Researchers, scientists and design engineers always try to know about first few modes of vibrations before finalizing any design of machine or structure so that they can save the money & time of the nation. For these two fold requirements i.e. money and time, they can get a lot of help from the contents of this book. Actually, deeply analysis of the graphs in each chapter or model can provide them the required or desired values of frequency parameters & surely, they can get the same with the proper choice of varying parameters i.e. thermal gradient, taper parameters, non-homogeneity constant, aspect ratio and skew angles. I assure them that if they use the findings of these models in practical manner, it will definitely help to provide more reliable, durable and efficient mechanical structures or machines

In the modern epoch of science and technology; scientists, engineers & practitioners have become more interested to analyze the vibrational characteristics of structural components which are mainly operated under elevated temperature i.e. space technology, nuclear energy applications, high speed atmospheric flights etc. Due to the influence of temperature, the physical properties of materials change significantly. Further, the use of various plate type structural elements under high temperature environments, particularly in space vehicles and modern missiles, demand that non-homogeneity of the material should be taken into account to predict their natural frequencies. The book is mainly the collection of theoretical mathematical models about the effect of bi-dimensional temperature variations on vibrations of four sided clamped square plate with bi-dimensional thickness variability. Numeric values of natural frequencies are calculated for both the modes of vibrations for different values of taper parameters & thermal gradient. We hope that the content of the book will definitely help the readers as well scientists to fulfill their requirements.

In the course of time, the study of vibration of plates with thickness variability has acquired great importance in nuclear reactor technology, earth-quake resistant structures, aeronautical field, naval structures and machine designs etc. Plates of variable thickness are mainly used for two fold requirements of safety and economy. Therefore, it becomes the need of the hour to provide some authentic theoretical mathematical models for the help of mechanical/design engineers & scientists so that before finalizing any design or mechanical structure, they can use of them with practical approach and provide much better design and structures with more durability, reliability & strength. In the present work, some mathematical models on vibrational behaviour of visco-elastic homogeneous rectangular plates with bi-directional thickness variability are deeply discussed for fixed boundary conditions. Numeric values of time period, deflection and logarithmic decrement are given in tabular and graphical forms both. I hope that the engineers/scientists may develop a qualitative understanding of plate vibrational behaviour by using these findings.