Free Vibration Analysis of Nanorings Using The Nonlocal Elasticity

In this research article, a free vibration analysis study of nanorings using nonlocal elasticity has been attempted. Nanorings are involved in many areas of our lives. It is seen that nanorings are frequently used, especially in technological tools. In this study, firstly the studies carried out by scientists on nanorings are discussed comprehensively. In particular, these studies are related to the theory of nonlocal elasticity and the vibration of nanorings. In this context, after a literature review, an attempt was made to express the theory of nonlocal elasticity. By using the simplest equations of the nonlocal elasticity theory, the main equation of this theory was obtained. While obtaining the main equation, a number of mathematical functions are used from the simplest equations. It has been seen that the main equation obtained in the nonlocal elasticity theory for nanorings has been used in many studies. After the theory of nonlocal elasticity, the free vibration of nanorings is discussed. Here, mathematical equations are used in the environment embedded in elastic soil. In addition, new mathematical equations are crated by taking the ''"k" _"m" '' value to zero in order to get rid of the elastic embedded environment, that is, the ground effect. Here, the free vibration dimensionless frequency equation of nanorings is obtained. As a result, mathematical equations regarding the theory of nonlocal elasticity and the free vibration of nanorings are derived.