Nonlinear multibody dynamics and finite element modeling of occupant response: part I—rear vehicle collision

Abstract

With the rise in vehicle ownership, the need to reduce the risk of injury among vehicle occupants that arises from vehicle collisions is important to occupants, insurers, manufacturers and policy makers alike. The human head and neck are of special interest, due to their vulnerable nature and the severity of potential injury in these collisions. This work is divided into two parts: In Part I, we focus our attention to modeling rear collision that could lead to whiplash. Specifically, two multibody dynamics (MBD) models of the cervical spine of the 50th percentile male are developed using realistic geometries, accelerations and biofidelic variable intervertebral rotational stiffness. Furthermore, nonlinear finite element (FE) simulations of two generic compact sedan vehicles in rear collision scenario were performed. Using the acceleration profiles measured at the driver’s seat of the colliding vehicles, FE simulation of a seated and restrained occupant in rear collision was performed to determine the occupant response. The resultant accelerations, measured at the T1 vertebra of the occupant model, were used as an input to the MBD models to obtain their kinematic response. Validation of the MBD models shows great agreement with experimentally published data. Comparison between the MBD and FE simulations for a 32 km/h vehicle-to-vehicle impact shows similar trends in head trajectory. However, the MBD models reported less peak head displacements compared to the FE model. This is attributed to the failure of the anterior longitudinal ligament at the mid cervical spine leading to increased intervertebral rotation in the FE model.

Appendices

Appendices

1.1 Appendix 1: Population of matrices of 1 DOF model

$$ \left[ A \right] = \left[ {\begin{array}{*{20}l} {m_{{1 - {\text{n}}}} l_{1}^{2} + I_{1} } \hfill & {m_{2 - n} l_{1} l_{2} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & \cdots \hfill & {m_{n} l_{1} l_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill \\ {m_{2 - n} l_{1} l_{2} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & {m_{2 - n} l_{2}^{2} + I_{2} } \hfill & \cdots \hfill & {m_{n} l_{2} l_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {m_{n} l_{1} l_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill & {m_{n} l_{2} l_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill & \cdots \hfill & {m_{n} l_{n}^{2} + I_{n} } \hfill \\ \end{array} } \right] $$

$$ \left[ B \right] = \left[ {\begin{array}{*{20}l} 0 \hfill & {m_{2 - n} l_{1} l_{2} \sin \left( {\theta_{1} - \theta_{2} } \right)} \hfill & \cdots \hfill & {m_{n} l_{1} l_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)} \hfill \\ { - m_{2 - n} l_{1} l_{2} \sin \left( {\theta_{1} - \theta_{2} } \right)} \hfill & 0 \hfill & \cdots \hfill & {m_{n} l_{2} l_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ { - m_{n} l_{1} l_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)} \hfill & { - m_{n} l_{2} l_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)} \hfill & \cdots \hfill & 0 \hfill \\ \end{array} } \right] $$

$$ \left[ C \right] = \left[ {\begin{array}{*{20}l} {c_{1} + c_{2} } \hfill & { - c_{2} } \hfill & 0 \hfill & 0 \hfill \\ { - c_{2} } \hfill & \ddots \hfill & \ddots \hfill & 0 \hfill \\ 0 \hfill & \ddots \hfill & {c_{n - 1} + c_{n} } \hfill & { - c_{n} } \hfill \\ 0 \hfill & 0 \hfill & { - c_{n} } \hfill & {c_{n} } \hfill \\ \end{array} } \right] $$

$$ \left[ D \right] = \left[ {\begin{array}{*{20}l} {k_{1} \left( {\phi_{1} } \right) + k_{2} \left( {\phi_{2} } \right)} \hfill & { - k_{2} \left( {\phi_{2} } \right)} \hfill & 0 \hfill & 0 \hfill \\ { - k_{2} \left( {\phi_{2} } \right)} \hfill & \ddots \hfill & \ddots \hfill & 0 \hfill \\ 0 \hfill & \ddots \hfill & {k_{n - 1} \left( {\phi_{n - 1} } \right) + k_{n} \left( {\phi_{n} } \right)} \hfill & { - k_{n} \left( {\phi_{n} } \right)} \hfill \\ 0 \hfill & 0 \hfill & { - k_{n} \left( {\phi_{n} } \right)} \hfill & {k_{n} \left( {\phi_{n} } \right)} \hfill \\ \end{array} } \right] $$

$\left\{ Q \right\} = \left\{ {\begin{array}{*{20}l} { - m_{1 - n} l_{1} \left( {a_{x} \cos \theta_{1} + a_{z} \sin \theta_{1} } \right)} \hfill \\ { - m_{2 - n} l_{2} \left( {a_{x} \cos \theta_{2} + a_{z} \sin \theta_{2} } \right)} \hfill \\ \vdots \hfill \\ { - m_{n} l_{n} \left( {a_{x} \cos \theta_{n} + a_{z} \sin \theta_{n} } \right)} \hfill \\ \end{array} } \right\}$

$$ \left\{ {\ddot{q}} \right\} = \left\{ {\begin{array}{*{20}l} {\ddot{\theta }_{1} } \\ \vdots \\ {\ddot{\theta }_{n} } \\ \end{array} } \right\}\quad \left\{ {\dot{q}^{2} } \right\} = \left\{ {\begin{array}{*{20}l} {\dot{\theta }_{1}^{2} } \\ \vdots \\ {\dot{\theta }_{n}^{2} } \\ \end{array} } \right\}\quad \quad \quad \left\{ q \right\} = \left\{ {\begin{array}{*{20}l} {\theta_{1} - \theta_{01} } \\ \vdots \\ {\theta_{n} - \theta_{0n} } \\ \end{array} } \right\}\quad \left\{ {\begin{array}{*{20}l} {\phi_{1} } \\ \vdots \\ {\phi_{n} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {\theta_{1} - \theta_{01} } \\ {\left( {\theta_{2} - \theta_{02} } \right) - \left( {\theta_{1} - \theta_{01} } \right)} \\ \vdots \\ {\left( {\theta_{n} - \theta_{0n} } \right) - \left( {\theta_{n} - \theta_{0n} } \right)} \\ \end{array} } \right\} $$

$$ m_{1 - n} = \mathop \sum \limits_{i = 1}^{n} m_{i} \quad l_{1 - n} = \mathop \sum \limits_{i = 1}^{n} l_{i} \quad n = 8\quad {\text{for}}\; 1\;{\text{DOF}} $$

1.2 Appendix 2: Matrices population of 2 DOF model

$$ \begin{aligned} \left[ A \right] & = \left[ {\begin{array}{*{20}l} {I_{1} + m_{1 - n} l_{1}^{2} } \hfill & {m_{2 - n} l_{1} l_{2} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & \cdots \hfill & {m_{n} l_{1} l_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill \\ {m_{2 - n} l_{1} l_{2} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & {I_{2} + m_{2 - n} l_{2}^{2} } \hfill & \cdots \hfill & {m_{n} l_{2} l_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {m_{n} l_{1} l_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill & {m_{n} l_{2} l_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill & \cdots \hfill & {I_{n} + m_{n} l_{n}^{2} } \hfill \\ 0 \hfill & {m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)l_{2} } \hfill & \cdots \hfill & {m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)l_{n} } \hfill \\ { - m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)l_{1} } \hfill & 0 \hfill & \cdots \hfill & {m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)l_{n} } \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ { - m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)l_{1} } \hfill & { - m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)l_{2} } \hfill & \cdots \hfill & 0 \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} 0 \hfill & { - m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)l_{1} } \hfill & \cdots \hfill & { - m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)l_{1} } \hfill \\ {m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)l_{2} } \hfill & 0 \hfill & \cdots \hfill & { - m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)l_{2} } \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)l_{n} } \hfill & {m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)l_{n} } \hfill & \cdots \hfill & 0 \hfill \\ {m_{1 - n} } \hfill & {m_{2 - n} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & \cdots \hfill & {m_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill \\ {m_{2 - n} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & {m_{2 - n} } \hfill & \cdots \hfill & {m_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {m_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill & {m_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill & \cdots \hfill & {m_{n} } \hfill \\ \end{array} } \right] \\ \end{aligned} $$

$$ \begin{aligned} \left[ B \right] & = \left[ {\begin{array}{*{20}l} 0 \hfill & {m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)l_{1} l_{2} } \hfill & \cdots \hfill & {m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)l_{1} l_{n} } \hfill \\ { - m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)l_{1} l_{2} } \hfill & 0 \hfill & \cdots \hfill & {m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)l_{2} l_{n} } \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ { - m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)l_{1} l_{n} } \hfill & { - m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)l_{2} l_{n} } \hfill & \cdots \hfill & 0 \hfill \\ { - m_{1 - n} l_{1} } \hfill & { - m_{2 - n} l_{2} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & \cdots \hfill & {m_{n} l_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill \\ { - m_{2 - n} l_{1} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & { - m_{2 - n} l_{2} } \hfill & \cdots \hfill & {m_{n} l_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ { - m_{n} l_{1} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill & { - m_{n} l_{2} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill & \cdots \hfill & {m_{n} l_{n} } \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {m_{1 - n} l_{1} } \hfill & {m_{2 - n} l_{1} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & \cdots \hfill & {m_{n} l_{1} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill \\ {m_{2 - n} l_{2} \cos \left( {\theta_{1} - \theta_{2} } \right)} \hfill & {m_{2 - n} l_{2} } \hfill & \cdots \hfill & {m_{n} l_{2} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {m_{n} l_{n} \cos \left( {\theta_{1} - \theta_{n} } \right)} \hfill & {m_{n} l_{n} \cos \left( {\theta_{2} - \theta_{n} } \right)} \hfill & \cdots \hfill & {m_{n} l_{n} } \hfill \\ 0 \hfill & {m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)} \hfill & \cdots \hfill & {m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)} \hfill \\ { - m_{2 - n} \sin \left( {\theta_{1} - \theta_{2} } \right)} \hfill & 0 \hfill & \cdots \hfill & {m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ { - m_{n} \sin \left( {\theta_{1} - \theta_{n} } \right)} \hfill & { - m_{n} \sin \left( {\theta_{2} - \theta_{n} } \right)} \hfill & \cdots \hfill & 0 \hfill \\ \end{array} } \right] \\ \end{aligned} $$

$$ \left[ C \right] = \left[ {\begin{array}{*{20}l} {c_{r1} + c_{r2} } \hfill & { - c_{r2} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ { - c_{r2} } \hfill & \ddots \hfill & \ddots \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & \ddots \hfill & {c_{rn - 1} + c_{rn} } \hfill & { - c_{rn} } \hfill & 0 \hfill & 0 \hfill & {0\backslash } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & { - c_{rn} } \hfill & {c_{rn} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {c_{e1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {c_{e2} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ddots \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {c_{en} } \hfill \\ \end{array} } \right] $$

$$ \left[ D \right] = \left[ {\begin{array}{*{20}l} {k_{r1} \left( {\phi_{1} } \right) + k_{r2} \left( {s_{2} } \right)} \hfill & { - k_{r2} \left( {\phi_{2} } \right)} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ { - k_{r2} \left( {\phi_{2} } \right)} \hfill & \ddots \hfill & \ddots \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & \ddots \hfill & {k_{rn - 1} \left( {\phi_{n - 1} } \right) + k_{rn} \left( {\phi_{n} } \right)} \hfill & { - k_{rn} \left( {\phi_{n} } \right)} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & { - k_{rn} \left( {\phi_{n} } \right)} \hfill & {k_{rn} \left( {\phi_{n} } \right)} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {k_{e1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {k_{e2} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ddots \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {k_{en} } \hfill \\ \end{array} } \right] $$

$$ \left\{ Q \right\} = \left\{ {\begin{array}{*{20}l} { - m_{1 - n} l_{1} \left( {a_{x} \cos \theta_{1} + a_{z} \sin \theta_{1} } \right)} \hfill \\ { - m_{2 - n} l_{1} \left( {a_{x} \cos \theta_{2} + a_{z} \sin \theta_{2} } \right)} \hfill \\ \vdots \hfill \\ { - m_{n} l_{1} \left( {a_{x} \cos \theta_{n} + a_{z} \sin \theta_{n} } \right)} \hfill \\ { - m_{1 - n} \left( {a_{x} \sin \theta_{1} - a_{z} \cos \theta_{1} } \right)} \hfill \\ { - m_{2 - n} \left( {a_{x} \sin \theta_{2} - a_{z} \cos \theta_{2} } \right)} \hfill \\ \vdots \hfill \\ { - m_{n} \left( {a_{x} \sin \theta_{n} - a_{z} \cos \theta_{n} } \right)} \hfill \\ \end{array} } \right\}\quad \left\{ {\ddot{q}} \right\} = \left\{ {\begin{array}{*{20}l} {\left\{ {\ddot{\theta }_{i} } \right\}} \hfill \\ {\left\{ {\ddot{l}_{i} } \right\}} \hfill \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {\ddot{\theta }_{1} } \hfill \\ {\ddot{\theta }_{2} } \hfill \\ \vdots \hfill \\ {\ddot{\theta }_{n} } \hfill \\ {\ddot{l}_{1} } \hfill \\ {\ddot{l}_{2} } \hfill \\ \vdots \hfill \\ {\ddot{l}_{n} } \hfill \\ \end{array} } \right\} $$

$$ \left\{ {\dot{q}^{2} } \right\} = \left\{ {\begin{array}{*{20}l} {\left\{ {\dot{\theta }_{i}^{2} } \right\}} \\ {\left\{ {2\dot{\theta }\dot{l}} \right\}} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {\dot{\theta }_{1}^{2} } \\ {\dot{\theta }_{2}^{2} } \\ \vdots \\ {\dot{\theta }_{n}^{2} } \\ {2\dot{\theta }_{1} \dot{l}_{1} } \\ {2\dot{\theta }_{2} \dot{l}_{2} } \\ \vdots \\ {2\dot{\theta }_{n} \dot{l}_{n} } \\ \end{array} } \right\}\quad \left\{ {\dot{q}} \right\} = \left\{ {\begin{array}{*{20}l} {\left\{ {\dot{\theta }} \right\}} \\ {\dot{l}} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {\dot{\theta }_{1} } \\ {\dot{\theta }_{2} } \\ \vdots \\ {\dot{\theta }_{n} } \\ {\dot{l}_{1} } \\ {\dot{l}_{2} } \\ \vdots \\ {\dot{l}_{n} } \\ \end{array} } \right\}\quad \left\{ q \right\} = \left\{ {\begin{array}{*{20}l} {\left\{ \theta \right\}} \\ {\left\{ l \right\}} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {\theta_{1} - \theta_{01} } \\ {\theta_{2} - \theta_{02} } \\ \vdots \\ {\theta_{n} - \theta_{0n} } \\ {l_{1} - l_{01} } \\ {l_{2} - l_{02} } \\ \vdots \\ {l_{n} - l_{0n} } \\ \end{array} } \right\} $$

$$ \left\{ {\begin{array}{*{20}l} {\phi_{1} } \\ \vdots \\ {\phi_{n} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {\theta_{1} - \theta_{01} } \\ {\left( {\theta_{2} - \theta_{02} } \right) - \left( {\theta_{1} - \theta_{01} } \right)} \\ \vdots \\ {\left( {\theta_{n} - \theta_{0n} } \right) - \left( {\theta_{n} - \theta_{0n} } \right)} \\ \end{array} } \right\} $$

$$ m_{i - n} = \mathop \sum \limits_{i}^{n} m_{j} \quad l_{i - n} = \mathop \sum \limits_{i}^{n} l_{j} \quad n = 16\quad {\text{for}}\; 2\;{\text{DOF}} $$