(untitled) (Journal of Civil Engineering Researchers, 2025)
Beams are always noteworthy as an engineering structure due to their wide application in industry such as bridges, railway tracks, floors and ceilings of buildings and many other cases. Therefore, considering the wide application of these materials in industry, the analysis of this category of structures becomes important in the overall design process of these parts in a structure. With the increasing use of beams in industry and the need to increase their efficiency and ensure their proper functioning, the use of new materials such as functionally graded materials[1]has increased. The use of composite materials, shape memory alloys, piezoelectric materials, etc. and the expansion of the scope of use of these materials has led to increased efforts by researchers to achieve the construction and design of structures and parts with better efficiency and quality. On the other hand, conducting experimental analyses on these materials is associated with problems such as size, price, complexity of the laboratory model, etc., hence the presentation of general theoretical models. More suitable for ‌In this research, the vibration analysis of a nanobeam is considered. Unidirectional FGM (functional properties along the beam thickness) with carbon nanotubes and a layer‌ Metal and ceramic supports on various types of supports ‌Sometimes. ‌The solutions, including simple, complex, etc., were discussed. The solution method in question was the mixing method, and in line with the thickness of the numerical method, the differential function. It has been DQ based on this, relationships have been extracted regarding how to increase the natural frequency and the lowest natural frequency, as well as the length-to-thickness ratio, natural frequency changes, and dimensionless natural frequency changes in the beam.