STABILITY AND CONTROLLABILITY

MANOEUVRABILITY OPTIMISATION FOR THE NAVANTIA S-80 SUBMARINE PROGRAMME

2. STABILITY AND CONTROLLABILITY

LINEAR MODEL

The mathematical model of the submarine can be established following the Newton 2nd law and the kinetic moment theorem:

, . _G

ext M a

¦

F

, H N

d

Fext G

dt

Assuming a three axis orthogonal reference system moving along with the submarine where the origin is arbitrarily taken, the expression for the absolute

Warship 2008: Naval Submarines 9, Glasgow, UK

©2008: The Royal Institution of Naval Architects acceleration of a point of the submarine taken as rigid

solid, for instance the centre of gravity is as follows:

OG

O OG O

G

a ȍ V Į r ȍ ȍ r

a š š š š

Where:

^p,q,r

ȍ , is the instantaneous angular speed.

^p,q,r

Į , is the instantaneous angular acceleration.

^u,v,w

a _O , is the drag acceleration

^{u, v, w}

VO , is the drag speed

In the Linear approximation the expression can be reduced to:

O O

G a ȍ V

a š

The expression for the Kinetic moment is:

^ `

^I

^ `

^ȍ

HG

Where

^ ` ^I

is the inertia matrix and

^ ` ^ȍ

is the instantaneous angular speed vector

Finally using the simplified equation for the absolute acceleration, deriving the before written expression and avoiding second order inertia products the equations can be written as follows:

ȋ u

m ¦ (1)

^{v ru}

^Y

m ¦ (2)

^w-qu

^Z

m ¦ (3)

K p

I_xx ¦ (4)

M q

I_yy ¦ (5)

N r

I_zz ¦ (6)

This system of equations is the submarine linear equations of motion in movable axes.

The right term of the equations are the resultant of the total forces and torques applied on the submarine in all the three components of the reference axes, including:

Hydrodynamic forces and torques.

Propulsion Forces, (first equation).

Forces and torques due to control actions.

The horizontal plane equations are (2) and (6), and that for the vertical plane (3) and (5).

To analyze the equations it is necessary to properly describe the forces and moments in the second term. The most frequent approximation is based on the dimensional analysis where the inertia forces appear as proportional to the square of a characteristic speed (lift forces, drags, etc). The known approach used in the cases where the forces are basically defined as orthogonal to the characteristic speed, (the advance speed) is called that of the hydrodynamic derivatives.

Writing the equations in non-dimensional form and deleting the control terms, (terms with “G”), because the aim is to analyse the submarine natural (non-forced) response for the stability study, the result is:

Horizontal Motion:

x

^m'-Y'v

^v'-Y'v^v'

^m'-Y'r

^{r' 0} (7)

x

^I'zz^-N'r

^r'-N'v^v'-N'r^{r' 0} (8)

Vertical Motion:

x

^{m'- Z'}w

^-Z'w^w'

^{m' Z'}q

^{q' 0} (9)

x

⁰

u șg q' GB M' - w' M' - ' q M' -

I'_{yy q w q 2} (10)

Where:

Z´_i, Y´_i, M´_i, N´_i, etc. are the hydrodynamic derivatives of vertical and transverse forces, and the torques around transverse and vertical axes and in non dimensional form.

Solving both systems for instance, by using the method of D operator, the linear differential equations in constant coefficients are obtained:

Horizontal 0 cv v b v a

Characteristic equation:

0 bı c aı² Where:

x

a

I'zzN'r

m'Y'v

x

b

I'zzN'r

Y'vN'r

m'Y'v

x

^{c N'}_r^Y'_v

^m'-Y'_r

^N'_v

Warship 2008: Naval Submarines 9, Glasgow, UK

©2008: The Royal Institution of Naval Architects Vertical

0 D q C q B q

A Characteristic equation:

0 Cı D Bı

Aı^{3 2}

Where:

x

A

m'Z'w

I'yyM'q

x

B

>

Z'w

I'yM'q

M'q

m'Z'w

@

x

_«¬^ª

^GBg_»¼^º

u Z' m' m' Z' m' M' M' Z'

C _{w q w q w 2}

x

w ₂

u g m'GB Z'

D

As can be easily deduced the solution of the characteristic equation in the vertical plane depends of the submarine speed, due to the hydrostatic torque term.

2.1 STABILITY CRITERIA Horizontal plane

To obtain an inherently stable submarine in the horizontal plane, the solutions of the characteristic equation must be real and negative, for that the Routh- Hurwitz criteria (ref. [2]) establish:

c > 0 Vertical Plane

In this case the Routh-Hurwitz criteria are as follows:

B > 0 C > 0 BC > AD

Where A, B, C and D have been previously defined.

The characteristic roots are of two types:

Real negative at high speed.

Complex roots with negative real part at low speed.

At low speed the complex coefficient roots implies oscillatory motions with dampened amplitude, so a new coefficient is defined, the damping. Whose typical values are between 0.7- 1 or 0.6-0.8 depending on the authors.

2.2 STABILITY AND CONTROLLABILITY

INDEXES

Apart from the mathematical stability study, the best way to analyse the design decisions using the linear equations of motion is through the use of a set stability and controllability indices, a set of these are for instance those in the table 1 coming from ref (2).

The controllability indices compares the control forces and torques with the corresponding hydrodynamic forces and torques, so they are directly related to the response accelerations.

Also in the table are indicated the main influence of the controllability parameters, useful for addressing the design decisions.

2.3 DESIGN DECISSIONS

The early design decisions when sizing and arranging the submarine control surfaces of a submarine are of vital importance especially for saving time and cost especially during the validation study which starts necessarily with the corresponding towing tank manoeuvrability test and simulations.

In the case of the S80 the decisions were:

-Rudders and aft planes in cross configuration -Fwd planes in the fin

-Sufficient inherent stability which could guarantee successful depth keeping and course keeping with minimum actuation of planes, and so less noise generated, but in such way that the submarine responses will not be too stiff.

-Critical speed enough below the patrol speed -Robust Horizontal control movements

-Moderate vertical control movements, especially to avoid high depth excursions in case of an aft planes jam.

In table 1 the typical values for a conventional submarine and the design decisions related to them for the S80 are shown.

Denominat

ion Definition

Typical SSK figures

S-80 Design Decisio

ns

Commen t [Ref (2)]

Horizontal Index

N'r Y'v

N'v Y'r 1 m'

GH

< 0,1 > 0,3

Near to upper

limit Vertical

Index

Z'w M'q

Z'q wm' 1 M'

Gv

< 0,3 > 0,3

Above upper

limit Heave

coefficient

^{Z' m'}

10 L

Z'

3 w įs

~ 2,5 > 3

Nearer upper

limit Pitch

coefficient 3

^{q 'yy}

įs

I M' 10

L M'

~ 0,3 > 0,4

Above upper

limit Heave

coefficient

^{Z' m'}

10 L

Z'

3 w įb

~ 3 > 3

Typical increme

nt Pitch

Coefficient 3

^{q 'yy}

įb

I 10 M'

L M'

> - 0,2 > - 0,2 Intermed iate

Sway

coefficient 3

v

įr

Y' m' 10

L Y'

~ 4 > 4

Near to upper

limit Yaw

coefficient 3

r zz

įr

I' N' 10

L N'

~ 0,5 > 0,5

Nearer upper

limit Table 1. Comments for S80 figures from [Ref (2)].

Warship 2008: Naval Submarines 9, Glasgow, UK

©2008: The Royal Institution of Naval Architects X rudders configuration was not selected due the fact that

to this configuration is based on symmetry, but the final arrangement on the ship normally can not be completely symmetrical so added complexity in control to a more difficult arrangement of the submarine stern are expected, on the other hand it is considered the cross configuration as more intuitive from the point of view of the control surfaces movement and design.

Regarding the forward planes in the fin, the arrangement of the submarine and the inherent disadvantages of the forward planes located in the forward body, firstly gear and hydrodynamic noise which could produce interferences with the sonar, and vortices generation which could disturb the efficiency of the aft planes, or if shifted up or down from the horizontal centre waterline, the hydrodynamic radiated noise increases.

2.4 VALIDATION PROCESS

It is clear that for a successful strategy for the control surfaces design it is necessary the utilization of a linear and not linear mathematical model of the submarine motion. In such way the design decisions are checked against the corresponding manoeuvres simulations in real time.

For that purpose Navantia developed the “SIMUSUB”

code for the dynamics of the submarine, the code can work in either real time or in calculation time for speeding up the analysis. The code is briefly described in later on in this paper.

The linear approach to the motion equations is used for stability analysis and simulations with the automatic pilot model, and the non linear approach for more complex simulations like aft planes jam or flood recovery trajectories.

The “SIMUSUB” code was previously validated by using the results obtained in SSPA Towing Tank for test and simulations carried out for a Navantia submarine design called P650.

Finally the studies and submarine trajectory simulations carried out by QinetiQ on the Safety Operating Envelope (SOE) complete the validation process followed.

Figure 1 shows an example of Pitch Limited SOE S-80 diagrams (>20 degree Stern Hydroplane Angle Jam).

Figure 1 2.5 SIMULATED MANOEUVRES

The manoeuvrability qualities of main interest for submarine operations are:

Course keeping and depth keeping with minimum rudders and planes work.

Turning ability.

Fast Course and depth changes capability.

Depth keeping ability with minimum diving planes work

These qualities can be studied after the dimensioning process by means of the simulation of a series of standard manoeuvres representative of the manoeuvrability qualities, this simulation study made possible by the use of the “SIMUSUB” code has demonstrated to be a powerful tool to check the planes design against the requirements.

Warship 2008: Naval Submarines 9, Glasgow, UK

©2008: The Royal Institution of Naval Architects Figure 2: Vertical Meander, 10 knots.

Figure 3: Horizontal to 10º/10º zig-zag, 10 knots

Figure 4: Turning circle. 15 knots (plant view)

Figure 5: Turning circle. 15 knots (front view)

Figure 6: Vertical 10º/10º zig-zag, 15 knots

Figure 7: Depth change. 15 knots

Warship 2008: Naval Submarines 9, Glasgow, UK

©2008: The Royal Institution of Naval Architects 3. “SIMUSUB” CODE

The “SIMUSUB” code was designed in order to analyse qualities of primary interest for operation submarines in the coupled horizontal and vertical planes:

The ability to hold a straight course with minimum amount of rudder activity

The ability to turn tightly

The ability to initiate and change course or change depth quickly

So, in order to answer these questions, the simulator must have the possibility to analyse in a programmed way different standard manoeuvres in the horizontal and vertical plane such as zig-zag tests, turning circles, meander tests, stopping tests depth and heading changes, overweight’s reaction, etc… In a second phase of development of the code, and taking into account the importance of the emergency reaction against rudder jamming or uncontrolled flooding, the code was implemented to simulate the auto-pilot reaction including the normal and emergency blowing of its ballast tanks.

The general non-linear DTNSRDC equations of motion were implemented as the mathematical representation of the six degrees of freedom for the physical S-80 Class behaviour. In the final part of the article, figures related with the structure and presentations of this code are shown.

Figure 10 shows the algorithm block diagram to integrate the named non linear equation.

Figure 11 shows the Autopilot control algorithm block diagram, and some details about the main block are given in the subsequent paragraphs.

Figure 13 shows the algorithm block diagram for blowing and some detail about physical assumption for the non-linear-related integration.

Figure 14 shows a view of the interface running in a PC- station of the current “SIMUSUB” code.