# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### How Cartesian 3D Printer Kinematic Scheme Affects Its Dynamics  
Cartesian 3D printers consist of two independent parts: the frame with the printhead mounted on it and the printing surface (the printbed). These parts move relative to each other. The system requires the mobility of 3D printer parts along three orthogonal axes to be functional. Using the standard graphical notation of kinematic schemes [47], we can depict a kinematic structure of any Cartesian design as a very simple object with three translational degrees of freedom provided by three orthogonal groups of prismatic joints (redundant systems will not be considered). Examples of such schemes are given in Figure 3, left column. Figure 3a refers to the simplest structure used in very cheap 3D printers, Figure 3b refers to popular gantry kinematics with one closed kinematic chain, Figure 3c presents a variant of more stiff kinematics, containing three closed kinematic chains. These kinematic schemes figure out only how the structure is intended to move. It is not obvious from each kinematic scheme in the left column, which one is better in terms of vibration tolerance unless we consider an additional factor: a finite stiffness of joints between beams, linear guides and other elements of construction. As the simplest approximation, these joints can be considered spherical joints with internal stiffeners. Such a point of view is reflected in Figure 3d-f. From these extended schemes, the advantage of closed kinematic chains becomes obvious: the more spherical joints are involved in possible angular movement the higher the overall stiffness is. For better illustration, we introduce a simple formalism describing differences in relative axis mobility and axes structure for 3D printers and present all possible kinematic designs within this formalism in Appendix A.  
Figure 2: Experimental frequencies of the beam. (**Top**): one hidden bracket for fastening to a base is used, several peaks emerge because of the joint stiffness non-linearity [46]. (**Bottom**): two hidden brackets are used, the vibration is close to harmonic. Now, let us consider how the kinematics of the 3D printer affects its vibration characteristics and therefore printing quality. Denote the vector of angular rotations in spherical joints as \(\mathbf{\theta}=(\theta_{1},\theta_{2},\dots,\theta_{n})^{\top}\), and a vector of linear translations in prismatic joints in local coordinate systems associated with each direction of prismatic joints as \(\mathbf{q}=(x_{L},y_{L},z_{L})^{\top}\), where \(N\) is the number of spherical joints and \(n\leq N\) is the number of rotational degrees of freedom, and the number of translational degrees of freedom is 3. Denote the global coordinates of the 3D printer working point as \(\mathbf{x}=(x,y,z)^{\top}\). The overall dynamics of the 3D printer correspond to a linearized ODE:  
\[ \begin{cases}J(\mathbf{x})\mathbf{\theta}+C_{\theta}\mathbf{\theta}+K_{\theta}\mathbf{\theta}= \mathbf{f}_{\theta}(\mathbf{x},\dot{\mathbf{x}},\ddot{\mathbf{x}},\mathbf{q},\dot{\mathbf{q}},\ddot {\mathbf{q}}),\\ \mathbf{x}=L(\mathbf{q},\mathbf{\theta}),\\ M\ddot{\mathbf{x}}+C_{x}\dot{\mathbf{x}}+K_{x}\mathbf{x}=\mathbf{f}_{x}(\mathbf{x},\dot{\mathbf{x }},\ddot{\mathbf{x}},\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}}),\\ \mathbf{x}=L(\mathbf{q},\mathbf{\theta}),\\ \mathbf{e}=\mathbf{q}-\mathbf{x},\end{cases}  \tag{3}\]  
where \(J(\mathbf{x})\) is a diagonal matrix of moments of inertia, \(M\) is a diagonal matrix of masses, \(C_{\theta},C_{x}\) are diagonal matrices of damping coefficients, \(K_{\theta},K_{x}\) are matrices of linearized stiffness, \(\mathbf{f}_{\theta}\) is a vector of disturbance torques, \(\mathbf{q}\) is a vector of relative translations in local coordinate systems attached to the prismatic joints, \(\mathbf{f}_{x}\) is a vector of disturbance forces, \(\mathbf{x}\) is an absolute position of the 3D printer working point, \(\mathbf{e}\) is the positioning error.