This page contains a Flash digital edition of a book.
Trans RINA, Vol 152, Part A2, Intl J Maritime Eng, Apr-Jun 2010 and [C2] =


⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥


0


00 00


0 0


ur w vw uw uv vp w vp vq


0


v 0


wq wp uq qr rp wp pqq wr vp r pr r


pq δs uq ur vr δδ δr q r s


⎢⎥ ⎣⎦


00 0 . 00 0 0


cs s r s


θφ δ δ θ


0 3.2 (c) Third Mathematical Model


The third model considered retains only strictly linear terms. Thus,


[B3] =


⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥


00 0 0


ZZ Z KK K


00 0 00 0


⎣⎦ (20)


00 0 00 0


M and [C3] =


⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥


00 0 0 0 00 00 0 0 0 0 00 0 0 0 0 00 0 0 0 00 0 0 δδ δ


w vp w v q


rs rs r q


δ


⎢⎥ ⎣⎦


00 0 . 00 0 0


cs s r s


θφ δ δ θ


0


Other mathematical models such as those given in [14, 15] can also be cast in this form, requiring additional HDCs to be evaluated / estimated. These models have retained many additional (cross-coupled and non-linear) HDC’s, in addition to those obtained strictly by Taylor series expansions, based on the best-fit for measured forces in model testing.


4. TRAJECTORY SIMULATION 4.1 ALGORITHM FOR SIMULATION


The dynamic equations summarised in equation (7) can now be recast as:


{} [


uv w p qr A F F Adiag B C


     =+{ T ] − 1 } =+FI [] {{ [].[ ] { }}


{{ EI }} −1


(22)


0 r


NN N 0


ps r wq s vr r


0 δδ Gz W 0


δ δ


0 MM zGW 0


− 0


vr r wq s 00 0


YY Y 0


0 0


δ δ


0 0


0 0 0


0 0 0


0 (19)


The following transformation relations exist between the velocities in the body system oxyz and the inertial system OxOyOzO:


02 3 () (+ − 1 3 1 2 3 ) zsu=− +2 12 12


x c c u c s s sc v s s c sc w yc s u c c s s s s c v + c c w


  


φ θ ψ


 


02 3 () ( 1 3 1 2 3 ) =+ + 1 2 3 v s c c s s w


=+− 1 3 + 1 2 3 1 3


0


=+ + =−


(/ ) 12 2 (/ ) 12 12


sin ; sin ;


where


ss sin ;== s3 = sin cc sin ;== c3 = sin


φ θψ φ θψ


Equations (22, 23, 24) are (25) thus 12 ordinary linear


differential equation whose integration in time will determine the values of vehicle position XO, YO, ZO and orientation φ, θ, ψ with time. There are numerous integration schemes for integration of such ordinary linear differential equations, starting from simplest 1st order explicit schemes to high order implicit schemes such as Runge-Kutta


schemes, Predictor-corrector


schemes, etc. These (dynamic motion) equations are usually very robust and do not lead to numerical instability due to integration scheme if the time step size is sufficiently small.


(21)


By substituting the HDC values in the above equations of motion, the equations of motion can thus be integrated for known control inputs and speed, to calculate the accelerations, velocities and displacements (position) of the body as function of time. Thus, the trajectory of an underwater vessel can be simulated [5, 13].


4.2 TYPICAL RESULTS


A flexible trajectory simulation program was created, with scope for modifying the external force matrices [B] and [C], thus enabling the mathematical model to be readily changed.


Results of a 15/5 zigzag manoeuvre at 5 knots for the axisymmetric body and for SUBOFF geometry, both using the linear mathematical model (3.2(c) above) are shown in Figures 3 and 4. For the trajectory simulations shown in Figures 3 and 4, the main parameters of interest are shown in Table 1.


To examine the effect of varying the mathematical model,


the same manoeuvre was simulated using


different mathematical models (described in sub-section 3.2 above). Results are plotted for two of the models in Figure 5 (for axisymmetric body) and the parameter values obtained for all three mathematical models are given in Table 2.


=+ c( / )r


11 12 12


c


p s s cq c s cr cq sr sc q


( 1 2 2 ) / (24) +


+ +


(23)


©2010: The Royal Institution of Naval Architects


A - 75


Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |  Page 8  |  Page 9  |  Page 10  |  Page 11  |  Page 12  |  Page 13  |  Page 14  |  Page 15  |  Page 16  |  Page 17  |  Page 18  |  Page 19  |  Page 20  |  Page 21  |  Page 22  |  Page 23  |  Page 24  |  Page 25  |  Page 26  |  Page 27  |  Page 28  |  Page 29  |  Page 30  |  Page 31  |  Page 32  |  Page 33  |  Page 34  |  Page 35  |  Page 36  |  Page 37  |  Page 38  |  Page 39  |  Page 40  |  Page 41  |  Page 42  |  Page 43  |  Page 44  |  Page 45  |  Page 46  |  Page 47  |  Page 48  |  Page 49  |  Page 50  |  Page 51  |  Page 52  |  Page 53  |  Page 54  |  Page 55  |  Page 56  |  Page 57  |  Page 58  |  Page 59  |  Page 60  |  Page 61  |  Page 62  |  Page 63  |  Page 64