## 3 Utilized Design Optimization Algorithms
### Grey Wolf Optimizer (GWO)
\begin{array}{ll}F_{sx}&=sin\beta\ \{2(m_{A}+m_{B})r\omega^{2}\ sin\theta sin \beta\\ &-2(m_{A}+m_{B})r\alpha\ cos\delta sin\beta\\ &+2m_{B}\frac{r^{2}\omega^{2}}{l}\ sin2\theta sin2\beta+mg\frac{r^{2}\alpha}{l }\ cos2\theta sin2\beta\right\}\\ &+\ \omega^{2}sin\beta\ \{m_{cb}2r_{2}\ cos(\theta+\beta)\\ &-\ m_{cb}r_{1}\ cos(\theta-\beta)\}\\ &+\alpha sin\beta\ \{m_{cb}2r_{2}\ sin(\theta+\beta)\\ &-\ m_{cb}r_{1}\ sin(\theta-\beta)\}\\ &+cos\beta\ \{2m_{A}r\omega^{2}\ sin\theta cos\beta-2m_{A}r\alpha\ cos\theta cos \beta\}\\ &+\alpha cos\beta\ \{m_{cb}1r_{1}\ cos(\theta-\beta)+m_{cb}2r_{2}\ cos(\theta+\beta)\}\\ &-\omega^{2}cos\beta\ \{m_{cb}1r_{1}\ sin(\theta-\beta)\\ &+m_{cb}2r_{2}\ sin(\theta+\beta)\}\\ \end{array}  \tag{8}\]

\[ \begin{array}{ll}F_{sy}&=cos\beta\ \{2(m_{A}+m_{B})r\omega^{2}\ cos\theta cos\beta\\ &+2(m_{A}+m_{B})ra\ sin\theta cos\beta\\ &+2m_{B}\frac{r^{2}\omega^{2}}{l}\ cos2\theta cos2\beta+m_{B}\frac{r^{2}\alpha} {l}\ sin2\theta cos2\beta\right\}\\ &-\ \omega^{2}cos\beta\ \{m_{cb}1r_{1}\ cos(\theta-\beta)\\ &+m_{cb}2r_{2}\ cos(\theta+\beta)\}\\ &-\ \alpha cos\beta\ \{m_{cb}1r_{1}\ sin(\theta-\beta)\\ &+\ \ m_{cb}2r_{2}\ sin(\theta+\beta)\}\\ &+sin\beta\ \{2m_{A}r\omega^{2}\ cos\theta sin\beta+2m_{A}r\alpha\ sin\theta sin \beta\}\\ &+asin\beta\ \{m_{cb}2r_{2}\ cos(\theta+\beta)\\ &-m_{cb}1r_{1}\ cos(\theta-\beta)\}\\ &+\omega^{2}sin\beta\ \{m_{cb}1r_{1}\ sin(\theta-\beta)\\ &-m_{cb}2r_{2}\ sin(\theta+\beta)\}\\ \end{array}  \tag{9}\]The corresponding masses at distances \(r_{1}\) and \(r_{2}\) for the left and right banks are \(m_{cb1}\) and \(m_{cb2}\), respectively, in the previous formulae. In Equations (10) and (11) below, the overall shaking moments for the V-twin engine are articulated:

\[ \begin{array}{ll}M_{sx}&=cos\beta\left\{2(m_{A}+m_{B})r\omega^{2}\ cos\theta cos\beta \right.\\ &+2m_{B}\frac{r^{2}\omega^{2}}{l}cos2\theta cos2\beta\\ &+2(m_{A}+m_{B})ra\ sin\theta cos\beta\\ &\left.+m_{B}\frac{r^{2}\omega^{2}}{l}sin2\theta cos2\beta\right\}\cdot z\\ &-\omega^{2}cos\beta\left\{m_{cb1}r_{1}\ cos(\theta-\beta)\right.\\ &+m_{cb2}r_{2}\ cos(\theta+\beta)\}\cdot z\\ &-\arccos\beta\left\{m_{cb1}r_{1}\ sin(\theta-\beta)\right.\\ &\left.+m_{cb2}r_{2}\ sin(\theta+\beta)\right\}\cdot z\\ \\ M_{sy}&=sin\beta\left\{-2(m_{A}+m_{B})r\omega^{2}\ sin\theta sin\beta\right. \\ &-2m_{B}\frac{r^{2}\omega^{2}}{l}sin2\theta sin2\beta\\ &+2(m+m_{B})ra\ cos\theta sin\beta\\ &+m_{B}\frac{r^{2}\omega}{l}cos2\theta sin2\beta\right\}\cdot z\\ &+\omega^{2}sin\beta\left\{m_{cb1}r_{1}\ cos(\theta-\beta)\right.\\ &\left.-m_{cb2}r_{2}\ cos(\theta+\beta)\right\}\cdot z\\ &+asin\beta\left\{m_{cb1}r_{1}\ sin(\theta-\beta)-m_{cb2}r_{2}\ sin(\theta+ \beta)\right\}\\ &\cdot z\\ \end{array}  \tag{11}\]

The shaking torque is expressed in Equation (12). \[ \begin{array}{ll}T_{s}=&\frac{1}{2}m_{B}r^{2}\omega^{2}\big{[}\{sin\theta cos \beta-2 sin2\theta cos2\beta-\frac{3r}{l}sin3\theta cos3\beta\}\big{]}\hat{k} \end{array}  \tag{12}\]

where \(r\) is the crank length, \(\theta=\omega t\), \(m_{B}\) is the equivalent rotating mass, and \(\beta\) is the bank angle. ### Model 2: Shaking Forces and Road Loads (Combined Loads)

Both shaking forces related to imbalance inside the engine and loads due to irregularities in the road profile are considered in the model given here. Model 1 shows the governing equations for shaking loads. Through the tire patch, road loads are passed to the frame. The road loads that are transmitted to the frame might be either periodic or nonperiodic. The frequency content of these loads is determined using Fourier Transforms [21] after they are evaluated for specified displacement functions. Fast Fourier Transforms (FFT) in MATLAB were employed in this study. The loads are calculated using Equation (13) for a certain road profile. \[F_{y}=kx+c\dot{x} \tag{13}\]

In Equation (13), \(F_{y}\) is the vertical force component that is transmitted through the tire patch due to displacement \(x\) and the velocity \(\dot{x}\) generated from the road profile.