthropometric test dummies.

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Abbreviations

\( x \) :

Horizontal position

\( y \) :

Vertical position

t :

Time

\( a_{x} \) :

Horizontal whiplash acceleration

\( a_{y} \) :

Acceleration due to gravity

\( \theta_{i} \), \( \dot{\theta }_{i} \), \( \ddot{\theta }_{i} \) :

Linkage angle, angular velocity, and angular acceleration

\( \theta_{initial,i} \) :

Initial linkage angle

\( \gamma_{i} \) :

Joint rotation

\( \varDelta_{i} \) :

Intervertebral extension

\( l_{i} \) :

Linkage length

\( m_{i} \) :

Joint mass

\( k_{i} \) :

Joint stiffness

\( k_{overall} \) :

Overall cervical stiffness

\( c_{i} \) :

Joint damping

\( x_{head} \) :

Horizontal head center of mass position

\( y_{head} \) :

Vertical head center of mass position

\( \theta_{head} \) :

Head angle

\( I_{head} \) :

Head inertia

\( x_{\exp } \) :

Horizontal experimental head position

\( y_{\exp } \) :

Vertical experimental head position

\( \theta_{\exp } \) :

Head angle

\( T \) :

Kinetic energy

\( V \) :

Potential energy

\( Q_{i} \) :

Generalized force

\( \delta_{ij} \) :

Kronecker delta

\( Q_{damping,i} \) :

Generalized joint energy loss

\( Q_{accel,i} \) :

Generalized acceleration

\( headfit \) :

Head response objective function

\( extfit \) :

Intervertebral response objective function

\( fit \) :

Overall objective function

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Acknowledgments

This Research effort was made possible by NPRP Grant No. (5-020-2-009) from the Qatar National Research Fund (a member of Qatar Foundation). The study sponsors had no involvement in the research and presented material. The authors would like to thank Dr. Craig Simmons and members of the Mechanics and Aerospace Design Laboratory for their helpful discussions throughout the project.

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There are no conflicts of interest to report.

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  1. Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON, M5S 3G8, Canada

    J. Hoover & S. A. Meguid

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Correspondence to S. A. Meguid.

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Hoover, J., Meguid, S.A. Analytical viscoelastic modelling of whiplash using lumped-parameter approach. Int J Mech Mater Des 11, 125–137 (2015). https://doi.org/10.1007/s10999-015-9306-1

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