# Synthesis and Functional Optimization of a Vibratory Machine with a Parallel Mechanism Structure
## 3 Mathematical Model of the Structure of the Mechanism  
The vibratory motion of the working organ is characterized by the following equations:  
\[x=x_{a}.\cos(\omega.t+\phi_{x});\ \ y=y_{a}.\cos(\omega.t+\phi_{y}). \tag{1}\]  
where:  
\[\begin{array}{l}x_{a}=\frac{F_{0}.\cos\alpha}{m\sqrt{\left(p_{z}^{2}-\omega ^{2}\right)^{2}+4\pi_{2}^{2}\omega^{2}}};\ fg\phi_{x}=\frac{2n_{x}\omega}{p_{1}^ {2}-\omega^{2}};\\ y_{a}=\frac{F_{0}.\sin\alpha}{m\sqrt{\left(p_{z}^{2}-\omega^{2}\right)^{2}+4 \pi_{2}^{2}\omega^{2}}};\ fg\phi_{y}=\frac{2n_{y}\omega}{p_{2}^{2}-\omega^{2}}.\end{array} \tag{2}\]  
In which the following is noted:  
\[p_{x}=\sqrt{\frac{k_{x}}{m}};\ \ n_{x}=\frac{c_{x}}{2m};\ \ p_{y}=\sqrt{ \frac{k_{y}}{m}};\ \ n_{y}=\frac{c_{y}}{2m}. \tag{3}\]  
Figure 1: Kinematic scheme of the mechanical system: 12 rotation torques; 6 translation forces.  
where \(p_{x},p_{y}\) are the eigenpulsations in the Ox and Oy directions, respectively, \(n_{x},n_{y}\) are the damping factors in the Ox and Oy directions, respectively, \(k_{x},k_{y}\) are the stiffness constants in the Ox and Oy directions, respectively, \(c_{x},\ c_{y}\) are the damping coefficients in the directions of Ox and Oy, respectively, \(m\) is the mass of the working body, \(F_{0}\) is the amplitude of the exciting force, \(\varphi_{x},\varphi_{y}\) are the phase shifts of the motion in the directions of Ox and Oy, respectively, and \(\omega\) is the pulsation of the exciting force.  
The equation of the trajectory of the center of mass (CM) with respect to the center of oscillation (CO), due to the shape of Equation (1), with the same pulsation \(\omega\) in both directions, is a Lissajous curve of elliptic shape and has the following equation:  
\[\frac{x^{2}}{x_{a}^{2}}+\frac{y^{2}}{y_{a}^{2}}-\frac{2 xy}{x_{a}y_{a}}\cos( \phi_{x}-\phi_{y})=\sin^{2}(\phi_{x}-\phi_{y}) \tag{4}\]  
Angle of incline \(\alpha_{1}\) of the ellipse with reference to the \(Ox\) axis is given as follows:  
\[tg2\alpha_{1}=\frac{2x_{a}y_{a}\cos(\phi_{x}-\phi_{y})}{x_{a}^{2}-y_{a}^{2}} \tag{5}\]  
Amplitudes of vibration in directions \(Ox,\ Oy,\) respectively, are given as follows:  
\[a=\frac{x_{a}y_{a}|\sin(\phi_{x}-\phi_{y})|}{\sqrt{\frac{1}{2}(x_{a}^{2}+y_{a} ^{2})-\frac{1}{2}(x_{a}^{2}-y_{a}^{2})\cos 2\alpha_{1}-x_{a}y_{a}\cos(\phi_{x}- \phi_{y}).\sin 2\alpha_{1}}} \tag{6}\]  
\[b=\frac{x_{a}y_{a}|\sin(\phi_{x}-\phi_{y})|}{\sqrt{\frac{1}{2}(x_{a}^{2}+y_{a} ^{2})+\frac{1}{2}(x_{a}^{2}-y_{a}^{2}).\cos 2\alpha_{1}+x_{a}y_{a}\cos(\phi_{x}- \phi_{y}).\sin 2\alpha_{1}}} \tag{7}\]  
In Figure 2, the trajectory of the center of mass (CM) of the working element of the mechanical system, the amplitudes of the motions \(a,\ b,\) and the inclination \(\alpha_{1}\) relative to the direction of the generating force and the inclination \(\gamma\) of the plane of the working element (the plane of a site) are presented [22; 23].  
What needs to be noted for such mechanical systems is that they operate only in pre-resonant or post-resonant regime so that \(\omega\) is smaller or larger than \(p_{x,y}\) and the trajectory parameters depend on the intrinsic parameters of the system, as well as the stiffness coefficients \(k_{x,y}\) or the damping coefficients \(c_{x,y}\) and, in particular, on their rate, which determines the values for the angle \(\alpha_{1}\).