Melnikov Function

As previously seen in Figure 4.2, perturbation of the unforced and undamped system by excitation and damping leads to the separation of the stable and unstable manifolds of the heteroclinic separatrix. The separation d(t₀) between them to O(ϵ) is defined as the Melnikov function and can be shown to be equal to the expression

in (4.18)

d(t₀) = ϵM (t₀) + O(ϵ²) (4.17)

M (t₀) =

∫ _∞

−∞

y₀(t) {

− δ1y₀(t)− δ2y₀(t)|y0(t)|

−[

p₁(t + t₀)x₀(t) + p₃(t + t₀)x³₀(t) + p₅(t + t₀)x⁵₀(t) + ...]}

dt (4.18)

where x₀(t) and y₀(t)) are solutions representing the separatrices of the unper-turbed system (Eqns. (4.10) and (4.11)). The theoretical proof that the Melnikov function represents the separation distance to O(ϵ) can be found in classical texts on nonlinear dynamical systems [12, 120]. An alternate approach to derive the distance between the manifolds was investigated by Vishnubhotla [124] with an accuracy up to the first order of randomness. It was also shown from this method that considering up to O(ϵ) the expressions for the distance between the manifolds was equivalent to the Melnikov function.

The Melnikov function can be separated into two parts - the time invariant mean component M which depends on the damping and the time varying oscillatory part M (t˜ 0) which depends on the excitation.

M (t₀) = M + ˜M (t₀) (4.19)

where

M =

∫ _∞

−∞

y₀(t) {

− δ1y₀(t)− δ2y₀(t)|y0(t)| }

dt (4.20)

M (t˜ 0) =

∫ _∞

−∞

y0(t) {−[

p1(t + t0)x0(t) + p3(t + t0)x³₀(t) +p₅(t + t₀)x⁵₀(t) + ...] }

dt (4.21)

It can further be expressed as

M (t˜ ₀) = ˜M_p1(t₀) + ˜M_p3(t₀) + ˜M_p5(t₀) + ... (4.22)

where

M˜_p1(t₀) = −

∫ _∞

−∞

y₀(t)x₀(t)p₁(t + t₀)dt (4.23) M˜_p3(t₀) = −

∫ _∞

−∞

y₀(t)x³₀(t)p₃(t + t₀)dt (4.24) M˜_p5(t₀) = −

∫ _∞

−∞

y₀(t)x⁵₀(t)p₅(t + t₀)dt (4.25)

In the case of systems excited by only direct excitation the oscillatory compo-nent of the Melnikov function is a linear function of only a single compocompo-nent - roll excitation moment [14, 123, 16, 21]. However, for the case of parametric excitation, M (t˜ ₀) is a sum of stochastic processes as shown in (4.22). Note that the evaluation of the Melnikov function requires only the knowledge of the closed form solution of the unperturbed system (x₀(t), y₀(t)). As the degree of odd polynomial used to repre-sent the stiffness is increased, it becomes harder to obtain the unperturbed solutions (x₀(t), y₀(t)) in closed form [129]. A cubic restoring model is the most commonly used model for representing the stiffness in directly excited systems and has been applied and investigated well by Falzarano et al. [128] and Hsieh et al. [15] for the problem of ship rolling. However, only a few researchers have investigated the fifth and higher order restoring models [14, 130, 131]. Although up to 5^th order stiffness

it is still possible to get analytical expression for (x₀(t), y₀(t)), for higher orders of stiffness (x₀(t), y₀(t)) can only be evaluated numerically [14].

Ideally one would like to use a 9^th or 11^th order polynomial to represent the nonlinear roll stiffness. However, due to the lack of closed form expressions much of the literature is limited to 3^rd or at most 5^th order representations. While the 3^rd order polynomial approximation enables capturing a softening stiffness, a 5^th order representation is required to capture the initial hardening and eventual softening of a typical righting arm curve for wall sided ships. This effect of initial hardening and eventual softening is more enhanced when the righting arm is calculated for the condition of wave crest at midship and is shown in Figure 4.5 for the Pram hull form. The variation in GZ due to the location of wave crest relative to the ship hull is shown in Figure 4.6. As the wave crest moves away from the midship leading to the situation of trough at midship, the initial hardening effect is less pronounced. However to accurately capture both cases one must use at least a 5^th order representation. For this reason, in this investigation of Melnikov methods for the problem of parametric rolling, a 11^th order restoring term will be used to represent the nonlinear roll stiffness. Since no closed form solutions are available, the unperturbed solutions in this study are evaluated numerically.

Each of the components of the oscillatory part of Melnikov function ˜M_pi(t₀) for i = 1, 3, 5, ... are linear functions of the parametric excitations p_i(t) for i = 1, 3, 5, ... re-spectively. From chapter 3, it is known that the parametric excitations p_i(t) for i = 1, 3, 5, ... are related to the wave elevation through various orders of Volterra transfer functions. A comparison of the first and second order of GM variation obtained using the Volterra GZ formulation for irregular sea state characterized by Bretschneider spectrum with H_s = 4 m and T_p = 13 s is shown in Figure 4.7. It can be seen that the second order of variation of GM is significantly smaller in magnitude than the

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

0 10 20 30 40 50 60 70 80

GZ (m)

Roll Angle in degrees GZ Curve for crest at midship condition

GM (m) Hardening Stiffness

Softening Stiffness

Figure 4.5: GZ curve of Pram hull in case of wave crest at midship

0 10 20 30 40 50 60 70 80

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

Roll Angle in degrees

GZ in meters

Variation of GZ curve in Waves

Calm Water GZ Crest at Midship GZ Trough at Midship GZ

Figure 4.6: GZ curve variation in waves

2000 2020 2040 2060 2080 2100 2120 2140 2160 2180 2200

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

Time t in sec GM(1) (t) and GM(2) (t)

Comparison of orders of GM variation from Volterra GZ model for H

s = 4 m and T

p = 13 s

Volterra 1st Order Volterra 2nd Order

Figure 4.7: Comparison of first and second order of GM variation from Volterra GZ model for H_s= 4 m and T_p = 13 s

first order variation.

If the parametric excitations p_i(t) for i = 1, 3, 5, ... are evaluated using only the first order transfer functions, then by definition p_i(t) for i = 1, 3, 5, ... are all zero mean random processes. Under the assumption that wave elevation is an er-godic Gaussian process, it follows that p_i(t) for i = 1, 3, 5, ... evaluated using only the first order Volterra transfer functions will also be ergodic Gaussian processes.

Since ˜M_pi(t₀) for i = 1, 3, 5, ... are linear functions of the parametric excitations pi(t) for i = 1, 3, 5, ..., it also follows that ˜Mpi(t0) for i = 1, 3, 5, ... are ergodic Gaussian processes. From (4.22), ˜M (t₀) is a sum of Gaussian random variables M˜_pi(t₀) for i = 1, 3, 5, ... and hence has a Gaussian distribution as well. The

spec-trum of ˜M (t₀) expressed as SM ˜˜M(ω) is given by

where N_q is the order of stiffness considered in the Volterra Model (in this work a 11^th order stiffness is used) and the cross spectra SM˜_pjM˜_pk(Ω) are given by

SM˜_pjM˜_pk(Ω) = (2π)²T¯j(Ω)Tk(Ω)Spjp_k(Ω) (4.27)

= ϵ²(2π)²T¯j(Ω)Tk(Ω) ¯fj(Ω)fk(Ω)Sηη(Ω) (4.28)

Note that ¯( ) represents complex conjugate, Ω = _ω^ωe

n is the scaled encounter frequency and S_ηη(Ω) is the wave encounter spectrum. f_j represents the first order Volterra transfer function and is given by (3.61). The transfer function T_j is the Fourier transform of −y0(t)x^j₀(t) and is given by

T_j(Ω) = 1 2π

∫ _∞

−∞−y0(t)x^j₀(t)e^−iΩtdt (4.29) Since each of the components of the oscillatory part of the Melnikov function have zero mean, ˜M is also a zero mean ergodic Gaussian process. The mean square value of ˜M is given by

Since a Gaussian distribution of a process is characterized by its mean and

stan-dard deviation, the probability density function of ˜M is given by

pM˜(x) = 1

√2πσM˜

exp (

− x² 2σ²_˜

M

)

(4.31)