## 3 Utilized Design Optimization Algorithms
### Grey Wolf Optimizer (GWO)
\(k\) and \(c\) are the respective stiffness and damping of the rear wheel in the y-direction. The continuous time Fourier series (CTFS) for a periodic function is expressed in the equation below:

\[x(t)=\sum\limits_{m=-\infty}^{\infty}c_{m}e^{jm\omega_{o}t}\ ;\text{where}\ \omega_{o}=\frac{2\pi}{P} \tag{14}\]

In the above equation, \(c_{m}\) is the Fourier series coefficients that are determined as follows:

\[c_{m}=\frac{1}{P}\int\limits_{ \begin{subarray}{c}-2\\ -2\end{subarray} }^{\frac{p}{2}}x(t)\ e^{-jm\omega_{o}t}\ dt \tag{15}\]\(P\) is the fundamental period of the displacement function, which corresponds to the fundamental frequency \(\omega_{o}\), as shown in Equation (14). The discrete time Fourier series (DTFS) is written as

\[x[n]=x(nT)=\sum_{m=\langle N\rangle}c_{md}\ e^{jm\omega_{o}nT} \tag{16}\]

where \(\omega_{o}=2\pi/NT\) and \(c_{md}\) are the Fourier series coefficients that are determined using the following equation:

\[c_{md}=\frac{1}{N}\sum_{n=0}^{N-1}x[n]e^{-jm\omega_{o}nT} \tag{17}\]

In Equations (16) and (17), the sample period is denoted by \(T\). Equation (18) is used to derive the DTFS coefficients. When a restricted band displacement function is used and a suitable sample frequency is set using FFT, the following results are obtained:

\[c_{md}=\frac{X[m]}{N} \tag{18}\]

The continuous time Fourier transform (CTFT) of the displacement function is shown in Equation (19), and the discrete time Fourier transform (DTFT) is expressed in Equation (20) below:

\[X(\omega)=\int\limits_{-\infty}^{\infty}x(t)e^{-j\omega t}dt \tag{19}\]

\[X_{d}(\omega)=\sum_{n=-\infty}^{\infty}x(nT)e^{-j\omega nT} \tag{20}\]

In Equations (19) and (20), \(X(\omega)\) is the spectrum of \(x(t)\) that can be used for periodic and non-periodic displacement functions. The optimization problem for the model presented in Figure 2 is solved using a swarm intelligence optimization method (GWO), an evolutionary algorithm (GA), and a local optimization algorithm known as Sequential Quadratic Programming (SQP). The objective function is designed to lower transmitted forces while also considering the engine displacement constraints imposed by static and dynamic loads. ### Formulation of the Optimization Problems

The cost function used in this study is the weighted total of the transmitted force through each individual mount. The shaking loads inside the engine generate the transmitted forces through the mounts. The force that each individual mount exerts on the frame is calculated as follows:

\[f_{i}=[-k_{i}^{*}k_{i}^{*}\ \widetilde{r}_{i}] \begin{bmatrix}X_{ti}\\ X_{ri}\end{bmatrix}  \tag{21}\]

\(X_{ti}\) and \(X_{ri}\) are the translational and rotational displacements at the powertrain's center of gravity owing to shaking loads, and \(r_{i}\) is the skew symmetric from the position vector of the individual mount stated in Equation (1). \(k_{i}\) is the individual mount's local stiffness matrix. The objective function,\(f_{w}\), is obtained by adding the Euclidean norms of the individual forces passed by each mount. \[f_{w}=\sum_{j}\lambda_{j}\sum_{i}\|f_{i}\| \tag{22}\]

\(\lambda_{j}\) is the weighting parameter in Equation (22) that corresponds to various loading circumstances. Adding a deflection limitation to the optimization issue determines the maximum permitted engine weight. The static deflection \(X_{st}\) at the origin of the global coordinate system is calculated as follows:

\[X_{st}=K^{-1}F_{st} \tag{23}\]

In Equation (23), \(F_{st}\) is the static load on the system. The engine mount optimization problem can be expressed as follows:

\[\text{Minimize }f_{w}\left(k_{i},\ r_{i},\ \theta_{i}\right) \tag{24}\]

\[\text{subject to }g_{j}(k_{i},r_{i},\ \theta_{i})\ \leq 0\ j=1,\ldots,N\]

Mount stiffness, location, and orientation, \(k_{i},\ r_{i},\ \theta_{i}\), represent the design parameters in Equation (24) that are subjected to a total of \(N\) constraints \(g_{j}\).