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1

Application of isogeometric approach to dynamics of curved beam

Łasecka-Plura Magdalena, Białasik Jan, Kuczma Mieczysław, Tabrizikahou Alireza

Vibrations in Physical Systems

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2023

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Vol. 34, nr 1

art. no. 2023117

EN

In the paper dynamics of a free-form Timoshenko curved beam is investigated. The considered problem is solved using isogeometric analysis. Non-uniform rational B-spline (NURBS) basis functions are applied to describe both geometry and displacement field of the considered beam. The Timoshenko beam theory is used to derive the element stiffness and mass matrices. The application of the presented method is shown in numerical examples. The correctness of the presented approach is proved by comparing the obtained results to those available in the literature and calculated by the finite element method. Analysis of convergence is presented for different orders of NURBS basis functions.

2

Size-dependent coupled bending-torsional vibration of functionally graded carbon nanotube reinforced composite Timoshenko microbeams

Dehkordi Hamid Reza Balali, Beni Yaghoub Tadi

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 3

art. no. e186, 2023

EN

This paper investigates the free vibration of a carbon nanotube-reinforced composite Timoshenko microbeam considering the effect of axial load and bending-torsion coupling. The microbeam properties are developed based on the micromechanical model concerning the extended rule of mixtures. The governing equations of motion are derived using the modified couple stress theory and Hamilton’s principle. The uniform nanotube distribution and three functionally graded distributions are considered for the carbon nanotube-reinforced composite microbeam. The generalized differential quadrature method is applied to the governing equations for deriving the natural frequency under different boundary conditions. Next, the effects of different parameters, including nanotube distribution, geometric characteristics of microbeam, material length scale, and nanotube volume fraction, on the natural frequency are demonstrated through different tables and diagrams. Among obtained results is the significant effect of the carbon nanotube volume fraction on the natural frequency of the microbeam. Also, the nonconformity between the mass and elastic axes leads to the natural frequency reduction. The comparison between obtained results and results of other credible papers confirms the validity of obtained results.

3

Analytical and numerical analysis of injection pump (stepped) shaft vibrations using timoshenko theory

Noga Stanisław, Rejman Edward, Bałon Paweł, Kiełbasa Bartłomiej, Smusz Robert, Szostak Janusz

Acta Mechanica et Automatica

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2022

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Vol. 16, no. 3

215--224

EN

The free transverse vibrations of shafts with complex geometry are studied using analytical methods and numerical simulations. A methodology is proposed for evaluating the results of a natural transverse vibration analysis as generated by finite element (FE) models of a shaft with compound geometry. The effectiveness of the suggested approach is tested using an arbitrarily chosen model of the injection pump shaft. The required analytical models of the transverse vibrations of stepped shafts are derived based on the Timoshenko thick beam theory. The separation of variables method is used to find the needed solutions to the free vibrations. The eigenvalue problem is formulated and solved by using the FE representation for the shaft and for each shaft-simplified model. The results for these models are discussed and compared. Additionally, the usefulness of the Myklestad–Prohl (MP) method in the field of preliminary analysis of transverse vibration of complex shaft systems is indicated. It is important to note that the solutions proposed in this paper could be useful for engineers dealing with the dynamics of various types of machine shafts with low values of operating speeds.

4

Transient analysis of transversely functionally graded Timoshenko beam (TFGTB) in conjunction with finite element method

Khafaji Salwan Obaid Waheed, Al-Shujairi Mohammed A., Aubad Mohammed Jawad

Archive of Mechanical Engineering

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2020

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LXVII, nr 3

299--321

EN

In this work, transient and free vibration analyses are illustrated for a functionally graded Timoshenko beam (FGM) using finite element method. The governing equilibrium equations and boundary conditions (B-Cs) are derived according to the principle of Hamilton. The materials constituents of the FG beam that vary smoothly along the thickness of the beam (along beam thickness) are evaluated using the rule of mixture method. Power law index, slenderness ratio, modulus of elasticity ratio, and boundary conditions effect of the cantilever and simply supported beams on the dynamic response of the beam are studied. Moreover, the influence of mass distribution and continuous stiffness of the FGM beam are deeply investigated. Comparisons between the current free vibration results (fundamental frequency) and other available studies are performed to check the formulation of the current mathematical model. Good results have been obtained. A significant effect is noticed in the transient response of both simply supported and cantilever beams at the smaller values of the power index and the modulus elasticity ratio.

5

Investigation of boundary condition effects on the stability of FGP beams in thermal environment

Nasirzadeh R., Behjat B., Kharazi M., Khabazaghdam A.

Journal of Theoretical and Applied Mechanics

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2017

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Vol. 55 nr 3

1003--1014

EN

In this paper, stability and instability of Functionally Graded Piezoelectric (FGP) beams is investigated based on the Timoshenko beam theory. The material properties of the beam are considered to change gradually through thickness of the beam by a simple power law. By using the principle of minimum total potential energy, governing equations of the beam are derived. Stability behavior of the beam is predicted by solving the governing equations of the FGP beam. The results show that the homogeneity of boundary conditions plays a critical role in the stability of the FGP beam. While non-homogeneous boundary conditions lead to stable behavior of the FGP beam; homogeneous boundary conditions cause instability in the beam. By solving the eigenvalue equation of the FGP beam, the buckling load of the beam is obtained for the beams that have unstable behavior. Finally, the effects of various parameters on the buckling load of the unstable beam, such as power law index, temperature, applied voltage and aspect ratio are investigated, and the results are compared with the Euler-Bernoulli beam theory.

6

Dynamic stability of moderately thick beams and frames with the use of harmonic balance and perturbation methods

Obara P., Gilewski W.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 4

739--750

EN

The objective of this paper is to determine dynamic instability areas of moderately thick beams and frames. The effect of moderate thickness on resonance frequencies is considered, with transverse shear deformation and rotatory inertia taken into account. These relationships are investigated using the Timoshenko beam theory. Two methods, the harmonic balance method (HBM) and the perturbation method (PM) are used for analysis. This study also examines the influence of linear dumping on induced parametric vibration. Symbolic calculations are performed in the Mathematica programme environment.

7

Buckling analysis of short carbon nanotubes based on a novel Timoshenko beam model

Hosseini-Ara R., Mirdamadi H. R., Khademyzadeh H.

Journal of Theoretical and Applied Mechanics

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2012

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Vol. 50 nr 4

975-986

EN

In this paper, we present a novel method to investigate the buckling behavior of short clamped carbon nanotubes (CNTs) with small-scale effects. Based on the nonlocal Timoshenko beam kinematics, the strain gradient theory and variational methods, the higher-order governing equation and its corresponding boundary conditions are derived, which are often not considered. Then, we solve the governing differential equation and determine exact critical buckling loads using a linear polynomial plus trigonometric functions different from the purely trigonometric series. We also investigate the influences of the scale coefficients, aspect ratio and transverse shear deformation on the buckling of short clamped CNTs. Moreover, we compare the critical strains with the results obtained from the Sanders shell theory and validate them with molecular dynamic simulations which are found to be in good agreement. The results show that unlike the other beam theories, this model can capture correctly the small-scale effects on buckling strains of short CNTs for the shell-type buckling.

PL

W pracy zaprezentowano nową metodę analizy problemu wyboczenia krótkich, obustronne zamurowanych nanorurek węglowych (tzw. CTN – Carbon NanoTubes) z uwzględnieniem zjawisk małoskalowych. Na podstawie nielokalnego sformułowania kinematyki belki Timoszenki opracowano teorię gradientu odkształcenia oraz metodę analizy wariacyjnej, wyprowadzono równania konstytutywne wyższego rzędu i odpowiadające im warunki brzegowe, do tej pory z rzadka stosowane w tego typu zagadnieniach. Następnie rozwiązano równania modelu, z których wyznaczono dokładną wartość krytycznego obciążenia prowadzącego do wyboczenia. Użyto w tym celu kombinacji funkcji wielomianowych i trygonometrycznych zamiast szeregów wyłącznie trygonometrycznych. Zbadano również wpływ współczynników skali, proporcji oraz odkształcenia postaciowego na wyboczenie utwierdzonych nanorurek CNT. W trakcie symulacji numerycznych dynamiki molekularnej modelu wykazano dobrą zbieżność otrzymanych wyników z powłokowym modelem Sandersa. Potwierdzono, że – w odróżnieniu od innych teorii belek – zastosowany model dokładnie odzwierciedla efekty małoskalowe przy opisie powłokowego wyboczenia krótkich nanorurek CNT.

8

Comprehensive mathematicaI modelling of a transversely vibrating flexible link robot manipulator carrying a tip payload

Loudini M., Boukhetala D., Tadjine M.

International Journal of Applied Mechanics and Engineering

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2007

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Vol. 12, no 1

67-83

EN

The purpose of this paper is to describe the application of the Timoshenko beam theory (TBT) to the mathematical modelling of a planar one link flexible robot manipulator pinned at its actuated base and carrying a payload at its free end-point. The emphasis has been put on obtaining accurate and complete equations of motion that display the most relevant aspects of structural propenies inherent to the modelled lightweight flexible link. So, in addition to the classical effects of shearing and rotational inertia of the link cross-section, two imponant damping mechanisms: external viscous air damping and internal structural viscoelasticity effect (Kelvin-Voigt damping) have been included. Gravity, torsion, and longitudinal elongation have been neglected. Numerical simulations, performed to show the free vibrational behaviour of the modelled system, demonstrate the imponant effect of the carried payload on the amplitude and the frequency of vibrations.

9

Drgania belek z przegubami sprężystymi jako modelu belek ze szczelinami

Drewko J.

Zeszyty Naukowe Politechniki Rzeszowskiej. Mechanika

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1999

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z. 52 [174]

277-282

EN

A method for evaluation of eigenfrequencies of cracked beams is presented in the paper. The method is based on Timoshenko beam theory. The crack is replaced by an equivalent elastic hinge taking into consideration rotational inertia and shear deformations. Elastic hinge model is based on classical beam bending theory. Stiffness of the elastic hinge was determined for rectangular cross-section with one- and two-sided crack. Influence of crack localisation and depth on beam eigenfrequencies was analysed. The results were compared to the results obtained using NASTRAN for three prismatic beams.