# A Female Multi-Body Head-and-Neck Model and Validation for Rear Impact Simulations
## 3 Model Details
The 35 th percentile female head-neck model is built by using the multi-body dynamics software MSC Visual Nastran 4D 2001 R2. Revolute motors connect each pair of vertebrae, and these motors produce resistive torques similar to rotational springs and rotational dampers. The head-neck model is designed to allow motion in the sagittal plane only in the same way as the BioRID II dummy. The initial posture of the head-neck is a typical driving posture in which the head looks forward, as shown in Figure 3. The intervertebral joints are located at the intersections of the neck segments and the drawn arc, as shown in Figure 4. These segments have uniform density distribution, which is a viable approximation as applied successfully in other studies [3; 22]. The variation of torque as a function of intervertebral angular displacement is shown in Figure 5, where these torque values represent the rotational stiffness of each intervertebral joint. Figure 5 represents the quasi-static response of the neck without muscle activation. In Figure 5, the angles in the horizontal axis correspond to angular displacements at the intervertebral joints, which are measured from the initial configuration of the neck depicted in Figures 3 and 4. In all intervertebral joints, the same stiffness function is used for simplicity. In accordance with the EvaRID average female model [22], rotational stiffness values were scaled from those of the average male model in [PERSON] studies [3; 26]. The average male model's torque values were first multiplied by 0.65 or 0.40, which was \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Vertebra**} & \multirow{2}{*}{**Mass (kg)**} & \multirow{2}{*}{**Relative Orientation (deg)**} & \multicolumn{2}{c}{**Initial Velocities (m/s)**} \\ \cline{1-1}  & & & \(\mathbf{V_{x}}\) & \(\mathbf{V_{z}}\) \\ \hline C0 (Head) & 3.47 & \(-\)5.0 & 2.16 & \(-\)0.38 \\ \hline C1 (1 st cervical) & 0.0953 & 1.2 & 2.16 & \(-\)0.38 \\ \hline C2 (2 nd cervical) & 0.0953 & 3.8 & 2.16 & \(-\)0.38 \\ \hline C3 (3 rd cervical) & 0.0946 & 5.3 & 2.16 & \(-\)0.38 \\ \hline C4 (4 th cervical) & 0.0946 & 4.7 & 2.16 & \(-\)0.38 \\ \hline C5 (5 th cervical) & 0.0946 & 5.2 & 2.16 & \(-\)0.38 \\ \hline C6 (6 th cervical) & 0.0946 & 5.6 & 2.16 & \(-\)0.38 \\ \hline C7 (7 th cervical) & 0.0946 & \(-\)20.8 & 2.16 & \(-\)0.38 \\ \hline T1 (1 st thoracic) & - & 0 & 2.16 & \(-\)0.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The inertial properties and relative orientations of the female head–neck model. followed by re-shaping and fine-tuning the torque-angle curve. Figure 5 displays the torque-angle curves with scale factors 0.65 and 0.40, which provided biofidelic responses during the validation process. Details of model validation and the sensitivity of the model to scale factors are explained in Section 4 and in the discussion. The revolute motors at the intervertebral joints also incorporate viscous damping. The viscous damping coefficient varies with time, as shown in Figure 6. This damping function simulates both the viscoelastic behavior of soft tissues and the effects of active muscle contraction. The damping function has a great role in the simulation of active muscle response, and it is essential to reduce head rotation. The damping function in Figure 6 is an example from the simulations which were run to validate the head-neck model in this study. According to Figure 6, between 0 and 80 ms, active muscle response does not occur. Around 80 ms, the reflex of active muscle contraction begins. The maximum response occurs around 120 ms and remains until 150 ms. After this time, discharge of the muscles decreases until 180 ms and is maintained afterwards. The damping function variation, as shown in Figure 6, follows the pattern of electromyography (EMG) activity of the dominant muscles activated in rear impact conditions. This approach was successfully applied and validated in the former studies by [PERSON] [34,26]. In all intervertebral joints, the same damping function is used for simplicity.