# Modeling a Milling Dynamometer as a 3 DOF Dynamic System by Stiffness Identification
## 4 Model of Dynamometer as a System with Three Degrees of Freedom
### Motion of the Upper Plate  
To analyze the dynamic behavior of the considered dynamometer, we considered that the upper plate has a combined motion in both directions, longitudinal and transverse, with an o twist motion. The longitudinal and transverse motions are defined by positions denoted with \"x\" and \"y\", and the rotation is defined by the small angle \"\(\theta\)\" (Figure 14).  
Based on the scheme in Figure 14, the positions of the points in new coordinates given by the orthogonal reference system \(x_{0}O_{0}y_{0}\) are as follows:  
\[\begin{array}{l}\left\{\begin{array}{l}x_{01}\ =\ x_{CG}\ +\ x_{1}\cos\theta\ -\ y_{1}\sin\theta;\ y_{01}\ =\ y_{CG}\ +\ x_{1}\sin\theta\ -\ y_{1}\cos\theta;\\ x_{02}\ =\ x_{CG}\ +\ x_{2}\cos\theta\ -\ y_{2}\sin\theta;\ y_{02}\ =\ y_{CG}\ +\ x_{2}\sin\theta\ -\ y_{2}\cos\theta;\\ x_{03}\ =\ x_{CG}\ +\ x_{3}\cos\theta\ -\ y_{3}\sin\theta;\ y_{03}\ =\ y_{CG}\ +\ x_{3}\sin\theta\ -\ y_{3}\cos\theta;\\ x_{04}\ =\ x_{CG}\ +\ x_{4}\cos\theta\ -\ y_{4}\sin\theta;\ y_{04}\ =\ y_{CG}\ +\ x_{4}\sin\theta\ -\ y_{4}\cos\theta.\end{array}\right.\end{array} \tag{6}\]  
Figure 13: Model of upper plate: (**a**) the connection points and dimensions; (**b**) the mass–spring model of the upper plate and the rings.  
\begin{table}
\begin{tabular}{c c c} \hline
**Direction of Stiffness** & **Measured [N/m]** & **FEM [N/m]** \\ \hline Longitudinal & \(2.165\times 10^{9}\) & \(1.932\times 10^{9}\) \\ Transverse & \(2.137\times 10^{9}\) & \(1.911\times 10^{9}\) \\ \hline \end{tabular}
\end{table}
Table 8: The average values of dynamometer stiffness.  
In the hypothesis of small deformations, one can use the following approximations: \(\sin\!\theta\cong\theta\), and \(\sin\!\theta\cong 1\). Then, Equation (6) becomes:  
\[\left\{\begin{array}{l}x_{01}\ =\ x_{CG}\ +\ x_{1}\ -\ y_{1}\theta;\ \,\,\, _{01}\ =\ y_{CG}\ +\ x_{1}\theta\ -\ y_{1};\\ x_{02}\ =\ x_{CG}\ +\ x_{2}\ -\ y_{2}\theta;\ y_{02}\ =\ y_{CG}\ +\ x_{2}\theta\ -\ y_{2};\\ x_{03}\ =\ x_{CG}\ +\ x_{3}\ -\ y_{3}\theta;\ y_{03}\ =\ y_{CG}\ +\ x_{3}\theta\ -\ y_{3};\\ x_{04}\ =\ x_{CG}\ +\ x_{4}\ -\ y_{4}\theta;\ y_{04}\ =\ y_{CG}\ +\ x_{4}\theta\ -\ y_{4}\,. \end{array}\right. \tag{7}\]  
Considering the coordinates of the points given in (5), the new coordinates are as follows:  
\[\left\{\begin{array}{l}x_{01}\ =\ x_{CG}\ -\ a\ -\ b\theta;\ y_{01}\ =\ y_{CG}\ -\ x_{4}\theta\ -\ b;\\ x_{02}\ =\ x_{CG}\ +\ a\ -\ b\theta;\ y_{02}\ =\ y_{CG}\ +\ a\theta\ -\ b;\\ x_{03}\ =\ x_{CG}\ +\ a\ +\ b\theta;\ y_{03}\ =\ y_{CG}\ +\ a\theta\ +\ b;\\ x_{04}\ =\ x_{CG}\ -\ a\ +\ b\theta;\ y_{04}\ =\ y_{CG}\ -\ a\theta\ +\ b\,. \end{array}\right. \tag{8}\]  
Considering the initial position and the relations from (8), the displacements of each point become:  
\[\left\{\begin{array}{l}\Delta x_{1}\ =\ x_{01}\ -\ x_{1}\ =\ x_{CG}\ -\ b \theta;\ \,\,\,\Delta y_{1}\ =\ y_{01}\ -\ y_{1}\ =\ y_{CG}\ -\ a\theta;\\ \Delta x_{2}\ =\ x_{02}\ -\ x_{2}\ =\ x_{CG}\ -\ b\theta;\ \,\,\Delta y_{2}\ =\ y_{02}\ -\ y_{2}\ =\ y_{CG}\ +\ a\theta;\\ \Delta x_{3}\ =\ x_{03}\ -\ x_{3}\ =\ x_{CG}\ +\ b\theta;\ \,\,\,\Delta y_{3}\ =\ y_{03}\ -\ y_{3}\ =\ y_{CG}\ +\ a\theta;\\ x_{04}\ =\ x_{04}\ -\ x_{4}\ =\ x_{CG}\ +\ b\theta;\ \,\,\,\Delta y_{4}\ =\ y_{04}\ -\ y_{4}\ =\ y_{CG}\ -\ a\theta\,. \end{array}\right. \tag{9}\]  
The above relations (9) will be used in defining the motion equations, which are presented in the next sub chapter.