A simplified lower-bound criterion for stable rolling motion

r

nl rod u ct ion

A ship at rest in a calm sea ¡s an archetypal

example of a dynamical system in stable equilibrium. The system will return to this position, dynamically,

given some small perturbation due to the combined

effects of (positive) damping and restoring force. For small-amplitudes the comparison with the free response of a simple spring-mass-damper mechanical oscillator is clear. A further analogy extends to consider the motion of a ball rolling on a locally parabolic track, where the shape of the track represents the underlying potential energy of the system. Trajectories in the phase portrait spiral towards the equilibrium position, and all possible initial conditions cause transients which decay on lo

this point attractor [1].

The static stability of a floating vessel is typically limited by the angle of vanishing stability resulting from the softening spring characteristic of the GZ curve. For a symmetric hull geometry these

laránle unstable equilibria are represented by

saddle points in the phase portrait [2J. The unstable equilibria give rise to sparatrices which now divide

the plane of initial conditioThT domains of

attraction. Transient motion may no loh7 be captured by the attractor and initial conditions lying

outside of the domain of attraction or basin boundary

will lead to unstable, growing oscillations. This is a familiar scenario from ship dynamics where a floating Vessel inclined to a large angle will lose its stability

resulting in a rapid roll to an inverted position [3]. This is properly viewed in the two-dimensional phase

Portrait of displacement and velocity so that it can be Seen that ¡t is the combination of roll angle and roll Velocity that may

Abs tract

This paper develops an ad hoc criterion for stab/e rolling motion of a floating vessel ¡ri

regular waves under slowly changing conditions. The total energy associated with the angle of

vanishing stability ¡s compared with the total energy of the harmonic response. Incorporation of a factor of safety enables the determination of critical forcing parameters which delineate 'safe regions where stable rolling motion persists. An alternative measure of stability is also presented

which is applied to an idealised version of a biased floating vessel and the results are compared with

the dynamic instabilities obtained from numerical integration in previous studies.

Assistant Professor Scho0i of Engineering Duke University

Durham, _{NC 27706, USA.}

A SIMPLIFIED LOWER-BOUND CRITERION

FOR STABLE ROLLING MOTION

Lawrence N. Virgin1

45

-In this unforced (autonomous) case the effects Th rapidly applied loads, for example modeled by

impulse, pulse on step inputs, can be assessed. This has obvious relation lo ocean wave slam loading. ThIs

paper is restricted to the consideration of periodic excitation which might anise for example as a regular wave train from a distant storm. Furthermore, the

forcing parameters are assumed to change slowly so

that transient effects are negligible [4j.

Returning to the analogy of the ball rolling on

the potential energy surface, the angle of vanishing

stability is manifested as two 'hilltops' on either side of the stable 'well' at the origin. Under the action of

harmonic excitation, the steady-state forced response

will depend on the magnitude of the forcing amplitude

and frequency, including the familiar resonance feature [51. This is akin to oscillating the track on which the ball is rolling; a horizontal projection of

this nonautonomous behavior now occupying a

three-dimensional phase space with lime as the third axis. It

is clear that under certain circumstances, especially proximity to resonance, the forcing may be such that the ball will 'escape' from the potential well

corresponding to a capsize situation.

For large-amplitude motion the nonlinear restoring force causes a variety of complex instability

phenomena preceding escape [6]. The total energy of

the response increases with the amplitude and this is

used as a measure of proximity lo capsize, i.e. the

total energy (kinetic and potential) of the response is

required to be somewhat less than the total energy

associated with the angle of vanishing stability. A safety factor is incorporated to ensure that this criterion is satisfied in terms of the forcing

parameters. Previous studies have indicated where

typical ordinary differential equations lose their stability leading to escape [6,71. The present paper outlines a simplified ad hoc criterion based on

considerations of limiting the tota energy of a system, and deriving conditions under which stable rolling motion persists.

ree ro ¡ng motion

n order to convey the underlying principle of the new criterion consider the free roll motion of _a

floating body described by

O=f(O,O).

₍₁₎

Initially damping is ignored so that the ordinary differential equation of motion is of the form

OV'(0)=O

₍₂₎

where a prime denotes

differentiation with respect to

the roll angle O and _{a dot denotes differentiation} with respect lo time t.

A first approximation to a typical restoring force (GZ cwve)

experienced by a body inclined to an angle O is

V'(0)=O-a&

₍₃₎

This type of system _{is often referred to as a softening}

spring charijc.

_{lt can be seen}

that equilibrium

points correspond to stationary values of the underlying potential energy, i.e.

OeO,0e±

₍₄₎

The saddle points_{associated with the second} two

equ)bra of equation _{(4) correspond to}

teã?igle of

vanishing stability.

Returning to the analogy of the rolling ball, _the

underlying potential _{energy curve is given by}

V(9)=-O2-O4+c

(5)

and trajectories in the phase portrait are given by contours of constant total _{energy, i.e.}

E= +V

(0)

-4

+c

(6)

where the constant_{depends on the initial}

conditions. The value of the total energy at the saddle points is

E=

_{i (}

+ i

2

-where the arbitrary constant c is taken _{as zero, giving}

E=

4 (8)

The restoring _{force of equation (3) is shown plotted} in

figure i(a) _{together with the underlying potential}

energy function of equation (5) in figure _{1(b). The} phase trajectories for this conservative _System

describe ellipses _{as given by equation}

(6) and are illustrated in figure 1(c).

The safety factor p can be incorporated _{into the}

analysis to give a limiting total _{energy value of}

E=--4a

(9)

-

46

-V 1.0 O 1.0 F

Figure 1. (a) Restoring _{force; (b) potential}

energy;

and (c) phase portrait_{for a softening spring. The ad} hoc boundary for stable rolling is shown _{for p=O.8.}

(cr=l.0, ¡3=0.0)

and this is shown in figure 1(c) for _{p = 0.8. In other} words, the motion of

the vessel should remain at least

within the shaded region for stabìlity.

Forced rolling motion _(Linear)

Harmonically excited roll motion can be approximately described _by

where the forcing amplitude F and_{frequency u can be}

thought of as

sinusoidally perturbing the potential

energy function.

For relatively small _{amplitude motion the} nonlinearity in the restoring force is _{negligible, so} that the roll motion _{can be described by}

and the steady-state

response is given by

9=Asin(wt-,)

₍₁₂₎ where

(+L

₍₇₎ -LO 1.0 e

and

wj3

tanq=

- ('y.

The familiar amplification of the response near

resonance (w = 1) can be seen.

There are a number of ways in which this response can be compared with _{the static stability} features of the underlying GZ curve [4]. Firstly, the

maximum total energy of the response can be used as a 'measure of proximity_{to the total energy associated}

with the angle of vanishing stability. Later, the

amplitude and maximum_{velocity of the forced} response is used as a measure of proximity to the amplitude and maximum velocity associated with the ingle of vanishing stability.

Maximum total energy bound

The total energy associated with the forced response is

where O is given by equation _{(12). Therefore, for}

a

given F and f3 the total _{energy will depend primarily} on w, i.e.

E=-(A o)2cos2(ot) +-A2sin2(ut)

(14)

which, for low damping, _{is a maximum at}

resonance

(w = 1), giving a maximum total energy of

E4A2

.1 F2

2p

(15)

For example, for _{a damping coefficient of}

f3 = 0.1, nonlinearity parameter of_{CZ = 1, and a safety factor of} p = 0.8, a forcing amplitude of F = 0.0632 would result in an oscillation whose maximum total energy

is equal to a prescribed proportion of the total energy

associated with the angle of vanishing stability, i.e.

.L F2<P

2 _p 4cx (16)

A forcing frequency _{less than or greater than} the resonant value will result in an oscillation with less

total energy and therefore _{would require a greater} level of forcing to _{violate the condition of}

equation (16). With co=0.8 _{Fc=0.233 and with w=1.2,} Fc0 .2405.

Figure 2 shows the maximum total energy of the forced linear _{response (equation (14)) for ¡3}

= 0.1 and F = 0.05 _{, 0.07. Also plotted on this diagram} is the total energy associated with the angle of

Vanishing stability _{(equation (9)). This illustrates}

that the limiting _{condition is reached between} F=0.05 and F0.07, thus _{confirming the value of Fc = 0.0632} for equation

(16). The relative contributions of the displacement and velocity terms in equation (14)

depend on w but the light damping present in _the System results

in the symmetric nature of the curves.

(13)

47

-F0.07

F=0.05

However as the amplitude of the motion grows

then tile_effect of the noni

_{rriing force wHI}

grow. In this case the response must be based upon the

sotutions to equation (10) rather than the assumed form of equation (11). The criterion of equation ₍₁₆₎

can be summarized as

Energy (response) < p Energy (instability)

The next section obtains expressions for the total energy of the response based _{on nonlinear (i.e. larger}

amplitude) motion.

Forced rolling motion _(Nonlinear)

The use of small oscillation _{theory which leads} to the linear response of equation (12) results in amplitudes which would simply grow indefinitely with no instability. For consistency _{the forced response}

must be based on the same equation as the unforced_or

free response with the addition of external excitation: i.e. equation (10) must be solved to obtain the

nonlinear response which can then be compared with

the approach to instability. An approximate solution can be obtained to equation (10) _{using various} methods [8]. Here, a harmonic_{balance approach is}

used whereby a solution is assumed of the form

0=asinot+bcosot

(17)

which can then be substituted into equation (10) and

after equating sine nd cosine coefficients _{leads to the}

following algebraic equations

-ao)2- b o

+a-3-a

_{--a b2r0}

(18)

which can be rearranged to give

F2 =A2[ (i

- o - A2)2_{+ (}

o)2]

₍₁₉₎

where A is the amplitude of the response, i.e.,

A = Y a2 + b2 ₍₂₀₎

Given F and ¡3, equation (19) can be solved for

A in terms of w. Figure 3 shows typical amplitude

response diagrams with F=0.2 and F=0.3, _{and ¡3=0.6.}

0.5 _{0.8 1.2}

i 1.5

Figure 2. Variation of the maximum total energy with frequency for the linear response of a harmonically oscillating system. (/3=0.1).

E

max

0.2

0.4

0.2

The familiar softening spring effect can be clearly seen although with this relatively high level of damping the jump in resonance feature is not

observed. A

0.0

E 0.2 0.1

Figure 3. Amplitude response diagram for the ful/y nonlinear system. (/3=0.6).

Again the total energy associated with _the

response is given by equation (14) where _{now the} (nonlinear) frequency-amplitude _{relationship is} governed by equation (19). Figure 4 illustrates the total energy of the harmonic _{response for a=1, ]3=0.6} and u=0.8 through a complete cycle of motion. lt can

be seen that the maximum total energy occurs at

(o. = fl211 ₍₂₁₎ 0.0

_...r

0 2

/

F=0.31 4 6

t-

) rad

Figure 4. Variation of the total energy of the harmonic response through a complete cycle. (/3=0.6, a=I.0, a)=0.8 and F increments by 0.02 from 0.21 to 0.31). The curves in this figure start with F=0.21 and increment by 0.02 to the value of F=0.31 giving a maximum total energy greater than the E=0.2 obtained from equation (9). This is confirmed by the upper curve in figure 3 reaching _{amplitudes greater}

than

E=A2, i.e.

A =0.632

corresponding to a safety _{parameter of p=O.8.} In this way, the maximum total energy of the

nonlinear response_{can be obtained for different}

values of e given F _{and ¡3. This is shown in figure 5 for}

p=O.6 and p=0.8. lt _{can be seen that a larger forcing}

amplitude F can be tolerated for p=0.8 than for _p=O.6 for the total energy criterion to be met. _{The curves in} figure 5 thus represent levels of forcing _{which ensure}

that the magnitude of the response is_{constrained to}

remain somewhat below the loss of stability _[9].

48

-/

P=0.8 0.4

0.2 0.6 1.0

So far no mention has been_{made of the} mechanisms by which a system such as equation (10)

will typically lose its stability. _{This is a subject of} current research. lt is important for the curves of figure 5 to be beiow the forcing _{levels at which these}

nonlinear phenomena _{occur in order for the total}

energy criterion to be lower bound and _{useful [10]. As} mentioned earlier it is also possible to limit the

response by the amplitude and maximum velocity associated with the tota! energy of the angle of

vanishing stability. This approach _{has been} successfully used by the author _{and is briefly}

introduced here to show that such ad hoc criteria can

ensure stable rolling motion [4].

Amplitude and maximum velocity_bound An alternative measure of_{proximity to} instability was developed in _{reference [4]. The} separatrix joining the saddle points in figure 1 is

gé5equation (6) with

c=-1/4a. Rearranging equation (6) leads to

= ± [

. e +

2

(22) where p is again used as a safety margin.

From this alternative form for constant total energy it is possible to obtain the _{maximum velocity} as

_+p

max-2aJ

and the maximum displacement

I ±(1

The alternative criterion for stable oscillations _now requires that the maximum velocity and amplitude of

the forced response to be less than the maximum

velocity (equation(23)) and _{maximum displacement} (equation (24)) respectivey for the undamped,

unforced conservative system.

Assuming the linear forced rolling model of equation (11) leads to the following condition to limit the amplitude

FC-I-1-(

-

I ±( l-p) fl(1 (25) and the maximum velocity

Figure 5. Limiting values of the _{forcing parameters}

(/3=0.6, a=1.0).

p0.6

0.2 _0.8 1.4 0.3 0.2

rwhich

are plotted in figure 6(a) for _{=0.1, p=O.8 and} ct=1.0 (for the unforced system), cx=0.0 (for the

forced system). The response of the nonlinear equation

(10) leads to the condition limiting the amplitude as

0.8 F 0.6 0.4 0.2

F=

Fc=[

L 2aw2 0.5 1.0

and limiting the maximum velocity as

-L

I p ₍₍ p \2 2\

+(f3o)

L2ao)2

82J

which are plotted in figure 6(b) for 3=0.1 _{p=O.8 and}

cx=1 .0, now for both the unforced and forced systems. The effect of the softening spring can now be observed,

resulting in frequencies around w=0.75 _producing oscillations with the greatest energy for fixed forcing

and damping. The two conditions of maximum

displacement and maximum velocityare both required

because of the effect of the frequency on the response.

This s related to the relative contributions of the two terms on the right hand side of equation (14).

A static bias, due perhaps to a constant wind loading or shifted cargo has been shown to have a marked effect on the dynamics of a rolling ship. This effe,ct can be incorporated into the dynamic model with the addition of a constant term in the equation of motion. The criteria suggested in this paper can

Include this effect with relative ease. Bifurcational behavior

Considerable research has been conducted into

the loss of stability of forced mechanical oscillators

[1]. These _{inherently nonlinear phenomena can be} 1.5 (4, w

i-F={I±(1-p)(1o2-

1±(1p)))2+(Ú))2)]2 (26)

Figure 6. (a) Stability criterion based on the assumption of linear resonance. (ß=O.1, _p=O.B). (b) nonlinear resonance.

2.0

(27)

(28)

49

-investigated in one of two ways. Firstly, an analytical method such as the Routh-Hurwitz criterion or Floquet theory [9] may be used to study stability characteristics of small perturbations about the steady-state solutions as described by_{equation (17).}

For example, equation (10) has been shown to lose its

stability close to resonance in one of _{two generic ways:}

a jump to resonance at a saddle-node bifurcation; or a

flip bifurcation leading to subharmonic _oscillations initiating a sequence of period-doubling _bifurcations

leading to chaos and escape 16]. For slowly changing conditions both of these events have been shown to

occur after the critical conditions of equations (27) and (28) have been reached, with a suitable choice for

p.

The second approach to studying the dynamic

stability of nonlinear oscillators involves _extensive

numerical integration. As an example of an asymmetric mechanical oscillator consider the following equation:

O+3O+O-O2=Fsincot

(29)

which has a potential of

V(8)=--9--+c.

(30)

Solutions to this equation may 'escape _{over the lower}

of the Iwo hilltops analogous to figure 1(c). _{Figure 7} shows instabilities obtained by the numerical integration of equation (29) which lead to _{escape or} capsize, together with the proposed ad hoc _criterion

suggested here [4]. For a given damping ratio _and

safety factor, forcing parameters within the shaded region have been shown to lead to escaping

solutions[6J. The ad hoc criterion of equations ₍₂₇₎ and (28) is also plotted indicating _{a conservative} estimate of stable rolling motion. Instabilities _{such as} a relatively small jump in amplitude may occur below

these forcing levels but do not lead to escape for this

system. For further details on these instabilities see reference [4]. F 0.3 0.2 0.1 - - Numerical bifurcations Approximate -criterion p=O.8 ß=0.1

Figure 7. A comparison between the maximum

velocity and amplitude bounds and the numerically determined bifurcations leading to escape (foran unsymmetrical system). (ß=O.1, p-OS).

!ConclusioflS

research into a possible new criterionThis paper describes the early stages offor antifying

the stability of a rolli _{vessel in a regular and slowly}

cfanging environment. Although the suggested method isbased on simplistic equations and takes no account of

transient motion it is a useful way of introducing important dynamic effects: in contrast to much of the existing stability criteria based on static concepts [11]. A particular advantage of_{the proposed method is} the inclusion of a factor of safety _{which can be}

adjusted to different circumstances _{and improved} information. Further developments would include coupling with other degrees of _{freedom, improvements}

n the modeling process, more realistic forcing conditions and incorporation of transient motion effects to take account of_{wave conditions which are} not slowly changing.

Acknowledgement

The author would like _{to express his}

appreciation to Raymond Plaut of Virginia Polytechnic Institute and State University, _{and Michael Thompson} of University College London _{for useful discussions.}

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