# Optimization of Damping in a Semi-Active Car Suspension System with Various Locations of Masses
## 3 Results and Discussion
Figure 1: Dependence of (**a**) the optimized damping coefficient; (**b**) RMS of optimized vertical acceleration of the driver; (**c**) the difference between RMS of vertical accelerations of the driver when optimized and non-optimized; (**d**) RMS of non-optimized vertical acceleration of the driver; on-road profile waviness \(w_{1}\) and \(w_{2}\) when all masses are \(m_{1}=m_{2}=m_{3}=m_{4}=80\) kg and vehicle speed is 20 km/h, 50 km/h, 70 km/h, 90 km/h, or 130 km/h. Black dotted line shows values of major ticks. Black solid line is zero. The results of all other cases are given as supplementary material Tables S1-S6 or are presented as the difference between optimized and non-optimized values in Figures 2a, 3a, S3a, S4a, S5a and S6a. As seen in the results, as the values of \(w_{1}\) and \(w_{2}\) increase, the values of optimal damping increase, except in situations of speed = 20 km/h and increasing \(w_{1}\). Analogic dependencies occur for all seven mass cases. This could be explained in two ways. First, using Equation (3), we have a small decrease in the \(G_{\text{d}}(\Omega)\) values in the range from 0.16 \(\times\) 2\(\pi\) rad/m up to 0.21 \(\times\) 2\(\pi\) rad/m increasing \(w_{1}\). Second, due to the relationships between angular spatial frequency \(\Omega\), frequency \(f\), and velocity \(v\)\(\Omega=\frac{2\pi f}{v}\), we need higher angular spatial frequencies at decreasing speeds to excite the same minimal frequency. For example, to excite 1 Hz vibrations driving at 20 km/h, we need an angular spatial frequency of 0.18 \(\times\) 2\(\pi\) rad/m. The highest damping values occur when \(w_{1}\) = 6 and speed varies from 70 km/h up to 90 km/h. Comparing the optimized damping with its value in a passive suspension system 2700 Ns/m [65], one can say that all values are smaller, except in several situations (\(m_{1}\) = \(m_{2}\) = 80 kg, \(w_{1}\) = 1, \(w_{2}\) = 3; \(m_{1}\) = 160 kg, \(w_{1}\) = 1, 2, 4, 6, \(w_{2}\) = 3; \(m_{1}\) = 100 kg, \(m_{2}\) = 60 kg, \(m_{3}\) = 50 kg, \(m_{4}\) = 40 kg, \(w_{1}\) = 1, 2, \(w_{2}\) = 3). The surface area of the bigger values increases with increasing car speed and reaches maximum damping values when the speed is 70 km/h or 90 km/h and \(w_{1}\) = 6. This was observed as well for other locations of masses (see Figures 2a, 3a, S3a, S4a, S5a and S6a). The highest damping is needed when mass \(m_{1}\) = 160 kg (from 1.1 up to 1.43 times bigger compared with \(m_{1}\) = 80 kg or 1.06-1.47 times compared with \(m_{1}\) = \(m_{2}\) = \(m_{3}\) = \(m_{4}\) = 80 kg). Comparing differences in the optimized damping of various locations of masses of 80 kg, the variations are less than 100 except in situations when \(w_{1}\) = 6 or \(v\) = 90 km/h. Graphic representation of the original and optimized values is presented in Figure 1b-d. The dependencies of the optimized damping coefficient on vehicle speed (see Figures S1 and S2) show that:  
1. When fixed waviness index = \(w_{1}\): * Higher damping values are needed, increasing values of \(w_{2}\) at low speed (differences amount to several hundreds); * This difference decreases with increasing speed (down to several tens); * Higher \(w_{1}\) values significantly increase damping values when the speed is more than 50 km/h;  
Figure 2: Dependence of the difference between optimized and non-optimized (**a**) damping coefficient; (**b**) RMS of vertical accelerations of driver; on-road profile waviness \(w_{1}\) and \(w_{2}\) when \(m_{1}\) = 80 kg and vehicle speed is 20 km/h, 50 km/h, 70 km/h, 90 km/h, or 130 km/h. Black dotted line shows values of major ticks. Black solid line is zero. It reaches the maximum when \(w_{1}=6\) and the speed is 70 km/h or 90 km/h. 2. When fixed waviness index = \(w_{2}\): * The differences of damping amount to several tens at 20 km/h speed; * This difference increases with increasing speed and reaches the maximum at 70 km/h or 90 km/h speed.