An Analytical-Based Lightning-Induced Damage Model for an Aircraft Wing Exposed to Pressure Loading of Lightning

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Introduction
In recent years, air transportation has played a very important role in both global economic growth and traveling long distances in a shorter time.However, one of the common occurrences that any aircraft encounters during air transportation is lightning strikes.Moreover, the threat can be even more critical for aircrafts that must fly at low altitudes and weather-independent conditions, where the lightning risk is greatest.Past lightning events and aircraft accidents revealed that the impact of lightning can result in structural damage [1], operational disruptions [2], and loss of life and property [3].Furthermore, in recent years, it has been stated that the flying ring cost of lightning-caused disruptions (e.g., delays and cancelations) can be nearly five times the cost of replacing an existing facility [4].Therefore, the influence of the impact of lightning on aircraft structural materials must be investigated not only to determine and minimize the lightning-induced damages but also to reduce the cost caused by lightning, which contributes to the procedures for the design, production, and certification standards [5] of civil and commercial aircrafts.
A lightning event arises once a supercooled liquid and ice particles above freezing encounter.As a result of this event, large-and-distinct positive and negative electric charges in the clouds arise.Then, the lightning current discharge starts, and an intense current of around 30,000 amps' flows through the arc channel of lightning [6].If an aircraft is exposed to an intense current, a large amount of energy is transmitted to one of the front extremities of the aircraft (such as the radome [7], fuselage [8], wing tip [9], etc.) and then exits from another extremity (e.g., tail) of the aircraft.The air around the structural part of the aircraft exposed to lightning begins to ionize, and then the electromagnetic field intensity increases in this part, which generates a pressure impact.Thereafter, the wave of the pressure coming from the arc channel of the lightning propagates by acting as axial, lateral, and radial pressure on nearby structures [10].
Some authors [10][11][12][13][14][15] in the literature studied with the lateral pressure loading of lightning as a transverse force applied to a structure in the thickness direction of the structure by using different approaches.While some of them pointed out that lateral pressure loading is electromagnetic (or magnetic) pressure [10][11][12], others suggested that lateral pressure loading is a resultant pressure that consists of electromagnetic and acoustic pressure [13][14][15].The modeling approaches proposed by the authors are given in Table 1.The findings of the studies of the authors indicated that when the arc channel of lightning attaches to the surface of a structure, the radius of the arc channel expands over time, which supports that the radius of the arc channel is timevarying.Moreover, once any structural part of an aircraft is subjected to a lightning strike, while the arc channel of lightning stands in its initial position, the aircraft travels forward a significant distance during the flash lifetime [16].Therefore, although lightning hits to a single point on the aircraft structure, the lightning currents [17] and pressure effect [10] of lightning are distributed over the aircraft structure.As seen in Table 1, in the modeling approaches of the lateral pressure loading of lightning, some of the authors considered the distributed pressure effect of lightning but neglected the time-varying radius of the arc channel in the model.On the other hand, some took into account the time-varying radius of the arc channel but ignored the distributed pressure effect of lightning.Although these modeling approaches gave relatively acceptable and reasonable results, new modeling approaches are always possible to represent the natural behavior of lightning.For instance, introducing a model, that includes both the distributed pressure effect and timevarying radius of the arc channel of lightning, would be a good advance in modeling the pressure impact of lightning.Lightning is a multi-physical phenomenon [18,19], although it is almost impossible to fully understand its nature, improvements are always possible.
In Table 1, μ 0 is the magnetic permeability, μ 0 ¼ 4π × 10 −7 N=A; IðtÞ: is the lightning current; r c is the timeinvariant radius of the arc channel of lightning; r is the radial distance to the lightning strike attachment point; P 1 and P 2 are coefficients and τ 1 and τ 2 are time constants; μ r is the relatively permeability of the nonmagnetic lightning protection, μ r ¼ 1; R r ðtÞ: is the value of the radius of arc root, and t is time.
Since lightning strike is a threat to all aircraft structures made of metallic or composite material, numerous experimental and numerical studies were covered in the literature to investigate the damage behavior of aircraft structural materials exposed to pressure loading of lightning.Among the experimental studies, Damghani et al. [20] examined the effects of transverse impact loading of lightning on two laminate configurations to determine the shear-buckling behavior of the laminate.Boushab et al. [21] studied the lightning damage resistance of a carbon-epoxy panel by considering lightning arc channel expansion to find out the cause of the widespread surface damage.Guerrero et al. [22] analyzed a carbon-aluminum wing box subjected to the loading of lightning to improve the structural behavior of the wing box.Although experimentation is the safest way to solve an engineering problem, providing reasonably good experimental solutions to the lightning problems for aircraft structural materials in the appropriate physical environment and under ideal conditions can be quite burdensome, both financially and temporally.To overcome such difficulties, some numerical models of lightning-induced damages in aircraft structural materials were proposed in the literature.Lee et al. [23] predicted the mechanical damage caused by the impact of lightning by considering the shock wave overpressure and lightning arc channel expansion via the finite element method.Foster et al. [24] investigated the potential contribution of lightning pressure loading to composite sample damage in terms of the form and scale of damage.Bigand et al. [25] proposed a numerical model of the damage caused by the overpressure of a lightning strikes on an aircraft composite structure protected with a metallic mesh and painted.Qian et al. [26] realized the simulation of a turbofan engine of an aircraft encountering a lightning strike to investigate the electromagnetic effect of lightning strike on motor control cables by using CST software based on the transmission line matrix method in the simulation.In spite of progress that was made in the experimental studies to model lightning-induced damages in aircraft structural materials, a complete analysis is almost impossible under current technological conditions because of the multiphysical nature of lightning [27].Furthermore, despite the tremendous development of computers and numerical algorithms, it is also impossible to solve the lightning problem numerically today [28].However, by simplifying engineering structures under some assumptions, utilizing the power of analytical modeling can be a guide in solving engineering problems.Moreover, even if it may seem unfeasible to understand the relationships between different parameters of a structure and forces that influence a particular outcome, analytical modeling is an effective and reliable technique for turning many different variables and conditions into information one can use.
The present article is an extended version of the article published in [29].The goal of the present article is to investigate the lightning-induced damage behavior of an aircraft

Author
Formulation of pressure loading of lightning Kawakami [10] Pðr; tÞ: ¼ μ 0 I 2 ðtÞ: =8π 2 r 2 structural material from an analytical perspective.For this purpose, two analytical-based models are proposed.The first is for the pressure loading of lightning, which is named the improved electromagnetic pressure impact model (IEPIM); the latter is for the damage in an aircraft wing exposed to the pressure loading of lightning, which is named the damage model in the rest of the present article.In this context, first, the IEPIM is established.Second, with the help of the IEPIM, the damage model is established.In the solution of the damage model, free vibration analysis by using the differential transform method (DTM) and forced vibration analysis by using the modal expansion method are conducted.The results of the theoretical formulations developed in the present article are obtained with codes developed by the user in the MATLAB environment.Then, to verify the models established, the results of the models are compared with the results of some experimental studies borrowed from the open literature where comparison is possible.A diagram summarizing the stages of the present article is given in Figure 1.

Interaction between Lightning Strike and an Aircraft.
Due to unexpected hazards and serious accidents caused by lightning strikes on aircrafts, aviation regulatory bodies around the world have established certification standards to ensure that aircrafts can withstand lightning strikes and land safely at a suitable airport at the end of the flight.Considering the certification standards established, modern aircraft manufacturers have used various lightning protection systems to prevent and/or reduce potential direct and indirect hazards caused by lightning strikes.The main purpose of protecting an aircraft's fuselage and structures is to identify areas on the aircraft that are most vulnerable to lightning strikes.Thus, the surface of the aircraft can be classified as a function of the lightning threat, which results in the information about the intensity of lightning and the extent of damage in the aircraft.
Based on vulnerability to lightning, the division of surface of aircrafts into lightning strike zones may vary depending on some factors such as the aircrafts' intended use, geometry, and how often they are flown, etc.; however, the major standards are similar for most aircrafts.In the open literature, the lightning strike zones for aircrafts were determined according to the Aerospace Recommended Practice (ARP) 5414 standards of the Society of Aerospace Engineers (SAE) [13].The lightning strike zones for an aircraft wing are shown in Figure2 [30], and the descriptions of these zones are given in Table 2 [31].
Based on experimental studies, a lightning current was formulated as the following expression [11,12]  Mathematical Problems in Engineering where α and β are parameters, and their values change depending on the lightning current components that are A, B, C, and D, as shown in Table 3 [32].The durations and amplitudes of the components are shown in Figure 3 [10].In addition, the relationships between lightning current components and lightning strike zones are given in Table 4 [1, 10].

Constraints Induced by Lightning Strike on an Aircraft
Material.Aircrafts can trigger lightning while flying through a charged cloud.Once the trigger starts, the lightning process begins.One of the main stages of this process is the leading stage [30].The leading stage is the onset and development of a conductive arc channel of lightning.After the arc channel is developed, lightning discharges begin to flow through the arc channel.The lightning discharges are grouped depending on the beginning and ending of the discharges, which are intracloud, cloud-to-ground, and cloud-to-cloud discharges [33].
The most common type of lightning discharge is cloud-toground, which accounts for 90% of all lightning strikes [34].
The percentage of the form of lightning discharge varies depending on the storm in the current geography [35].However, according to the findings of a study [35], it was revealed that single-lightning flashes are much more common among lightning flashes recorded by lightning detection systems.
Even if a single lightning flash hits to an aircraft in flight, the lightning flash has an impact, including thermal [12], electrical [14], and mechanical [11] effects.These effects of lightning can be broadly divided into two categories: thermomechanical constraints and electromagnetic constraints.The thermal-mechanical constraints relate to the damage to the airframe materials and can lead to destructive effects such as vibrations that resonate the structure [36], rupture of the extremities of the aircraft [1], pitting [37], deflection [38], etc. Conversely, the electromagnetic constraints relate to the damage to any electronic system, causing interference to the electronic equipment on board and thus leading to computer control systems malfunction [39].

Mechanical Constraints Caused by Electromagnetic
Pressure Impact.Figure 4 shows the life cycle of damage caused by a single-lightning strike on an aircraft wing made of a metallic material.As shown in Figure 4, if the material of an aircraft wing exposed to a lightning strike has high thermal and electrical conductivity, the electrical and thermal energy generated by the lightning strike are swept from the wing and given back to the air [30]; and therefore, if negligible local thermal effects coming from the energy conversion are not considered, only mechanical constraints caused by the pressure impact of lightning occur in the area struck by lightning.At this stage, the arc channel of lightning interacts with the surface of the aircraft wing, and then the radius of the arc channel expands over time   Mathematical Problems in Engineering  Mathematical Problems in Engineering 5 [5,[11][12][13][14][15].During this time, the magnetic interaction occurs between the current flow in the arc channel and the current flow on the surface of the aircraft wing.Then, this interaction exerts an impulsive force on the surface of the aircraft wing [13], which generates the pressure waves propagating from the arc channel by acting on nearby structures [10].The lateral pressure waves arising in the expanding arc channel contribute significantly to the mechanical constraints [13], including elastic displacements, deflection, etc.One of the factors affecting lightning-induced damage in a structure is the slenderness of the structure.For thinner structures with a high slenderness ratio, such as cables, the effect of the pressure enhances by the slenderness ratio, so that even low-pressure states far from the buckling stage may have a significant impact on the dynamics of the structure [40].For instance, when an overhead ground wire that has a high slenderness ratio (almost all 1D structure) and is generally made of an aluminum-clad steel strand [41] is struck by lightning, thermal ablation damage [42] or rupture accident [43] can occur in the structure.However, due to mission requirements, aircrafts present high-aspect-ratio wings that result in relatively flexible structures [44].Therefore, the high flexibility and large aspect ratios can cause large deflections of up to approximately 25% of the wing half-span [45] rather than rupture hazards or thermal ablation.
During lightning strikes, another source of mechanical constraints is acoustic overpressure [13].Lightning releases substantial amounts of energy into a quite narrow volume of air in a very short period [46].Thereby, the air in the ionized lightning arc channel is heated to extremely high temperatures almost instantly [47], hence arising high pressures of the expanding plasma in the arc channel [10].This pressure generates radial pressure and resulting shock waves in the heated channel [48].Then, the shock waves expand and propagate radially outwards from the center of the discharge channel, leading to mechanical constraints in the structures, especially in the structures that are more sensitive to breakage (e.g., composite skin) [13].In this article, the acoustic pressure propagating in radial directions of the ionization channel is neglected, and only the lateral pressure effect (i.e., EPI) is considered due to the subject of the article.

Establishment of an Improved Electromagnetic Pressure Impact Model (IEPIM).
According to some studies [49], the radius of the arc channel of lightning can be modeled as a function of response time as follows: where r c ðtÞ: is radius of the arc channel (m) expanding in time, α 0 is constant, α 0 ¼ 0:102, ρ 0 is the air density at atmosphere pressure, and ρ 0 ¼ 1:29 kg=m 3 , IðtÞ: is lightning current.Muñoz et al. [13] considered the distributed pressure effect of lightning with a time-invariant radius of the arc channel in the pressure model.To propose an IEPIM involving not only the distributed pressure effect of lightning but also the time-dependent radius of the arc channel, the time-dependent radius of the arc channel given in Equation ( 2) is combined with the pressure model of lightning, which is also given in Table 1, in [13].The radial distance relative to the lightning strike point, r, can be reexpressed in terms of the strike point on the wing as L − x in the formulation of the pressure.Therefore, an IEPIM of lightning is obtained as follows: where x denotes any point of a structure subjected to the pressure loading of lightning, L is the length of the structure, and r c is the maximum point reached by the radius of the arc channel of lightning.
In the next section, by using the IEPIM expressed in Equation ( 3), an analytical model of the damage caused by the pressure loading of lightning in an aircraft wing is established to investigate the lightning-induced damage behavior of the aircraft wing material.In the damage model to be established, a single cloud-to-ground type lightning strike is considered because it is more common and more observable in nature [34,35].

2.
3. An Analytical-Based Lightning-Induced Damage Model 2.3.1.Establishment of the Model.One of the most vulnerable locations to lightning strikes is the wing of most aircrafts [50].In the construction of an aircraft wing and examination of its resistance deviation against static and dynamic loads, beam structures can be used [15,18].Especially, beams, which are clamped at the origin of the body axes and have coupled bending and torsional behavior, are good approaches to aircraft wing structures [17].For this reason, in the establishment of an analytical model of the lightninginduced damage in an aircraft wing, an average uniform aircraft wing structure made of a material with high thermal and electrical conductivity is considered a bending-torsion coupled beam while the pressure loading of lightning is modeled as a flexural loading applied to the beam.
It should be noted that when lightning hits to an aircraft wing material, the constraints caused by the lightning strike in the material occur in a time interval of the order of microseconds [51]; for this reason, the effects of the aerodynamic forces, which are the forces acting on the body moving through the air such as lift, drag, and downforce in the air, can be negligible in this process.Moreover, the local heat effects caused by the energy conversion are not included in the model since these effects are quite small when compared with the mechanical effects in an aircraft wing.Furthermore, due to the subject of the study, the other parts (such as rips, spars, engines, or avionics systems) of the wing structure are not included in the damage model.
The partial differential equations and boundary conditions of the damage model in an aircraft wing subjected to the pressure loading of lightning are given in Equations ( 4)- (8).Here, uðx; tÞ: is bending translation in the vertical direction and ψðx; tÞ: is torsional rotation about the elastic axis of the wing, where x 6 Mathematical Problems in Engineering denotes distance from the origin and t denotes time.Moreover, EI, GJ; m; x α , and I α are, respectively, bending rigidity, torsional rigidity, the mass of the wing per unit length, bending-torsion coupling term, and mass moment of inertia per unit length of the wing about the elastic axis; the coefficients c 1 and c 2 are the linear viscous damping terms of per unit length of the wing in bending deformation and torsional deformation, respectively; ð 0 Þ: and ðÞ : denote, respectively, differentiations with respect to space x and time t.
Solution of the Model.
(1) Free vibration analysis.For undamped free vibration analysis, the external distributed force P G ðx; tÞ: and the damping coefficients c 1 and c 2 are set to zero to determine the dynamic characteristics (i.e., vibration frequency and vibration mode shapes) of the aircraft wing model stated in Equations ( 4)- (8).
In harmonic oscillation, uðx; tÞ: and ψðx; tÞ: are approximated by a sinusoidal variation with circular vibration frequency ω n as follows: where U n ðxÞ: and Ψ n ðxÞ: are, respectively, amplitudes of the sinusoidal variation of the bending displacement and torsional rotation of the wing, and n ¼ 1; 2; 3; ….Substituting Equations ( 9) and (10) into Equations ( 4)-( 7) gives the ordinary differential equations and related boundary conditions as follows: where ð 0 Þ: ¼ d=dx and the coefficients a; b; c; and d are as follows: Then, DTM is applied to solve the ordinary differential equations with associated boundary conditions given in Equations ( 11)- (14).According to the theory of DTM [52], an analytical function f ðxÞ: is expanded to a power series with the center η 0 in the domain of f ðxÞ: .Thus, the differential transform and inverse transform of f ðxÞ: are stated by F D ½k: and f D ½x: , respectively, as follows [52]: The transformation rules of DTM for different functions and operations are given in [53].After applying the transformation rules to both the ordinary differential equations and the boundary conditions, a set of algebraic equations is obtained as follows: where U D ½k: and Ψ D ½k: are, respectively, the differential transform of U n ðxÞ: and Ψ n ðxÞ: : In addition, in Equations ( 22)-( 24), N denotes the number of terms included in the application of DTM, and its value is determined depending on the convergence of the vibration frequencies.Subsequently, using the inverse transform formulation given in Equation ( 18), the vibration mode shapes of the wing are obtained as follows: The Equations ( 25) and ( 26) are the conjunction with the boundary conditions.Furthermore, the orthogonality condition Mathematical Problems in Engineering can be stated for different vibration mode shapes as follows [54]: where μ n is the generalized mass in the nth mode and δ mn is the Kronecker delta function.The pseudocode related to the application steps of DTM is given in Algorithm 1.
(2) Forced vibration analysis.In the forced vibration analysis, the modal expansion method is implemented.In this method, the solutions of the Equations ( 4)-( 8) are approximated as follows: and the external flexural loading is assumed in the following form: where U n ðxÞ: and Ψ n ðxÞ: are the vibration mode functions, q n ðtÞ: is a time-dependent generalized coordinate for the nth mode; DðxÞ: denotes the spatial distribution, and f ðtÞ: denotes the time-dependent function factor of the external flexural load.Substitution of Equations ( 28) and (29) into Equations ( 4) and ( 5), multiplication the final equations by U m ðxÞ: and Ψ m ðxÞ: , addition the last two equations and then integration the last equation from 0 to L, along with the use of the orthogonality property of the mode functions given in Equation ( 27) results in the following equation in the modal coordinates: where ξ n is the nondimensional damping coefficient in the nth mode and defined by the following: here, r g is the radius of gyration defined by ffiffiffiffiffiffiffiffiffi I α =m p ; and F n ðtÞ: is calculated as follows: Step 1. Read the data EI; GJ; m; x α ; I α ; L: Step 2. Compute coefficeints a, b, c, d.
Step 4. Solve the following system of equations: Step 6. Select the real parts of the vibration frequencies obtained in Step 5 as the resulting vibration frequencies.
Step 7. Compute the vibration mode shapes U n ðxÞ: and ψ n ðxÞ by using inverse transformation functions.ALGORITHM 1: The pseudocode for application of DTM (ε : is a tolerance value, BC: boundary condition).

8
Mathematical Problems in Engineering where x 0 is the point where lightning attaches to the aircraft wing exposed to lightning.By the Duhamel's integral, the solution of Equation ( 31) is obtained as follows: where A n and B n are coefficients related to the initial conditions and ω nd is defined as follows: Subsequently, substitution of Equation ( 34) into Equations ( 28) and ( 29) gives the general solution for the bending displacement (i.e., bending deflection) and torsional displacement (i.e., torsion deflection) in the aircraft wing with the following forms: The MATLAB/Simulink model of the analytical models established is given in Figure 5.

Verification of the Models Established.
In the verification of the models established (i.e., the IEPIM and the damage model), the following steps were realized: (1) The results of the IEPIM were evaluated by using the results of two different experimental studies, borrowed from the open literature, in terms of the relative error.(2) The results of the damage model were evaluated by using the results of a relevant experimental study, taken from the open literature, in terms of the relative error.In this context, first, the dynamic characteristics of the aircraft wing were obtained in the free vibration analysis.Then, by using the dynamic characteristics, deflections that occurred in the aircraft wing exposed to pressure loading of lightning were obtained in terms of bending deflections and torsion deflections in the forced vibration analysis.
As the material properties of an aircraft wing, the properties of the Goland wing, which were given in Table 5 [55,56], were considered.

Mathematical Problems in Engineering
The graphics and results of the models established were realized in the MATLAB/Simulink environment with the codes developed by the user.

Results and Discussion
3.1.Verification of the IEPIM.Martins [11] provided an experimental database to understand the direct effects of lightning strikes and validate the lightning arc physical model.The author obtained the value of the electromagnetic pressure impact (EPI) of lightning for 100 kA lightning current by using a current generator.The author showed that for 100 kA lightning current, the radius of the arc channel is about 3.5 mm nearly at t ¼ 2:6 μs.About 100 kA lightning current is regarding the component D waveform of lightning as shown in Figure 3. Thus, by using Equation ( 2), the radius of the arc channel of lightning was found to be reached about 3:6 mm at t ¼ 2:6 μs, as shown in Table 6, in the present article.This result proves that the database provided by Martins and the model of the radius of the arc channel of lightning given in Equation ( 2) are consistent.
In another experimental study, Martins et al. [57] obtained the value of the pressure loading of lightning for different lightning currents by using EMMA and SuperDICOM (SDICOM) lightning high current generators at four instants, t ¼ 6; 9, 14; and 26 μs: In this experimental study, the generators were permitted to transmit intense currents by a conduction electrical arc on a test sample connected to a rig made of aluminum plate.The magnitudes of the pressure loading models obtained in the experimental studies and the magnitudes of the IEPIM were evaluated for 100 kA lightning current in Table 7 and for 200 kA lightning current in Table 8, in terms of relative errors at the instants t ¼ 6; 9, 14; and 26 μs.As indicated in Tables 7 and 8, it was found that while the results of the IEPIM for 100 kA lightning current are more consistent with the results of certain generators at certain time points, the results of the IEPIM for 200 kA lightning current are more consistent with the results obtained using the generator so-called SDICOM.More precisely, the results of the IEPIM for 100 kA lightning current are closer to the results of SDICOM at t ¼ 6 and 9 μs and closer to the results of EMMA at t ¼ 14 and 26 μs, as shown in Table 7.On the other hand, the results of the IEPIM for 200 kA lightning current are  In the free vibration analysis, first, the convergence of the vibration frequencies of the aircraft wing was determined, as shown in Figure 6.In Figure 6, it can be seen that only the first thirty terms of DTM are required to get the first six convergent vibration frequencies.Subsequently, the vibration frequencies obtained were compared with the vibration frequencies found in [55] in terms of relative error ε.It was found that the results of DTM were found to converge to the results of the vibration frequencies found in [55] with almost zero relative error, as given in Table 9.Then, by using the vibration frequencies, the vibration mode shapes of the aircraft wing were obtained, as depicted    in Figure 7. Compared with the vibration mode shapes obtained in [55], a good agreement was found between the present article and the reference.

3.2.2.
Results of the Forced Vibration Analysis.Martins, in the experimental study [11], found that when the pressure impact of lightning, including 100 kA lightning current, was applied to the center of a pure aluminum panel, the maximal amplitude of the deflection in the lateral direction was found about 3:1 mm approximately at t ¼ 1 ms, which is larger than the duration of mechanical action.Then, the author explained that the deflection at the center of the panel continues to increase without arc.This suggests that deflection at the center of the panel arises after the mechanical action time.In the present article, the center of the aircraft wing is the mid-wing (x 0 ¼ 3 m), which is in Zone 2A associated with approximately 100 kA lightning current, as shown in Figure 2 and Table 4.In the defection model, when the impact point for 100 kA lightning current was taken as x 0 ¼ 3 m (i.e., the mid-wing) from the leading edge, the bending deflection was found about 3:1 mm at t ¼ 20 μs depending on the value of the damping coefficient, as shown in Table 10.This proves that the result of the damage model for the mid-wing is consistent with the result of the experimental study provided by Martins.Furthermore, when the impact point for 100 kA lightning current was also taken as x 0 ¼ 0:5 (wing-root) and x 0 ¼ 4:5 (wing tip), that are in Zone 2A, the deflections occurred in these regions were about 0:2 and 9:5 mm at t ¼ 20 μs, respectively, as shown in Table 10.This indicates that when the impact point approaches to the wing tip, the deflection occurred in the wing increases; on the other hand, when the impact point approaches to the wing root, the deflection occurred in the wing decreases.In addition, in Table 10, it was seen that when the lightning strike point (x 0 ) is fixed, and the damping coefficient (ξ) is increased in the range of ½0; 2ξ: , the amount of deflection decreases as the amount of damping coefficient increases.
Regarding the torsion deflections in the aircraft wing, Table 11 shows that the torsion deflection for 100 kA lightning current is about 0.8 mm in the wing root, is about 1.9 mm in the mid-wing and is about 2.3 mm in the wing tip at t ¼ 20 μs.Furthermore, as in Table 10, when the impact point moves to the wing tip, the deflection arisen in the wing increases; by contrast, when the impact point moves to the wing root, the deflection arisen in the wing decreases.In addition, when the damping coefficient is increased in the interval ½0; 2ξ: , the amount of deflection in the wing decreases.Considered Tables 10 and 11, it can be concluded that when lightning, including 100 kA current, hits to the wing root of an aircraft, the pressure impact of the lightning causes more torsion deflection than bending deflection at the wing root; conversely, when lightning with a current of 100 kA hits to the mid-wing or wing tip of an aircraft, the pressure impact of the lightning causes more bending deflection than torsion deflection at the mid-wing or wing tip.
The deflected shapes of the aircraft wing in the form of bending displacement (bending deflection) and torsional displacement (torsion deflection) were given for the wing root, mid-wing, and wing tip, respectively, in Figure 8.As shown in Figure 8(b), the deflected shape of the wing exposed to a lightning strike at the mid-wing is in the form of a damped sine wave, as in the experimental study [11].(1) The results of the lightning pressure model proposed (i.e., IEPIM) are quite consistent with the results of two experimental studies for 100 kA lightning current and 200 kA lightning current.(2) Although lightning hits to regions with the same characteristics on an aircraft wing in terms of the lightning strike zone, the amount of deflection caused by the pressure impact of lightning on the wing increases as the impact point approaches the wing tip and decreases as it approaches the wing root.(3) When the lightning strike point (x 0 ) is fixed, and the damping coefficient (ξ) is increased in the range of ½0; 2ξ: , the amount of deflection decreases as the amount of damping coefficient increases.(4) When lightning including 100 kA current hits to the wing root of an aircraft, the pressure impact of the lightning causes more torsion deflection than bending deflection at the wing root; in contrast, when lightning including 100 kA current hits to the mid-wing or wing tip of an aircraft, the pressure impact of the lightning causes more bending deflection than torsion deflection at the mid-wing or wing tip.
Based on the results obtained in this article, it was found that the theoretical results of the article are in good agreement with the results of the experimental studies in the literature.However, considering today's laboratory conditions and the current state of advanced computer simulations, it is recommended that the models established in the present article be retested before implementation.

FIGURE 2 :
FIGURE 2: Dividing an aircraft wing into zones based on its sensitivity to lightning strikes.

FIGURE 3 :TABLE 4 : 4 FIGURE 4 :
FIGURE 3: Duration and amplitude of the lightning current components.

1 FIGURE 5 :
FIGURE 5: MATLAB/Simulink model related with the models established.

ϖ 1 =FIGURE 6 :
FIGURE 6: Convergence of the first six vibration frequencies of the aircraft wing with respect to N (N: number of terms included in DTM).

FIGURE 7 :
FIGURE 7: The first six vibration mode shapes of the aircraft wing.

TABLE 1 :
Pressure loading of lightning in the existing literature.

TABLE 2 :
Descriptions of lightning strike zones on an aircraft wing.ZonesDescriptions of lightning strike zones Zone 1AInitial attachment point with low possibility of lightning channel hang-on Zone 1BInitial attachment point with high possibility of lightning channel hang-on Zone 2ASwept stroke zone with low possibility of lightning channel hang-on Zone 2BSwept stroke zone with high possibility of lightning channel hang-on Zone 3Areas not in Zone 1A, 1B, 2A, or 2B, where damage least probable

TABLE 3 :
Values of the lightning current parameters.

TABLE 5 :
Material properties of an aircraft wing.

TABLE 6 :
Radius of the arc channel at some instants.
Bold values to highlight the validation by an experimental study that was mentioned in the manuscript.

TABLE 7 :
[10]arison of the magnitudes of the pressure models for 100 kA lightning current.result of EMMA at t ¼ 6 μs and closer to the results of SDICOM at t ¼ 9; 14; and 26 μs, as shown in Table8.Here, it can be noticed that the results of the IEPIM are quite consistent with the results of two experimental studies up to the first 26 μs for 100 kA lightning current.At the 26th microsecond, the amount of the relative error was found to be relatively higher.Since 100 kA lightning current is associated with a shortduration component D waveform with a rise time of up to 25 μs[10], the relative error at t ¼ 26 μs for 100 kA lightning current can be neglected.This proves that the component D waveform is valid before 26 μs.In contrast, it is seen that the results of the IEPIM are quite consistent with the results of the experimental study for 200 kA lightning current because 200 kA lightning current is associated with component A waveform, as shown in Figure2, with a rise time of less than 50 μs[10].

TABLE 8 :
Comparison of the magnitudes of the pressure models for 200 kA lightning current.

TABLE 9 :
The first six vibration frequencies of the aircraft wing.

TABLE 11 :
Amount of torsion deflection in an aircraft wing at different locations (x 0 ) with respect to damping coefficient (ξ).

TABLE 10 :
Amount of bending deflection in an aircraft wing at different locations (x 0 ) with respect to damping coefficient (ξ).