Static Analysis and Design of a Flight Control with Oleopneumatic Centering for a Sidestick CPT/FO in A320 Simulators

This new design will be composed of oleo-pneumatic dampers with self-centering to regulate the point of origin. The design includes pieces anchored to the frame that provide degrees of freedom to the system. The selection of dampers must guarantee their specific resistance and the force of resistance to movement.

The required damping coefficient, distances and forces applied to the parts of the system were determined. Through a static analysis of each model obtained in SolidWorks, the value of the stresses and deformations present in each of the parts of the system was determined, by numerical methods, to validate the capacity of the device to resist the stresses present during its operation. Figures 1 and 2 show the functional and non-functional requirements of the product.

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Figure 1. Non-functional requirements diagram

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Figure 2. Functional requirements diagram

A36 steel was selected as the material for the basic and supporting structural elements of the Sidestick, given the moderate load conditions that act in a flight simulator, significantly lower than those in real aeronautical environments. This makes it suitable to withstand these loads without oversizing the design, ensuring a favorable balance between mechanical strength, durability, ease of manufacture and costs. For the control stick, 304 stainless steel was chosen, a choice justified by its excellent mechanical properties, corrosion resistance and its adequate response to continuous use, meeting the functional and structural requirements of the FSTD A320 flight simulator.

A fixed base was designed (see Figure 3), with the main function of anchoring the general system of degrees of freedom to the frame of the box, using pins located perpendicular to the exterior faces, the same hole that will be used to assemble the internal parts of the system.

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Figure 3. External part of the degrees of freedom mobility system

Figure 4 shows the intermediate piece of the degrees of freedom mobility system, which fulfills the main function of granting the first degree of freedom in the movement of the lever through rotation in the "z" axis through the two central holes located on the front and rear faces in the working position. In addition, this piece has the required anchors for the external couplings of the dampers.

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Figure 4. Intermediate piece of the degrees of freedom mobility system

Figure 5 shows the internal part of the degrees of freedom mobility system, which has the main function of granting the second degree of freedom in the movement of the lever, through its rotation around the "x" axis with pins exposed on the faces right and left sides. This piece also has the spherical coupling for the end of the damper and the cylindrical shaft that connects to the lever.

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Figure 5. Internal part of the degrees of freedom mobility system

The design of the frame where the entire system was located (see Figure 6) was made by considering the original measurements of the Sidestick from the Latin American Aeronautics – Vortex Airspace company. Which ensures minor changes to the general structure of the Sidestick.

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Figure 6. Model of the box and the degrees of freedom mobility system

Figure 7 depicts the force vector applied by the pilot during a maneuver and that requires to be reversed by the damping through self-centering. This determines a specific damping resistance of 7 lb, according to the Sidestick technical sheet, and the standard of the Airbus A320 commercial aircraft.

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Figure 7. Indication of the force vector exerted by the pilot

2.1 Damping factor calculation

The damping factor (ξ) measures the level of energy dissipation of the system compared to the critical damping, and indicates the response of the system to disturbances and is determined by the equation:

$\xi=\frac{c}{C c}$               (1)

where,

c = linear damping coefficient

Cc= critical damping coefficient

Every mechanical damping device has a damping constant (c) that governs the behavior of the mechanism when an external force is exerted on it and that can be determined from the technical data of the selected damper and the operating conditions. The critical damping (Cc) is the value that allows the system to return to equilibrium without oscillation and in the shortest possible time.

Next, the values of the actual linear damping coefficient and the Cc value will be determined.

2.1.1 Calculation of the damping coefficient

In this case, the 89U076214TT shock absorber from Industrial Gas Springs, Inc. was selected, whose data are indicated in Table 1, which respond to the dimensional and operational requirements of the Sidestick CPT/FO.

Table 1. Damper data sheet

Source: Industrial Gas Springs, Inc.

Assuming that the damping specification behaves linearly for small variations in force values, it was determined that to damp a force value of 10 lbf (44.482 N), a dampening specification of half a second per inch, from the relation:

$\frac{\frac{1 \mathrm{s}}{1 \mathrm{in}}}{20 \mathrm{lbf}} \rightarrow \frac{X}{10 \mathrm{lbf}}$              (2)

Which corresponds to a value of $0.01969 \frac{{s}}{\mathrm{~mm}}$ when expressing the time in “s” and the distance traveled, in mm. The inverse of this value corresponds to the magnitude of the speed, expressed in mm/s, at which the damper stem expands and compresses, as follows:

$V=\left[0.01969 \frac{{s}}{\mathrm{mm}}\right]^{-1}=50.8 \frac{\mathrm{mm}}{{s}}$                (3)

From the value obtained of the speed and the force that the damper must resist, its linear damping constant is determined, which is a parameter that indicates its capacity to dissipate the energy absorbed by the rod, and is calculated using the expression:

$F=c * v$               (4)

where,

F: damper resistant force

c: real linear damping constant

v: relative velocity between association points

Therefore:

$c=\frac{F}{v}=\frac{44.482 \mathrm{N}}{50.8 \frac{\mathrm{mm}}{{s}}}=0.88 \frac{\mathrm{N}^* \mathrm{s}}{\mathrm{mm}}$

2.1.2 Calculation of Cc

Cc is calculated by the expression:

$C c=2 \sqrt{k . m}$               (5)

where,

k: shock absorber stiffness constant (see Table 1)

m: effective mass of the shock absorber (see Table 1)

The stiffness constant (k) for a system such as the Sidestick CPT/FO with oleopneumatic shock absorber, represents the relationship between the applied force and the displacement of the rod. The critical damping according to Eq. (5) would be:

$C c=2 \sqrt{1.3 \times 0.1644}=0.925 \mathrm{~N} \mathrm{~s} / \mathrm{mm}$

Finally, by substituting the values obtained from the coefficients (c) and (Cc) in equation Eq. (1), the value of the damping factor (ξ) is obtained.

$\xi=\frac{0.88}{0.925}=0.95$

2.2 Calculation of distances and travel of the damper stem

From the technical specifications of the oleo-pneumatic shock absorber (Table 1) and the extreme positions of the rod shown in Figure 8, it can be deduced that the piston rod will have a maximum travel in extension of 217.93 mm and in compression a minimum travel of 141.73 mm. So the actual working stroke of the stem is 76.2 mm. The cylinder diameter is 18 mm and the stem diameter is 8 mm, the dimensions of the end couplings are not considered.

The average piston rod travel value (38.1 mm) is half of the total piston rod travel (76.2 mm). When this is located in its mid-travel position, it will have the ability to compress and expand when the required force is applied.

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Figure 8. Damper piston rod travel diagram

From the value obtained in the average stroke and the value of the distance of the damper with the piston rod in extension, the calculation of the variable “a” was carried out (see Figure 8).

$\begin{gathered}a=\text { shoulder length-average stroke } \\ a=217.93 \mathrm{~mm}-38.1 \mathrm{~mm} \\ a=179.83 \mathrm{~mm}\end{gathered}$                (6)

Since a right triangle is formed by locating each damper with respect to its base inside the Sidestick box, a value for the installation angle $\beta_{ {inst }}$ de 40° was set, which will be modified during Sidestick operation in expansion or compression (see Figure 9).

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Figure 9. Scheme for geometric calculations of the location of the dampers

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Figure 10. Dimensions of the intermediate piece of the degrees of freedom mobility system

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Figure 11. Static diagram of the rigid body

The value of leg “b” was determined from the trigonometric sine function of $\beta_{{inst}}$, knowing the value of the hypotenuse “a”. That is the distance to set the position of the damper in the Y axis and its value is 115.59 mm. Similarly, by applying the trigonometric cosine function of $\beta_{{inst}}$, knowing the value of the hypotenuse “a”, the value of leg “c” was determined, and it was assumed as the location distance of the damper on the X axis and its value is 137.757 mm.

Figure 10 shows the intermediate piece of the mobility system of degrees of freedom to which the respective moment calculations were carried out, in order to know what force the shock absorbers exert when applying the required force at the pivot point.

The rigid body diagram of the intermediate piece of the mobility system of degrees of freedom was made (see Figure 11), in which the forces and reactions involved are indicated with their respective distances. The clockwise direction was assumed as positive for the calculation of moments.

$\begin{gathered}M=F * d \\ F(70 \mathrm{~mm})-R_{1 y}(56 \mathrm{~mm})-R_{2 y}(56 \mathrm{~mm})=0 \\ \text { for } R=R 1 y=R 2 y \\ \text { then: } F(70 \mathrm{~mm})-2[R(56 \mathrm{~mm})]=0 \\ R=\frac{-F(0.070 \mathrm{~m})}{-0.112 \mathrm{~m}} \\ R=\frac{(-31.1374 N)(0.070 \mathrm{~m})}{-0.112 \mathrm{~m}} \\ R=19.46 \mathrm{~N}\end{gathered}$                  (7)

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Figure 12. Scheme for calculating the real force on the piston rod

Using trigonometric equations, the right triangle was solved with the two data presented. For this, the sine operation of the beta installation angle was used to obtain the value of the force of the piston rod (see Figure 12).

Data:

$\beta_{{inst }}=40^{\circ} ; R=19.46 {N}$

Calculation of the force on the piston rod (F_v).

$\begin{aligned} & \sin \beta_{ {inst }}=\frac{R}{F_v} \\ & F_v=\frac{19.46 {N}}{\sin 40^{\circ}}\end{aligned}$                  (8)

where,

$\begin{gathered}F_v< { maximum \,\, strength } \\ F_v=30.274 {~N} \\ 30.274 {N}<44.482 {N}\end{gathered}$

The calculation of the variation of the piston rod force, according to the installation angle was carried out, based on the diagram made in Figure 12, $\beta_1$ is the smallest possible angle of the damper and $\beta_2$ is the largest possible angle of the damper, the values of these angles were taken from the assembly made in SolidWorks, where:

$\beta_1=33.73^{\circ}; \beta_2=46.27^{\circ}$

Calculation of the maximum outside of the piston rod.

$\begin{gathered}F_v \,max =\frac{R}{\sin \beta_1} \\ F_v \,max =\frac{19.46 N}{\sin 33.73^{\circ}}\end{gathered}$                    (9)

where,

$\begin{gathered}F_v \, max <{maximum} \, { strength } \\ F_v \, max =35.045 \,{N} \\ 35.045 \, {N}<44.482 \,{N}\end{gathered}$

Calculation of the minimum outside of the piston rod.

$\begin{gathered}F_v \, min =\frac{R}{\sin \beta_2} \\ F_v \, min =\frac{19.46 {~N}}{\sin 46.27^{\circ}} \\ F_v \,min =26.930 {~N}\end{gathered}$            (10)

The choice of the commercial damper made is correct, since the selected damper delivers, by design, up to a force of 44.482 N, and the maximum force of the stem is 35.045 N.