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Research DETERMINATION OF SUITABLE OPENING ANGLE OF HYDRAULIC SUPPORT LEGS FOR DESIGNING M46-130mm SELF-PROPELLED ARTILLERY Pham Quoc Hoang, Phan Hoang Cuong*, Phung Tuan Anh, Vuong Tien Trung Abstract: This paper determines a suitable opening angle of the hydraulic support legs of M46-130 mm self-propelled artillery. It is a combination of M46-130 mm field gun, mounted on KrAZ-255B 6x6 military truck chassis. Based on results of investigating the changes of forces and moments acting on artillery system in static condition when changing the opening angle of hydraulic support legs at critical fire angles, a suitable angle is selected at ψ = 400 and γ = 200. From that angle, this paper shows a table consists of geometric parameters, forces and moments acting on system. The results of this research are the input parameters to calculate, design and optimize structures of self-propelled artillery. Keywords: M46-130 mm self-propelled artillery; Hydraulic support legs; Opening angle of hydraulic support legs; Static condition 1. INTRODUCTION For medium and heavy artillery with long barrel like M46-130mm, 152mm-52 when mounted on a motorized wheeled or tracked chassis then the general structure of self-propelled artillery such as Caesar, Archer, Atmos 2000 artillery utilizes two hydraulic support legs at the rear back as shown in figure 1. (a) (b) Figure 1. Archer (a) and Atmos 2000 (b) self-propelled howitzer [1]. During the process of firing, the middle and rear axles are clear of the ground by two hydraulic support legs to release the suspension system of the vehicle. Therefore, hydraulic support legs receive almost the whole firing impulse force and the weight of artillery system. The opening angle of the hydraulic support legs directly effects on static balance of artillery system, stroke length of pistoncylinder, length of hydraulic support legs, and forces and moments in hydraulic support legs. Previous research is mainly concentrated on general information of theory and design of gun and ammunition [2-8]. [9] investigates the kinetic of guns which integrates on military trucks. In addition, [10] shows the basic theory of military engineering vehicle. However, the analysis related to opening angle of the hydraulic support legs of self-propelled artillery has not been investigated. It is needed to find out of the opening angle of hydraulic support legs in order to determine the input parameters in design stage of M46-130mm self-propelled artillery. This paper presents steps to examine the changes of forces and moments Journal of Military Science and Technology, Special Issue, No.66A, 5 - 2020 117 Mechanics & Mechanical engineering depending on the opening angle of hydraulic support legs to determine suitable values to meet given requirements when applying maximum recoil resistance force on system in the static equilibrium condition. In section 2 a model of M46-130mm self-propelled artillery is presented to establish static equilibrium equations. Based on some initial conditions, investigating the changes of forces and moments depends on the opening angle of hydraulic support legs and discussing the results in section 3. Finally, this paper is concluded in section 4. 2. MECHANICAL MODEL OF THE SYSTEM In design stage, it is necessary to determine a suitable value of opening angle of hydraulic support legs to meet the following requirements: - The artillery system must to satisfy static stable conditions in firing period. - The compatibility between axial forces and bending moments in hydraulic support legs so that the maximum stress does not exceed the allowable value. - The dimensions and weight of artillery system are within allowable limits. 2.1. Assumptions - Rotary moment of projectile when moving in bore is ignored. - Flexural rigidity of chassis, artillery structures, hydraulic support legs and ground are infinite. In addition, there is no gap between joints and assemblies. - Artillery system is placed on a horizontal plane. 2.2. Mechanical model Mechanical model of M46-130mm self-propelled artillery is shown in figure 2. Figure 2. Model of M46-130mm self-propelled artillery. 118 P. Q. Hoang, …, V. T. Trung, “Determination of suitable … self-propelled artillery.” Research Geometric characteristics and forces, moments acting on artillery system in the static condition are listed in table 1 and table 2. Table 1. Geometric characteristics of M46-130mm self-propelled artillery. Notations Descriptions C Center of gravity of artillery system Center of gravity of the recoiling parts before firing and at C1 , C1' any instant A, A ' Center of contact of left and right front wheels and ground Center of contact area between first and second spades D, D ' hydraulic support leg (left and right hydraulic support leg) and ground Angle of elevation (angle between the line through the axis of  the bore and XOY plane) Angle of traverse (angle between the line through the axis of  the bore and XOZ plane) Angle between the projection of the hydraulic support legs on  the YOZ plane and OZ axis Angle between the projection of the hydraulic support legs on  the XOZ plane and OX axis Angle between the projection of the hydraulic support legs on  the XOY plane and OX axis Angle between the projection of lines connect center of gravity of recoiling parts to spades on the XOY plane and the 1, 2 axis of the bore Distance from the center of gravity of the recoiling parts to e the axis of the bore Displacement from beginning of counter recoil of recoiling X parts with respect to guides (C1C1’) Distance from the midpoint of the center of two spades to l front axle Horizontal distance from AA’ to C l1 Horizontal distance from C to the center of gravity of recoiling l2 parts Distance from the midpoint of the center of two spades to the l03 center of gravity of recoiling parts before firing and at any l3 = l03 - X coscos instant h01 Vertical distance from ground to the center of gravity of h = h01 – X recoiling parts before firing and at any instant sincos h1 Vertical distance from ground to KK’ a, b and c Horizontal distance from A to A’, K to K’ and D to D’ Distance between the projection of the center of gravity of y recoiling parts on the XOY plane and OX axis before firing Journal of Military Science and Technology, Special Issue, No.66A, 5 - 2020 119 Mechanics & Mechanical engineering Notations F  ( Fx , Fy , Fz ) RA , RA' P mr R1  ( R1x ; R1 y ; R1z ) R2  ( R2 x ; R2 y ; R2 z ) Plg Table 2. Forces acting on artillery system. Descriptions Recoil resistance force Normal reactions of ground on left and right front wheel at A and A’ Weight of artillery system Mass of the recoiling parts Component reactions on spade of hydraulic support leg 1 (D) Component reactions on spade of hydraulic support leg 2 (D’) Total pressure on breech The relation between  2 , 1 and other geometric parameters Figure 3. Figure used to determine  2 , 1 . From figure 3, we have the geometric relation to determine  2 , 1 .  1   2      2  1   (1) c   y  X cos  sin   2 ) 1  arctan( l3   c   y  X cos  sin    arctan( 2 )  2 l3  (2) with 1 ,  2 is calculated by 120 P. Q. Hoang, …, V. T. Trung, “Determination of suitable … self-propelled artillery.” Research where c  b  2h1 tan  There are the following relations among geometrical parameters ψ, δ, γ, θ and ε tan     arctan( tan )    arctan(tan  .cos )  sin    arctan( ) tan   The length of each hydraulic support leg is calculated as: h tan  L 1 sin  (3) (4) Figure 4. Geometric relationship of ψ, δ, γ, θ and ε. 2.3. Equations of static equilibrium During the process of firing, the static stable condition should be satisfied the following conditions:  RA  0   RA '  0  0 0  R1z  0 (   25 ) or R2 z  0 (   25 ) (5) Suppose the forces have directions as shown in figure 2. According to [2-9] the static equilibrium equations of forces and moments when applying the maximum value of recoil resistance force on artillery system can be expressed as Journal of Military Science and Technology, Special Issue, No.66A, 5 - 2020 121 Mechanics & Mechanical engineering  X Fx  R1x  R2 x  0   Y R1 y  R2 y  Fy  0  Z R  R  R  R  P  F  0 A A' 1z 2z z   M DD ' [P (l  l1 )  mr gX cos  cos ]  Fz l3  Fx h  ( RA  RA ' )l  Plg e cos   0  c c a   M D (YOZ ) Fy h  ( P  mr gX cos  sin  )  Plg e sin   RA ' (  ) (6)  2 2 2  c a c  RA (  )  R2 z c  Fz (  X cos  sin   y )  0 2 2 2   c  M C ( XOY ) ( R2 y  R1 y )l3  R1 x (  y  X cos  sin  ) 2  c   R2 x ( 2  y  X cos  sin  )  Plg e sin   0 where Fx  F .cos .cos , Fy  F .cos .sin , Fz  F .sin  are the components of 1 the recoil resistance force in x, y and z directions. There are 6 equations but 8 unknown quantities, thus it is needed to establish requiring additional conditions. According to [3] the relation between  2 , 1 and R1 y , R2 y is described as R1 y R2 y  sin 2 sin 1 (7) Forces RA , RA ' depend on β (if β=0 then RA  RA ' , if β≠0 then RA  RA ' ); α; X; a and c; l and the center of gravity of artillery system. Among the factors mentioned above, β is the most influent on RA , RA ' so that this is considered, others are ignored. RA   R A ' ;   (0;1] (8)   1 if   0    0 if   1 Equations (6)-(8) allow determining 8 unknown quantities. Axial forces and bending moments on planes KND and KMD of hydraulic support legs:  Ni  Riz cos  Rix sin cos  Riy sin   M iKND  ( Rix cos cos  Riy sin cos   Riz sin  ) L (i  1  2)   M iKMD  ( Riy cos   Rix sin sin   Riz cos sin  ) L (9) 3. DETERMINATION OF THE SUITABLE OPENING ANGLES According to [9], it is clear that at critical fire angles   00 ,   250 the system is most unstable and at   450 ,   250 the recoil resistance force is 122 P. Q. Hoang, …, V. T. Trung, “Determination of suitable … self-propelled artillery.” Research maximum, therefore. Due to the symmetry, considering only   00 ,   250 and   450 ,   250 . Theoretically, the values of angles  ,  can be assigned as:  (00    900 )  0 0  (0    90 ) (10) However, in fact the suitable opening angles  ,  is constrained by the allowable dimensions and total weight of artillery system, constraint forces and the position of hydraulic support legs connect with frame of KrAZ-255B. In preliminary stage,  ,  is determined at range of   200  600 ,   100  400 . The problem is examined by the Matlab software with inputs are shown in Tab.3. Table 3. The input parameters. No. Notations Values Units No. Notations Values Units 0 o  1 X (α=0 ) 1,25 m 8 20÷60 0 o  2 X (α=45 ) 0,775 m 9 10÷40 3 h01 2,785 m 10 P 240345 N 4 h1 1,719 m 11 Plg 4400000 N 5 e 0,015 m 12 mr 3029 kg o  6 25 13 (Fmax)α=0 260000 N o  7 0; 45 14 (Fmax)α=45 350000 N 3.1. Results a. Firing at angles   00 ,   250 (a) (a’) (b) (b’) Journal of Military Science and Technology, Special Issue, No.66A, 5 - 2020 123 Mechanics & Mechanical engineering (c) (c’) (d) (d’) (e) (e’) Figure 5. Results of firing at angles   00 ,   250 . b. Firing at angles   450 ,   250 (a) 124 (a’) P. Q. Hoang, …, V. T. Trung, “Determination of suitable … self-propelled artillery.” Research (b) (b’) (c) (c’) (d) (d’) (e) (e’) Figure 6. Results of firing at angles   450 ,   250 . Journal of Military Science and Technology, Special Issue, No.66A, 5 - 2020 125 Mechanics & Mechanical engineering 3.2. Discussions As can be seen from the figure 5 and figure 6 that: At firing angles   00 ,   250 , axial forces and bending moments of the right hydraulic support leg are larger than the left one. This is because of  2 is smaller than 1 so that forces and moments on the right leg is larger than the left leg. Conversely, if   00 ,   250 , axial forces and bending moments of the left hydraulic support leg are larger than the right one. - When  ,  increase then RA, RA’ in figure 5a, figure 5a’, figure 6a, figure 6a’ and R1z in figure 5b’, figure 6b’ also increase, hence the artillery system is more static stability. This is interpreted that when  ,  increase then the distance from DD’ to C and the distance between points D and D’ also increase, that results in RA, RA’ and R1z increase. - When  increases then axial force of hydraulic support legs decreases in figure 5c, figure 5c’, figure 6c and figure 6c’. This is illustrated by the fact that when  increases then the distance from DD’ to AA’ increases, thus RA, RA’ increase but the axial force in hydraulic support legs decrease. - When  increases then axial force of hydraulic support legs also increases in figure 5c, figure 5c’, figure 6c, and figure 6c’. - When  ,  increase then bending moments on planes KND and KMD decrease and get the minimum values in the vicinity of   400 ,   200 as shown in figure 5d, figure 5d’, figure 5e, figure 5e’, figure 6d, figure 6d’, figure 6e, and figure 6e’. From the analysis above, there are many opening angles of hydraulic support legs meet the given requirements, but the opening angle of hydraulic support legs is chosen at   200 ,  400 . This value is approximately with the values of some self-propelled howitzers such as Archer, Atmos 2000. The input parameters of self-propelled howitzer M46-130mm at   200 ,  400 is given in table 4. Table 4. Results at   200 ,  400 . No. Notations   00 1 R1z 2   450 Units 9026 119840 N R2 z 187510 260870 N 3 RA 12516 30606 N 4 RA ' 31291 76514 N 5 N1 99478 156040 N 6 N2 305240 351230 N 126 Values P. Q. Hoang, …, V. T. Trung, “Determination of suitable … self-propelled artillery.”