## 2 Materials and Methods
### Moment of Inertia Measurement  
The focus of the investigations concerning the moments of inertia was the yaw moment of inertia \(I_{z}\), which is needed for most vehicle lateral dynamic models such as single track models. The rotational test bench shown schematically in Figure 4 was developed for the determination of this value. The test bench and test object were driven via a rope drum by a driving mass \(m_{D}\). A photoelectric switch served as incremental encoder for the angle of rotation measurements. Neglecting frictional influences and the masses and inertias of rope and deflection pulleys, the equilibrium shown in Equation (5) resulted for the system.  
\[\ddot{\varphi}_{Meas}(I_{TB,z}+I_{V,z})=\left(m_{D}g-m_{D}\ddot{\varphi}_{Meas }r\right)r \tag{5}\]  
The inertia is composed of the partial inertias of test bench (\(I_{TB,z}\)) and vehicle or test object (\(I_{V,z}\)) and can be determined from the measured angular acceleration \(\ddot{\varphi}_{Meas}\) considering the drum radius \(r\) and the drive mass \(m_{D}\). Tests with a geometrically defined test object have shown that frictional influences lead to an overestimation of the moment of inertia by about 10%. The effective driving torque is reduced by a frictional torque \(M_{fr}\). As the measurement setup did not allow a direct measurement of the friction torque, it had to be estimated from a spin-out test. Considering the friction torque, Equation (6) applies to the driven test bench.  
\[\ddot{\varphi}_{Meas}(I_{TB,z}+I_{V,z})=\left(m_{D}g-m_{D}\ddot{\varphi}_{Meas }r\right)r-M_{fr} \tag{6}\]  
The spin-out tests were performed without drive mass and rope so that their frictional components were not included. It was assumed that they were negligible compared with the friction of the test bench bearings. Equation (7), therefore, applies to the spin-out test. It was assumed that the frictional torque would remain constant during driving and spin-out. By decoupling the drive mass, a spin-out test directly followed a driven test in each case.  
\[\ddot{\varphi}_{spin-out}(I_{TB,z}+I_{V,z})=-M_{fr} \tag{7}\]  
Figure 4: Rotational inertia test bench.  
Equations (5) and (6) can be used to obtain Equation (8) for determining the combined moment of inertia of the vehicle and test bench.  
\[I_{TB,z}+I_{V,z}=\frac{\big{(}m_{D}\underline{g}-m_{D}\bar{\varphi}_{Meas}r \big{)}r}{\bar{\varphi}_{Meas}-\bar{\varphi}_{spin-out}} \tag{8}\]  
In cases where the center of gravity could not be positioned directly above the test bench axis of rotation due to spatial restrictions, the Steiner component was also taken into account. To determine the angular accelerations \(\bar{\varphi}_{spin-out}\) and \(\bar{\varphi}_{Meas}\), the measured angular-time curve was fitted to a second-order polynomial using a least-squares method. At higher rotation speeds, the nonlinear influence of the air/fan resistance increased strongly. The simplification to a constant friction torque was therefore only valid for low speeds. Each test was repeated five times. The mean values \(\overline{\overline{\varphi}}_{Meas}\) and \(\overline{\overline{\varphi}}_{spin-out}\) were used to calculate the rotational inertia of the entire setup according to Equation (8).  
Based on tests with various geometrically determined test bodies, the approach presented above showed errors of about 2% of the theoretical value for different test bodies.