## AUTHORS

- Colorio Gian Marco gianmarco.colorio@gmail.com
- Vivian Gianluca gianluca.vivian@gmail.com

## INTRODUCTION

In this work, we have modeled with simply shape the landing gear mechanisms of a small aircraft with retraction system. The dimensions of geometrical model was concerned to a CESSNA Citation Jets represented in the following figure.

We thought that was interesting to study the behavior of these systems when the airplane reaches the ground and starts to decrease the speed.

PROJECT AIM

The aim of the project is to create a model of the landing gear system of the aircraft, and to observe what happen when the airplane lands and treads the airstrip, and focus the analysis to the contact of the wheel with ground and to the vibration of front wheel around its rotation (steering) axis (shimmy).

PHASE 1: GEOMETRIC MODELING

We have begun our work building the model of the aircraft with the CAD features of LMS Virtual.Lab. The first part was the principal structures of the aircraft: the frame, composed by nacelle, wings and tail. This part isn’t so important in the dynamic analysis of the landing gear, but it was important to have a geometrical reference and some approximate inertia values to create the final model: in fact the frame gives the geometrical parameters to place the subsystems in correct way and it contains also important dynamics information, as the mass and the center of gravity of the plane. Then we have created the front land gear, composed by three parts: the wheel, made by rim and tire, that was linked by a fork with a revolute joint. The last part is the body of the damper, where slides the fork. Fork has a cylindrical joint to be coupled to the damper, so it can translate in vertical direction. To avoid fork run away from damper, we have inserted a limit: we used two spheres that create a contact force when they hit, so the fork can get a limit position without exit by damper. To describe the effect of spring and damper we inserted in this subsystem a TSDA force between fork and damper body. The last sub mechanism included in the model is the rear land gear: in this element, the wheel is fixed to an upright that allows tire to move in a circular path around the pin of the bracket. This bracket is fixed to the body of the aircraft. The spring-damper element links the upright with the bracket.

To model the tire we use the complex tire forces, implemented in Virtual.Lab. Below it’s represented the complete CAD model of the aircraft with landing gear system.

## PHASE 2: SIMULATION

The first step in our work was to set up the aircraft’s model: we create the tire forces with some common values of friction, rolling resistance and cornering stiffness described in the following table. To adjust vertical stiffness the designer should consider tire pressure; instead we choose medium values.

Tire |
Front |
Rear |

Radius |
270 mm |
272.5 mm |

Rolling Resistance |
0.05 |
0.05 |

Carcass Mass |
5 kg |
7 kg |

Section Height |
110 mm |
85.5 mm |

Section Width |
180 mm |
187 mm |

Friction |
0.8 |
0.8 |

Vertical Stiffness |
1*10 |
1.4*10 |

Damping |
8000 kg*s |
10000 kg*s |

Cornering Stiffness |
260000 m*kg*s |
260000 m*kg*s |

To set the suspensions we had to check that the normal forces on the all tires do not become null: if this happens, it means that the tire heaves from the ground.

In this representation, we can see the moment when the tires touch the ground, when the forces start to increase, and the effect of the springs and the dampers. It’s interesting to observe that when normal forces on front tire get the highest value, the rear wheels are unloaded, in others terms we find a load transfer. The forces of left and right tire should be the same for the symmetry of the model.

To simulate the landing, we analyzed the phenomenon: at first there is the controlled fall, where gravity it’s equipoise by the aerodynamic forces on the wings, and where we assigned a vertical speed to the aircraft; the vertical speed used was 3.05 m*s^{-1}. When the aircraft stays in the air, it proceeds with constant speed, and when it reaches the ground it can start to brake. To simulate this behavior, we have inserted in our model a engine force to accelerate the body until the horizontal landing speed, when the airplane is still over the ground. The engine force is a horizontal force applied on the tail, for which we used an expression that can keep the speed constant, and overcomes the drag aerodynamic force. In this phase the pitch angle is controlled (about 6°). In the second part the airplane lands and, first the rear tires get down; then also front tire can drop. After that the tires have touched the airstrip, the aircraft can decelerate.

We set the suspension with following values:

Mechanism |
Free length |
Spring Constant |
Damper Constant |

Front damper |
350 mm |
3*10 |
6*10 |

Rear damper |
410 mm |
6*10 |
1.1*10 |

We ran various simulation changing initial condition (relax length of front tire, steering damper and spring, caster angle of front suspension, trail of front wheel that is the distance between the center of the wheel and the rotation center of axis system) to analyze the behavior of the aircraft, focused on the front wheel’s steering axis rotation.

Animation of landing of the aircraft’s model

Animation of landing from front view

## PHASE 3: ANALYSIS OF RESULTS

#### ANALYSIS OF SIMPLIFIED MODEL

Before running the complex model, we have created a simplified model of only front land gear system to verify that all the parameters inserted worked in right way. We used only the front sub-mechanism without the frame of the aircraft, it was constrained to translate only in horizontal direction along the road and we put the wheel on the road with constant horizontal speed; we also assigned to the fork an harmonic rotation (with a Joint Position Driver inserted on the Cylindrical Joint between the damper and the fork).

Animation of front fork with harmonic rotation

Then we could read from result’s data the lateral force on the tire and, changing the relax length, we can verify if the phase shift between the lateral force and the steer angle exist.

The fact that the peak of harmonic rotation of steer axis isn’t always 1 is due to the way we used to defined the function and to the selected value of sampling time we set up in the simulations.

In these graphs we can see the situation of the simplified model: the constants are the normal force on the wheel equal to 26000 N, which was about the value computed in the complete model, and the trail, -20 mm, so the wheel center is behind the rotation center of steering axis. We can observe that when a low angular speed is imposed, the shift between the lateral force of 0 relax length and lateral force computed with a relax length of 450 mm is so small that we have to change the graph scale if we want to read it. This happens because the wheel turns very slowly, so the force can follow it (when relax length is null the tire generates the force immediately when the variation of steer angle happens). If the angular velocity increases, the variation of steer angle is faster, so the tire needs some time to adapt, and we can see that the shift rises.

So we can conclude in the complete model we have to found more difference between simulations where the modes of vibration has higher frequencies, by changing the relax length, and if there are vibrations that arises with low frequencies, if we modify this parameters, these should not give different results.

Another important thing to consider is also using a high value for the relax length (450 mm is the limit we used in next tests), and giving the wheel a horizontal speed of 50 m/s, the shift that we can read in these graphs is very low, so it must be considered with attention during the simulations. The last plot was obtained by changing horizontal speed to 20 m/s; the phase shift increases with respect to the simulations run with horizontal speed of 50 m/s, so we can say that in the complex model the effect of relax length should appear when the aircraft reaches low speed.

#### ANALYSIS OF COMPLETE MODEL

We focused our attention on the front wheel, because it is the only that can turn. At the first step, we ran some simulations in which we have changed the value of relax length and trail. In particular, it was interesting to observe what happens when the center of the front wheel is moved back respect to the center of steering axis of front fork, to understand the conditions when the steering vibration becomes unstable (if the wheel center is moved forward, the system is always unstable; it can be simply verified also with this model).

In the previous plot it’s represented the horizontal speed of the airplane on the ground: at first it arrives with constant velocity (about 60 m/s) and, after all the tires touches the airstrip, it starts to decelerate. In this graph and in the next, the horizontal axis represents the simulation time. It start from 45 s because in each simulation we made the aircraft accelerates and travels some distance in air, so it reaches the airstrip and start to brake after some moments.

In the previous graph, we can see the rotational movement of front wheel during the simulations with variable relax length and trail = -20 mm (in fact the center of the wheel is in opposite side respect to the direction of horizontal speed). we can say that the vibration become unstable when the relax length is 450 mm, limit value taken for these tire. With a more accurate analysis, we can observe the steer angle, when relax length is lower than 450 mm, isn’t null, but the maximum amplitude of oscillation is about 0.01°, and it can be neglected in this analysis.

If we put the trail at -40 mm we see the situation change. The instability arises when relax length is over 200 mm, a values lower than 450 mm found when the trail was -20 mm. The stability of this system is worse than the previous. We can also say that the instability appears when the aircraft runs with low speed: the align gyroscopic moment on the wheel is high when the rotational speed of the wheel is high too: when the velocity of the airplane is at the maximum value, a rotation of axis system generate an align moment that straightens the wheel. With these parameters, we also plot the steer angle and the lateral force on front tire. To compare parameters with different units, we normalized the data by maximum value.

The previous data was normalized with respect to maximum of each parameters:

- Max Value Steer Angle (with rl = 120 mm) =0.02852 deg
- Max Value Steer Angle (with rl = 250 mm) =179.7 deg
- Max Value Lateral Force (with rl = 120 mm) =0.02097 N
- Max Value Lateral Force (with rl = 250 mm) =10231 N

We can make some consideration: when the relax length is 120 mm the vibration is stable, but there are two components of rotation: one with high frequency, and one with lower frequency. When we increase the relax length, the two components of the rotation of steering axis disappear, but the system becomes unstable when speed gets close to 0. This phenomenon was already observed in the simplified system with only front land gear, because we have computed that the phase shift raised by decreasing the horizontal velocity, and so the lateral force can get fast the value to align the wheel, and so the tire start to turn around the steering axis.

#### CONCLUSIONS

Analyzing the results of our simulations, we have verified that the front wheel is a critical component for an airplane, because it can be subjected to phenomena of instability. At last, after we have observed that changing the parameters, the conditions of stability vary too, we can build a table when we put in relation the physical of relax length and trail of front wheel. The limit values that can make the system unstable were computed by trying the complete system with a constant trail assigned whit increasing value of relax length, until we found the instability. We obtained the following results:

Trail [mm] | Relax Length [mm] |

10 | 735 |

20 | 410 |

30 | 265 |

40 | 205 |

50 | 190 |

60 | 175 |

From this table we can extrapolate a simple graph to summary the couple of parameters that are critical for the system, and it can be used to verify rapidly if the front wheel become unstable or not.

This simple model leave open a lot of way to new develop, such as for example the introduction of control system to describe the action of the pilot on the airplane. Also the parameter of the wheel and the suspensions can be improved with some experimental data, to verify the accuracy of the model with respect to a real system.