Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

Sustainable Marine Structures

http://ojs.nassg.org/index.php/sms/index

ARTICLE
Non-linearity Analysis of Ship Roll Gyro-stabilizer Control System
Sathit P.1*　Chatchapol C.2　Phansak I.3
1. Department of Maritime Engineering, Faculty of International Maritime Studies, Kasetsart University, Chonburi, 20230,
Thailand
2. Department of Mechanical Engineering, Faculty of Engineering, Kasetsart University, Bangkok, 10900, Thailand
3. Department of Nautical Science and Maritime Logistics, Faculty of International Maritime Studies, Kasetsart University,
Chonburi, 20230, Thailand

ARTICLE INFO ABSTRACT

Article history A gyro-stabilizer is the interesting system that it can apply to marine vessels
Received: 19 January 2021 for diminishes roll motion. Today it has potentially light weight with no hy-
drodynamics drag and effective at zero forward speed. The twin-gyroscope
Accepted: 24 May 2021 was chosen. Almost, the modelling for designing the system use linear
Published Online: 30 May 2021 model that it might not comprehensive mission requirement such as high
sea condition. The non-linearity analysis was proved by comparison the re-
Keywords: sults between linear and non-linear model of gyro-stabilizer throughout fre-
Active gyro-stabilizer quency domain also same wave input, constrains and limitations. Moreover,
they were cross checked by simulating in time domain. The comparison of
Twin gyro-stabilizer interested of linear and non-linear close loop model in frequency domain
Ship large roll motion has demonstrated the similar characteristics but gave different values at
System identification same frequency obviously. The results were confirmed again by simulation
in irregular beam sea on time domain and they demonstrate the difference
Inverse problems
of behavior of both systems while the gyro-stabilizers are switching on and
Non-linear damping moment off. From the resulting analysis, the non-linear gyro-stabilizer model gives
Non-linear restoring moment more real results that correspond to more accuracy in a designing gyro-sta-
bilizer control system for various amplitudes and frequencies operating
condition especially high sea condition.

　
1. Introduction gyro, u-tube, sea-ducted, variable angle fins, hydro-foil
keel fin and rotating cylinder etc.
Of the six modes of motions of marine vessel, roll motion Since 1995, Chadwick [1] had gathered types of roll
is the important mode to be realized. It is the greatest rea- stabilizers and it has been classified by control method to
son to capsize also affect to operation of crews, passengers be passive and active control. The passive control is the
comfortable and cargos damage, when a vessel is excited by control system does not require any external power source
wave load at high sea. In order to diminish amplitude of roll to operate the control device but the active control system
motion, roll stabilizer system becomes an important role. does opposite way [2]. Some types of stabilizer are only
More than 100 years, many types of roll stabilizer have passive control as bilge keels and sloshing. Some types
been engineered by many researchers and designer. The are only active control such as variable angle fins and ro-
following example, bilge keels, sloshing, sliding weight, tating cylinder. And some types are both as sliding weight,

*Corresponding Author:
Sathit P.,
Department of Maritime Engineering, Faculty of International Maritime Studies, Kasetsart University, Chonburi, 20230, Thailand;
Email: sathit.pon@ku.th

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

u-tube, sea-ducted and gyro. researches of gyro-stabilizer with novel control methods.
Another way, Haghighi and Jahed-Motlagh [3] have men- Townsend et al. [4] published a new active gyrostabiliser
tioned classification of system control types that can be clas- system for ride control of marine vehicle. McGookin et al.
[22]
sified as either external or internal control systems. The ex- published application of MPC and sliding mode con-
ternal control system is systems generate resisting load (forces trol to IFAC benchmark models. Perez and Steinmann [23]
and moments) outside a ship hull and the internal control demonstrated analysis of ship roll gyrostabilizer control
system is systems generate resisting load inside [4]. The prin- that revisited the modelling of coupled vessel-gyrostabilizer
cipal advantages of internal systems are not hydrodynamic and also describes design trade-offs under performance lim-
drag and effective zero forward but it has heavyweight, itation. Haghighi and Jahed-Motlagh [3] proposed ship roll
volume penalty and there are limited in stabilisation capa- stabilization via sliding mode control and gyrostabilizer.
bility. The external systems have lightweight but it creates For designing ship roll gyro-stabilizers, a preliminary
hydrodynamic drag and ineffective at zero forward speed [5,6]. design is important thing. In order to design it, designers
The following above mention and nowadays technology, a need to know wave loads, ship motion (ship model) and
gyro-stabilizer become to be an interesting system at present. actuator characteristics (gyro-stabilizer model). Almost,
Because it has a combination of internal and external control modelling of designing uses an equation of motion in lin-
system advantages: potentially light weight with no hydrody- ear model that it may not comprehensive mission require-
namic drag and is effective at zero forward speed. ment, such as large roll motion from both high amplitudes
A gyro-stabilizer system is to use resisting moments, and frequencies of waves. Normally, these have to be
which moments are the cross product between angular non-linear model under limitations that have more correc-
momentum vector of flywheel and angular velocity vector tion and accuracy also failure cases from system instabili-
around precession axis. The resisting moment is applied ty. The failure cases may not occur in linear modelling but
to vehicle or other to resist external excitation moments it was found in non-linear. From these reasons, it becomes
that are keeping minimal oscillatory amplitude of rolling the motivation of this study. Both linear and non-linear
vehicle. However, this paper focus on a gyro-stabilizer is model of gyro-stabilizer and ship were concerned.
applied to small ships. Non-linear system modelling of gyro-stabilizer was
Background stories of gyro-stabilizers, it has been comprised of ship rolling model and single axis gimbal
applied to various inventions. The first gyro-stabilizer ap- gyroscope model (non-linear equation of motion). Nor-
plication was invented by Brennan [7]. Brennan used twin mally, non-linearity of gyro-stabilizer model appears
gyro that it had counter-rotating flywheels to stabilize in restoring term and exciting moment term. However
unstable vehicle (two-wheel monorail car). This invention non-linearity of ship roll model able to appear all terms
had similar patents [8,9,10]. More researches, gyro-stabiliz- of equation of motion (term of inertia, damping, restoring
ers were applied to stabilize to two-wheel vehicles such as and exciting moment).
a bicycle [11] and motorcycles [12]. In 1996, Brown and his In order to formulate non-linear ship roll model, the
team published about the using gyro-stabilizer that pro- system identification method is used to find non-linear
vides mechanical stabilization and steering a single-wheel coefficients of added inertia, damping and restoring term
robot [13]. Another field, NASA used the advantage of of ship. The system identification methods of ship roll
gyro-stabilizer to control attitude of large space structures motion can be found in many papers such as Masri et al.
or satellite [14]. In additional, gyro-stabilizers have been (1993) [24], Chassiakos and Marsi (1996) [25], Liang et al.
applied to maritime field, e.g., autonomous under water (1997) [26], Liang et al. (2001) [27], Jang et al. (2009) [28],
vehicle [15,16], torpedo [17] and free surface vehicle etc. The Jang et al. (2010) [29], Jang (2011) [30] and Jang et al. (2011)
[31]
first record of a gyro-stabilizer in marine vehicles was etc. However, in this paper uses method of Pongdung
found accidentally, Howell torpedo, it was installed rapid et al. [32] which is the novel method and able to find all of
rotation of fly-wheel. There was 16 inches a steel wheel non-linear terms of ship model.
diameter and was spun up to 16,000 rpm. The torpedo The objective of this presentation is to analyse non-lin-
was experimented for locking target on U.S. Navy boat [17]. earity of ship roll twin gyro-stabilizer control system
For the first time of free surface vehicle, a gyro-stabilizer under limitations of wave load and precession angle via
device was passive system that it was utilized to diminish frequency domain analysis and time domain simulation.
roll motion [18,19,20]. Active gyro-stabilizer systems were
developed from passive systems. The first system was 2. Principals and Theories
proposed by Elmer Sperry in 1908 [21].
Analysis of gyro-stabilizer system has three parts
Recently, many researchers have proposed related new
that is realised. It comprised of water wave model, ship

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

model and gyro-stabilizer model. In order to reach the 2.1 Full Non-linear Gyro-stabilizer Model
present objective, the regular and irregular deep water
wave models are selected to set simulation cases. The ship The prediction of gyro-stabilizer performances was
and gyro-stabilizer models is concerned both linear and modelled via the two equations of motions. The first equa-
non-linear model to observe and analyze effects of system tion is the ship model and the second equation is gyro-sta-
non-linearity from simulation results. bilizer model.
The general principle of gyroscopic stabilization, its Consideration a ship motion, while it is excited by wa-
torque is produced by gyro-stabilizer that installed in a ter wave that is demonstrated in Figure 3. Instantaneous,
ship opposes roll exciting moment from water wave. This the ship is rolled by moment of inertia in counter clock-
exciting moment disturbs the angular momentum of fly- wise and is acted by wave, which wave free surface has
wheel such that develops precession motion. The cross β (wave slope angle) against horizontal line. The y axis
product of flywheel angular momentum and precession of body fix frame has φ (roll angle) against horizontal
rate induces moment to resist the exciting moment in op- line. Thus θ is relative angle between roll and wave slope
posite direction [33]. Figure 1 explains working principle of angle. The following Newton’s second law, the equation
gyro-stabilizer that is installed in a marine vessel. At pres- of motion of ship can be written as
ent, twin-flywheels are selected, and there are spinning ¨ ¨
and precession angle rotate in opposite direction. Its result 0 (1)
( I 44 + I 44 a (θ )) θ + B44 (θ )θ + C44 (θ )θ =
cancels the side effect of gyroscopic moments in the other
directions (normally in pitch and yaw of ship). Figure 2 where I 44 is moment of inertia of ship that is a constant
displays the working of twin gyro-stabilizer. value. I 44 a (θ) is non-linear added moment of inertia
function. B44 (θ) is non-linear damping moment function
and C44 (θ ) is non-linear restoring moment function. In
order to find non-linear functions, non-parametric system
identification method is used.
Briefly, Pongdung’s method [32] is chosen to determine
non-linear functions because it able to find all non-linear
functions in Equation 1 synchronously. The method needs
measured motion data from free roll decay experiment or
CFD (Computational fluid dynamics) to formulate inverse
problem. Actually, the responses are outputs and are calcu-
lated via equation of motion that the non-linear functions of
each term are known variable values. On the other hand, the
Figure 1. Illustration single gyro-stabilizer installation responses become to input (measured data) in inverse prob-
and its working principle lem and the non-linear functions become to output (unknown
variables). Each moment terms are solved by inverse prob-
lem formalism and stabilized by Landweber’s regularization
method. Its solutions are chosen the optimal solution through
L-curve criterion. Finally, the zero-crossing detection tech-
nique of measured data is compared with that solution for
identifying each moment function and reconstruction them.
For more detail can see in Pongdung et al. [32].

Figure 2. Demonstration working principle of twin gy-

ro-stabilizer and elimination of its side effect of gyro-sta- Figure 3. Rolling of ship on water wave free surface
bilisation moment
When all non-linear functions is known, substituting

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

θ= φ − β into Equation 1, it becomes ¨

I g α + Bg α + C g sin α =τ s − τ p (10)
¨ ¨ ¨ ¨
[ I 44 + I 44 a (φ− β )](φ− β ) + [ B44 (φ − β )](φ − β )
(2) where α , α and α in Equation 9 are precession angle,
+[C44 (φ − β )](φ − β ) = 0 precession rate and precession acceleration respectively.
The following Equation 10 I g , and Cg are moment of
Rearranging Equation 2, it yields inertia, damping and restoring coefficient of gyro-stabiliz-
er about precession axis.
  ¨ ¨  ¨   
 I 44 + I 44 a  φ− β   φ+  B44 φ − β  φ
 
( ) Equation 9 represents the full non-linear ship roll dy-
 namics, while Equation 10 represents the non-linear dy-
¨ ¨ ¨ (3) namics of gyro-stabilizer about the precession axis. The
+ C44 ( φ − β )=
 φ [ I 44 + I 44 a (φ− β )] β
following Equation 9 and Equation 10 associate coupled
+[ B44 (φ − β )] β + [C44 (φ − β )] β system, the wave-induce roll moment ( τ w ) excite the ship
rolling. When roll motion develops, the roll rate induces a
Equation 3 is the full nonlinear ship motion: left-hand moment about the precession axis of spinning wheels ( τ s
side is ship moment and right-hand side is exciting mo- ). And then the spinning wheels develop precession, its re-
ment. The Equation 3 has the same coefficient function in action moment resists on the ship with opposes direction
the same term of inertia, damping and restoring: the roll of the wave-induce moment ( τ g ).
angle and wave slope are equal values at steady state on
time domain, but at transient are not. τ= K g φ cos α (11)
s
Thus, define

¨ ¨
[ I 44 + I 44 a (θ )] β + [ B44 (θ )] β + [C44 (θ )] β (4)
τw = τ g = nK g α cos α  (12)

The wave slope β is determined from Equation 5, w h e r e Kg i s s p i n n i n g a n g u l a r m o m e n t u m

which it is regular linear wave (deep water wave). ( K g = ωspin I spin )
The roll stabilization moment for passive system can
η ( x, t ) η0 sin ( kx − ωt ) (5)
= be modified the precession damping and stiffness as well
as leave the gyro to free work. For an active system, it
where η0 is wave amplitude, k is wave number, x is is controlled through the precession dynamics via the
distance in x direction, ω is angular frequency and t is precession control moment that is PD controller and ex-
time. When differentiate Equation 4 with x , it becomes pressed follow
to wave slope equation follow as Equation 6. Then wave
slope velocity and acceleration are Equation 7 and 8 re- τ p K pα + K dα (13)
=
spectively.
where K p is proportional control gain and K d is deriv-
dη ( x, t ) ative control gain. The advantage of this control law is no
= β= η0 k cos(−ωt ) (6)
dx needing ship roll sensors.

=β η0 kω sin(−ωt ) (7)

β =
−η0 kω 2 cos(−ωt ) (8)

Thus, a model for motion of the ship in roll together

with and n -spinning-wheel gyro-stabilizer can be ex-
pressed follows as block diagram in Figure 4. Then it can
be formulated in equation of motion follow as Equation 9
and 10.

¨ ¨
I 44 + I 44 a (θ ) φ+ B44 (θ )φ + C44 (θ )=
φ τ w − τ g (9) Figure 4. Block diagram of full non-linear twin gyro-sta-
bilizer model

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

2.2 Linear Gyro-stabilizer Model From zero condition, thus the open loop transfer func-
tion is
In the past, analysis any control systems via equation
of motions were treated to be linear differential equation. φol ( s ) 1
There are reduced complexity and able to transform to ol ( s )
H= = (22)
τ w ( s ) ( I 44 + I 44 a ) s 2 + B44 s + C44
s-domain, and then change s-domain to be frequency do-
main. At steady state, the analysis control systems through The close loop transfer function is
frequency domain are proper. This section uses almost  (23)
equation from Perez and Steinmann (2009) [23] where φol ( s ) and φcl ( s ) are Laplace transforms open
From Equation 9 and Equation 10, the models are lin- and close loop roll angle respectively. And the transfer
earized: for small angle of roll and precession, the coef- function of precession angle to roll angle is
ficients of left-hand side are constant value. However, let
define, φcl ( s ) I g s 2 + Bg' s + C g'
cl ( s )
H= =
φ cos α ≈ φ (14)
τw ( s ) (I 44 s
2
+ B44 s + C44 )( I gs
2
)
+ Bg' s + C g' + nK g2 s 2

α ( s ) α ( s ) Kg s
( s)
H pr= = = (24)
φ ( s ) φ ( s ) I g s + Bg' s + C g'
2
α cos α ≈ α (15)

and where

'
sin α ≈ α (16) B=
g Bg + K d (25)

Hence, the linear equation expressed as follow:

'
¨
C=
g C g + K p (26)
( I 44 + I 44a ) φ+ B44 φ + C44=
φ τ w − τ g (17)
The transfer function of precession angle to wave ex-
citing roll moment that is the result from Equation 23 and
¨
I g α + Bg α + C g α =τ s − τ p (18) 24 is

α ( s)
H pw
= ( s) = H pr ( s ) H cl ( s )
¨ τw ( s )
τ w =( I 44 + I 44 a ) β + B44 β + C44 β (19) (27)
Kg s
=
(I 44 s
2
+ B44 s + C44 )( I gs
2
)
+ Bg' s + C g' + nK g2 s 2
τ=
s K g φ (20)
Rearranging Equation 24, it yields

τ g = nK g α (21)  
 
 Kg   s 
Note that, τ p no changes in linear model. And linear H pr ( s ) =    (28)
 Ig  ' '
gyro-stabilizer system is demonstrated in Figure 5.    s 2 + Bg + C g 
 Ig I g 


and the roots are

2
Bg'  Bg'  C g'
p1,2 =
− ±   −4 (29)
Ig  Ig  Ig
 

Both roots are negative real roots (stability condition)

if and only if, Bg′ > 0 , Cg′ > 0 also constraint of
Figure 5. Block diagram linear twin gyro-stabilizer model

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

2 Another form of roll reduction function is

 Bg'  C g'
  >4 (30)
 Ig  Ig  H ( jω ) 
 
)  1 − cl
RR ( ω=  (39)
 H ol ( jω ) 

The following Equation 13, the proportional term is
set during design due to centre of mass location, normally Note that, the roll reduction values possible to be neg-
locate below the precession axis, which like a pendulum, ative or positive values but less than 1 along interested
the Cg′ value is fixed. Thus, the control moment becomes range of frequency. The meaning of negative value is
roll amplification. The meaning of positive value is roll
τ p = K d α (31)
reduction. Then the roll reduction close to 1, it has better
performance.
From Equation 30, it able to formulate the condition of
two poles to be real root: 2.4 Constrained Performance

Bg' > 4C g' I g (32) The additional constrain is cause by the precession
angle limiting due to mechanical design. If the precession
It can set as angle reaches this limit, the device may get damage or de-
teriorate. Additionally, it may causes of roll amplification
rather than roll reduction of ship: phase of resisting mo-
Bg' = γ 4C g' I g , r > 1 (33)
ment cannot eliminate wave induce roll moment.
Then substitute Equation 33 in to Equation 25, it be- For a regular wave of frequency ω 0 and wave height
H s , it induces roll exciting moment amplitude τ w0 . The
0
come
following Equation 27, it can be obtained the amplitude of
precession angle:
=K d γ 4C g' I g − Bg , r > 1 (34)

This derivative control gain is used through under con-

0
(
a = H pw jω0 τ w (40)) 0

strained performance follow section 2.4

The given constrain is the maximum precession angle
2.3 Control Performance and Limitations α max , it can be obtained the optimal of Bg′ that it take the
precession angle close to its limit:
In order to observe the objective performance, the out-
put sensitivity function is defined as:
= (
γ * (ω 0 ) arg min α max − H pw ( jω 0 , Bg′ ( γ ) ) τ w0 (41)
γ >0
)
φcl ( s )
S ( s)  (35)
φol ( s )
3. Cases Study Configurations and Numerical
The Bode’s integral constraint is Experiment Setup

∞ To carry out the aim of this presentation, the non-linear

gyro-stabilizer system is validated with linear gyro-stabi-
∫ log S ( jω) dω = 0 (36)
0
lizer system to observe and analyze its performance at the
same environment and limitation. The vessel was set to
and the roll reduction (complementary sensitivity func- zero speed and moved against beam sea direction. In order
tion) is defined as to determine the system design point, assume that the sys-
tem was designed for deep water wave, significant wave
φol ( jω ) − φcl ( jω ) height of 0.04 m, all frequency of its.
RR ( ω ) =
1 − S ( jω ) = (37)
φcl ( jω ) In order to observe and analyze non-linearity of a
gyro-stabilizer system, this present, V-hull section was
the integral constrain becomes selected because it is a general section profile of high-
speed boat and small ship that are suited. The dimensions
∞
of selected V-hull are demonstrated in Figure 6. Both
∫ log 1 − RR ( jω) dω =
0
0 (38)
linear and non-linear equation of motion was found via

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

measured data that were the result from CFD method, tial function the linear ship model is
XFLOW commercial program. The simulations are set to
¨
be unsteady flow. There is free roll decay method, which 0.3255 φ+ 0.0494φ + 19.7321φ = 0 (42)
was set initial condition of roll angles are 5, 10, 15, 20, 25
and 30 degrees. According to CFD simulation results, the Note that, in free roll motion test θ = φ , there are not
example of measured data from simulation case, which have relative motion between free surface and roll motion
was set 30 degree of initial condition was demonstrated in angle and I 44 = 0.26 .
Figure 7.

Figure 6. The V-hull Geometry and dimension for simula-

tion
Figure 8. Determination of exponential function from roll
angle data of 5 degree initial condition
The non-linear ship model was formulated by re-
construction non-linear damping coefficient and added
moment of inertia via systems identification method that
used all of measured data (roll angle, angular velocity and
angular acceleration) and all initial condition from CFD
method.
The non-linear restoring coefficient function was
known via inclination calculation. The calculation result
was fitted by polynomial function follow as Equation 43
and its curve was shown at the top of Figure 9.

Figure 7. Measured Data from CFD of Initial Condition C44 ( φ ) =−19.7321φ18 + 108.8680φ16 − 258.2096φ14
30 Degree
+344.3239φ12 − 283.2101φ10 + 148.2937φ8 (43)
The linear ship model was formulated from logarith- −49.4689φ6 + 10.4220φ 4 − 1.4402φ 2 + 0.1454
mic decrement method. The method appropriates to small
roll motion less than 8 degree. It requires only roll angle The non-linear damping coefficient function was found
data (measured data from CFD) to detect maxima or min- by accumulating the damping moment data point from
ima values, where are used to find estimated exponential system identification method, and there were taken by
function. The function leads to determine the damping curve fitting method which is shown in the middle of Fig-
ratio and natural frequency. There can be converted to add ure 9. Then the moment function was divided by φ , thus
moment of inertia, damping coefficient. However, restor- the non-linear damping coefficient is
ing coefficient can be found from inclination calculation.
According to this method, the details of calculation were ( )
φ
B44 = 0.1588φ 2 + 0.1391 (44)
omitted in this presentation.
For this paper, the simulation of 5 degree of initial con- From the same procedure of formulating non-linear
dition was used. The estimating of exponential function damping moment, the added moment of inertia fitting
was shown on Figure 8. According to estimated exponen- curve is shown at the bottom of Figure 9 and its average

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

value is The non-linear and liner gyro-stabilizer model was

used according to Equation10 and Equation 18 respec-
¨ tively. There have same characteristic coefficients. The
I 44a  φ  =0.066 (45)
  gyro-stabilizer was assumed that its speed is constant at
300 rpm ( ωspin ), the moment of inertia about spin axis
The full non-linear ship model was simulated on free is 0.0125 kg-m2 ( I spin ) and the moment of inertia about
decay motion with 30 degree initial condition to verify its precession axis is 0.0005 kg-m2 ( I g ). The damping coef-
accuracy. The simulated result of roll angle via Equation 1 ficient is zero Bg = 0 , there is on friction about precession
was plotted in Figure 10 also result from linear model and axis. The restoring coefficient is zero; the center of gravity
measured roll angle from CFD. It proves that the non-lin- position was located at middle of spin and precession axis
ear model is better accuracy more than linear model, when (no effect of pendulum).
both results were compared with measured roll angle from The following Equation 13, the proportional gain K p
CFD. Hence, the non-linear ship model has enough accu- is fixed value of 0.1. The derivative gain was determined
racy and can be used in this presentation. via Equation 34 and Equation 41.
Both linear and non-linear systems were determined
through frequency domain and simulated with regular
waves, which the range of frequencies is 0 to 14.84 rad/
s. The simulation cases of linear system were varied wave
amplitudes from η0 = 0.02 m up to 1.5η0 . The linear sim-
ulation case of η0 was set to be reference case. The γ of
*

each wave amplitude of simulation cases follow Equation

41 were gathered, and then they were used in simulation
cases of full non-linear system. The full non-linear system
was simulated in time domain with only amplitude of η0
. Moreover, in order to find a design trade-off, the γ of
*

each wave amplitude of linear system were used. The

name of simulation cases are described in Table 1.

Table 1. The description of simulation cases

Case Name Description Gyro-model Ship-model

LSOL Linear open loop system - Linear

Figure 9. The Estimating functions of restoring moment LSCL Linear close loop system Linear Linear
(top), damping moment (middle) and added moment of Linear close loop system that de-
LSCL
termined with wave amplitudes of
inertia (bottom) @ cη0
Linear Linear
cη0 ( c =1.1, 1.2, 1.3, 1.4 and 1.5)
FNSOL Full non-linear open loop system - Non-linear
FNSCL Full non-linear close loop system Non-linear Non-linear
Full non-linear close loop system
FNS- with the γ* values of each wave
Non-linear Non-linear
CL@ cη0 amplitude of LSOL ( c =1.1, 1.2,
1.3, 1.4 and 1.5 )

4. Results and Discussions

In order to examine the characteristics of performances
of non-linear gyro-stabilizer system, the linear system was
determined first. The following Figure 11 – 14, they show
Figure 10. The comparison of simulations between results the results of linear gyro-stabilizer characteristics.
of non-linear ship model (system identification method) Figure 11 show exciting moment (input), the result
and linear ship model (logarithmic decrement method) from Equation 19. The graph was also plotted Stokes lim-
with measured data from CFD it; the exciting moment cannot exceed this line at upper

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

side. Stokes and exciting moment line of amplitude 0.02

m intersect at 14.84 rad/s, which it was set to be maxi-
mum value of frequency range. All lines have tough at
ω = 7.786 rad/s that is natural frequency of linear vessel
model because it has the same coefficients. And then,
when frequencies increase, the exciting moment rapidly
increase. It is obviously the increasing of wave amplitudes
correspond to increasing exciting moment.
Figure 12 and 13 are they were γ values and preces-
*

sion angles of each wave amplitude respectively. They

simultaneously determined via Equation 27 and 41.
Figure 12 shows the minimum γ values of each wave
*

amplitude. All wave amplitudes gave γ values of 1 until

*

the system gets the exciting frequency that makes the pre-
Figure 12. Estimating of γ * values each wave amplitude
cession angle reach its limit value (60 degree, also observe
from Equation 41
Figure 13). It has been called critical point. From this point
γ * value rapidly increasing to keep precession angle does
not exceed 60 degree. As the higher wave amplitude, the
critical points have appeared at lower frequency.
Figure 13 shows the precession angles in frequency do-
main. The trend of precession angle resembles the pose of
exciting moment in Figure 11. However, the crests of all
lines about exciting frequency value of 2 were affected from
turning proportional gain K p that was fixed value of 0.1.
And the higher wave amplitude gives the higher precession
angle. The flat band at high exciting frequency that has con-
stant value of 60 degree is explained as above paragraph.
Figure 14 shows the roll angle responses. At the lower
exciting frequency from critical point, the result trends
resemble precession angle but it rapidly increases when Figure 13. Precession angles of linear gyro-stabilizer
the exciting frequency increase from critical point. The model so that determined via Equation 27 and 41 with the
increasing rapidly of roll angle from the critical point is following γ * values each wave amplitude in Figure 12
caused by the precession angle reach its limit, which it
cannot have more precession rate to create resisting mo-
ment for cancel exciting moment. Moreover, at the higher
wave amplitude gives the higher roll angle response.

Figure 14. Roll angle responses of linear gyro-stabilizer

model each wave amplitude via Equation 23
The following linear model results consistent and rea-
Figure 11. Exciting moment of the linear gyro-stabilizer sonable to each other. They can be used to compare with
model the non-linear system in the same wave condition. How-

16 Distributed under creative commons license 4.0 DOI: http://dx.doi.org/10.36956/sms.v3i1.316

 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

ever, according to section 3, the non-linear gyro-stabilizer roll angle responses values are above the linear line. The
system model was assumed as it was designed to operate closed-loop non-linear system uses more precession angle
in beam sea that it has significant wave height ( H1 3 ) of than linear system but they give more roll angle responses
0.04 m ( η0 = 0.02 m). But it cannot directly determine γ than linear system. After the critical point of linear closed
*

through transfer function like linear system model. Thus loop system, all lines of roll angle responses increase and
the γ values each wave amplitude from linear model
*
approach to open-loop systems because precession angles
were applied to non-linear model. The non-linear system were forced and reduced to constant in linear model and
needs to simulate in time domain and collected the ampli- non-linear model respectively. This reason refers to ineffi-
tude simulation results at steady state. Hence, Figure 15 - ciency of the systems, when the precession angle reaches
18 illustrate the comparison between linear and non-linear the limited angle.
model. Some data of non-linear model disappear because Figure 18 explains the responses in view of a relative
of its solutions became unstable (no steady state) when roll angle response and can say that are the inverse behav-
the precession angle reached the limit angle. ior of a roll angle response. While the stabilizer attempt to
Figure 15 shows the exciting moments both open and keep roll angle approach to zero, the difference between a
close loop of linear and non-linear system. They have the wave slope and roll angle increase (see relation in Figure
trend like exciting moments of linear system. Linear open 3).
loop and close loop exciting moments are same line be-
cause they have same coefficients of vessel model. The all
non-linear exciting moment lines are above liner exciting
moment at high frequencies from around the natural fre-
quency. The difference of exciting moment values causes
of the coefficient functions follow as Equation 4 that relat-
ed to θ , θ and θ .
Figure 16 shows the precession angle result that all
non-linear precession angles have the same line until
they reach to a limited angle at the exciting frequency
value of 11.85 rad/s. And almost values are above linear
system. The cause of all non-linear precession angles
has the same values. They used same γ =1 as following
*

Figure 12. After they reach the limited angle at higher

frequencies, the data lost because the system became Figure 15. Comparison exciting moments between
unstable so that it now has steady state. However, the non-linear and linear gyro-stabilizer models; the linear
non-linear system back to stable at the end of lines be- model of η0 was set to be the reference
cause of the increasing of γ value (see Figure 12) able
*

to reduce the precession angles below limited angle.

And then, they obviously show the higher γ each wave
*

amplitude gave lower precession angle at the higher fre-

quencies.
The following Figure 17, roll angle responses of the
open and closed-loop of linear and non-linear systems
are shown. The comparison between open-loop of lin-
ear and non-linear system obviously difference behavior
along frequencies range; the linear open-loop system is
fair curve, while the non-linear closed-loop system has
the apex at the frequency value of 6 rad/s. This is the
important cause to deep studying throughout non-linear
gyro-stabilizer system; a difference response behavior
will give the difference a system design point. The linear Figure 16. Comparison precession angle responses be-
and non-linear closed-loop systems have a same trend. tween non-linear and linear gyro-stabilizer models; the
All non-linear roll angle responses have same values linear model of η0 was set to be the reference
until they reach the frequency of 11.85 rad/s and almost

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

The non-linear behaviors were analysed and explained

throughout frequency domain as they were mentioned
above. However, in reality, a γ value cannot adjust base
*

on frequency domain as following Figure 12; a frequen-

cy cannot know immediately in time domain. Thus a γ
*

will be selected only one value from Figure 12 so that

appropriate with operation requirements. The γ value of
*

10.6 was selected. It locates at the right end of line LSCL

@ η0 . The value is the highest value, which forces the
precession angle to work do not exceed the limited angle
throughout the frequency range. Then the selected value
was used for both linear and non-linear gyro-stabilizer
system. The results were plotted follow as Figure 19 - 22
Figure 17. Comparison roll angle responses between in frequency domain. In order to examine the effect of
non-linear and linear gyro-stabilizer models; the linear non-linearity, linear and non-linear gyro-stabilizer system
model of η0 was set to be the reference model were simulated in irregular wave model of Bret-
schneider’s [34] method that it has significant wave height
of 0.04 m and the results were plot in Figure 23.
The following Figure 19, the precession angles of
non-linear system work under the limited angle through-
out frequency range. It has same characteristic of linear
system but has different values. Hence, the roll angle re-
sponse in Figure 20 is consequence of precession angle.
The roll angle response of non-linear system has same
characteristic of linear system but has different values as
well.
Figure 21 shows the frequency response results. The
peak of frequency response of non-linear open-loop sys-
tem that refers to the natural frequency of vessel is dif-
ferent from linear open-loop system. The curve of linear
Figure 18. Comparison relative roll angle responses closed-loop system is fair curve and approach to zero
between non-linear and linear gyro-stabilizer models; throughout the frequency range. The curve of non-linear
the linear model of wave amplitude η0 was set to be the closed-loop system has same behaviour of open-loop and
reference its peak is lower. According to Figure 22, the reduction
rates of linear and non-linear system are shown. The re-
duction rate refers to the efficiency of the stabilizer sys-
tem. The non-linear system has the reduction rate values
near linear system at the frequencies lower than 6 rad/s.
But at the higher frequencies value, the reduction rate val-
ues of non-linear system are lower.
According to the previous results, the γ value was
*

selected and fixed. They are made to clearly understand

and proved that the non-linearity of non-linear model give
a lot of different results at all. Thus, the designing base
on linear model may give the wrong respond in reality. In
order to prove again, the linear and non-linear gyro-sta-
bilizer system models were simulated in same irregular
beam sea on time domain. Assume that, precession angle
Figure 19. Precession angle responses of linear and is limited by mechanism at 90 degree but the γ value was
*

non-linear gyro-stabilizer model of wave amplitude η0 selected from the limited angle at 60 degree of regular
and fix γ =10.6 wave amplitude in order to let it has margin to prevent a

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

damage when the gyro-stabilizer system get higher ampli- In the following Figure 23, the setting of gyro-
tude in irregular wave. stabilizer simulation cases (linear and non-linear
system) were simulated in irregular wave model. The
gyro-stabilizers were switched off first, and then be-
gin to switched on at 20 second of simulation time to
observe the difference of behaviors when the gyro-sta-
bilizer models were switch off and on. The systems
performances were gathered and shown in Table 2 in
root mean square. At the top of Figure 23, the exciting
moments of linear and non-linear system were shown.
They were induced from the same irregular wave mod-
el but it gave exciting moment amplitude and phase
shift slightly difference. The RMS of exciting moment
of linear model has the value less than non-linear
Figure 20. Roll angle responses of open and close loop model so that accord to Figure 15. When the stabilizer
condition of linear and non-linear gyro-stabilizer model of switched off, the vessel did not stabilize. The roll angle
wave amplitude η0 that the close loop model was set the response of linear model shown at the middle of Fig-
fix value of γ =10.6 ure 23 has the RMS value more than non-linear model
slightly. These results accord to Figure 20. When the
gyro-stabilizers were switched on, the precession began
to move for stabilize the vessel, the roll angle respons-
es were reduced. The precession angle responses were
shown at the bottom of Figure 23. As the gyro-stabi-
lizer switched on, the precession angle of linear model
has RMS value more than non-linear accord to Figure
19. And the roll angle response of linear model has
RMS value lower than non-linear model. However, the
reduction rate (RR) of linear system has RMS value
more than non-linear system and so very different.
As the mentioned these results, they clearly show
that the designing via the linear model cloud makes the
Figure 21. Comparison of frequency responses of open
gyro-stabilizer system miss the design point of mission
and close loop condition of linear and non-linear gyro-sta-
requirements. On the other hand, the non-linear system
bilizer model of wave amplitude η0 that the close loop
model cloud gives more approach to reality for designing
models were set the fix value of γ =10.6 follow mission requirements.

Table 2. The root mean square values (RMS) of gyro-sta-

bilizer system performances in irregular beam sea accord-
ing to Figure 23; the significant wave height and average
wave frequency is 0.04 m and 10.68 rad/s respectively.

Model τw [Nm] α [deg] φ [deg] RR [-]

LSOL 14.5 - 21.56 -

FNLSOL 15.38 - 18.57 -

Figure 22. Comparison of reduction rate between linear LSCL 14.5 32.03 4.75 0.78
and non-linear gyro-stabilizer model of wave amplitude
η0 that the close loop models were set the fix value of γ FNLSCL 15.05 31.13 11.46 0.38
=10.6

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 Sustainable Marine Structures | Volume 03 | Issue 01 | January 2021

operating condition especially high sea condition.

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