An analysis was carried out to investigate the time period of the thermally induced vibration of clamped and simply supported circular plates with circular variation both in thickness and density. Prior to this study, the variations considered were either linear, quadratic, parabolic, or exponential in nature. To study thermal effect, one-dimensional linear temperature variation on the plates is taken into consideration. Rayleigh–Ritz method is applied to compute the time period of the first three modes of vibration for both plates by varying tapering parameter, thermal gradient, and density. Convergence study of frequency modes for both plates conducted suggests that the convergence rate in case of circular variation is faster than the other studies done. A comparison of time period with the available published results is done. The comparison done concludes that time period obtained in the present study by varying thermal gradient and tapering parameter is found to be less than the other studies done for the same set of parameters. This study helped to establish the fact that, by using circular variation in plate parameters, we can get less time period of frequency modes in comparison to other variations considered till date.

The present study analyzes the natural vibration of non homogeneous visco elastic skew plate (parallelogram plate) with non uniform thickness under temperature field. Here non homogeneity in the plate’s material arises due to circular variation in Poisson’s ratio. Also the circular variation in thickness causes non uniformity in the shape of the plate. Bi linear temperature variation on the plate along both the axes is being viewed. The equation of motion related to frequency modes are solved by Rayleigh Ritz method. The findings of the present analysis are presented with the help of tables.

Time period of natural transverse vibration of a nonhomogeneous skew (parallelogram) plate with variable thickness and temperature field has been investigated on clamped CCCC and combination of clamped and simply supported CSCS edge conditions. The thickness variation on the plate is assumed to be linear in two dimensions, and the temperature variation on the plate is considered to be parabolic in two dimensions. For nonhomogeneity, authors considered circular variation in density. The Rayleigh–Ritz technique is applied to solve the differential equation of motion. A comparative analysis of frequency modes of the present study with the available published result is also given to support the present findings. The convergence study of obtained results is also presented with the help of figures.

A study is carried out to analyze the natural vibration of non-uniform skew (parallelogram) plate made up of non-homogeneous material on different edge conditions, namely clamped (C), simply supported (S) and free (F), using Rayleigh Ritz method. The thickness and Poisson’s ratio of the plate varies circular in one dimension, while the temperature variation on the plate is viewed to be parabolic in nature. Natural frequencies have been obtained for different combinations of structural parameters involve in the study. The results are compared with the available published results from the open literature.

In this article, we mainly discuss the existence and uniqueness of fixed point satisfying integral type contractions in complete metric spaces via rational expression using real-valued functions. We improve and unify many widely known results from the literature. Among these, the work of Rakotch (1962), Branciari (2002), and Liu et al. (2013) is extended. Finally, we conclude with an example presented graphically in favour of our work.