ISSN 1068-7998, Russian Aeronautics Aeronautics (Iz.VUZ), 2012, Vol. Vol. 55, No. 4, pp. 348–352. 348–352. © Allerton Press, Inc., 2012. 2012. Original Russian Text © S.A. Mikhailov, Mikhailov, L.V. Korotkov, D.V. Nedel’ko, 2012, published in Izvestiya VUZ. Aviatsionnaya Aviatsionnaya Tekhnika, 2012, No. 4, pp. 15–18.
STRUCTURAL MECHANICS AND STRENGTH OF FLIGHT VEHICLES
Analysis of Static Loading for a Helicopter Tubular Skid Landing Gear a
b
b
S. A. Mikhailov , L. V. Korotkov , and D. V. Nedel’ko a
Tupolev Kazan National Research Technical University, Kazan, Russia b
OAO Kazan Helicopters, Kazan, Russia Received July 6, 2011
Abstract—We Abstract—We compare the results of analyzing the stress strain state for the tubular skid landing gear with regard for the physically and geometrically nonlinear scheme of deformation; the analysis is carried out by the finite element method and with the aid of a special beam-type design model. The analysis has been performed for a real helicopter design. The peculiarities of deforming the aircraft structures of the type under study are show n based on the comparison results. DOI: DOI: 10.3103/S1068799812040046 Keywords: Keywords: helicopter, skid landing gear, landing gear spring, elastoplastic strains, nonlinearity, large displacements.
The objective of the work has been to verify the design model of the helicopter skid landing gear developed earlier for studying the dynamic loading of a structure at emergency landing. The main theoretical principles for constructing the design model are presented in [1]. As was mentioned in [1], the model is based on the geometrically nonlinear theory of the spatially deformable airfoil beams [2, 3]; in this case, as a fundamental basis, use is made of the Kirchhoff–Clebsch theory developed for analyzing the beams that are nonlinear before deformation. To take into account the physical nonlinearity of elastoplastic deformation of the landing gear material, use is made in the design model of the physical relations connecting the bending moments in each design section of the tubular beam and the curvature vector components based based on the hypothesis of plane plane sections [4]. The system of resolving resolving equations is discretized by the general method of integrating matrices [5]. Figure 1 presents the general view of a design model for the skid landing gear. In disclosing the statically indeterminate design scheme by the force method, the imaginary cut of the right-hand skid structure (in helicopter flight) in the cross-section corresponding to the points K 1 and ˆ and unknown moments Xˆ ,..., X ˆ are K (Fig. 1a) is considered. Three unknown forces Xˆ ,..., X 2
1
3
4
6
introduced at the imaginary cut location. The “redundant” reactions R x 2 , R y 2 , R z 2 , R x 4 are also ˆ . The unknown forces X ˆ , i 1,...,10 are determined from the replaced by the unknown forces Xˆ 7 , ..., X 10 i strain compatibility conditions at the imaginary cut location or the condition of displacements in the supports A2 and A4 being equal to zero. =
In addition to the material presented in [1], let us consider in more detail the distinction in modeling the landing gear springs and skids. The landing gear springs are modeled in the framework of the geometrically nonlinear theory of the spatially deformable airfoil beams taking into account all peculiarities of their work in the structure. In this case, the skids in the design model realize the geometrical connection between the forward and back springs, taking part in redistributing the internal force factors between as a part of the statically indeterminate structure. Skids do not by themselves receive large displacements, which is confirmed by the experience of analyzing the structures of this type, and their elastic strains remain within the scope of Hooke’s law. On these grounds, the approximate scheme of taking into account the elastic strains of skids is introduced into the design model. This scheme is similar to that presented in [6]. 348
349
ANALYSIS OF STATIC LOADING FOR A HELICOPTER TUBULAR SKID R y4 Y
R x 4 X R y2 A
B 4
R y3 B3 3 P
R x 3 M A3
R y1 R x 1 O1 R 1 A1 B1 Z 1 M
P
R x 2 R 2 A2 B2 2
4
P M
X 10 X
Y
P
K 1
X 8 X 7 X 9
X 2 X X 3 X 6 X 5 1 X 4 X X 1 4 X 5 X 6 X 3 Z X 2
M M
P
O1
K 2 (a)
P
M
P
M
(b)
P M
Fig. 1. Design scheme of the tubular skid landing gear structure.
Practically for all structures of the landing gear types being considered the skids are the circular metals tubes of high stiffness, for which it is easy to determine the stiffness parameters: EJ x EJ y EJ S ; =
GJ torq
=
=
2GJ S , where J S is the moment of inertia for the tube cross-section.
ˆ , i By the action of forces and moments X i
=
1,...,6 concentrated in the imaginary cut cross-section, it
is easy to determine the increments of the rotation angles for the right-hand skid cross-section at the point K 1 , if it will be considered as a cantilever restrained beam [7]: Qi ( l 2 )
2
Qi l 2
=
;
Mi l 2
Mil
∆ϕ Mbend =
=
;
M il 2
M il
, EJ S 2 EJ S 8 EJ S 2 EJ S 2GJ S 4GJ S where Qi , M bend. i , and M torq.i are the corresponding shearing forces, bending and torque moments. ∆ϕQ =
∆ϕ M torq =
=
(1)
Under the assumption that the spatial angles ∆ϕ1 , ∆ϕ2 , ∆ϕ3 increments are small, we can determine ˆ1, ϕ ˆ2, ϕ ˆ 3 with regard for relations the total values of the cross-section rotation angles at the point K 1 ϕ ˆ , i 1,...,6 : (1) and forces X i =
ˆ 1 = ϕ1 + ∆ϕ1 ≈ ϕ1 + ϕ
where
ϕ1 , ϕ2 , ϕ3
Xˆ 5l
2 EJ S
−
Xˆ 3 l 2
8EJ S
;
ˆ 2 = ϕ2 + ∆ϕ2 ≈ ϕ2 + ϕ
Xˆ 6 l
2 EJ S
+
Xˆ 2 l 2
8EJ S
;
ˆ 3 = ϕ3 + ∆ϕ3 ≈ ϕ3 + ϕ
Xˆ 4 l
4GJ S
, (2)
are the spatial angles of the back spring cross-section rotation at the point B1 (see Fig. 1a)
taking into account their orientation in the local axes of the skid cantilever being considered, l is the skid length. Then the coordinates of the point K 1 of the imaginary cut in the global axes O1 XYZ of the design model can be calculated according to the formulas given in [6]: l l l ˆ 2 cos ϕ ˆ 3 ; YK 1 = YB1 + cos ϕ ˆ 2 sin ϕ ˆ 3 ; Z K1 = Z B1 − sin ϕ ˆ 2. X K 1 = X B1 + cos ϕ (3) 2 2 2 The similar conclusions can be made for the right-hand skid cantilever from the side of the forward spring (point B2). Figure 2 presents the bending moment distribution diagrams for the landing gear skids in the horizontal plane that, in the general case, are obtained in the following way: ˆ , —for the right-hand skid it is obtained by summation of bending moment diagrams for each force X i
i
1,...,6 in the corresponding plane and with the proper rule of signs accepted; —for the left-hand skid it is obtained by constructing the diagrams with account for the bending moments acting along the edges of this skid { M x 3 , M y3 , M z3 } and { M x 4 , M y 4 , M z 4 } . =
It should be noted that for the left-hand skid use is made of the similar scheme of taking into account its elastic strains and geometrical mating at the points K 3 and K 4 (mating can be also made at the point
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B4 —this question is not a matter of principle for this problem). To provide the geometrical mating of the left-hand skid, we introduce the unknown boundary conditions Cˆ , Cˆ , Cˆ , Cˆ , Cˆ , Cˆ that are the
{
x
y
z
ϕ1
ϕ2
ϕ3
}
integration constants of the main resolving equations for the forward spring [1], into the general number of the main unknowns of the problem. The finite element method was used to verify the design model developed and the corresponding finite element model was composed, the fragment of which is shown in Fig. 3. The finite element model of the skid landing gear was developed by V.P. Timokhin by using the MSC.Sofware (MSC.Patran 2008r1 preprocessor and MSC.Marc solver). The basic principles of the finite element simulation for the given problem are presented in [8]. In this case, the landing gear springs are simulated by the two-dimensional elements made of the elasto-plastic material with the failure criterion with respect to the maximum permissible elongation. The finite element dimensions were chosen as a result of calculations, in which we considered the influence of the element number in the circumferential and longitudinal directions on the accuracy of the solution being obtained.
Y
M x 4 M y4 M z4 B4
K 4 K 3 B3 M x 3 M y3 M 3
A4
X A2 M x 2 M y2
B2
A3 O1 A1 M x 1 Z M y1 B1 M z1
M 2 K K 1 2
Fig. 2. Scheme of taking into account skid elasticity.
Fig. 3. The general view of the finite element model of the skid landing gear.
To perform verification, we formed two abstract design cases, the loading schemes of which are presented in Fig. 4: 1) each of the points B1 ,..., B4 is acted upon by the identical vertical conservative forces P y1 ,..., Py 4 equal to 2600 daN; 2) each of the points B1 ,..., B4 is acted upon by the identical vertical conservative forces
P y1 ,..., Py 4
equal to 2600 daN and the longitudinal conservative forces Q x1 ,..., Qx 4 equal to 1000 daN.
(a)
(b) Fig. 4. Scheme of applying the external loads.
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The equal values of external loads on the forward and back springs of the landing gear to obtain the substantial difference of their proper displacements are specified that is due to distinctions in their stiffness characteristics. The external loads are specified in the global axes O1 XYZ of the design model. Figures 5a and 6a present the calculation results according to the technique developed (curves) and the finite element method (graphical marks) for design case 1. Comparison reveals the satisfactory agreement of the results obtained.
(a)
(b)
Fig. 5. Vertical displacements of the landing gear cantilever springs for design cases l (a) and 2 (b).
(a)
(b)
Fig. 6. Maximum stresses in the landing gear springs for design cases l (a) and 2 (b).
According to the technique developed, in the zone of elastic strains of landing gear springs and at the beginning of the development zone of their plastic strains, we have a slightly less values of the spring cantilever displacements (Fig. 5a) in the O1Y axis direction. This fact takes place because this technique does not take into account the influence of the shear strains on displacements of the design structure cross-sections but the difference in the displacement values does not exceed 1–2 %. The satisfactory coincidence of the bending stress values in the most loaded spring cross-sections (near the undercarriage units A1 and A2) is also seen in Fig. 6a. Figures 6b–8 present the calculation results according to the technique developed and the finite element method for design case 2 (notation is the same as for design case 1). Comparison reveals the satisfactory agreement of the results obtained. A great number of design parameters is compared for the given design case, since there is a great number of external loads directed along the different coordinate axes. RUSSIAN AERONAUTICS
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(a) Fig. 7. Longitudinal displacements of the landing gear spring cantilever for design case 2.
(b)
Fig. 8. Vertical (a) and horizontal (b) reactions at undercarriage units A1 and A2 for design case 2
The slight disagreement of the calculation results is obtained for bending stresses (Fig. 6b), displacements of the point B2 in the direction of the O1 X axis (Fig. 7) and the horizontal reactions R x values (Fig. 8b) that is explained to a greater degree by the approximate scheme used to consider the elastic strains of the landing gear skids. It should be noted that the results obtained are characterized by the quite adequate accuracy of engineering calculations. Thus, the design model of the skid landing gear has been verified and can be used in further analysis. ACKNOWLEDGMENTS This work was supported by the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” for 2009–2013 years (state contract no. P665 of May 19, 2010).
REFERENCES 1. Mikhailov, S.A., Korotkov, L.V., and Nedel’ko, D.V., Analysis of Static Loading of a Helicopter Tubular Skid Landing Gear, Izv. Vuz. Av. Tekhnika, 2010, vol. 53, no. 4, pp. 3–6 [Russian Aeronautics (Engl.Transl.), vol. 53, no. 4, pp. 369–374]. 2. Mikhailov, S.A., Geometric Nonlinearity in Statics and Dynamics in the Calculations of Helicopter Main Rotor Blades, Cand. Sci. (Tech.) Dissertation, Kazan: Kazan Aviation Institute, 1983, p. 16. 3. Pavlov, V.A., Mikhailov, S.A., and Gainutdinov, V.G., Theory of Large and Finite Beam Displacements, Izv. Vuz. Av. Tekhnika, 1985, vol. 28, no. 3, pp. 55–58 [Soviet Aeronautics (Engl.Transl.), vol. 28, no. 3, pp. 61–64]. 4. Mikhailov, S.A., Korotkov, L.V., and Nedel’ko, D.V., Simulation of Elastoplastic Deformation of Helicopter Skid Landing Gear Springs, Izv. Vuz. Av. Tekhnika, 2010, vol. 53, no. 1, pp. 8–12 [Russian Aeronautics (Engl.Transl.), vol. 53, no. 1, pp. 9–15]. 5. Vakhitov, M.B., Integrating Matrices as an Apparatus of Numerical Solution of Differential Equation in Structural Mechanics, Izv. Vuz. Av. Tekhnika, 1966, vol. 9, no. 3, pp. 50–61 [Soviet Aeronautics (Engl.Transl.), vol. 9, no. 3]. 6. Nedel'ko, D.V., Calculation of Helicopter Skid Landing Gear with Regard for Geometric, Structural and Physical Nonlinearity, Cand. Sci. (Tech.) Dissertation, Kazan: Kazan State Technical University, 2001, p. 20. 7. Astakhov, M.F., Karavaev, A.V., Makarov, S.Ya., et al., Spravochnaya kniga po raschetu samoleta na prochnost’ (Reference Book on Aircraft Strength Anaysis), Moscow: Oborongiz, 1954. 8. Aleksandrin, Yu.S. and Timokhin, V.P., Technique and Some Results of Studying the Special Features of Helicopter Landing Impact Characteristics with Regard for Landing Pad Surface Properties, in Prochnost’, kolebaniya i resurs aviatsionnykh konstruktsii i sooruzhenii (Strength, Vibrations, and Service Life of Aircraft Structures and Constructions), Moscow: TsAGI, 2007, pp. 148–156.
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