Vibration Response Analysis of a Main Landing Gear System for High In-Flight Dynamic Loadings

The prediction of aircraft vibrations is necessary for identifying possible design optimization points of each on-board system. In this context, the authors investigated the dynamic response of a Main Landing Gear (MLG) conceived for a fast helicopter when exposed to flight vibrations arisen from the engine propellers. The research activity falls within the Racer program of Clean Sky 2 framework, which aims to develop a novel high-speed rotorcraft. Relying on Airbus Helicopters operative requirements, this paper deals with the numerical procedure description of the MLG dynamic response assessment (resonance frequencies, accelerations amplitude, generalized masses) with respect to the expected in-flight vibrations levels. In particular, an equivalent combined load (sine and random spectrum) based on the normal modes and the corresponding modal mass distribution is employed for investigating the relevant effects on structural endurance. The central themes focus on the possible numerical modelling strategies and vibration loads analysis, which led safely to the qualification. This method could allow for performing sensitivity dynamic analyses in case of further design stiffness or weight changes.


Introduction
Sustained oscillations of the mechanical parts in the aircraft can influence the structural airframe and Flight Control System (FCS) design.The periodic forced vibrations originating from the power plant are one of the major disturbance sources in the aircraft, [1][2].Consequently, an efficient dynamic analysis is of course a central feature for looking to a more fault-tolerant, high-efficiency and comfortable architecture during the initial stages of the vehicle development, [3][4].Fault modelling strategies based on Kalman filtering and data-driven approaches are discussed in [5].A Monte Carlo statistical approach to detect and diagnose particular Limit Cycle Oscillations (LCOs) of the control surfaces is proposed in [6].Vibrations affect generally all the locations of the aircraft.The research in the field of vibrations is really huge in the aircraft wings applications, [7][8][9][10][11].Flutter mechanisms ensuing from oscillatory aerodynamic condition with large deformations are actually of specific attention for assessing the dynamic wings behaviour.Monitoring the vibratory response of engines is necessary to identify possible in-service anomalies.Fault diagnosis for excessive vibration of civil aviation engine is analysed in [12].The paper [13] presents a methodology from Safran Aircraft Engines for monitoring large-scale vibration civil aircraft engines, based on learning algorithms and flight data collection.The development of the drive train vibration monitoring system (VMS) to support the on-condition maintenance of V-22 tiltrotor mechanical components (rotors, blades, pylon shafts, nacelles) is discussed in [14].The landing systems inspired a considerable interest for the study of the dynamic stability of the aircraft on ground, especially for shimmy and gear walk phenomena, [15][16][17][18][19][20][21][22][23][24][25]; on the other side, less literature references are found on the analysis of landing gears modal interaction with respect to the in-flight vibration spectra.In this sense, the present work provides novel key aspects on the dynamic response analysis to be considered during the development of propeller vehicles.The study focues in particular on the Main Landing Gear (MLG) designed for the Racer rotorcraft of Airbus Helicopter, [26][27].It describes an effective methodology for the evaluation of the structural integrity using a quasi-steady loading approach which combines the propeller sine tones and broadband spectrum; a rational procedure can prove suitable in reducing the systems qualification risks thus speeding up the in-service start up.The associated computational strategies were in fact planned to drive the full-scale tests on the shaker table.The landing gears (LGs) have been delivered by Clean Sky 2 ANGELA consortium, comprising partners as Magnaghi Aeronautica SpA, CIRA (Centro Italiano Ricerche Aerospaziali), AVIATest Lab and other partners.The work is a continuation of what explored by the authors with regard to the dynamic evaluation of MLG door system, [28].The paper is organized introducing the topic with some dynamics theoretical points and subsequently providing the explanation of equivalent static loading method; full depiction of the LG system and numerical model used as benchmark for this application are described subsequently.

Vibration loads methodology
Steady and random responses are often investigated to verify the compliance with structural requirements and stiffness constraints of space structures during launch, [29][30][31].A common preliminary approach from design perspective consists in joining the dynamic excitations through an equivalent steady loads set; it represents a proper engineering way alternative to the use of transient or spectral methods much more demanding as computational cost.In the case of propeller vehicles, a combination of random and sinusoidal signals is generally used to replace the operational vibration spectrum; in this field, the sine-on-random procedure as per RTCA/DO-160G Part 8.0 is a wellrecognized reference for demonstrating equipment vibrations compliance both for fixed wing and helicopters installations, [32].Even if the LG structure is sized for impulsive landing loads with amplitudes well above the regulated vibration levels, it may be however worthy investigating the system dynamic behaviour especially in case of a possible modal coupling with the operative frequencies is expected.In such a condition the equipment dynamic response and stress levels also would be certainly augmented.In the hypothesis of a linear system, the harmonic load due to the input acceleration field (ain) can be expressed as function of a transmissibility factor Q = 1/2ξ, which takes into account the dynamic amplification at the possible resonances: where ξ is the viscous damping ratio and mg the respective generalized mass participating at a particular mode shape.Assuming in a first instance the LG system as a metallic structure with joints, the damping coefficient ξ is comprised between 0.03 and 0.07 ( [33]) (maximum transmissibility factor: Qmax = 1/2ξmin = 16.67 ≈ 20.0, conservatively).An approach based on Miles equation is instead used for the evaluation of broadband loads, [34][35][36].In particular, it allows to calculate the root-meansquare (r.m.s.) response when an elastic single-degree-of-freedom (SDOF) system is excited by random enforced accelerations.The mass acceleration of the SDOF system associated to the significant modal frequencies is given by the equation ( 2): This relationship denotes an equivalent steady acceleration where fn is the resonance frequency and  ̈ the relevant auto-power spectral density (ASD) of the random enforced acceleration at the i-th mode shape.The 3σ values of the random load factors are then applied for considering an equivalent static peak-load in view of a structural strength assessment as per equation ( 3): The corresponding quasi-steady load is given by relationship (4):

Racer Main Landing Gear
3. 1 LG system and numerical model description The Racer LGs system comprises a tricycle wheeled type and oleo-pneumatic shock absorber for dissipation of the energy during landing.In particular, the MLG is a telescopic type integrating one W&B (wheel & brake) unit, supported laterally by hydraulic side-brace actuator (SBA), Figure 1.The main fitting is then hinged by a pair of pintle pins at the H/C bay attachment brackets.The shock absorbers (S/A) is a single stage nitrogen-oil type with a separator piston which divides the oil and nitrogen (N2) phases.The FE models adopted to analyse the static structural sizing were used for the vibration analysis too.
The mesh criteria were calibrated to well get the stress-strain gradients for correlating the experimental results from qualification static strength test carried out on the MLG.The numerical model representative of MLG leg and SBA is provided in Figure 2; the fully extended condition is assumed for the present study.Material references of are derived from the Metallic Materials Properties Development and Standardization (MMPDS) Handbook, [37].The FE model involves mainly 3D mesh parts considering the actual nonlinear contact between the interfacing surfaces (i.e.lugs, holes, bearings, bushings, etc.) according to MSC Nastran ® guidelines, [38].

Boundary conditions definition
The vibration performance and endurance H/C test profiles provided by Airbus Helicopters are in the form of a Power Spectral Density (PSD) of the random noise distribution including four superimposed rotors sine g-tones (normalized respect to the maximum value for confidentiality), as indicated in the Table 1.The linear accelerations are defined in terms of spectral bands and central frequency fc.The point with maximum modal displacement (αx,y,z) based on the LG mode shapes was chosen for the static loads application.In view of its cantilevered architecture, at very low frequency modescorresponding to a greater LG deformation too -such point will be plausibly near the MLG wheel as numerically demonstrated in Figure 3(a) and 3(b).The load components (Fx, Fy, Fz) for the n natural frequencies are given in the relationship ( 5): The actuator for its design boundary conditions (pinned-pinned) will exhibit a first flexural mode with maximum midspan elongation, Figure 3(c).The normalized modal displacements for following load calculation are reported in Table 2 and applied according to the scheme of Figure 3(d).The normal modes extrapolation is limited to 50 Hz in order to consider just the most contributing deformations with highest modal masses for the structural verify.The first MLG and actuator resonance frequencies fall inside the range of the second excitation tone (26.28 -32.12 Hz); the second MLG mode shape is slightly outside of the range but anyway considered in the stress analysis.

Stress results
The Von Mises stress levels for the most loaded components are below the yield strength (fty) of the materials [37], Figure 4.The results of the MLG include conservatively the two calculated mode shapes even if the second on is outside the excitation range.The Max Principal stress is also represented in order to identify possible high-tension points, source of crack propagation due to the vibrations cycles, Figure 4(a),(b),(c).In the case of the SBA designed for working mainly as a strut rod, the oscillations lead in fact to additional bending loads, which could imply local contact stress concentrations.For this reason, the Min Principal component is additionally provided, Figure 4(d).

Experimental outcomes
The Table 3 summarizes some trial test results from the qualification campaign.The results demonstrate that the experimental frequencies are very close to what assessed in the numerical model.A slight discrepancy is found on the retraction actuator bending frequency probably due to a possible influence of the actual stiffness constraints.The two test articles were tested separately on the shaking table along all reference axes according to the spectra reported in Table 1.Different points were chosen to monitor with the accelerometers.The results reported are indicative of the stations exhibiting maximum amplitude of the dynamic response (at the MLG wheel and at the actuator midpoint respectively).The test demonstrated that the maximum amplification factor is enveloped by the engineering assumption done for the structural verify.

Discussion and conclusions
The present activity falls in the scenario of Clean Sky 2 Racer programme aimed to develop a new generation fast helicopter.In particular, the authors face the issue to predict the MLG dynamic behaviour with respect to the vibration loads coming from the H/C excitation sources.In this paper, the equivalent static loads due to the combination of sine tones and random spectrum (as per standard RTCA/DO-160G) have been calculated upon MLG modal parameters: resonance frequencies and generalized masses.This philosophy of loads combination can be useful as a preliminary way for estimating dynamic response when a database of numerical analyses (i.e.transient, spectral) with higher computational cost is not available.The results can be used to lead the critical phases of the qualification test to be held on the vibration table facility.A subsequent tuning phase is expected to be performed in order to include the actual amplification factors -currently assumed with a conservative choice -in the simulation environment.A modal analysis with a controlled force loop would be a proper feature to correlate the modal participating masses too.

Funding
This research was funded by Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation program under grant agreement No. CS2-GAM-FRC-2014-2015 and the following extensions.

Figure 1 .
Figure 1.Design details of Racer MLG system.

Figure 2 .
Figure 2. FE model overview and material data.

Table 2 .
Modal results for loads calculation.

Table 3 .
Comparison with vibration test on shaker table.