# Finite Element Analysis of Convergent Nozzle of An Aircraft

# 1. Introduction

The force needed to keep the petals in place should be determined when designing the actuators for a jet engine’s convergent nozzle. To complete this job, perform a Finite Element Analysis of the actuator system to evaluate the mechanism's reaction forces and deflection. This task's fundamental goal is to determine the minimum thickness of the petal with a maximum deflection of 1 mm and equate the maximum and minimum stress values at this stage to the ‘Factor of Safety.’ It is, therefore, necessary to calculate the reaction force at the arm ends, as well as the displacement of the element during minimum thickness. The 2D idealised structure should be modelled, consisting of a petal and arm to perform the task. Fig. 1 shows the detailed 2d description of the ANSYS model. Both components are made of an Inconel alloy.

# 2. Model Description

# 2.1 Element Type

The structure consists of two parts and three joints; Beam is the element used to model the parts.

Beam 2 — node 188 is a linear or quadratic element appropriate for linear, significant rotation, and/ or considerable strain nonlinear applications. The 3 — node 189 beam element is just a** quadratic** **beam element**, which is the only distinction between it and the 188-beam element; hence the 2 — node beam element is chosen.

In this model, the beam element should behave (K3) in ‘quadratic form’. Beam element can select this from the options of the type of element. “Quadratic elements are generally preferred because they capture the curvature in the solution along with the element edges and across the element faces” (Peteroznewman, 2018). Since the stress was requested in the output, the beam element ‘stress/strain’ (K9) is treated as an “All section node” in the cross-section.

# 2.2 Material Property

The Young’s modulus, Poisson’s ratio, Density, Yield power, and Ultimate tensile strength of both the petal and arm components are 180 (GPa), 0.28, 8220 Kg/m3, 1090 MPa, and 1340 MPa, respectively. As a result, the given values of Young’s modulus and Poisson’s ratio are used in the material property “Linear Isotropic Material Properties” for material number 1 (Petal, beam element). The radius of the arm cross-section is 5 mm, and the petal width is 0.15 m. Due to the tensile effect in this application, this petal width's value cannot be a constant, and this value is given based on the petal’s shown condition.

# 2.3 Section Property

The beam factor is used in this simulation for petal (Material 1) with the job name ‘Rectangle’ and the rectangle's subform. This petal width is considered ‘B’ 0.15 m, and thickness must be determined by trial and error. The thickness is initially given as 0.001368 m. The beam tool is used for arm (Material 2) in the same way as material 1, but with the solid circle's sub-type and a radius of 5 mm (0.005 m).

# 2.4 Boundary Conditions

The boundary conditions are applied to the arm end hinge pin joint to determine reaction forces; since there are no rotational degrees of freedom at this stage, it is appropriate to fix all degrees of freedom; however, in this situation, setting the displacement helps determine whether the configuration is robust enough to withstand the force. The petal deflection due to the pressure load should not exceed more than 1 mm (0.001 m). It is fixed for all degrees of freedom at the petal end hinge point. When both the petal and the arm serve as beam elements, the junction cannot be a pin joint, so both parts of the beam can be welded. Since certain degrees of freedom are not associated between the petal and the arm, the command ‘End — release’ must be issued to ANSYS.

*2.4.1. Loading Conditions*

The loading is caused by a pressure difference of 34000 N/m2 between the jet exhaust gases and the nozzle's interior. The pressure load is added to the petal beam portion as a distributed load, as stated in the briefing. We are converting the weight to N/m and adding it to the petal as a uniformly distributed load of 17000 N/m.

# 2.5 Geometry

The requisite parameters are mentioned in section 2.2, and the necessary geometry of the 2D model is shown in fig.2.

The key points to be used in ANSYS are determined using the Pythagorean theorem, as seen in fig.2. A comparison diagram of the measurement concept is shown in fig.3.

Key points are calculated and listed:

Away from the actual model, ‘Keypoint 4’ was developed to decide the beam's cross-section.

*2.5.1**Meshing*

“Meshing is a crucial component of the computer-aided engineering simulation process. The mesh influences the solution’s precision, convergence, and speed.” In mesh attributes selecting the ‘Pick Orientation Key points’ allow us to pick the key — points to define the orientation. The number of openings in one square inch of a screen is known as mesh size; the higher the mesh number, the smaller the screen opening and the smaller the particle that passes through. The number of element divisions for the petal is ‘20,’ while the number of element divisions for the arm is ‘1.’ If a significant element of division is used, the component will not withstand lateral loading, so the element division for the arm will be only one.

# 3. Implementation

Based on the model described in section 2, the relevant material properties and the key points are given to the ANSYS. The line command then links all of the necessary vital points to create the model. The model is then given the boundary conditions. Finally, the mesh command is executed to obtain a precise result. The load is now defined as the model's pressure, and the model is solved to get the desired results.

# 4. Result

The simulated findings for displacement, reaction forces, and stress are seen in this section, along with an explanation of how these FEA values were extracted. The minimum thickness of the petal necessary to maintain the element’s optimum displacement and structural integrity is also specified.

# 4.1 Displacement

The maximum allowed displacement is 1 mm at this stage, and the thickness of the part is measured using the trial and error method to ensure that it does not distort its shape. In this simulation, we obtained a **deflection of 9.89 mm** (0.996E-03 Meters). The **thickness is taken to be 13.65 mm** (0.01365 Meters) at this stage, as seen in fig.4.

Trial and error table:

# 4.2 Reaction Force

The reaction force is taken from the general post processor, list result, and the simulation's reaction force. Fig. 5. illustrates the reaction force values.

# 4.3 Stresses

The maximum and minimum stress are shown in Fig.6. the unit is in N/m. The minimum stress value is -0.113x1011 (Pa), and the maximum stress value is 0.113x1011 (Pa).

# 4.4 Structural Integrity

Structural integrity is a branch of engineering that ensures that a structure or structural part is fit for use under standard operating conditions and is stable even if requirements surpass those specified in the original specification. **(Anon., 2021)**. Formula to find the safety factor is ultimate tensile strength over stress; this is clearly shown in fig.7.

The potential stress is determined as 0.89 using the formula (GPa). To be suitable for use in the above application, the simulated stress must be equal to the actual stress.

However, the calculated stress does not match the actual stress. This is since the ‘Factory of Safety’ Is an illusion and somewhat of a theoretical meaning. Simultaneously, the virtual stress is composed of structural properties and real-life boundary conditions, resulting in a disparity.

And one of the explanations for this significant distinction will be simulation mistakes and human errors.

# 5 Conclusion

The minimum thickness of the petal part is 13.65 mm, with a maximum acceptable deflection of 9.89 mm. At this point, the maximum stress is 0.113x1011 (Pa), while the minimum stress is -0.113x1011 (Pa). The reaction force in the arm end is -1906.5 on the x-axis, and it is simulated as -4935.8 on the y-axis, as seen in fig.5 in section 3.2.

As a result, to expand the load’s withstanding property by increasing the load distribution, the convergent nozzle must change each element’s length geometry. Also, attempting to ignore any hinge points would have a significant effect on the element’s deflection. The design recommendations include upgrading the material to provide improved material properties that can effectively support the load.

# References

Anon., 2021. *Twi Global. *[Online]

Available at: https://www.twi-global.com/technical-knowledge/faqs/structural-integrity

[Accessed 24 March 2021].

Peteroznewman, 2018. *Forum ANSYS. *[Online]

Available at: https://forum.ansys.com/discussion/1830/changing-mesh-type#:~:text=Quadratic%20means%20there%20are%20mid,Sometimes%20Linear%20elements%20are%20required.

[Accessed 21 March 2021].