## 3 Utilized Design Optimization Algorithms
### Grey Wolf Optimizer (GWO)
### Gradient-Based Algorithms

Gradient-based algorithms are one type of deterministic optimization technique. When the gradient is easily defined, it can deal with problems involving unimodal and convex functions. The gradient of the goal function and the initial guess or starting search point are crucial in these algorithms. They resemble a ball shot aloft that is retracted downhill by gravity until it lands in a stable spot. SQP methods solve a sequence of optimization sub-problems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints. The downside of these algorithms when dealing with multi-model objective functions is that they are easily locked in local optimum rather than global optimum. SQP (Sequential Quadratic Programming) [24] is an example of such algorithms that will be used to compare and explain whether gradient-based methods succeed or fail in tackling MPS design optimization in this study. The failure of gradient-based algorithms to converge to global solutions or handle complex optimization problems is widely known. It is used in this research to show that such algorithms fail horribly in complex engineering issues such as PMS. Some researchers, including [PERSON] in [40], employed a gradient-based technique and were pleased with the results; however, some parameters, such as the initial search point, needed to be changed or adjusted in order for the algorithm to converge to an optimal solution. The pseudo code of SQP (Algorithm 2) can be summarized as follows:
```
0.3.1 begin Choose a starting point \(Xo\) and approximation \(Ho\) to the Hessian repeat\(i=1,2,\ldots\) Solve a QP sub-problem QP\({}_{i}\) to obtain the search direction \(S_{i}\) Given \(S_{i}\), find alpha so to determine \(X_{i+1}\) Update the approximation Hessian \(H_{i+1}\) using the BFGS scheme \(i=i+1\) Until reaching stopping criterion end
```

**Algorithm 2** Sequential Quadratic Programming

## 4 Motorcycle Engine Mount Design Models

One of the most common challenges engineers face when it comes to vibration isolation is motion isolation. This is a burden when external loads are transmitted to the engine. These loads, which are caused by road profile faults, are transferred to the frame via the tire patch. The typical engine mounting system consists of an engine, rubber engine mounts, and a base, which is represented by the motorcycle frame or body. As can be seen in Figure 1, the engine is modeled as a twelve DOF rigid body that can translate and rotate along/around three Cartesian axes. According to the pattern of excitations, the powertrain is allowed to oscillate on its mounts. Because the engine's inherent frequency is substantially higher than that of the mounting mechanism, it should be treated as a stiffbody. In each of the three major directions, the mounts are represented as springs with stiffness coefficients and hysteresis or viscous damping. Figure 2 depicts a 12-DOF model with a shaft assembly connecting the powerplant to the swing arm. This consists of two rigid bodies, one for the powertrain and the other for the swing-arm assembly, connected by a coupler shaft. This section shows the first of two twelve-degree-of-freedom (DOF) models for the motorcycle mounting system, with the first considering the influence of road loads. Road loads must be addressed when building a motorcycle mounting system because of the nature of motorcycles. This ensures the greatest possible handling. In this work, two twelve-DOF motorcycle engine mount models were modified and improved. The first model has twelve degrees of freedom but ignores road loads, whereas the second model has twelve degrees of freedom but takes into consideration both road and shaking loads. Figure 1: Powertrain and swing-arm layout [40].