Approximate Series Solutions for Nonlinear Free Vibration of Suspended Cables

This paper presents approximate series solutions for nonlinear free vibration of suspended cables via the Lindstedt-Poincare method and homotopy analysis method, respectively. Firstly, taking into account the geometric nonlinearity of the suspended cable as well as the quasi-static assumption, a mathematical model is presented. Secondly, two analytical methods are introduced to obtain the approximate series solutions in the case of nonlinear free vibration. Moreover, small and large sag-to-span ratios and initial conditions are chosen to study the nonlinear dynamic responses by these two analytical methods. The numerical results indicate that frequency amplitude relationships obtained with different analytical approaches exhibit some quantitative and qualitative differences in the cases of motions, mode shapes, and particular sag-to-span ratios. Finally, a detailed comparison of the differences in the displacement fields and cable axial total tensions is made

Research in structural and solid mechanics, 1982

Advances in structural and solid mechanics, including solution procedures and the physical investigation of structural responses are discussed

Large-Amplitude Vibration of Imperfect Rectangular, Circular and Laminated Plate with Viscous Damping

Large-amplitude vibration of thin plates and shells has been critical design issues for many engineering structures. The increasingly more stringent safety requirements and the discovery of new materials with amazingly superior properties have further focused the attention of research on this area. This thesis deals with the vibration problem of rectangular, circular and angle-ply composite plates. This vibration can be triggered by an initial vibration amplitude, or an initial velocity, or both. Four types of boundary conditions including simply supported and clamped combined with in-plane movable/immovable are considered. To solve the differential equation generated from the vibration problem, Lindstedt\u27s perturbation technique and Runge-Kutta method are applied. In previous works, this problem was solved by Lindstedt\u27s Perturbation Technique. This technique can lead to a quick approximate solution. Yet based on mathematical assumptions, the solution will no longer be accurate for large amplitude vibration, especially when a significant amount of imperfection is considered. Thus Runge-Kutta method is introduced to solve this problem numerically. The comparison between both methods has shown the validity of the Lindstedt\u27s Perturbation Technique is generally within half plate thickness. For a structure with a sufficiently large geometric imperfection, the vibration can be represented as a well-known backbone curve transforming from soften-spring to harden-spring. By parameter variation, the effects of imperfection, damping ratio, boundary conditions, wave numbers, young\u27s modulus and a dozen more related properties are studied. Other interesting research results such as the dynamic failure caused by out-of-bound vibration and the change of vibration mode due to damping are also revealed

Nanotalade võnkumise numbriline analüüs

Väitekirja elektrooniline versioon ei sisalda publikatsiooneKäesolevas 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.https://www.ester.ee/record=b550871

Нелінійна динаміка — 2013

The book of Proceedings includes extended abstracts of presentations on the Fourth International conference on nonlinear dynamics

Advances in Vibration Analysis Research

Vibrations are extremely important in all areas of human activities, for all sciences, technologies and industrial applications. Sometimes these Vibrations are useful but other times they are undesirable. In any case, understanding and analysis of vibrations are crucial. This book reports on the state of the art research and development findings on this very broad matter through 22 original and innovative research studies exhibiting various investigation directions. The present book is a result of contributions of experts from international scientific community working in different aspects of vibration analysis. The text is addressed not only to researchers, but also to professional engineers, students and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and what kind of open problems are in this area

Nonlinear Dynamic Responses of a Honeycomb Sandwich Plate Subject to Transverse Excitations

Nonlinear dynamic behaviors of a simply supported honeycomb sandwich plate subjected to the transverse excitations are investigated in this paper. Based on the classical thin plate theory and Von Karman large deformation theory, the governing equation of motion for the honeycomb sandwich plate is established by using the Hamilton principle. The nonlinear governing partial differential equation is discretized to the ordinary differential equations by differential quadrature method and then solved by Runge-Kutta-Fehlberg method. Based on the numerical simulations, combined with nonlinear dynamic theory, the influences of the frequency and amplitude of the transverse excitation are investigated respectively by using the bifurcation diagrams, Poincare maps and phase portraits. The results exhibit the existence of the period-1, period-2 and chaotic responses with the variation of the excitations, which demonstrate that those motions appear alternately

Нелінійна динаміка — 2013

The book of Proceedings includes extended abstracts of presentations on the Fourth International conference on nonlinear dynamics