Impacts of various boundary conditions on beam vibrations

In real life, boundary conditions of most structural members are neither totally fixed nor completely free. It is crucial to study the effect of boundary conditions on beam vibrations . This thesis focuses on deriving analytical solutions to natural frequencies and mode shapes for Euler-Bernoulli Beams and Timoshenko Beams with various boundary conditions under free vibrations. In addition, Green\u27s function method is employed to solve the close-form expression of deflection curves for forced vibrations of Euler-Bernoulli Beams and Timoshenko Beams.;A direct and general beam model is set up with two different vertical spring constraints kT1, k T2 and two different rotational spring constraints kR1, kR2 attached at the ends of the beam. These end constraints can represent various combinations of boundary conditions of the beam by varying the spring constraints. A general solution for the Timoshenko beam with this various boundary conditions is derived, and to the best of our knowledge, this solution is not available in the literature. Numerical examples are presented to illustrate the effects of the end constraints on the natural frequencies and mode shapes between Euler-Bernoulli beams and Timoshenko beam. The results show that Euler-Bernoulli beams have higher natural frequencies than Timoshenko beams at different modes. The ratio of the natural frequencies for Timoshenko beams to the natural frequency for Euler-Bernoulli beams decreases at higher modes. Natural frequencies at lower modes are more sensitive to boundary constraints than natural frequencies at higher modes

Continuum Modeling on Size-dependent Properties of Piezoelectric Nanostructures

Piezoelectric beam- and plate-based nanostructures hold a promise for device applications in the nanoelectromechanical systems (NEMS) due to their superior mechanical and electromechanical coupling properties. “Small is different”, nanostructured piezoelectric materials exhibit size-dependent properties, which are different from their bulk counterparts. For predicting the unique physical and mechanical properties of these novel nanostructures, continuum mechanics modeling has been regarded as an efficient tool. However, the conventional continuum models fail to capture the size effects of nanostructures and thus are not directly applicable at the nanoscale. Therefore, it is necessary to develop modified continuum models for piezoelectric nanostructures by incorporating the size effects and investigate the size-dependent properties of piezoelectric nanostructures based on the developed models. Nanoscale structures are characterized by a high surface to volume ratio. The atoms in the surface layers of a structure are exposed to a different environment compared to those in the bulk of the structure. Thus, surface has a considerable influence on the physical and mechanical behaviors of nanoscale structures and is believed to be responsible for their size-dependent properties. In addition, for nanostructured piezoelectric materials, the strain gradient induced flexoelectricity could be significant and contribute to their size-dependent properties. In this thesis, the influence of the surface effects and flexoelectric effect on the mechanical and electrical properties of piezoelectric nanostructures is investigated through modified continuum models. Firstly, based on a surface piezoelectricity model and the generalized Young-Laplace equations, modified continuum models with surface effects are developed to investigate the bending, vibration, buckling behaviors and electromechanical properties of piezoelectric nanobeams and nanoplates with different boundary conditions. Next, by accounting for the flexoelectric effect through the extended linear theory of piezoelectricity and conventional beam models, the static and dynamic responses of piezoelectric nanobeams are presented. It is demonstrated from this study that the size effects prominently influence the mechanical behaviors and the electroelastic responses of piezoelectric nanostructures. This research carries out a theoretical methodology to predict the static bending, electroelastic field distribution, resonant frequencies of vibration and critical electric potential for the mechanical buckling of piezoelectric nanostructures with different structure geometries, loading conditions and boundary conditions, which is expected to provide a fundamental understanding on the electromechanical coupling behavior of piezoelectric structures at the nanoscale. It is helpful for understanding the size-dependent properties of nanostructured piezoelectric materials and performance improvement of the beam- and plate-based electronic devices in NEMS

Haari lainikute meetod omavõnkumiste analüüsiks ja parameetrite määramiseks

Tala on konstruktsioonielement, mille ülesandeks on vastu pidada erinevatele koormustele. Projekteerimisel alahinnatud koormused, ebatäpsused tootmisel, söövitav keskkond, konstruktsiooni vananemine ekspluatatsiooni käigus võivad talasid kahjustada ning põhjustada kogu konstruktsiooni purunemist. Seetõttu talade dünaamilise käitumise modelleerimine ja ekspluatatsiooni jälgimine on jätkuvalt aktuaalne teema konstruktsioonide mehaanikas. Käesolev väitekiri on suunatud süstemaatilisele lähenemisele võnkumiste analüüsimiseks ja purunemise parameetrite määramiseks Euler-Bernoulli tüüpi talades. Töös pakutakse välja Haari lainikute meetod sageduste arvutamiseks ja andmete töötlemiseks. Nimelt, väitekirja esimeses osas on Haari lainikuid ja nende integreerimist rakendatud vabavõnkumise ülesannete korral, kus lahendatavaks võrrandiks on muutuvate kordajatega diferentsiaalvõrrand, millel puudub analüütiline lahend (näiteks ebaühtlase ristlõikega tala, materjali funktsionaalse gradientjaotusega tala). Arvutused kinnitasid, et pakutud lähenemisviis on kiire ja täpne vabavõnkumiste sageduste arvutamisel. Väitekirja teine osa käsitleb vabavõnkumisega seotud pöördülesandeid: pragude, delaminatsioonide, elastsete tugede jäikuse, massipunktide parameetrite määramist modaalsete omaduste kaudu. Kuna purunemise asukoha ja ulatuse arvutamine võnkumise diferentsiaalvõrrandist ei ole analüütiliselt võimalik, kasutatakse antud töös tehisnärvivõrke ja juhumetsi. Andmekogumite genereerimiseks lahendati võnkumise võrrand ning tulemusi töödeldi Haari lainikute abil. Arvutused näitasid, et Haari lainikute abil genereeritud andmekogumite arvutamiseks kuluv aeg oli üle kümne korra väiksem kui vabavõnkumiste sagedustele põhinevate andmekogumite arvutusaeg; Haari lainikute abil genereeritud andmekogumid ennustasid paremini purunemise asukohta, samas vabavõnkumiste sagedused olid tundlikumad purunemise ulatuse suhtes; enamikel juhtudel andsid tehisnärvivõrgud sama täpseid ennustusi kui juhumetsad. Töös pakutud meetodeid ja mudeleid saab kasutada teistes teoreetilistes ülesannetes vabavõnkumiste ja purunemiste uurimiseks või rakendada talade purunemise diagnostikas.A beam is a common structural element designed to resist loading. Underestimated loads during the design stage, looseness during the manufacturing stage, corrosive environment, collisions, fatigue may introduce some damage to beams. If no action is taken, the damage can turn into a fault or a breakdown of the whole system. Hereof, the entirety of beams is a crucial issue. This dissertation proposes a systematic approach to vibration analysis and damage quantification in the Euler-Bernoulli type beams. The solution is sought on the modal properties such as natural frequencies and mode shapes. The forward problem of the vibration analysis is solved using the Haar wavelets and their integration since the corresponding differential equations do not have an analytical solution. Multiple numerical examples indicate that the proposed approach is fast and accurate. Damage quantification (location and severity) of a crack, a delamination, a point mass or changes in the stiffness coefficients of elastic supports on the bases of the modal properties is an inverse problem. Since it is not analytically possible to calculate the damage parameters from the vibration differential equation, the task is solved with the aid of artificial neural networks or random forests. The datasets are generated solving the vibration equations and decomposing the mode shapes into the Haar wavelet coefficients. Multiple numerical examples indicate that the Haar wavelet based dataset is calculated more than ten times faster than the frequency based dataset; the Haar wavelets are more sensitive to the damage location, while the frequencies are more sensitive to the damage severity; in most cases, the neural networks produce as precise predictions as the random forests. The results presented in this dissertation can help in understanding the behaviour of more complex structures under similar conditions, provide apparent influence on the design concepts of structures as well as enable new possibilities for operational and maintenance concepts.https://www.ester.ee/record=b539883

The use of positive and negative penalty functions in solving constrained optimization problems and partial differential equations

The Rayleigh-Ritz Method together with the Penalty Function Method is used to investigate the use of different types of penalty parameters. The use of artificial springs as penalty parameters is a very well established procedure to model constraints in the Rayleigh-Ritz Method, the Finite Element Method and other numerical methods. Historically, large positive values were used to define the stiffness coefficient of artificial springs, until recent publications demonstrated that it is possible to use negative values to define the stiffness coefficients of the springs. Furthermore, recent publications show that constraints can be enforced using positive and negative mass or inertia in vibration problems and in a more generic sense using eigenpenalty parameters which are penalty parameters in the matrix associated with the eigenvalue. Before the commencement of this thesis, solutions using artificial inertia were published only for beams and simple spring-mass systems. In this thesis the use of all possible types of penalty parameters are investigated in vibration problems of Euler-Bernoulli beams, thin plates and shallow shells and in elastic stability analysis of Euler-Bernoulli beams, including penalty parameters associated with the geometrical stiffness matrix. The study includes the use of penalty parameters for both enforcing support boundary conditions and continuity conditions along structural joints. This investigation started with the selection of the set of admissible functions that would: (a) allow modelling of beams, plates and shells in completely free boundary conditions; (b) not present any limitation in the number of functions that can be used in the solution. This gives the possibility to converge to the constraint solution and to model any type of boundary conditions. The procedure proposed in this work combines several advantages: accuracy of the results, relative fast convergence, simplicity of the set of admissible functions and flexibility to define boundary conditions. While there are other procedures that may give better accuracy for specific cases, the proposed method is more widely applicable. The procedure used in this work also includes a way to check for round-off errors and ill-conditioning in the results; as well as a way to bracket the exact solution with upper and lower-bound results

Identification of Non-Classical Boundary Conditions with the Aid of Artificial Neural Networks

Käesolev magistritöö uurib mitteklassikaliste kinnitustingimustega elastsete Euler-Bernoulli talade vabavõnkumise resonantssagedusi. Eesmärgiks on vaatluse all olevate tala mudelite korral hinnata ning võrrelda tehisnärvivõrkude abil identifitseeritud jäikuse parameetreid elastsete kinnitustingimuste korral. Vaatluse all on kahte tüüpi talad: tala elastse otsakinnitusega ning tala vahepealse elastse toega. Mõlema variandi kohta töötatakse läbi rida näiteid erinevate kinnitustingimustega. Kuna kinnituste jäikusparameetrite arvutamine võnkumise diferentsiaalvõrrandist ei ole analüütiliselt võimalik, siis on mõistlik otsida sellele alternatiivi. Ühe variandina pakutakse käesolevas töös välja tehisnärvivõrkude rakendamine. Tehisnärvivõrgud põhinevad bioloogilistel närvivõrkudel, nagu näiteks inimese aju. Tehisnärvivõrgu peamiseks eeliseks teiste meetodite ees on tema võime olemasolevate näidete põhjal õppida, mis tähendab, et närvivõrke on võimalik treenida sisendi abil soovitud tulemusi produtseerima. Seega, vajaliku ülesande lahendamiseks pole enam tarvis ise kõiki parameetrite koefitsiente arvutada, vaid piisab, kui meil on olemas teatud hulk näiteid oodatavate koefitsientide kohta, ning nende näidete abil treenitud tehisnärvivõrk on suuteline ülejäänud tulemusi ise identifitseerima. Käesolevas töös antakse ülevaade võnkuvatest Euler-Bernoulli taladest ja nende võimalikest kinnitustingimustest, ning tutvustatakse tehisnärvivõrkude peamisi omadusi. Töö peamine rõhk on asetatud praktilisele osale, kus uuritakse kahte tüüpi elastseid talasid (elastsete otsakinnitustega ja elastse vahekinnitusega) ning analüüsitakse tehisnärvivõrkude abil saavutatud ennustuste tulemusi erinevatel juhtudel. Lisaks erinevatele kinnitustingimustele võrreldakse tulemusi erineva sisendsageduste arvu (kolm, neli, viis, kuus või üheksa sagedust) korral. Saadud tulemusi analüüsitakse ja võrreldakse teatud täpsusparameetrite põhjal. Läbiviidud arvutuste ning analüüsi põhjal selgub, et enamikel juhtudel on ennustuse teel saavutatud tulemused üpris ligilähedased oodatavatele tulemustele, seega on võnkuvate Euler-Bernoulli talade kinnitustingimuste jäikusparameetrite ennustamisel närvivõrkude rakendamine mõistlik.In the present thesis, an overview of the Euler-Bernoulli beam theory and the basics of artificial neural networks were presented. The main emphasis was on the practical implementation of training the artificial neural networks for predicting the stiffness parameters of the support conditions of the vibrating beams. The main purpose of the current paper was to study the frequencies of vibrating Euler-Bernoulli beams with different non-classical support conditions, and to analyze the efficiency of predicting the support condition coefficients (either translational or rotational). The calculated natural frequencies of the vibrating beams were used as the input for training the neural networks. The results were computed for various cases, using different numbers of input frequencies (three, four, five, six, or nine) besides the different support conditions. The results of the predictions were analyzed in two different parts: the efficiency of prediction in case of beams with elastic support at the boundaries, and the efficiency of prediction in case of beams with intermediate elastic support. The analysis of the efficiency of prediction in case of beams with elastic support at the boundaries showed that the overall efficiency of the predictions was substantially high and the identified results were quite similar to the expected outcomes. The best average results among all conditions were received with the beam clamped or simply supported at left end and translationally and rotationally restrained at right end. But even in the worst cases, most of the results were considerably nice. The analysis of the efficiency of predicting the rotational coefficient at the intermediate support in case of beams with intermediate elastic support showed that the results greatly depend on the generation of the training and test sets. If the training data contains noise, then the efficiency of the prediction is rather low, but it could be improved by modifying the training and test data sets. Also, alternative methods should be elaborated to extract features for parameter identification of vibrating systems

Dynamic Analysis of a Rotating Shaft Subject to the Double Cutting Force and Time-varying Mass Effects of the Rod

AbstractThis paper investigates the dynamic behaviors of a rotating shaft subjected to the double cutting force and time-varying mass effects. The Timoshenko beam theory is used to model the rotating shaft, and the general boundary condition is assumed as the clamped-hinged supports. This system is used to simulate the manufacture process of the double turret CNC lathes, and the mass of the rod which is reduced gradually in cutting process. The system equations of motion are derived based on the global assumed mode method, and the dynamic responses of the system are obtained by Runge-Kutta numerical method. The transformation matrix is derived to make the equation of motion completing the boundary geometric constraints. The numerical results compare the dynamic response in different moving speeds and skew angles of the cutting forces with/without the time-varying mass effects. Additionally, this paper compares the response with single cutting force and double force. The results show that the double moving force system can reduce not only the machining time but also the amplitude of shaft vibration

Performances of passive electric networks and piezoelectric transducers for beam vibration control

This thesis is focused on beam vibration control using piezoelectric transducers and passive electric networks. The first part of this study deals with the modeling and the analysis of stepped piezoelectric beams. A refined one-dimensional model is derived and experimentally validated. The modal properties are determined with four numerical methods. A homogenized model of stepped periodic piezoelectric beams is derived by using two-scale convergence. The second part deals with the performance analysis of three passive circuits in damping structural vibrations: the piezoelectric shunting, the second order transmission line and the fourth order transmission line. The effects of uncertainties of the electric parameters on the system performances are analyzed. Theoretical predictions are validated through different experimental setup