# Precision Analysis and Design of Rotating Coil Magnetic Measurements System
## 2 Mathematical Modeling
### Mathematical Model of a Rotating Coil Rod  
The structure of the cylinder rod that contained the search coil was made from G10 material (composite material) [26]. This has many advantages: (a) it does not affect the measurement of the electromagnetic field; (b) it is designed to look like a round shaft, has a diameter of 80 mm, length of 500 mm, can support the weight of 2 bearings, around itself at a constant speed, and can estimate the mathematical model such like that of the Jeffcott rotor model as shown in Figure 5.  
Figure 4: Diagram of the armature cycle of a DC motor.  
Figure 5: Jeffcott rotor model.  
From [PERSON]'s model, the problem of the cylinder rod's movement results from the weight of the whirling, which causes an unbalance in its rotation. The imbalance in the rotation of the cylinder rod affects the harmonic force, which is a forced tremor. The motion model, due to the fact of an imbalanced rotating coil, is shown in Figure 6.  
Figure 6 displays a model of rotational unbalance, where \(m\) is shaft weight and \(F_{C}\) is forced unbalanced as shown in Equation (11).  
\[F_{C}=me\omega^{2}, \tag{11}\]  
where \(e\) is the whirling of shaft (\(m\)) and \(me\) is the mass with distance causing the unbalancing of rotation and affecting the whirling of the cylinder rod as shown in Figure 7.  
Figure 7 shows the whirling of the shaft due to the presence of weight that may cause movement in various ways around the rotational axis in a circle or an ellipse as shown in Figure 8.  
Figure 6: Model with dampers and springs.  
Figure 7: Model of a rotating coil.  
In this research, the whirling of the cylinder rod is studied and analyzed by considering the bearing that supports it as a strong object, while the cylinder rod supports it as a flexible object. In practice, two axes were considered: vertical axis and horizontal axis. The equation of movement at the vertical axis and horizontal axis is shown in Equations (12) and (13).  
Movement of vertical axis equation, \(y\):  
\[m\ddot{x}+c_{y}\dot{y}+k_{y}y=me\omega^{2}\sin\omega t, \tag{12}\]  
where \(m\) is the mass (kg), \(c_{y}\) is the damping coefficient in the \(y\) direction (N-s/m), and \(k_{y}\) is stiffness in the \(y\) direction (N/m).  
Movement of the horizontal axis equation, \(x\):  
\[m\ddot{x}+c_{x}\dot{x}+k_{x}x=me\omega^{2}\cos\omega t, \tag{13}\]  
where \(c_{x}\) is the damping coefficient in the \(x\) direction (N-s/m), and \(k_{x}\) is stiffness in \(x\) direction (N/m).  
The amplitude of the centrifugal force exerted by the unbalanced rotating of the cylinder rod was \(me\omega^{2}\). The amplitude of force directly converted with the rotation (\(\omega\)) of the cylinder rod, where \(y\) and \(x\) were amplitudes at vertical and horizontal. The whirling distances at the vertical and horizontal are shown in Equations (14) and (15).  
Vertical amplitude, \(y\):  
\[y=\frac{me\omega^{2}}{\sqrt{\left(k_{y}-m\omega^{2}\right)^{2}+\left(c_{y} \omega\right)}}, \tag{14}\]  
Horizontal amplitude, \(x\):  
\[x=\frac{me\omega^{2}}{\sqrt{\left(k_{x}-m\omega^{2}\right)^{2}+\left(c_{x} \omega\right)}}, \tag{15}\]  
From Equations (13) and (14), the vertical and horizontal whirling distances can be analyzed and calculated using the equation of movement of cylinder rod orbit while rotating. This indicates that the movement is in ideal, horizontal ellipse, and vertical ellipse feature, etc.  
Figure 8: Shaft orbit: (**a**) circular motion; (**b**) elliptical motion.