## 1 Introduction
### CAx and Digital Twin Methods  
Developing the final machine is a time-consuming and often inefficient process. Today, machine tool manufacturers cannot afford the time and cost of manufacturing and testing physical prototypes. For these reasons, it is suitable to use virtual prototyping tools and CAx tools in general to predict weak points and optimise machine design [23; 24; 25]. CAx methods include computer-aided design (CAD), computer-aided engineering (CAE) [26], and computer-aided manufacturing (CAM) [27]. These tools allow to simulate the complex behaviour of a real machine in the pre-production and production phases and to design and compare individual variants of a technical system. Using these tools (numerical analysis of the virtual prototype) is suitable to verify each step of the design to ensure that the designed machine meets all requirements. This places high demands on HW and SW equipment. In particular, advanced numerical simulations (stress, static stiffness, and modal analyses to calculate eigenmodes and natural frequencies) and various types of optimisation (geometric, mass, topology [28; 29], and optimisation of the wall thickness of the part) can be used in the structural design of machine tools. For the complex prediction of the behaviour of a technical system, a combination of these procedures (CAx) with the principle of using the digital twin of the machine seems to be suitable. The exact procedure is developed in the form of a methodology and is described in the following section. The concept of a digital twin appears to be the most effective method of optimising manufacturing and logistics processes. It fits fully into the areas of Industry 4.0 and Industry 5.0, which aim to integrate digital technologies into industrial processes (manufacturing processes) and which are described in the introduction of this research. There are several types of digital twins, which differ according to the phase of the product life cycle at which the digital twin model is created. A digital twin can be created before the actual manufacturing starts or after the (real) machine has been built. In the first case, the digital twin will detect defects before the machine is manufactured or commissioned. It results in significant cost savings and a reduction in the time of the entire production process. In the second case, the behaviour of the manufactured (real) machine can be optimised or its virtual commissioning can be performed. This brings significant cost savings, because any critical conditions can be debugged on a virtual machine, not a real one. If the digital twin extends to all phases of the product life cycle, we talk about the so-called digital thread. In general, digital twins are divided into:  
* Digital twin of a product--This is a computer model of a product (for example, a real machine tool) on which its behaviour can be simulated. This means not only its stiffness but generally its functioning under certain conditions, specifically kinematic links between bodies, movements, communication (signals), or control system. Based on the data obtained from the virtual digital twin, it is possible to evaluate the behaviour and performance of the real machine. * Digital twin of a process--It simulates complex processes such as a manufacturing plant, power grid, etc. It allows to analyse processes in real time. * Digital twin of a system--It simulates entire systems, integrating several digital twins of products or processes together. For example, this includes simulations of transport networks, cities, etc. Case studies for machining a specific part with a specific tool technology can be performed using the digital twin of a machine tool. In this way, it is possible to determine machine behaviour, determine manufacturing times (main and secondary times), and predict errors and possible collisions in the workpiece-machine tool system, thus eliminating the cost of correcting them in real operation.