The rotation of a stored cylinder body by an outer rotating structure.

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The rotation of a stored cylinder body by

an outer rotating structure.

Christopher Vestman

Space Engineering, master's level 2019

Luleå University of Technology

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Abstract

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List of Symbols

α Angular acceleration

µ Frictional coefficient

ω Angular velocity

abarrel Acceleration inside barrel

fw Warhead frequency

HEAT High-Explosive Anti-Tank

I Moment of inertia

ID Inner Diameter

Krot Rotational kinetic energy

M Momentum

r Radius from centre of mass to edge of the cylinder bodies

tacc Time of acceleration

tf ly Time of flight

v0 Muzzle velocity

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List of Figures

1.1 Initial draw-up. . . 7

2.1 Cross section of CAD Model of the layout with bearings. . . 10

2.2 Proposed geometry for bearings. (Nils Manne, personal communica-tion, December 2018) . . . 11

3.1 Side-view of warhead model. . . 12

3.2 Side-view of model without hull. . . 13

4.1 Projectile moment in barrel phase. . . 15

4.2 Warhead moment in barrel phase. . . 16

4.3 Warhead spin during barrel phase. . . 16

4.4 Warhead spin during flight phase. . . 17

4.5 Warhead frequency versus projectile frequency. . . 17

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List of Tables

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Chapter 1 Introduction

1.1 Background / SCOPE

One approach to minimise the loss of soldiers is to have them further away from the battle. However with some types of weapons soldiers will have to risk their life’s while trying to get closer to their objectives. Such a weapon might be the HEAT-grenade which is a HEAT-grenade using the shaped charge technology for penetration of thick armour and structures. A technology that focuses all of its energy in one single beam [1].

To be able to fire such grenades with shaped charge technology from a distance is somewhat a dream hoping to be realised to minimize the risks of soldiers and at the same time be more precise and increase the efficiency of the shaped charge. Additionally, the limit of frequency for a shaped charge beam in rotation is fw ≤ 15

Hz. Rotation above this frequency is not allowed since the beams efficiency will decrease above this value [2].

1.2 Earlier Studies

There are strictly limited earlier studies in this particular subject and the studies one do find are locked behind access. In the fluid dynamic field there are numerous simulations with both eccentric and concentric cylinders and their rotation. How-ever, in these cases one of the cylinders are held stationary while letting the other body rotate, usually the outer cylinder. In this project the problem is inverted.

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1.3 Formulation of the problem

There are two ways and two ways only to stabilise a projectile in flight:

• Rotation • Wings

The advantage to use a projectile stabilised by rotation is that the fire distance and accuracy is greatly improved compared with a wing stabilised projectile. The problem is that a wing-stabilised projectile using shaped charge technology can not handle the high rotation frequency due to the beam precision and reliability to penetrate the target of the shaped charge is lost [5]. Hence, Saab Dynamics are interested in studying the possibilities of using a shaped charge in a rational stabilised projectile. The thesis is done to analyse how the outer structures rotation affects an inner stationary cylinder-body using reasonable bearings for transferring momentum. The main question of the thesis is to answer how long time it takes for the inner warhead to reach the frequency limit of 15 Hz.

Figure 1.1: Initial draw-up.

Initial parameters:

Hull Warhead Mass [kg] 1,2 0,8 Radius [mm] 40 36 Frequency [Hz] 100 TBD

Table 1.1: Initial values, frequency when precisely leaving the barrel

In addition to the values from table 1.1, it is given that the arrangement is accelerating from abarrel = 0 − 100 Hz in tacc ∼ 4 ms inside of the barrel, and have

a total fly-time of tf ly = 6 s, with a yet to be defined rotation-declination. The

velocity of the projectile directly out of the muzzle is v0 = 200 m/s and declining

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1.4 Limits

Other limits to have in mind

• Efficiency is not to be lost - the grenade still have to be able to penetrate its objective and target.

• Mass - the grenade is not to loose its aerodynamic properties.

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Chapter 2 Theory

2.1 Governing Equations

Initially, the study analysed the differences between the rotations. By simplifying the rotation from 3D to 2D, leaving the transitional movement out, examining the pure rotational movement one can use the following formula for the rotational energy:

KRot =

1 2Iω

2 _(2.1)

where I is the moment of inertia and ω is the angular velocity. Further

vRot = rω (2.2)

with r being the radius to the edge of the cylinder-bodies.

From the two equations above, 2.1 and 2.2, one is able to calculate the highest rotational energy and velocity allowed for the inner cylinder, making that an upper limit.

The hull and inner cylinder have to be connected by the use of ball bearings. To calculate the frequency the inner cylinder-body will be sped up to, one have to make use of the momentum equation

M = Iα (2.3)

where the inertia I will be acquired from CAD-modelling and α, the angular ac-celeration, will have to be calculated. From there, one can calculate the frequency as f = ω 2π = α · dt = Z M (t) I · dt. (2.4) When the outer cylinder-body is spinning, a radial force will be targeted inwards toward the centre of rotation according to the centripetal force, pressuring down on the inner cylinder such that

Frad = Fcp =

mv2

r = mrω

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2.2 Bearing Selection

For bearing selection, one of the limiting factors was to maximise the diameter of the inner cylinder body. Bearings are selected from the SKF Rolling Bearings Catalogue [6]. The selected needle bearing K 72x80x20 does suit the cause due to great fitting dimensions to the inital draw-up and a very low friction of µ = 0.0045. The small diameter of the needles allows the internal body diameter to be as large as possible. The ball bearing selected was SKF 6405 as it is the only one that fits somewhat within the limitations.

Figure 2.1: Cross section of CAD Model of the layout with bearings.

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Figure 2.2: Proposed geometry for bearings. (Nils Manne, personal communication, December 2018)

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Chapter 3 Method

3.1 CAD Modelling

The modelling were done in Autodesk Fusion 360 and 3DExperience. CAD-models of ball and needle bearings are provided from the bearings respective product-page from SKFs website. The drawing is completed by assuming an ideal construc-tion, i.e. not considering how other components are attached within. The material chosen was stainless steel. With the drawing and material chosen, Fusion360 would output a moment of inertia at the centre of mass. This inertia is then used for the rest of the calculation and analysis.

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3.2 Analysis

For the theoretical analysis of the problem the calculations were done in MATLAB. The code used is to be found in the Appendix A. The code is valid given an input of typical moments of known earlier tests data, which are also found in the Appendix above. We need to keep in mind that the moments given are while the grenade is in the acceleration phase in the barrel.

From these previous values one calculates the moment of the outer body, hull, to compare with the reference values from the earlier tests. Continuing from here the analyse is divided in 2 sections. The first for the phase in which the projectile is located in the barrel and accelerates, and the second for the flying phase. In the acceleration phase, the majority of forces moments and accelerations occur.

Since the launching barrel is rifled, when the projectile accelerates it will be forced into a moment and given a rotation. This moment would mainly apply to the outer body and then transferred to the inner warhead with the use of ball bearings [7]. Theoretically the low friction coefficient will give the warhead a spin lower then the hulls rotational speed. This also mean that less momentum should be acquired directly on the inner warhead since some will be used by the bearings.

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Chapter 4 Results

Running the written code initially produces results in form of two plots on how the hull and warhead follows the reference plot of the moments. The values of the reference etc. are all included in Appendix A. This was done to verify that the moments are following each other.

Figure 4.1: Projectile moment in barrel phase.

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Figure 4.2: Warhead moment in barrel phase.

Figure 4.3: Warhead spin during barrel phase.

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during the 4 millisecond acceleration the warhead will be spun up to a rotation of roughly 0.35 Hz.

Figure 4.4: Warhead spin during flight phase.

In the flight phase the frequency is increasing linearly to just over 45 Hz.

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In figure 4.5 the warhead frequency vs projectile frequency is shown. The de-crease of the projectile is from aerodynamic properties such as drag. It was found that the retardation of the hulls rotation is around 2% per 100 m.

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Chapter 5 Discussion and Summary

The analysis propose that the inner warhead will acquire a frequency low enough such that, it most likely during the fly time will not reach the upper frequency limit. Giving an continuation for further research.

The linear increase of frequency is an unexpected result. One expects an expo-nential curve for the rotations, higher frequency as the time increases. A contradic-tion to the momentum’s shown in the plots.

The results might suggest that it is possible as it is. However, since we based on calculations without bearings from [7], when we take that into mind the results will be skewed in a positive way. As in even lower frequency. While the results are positive and seems to agree with solutions in [3], with the limitations of the vague earlier research and the single reference data set there is a hard time to verify the results mainly due to the secrecy and gap of knowledge. However even though there is only a single set of reference data, they are valid since they are used in real simulations as well.

Generic bearings are selected from a current product line [6] instead of having proper bearings produced for this application. Also applicable to how the projectile is to be constructed. The generic bearings might be both impractical and are made to have a long lifetime. Custom made bearings might hold back the construction cost with its fraction of lifetime.

What was sought after was how long it would take for the warhead to reach a frequency of 15 Hz. From figure 4.4 one can see that it takes approximately 2 seconds for the warhead to reach said limit. However the frequency of the warhead is increasing faster than the retardation of the projectile as seen in figure 4.5. This seems very illogical due to it is a contradiction since the projectiles inner rotation is based of the projectiles outer rotation.

Comparing figures 4.2 and 4.1 one can see that the moment in the warhead is around 200 times lower than the outer hulls momentum. The spin acquired from that momentum only reaches 0.35 Hz from 4.3, which serve as a reasonable base frequency for the rest of the analysis with the assumption of majority of forces and accelerations occurs in the barrel-phase stated in section 3.2.

One major aspect to take into account is that while the projectile is in the barrel there will be a load onto all of the ball bearings. As the projectile leaves the barrel the loads will be no more. If there is no load onto the bearings there will not be any transfer of momentum onto the warhead and the frequency should be lower.

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Appendix A

Code

clear;close all;clc

%% Draft1 v7 Output f r n Fusion360 % Area 1.240E+05 mmˆ2

% Density 0.008 g / mmˆ3 % Mass 3168.406 g

% Volume 3.961E+05 mmˆ3

% Physical Material Stainless Steel

%% Frictional Moment - SKF Lager 6405

M rr = 226.5; %Nmm M sl = 582.2; %Nmm M tot = 808.7; %Nmm M start = 1746.6; %Nmm %% Exempel p momentkurva t ex = [0.000000 0.141945 0.482614 0.794894 0.908450 1.022006... 1.135562 1.220730 1.277508 1.362675 1.760122 2.214347... 2.469848 2.867295 3.520243 3.633799 3.690577]*1e-3; %ms M ex = [0.000000 30.435872 210.992297 480.229089 524.028012... 541.877760 553.319939 563.294823 567.407182 554.854307... 427.943293 323.131505 280.073107 249.088956 229.022165... 28.062378 26.468634]; %Nm % T r g h e t f r HELA MODELLEN

% Endast moment runt z axel som r intressant? OBS I X I MODELL

I Proj Tot = [2.443E+06 -1.962E-04 -2.215E-09; -1.962E-04 3.363E+06 -1.817E-09; -2.215E-09 -1.817E-09 3.363E+06].*10e-9; % [g mmˆ2] COM

I Projectile = I Proj Tot(1,1);

for i=1:length(M ex)

Alfa Proj = M ex ./ I Projectile; Omega Proj = Alfa Proj.*t ex;

end %% Inner WARHEAD % Area 2.766E+04 mmˆ2 % Density 0.008 g / mmˆ3 % Mass 2655.657 g % Volume 3.320E+05 mmˆ3

% Physical Material Stainless Steel

% Moment of Inertia at Center of Mass (g mmˆ2)

I Warhead Tot = [ 1.674E+06 -1.965E-04 0.00; -1.965E-04 2.378E+06 -7.276E-10; 0.00 -7.276E-10 2.378E+06].*10e-9;

I Warhead = I Warhead Tot(1,1);

M Warhead Out = I Warhead.* Alfa Proj;

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Poly launch = polyfit(t ex ,M ex ,6); time barrel = 0:0.0001:0.0037;

M hull launch poly = polyval(Poly launch,time barrel);

%% E l d r r s f a s

Friction = 0.0045; % SKF Lagerfriktion f r n l l a g e r .

M Warhead In = M Warhead Out * Friction; Alfa Hull = M hull launch poly / I Projectile; M Warhead Launch = Alfa Hull * Friction * I Warhead; Alfa Warhead Launch = M Warhead Launch ./ I Warhead;

timestep launch = 0.0001; Omega Warhead Launch(1) = 0; Frequency Warhead Launch(1) = 0;

for i launch = 2:length(Alfa Warhead Launch)

Omega Warhead Launch(i launch) = Omega Warhead Launch(i launch-1) + timestep launch * (Alfa Warhead Launch(i launch-1) +

Alfa Warhead Launch(i launch))/2;

end

Frequency Warhead Launch = Omega Warhead Launch / (2*pi);

%% Flygfas

timestep fly = 0.1;

time fly = 0:timestep fly:6; % 6s flygtid

Omega Warhead Fly(1) = Omega Warhead Launch(length(Omega Warhead Launch)); Alfa Warhead Fly = (M tot*1e-3)/I Warhead;

for i fly = 2:length(time fly)

Omega Warhead Fly(i fly) = Omega Warhead Fly(i fly-1) + timestep fly * Alfa Warhead Fly;

end

Frequency Warhead Fly = Omega Warhead Fly / (2*pi);

%% Plots

figure(1)

plot(t ex ,M Warhead Out,t ex ,M ex) legend(' M {Projectile}',' M {Reference}') ylabel('Momentum [Nm]')

xlabel('Time [s]')

figure(2)

plot(t ex ,M Warhead In) legend(' M {Warhead}') ylabel('Momentum [Nm]') xlabel('Time [s]')

figure(3)

plot(time barrel,Frequency Warhead Launch) legend('Warhead')

ylabel('Frequency [Hz]') xlabel('Time (s)')

figure(4)

plot(time fly,Frequency Warhead Fly) legend('Warhead')

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Appendix B

Concept for construction

There was one concept of design implementated for when not using ball bearings. It is proposed to make use of epicyclic (also known as planetary) gears. There are many pros such as a high torque capabilities and efficiency and clear ratios. Problem is that the gearteeth cannot handle the high instantanous loads from the acceleration phase together with an complex design.

Figure B.1: Design of a epicyclic gear.

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Bibliography

[1] Dennis Baum. “Shaped charges pierce the toughest targets”. In: Science & Technology Review 6 (1998), pp. 17–19. doi: https://str.llnl.gov/str/ pdfs/06_98.3.pdf.

[2] Andrew Medin Gunnar Medin. “Rotationens inverkan p˚a RSV-laddningar med svarvade genomslagskroppar”. In: FOA Rapport (1975).

[3] Rheinmetall GmbH. Handbook on Weaponry. First English Edition. 1982. Chap. 11.2.3.3.2, p. 524.

[4] R. Ogorkiewicz. Tanks: 100 years of evolution. Bloomsbury Publishing, 2015. isbn: 9781472813053. url: https://books.google.se/books?id=ZICXCwAAQBAJ. [5] Kurt Andersson et al. L¨arobok i Milit¨arteknik, vol. 4: Verkan och skydd. 2009.

isbn: 978-91-89683-08-2.

[6] SKF. SKF Rolling Bearing Catalogue June 2018.

[7] H.S. Lipinski et al. Spin-stabilized projectile with non-rotating shaped charge. Patent number - 2,981,188. Apr. 1961.

[8] Milan JOZEFEK Peter LIPT ´AK. MOMENTS HAVING EFFECT ON A FLY-ING MISSILE.

The rotation of a stored cylinder body by an outer rotating structure.