## 2 Calculation of Forces on a Motorcycle
### Braking Forces and Tire Friction  
As found in Equation (1), the total braking force (\(S_{tot}\)) is:  
\[S_{tot}=S_{f}+S_{r}=md \tag{6}\]  
A condition that should be avoided to maintain driver safety during braking is tire skidding, resulting in a loss of tire grip with the road [21]. Tire traction limit \(D\) is based on tire characteristics and is related to the ratio \(\mu\), which is the friction force (or otherwise the normalized braking force), effected by the braking force \(S\) and the vertical load \(N\). In order to maintain road grip during braking, this ratio should not exceed a maximum available value of \(D\), for the front (\(D_{f}\)) and rear (\(D_{r}\)) tire traction, respectively:  
\[\mu_{f}=\frac{S_{f}}{N_{f}}<D_{f} \tag{7}\]  
\[\mu_{r}=\frac{S_{r}}{N_{r}}<D_{r} \tag{8}\]  
As the total brake force (\(S_{tot}\)) increases, tire skidding may occur either in the front or in the rear tire depending on braking force distribution [21]. In order to examine this phenomenon, the ratio of braking balance (\(\rho\)) is introduced, linking the braking force on the rear wheel and the total braking force:  
\[\rho=\frac{S_{r}}{S_{f}+S_{r}}=\frac{S_{r}}{S_{tot}} \tag{9}\]  
By using Equations (1)-(3) and (7)-(9), normalized braking force \(\mu\) may be expressed as a function of the deceleration \(d\) and the braking balance \(\rho\), as follows [21]:  
\[\mu_{f}=(1-\rho)\frac{(a+b)d}{ag+hd} \tag{10}\]  
\[\mu_{r}=\rho\frac{(a+b)d}{ag-hd} \tag{11}\]  
When \(\rho=0\) then \(\mu_{r}=0\) meaning only the front brakes are applied, while if \(\rho=1\) just the rear brakes are applied.  
In order to calculate the brake ratio at the verge of skidding, the actual deceleration at this point must be known [22]. Assuming the tires (front and rear) have the same coefficient of friction, the total braking force at the point of skidding is calculated by Equation (2):  
\[F_{skid}=\Big{(}N_{f}+N_{r}\Big{)}\mu=mg\mu \tag{12}\]  
By equating (6) with (12), the expression for maximum deceleration is obtained:  
\[F_{skid}=S_{tot}=>d=-\mu g \tag{13}\]  
The negative sign is an indication that the vehicle is in a state of deceleration.  
In general, tire friction data are linked to road conditions (dry, wet, ice), as also on different road material types (such as asphalt, gravel etc.) [7; 9; 27]. Nevertheless, motorcycle tires are different than car tires and have different shapes of contact patch on the road [27; 28]. Experimental results focused on motorcycle tire friction coefficient (\(\mu\)) calculation, indicate values close to \(\mu\) = 0.8 [22]. Experimental and theoretical results for car tires in contact with a road made from tarmac in dry conditions can be found in [23; 27], presenting a range of values \(0.7<\mu<1.0\). Research on motorcycle tires can also be found [22; 27], indicating that for a dry road \(\mu\) = 0.8. Considering all the above information, for our calculations the reference coefficient of friction used for the front and rear tires is \(\mu_{f}=\mu_{r}\) = 0.8.  
Therefore, based on Equation (13) and assuming \(\mu=0.8\) as a reference value, the maximum theoretical deceleration so that no tire skidding occurs is calculated at \(d=0.8\) g.