# Improved design of an active landing gear for a passenger aircraft using multi-objective optimization technique

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## Abstract

One of the major subsystems of each airplane is the landing gear system which must be capable of tolerating extreme forces applied to structure during ground maneuver to improve vibration absorbing performance. The traditional landing gear system performs this function well under normal condition, whereas with varying condition of landing and situation of the runway for the airplane, performance of this system decreases noticeably. In this research, for overcoming this problem, the coefficients of controller, the parameters of hydraulic nonlinear actuator added to the traditional shock absorber system, and the vibration absorber are optimized simultaneously by the bee intelligent multi-objective algorithm. As well as, for proving adaptability of this algorithm, this paper presents the sensitivity analysis of three point landing due to the additional payload and the touchdown speed and the robustness analysis of one and two point landings due to the wind conditions as emergency situation on the runway as an innovated work. In order to evaluate the effectiveness and the efficiency of proposed method, the flight dynamic differential equations of an Airbus 320–200 vibrational model during the landing phase are derived and through the numeric technique are solved. The results of numerical analysis for this large-scale airplane model with six degrees of freedom demonstrate that the active shock absorber system in accordance with two types of the bee multi-objective algorithm has good performance in comparison with the passive approach to minimize the bounce displacement and momentum, the pitch displacement and momentum, the suspension travel and impact force in time-domain and frequency-domain by using signal processing, that results in improvement of passenger ride comfort importantly. As well as, enhancement of structure’s fatigue life is a likely case as a consequence of study applicable to the industry.

## Keywords

Airbus 320–200 vibrational model Optimal active hydraulic nonlinear actuator Bee swarm-based multi-objective algorithm Robustness and sensitivity analysis Signal processing by fast Fourier transform## Nomenclature

*e*_{n}(*t*),*e*_{l}(*t*),*e*_{r}(*t*)Error signals based on PID control logic for the nose, left, and right gears

*k*_{a},*k*_{b}Hydraulic coefficients of active control unit

*q*(*t*),*C*_{d},*w*,*l*,*ρ*,*p*_{sh},*p*_{sl}Flow fluid flow quantity from servo valve, coefficient of discharge, gradient area of servo valve, displacement of servo valve, density of hydraulic fluid, high pressure in accumulator, low pressure in reservoir

*M*Sprung mass of the aircraft body

*I*_{xx}Mass inertia moment about pitch axis

*I*_{yy}Mass inertia moment about roll axis

*m*_{1}Un-sprung mass of nose landing gear

*m*_{2}Un-sprung mass of rear left landing gear

*m*_{3}Un-sprung mass of rear right landing gear

*ks*_{1},*ks*_{2},*ks*_{3}Nose gear sprung mass stiffness rate, rear left gear sprung mass stiffness rate, rear right gear sprung mass stiffness rate

*cs*_{1},*cs*_{2},*cs*_{3}Nose gear sprung mass damper rate, Rear left gear sprung mass damper rate, rear right gear sprung mass damper rate

*kt*_{1},*kt*_{2},*kt*_{3}Nose gear un-sprung mass stiffness rate, rear left gear un-sprung mass stiffness rate, rear right gear un-sprung mass stiffness rate

*f*_{1},*f*_{2},*f*_{3}Active control forces based on PID control logic for the nose, left, and right gears

*k*_{p},*k*_{i},*k*_{d}PID controller coefficients

*z*,*θ*,*φ*,*z*_{1},*z*_{2},*z*_{3}Vertical displacement, pitch displacement, roll displacement of sprung mass and vertical displacement of un-sprung masses

- \( \dot{z},\dot{\theta},\dot{\varphi},{\dot{z}}_1,{\dot{z}}_2,{\dot{z}}_3 \)
Vertical velocity, pitch velocity, roll velocity of sprung mass and vertical velocity of un-sprung masses

- \( \ddot{z},\ddot{\theta},\ddot{\varphi},{\ddot{z}}_1,{\ddot{z}}_2,{\ddot{z}}_3 \)
Vertical acceleration, pitch acceleration, roll acceleration of sprung mass and vertical acceleration of un-sprung masses

*rd*_{f},*rd*_{rl},*rd*_{rr}Relative displacement for nose, left and right gears

*rv*_{f},*rv*_{rl},*rv*_{rr}Relative velocity for nose, left and right gears

*ra*_{f},*ra*_{rl},*ra*_{rr}Relative acceleration for nose, left and right gears

*Fs*_{1},*Fs*_{2},*Fs*_{3}Suspension impact force imparted to the fuselage from nose, left and right gears

*Ft*_{1},*Ft*_{2},*Ft*_{3}Tyre impact force imparted to the nose, left and right gears from ground

- \( {\left\{ ITAE\right\}}_z,{\left\{ ITAE\right\}}_{\theta },{\left\{ ITAE\right\}}_{\varphi },{\left\{ ITAE\right\}}_{z_1},{\left\{ ITAE\right\}}_{z_2},{\left\{ ITAE\right\}}_{z_3} \)
The objective function for the displacements of body, nose, left and right gears

- \( {\left\{ ITAE\right\}}_{\dot{z}},{\left\{ ITAE\right\}}_{\dot{\theta}},{\left\{ ITAE\right\}}_{\dot{\varphi}},{\left\{ ITAE\right\}}_{{\dot{z}}_1},{\left\{ ITAE\right\}}_{{\dot{z}}_2},{\left\{ ITAE\right\}}_{{\dot{z}}_3} \)
The objective function for the velocities of body, nose, left and right gears

- \( {\left\{ ITAE\right\}}_{\ddot{z}},{\left\{ ITAE\right\}}_{\ddot{\theta}},{\left\{ ITAE\right\}}_{\ddot{\varphi}},{\left\{ ITAE\right\}}_{{\ddot{z}}_1},{\left\{ ITAE\right\}}_{{\ddot{z}}_2},{\left\{ ITAE\right\}}_{{\ddot{z}}_3} \)
The objective function for the accelerations of body, nose, left and right gears

- \( {\left\{ ITAE\right\}}_{F{s}_1},{\left\{ ITAE\right\}}_{F{s}_2},{\left\{ ITAE\right\}}_{F{s}_3} \)
The objective function for the suspension impact force

- \( {\left\{ ITAE\right\}}_{F{t}_1},{\left\{ ITAE\right\}}_{F{t}_2},{\left\{ ITAE\right\}}_{F{t}_3} \)
The objective function for the tyre impact force

- [
*ITAE*]_{Type1} The multi-objective function type1 for the sum of displacements, velocities and accelerations

- [
*ITAE*]_{Type2} The multi-objective function type2 for the sum of suspension and tyre impact forces

*D*_{N},*D*_{M},*V*_{N},*V*_{M}Drag aerodynamic forces and vertical forces

*L*,*W*,*T*Lift aerodynamic force, weight of aircraft and engine force

*V*,*α*,*β*Initial vertical velocity when landing, pitch and roll angles of aircraft

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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