Determination of the optimal arrangement of engines along the wingspan

. A modern gas turbine engine, in the layout of the designed aircraft, allows us to consider fundamentally different placement options. The contradictory influence of the type, number and location of engines on the safety and efficiency of flight leads to the need to study these tasks at the early stages of design. This article discusses an approach to determining the optimal position of engines by wingspan at the stage of preliminary design. The calculation features consist in taking into account various factors: strength, for different design cases, mass-inertial, operational-technological and others. In the course of the study, the aircraft wing power set was simulated with its subsequent calculation in the NX system. As a result of the study, the optimal arrangement of engines for a four-engine aircraft with known geometric characteristics of the wing and its shape was obtained. An approach is shown, as a result of which it is possible to determine the optimal position of the engines, both in terms of the magnitude of bending moments in the root part of the wing and moments of inertia. The same method can be used to prove or correct the already accepted position of the engines on the designed aircraft.


Introduction
During the layout of a competitive aircraft, the choice of the location of the engines is determined taking into account the independent operation of the power plant [1]. These conditions are typical for all stages of the life cycle of subsonic, supersonic [2,3], as well as remotely piloted aircraft of various configurations and masses [4][5][6]. A gas turbine engine, when assembling an aircraft, allows you to consider fundamentally different placement options. The placement of engines on pylons under the wing is widely used on modern aircraft.
Taking into account the fact that the mass of the wing structure depends on the maximum bending moment acting on it, then it should be sought to minimize it when modeling loading processes [7,8]. The engine is a concentrated load and unloads the wing in flight, so it is logical to position it as far as possible from the longitudinal axis of the fuselage. However, when parked, the engines load the wing, which is why, on the contrary, it is necessary to reduce the distance between the engines and the longitudinal axis of the fuselage. At the same time, other factors must be taken into account: the distance of the engine from the fuselage contributes to an increase in the longitudinal moment of inertia, which affects the control system and the mass-inertial appearance of the aircraft as a whole.

Problem statement
Aerodynamic shape is formed at the initial (first) stages of a life-cycle [9]. The placement of engines along the wingspan has a number of advantages. At the same time, it is necessary to determine the optimal, taking into account the limitations, the position of the engines, which would lead to a decrease in the weight of the wing structure and minimally affect the dynamic characteristics of the aircraft [10].

Solution method
The problem is solved for a four-engine aircraft with known geometric parameters of the wing (wingspan = 81052 ). Since the mass of the wing structure is proportional to the bending moment, the bending moment is accepted as a criterion for evaluating the optimal arrangement of the engines along the span. In this case, the change of position relative to the wing will be carried out for the extreme engine at a fixed distance between adjacent engines.
As a calculated flight case, we take A'curved flight at small positive angles of attack with the maximum velocity pressure of a vertical dive. In this case: ′ э = э = 2,5; ′ = . We will plot the distributed aerodynamic load, as well as the load from the mass of the structure and the mass of the fuel, which we will use in further calculations.
Distribution of aerodynamic load by wing span and chord The distribution of the aerodynamic load over the span of a straight flat wing is proportional to the relative circulation Г ̅̅̅̅̅ [11]: The presence of a sweep changes the law of distribution of aerodynamic load, loading the ends and unloading the root sections of the wing at a positive sweep angle and vice versaat a negative angle. The influence of the sweep is taken into account using the formula [1] where ° -sweep angle along the line of the ¼ chord; Г 45°̅ ̅̅̅̅̅̅change in relative circulation at °= 45°. When flying at low angles of attack (cases A', B, D'), it is also necessary to take into account the redistribution of aerodynamic load caused by the engine nacelles and fuselage [1]: Approximately, the total reduction in the area of the load plot over the fuselage and nacelles can be considered equal in area to a triangle with a catheter equal to the half-span of the wing. The load redistribution is further taken into account by summing the initial plot with "dips" over the fuselage and engine nacelles with the corresponding coordinates of the constructed triangle.
Distribution of inertial loads by wing span and chord Distributed loads from the own weight of the structure in the span proportional to the chords where wing weight; the chord in this section. Similarly, the distribution of the load from the weight of the fuel is determined by the formula: where fuel weight; tank area; the chord of the tank in this section. The total distributed load is equal to: Figures 1-3 show the plots of the distributed aerodynamic load, the distributed load from the wing's own weight and fuel weight, as well as the distributed total load.  To carry out the study, the wing with the main power set was simplified in Siemens NX (Figures 4 and 5).  The engines were modeled as cylindrical bodies. At the same time, five different models were created with different engine locations in terms of wingspan ( Figure 6). Further, on the basis of these PART-models, finite element models were created ( Figure  8-9). All elements were modeled as 2D bodies. The engines were modeled as 0D elements in the geometric center of the engine models, which were connected to the main structure via 1D connections. Next, a SIM model was created ( Figure 10). To apply loads, RBE elements were created, connected to the upper and lower surfaces of the wing and oriented parallel to the flow. Aerodynamic loads were applied to these elements along their entire length, as well as loads from the mass of the wing structure and fuel. Concentrated loads from the engines were applied in concentrated masses. Restrictions were imposed as a seal at the points of connection of the glasses with the center section.
During the calculations, the bending moment values were taken at eight nodes, from which the maximum value for this calculation case was selected ( Figure 11). The values of the bending moment in the nodes, depending on the design case, are summarized in Table  1.  The values of the bending moment in the nodes, depending on the design case, are summarized in Table 1. The maximum values of bending moments for each design case at different engine positions are given in Table 2.  The resulting dependencies are plotted on one graph (Figure 12). Since the obtained dependences do not have an explicit intersection, we use the given values to find the optimal position of the engines:  It is accepted here 1 = 2 − . According to this dependence, the moment of inertia of the power plant relative to the longitudinal axis OX of the aircraft depends on the coordinate of the engine squared ( Figure  14). Since the bending moment and the moment of inertia have different dimensions, it is impossible to compare them directly. But you can compare them qualitatively by using the following criterion: where 1 = 3,5; 2 = 2 * 10 −7 Let's build this dependence using the already available values ( Figure 15): According to this dependence, the optimal position of the engine will be at a distance of = 19900 .

Research results
During the preliminary design, using the similarity parameter for the prototype aircraft, the position of the external engine = 19572 was assumed. In the course of the study, the optimal position of the engine from the point of view of bending moments was found, equal to = 20973 . At the same time, the maximum error between the received and accepted positions is ∆= 6,680%, which allows us to conclude that there is some optimal position that is in the specified interval.