Digital filtering technique in the analysis of zig-zag manoeuvre tests

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15 IMJ 1974 _Lab.

_{y. Scheepsbouwkun&}

ec nisc Ï HpgesdJ

DIGITAL 'ILTEiiIG TCHiIU 11 TML. A1i'I

Delfi

ZIG-ZA MA1OEUWRE PLTS.

y K. 1.omoto.

fi. n

PREFACE : In the analysis of zig-z.g tests it Is often

desirable to eLiminate the noise

involved

i.n the

recorded ship motion.

It is particu1arly so at, the

phase-plane analysis to ob.t.ifl the charat.erStic parameters of a ship. in question, includiflg the. non-linear element, since

in such analysis a

numericl diffeientiatiOfl of

thenieasured z-rate ta needed.

At the Zagreb meettr

in

97O the writer presented a

numerical technique for this urpose, i.e. "seveb points stnoothin

etnp3,oying 3rd. power ppl1n9ialS.

This did apparently wori well

bt later lt beBrrze

Iear that it has a risk of

-introduing high

freq.ueflcy noisè. cC. RINA paper

by C1rke and Glansdor.p, 1972.

The present note relates to a new

nwtéril filtering

teca-nique to smoothen

the measured. yaw-rate and

at the sanie time

to give ita differentiation

with m.inimuìn possible noise. 2. LOW-PA S.S FILTER : First. consider the ideal low-pass

filter whose frequency response is:

..1L

i

-J

t

amplitude

ratio

f

,'=

»

fo

Wo

t,o

-

phase

._

foi'

4,1/

The wighting function of this

filter, i.e.,

i.tS

repOflBe to an

unit impul8e input applied at. t = O, is

S/ri.,

'c-C

This is shown in Fig. 1. It ShoUld be notedthat the

response

to an impulse at t = O does

exist

even at t < This ta

Impossible in any physical

filter, in

at-her words, the

ideal low-pass filter is not "physically realizable".

In computation, however, the ideal low-pass filtering is

posstb..e. To make .a oonvolutiofl integral of

the weighting

function and any given input signa].

-iS

the answer. That is,

t

T-

r) dc

where

»t/denotes the given inut.as

a. function of time:

ad

- t)

the filtered signai.

bliotheek y

OnderafdeI . -.-- . eepsbouwkunde

niçh ,Hoaeschoo

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A practical difficulty is introduced here from th.e poor ;convergence. (or decay) of &-) , which is remedied b.y modifyiag

the ideal ¿.V) by multiplying with "Lag-window" function

'*

:tL is to be adjusted depending upon situations. According to

.the experience tL= 30 sec. is more than adequate for actual ships with time interval for computation (sampling time) of 0.5

to i sec.; even t= 10 did apparently work very welL.

Next question is what is the best choice with

&c

,. the

cut-off frequency. If u)a is too high, much iloise will remain

and if too low, quick change of ship motion may be distorted, which usually happens at rudder execution. k compromise kiss to

be done any way. We will come back to this point later.

3. DIFFREMIAL FILTER : The same principle as the low-pass filter can be used to design a numerical differentiating filter to obtain the yaw acceleration from the measured yaw-rate with minimum possible noise.

The frequency response of the ideal filter in this case is:

F)

for

'w

o

"C-,

4

DP

4//

4-,.

T.he resulting weighting function is

Z-'Z

C'OS -ct

S/'v

,Tie lag window,

±(*

is again called for to improve the convergence.

j;L1.. CiWIC IN CUT-OFF FREQUENCY ; Fig. 2. shows the effect of the cut-off frequency of

:the low-pass and differential plus low-pass' filters mentioned previous1y, upon the. phase plane tra jectôry for a yaw-rate zig-zag manoeuvre. This is not an experimental result but a computed ,one and makes it clear how the cut-off frequency d.istort.s the

t.rajectory. The, mathematical model employed is shown in the

:fjgure.

The result tells us that the cut-off frequency less than :;O.l c/s. will introduce an appreciable distortion at-the yaw

acceleration peaks, which results in. an erroneous evaluation of

and

.

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On the other bend an experimental result of an actual -tanker as shown on Fig. 3 indi.te6 still some noise with the cut-off frequency of 0.167 c/s. The ship si.z.e is much the same as the mathematical model.

The optimum cut-Off frequency depends considerablY upon the size and speed of a ship in question and should be kept for further survey, but 0.07 to 0.1 c/s. may be a reasonable cholse

for large tankers fully loaied.

Finally an emphasis should e laid. upon the adwantage of the zig-zag manoeuvres of an established limit cycle. For this kind of motion it is possible to get a nice phase-Plane portrait by averaging over a feví cycles and with a relativelY high cut-off frequency. From this point of vie& the yaw-rate zig-zag test as proposed at 13th ITTC is perhaps most suitable for this sort