## 3 Experimental Research and Discussion
### Static Characteristic
#### 3.3.2 Experimental Analysis of FAR  
The FAR is an important performance indicator in the HFAM, which expresses the ratio between the output force and the input force, and the expression of the FAR \(A\) is  
\[A=\frac{F_{2}}{F_{1}} \tag{26}\]  
where F\({}_{1}\) and F\({}_{2}\) are the input force and the output force of the HFAM, respectively.  
During the movement of the VCM, there is a hysteresis loop between the shafts and the linear bearings. By calculating the average value of the HFAM input force and output force during the forward and reverse motion of the VCM, the three-dimensional diagram of the FAR is obtained through Equation (26), as shown in Figure 18a. Figure 18a is projected onto the YZ plane, and the FAR changes with the displacements of the diaphragms, as shown in Figure 18b.  
When the displacement of the RD is less than 2 mm, input and output forces are small. They are affected by friction forces, resulting in obvious fluctuations in the FAR. Meanwhile, when the displacements of the RD exceed 2 mm, in most working conditions, the FARs increase with the increase in the RD displacement but decrease with the increase in the CD displacement. When the displacement of the CD is 0 and that of the RD is 6 mm, the maximum FAR of the HFAM is 2.5.  
Figure 18: _Cont._  
When the set displacement of the CD remained unchanged, the working conditions with the CD set displacements of 0, 0.8, and 1.2 mm were selected, as shown in Figure 18b. When the set displacement of the CD is 0, the displacement of the RD increases from 1 mm to 2 mm, and the FAR decreases from 2.32 to 2.14. When the displacement of the RD increases from 2 mm to 6 mm, the FAR increases from 2.14 to 2.5. When the set displacement of the CD is 0.8 mm, the displacement of the RD increases from 1 mm to 3 mm, and the FAR decreases from 1.2 to 0.3. When the displacement of the RD increases from 3 mm to 6 mm, and the FAR increases from 0.3 to 1.61. When the set displacement of the CD is 1.2 mm, the displacement of the RD increases from 1 mm to 4 mm, and the FAR decreases from 1.27 to 0.18. When the displacement of the RD increases from 4 mm to 6 mm, and the FAR increases from 0.18 to 1.01.  
The reason why the FAR first decreases and then increases is that when the displacement of the RD is small, the deformation of the CD is small, so the displacement of the pilot shaft of the CD is not enough to contact the micrometer. At this time, although the pilot shaft is not in contact with the micrometer, the little displacement causes friction on the pilot shaft, which is recorded by the force sensor. Meanwhile, due to the significant deformation of the RD, the force of the RD increases significantly, resulting in a decrease in the FAR. When the pilot shaft is in contact with the micrometer, the force of the CD increases significantly after being amplified by the HFAM. The larger the displacement of the RD, the greater the FAR.  
The three planes with FARs of 1, 1.5, and 2 intersect with the curved surface of the FARs to form three intersection lines. These lines represent the critical diaphragm displacement curves of the corresponding FARs, as shown in Figure 19. Through least squares polynomial fittings, three critical curves with FARs of 1, 1.5, and 2 were fitted, and the expression is  
\[\left\{\begin{array}{l}A=1:x_{2}=0.01885x_{1}^{4}-0.3043x_{1}^{3}+1.788x_{1}^{ 2}-4.314x_{1}+4.021\\ A=1.5:x_{2}=0.008885x_{1}^{2}+0.07746x_{1}+0.009964\\ A=2:x_{2}=0.006457x_{1}^{3}-0.03382x_{1}^{2}+0.0583x_{1}-0.03243\end{array}\right. \tag{27}\]  
Figure 18: FAR diagram of HFAM. (**a**) Three-dimensional diagram of FARs. (**b**) Two-dimensional diagram of FARs.