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Numerical investigation of geometrically nonlinear clamped uniform rods and rods with sections varying exponentially free vibration

Abdeddine E., Majid A., Bouzida N., Bouzida Z., Zarbane Kh.

Journal of Achievements in Materials and Manufacturing Engineering

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2022

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Vol. 112, nr 2

49--63

EN

Purpose: The present paper is intended to investigate the problem of linear and non-linear longitudinal free vibration of uniform rods and rods whose cross-sections vary exponentially at large vibration amplitudes. Design/methodology/approach: The method adopted consists in discretizing the energy term on linear kij and non-linear rigidity tensor bijkl, as well as the mass tensor mij. Therefore, the formulation of this structure is based on Lagrange equations and the harmonic balance method so as to obtain the nonlinear algebraic equations. These latter are solved numerically and analytically through the explicit and linearized method. Findings: The response of Clamped-Clamped uniform and non-uniform rods on our structure are highlighted in the amplitude frequency and associated first three mode shapes. Moreover, this research leads to study the influence of the exponential slope on the maximum displacement, thus emphasizing the non-uniform bars usefulness. The obtained results are then compared with the available literature with a view to validating this theory. Research limitations/implications: As a perspective, the method used in this paper would be pushed to study the FDM material, taking into account other parameters related to additive manufacturing, and later to be validated experimentally. Practical implications: Longitudinal vibrations are important in mechanical structures; therefore, the determination of their dynamic behaviour needs to be understood. In the present study, the effect of the displacement amplitude on the exponential slope of the structure was analysed, which led to the determination of the reduction range of the vibration amplitude under resonance. However, this should be taken into account in the design process. Besides, the usefulness of the non-linearity geometric effects was demonstrated to examine these structures by considering all the parameters involved. Originality/value: A linearized procedure is used to solve a nonlinear algebra equation. The use of this method leads to reduce calculation time contrary to iterative methods.

2

Modelowanie ruchu niejednorodnych brył z wykorzystaniem metod sztucznej inteligencji

Rykała Ł.

Przegląd Mechaniczny

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2018

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nr 5

34--39

PL

W artykule zaprezentowano problem modelowania ruchu niejednorodnych brył poruszających się po równi pochyłej za pomocą równań Lagrange'a II rodzaju z mnożnikami. Zastosowanie wymienionego formalizmu pozwoliło określić przebiegi nieznanych mnożników Lagrange’a, które następnie aproksymowano z wykorzystaniem sztucznych sieci neuronowych. Przedstawiono również wyniki badań numerycznych analizowanego układu.

EN

The article presents the problem of modeling the movement of non-homogeneous solids moving along an inclined plane using Lagrange equations of the second kind with multipliers. The use of the mentioned formalism allows to obtain the courses of unknown Lagrange multipliers, which were then approximated using artificial neural networks. The numerical results of the analyzed system are also presented.

3

Modelowanie i sterowanie układu kula-belka

Burghardt A., Giergiel J.

Modelowanie Inżynierskie

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2009

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T. 6, nr 37

55-62

PL

Zaprezentowane podejście do procesu modelowania obiektu typu US z wykorzystaniem równań Lagrange'a umożliwiło otrzymanie modelu matematycznego w formie umożliwiającej jego wykorzystanie w procesie syntezy adaptacyjnego algorytmu sterowania. Ponadto otrzymany model matematyczny zastosowano do budowy emulatora obiektu kula-belka, który był wykorzystywany podczas badań symulacyjnych. Zaproponowany algorytm sterowania pozwolił na stabilizację pozycji kuli na belce, co wykazano w badaniach symulacyjnych oraz podczas weryfikacji na obiekcie rzeczywistym. Dalsze prace wykorzystujące zaproponowany model będą prowadzone w kierunku nie tylko stabilizacji pozycji kuli, ale również w kierunku sterowania nadążnego ruchem kuli po belce.

EN

The ball-and-beam problem is a benchmark for testing control algorithms. This paper we presents analysis and implementation adaptive control schemes on ball and beam system. Dynamic model for the system was derived using Lagrange equations. Control algorithm solves a stabilization under-actuated ball and beam system was created by using Lyapunov theory. The controllers designed for the system have been simulated by Matlab/Simulink.

4

Methodology of the stepper motor rotational motion investigations

Wawszczak W., Towarek Z., Jagiełło B.

Mechanics and Mechanical Engineering

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2004

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Vol. 7, nr 1

87-95

EN

Advantages and disadvantages of separate studies on dynamic systems using either theoretical or experimental methods have been emphasized, and a stand point has been expressed that both methods must definitely be combined. Measurements of vibration in the rotational motion by means of an RWJ instrument served as the methodology of experimental studies, while Lagrange quadratic equations were used for theoretical analysis. The combined experimental-theoretical methodology was adopted for the nonstationary motion of a stepper motor. The analysis and reduction of the system of equations of dynamics for an inverse issue have been presented and selected experimental solutions for the motion of the drive system of a stepper motor has been quoted.

5

Sformułowanie problemu drgań wzdłużnych pręta z uwzględnieniem ruchu unoszenia

Buchacz A.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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2003

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z. 20

46-51

PL

Wyprowadzono równania ruchu pręta drgającego wzdłużnie, z uwzględnieniem jego ruchu unoszenia, stosując równania Lagrange'a II rodzaju z uwzględnieniem transformacji tensora rotacji układów współrzędnych.

EN

The motion equations were formulated by the second type Lagrange equations with taking into consideration its transportation and transformation by the rotation tensor of coordinate system.