# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Flexible Joint Model  
The approach described in the previous section based on the flexible joint model is applicable to any construction made of rigid metal beams, in which natural frequencies are much higher than the natural frequencies emerging due to joint imperfectness. Unless other is claimed, we assume the system is strictly linear and symmetric and stiffness matrices in (3) are diagonal, so vibrations, which would occur in at least two orthogonal planes (denote them \(\theta_{x}(t)\) and \(\theta_{y}(t)\)) are independent and similar in case of similar initial conditions. Therefore, we first consider only a planar motion in one selected plane (denote the angle \(\theta(t)\)) and only one corresponding frequency, and then extend this approach to more complex cases. _Vibration model of the beam._ A joint model based on a linear flexible joint is proposed to find the natural frequency of vibrations, a. When the length of the beam is relatively small, it can be assumed that the structural profile is rigid. Then, according to [PERSON]'s second law, the motion of the beam is expressed in the following linear differential equation:  
\[J\ddot{\theta}+k\theta=0, \tag{6}\]  
where \(J=\frac{1}{3}M_{T}L^{2}\) is the beam moment of inertia, \(M_{T}\) is the total mass of the beam, \(L\) is the beam length, and \(k\) is the torsional stiffness of the joint, \(\theta\) is the angular displacement (see Figure 5). The analytical solution of Equation (6) has the form:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]  
Substituting (7) into (6), we obtain:  
\[-J\omega^{2}+k=0,\]  
hence, the vibration frequency of the extruded profile beam is expressed as:  
\[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]  
Equation (6) can be used to determine the stiffness of the joint by empirically determining the first mode frequency of the system and expressing the stiffness as a ratio:  
\[k=\frac{1}{3}\omega^{2}L^{2}M_{T}. \tag{9}\]  
Figure 5: Model of the beam. In the case of linear stiffness, (9) provides a simple way to determine \(k\) from experimentally estimated \(\omega\). In the case of nonlinear stiffness, the identification procedure should involve the waveform shape and would be more complicated [46]. _Vibration model of the gantry structure._ To find the natural vibration frequency of the gantry structure shown in Figure 6. Let us determine two independent mechanical stiffness of the joints. The stiffness of the lower joint is denoted by \(k_{1}\), and the stiffness of the upper joint is denoted by \(k_{2}\). The mass of the vertical beam is indicated \(M_{1}\), and the mass of the horizontal beam is represented by \(M_{2}\). The moment of inertia of the beam is equal to \(\frac{L_{1}}{2}\):  
\[J_{1}=\frac{M_{1}L_{1}{}^{2}}{3}\]  
The kinetic energy of motion of the gantry structure in this case is equal to:  
\[K=\frac{2J_{1}\dot{\theta}^{2}}{2}+\frac{M_{2}(\dot{\theta}L_{1})^{2}}{2}= \frac{M_{1}L_{1}^{2}}{3}\dot{\theta}^{2}+\frac{M_{2}L_{1}^{2}}{2}\dot{\theta} ^{2},\]  
the potential energy:  
\[P=\left(\frac{2k_{1}\theta^{2}}{2}+\frac{2k_{2}\theta^{2}}{2}\right)=(k_{1}+ k_{2})\theta^{2}.\]  
The Lagrange function, where \(\mathcal{L}\) is the Langrangian, is equal to the difference of kinetic and potential energy:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]0  
the Euler-Lagrange equation is written as:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]1  
from where  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]2  
and  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]3  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]4 Model of the portal. Using (7), we find the equation of the frequency of natural vibrations of the gantry structure:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]5  
It should be noted that the length of the horizontal beam does not affect the frequencies of the gantry structure. _Vibration model of a parallelepiped._ For simplicity, in the structure shown in Figure 7, the stiffness of the joints can be assumed and equal to \(k_{1}\) and \(k_{2}\).