Vibration of FG shell rested on Winkler foundation

Love’s first approximation theory is employed with the combination of Winkler term for the vibration of functionally graded cylindrical shell. MATLAB software is utilized for the vibration of functionally graded cylindrical shell with elastic foundation of Winkler and the results are verified with the open literature. For isotropic materials, the physical properties are same everywhere where the laminated and functionally graded materials, they vary from point to point. Here the shell material has been taken as functionally graded material. The influence of the elastic foundation, wave number, length- and height-to-radius ratios is investigated with different boundary conditions. The frequencies of length-to-radius and height-to-radius ratio are counter part of each other. The frequency first increases and gain maximum value in the midway of the shell length and then lowers down for the variations of wave number. It is found that due to inducting the elastic foundation of Winkler, the frequencies increases.


Introduction
The shell material is organized by various techniques and their applications are seen in dynamical elements such as plates, beams and shells. Moreover, these materials are also More the shell material sustains a load due to physical situations, more the shell is stable. Any predicted fatigue due to burden of vibrations is evaded by estimating their dynamical aspects.
Addition of more physical parameters may give rise more instability in a system of a submerged cylindrical shell (CSs). More than one type of materials is used to structure the functionally graded (FG) materials and their physical properties vary from one surface to the other surface. In these surfaces, one has highly heat resistance property while other may preserve great dynamical perseverance and differs mechanically and physically in regular manner from one surface to other surface, making them of dual physical appearance. All these materials have changeable outer and inner sides and their physical properties greatly differ from each other [1]. Loy and Lam (1997) investigated shell vibrations with ring supports that restricted the motion of cylindrical shells in the transverse direction [2]. This influence was inducted by the polynomial functions. Chung et al. (1981) investigated the vibrations of CSs and presented an analysis of experimental and analytical investigation [3]. Jiang and Olson (1994) recommended the characteristics of analysis of stiffened shell using finite element method to diminish large computational efforts which are required in the conventional finite element analysis [4].
A large use of shell structures in practical applications makes their theoretical analysis an important field of structural dynamics. Since a shell problem is a physical one, so their vibrational behaviors are distorted by variations of physical and material parameters. To elude any complications which may risk a physical system their analytical investigation was done.
Pankaj et al. (2019) studied the functionally graded material using sigmoid law distribution under hygrothermal effect [5]. The Eigen frequencies are investigated in detail. Frequency spectra for aspect ratios have been depicted according to various edge conditions. Ergin and Temarel (2002) did a vibration study of cylindrical shells [6]. The shells lied in a horizontal direction and contained fluid and submerged in it. Sewall and Naumann (1968) considered the vibration analysis of CSs based on analytical and experimental methods [7]. The shells were  [15]. A comprehensive research presented by Salvatore Brischetto (2015) to analyze the vibration characteristic of double-walled CNT by considering shell continuum model [16]. The findings of article were evolved around effects of van der Waals interaction in terms of frequency ratio. Recently some researcher used different methods for nonlinear modeling [17][18][19][20][21] and for other structures [22][23][24][25][26][27][28][29][30][31].
This current paper describes the vibration characteristics of FG-CSs with Winkler foundation using Love's first approximation theory. The frequency behavior is investigated versus circumferential wave number, length-to-radius and height-to-radius ratios. Moreover, frequency pattern is found for the various values of Winkler foundation. The frequency first increases and gain maximum value in the midway of the shell length and then lowers down for wave number
The element of matrix is tabulated in Appendix-I.

Results and discussions
Some numerical results are evaluated for isotropic shell for comparing with existing results found in the literature. The present model can be easily reduced to the isotropic one by considering suitable material parameter for isotropic shell. Hence the present model holds good agreement with the existing results [32][33][34] for isotropic shell as seen in Tables 1-3 Table 6 indicates that the frequency values versus circumferential wave number. It is observed that the frequencies first increases and after decreases and pronounces again on enhancing wave number. It is due membrane and flexural stiffness of the shell. In Table 7, natural frequencies (Hz) with thickness to radius (h/R) for Winkler elastic foundation K = 2× (N-m) for Type and Type-II. With increase in values of h/R, the frequency increases fast in the beginning but gets slower as the shell gets thicker. It is noted that with Winkler foundation, on increases h/R frequencies increases as for other cases.
In Table 7, natural frequencies (Hz) with thickness to radius (h/R) for Winkler elastic foundation K = 1× (N-m) for Type and Type-II. It is noted that with Winkler foundation, on increases h/R frequencies increases as for other cases.      . It is observed that from these Figs, the frequencies of Type-I is greater than Type-II with Winkler foundation K. The frequencies increases on increasing the value of foundation K, from 1x10 6 (N-m) (See Fig. 4) to 2x10 6 (N-m) (Fig. 5). It is noted that shell frequencies lower down as L/R is enhanced i.e., as the shell becomes longer. The C-F conditions have low frequencies than other conditions. Tables 8 and 9 shows the frequencies with the variation of Winkler foundation K =1x10 6~1 0x10 6 of FG-CSs with BCs C-S and SS-SS. The frequencies in Table 8 are tabulated with Type-I and Type-II. These variations of frequencies are drawn with two types of end conditions. In these Tables, the SS-SS (See Table   9) are lower than that of C-F (See Table 8 than that of C-F conditions. For K = 4x10 6~ 10x10 6 , a symmetrical behavior for natural frequencies is seen with proposed boundary conditions. It can be seen that Type-II frequencies are smaller than that of Type-I. It is due to the inducting of material in the shell vibration. The frequencies are effected on inhaling the foundation in the cylinder.

Conclusion
Love's first approximation theory is utilized for vibrations of functionally graded cylindrical shells with Winkler elastic foundation. The frequency behavior is investigated for circumferential wave number, height and length-to-radius ratios. Also the variations have been plotted against the different values of Winkler foundation. The frequency pattern is found for the increasing and decreasing for height and length-to-radius ratios. The frequency first increases and gain maximum value with the increase for circumferential wave mode. It has been investigated that the frequencies get higher on implicating the elastic foundation of Winkler. For future concerns, the present model can be done for investigating the rotating FGshells with Winkler model.

Declarations
Competing interests: The author declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding: Not applicable
Authors' contributions: The whole work is done by a single author