Numerical analysis of vibrations of nanobeams

Abstract

Käesolevas väitekirjas uuritakse nanomaterjalist valmistatud talade omavõnkumisi mitmesuguste kinnitusviiside korral. Väitekirjas on välja töötatud meetodid nanotalade omavõnkesageduse määramiseks astmelise nanotala jaoks erinevate kinnitustingimuste korral; kusjuures astmete nurkades asuvad stabiilsed praod või prao-tüüpi defektid. Prao mõju võnkesagedusele modelleeritakse nn kaalutu väändevedru meetodil. Selle meetodi kohaselt tuleb reaalne astmega tala asendada kahest elemendist koosneva süsteemiga, kus elemendid on omavahel ühendatud väändevedruga, mille jäikus on pöördvõrdeline pinge intensiivsuse koefitsiendiga prao tipu juures. Kuna pinge intensiivsuse koefitsiendi väärtused on leitavad kataloogidest, siis see meetod võimaldab omavahel siduda nanotala omavõnkesageduse ning prao pikkuse ja laiuse. Väitekiri koosneb sissejuhatusest, viiest peatükist ning kirjanduse loetelust, mis sisaldab 82 nimetust. Sissejuhatus kujutab endast esimest peatükki. Teises peatükis on toodud põhivõrrandid ning põhieeldused. Esimesed kaks peatükki on referatiivsed, ülejäänutes esitatakse originaalseid tulemusi. Kolmandas peatükis esitatakse nanotalade võnkumise võrrandid, mis arvestavad tala elementide pöördeinertsi. Need on Euler-Bernoulli võrrandite üldistuseks juhule, kui pöördeinertsi arvestamine on kohustuslik. See süsteem on lahendatav ka muutujate eraldamise teel. Neljandas peatükis lahendatakse põhivõrrandite süsteem numbriliselt. Näidatakse muuhulgas, et süsteemi saab hõlpsasti lahendada Maclaurini rea abil. Viies peatükk on pühendatud nanotalade võnkumise uurimisele juhul, kui nanotala on kinnitatud elastsete tugede abil st. toed ei ole jäigad. Kuuendas peatükis uuritakse pragudega nanotalade võnkumisi arvestades termilisi mõjutusi st. temperatuuripingeid. Väitekirjas saadud tulemusi on võrreldud erijuhtudel kirjandusest leitavate tulemustega ning veendutud, et väitekirjas esitatud tulemused on heas kooskõlas teiste uurijate poolt saadud tulemustega. Väitekirjas saadud tulemuste põhjal on avaldatud koos juhendajaga 10 teadusartiklit.

In this dissertation, an analysis of the dynamic behavior of nanobeams with different physical and geometrical properties using several numerical techniques is presented. Euler-Bernoulli beam theory and nonlocal theory of elasticity are used to simulate the nanobeam. Nanobeams are considered with some special requirements such as tapered, axially graded, and double beams. First of all, in a tapered beam, the width of the beam is varying exponentially along the x-axis from one end to another end. The properties of the tapered beam are to reduce material consumption and provide the cross-sectional area according to the moment distribution. Secondly, in an axially graded beam, material properties such as elasticity and density are varying exponentially from one end to another end. The axially graded beam can be considered as a non-homogeneous as well as a composite beam. In this beam, the material properties can be distributed according to the requirement. The axially graded beam overcomes the limitation of conventional composite. Finally, in a double beam, two identical nanobeams are connected by a Winkler-type spring layer. Double beams are used for absorbing the vibration. It reduces deflection and vibration. The double beam is modeled by the coupled differential governing equations. Some adverse effects such as cracks and the influence of the temperature are considered. Cracks are common defects in nanostructures. Single and multiple cracks are considered in this analysis. According to the model, the crack is replaced by a rotational spring where the crack divides the beam into two segments that are connected to each other by the spring at the crack position. Cracks reduce the overall stiffness of the beam. The effect of temperature is significant for the vibration of nanobeams. The thermal load is compatible with the mechanical load where the thermal load is modeled as an axial load. It reduces the natural frequency. The main objective of this research is to find suitable techniques for a reliable, cost-effective design that is able to fulfill the desired requirements. That is why the important feature of this research is to apply numerical techniques for solving these problems. Three different approximation techniques such as homotopy perturbation technique, power series method, and Maclaurin series method are used for solving these problems. These techniques are useful for solving linear and non-linear differential equations. However, these techniques are rare to analyze the nano-material. These techniques are applied effectively to scrutinize the model of nanobeams. Obtained results are verified with the results of other researchers in the existing literature. This analysis can be used to design nano-electromechanical devices effectively.