## 1 Introduction  
The demand for high-speed and high-precision machine tools has increased due to the increasing demands from different industries, including defense, aircraft, computers, communications, and consumer electronics. Tool manufacturing design is particularly important to achieve high speed and precision. Presently, computer-aided design is usually used in designing machine tools because it does not necessitate producing, processing, assembling the casting, or identifying areas of the structure that contribute to reduce the cost of development, shortening the development time. CNC machine tools are the most important tools in modern manufacturing, and they are widely utilized in the automobile, ship, aircraft, and other equipment manufacturing industries. The issue of machining precision has always been an active and difficult research problem [1; 2; 3]. [PERSON] et al. [4] suggested a computationally efficient multi-body dynamics model for five-axis machine tools that proved to be 90% faster than previous structural multi-bodymodels. [PERSON] et al. [5] proposed a topology optimization method, which was used to optimize the jacket structure by changing the structural layout to improve its design, minimize weight, and reduce cost. The maximum stress was reduced by 9.6%, and the stress distribution was more uniform, effectively reducing the stress concentration. [PERSON] et al. [6] presented a numerical study of a solar air collector with and without thermal storage. The collector's performance was evaluated using a validated mathematical model, and the results showed that the collector's efficiency increased with an increase in the mass flow rate and the size of the collector. The thermal storage system improved the collector's thermal performance by increasing the outlet air temperature, and it also helped to maintain the outlet air temperature when solar radiation was low. The study found that the maximum efficiency achieved for the collector with thermal storage was 80.9%, while that of the collector without storage was 64.5%. The method for optimizing the topological design of composite structures is presented by [PERSON] et al. [7]. In this method, the stiffness and vibration modes, among others, can both be optimized at the same time by using a coupled level set and a moving morphable component approach. The performance of a cantilever beam and a wing structure was significantly enhanced after being designed using the proposed method. Both the stiffness and the first natural frequency of the cantilever beam were improved by 34% and 20%, respectively. Both the torsional stiffness and the first natural frequency of the wing's structure were improved by 17% and 14%, respectively. The results of the research showed that the proposed method was useful for designing high-performing multifunctional composite structures. [PERSON] et al. [8] proposed a topology optimization method to minimize the maximum stress of multi-material structures with graded interfaces. The topology results revealed that the maximum stress could be effectively reduced compared to that using the stiffness design. [PERSON] et al. [9] suggested an integrated optimization design method for structural topology and embedded heat source layout. By using the velocity field-level set method, multiple constraints and two design variable problems were solved using a general optimizer. The proposed multi-objective optimization obtained a reasonable compromise between the two related physical properties. Another important thing to think about when designing the structure of a machine tool is its dynamic performance. A modal analysis finds the shapes of the fundamental vibration modes and the frequencies that go with them. Moreover, natural frequency is a measure of how stiff a structure is when it is in motion [10]. In the classical precision machine tool design, the machine operation frequency is usually higher than the first natural frequency [11].