# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Problems of Modal Analysis for Constructions with Bolted joints  
Knowing its dimensions and material, the vibration characteristics of any monolithic component can be calculated with one of the well-established approaches. In the case of beams, the basic tool for their normal mode estimation is the elastic beam theory, including the Euler-Bernoulli, Rayleigh, Timoshenko, and other models [35]. However, stiffness calculation for the composite structures with this approach is a nontrivial problem [36]. Usually, estimation of vibration frequency distinguishes several types of boundary constraints for beam couplings: hinged, fixed, sliding, and free. They imply an absolute coupling stiffness. In practice, components are usually coupled with different bracing (bolts, angles, screws, etc.). In such systems, joints, backlashes, surface friction, and nonlinear deformations play a huge role, making the problem of normal mode estimation extremely complex. Only very recent studies deal with efficient calculations of natural frequencies of bolted structures, given their nonlinear character. For example, the work [37] proposes a machine learning approach to nonlinear modal analysis. The paper [38] describes an artificial neural network for predicting parameters of normal contact stiffness, penetration limit, and contact. Recent work [39] proposed an efficient algorithm for estimating damping in bolted joints providing good accuracy compared to simulations in commercial finite element software. Nevertheless, this solution is still not implemented in commercially available engineering packages.  
In several papers, the determination of natural frequencies for bolted structures using FEM analysis shows unsatisfactory results, with errors ranging from 9 to 46.9% in determining natural frequencies [40]. A technique of using equivalent material can be applied to improve the accuracy[41]. Equivalent lower-dimensional models can also be efficient for simulating bolted constructions [42]. In civil engineering, to avoid detailed analysis of each element joint, a concept of effective stiffness is introduced reflecting the overall stiffness of a joint [43].  
In our previous study, we did not find low-frequency vibrations for the 3D printer Anycubic i3 Mega using finite-elements analysis [44] since we did not simulate joints, which played a leading role in the occurrence of this low-frequency mode.  
Therefore, we will consider an approach based on the equivalent model of the entire construction. Since we consider the preliminary design stage, we will omit any implementation details and focus on the effective stiffness of the joint. We will use the term \"stiffness\" for simplicity, meaning exactly \"effective stiffness\". Nevertheless, the proposed approach is valid only if the beams can be treated as absolutely stiff. Therefore, we must ensure that at least their first normal modes are far above the normal modes of the entire setup, i.e., the condition must be satisfied  
\[w_{b}>>w, \tag{1}\]  
where \(w_{b}\) is the beam first normal mode, \(w\) is the setup lower frequency. Elastic beam theory provides a well-established tool for verifying this assumption via estimating normal beam modes. The following section briefly recalls the main concepts of the Euler-Bernoulli beam theory and its application to our problem.