Mechanical Vibration Analysis of An Uav Wing Spar

This paper presents the mechanical vibration analysis of a UAV (Unmanned aerial vehicle) wing spar. Theoretical and numerical calculations are performed by considering the spar as a cantilever beam. The model is created and modal analyses are performed by using the MATLAB software. The natural frequencies and the related mode shapes are obtained. The results of theoretical calculations linear, nonlinear and random vibration of both single and multiple degree of freedom (DOF) system are graphically presented. The study aims to illustrate vibration tendencies of the wing during flight.


Introduction 1.1
Wing structure A wing structure of an aircraft is a crucial component of flight that together with the help of airfoil profile that generates lift by the vehicle's forward airspeed and the shape of the wings.The internal structures of most wings are made up of spars and stringers running spanwise and ribs and formers or bulkheads running chordwise (leading edge to trailing edge).The spars are the principle structural members of a wing [1].They support all distributed loads, as well as concentrated weights such as the fuselage, landing gear, and engines.The skin, which is attached to the wing structure, carries part of the loads imposed during flight.It also transfers the stresses to the wing ribs.The ribs, in turn, transfer the loads to the wing spars.lateral axis of the aircraft, from the fuselage toward the tip of the wing, and are usually attached to the fuselage by wing fittings, plain beams, or a truss.
A wing has two spars.One spar is usually located near the front of the wing, and the other about twothirds of the distance toward the wing's trailing edge Spar run parallel to the lateral axis of the aircraft, from the fuselage toward the tip of the wing, and are usually attached to the fuselage by wing fittings, plain beams, or a truss.Therefore the spar beams can be considered as a cantilever beam for the design purpose.
Spars may be made of metal, wood, or composite materials depending on the design criteria of a specific aircraft.They can be generally classified into four different types by their cross sectional configuration as shown in Figure 2.They may be solid, Box shaped, partly hollow and I-beam spar.The top and bottom of the I-beam are called the caps and the vertical section is called the web.The entire spar can be extruded from one piece of metal but often it is built up from multiple extrusions or formed angles.The web forms the principal depth portion of the spar and the cap strips are attached to it.Together, these members carry the loads caused by wing bending, with the caps providing a foundation for attaching the skin.

Figure 2: Types of Spar Configurations
In this paper, the aircraft spar is considered as a cantilever with a decreased load distribution from the root to tip.This decrease is all the same whether the spar is a rectangular, I-section, tapered Made of Aluminum alloy 2024-T4:2024-T351 characteristics.
In the present paper, linear, nonlinear and random vibration[2] was performed to both single and multiple degree of freedom model on an fixed wing UAV (Figure 3).Linear vibration on continuous model was also obtained.The simulation and calculations were performed by MATLAB and later compared to obtain a final analysis of the study.The material considered in the study is Aluminum alloys, valuable because they have a high strength-to-weight ratio, lightweight, corrosion resistant and comparatively easy to fabricate.Figure 3: Fixed wing UAV

LINEAR VIBRATION OF A SINGLE DEGREE OF FREEDOM
In linear vibration of a single degree of freedom analysis we will analyse free, forced and random vibration of the models considering both damped and undamped system and obtain a natural frequency and response.
2.1 Free vibration of a viscously damped single degrees of freedom system.When a system, after an initial disturbance, is left to vibrate on its own, the ensuing vibration is known as free vibration [3].No external force acts on the system.The oscillation of a simple pendulum is an example of free vibration.In this case the system is viscously damped so that was taken that into consideration.
A spar considered as a cantilever beam, mathematical models of the system was developed to investigate vibration in the horizontal direction.Consider the elasticity of the spar itself and introducing a damping and then mass (m) of the whole system is considered to be lumped at the end of the beam.Figure 1 shows a below is the mathematical model of a spar; The viscous damping force F is proportional to the velocity or v and can be expressed as Where c is the damping constant or coefficient of viscous damping and the negative sign indicates that the damping force is opposite to the direction of velocity.If x is measured from the equilibrium position of the mass m, the application of Newton's law yields the equation of motion: To solve the above equation we assume a solution in the form Where C and s are undetermined constants Introduce critical damping constant and damping ratio.For any damped system, the damping ratio z is defined as the ratio of the damping constant c to the critical damping constant Substituting equation (ii) and (iii) into (ii) we obtain two roots 1,2 = −  ±   √ 2 − 1 (6) Thus the solution, can be written as () =    (−+ √   −)   +    (−− √   −)   (7) The nature of the roots will determine behavior of the solution, depending upon the magnitude of damping. .C1 and C2 can be obtained through initial conditions.
To find the free-vibration response of a viscously damped wing spar system, MATLAB program was used to find the response of a system with the Following data: Lumped Mass (m) of the system = 62.096kgStiffness (K) = 3.046 x 10 6 N/m Damping constant (c) = 1.5x10 3 N.s/m Initial displacement  0 = 0.4321 m Initial velocity 0 =1.05m/sThe natural frequency   = 221.479and z = 0.005

Results
To find the free-vibration response of a viscously damped wing spar system, MATLAB program was used to find the response of a system with the  In this case, the spar is an undamped system and is subjected to a harmonic force, so the damping equation; ̇= 0 (9) If a force acts on the mass m of an undamped system, the equation of motion reduces to; The homogeneous solution of this equation is given by; Where   is the natural frequency of the system.Because the exciting force (t) is harmonic, the particular solution   is also harmonic and has the same frequency  Thus we assume a solution in the form; For this case;  ≠   (13) So we can obtain C1 and C2 as; So the general solution for the system is; The initial conditions (0) =  0  (0) = ̇0 results to; the response of the system under harmonic vibration will be:

Linear Vibration of a Multi Degree Freedom System
Systems that require two independent coordinates to describe their motion are called two degree of freedom systems.Considered a spar as a cantilever beam with the mass of a rib creating to masses with two independent coordinates creating a two degree of freedom system (DOF) and in this case we considered the free and forced vibration under damped and undamped condition.

3.1
Free vibration of a viscously damped two degrees of freedom system Consider a viscously damped two-degree-of-freedom spring-mass system, shown in the Figure below; The MATLAB code to find the free-vibration response of the system and modal nodes and response of the system is as below;

Forced Vibration of Undamped Two Degrees of Freedom System
In this case harmonically excited two degree of freedom system of a wing spar was analysed.Since damping was disregarded; c =0, the below equations were utilized to obtain a steady state response and frequency-response curve was plotted.We can write the steady-state solutions as Where X1 and X are, in general, complex quantities that depend on  and the system parameters.Substitution of Equation.(29) And (30) into Equation of motion leads to;

Non Linear Vibration 4.1 Lindstedt's perturbation method
In the case of non linear vibration analysis, we considered the pertubation.The perturbation method is applicable to problems in which a small parameter associated with the nonlinear term of the differential equation [4].The solution is formed in terms of a series of the perturbation parameter,, the result being a development in the neighbourhood of the solution of the linearized problem.If the solution of the linearized problem is periodic, and if , is small, we can expect the perturbed solution to be periodic also.We can reason from the phase plane that the periodic solution must represent a closed trajectory.The period, which depends on the initial conditions, is then a function of the amplitude of vibration.Consider the free oscillation of a mass on a nonlinear spring, which is defined by the equation; ̈=     +   (36) With initial conditions () = , ̇() =  When  = 0, the frequency of oscillation is that of the linear system,  = √   .We seek a solution in the form of an infinite series of the perturbation parameter p. as folIows:  =    +    +      (37) Furthermore, we know that the frequency of the nonlinear oscillation will depend on the amplitude of oscillation as well as  on.We express this fact also in terms of a series in;   =    +   +     (38) Where the   are as yet undefined functions of the amplitude, and  is the frequency of the nonlinear oscillations.We consider only the first two terms of Equations ( 37) and (38), which will adequately illustrate the procedure.Substituting these into Eq.(i), we obtain; ̈ + ̈ + (  −   )(  +   ) + (   +      + ⋯ ) (39) Because the perturbation parameter  could have been chosen arbitrarily, the coefficients of the various powers of  must be equated to zero.This leads to a system of equations that can be solved successively: ̈ +

Random Vibration of a Single Degree Of Freedom
The type of functions that were considered to this point can be classified as determinisic i:e mathematical expressions can be written that will determine their instantaneous values at any time t.However, in real life situation, there are a number physical phenomena that results in non deterministic sense like in the case we consider forces acting on aircraft as it flies in air.Random vibration of a viscously damped system by arbitrary force was determined.

Conclusion and Summary
This study was performed for an UAV aircraft wing spar modeled to analyse different vibration patterns.
The study was done for linear, nonlinear and random vibration considering the wing spar as a cantilever beam for both damping and undamped scenarios.
Responses for all the vibration analysis were plotted using MATLAB software to analyse the behaviour of the beam when subjected to different vibration phenomenon.The natural frequency for all single of freedom vibration analysis is constant meanwhile the frequencies for the two degree of freedom and continuous vibration showing some variations.For future study, other forces acting on the aircraft due to the speed will be incorporated in the analysis to better observe conditions of vibration.

Figure 6 :
Figure 6: Results of viscously damped single degrees of freedom system

Figure 7 :
Figure 7: Results of harmonically excited single degrees of freedom system

Figure 10 :
Figure 10: Results of a 2 DOF viscously damped system

Figure 12 :
Figure 12: Results of Forced Vibration of Undamped Two Degrees of Freedom System =  (40) ̈ +     =     −    (41) The solution to the first equation, subject to the initial conditions, () = , ̇() =  is;   =  (42) By substituting this into the equation above we get; ̈ +     =    −      (used.We note he re that the forcing term  would lead to a secular term  in the solution for   (i.e., we have a condition of resonance).Such terms violate the initial stipulation that the motion is to be periodic; hence, we impose the condition; term  eliminated from the right side of the equation, the general solution for is     =    +    + imposing the initial conditions () = , ̇() =  , constants      are   =     = vibration response equation can therefore be obtain by; obtained after analysis.

Figure 13 :
Figure 13: Results of Nonlinear Vibration by Perturbation Method

Figure 14 :Figure 15 :
Figure 14: Random vibration of a viscously damped system by arbitrary force

Figure 17 :
Figure 17: Results of the first three modes of the Continuous Vibration Analysis of a Wing Spar Ms. Merina Amon Mwasandube is an Assistant Lecturer, Researcher and Consultant in Aeronautical Engineering at the National Institute of Transport (NIT).She holds a Master's degree in Aircraft Design, from Nanjing University of Aeronautics and Astronautics and a Bachelor's degree in Aeronautical Engineering (Aircraft Manufacturing) from Shenyang Aerospace University both from China (2022).Also, she is a registered Graduate Engineer at the Engineers Registration Board (ERB).She has published three research articles in peer-reviewed journals.Her research interest is on aeronautical engineering.She is currently a head of Department at the National institute of Transport in Tanzania.9. References 1. V. Saran, V. Jayakumar, G. Bharathiraja, K. S. Jaseem, and G. Ragul, "Analysis of natural frequency for an aircraft wing structure under pre-stress condition," Int.J. Mech.Eng.Technol., vol.8, no.8, pp.1118-1123, 2017.2. "Theory of Vibration with Applications (5th).pdf." 3. S. A. Q. Siddiqui, M. F. Golnaraghi, and G. R. Heppler, "Large free vibrations of a beam carrying a