## 3 Utilized Design Optimization Algorithms
### Grey Wolf Optimizer (GWO)
### Model 1: Deriving Matrices for Stiffness and Damping Coefficients

In the model presented herein [40], the equation of motion (EOM) is represented as follows:

\[M\,\ddot{X}+C\,\dot{X}+K\,X=F\,e^{j\omega t} \tag{1}\]

For mass, damping, and stiffness, \(M\), \(C\), and \(K\) are 12 \(\times\) 12 matrices. \(F\) denotes the input force vector coming from engine imbalance and/or road loads produced by road profile defects. The term \(e^{j\omega t}\) is called a harmonic vector in which \(\omega\) is angular frequency and \(t\) is time. \(X\) represents both the swing-arm and the powertrain's displacement vector, which encompasses both translational and rotational DOFs. Equation (2) demonstrates how to represent the vector \(X\). \[X=[x_{sa}\quad y_{sa}\quad z_{sa}\quad\alpha_{sa}\quad\beta_{sa}\quad\gamma_{sa }\quad x_{e}\quad y_{e}\quad z_{e}\quad\alpha_{e}\quad\beta_{e}\quad\gamma_{e} ]^{T} \tag{2}\]

The subscripts '\(s_{a}\)' and '\(e\)' in Equation (2) denote the parameters for the wing-arm and powertrain assemblies, respectively. \(x_{sa}\); \(y_{sa}\); and \(z_{sa}\) represent translation motion along \(x\), \(y\), and \(z\), and \(\alpha_{sa}\); \(\beta_{sa}\) ; and \(\gamma_{sa}\) represent rotation motion for the wing arm. The same applies for powertrain assemblies where \(x_{e}\), \(y_{e}\); and \(z_{e}\) represent translation motion along the \(x\); \(y\); and \(z\) axes, and \(\alpha_{e}\); \(\beta_{e}\); and \(\gamma_{e}\) represent rotational motion around the \(x\), \(y\), and \(z\) axes. The system's total mass matrix is calculated as stated in Equation (3) below:

\[M= \begin{bmatrix}M_{swingarm}&Z_{6}\\ Z_{6}&M_{engine}\end{bmatrix}  \tag{3}\]

\(M_{engine}\) and \(M_{swingarm}\) are the 6 \(\times\) 6 mass matrices for the powertrain and swing-arm assemblies, respectively. \(Z_{6}\) is a 6 \(\times\) 6 zero matrix as a result. The wing-arm and engine mass matrices are equivalent, and they are defined as shown in Equation (4). The inertia matrices for the engine and swing-arm are set at their respective local centers of gravity. \[M_{e}= \begin{bmatrix}m_{e}&0&0&0&0&0\\ 0&m_{e}&0&0&0&0\\ 0&0&m_{e}&0&0&0\\ 0&0&0&I_{xxe}&-I_{xye}&-I_{xze}\\ 0&0&0&-I_{xye}&I_{yge}&-I_{yze}\\ 0&0&0&-I_{xze}&-I_{yze}&I_{zze}\end{bmatrix}  \tag{4}\]

Figure 2: Schematic diagram of the motorcycle engine mount model. The stiffness matrix for the 12-DOF system is determined in Equation (5), whereas the damping matrix is defined in Equation (6):

\[K= \begin{bmatrix}K_{sa}+K_{c}&-K_{c}\\ -K_{c}&K_{e}+K_{c}\end{bmatrix}  \tag{5}\]

\[C= \begin{bmatrix}C_{sa}+C_{c}&-C_{c}\\ -C_{c}&C_{e}+C_{c}\end{bmatrix}  \tag{6}\]

\(K_{e}\) and \(K_{se}\) and \(C_{e}\) and \(C_{sa}\) are the stiffness and damping matrices for the powertrain and swing-arm, respectively, in Equations (5) and (6). The stiffness characteristics of the two rear shock springs linking the swing-arm to the frame are used to create the swing-arm's stiffness matrix. The coupler's stiffness and damping matrices, \(K_{c}\) and \(C_{c}\), are both 6 \(\times\) 6. The frame is indefinitely stiff in this model. This assumption implies that the connection points between the engine mounts and the frame, as well as the frame and the rear suspension, exhibit zero deflection. Two situations will be discussed using the EOM formulation introduced in the previous section. The first example addresses the engine's shaking forces as they are communicated to the frame (Model 1). The second scenario examines the loads sent to the frame by shaking forces, as well as the loads transmitted to the frame by the road (Model 2). ### Model 1: Shaking Forces

In Equation (7), the shaking force vector is a 6 \(\times\) 1 vector. It is worth noting that the shaking force vector development is for a 45\({}^{\circ}\)-bank angle V-twin engine. \[F_{s}= \begin{bmatrix}F_{sx}&F_{sy}&F_{sz}&M_{sx}&M_{sy}&T_{s}\end{bmatrix} ^{T} \tag{7}\]

The overall shaking forces of the V-twin engine are shown in Equations (8) and (9)

\[