Mathematical Modelling of Response Amplitude Operator for Roll Motion of a Floating Body: Analysis in Frequency Domain with Numerical Validation

Samir K. Dasand Masoud Baghfalaki

Mathematical Modelling of Response Amplitude Operator for Roll Motion of a Floating Body: Analysis in Frequency Domain with Numerical Validation

Samir K. Das1＊and Masoud Baghfalaki2

1. Department of Applied Mathematics, Defence Institute of Advanced Technology, Girinagar, Pune-411025, India
2. Department of Mathematics, Islamic Azad University, Kermanshah branch, Iran

This paper investigates mathematical modelling of response amplitude operator (RAO) or transfer function using the frequency-based analysis for uncoupled roll motion of a floating body under the influence of small amplitude regular waves. The hydrodynamic coefficients are computed using strip theory formulation by integrating over the length of the floating body. Considering sinusoidal wave with frequency (ω) varying between 0.3 rad/s and 1.2 rad/s acts on beam to the floating body for zero forward speed, analytical expressions of RAO in frequency domain is obtained. Using the normalization procedure and frequency based analysis, group based classifications are obtained and accordingly governing equations are formulated for each case. After applying the fourth order Runge-Kutta method numerical solutions are obtained and relative importance of the hydrodynamic coefficients is analyzed. To illustrate the roll amplitude effects numerical experiments have been carried out for a Panamax container ship under the action of sinusoidal wave with a fixed wave height. The effect of viscous damping on RAO is evaluated and the model is validated using convergence, consistency and stability analysis. This modelling approach could be useful to model floating body dynamics for higher degrees of freedom and to validate the result.

RAO; roll motion; hydrodynamic coefficient; Froude-Krylov force; added mass; damping; floating body; frequency domain

1 Introduction1

An accurate prediction of ship motions under waves and wave-induced loads assumes enormous importance for initial stage of ship design, seaworthiness, stability and safety. Understanding roll and associated motion is important for efficient cargo-handling operation and passenger’s comfort. Investigations to understand the hydrodynamic behavior and motion response of a floating body were first started by Froude (1861) with an initial study of roll motion. Several researchers have investigatedthe problem by considering the ship motion to be single or multiple degrees of freedom (S/M-DOF).

Early research in ship hydrodynamics was developed primarily in calm water conditions till the pioneering work of Weinblum and St. Denis (1950) which focused attention on sea environment and subsequently extensive work followed by various researchers (St. Denis and Pierson, 1953; Korvin- Kroukovsky, 1955; Korvin-Kroukovsky and Lewis, 1955). Tick (1959) in his classical paper has shown that the governing equation of this system described by integro-differential equation can be approximated to ordinary differential equations in time domain under some conditions. The motion analysis in time domain was first reported by Cummins (1962) and subsequently Tasai (1967) developed a computational method for strip theory that is applicable for zero forward speed. Salvesen et al. (1970) presented a new strip theory for determining heave, pitch, sway, roll and yaw motions for a ship in arbitrary heading waves with constant forward speed. The motion of a floating horizontal cylinder in a uniform inviscid fluid at irregular wave frequencies was studied by Ursel (1955), considering it as a classical potential flow problem.

In the recent past, considerable amount of research works has been taken up to analyze the roll motion and system stability in time as well as frequency domain by introducing different solution techniques (Clauss et al., 1992). Faltinsen et al. (1995) investigated nonlinear wave loads on a vertical cylinder and Holappa and Falzarano (1998) examined roll motions in time domain considering frequency dependent hydrodynamic coefficients (HC). Using the analytical and numerical models in time domain corresponding to single degree of freedom (1-DOF), two degrees of freedom (2-DOF) and three degrees of freedom (3-DOF), considering zero or non-zero forward speed Das et al. (2005, 2006, 2008) investigated harmonic response of a floating body. Subsequently, Das et al. (2010) modeled sway, roll and yaw motions based on order wise analysis to determine coupled characteristics and computed restoring moment’s sensitivity analysis using numerical simulation. Baghfalaki et al. (2012) developed analytical models in frequency domain corresponding to roll and yaw motions. Baghfalaki and Das(2013) determined response amplitude operator (RAO) for roll motion in frequency domain and established an analogy with the free damped vibration.

In this paper, we present an analytical-numerical approach for the computation of uncoupled roll motion of a floating body in time and frequency domains under small amplitude unidirectional regular sinusoidal beam waves. The aim of the present study is to perform order-wise analysis based on relative magnitude of the HC, like added mass and damping for the roll motion. These coefficients are determined by integrating the two-dimensional sectional coefficients along the longitudinal axis of the ship, using the strip theory approach of Salvesen et al. (1970). The governing equations comprising of second order ordinary differential equations arise after balancing between the hydrodynamic forces and the external exciting forces. The paper is organized as follows: Section-2 provides the mathematical formulation. Section-3 discusses transfer function for roll motion. Section-4 deals with the analytical solution of governing equation and frequency based analysis. Section-5 provides numerical solution with convergence, consistency and stability analysis. Results and discussions are mentioned in Section-6.

2 Mathematical formulations

A right-handed Cartesian co-ordinate system O0(x0, y0, z0) fixed in space is considered where (x0, y0) plane lies in the still water surface, x0is wave propagation direction and z0is upward direction. Considering a right-handed moving co-ordinate system O(x, y, z) with a constant forward speed V of the floating body where x is the direction of the forward speed, y and z are transverse and vertically upward directions respectively. The (x, y)-plane lies in the still water surface and the origin O lies vertically above or under time-averaged position of the center of gravity G. Assuming that the floating body is supposed to carry out oscillations around this moving O(x, y, z) co-ordinate system, a third right-handed co-ordinate system OG(xG, yG, zG) is connected with its origin at G, the center of gravity of the floating body. Here, xGis longitudinal forward direction, yGis lateral direction and zGis upward direction and (xG, zG)-plane is parallel to the still water surface. For positive moving wave direction (x0) with an angle β relative to the speed vector (V), the wave profile can be expressed in the form of the water surface (ζ) can be expressed as a function of both x0and t as shown in Figs. 1.

whereaζ is the wave amplitude. Accordingly,0x can be expressed as in terms of ship′s speed V (Journée and Adegeest, 2003).

Using the relation between the frequency of encounter wave (eω) and the wave frequency (ω), we get

Fig. 1 Definition of co-ordinate systems and incident-wave directions

After substituting Eqs. (2) and (3) in Eq. (1), one can obtain

Considering encountering waves act perpendicular to the longitudinal axis of the floating body (90β=°), the frequency of encountering wave (eω) and wave frequency becomes identical (eωω=). For zero forward speed, both of co-ordinate systems O(x, y, z) and O0(x0, y0, z0) are same.

Under the action of waves, a floating body can exhibit motions of six degrees of freedom (6-DOF) as shown in Fig. 2.

Fig. 2 Schematic diagram of a floating body with sign convention

The wave induced motions of the body can be described as translatory displacements along x, y and z directions, which are known as surge (η1), sway (η2) and heave (η3) respectively, and angular displacements about the same set of axes are known as roll (η4), pitch (η5) and yaw (η6) respectively. Due to the restoring force, only three motions, i.e., heave, roll and pitch are purely oscillatory in nature. For constructing the governing equations, the following assumptions are considered: (1) the floating body has lateral symmetry; (2) incident wave is sinusoidal in form; (3) the responses are linear and harmonic;(4) force components generated by the propeller, wind and current are not considered. The equations of motion in frequency domain representing for six degrees of freedom (6-DOF) for coupled conditions can be written as (Tick, 1959)

where [Mjk], [Ajk(ω)], [Bjk(ω)], [Cjk(ω)] and [Fj(ω)] are the matrix representation of the coefficients for mass, added-mass, damping, restoring and wave force/moment respectively and Djis the wave amplitude for j-th mode of motion. The added-mass and damping are determined by integrating the respective two-dimensional sectional coefficients along the length of the body using new strip theory approach of Salvesen et al. (1970) as shown in Fig. 2.

Fig. 2 Schematic diagram of strips of a floating body

From Eq. (5), governing equations for uncoupled roll motion can be written as

where,

Here, integrations are performed over the length of the ship,a44and b44are the sectional added mass and damping coefficients. The roll restoring coefficient C44is given by

where ? is the displaced volume of the floating body in calm water,GMis the meta-centric height and ρ is the mass density of water. The wave exciting moment F4(ω), on ship hull due to wave of frequencyω, can be expressed as Salvesen et al. (1970).

where4AF is the amplitude of the roll exciting moment corresponding to the wave encountering frequency ω and phase angle ε. The integration has been performed over the length of the body; α is the amplitude of the incident wave;4f and4h represent the sectional Froud-Krylov force and sectional diffraction force respectively corresponding to the wave encountering frequency (ω). For zero forward speed, the term containing U does not appear.

3 Transfer function for roll motion

Considering Eq. (5) for uncoupled roll, one can define

where Zj(ω,θ) is the complex amplitude of the body motion in the j-th mode in response to an incident wave of unit amplitude, frequency ω, and direction θ. This ratio is known as the transfer or the response amplitude operator (RAO). Considering that the response of a vessel to any individual regular wave component to be a linear function of the amplitude and the effect of any individual wave force component is independent of its response to any other wave component, the uncoupled roll transfer function in single degree of freedom (1-DOF) for beam waves can be expressed as

where4X is the roll motion in frequency domain and4D is the responding wave amplitude. For asymptotic cases, i.e, for very small and very large wave frequencies, the expression for RAO for uncoupled roll Eq. (11) becomes

This shows that as the wave frequency approaches to zero or attains large value, the corresponding transfer function for roll becomes either unity or zero respectively. One can also obtain the order of transfer function from the order relation of wave exciting moment.

The order of analytical expression obtained in Eq. (14) becomes identical with the Newman’s formulation (1977) for uncoupled case. To obtain the system frequency, transfer function for intermediate frequencies is considered by using Eq. (11). If the denominator is not zero, a unique solution can be obtained. Using the characteristic equation, one can obtain mathematical expression for system frequency as

To obtain the effect of viscous damping in RAO, we add linear viscous damping termB4v4in Eq. (6) and accordingly mathematical expression for linearized viscous damping can be obtained as

where,

where ξ is the damping ratio and for j,k = 4,440C≠. Accordingly, the transfer function for the viscous roll damping can be expressed as

4 Analytical solution

We define x4(t)=X4(ω)eiωtand f4*(t)=F4(ω)eiωtand substitute in Eq. (6), one can get

After dividing by added mass

where

Considering the following initial conditions

The solution of Eq. (20) can be obtained as

where

We first compute hydrodynamic coefficients (HC) from Eq. (21) and based on relative order of magnitude various cases are obtained and classify them based on three cases. The coupling effects are investigated for three sets of ordinary differential equation (ODE) and designate them as Case-A, Case-B and Case-C. Here Case-A indicates very weakly coupled condition, Case-B indicates weakly coupled condition and Case-C indicates moderately coupled condition. The coupled condition (HC ＞ 0.01) and fully coupled condition (HC ＞ 0.001) is not considered here and for which the solution can be obtained from Das et al. (2010).

Case-A: HC?1.0.

The solution of Eq. (25) can be obtained as

Case-B: HC ＞ 0.1.

Group-II: ω= 0.7, 0.8, …, 1.2

The solution of Eq. (28) can be obtained from Eqs. (23) and (24), and the solution of Eq. (27) can be obtained by putting440b= in Eqs. (23) and (24).

Case-C: HC ＞ 0.01.

Group-I: ω= 0.3, 0.4, …, 1.2

5 Numerical solution

By putting y4=x˙4and theny˙4=˙x4, we converted the second order ordinary differential Eq. (20) into the system of first order ordinary differential equations with the following initial conditions (I.C)

After applying fourth order Runge-Kutta method, the following expressions can be obtained

By putting in equations

One can obtain

where

Return to the initial-value problem Eq. (30), let us define xΦ andyΦ a one-step method and accordingly1iη+,1iμ+and1it+can be obtained as (Stoer and Bulirsch, 1993)

5.1 Convergence analysis

From Eqs. (23) and (24) for4y, with the same coefficients, one can write

The global discretization error for the initial-value problem in Eq. (30), for fixed t when

can be obtained by using Eqs. (23), (24), (39) and procedure defined by Eq. (36) as

Accordingly, one can show that

Therefore, one-step method defined with Eq. (38) is convergent. From Eq. (36) it can be noticed that

5.2 Consistency analysis

Considering the initial-value problem (30), with exact solutions ()xt and ()yt obtained from the analytical method. We define the local truncation error at i-th step(stage) by

where k is the number of time steps. Using Eq. (43), one can show that

It means that the one-step method of Eq. (38) is consistent with the differential Eq. (30).

5.3 Stability analysis

If the one-step method in Eq. (38) is executed in floating-point arithmetic (d-decimal digits) with relative precision eps = 5×10-d, then instead of the one obtains other number, which satisfy a recurrence formula of the form

where the total rounding error εηi+1and εμi+1, in first approximations are made up of three components in the relative rounding error committed in the floating-point computation of

(1)xΦ andyΦ;

(2) the productxhΦ andyhΦ;

(3)ixhηΦ+andiyhμΦ+.

We now estimate the total influence of all rounding errors εηi+1and εμi+1.

Let η～iandμ～ibe the approximate values of ηiand μiactually obtained in d-digit floating-point arithmetic which satisfy the following relations

For simplicity, we also assume

From Eq. (38), one can obtain

ThereforexΦ andyΦ satisfied a Lipschitz condition respect to both of variables x and y, with Lipschitz’s constant as

From Eqs. (48) and (49), one can write

By subtraction of Eq. (38) from Eq. (47) and using Eq. (51), we obtain

We define rounding error

After substituting Eq. (53) in the Eq. (52) gives

The corresponding matrix can be expressed as

By using Eq. (55) and the mathematical induction in matrix form, from Eq. (54), one can show

where I2indicative the identity matrix. Since

we have

Now return to Eq. (37), we assume

According to the property of matrix multiplication, from Eq. (55), one can obtain

Therefore

From Eqs. (58) and (60), we get

If 0h→ then

This means thatrηi+1

andrμi+1are a linear function of εηand εμ. Thus the growth of the rounding error is linear and the above procedure could be sufficient for stability analysis of Eq. (38) using this method.

6 Results and discussions

We consider a floating body of length 150 m, beam 20.06 m, draught 9.88 m and mass of 19 190 tons for which beam-draught ratio is nearly equal to two. The z-coordinate of ship’s center of gravity and meta-centric height is considered at ?3.83 m and 4.0 m respectively. Using the strip theory formulation (Fig. 2 and 3), we compute the added mass, damping and wave force of the floating body based on experimental results of Vugts (1968) and close-fit curve (Frank and Salvesen, 1970). The relative magnitude of HC and their classification is shown in Table 1.

Table 1 Normalized hydrodynamic coefficients

In this procedure, where the respective term does not appear, we mark as ‘Abs’ (Absent). In order to validate the model result we have considered Panamax container ship (Wang, 2000) under the action of sinusoidal wave with periodicity 11.2 s (Fig.4). Table 2 shows the particulars of Panamax container ship for the roll motion and Fig.5 showsthe comparison of present simulation and results obtained by using the Panamax container ship data. The present result agrees well after time t ＞ 60 s, although some minor difference in profile is noticed during the period 30-60 s. Here, the initial fluctuations could be due to the initial condition of roll motion and angular roll moment. However, this effect is not noticed for the longer period.

Fig. 4 Body plan of a Panamax container ship

Table 2 Particulars of Panamax Container ship for roll motion

Fig. 5 Comparison of present simulation with Panamax container ship result

Fig.6 shows sinusoidal wave with frequency ω=0.56 rad/s at t = 20 s for x = 0 to 50 m and y = 0 to 20 m at β= 45ofixed angle in three-dimensional domain. The wave profiles for various angles (β= 0o, 30o, 60o, 90o, 120oand 180o) with frequency ω = 0.56 rad/s and x = y = 5 m for t = 200 s are shown in Fig. 7. For the above mentioned angles, it is observed that magnitude of the wave profiles remains unchanged manifesting periodic nature with changing time period. The time period is observed to be maximum at β = 60oand for angles β= 30oand 90otime periods are almost similar. At the angles β= 120oand 180o, the wave profiles manifest oscillatory characteristics with different beating period forming envelopes and this beating period indicates that along the time axis difference exists in the magnitude of the wave profile in each time step which is noticed only in Figs. 7(e)-(f). However, this difference is not noticed in the Figs. 7(a)-(d).

Fig. 6 Wave profile x= 0 to 50 m, y = 0 to 20 m for β = 450and ω = 0.56 rad/s

Fig. 7 Wave profile for different angles in frequency ω = 0.56 rad/s, at x = y =5 m

The roll exciting moment in frequency domain for time t= 5.23 s, 8.97 s and 20.93 s are shown in Fig. 8(a) and is found to maximum for t = 8.97 s with oscillatory characteristics. Fig. 8(b) shows the profile of roll exciting moment in frequency domain with phase angles φ=0, π/2and π/4 for time t = 8.97 s. The corresponding roll exciting moment in time domain for different frequencies ω = 0.3 rad/s, 0.7 rad/s and 1.2 rad/s are shown in Fig. 8(c) and it is observed that at ω= 0.7 rad/s roll exciting moment is maximum. The comparisons of roll wave exciting moment in frequency and time domain result indicates that magnitude remains almost unchanged for time t = 8.97 s corresponding to the wave frequency ω= 0.7 rad/s. However with the increase or decrease in time and frequency significant deviations are noticed in the profiles of roll wave exciting moment for frequency as well for time. Baghfalaki et al. (2012) has shown that natural frequency of the roll motion is equal to 0.7 rad/s. The behavior of frequency of encounter (eω) corresponding to change in angles are shown in Fig. 8(d) for ω= 0.3 rad/s, 0.7 rad/s and 1.2 rad/s. The magnitude of frequency of encounter is found to be maximum at ω= 1.2 rad/s and decreases with the decreases of frequency.

Fig. 8 Roll exciting moment and frequency of encounter

The schematic diagram for analytical and numerical modeling process is shown in Fig.9. We first compute hydrodynamic coefficients and wave exciting moment and analyze in frequency domain. The frequency based analysis are carried to study relative importance of the HC (based on normalization procedure) and three cases are obtained (1) Case-A, HC ＞ 1.0 (2) Case-B, HC ＞ 0.1 and (3) Case-C,HC ＞ 0.01. The Eq. (20) indicates the full form of the governing equation and corresponding analytical solution is shown in Eq. (23). Using the initial conditions x4(0)=0.5 and(0)=0.5 numerical solution is obtained using fourth order Runge-Kutta method as shown in Eq. (38).

Fig.9 Schematic diagram for analytical and numerical models

The real part, imaginary part and norm of roll transfer function corresponding to viscous and non-viscous cases is shown in Figs. 10 (a), (c) and (e). We notice that the real part of uncoupled roll transfer function is maximum for the frequency 0.6 rad/s. However, imaginary part and norm of roll transfer function is maximum for frequency 0.7 rad/s. In all the cases, the magnitude of viscous roll transfer function is lower. After decomposing the characteristic equation into real and imaginary parts, plots of real part, imaginary part and norm of natural frequency corresponding to viscous and non-viscous cases corresponding to the natural frequency (eω) are obtained (see Figs.10 (b) (d) and (f)). Marginal difference in the real part of natural frequency is noticed for viscous and non-viscous cases with the increase of frequency whereas for imaginary part this difference is significant (see Figs. 10 (b) and (d)). However, the norm of natural frequency becomes identical for viscous (v) and non-viscous (N.V) case (see Fig. 10(f)).

Fig. 10 Real part, imaginary part and norm of roll transfer function/natural frequency corresponding to iscous and non-viscous cases

Fig. 11 Roll amplitude for the Case A

Fig. 12 Roll amplitude for Case B

Fig. 11 shows roll amplitude for ω = 0.3, 0.56, 0.7, 0.74, 0.9 and 1.2 rad/s which is obtained from Eq. (26) for Case-A at t = 100 s. The roll amplitude is found to be maximum for ω= 0.3 rad/s and minimum for ω= 1.2 rad/s. The corresponding roll amplitude for Case-B for t = 100 s are obtained from Eq. (23) for the frequencyω= 0.3, 0.56 rad/s (Group-I, Figs. 12). For other frequencies ω= 0.7, 0.74, 0.9 and 1.2 rad/s roll amplitude corresponding to Case-B and Case-C are same.

Fig. 13 shows analytical solution (A.S) and numerical solution (N.S) for roll amplitude and a very good agreement is obtained while comparing both the methods. The behavior of discretization error for x4at frequencies ω = 0.3, 0.56, 0.7, 0.74, 0.9 and 1.20 rad/s for step length h = 0.01 is shown in Fig. 14.

Fig. 13 Roll amplitude for the Case C (analytical solution (A.S) by Eq. (23) and numerical solution (N.S) by Eq. (38)

Fig. 14 Discretization error for roll (x4)

Fig. 15 Rounding error for η when Eps = 0.000 01 at ω = 0.56 and ω = 0.74

We notice that for frequency 0.30 rad/s, the discretization error reach to zero. But the discretization error in the other frequencies does not tend to zero and is very small with acceptable limit. The maximum discretization error value is less than 0.005. Thus the numerical method corresponding to Eq. (38) has converged to four decimal digits. We compute rounding error for step length, h = 0.5, 0.05 and 0.01 as shown in Fig. 15. These figures show that the behavior of rounding error for η in frequencies 0.56 rad/s and 0.74 rad/s are almost same. It is noticed that for h = 0.5 the numerical method in Eq. (38) is not stable because the growth of a rounding error is of exponential in nature. The comparison of rounding errors for all three cases show that h= 0.01 gives best result. However, one can also choose h = 0.05 in order to reduce the step size for which maximum error value is 0.0016 which can be considered and within the acceptable range.

7. Conclusions

The main objective of this paper is to develop mathematical model to determine RAO for uncoupled rollmotion in the frequency domain which can be further investigated to solve the problem of higher degrees of freedom. The relative importance of the hydrodynamics coefficients through the group based classification is analyzed and the solution is validated through numerical experiments. Using the analysis of convergence, consistency and stability the effects of viscous damping on RAO is also analyzed. This modelling technique provides some guidelines to compute and validate the model when analytical solution is difficult to obtain for floating body motion of higher degrees of freedom.

Acknowledgements

The second author gratefully acknowledges the financial grant of Islamic Azad University Kermanshah branch, Kermanshah, Iran for this research work and its support for the Digital Library.

Nomenclatures

AijCross coupling coefficients

BijCross coupling coefficients for damping

CijHydrostatic restoring cross-coupling coefficients

DjWave amplitude for j-th mode of motion

FjExciting force/moment due to waves

FjAAmplitudes of exciting force and moment

GM Meta-centric height

IjMoment of inertia

IjkProduct of inertia

M Mass of the body

MjkGeneralized mass matrix for the body

U Speed of the body

aijCoefficient of two-dimensional sectional added mass

a4A4aijfor aftermost section

bijTwo-dimensional sectional damping coefficient

b4A4bijfor aftermost section

c44Roll restoring coefficient

fjSectional Froude-Kryloff force

g Acceleration due to gravity

hjSectional diffraction force

x, y ,z Coordinate system

Δ Displacement volume of ship

? Displacement of ship

α Incident wave amplitude

β Un-damped natural frequency of the damped system

x Displacement

x˙First order derivative of x with respect to time

˙xSecond order derivative of x with respect to time

θ Phase angle

ρ Density of water

ξ Damping ratio

ω Wave encountering frequency

ωnNatural frequency

kjk,λjkFormula in Runge-Kutta method

ηi,μiFinal expressions in Runge-Kutta method

τxi,τyiLocal truncation errors

rηi+1,rμi+1Rounding of errors

εηi+1,εμi+1Total influence of all rounding of errors

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Author biographies

Samir K. Das is Professor and Head in the Department of Applied Mathematics, Defence Institute of Advanced Technology. Prof. Das has obtained his Ph.D from Indian Institute of Technology (IIT) - Kharagpur in the year 1987. He has completed 24 client-sponsored engineering project of national and international importance that includes World Bank sponsored project as well. His research focus and area of interest are theoretical, computational and environmental fluid dynamics, mathematical modelling of engineering problems and floating body motion in waves. He has already supervised several PhD students and guided nearly 50 Masters thesis (M.Sc/M.C.A/M.Tech). He has published 75 research papers in various journals and conferences that also include 5 contributed article in books. He is a reviewer of several national and international journals of repute and also in the Editorial Board of 10 national/international journals. He has delivered invited/guest lectures at various national and international conferences and symposiums in India and abroad.

Masoud Baghfalaki is currently Head of the Department of Mathematics, Payame Noor University of Kermanshah, Kermanshah branch, Kermanshah, Iran. Earlier, he was a faculty of Islamic Azad University, Kermanshah branch, Kermanshah, Iran. He has also worked for various academic and administrative positions. He has submitted his Ph.D thesis, University of Pune, India. He has published five research papers and communicated few more papers in the international journals.

1671-9433(2014)02-0143-15

date: 2013-11-21.

Accepted date: 2014-04-21.

The financial grant of Islamic Azad University Kermanshah branch, Iran (Grant No: 35/3/622281, 7-9-2009)

＊Corresponding author Email: samirkdas@diat.ac.in

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014