Contents lists available at ScienceDirect Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng Wavelet based spectral algorithm for nonlinear dynamical systems arising in ship dynamics G. Hariharan ⁎ , R. Rajaraman, D. Sathiyaseelan Department of Mathematics, School of Humanities & Sciences, SASTRA University, Thanjavur 613401, Tamilnadu, India ARTICLE INFO Keywords: Vibrating circular membrane Damped spring mass system Ship oscillatory motion Chebyshev wavelets Operational matrices Nonlinear equation ABSTRACT In this paper, we have applied an efficient shifted second kind Chebyshev wavelet method (S2KCWM) to vibrating dynamical models arising in mechanical systems such as vibration of circular membrane, damped spring mass system and ship oscillatory motions. To the best of our knowledge, until now there is no rigorous wavelet based solution has been reported for the vibrating dynamical models. The power of the manageable method is confirmed. The wavelet solutions are compared with numerical simulations by MATLAB. Good agreement between the solutions is presented in this paper. Some numerical examples are given to demonstrate the validity and applicability of the proposed method. Moreover the use of Chebyshev wavelets is found to be simple, efficient, flexible, convenient, less computation costs and computationally attractive. 1. Introduction In recent years, membrane dynamics model is a classical problem in mechanical vibrations. Among the several types of membranes, circular membranes are the most widely studied due to their numerous applications in engineering (Agarwal and D.O’Regan, 2003; Shin, 1995; Javidinejad, 2013; Ji- Ping and Xin-LI, 2006; Siedlecka et al., 2012; Civalek and Gürses, 2009; Hsu, 2007; Chapra and Canale, 2002). From the study of musical notes of percussion instruments, circular membranes have been used to design diaphragms for condenser microphones, model the dynamics of the human ear (Alsahlani and Mukherjee, 2013), understand the vibration characteristics of mem- brane mirrors and gossamer structures (Alsahlani and Mukherjee, 2013), measure surface tension (Alsahlani and Mukherjee, 2013; Mgharbel et al., 2009), and design ink-jet printers (Alsahlani and Mukherjee, 2013). The similarity between the differential equations of membranes and waveguides motivated the study of circular mem- branes with constraints in the 1970s and 1980s (Krenk and Schmidt, 1981). Sen et al., (2006) established the interpolation for nonlinear boundary value problems (BVPs) in circular membrane with known upper and lower solutions. Recently, Alsahlani and Ranjan Mukherjee (Alsahlani and Mukherjee, 2013) had introduced the dynamics of a circular membrane with an eccentric circular areal constraint. Considerable attention has been directed toward the chaos, chaotic systems and solutions of nonlinear oscillator differential equations since they play crucial role in natural and physical simulations. Surveying the literature shows that a variety of solution methods have been developed so far to solve the duffing oscillator equation (Cvetićanin, 2009; Trueba et al., 2000; Huang and Zhu, 2012; Kim and Park, 2015; Nourazar and Mirzabeigy, 2013; Cveticanin, 2011; Joseph and Minh-Nghi, 2005; Sharma et al., 2012; Zhu, 2014; Kaur et al., 2014). Some researchers in their studies consider damping into the duffing oscillator. When the duffing oscillator involves damping, the amplitude of oscillation reduces over time and we have a non- conservative system. Most analytical methods are unable to handle non conservative oscillators. Our aim in the present study is to obtain the solution of the duffing oscillator free response considering different damping effects and with different initial conditions by the second kind Chebyshev wavelet method and comparing the results with the results of a numerical solution using the MATLAB. Roll motion is a major concern of ship and offshore operators. The major technical difficulties related to the roll motion of a floating body are the nonlinear effects of roll damping (Huang and Zhu, 2012; Kim and Park, 2015; Bulian, 2004). A highly nonlinear characteristic is strongly involved in the ship roll motion models. It is necessary that the dynamic stability of ships in realistic sea is dependent on its rolling motion and therefore the investigation of ship's roll dynamics is most crucial unlike other degrees of freedom of ship motion. For this purpose, it is generally required to investigate ship's roll damping for accurate and efficient prediction of its response to various loading environments and development of control strategies: this is essential for the design of ship-shaped structures. However, the determining of the roll damping is difficult because of its strong nonlinearity. Wavelet analysis has found their way into many different fields in http://dx.doi.org/10.1016/j.oceaneng.2016.09.022 Received 27 December 2015; Received in revised form 8 September 2016; Accepted 12 September 2016 ⁎ Corresponding author. E-mail addresses: hariharan@maths.sastra.edu (G. Hariharan), rraja@maths.sastra.edu (R. Rajaraman), sathyaseelan86@gmail.com (D. Sathiyaseelan). Ocean Engineering 126 (2016) 321–328 0029-8018/ © 2016 Elsevier Ltd. All rights reserved. crossmark