## 3 Vertical Speed Indicator (V.S.I.) (Analog Instrument and Computer Simulation)  
This instrument is also called the variometer or rate-of-climb indicator. It senses changes in air pressure but not the absolute variation. The dial of this instrument indicates if the aircraft is ascending or descending. This instrument also takes pressure information from the pitot tube. The V.S.I. shows the instantaneous climbing/sinking rate, as measured directly based on the changing air pressure. This instrument is very important during navigation because it is difficult for human pilots to distinguish between a light or strong lift or sink. The V.S.I. contains a bottle connected to the external atmosphere through a sensitive air flowmeter. When the aircraft modifies its altitude, the atmospheric pressure changes, affecting the pressure inside the bottle [4; 5; 6; 27; 28; 29; 30; 36; 40; 41; 43].  
The V.S.I. is a circular dial with two symmetrical main divisions originating in the flat horizontal left position. The main division is \(\pm 135^{\circ}\). Normally, the vertical speed is not as high as the horizontal traveling or cruising speed. The scales used to measure the vertical speed are shorter. In the instrument we developed, the main divisions are 0 to 50 (km/h). The V.S.I. also includes secondary divisions of 10 short lines to improve its visual precision to 1 km/hr. The V.S.I. takes the information from the variable \(\mathrm{v_{z}}^{\mathrm{t+\Delta t}}\) to display the corresponding information and only uses one indicator, a large needle, with the following assumptions:  
1. If the indicator does not move or change its horizontal position, the aircraft is static, or it may be that the aircraft is in a straight and leveled flight; consequently, the altitude does not change.
2. If the indicator moves clockwise over the horizontal line, the aircraft is ascending because the vertical speed is increasing. This vertical speed is considered positive (\(\mathrm{v_{z}}^{\mathrm{t+\Delta t}}>0\)).
3. If the indicator moves counterclockwise over the horizontal line, the aircraft continues to ascend, but its vertical speed decreases.
4. If the indicator moves below the horizontal line, the vertical speed is negative and the aircraft is descending. If the condition \(\mathrm{v_{z}}^{\mathrm{t+\Delta t}}<0\) is true, the aircraft descends.  
Equations (6)-(8) calculate the aircraft's speed on the z-axis and provide the instantaneous speed at each step time (\(\mathrm{t+\Delta t}\)) during the simulation. After this, Equation (9) calculates the angle position for the V.S.I. Here, \(\mathrm{v_{z}}^{\mathrm{t+\Delta t}}\) is the aircraft's real speed along the vertical axis. Figure 2a,b illustrate the partial segmentation of the dial, the movement regions of the indicators, and the V.S.I.'s computational display version.  
\[\upsilon_{z}^{t+\Delta t}=\upsilon_{z}^{n-1}+\upsilon_{z}^{\Delta t} \tag{6}\]  
\[\upsilon_{z}^{t+\Delta t}=\upsilon_{z}^{n-1}+(\upsilon_{z}^{n}-\upsilon_{z}^{n -1}) \tag{7}\]  
\[\upsilon_{z}^{t+\Delta t}=\upsilon_{z}^{n-1}+(\upsilon_{z}^{n}-\upsilon_{z}^{n -1})\bigg{(}\frac{t_{n}}{n\mathrm{s}}\bigg{)} \tag{8}\]  
\[\mathrm{a_{VSI}}=\upsilon_{z}^{t+\Delta t}\bigg{(}\frac{135^{\circ}}{50(km/ l\mathrm{r})}\bigg{)} \tag{9}\]