## 2 Model Development
### Emission Module
#### 2.2.2 Driving Dynamics  
Figure 2 shows that as a first step of the propulsion noise simulation, the driving dynamics of the car are calculated in order to obtain the instantaneous engine speed \(n(t)\) and engine load \(M(t)\). The engine speed in engaged mode reads [34]:  
\[n(t)=60\cdot i_{\text{gear}}(t)\cdot i_{\text{ax}}\cdot\frac{v(t)/3.6}{2\pi r _{\text{tire,dyn}}}\quad\text{[rpm]} \tag{7}\]  
with the instantaneous vehicle speed \(v\) given in km/h, the gear ratio \(i_{\text{gear}}\), the axle ratio \(i_{\text{ax}}\) and the dynamic tire radius \(r_{\text{tire,dyn}}\approx 0.3\) m. The traction \(F_{T}\) is modeled by: [34; 35]  
\[F_{T}(t) = F_{B}(t)+\tilde{e}m\cdot a(t)+mg\sin(\alpha)\;\;\text{[N]} \tag{8}\] \[F_{B}(t) = F_{0}+F_{1}\cdot v(t)+F_{2}\cdot v^{2}(t)\] (9) \[a(t) = \frac{dv(t)/3.6}{dt} \tag{10}\]  
with the vehicle mass \(m\), gravity \(g\), the inclination angle of the road \(\alpha\), the translational acceleration \(a\) of the car and a mean equivalent mass factor \(\tilde{e}=1.15\) for the rotational accelerations for each individual gear. The basic driving resistance \(F_{B}\) (consisting of rolling resistance and aerodynamic drag) is modeled by the coast-down parameters \(F_{0}\), \(F_{1}\) and \(F_{2}\) with units N, N/(km/h) and N/(km/h)\({}^{2}\), respectively. These parameters have to be provided by the manufacturer during the type approval procedure. The engine load (torque) is formulated by [34]:  
\[M(t)=\frac{r_{\text{tire,dyn}}\cdot F_{T}(t)}{\eta\cdot i_{\text{gear}}\cdot i _{\text{ax}}}\quad\text{[Nm]} \tag{11}\]with a globally-set efficiency factor \(\eta=0.9\) for the power transmission from the engine to the wheels. The engine load in percent is defined by [36]:  
\[\Gamma(t)\equiv\frac{M(t)}{M_{\text{max}}(n(t))}\cdot 100\ \ \ \text{[\%]} \tag{12}\]  
with \(\Gamma=100\%\) at full load. At idling engine, \(M=0\) Nm and \(\Gamma=0\%\). In engine overrun operation (e.g., while engine braking), the engine delivers a negative torque to the crankshaft, which means that the engine load \(M\) becomes negative. In the model, this state is approximated as idle, _i.e._, \(M\) is set to zero. Gearbox shifts are modeled by three consecutive processes: the clutch is disengaged; a new gear is put in (at idling engine); and the clutch is engaged again. In dependence of the driving style, these processes vary in their respective durations. In the model, for a sporty driving style, the total gear change takes 0.6 s, whereas for a cozy, economic driving style, the gear change takes 1.3 s. Furthermore, the moments of a gear change strongly depend on the driving style and can be formulated as a function of engine speed and engine load, which is also the basic working principle of an automatic gearbox. Figure 3a and 3b show two simulated engine condition courses within an engine load _vs._ engine speed diagram. The black lines show the simulated temporal progression of the engine condition during a virtual pass-by. Both simulations start at the same initial engine condition of 900 rpm and 3 Nm, marked by green stars. In both cases, the passenger car starts in first gear and accelerates from \(v=7\) km/h to 50 km/h, however with differing accelerations \(a\) and driving styles. For the medium driving style and an acceleration of \(a=1\) m/\(s^{2}\) (Figure 3a), three gear changes occur at around 2000 rpm (Sample 1); whereas for the sporty driving style and an acceleration of \(a=2\) m/\(s^{2}\) (Figure 3b), only two gear changes happen, but at higher engine speeds of 3000 to 4000 rpm (Sample 2). The temporal behaviors of the engine states for these two examples are published as supplementary data (see the videos in Supplementary File). Figure 3: Simulation results: The upper graphs (**a,b**) show two simulated engine condition courses of an accelerating Ford Focus 1.8i with different accelerations and driving styles. The gray triangles show the interpolation grid spanned by the measuring points marked as circles, as introduced in Section 2.2.3.