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1
Content available Two approaches in the analytical investigation of the spring pendulum
Sypniewska-Kamińska G., Starosta R., Awrejcewicz J.
Vibrations in Physical Systems
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2018
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Vol. 29
art. no. 2018005
EN
Dynamics of the nonlinear spring pendulum is analysed using two asymptotic approaches. The multiple scale method is commonly applied with using two time scales. The purpose of the research is to justify the introduction of an additional third scale. Results of the analysis clearly show that introducing the third scale improve correctness of the approximate analytical solution. The obtained results allow for qualitative and quantitative analysis of the behavior of the studied system with a high accuracy. Calculations are made both in the neighbourhood of the resonance and also far from it.
2
Content available Nonlinear vibrations of orthogonal mechanism of shaking table
Bissembayev K., Iskakov Z.
International Journal of Applied Mechanics and Engineering
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2014
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Vol. 19, no. 3
487--501
EN
The dynamics of orthogonal mechanism in the presence of a shaking table of fixed cargo is studied. The focus is on the vibrational motion of the mechanism in slow-speed motor rotation. The analysis of the solutions for motion nonlinear equations showed that the angular velocity of the guiding link oscillates according to the harmonic law about a mean value. The character of changing its amplitude, depending on the weight of the cargo and the length of the guiding link, is determined. Dependence of the mean angular velocity on the motor torque is set.
3
Content available remote On some aspects of multiple scale method in problems of nonlinear dynamics
Awrejcewicz J., Starosta R., Sypniewska-Kamińska G.
Vibrations in Physical Systems
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2012
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Vol. 25
53--58
EN
Nonlinear vibrations of the two degree-of-freedom system near resonances are studied. The system is externally and kinematically driven. The dynamical problem is solved by an analytical multiple scales method (MS). This analytical approach gives very good results in solving problems of nonlinear dynamics and is more and more popular in last decades. The investigations are focused on correctness of MS method using various number of considered time scales. Namely, we show that in some cases the use of only two time scales is insufficient to detect all possible resonances exhibited by the studied system.
4
Content available remote Multiple-scale method analysis of nonlinear dynamical system
Starosta R., Awrejcewicz J.
Vibrations in Physical Systems
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2008
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Vol. 23
335--340
EN
The paper concerns analysis of nonlinear vibration of smooth autoparametrical non-autonomous systems of two-degree-of-freedom. A multiple scales method of investigation of small vibrations is applied to the analysis of resonance. The obtained results have been confirmed numerically. Analytical calculation was made with the use of Mathematica.
5
Content available Hybrid asymtotic method for the effect of local thickness defects and initial imperfections on the buckling of cylindrical shells
Gristchak V. Z., Golovan O. A.
Journal of Theoretical and Applied Mechanics
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2003
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Vol. 41 nr 3
509-520
EN
A double asymptotic method for the effect of local thickness defects and initial imperfections on the buckling problems of some mechanical systems is proposed. asymptotic formulas for the critical buckling load parameter are obtained with the perturbation technique.
PL
W pracy zaprezentowano podwójnie asymptotyczną metodę szacowania wpływu początkowych zaburzeń strukturalnych i lokalnych uszkodzeń zmieniających grubość na wyboczenie wybranych układów mechanicznych. Zastosowane wyrażenia asymptotyczne pozwalające na określenie krytycznych parametrów wyboczenia otrzymano techniką perturbacyjną.
6
Content available remote Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 2, The strongly curved shells
Hamdouni A., Millet O.
Archives of Mechanics
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2003
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Vol. 55, nr 2
177-219
EN
In the first part of this paper we have deduced a classification of asymptotic shallow shell models with respect to the level of applied forces, from the non-linear three-dimensional elasticity. We have used a constructive approach based on a dimensional analysis of the non-linear three-dimensional equilibrium equations, which naturally makes appear dimensionless numbers characterizing the applied forces (... and ...) and the geometry of the shell (... and C). To limit our study to one-scale problems, these dimensionless numbers are expressed in terms of the relative thickness ... of the shell, considered as the perturbation parameter. In the first part, we have studied the case of shallow shells corresponding to C=.... In the second part of this paper, we will study the case of strongly curved shells for which C=.... The classification that we obtain is then more complex. It depends not only on the force levels, but also on the existence of inextensional displacements which keep invariant the metric of the middle surface of the shell.
7
Content available remote Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 1, The shallow shells
Hamdouni A., Millet O.
Archives of Mechanics
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2003
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Vol. 55, nr 2
135-175
EN
The purpose of this paper is to construct a classification of asymptotic shell models (inferred from the non linear three-dimensional elasticity) with respect to the applied forces and to the geometrical data. To do this, we use a constructive approach based on a dimensional analysis of the nonlinear three-dimensional equilibrium equations, which naturally gives rise to the appearance of dimensionless numbers characterizing the applied forces and the geometry of the shell. In order to limit our study to one-scale problems, these dimensionless numbers are expressed in terms of to the relative thickness ... of the shell, which is considered as the perturbation parameter. This leads, on the one hand, to distinguish shallow shells from strongly curved shells which have a different asymptotic behaviour, and on the other hand, to fix the applied force level. For each of these two classes of shells, using the usual asymptotic method, we propose a complete classification of two-dimensional shell models based on decreasing force levels, from severe to low. In the first part of this paper, we present the classification for shallow shells. We obtain successively the nonlinear membrane model, another membrane model, Koiter's non linear shallow shell model, and the linear Novozhilov-Donnell one, respectively for severe, high, moderate and low forces.
8
Fredholm - Volterra integral equation in three dimensional problem with Macdonald kernel
El-Bary A.
Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka
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2002
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z. 26 [199]
59-65
EN
Regular and singular asymptatic method are applied to three dimensional Fredholm-Volterra integral equation of the first kind. In this paper we constract Fredholm-Volterra integri equation in three dimensional contact problem with Macdonald kernel in many cases of contact problem.
9
Content available remote A dynamic asymptotic model of linear elastic orthotropic plates: first and second-order terms
Sławianowska A., Telega J. J.
Archives of Mechanics
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2001
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Vol. 53, nr 4-5
541-564
EN
By using the asymptotic method combined with variational approach a new dynamic model of elastic orthotropic plates is derived. This model accounts for rotational inertia and takes into account second order terms ... and ... in the asymptotic expansions of displacements and stresses. To avoid the boundary layer analysis, the boundary conditions for ... are satisfied in a suitable averaged sense. Convergence theorem is formulated. Influence of the second-order terms on plate response in the static case is investigated.
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