Free and forced large amplitude vibrations of periodically inhomogeneous slender beams

• Autorzy: Domagalski Ł.

Identyfikatory

DOI

10.1016/j.acme.2018.06.005

• Języki publikacji: EN

Abstrakty

EN

Considered are free and forced transverse vibrations of slender periodic beams of finite length. It is assumed that the vibration amplitude is of the order of cross-section dimensions, still much smaller than the beam length. An averaged non-asymptotic model is proposed as a tool in analysis. The description is based on the tolerance approach to averaging of differential operators, using the concept of weakly slowly-varying function. The resulting differential equations with constant coefficients involve the effect of periodicity cell length. The model is verified by comparison of linear frequencies and mode shapes with Finite Element Method results, and then applied in analysis of free and forced vibrations of beam with variable cross-section. The method employed in obtaining the solution involves Galerkin orthogonalization and Runge–Kutta (RKF45) method. The results of nonlinear vibrations analysis are presented by backbone and amplitude-frequency response curves, time series, Poincare sections and bifurcation diagrams.

Słowa kluczowe

EN

periodic beams; nonlinear vibrations; tolerance modelling

PL

wibracje nieliniowe; modelowanie tolerancji; belka

• Wydawca: Springer ; Wrocław University of Science and Technology
• Czasopismo: Archives of Civil and Mechanical Engineering
• Rocznik: 2018
• Tom: Vol. 18, no. 4
• Strony: 1506--1519
• Opis fizyczny: Bibliogr. 26 poz., rys., tab., wykr.

Twórcy

autor

Domagalski Ł.

• Department of Structural Mechanics, Łódź University of Technology, al. Politechniki 6, 90-924 Łódź, Poland

Bibliografia

• [1] J. Awrejcewicz, A.V. Krysko, M.V. Zhigalov, O.A. Saltykova, V. A. Krysko, Chaotic vibrations in flexible multi-layered Bernoulli–Euler and Timoshenko type beams, Latin Am. J. Solids Struct. 5 (2008) 319–363.
• [2] Ł. Domagalski, J. Jędrysiak, On the tolerance modelling of geometrically nonlinear thin periodic plates, Thin-Walled Struct. 87 (2015) 183–190.
• [3] Ł. Domagalski, J. Jędrysiak, Geometrically nonlinear vibrations of slender meso-periodic beams. The tolerance modelling approach, Compos. Struct. 136 (2016) 270–277.
• [4] Ł. Domagalski, J. Jędrysiak, Nonlinear vibrations of periodic beams, J. Theor. Appl. Mech. 54 (2016) 1095–1108.
• [5] B.A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972.
• [6] L. Han, Y. Zhang, X. Li, L. Jiang, D. Chen, Flexural vibration reduction of hinged periodic beam–foundation systems, Soil Dyn. Earthquake Eng. 79 (2015) 1–4.
• [7] W.M. He, W.Q. Chen, H. Qiao, Frequency estimate and adjustment of composite beams with small periodicity, Composites: Part B 45 (2013) 742–747.
• [8] J.S. Jensen, Phononic band gaps and vibrations in one- and two-dimensional mass–spring structures, J. Sound Vib. 266 (2003) 1053–1078.
• [9] J.S. Jensen, N.L. Pedersen, On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases, J. Sound Vib. 289 (2006) 967–986.
• [10] A.G. Kolpakov, Application of homogenization method to justification of 1-D model for beam of periodic structure having initial stresses, Int. J. Solids Struct. 35 (1998) 2847–2859.
• [11] T. Kopecki, Ł. Święch, Experimental and numerical analysis of post-buckling deformation states of integrally stiffened thin-walled components of load-bearing aircraft structures, J. Theor. Appl. Mech. 52 (2014) 905–915.
• [12] R. Lewandowski, Nonlinear fee vibrations of multispan beams on elastic supports, Comput. Struct. 32 (1989) 305–312.
• [13] T. Lewiński, J.J. Telega, Singapore, in: Plates, Laminates and Shells, World Scientific Publishing Company, 2000.
• [14] O.H. Mahrenholtz, Beam on viscoelastic foundation: an extension of Winkler's model, Arch. Appl. Mech. 80 (2010) 93–102.
• [15] K. Mazur-Śniady, P. Śniady, Dynamic response of a microperiodic beam under moving load – deterministic and stochastic approach, J. Theor. Appl. Mech. 2 (2001) 323–338.
• [16] N. Olhoff, B. Niu, G. Cheng, Optimum design of band-gap beam structures, Int. J. Solids Struct. 49 (2012) 3158–3169.
• [17] P. Ostrowski, B. Michalak, A contribution to the modelling of heat conduction for cylindrical composite conductors with non-uniform distribution of constituents, Int. J. Heat Mass Transfer 92 (2016) 435–448.
• [18] V.S. Sorokin, J.J. Thomsen, Eigenfrequencies and eigenmodes of a beam with periodically continuously varying spatial properties, J. Sound Vib. 347 (2015) 14–26.
• [19] V.S. Sorokin, J.J. Thomsen, Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam, Proc. R. Soc. A: Math. Phys. Eng. Sci. 472 (2016) 20150751.
• [20] Y. Sun, N. Pugno, Hierarchical fibers with a negative Poisson's ratio for tougher composites, Materials 6 (2013) 699–712.
• [21] B. Tomczyk, A new combined model of dynamic problems for thin uniperiodic cylindrical shells, in: M. Kleiber, et al. (Eds.), Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues, CRC Press Taylor & Francis Group, London, 2016 581–585.
• [22] C. Woźniak, et al., Mathematical Modelling and Analysis In Continuum Mechanics of Microstructured Media, Silesian University of Technology Press, Gliwice, 2010.
• [23] Y. Xiao, J. Wen, D. Yu, X. Wen, Flexural wave propagation In beams with periodically attached vibration absorbers: bandgap behaviour and band formation mechanisms, J. Sound Vib. 332 (2013) 867–893.
• [24] H.J. Xiang, Z.F. Shi, Analysis of flexural vibration band gaps In periodic beams using differential quadrature method, Comput. Struct. 87 (2009) 1559–1566.
• [25] T. Zheng, T. Ji, Equivalent representations of beams with periodically variable cross-sections, Eng. Struct. 33 (2011) 706–719.
• [26] A. Żak, M. Krawczuk, M. Palacz, Ł. Doliński, W. Waszkowiak, High frequency dynamics of an isotropic Timoshenko periodic beam by the use of the time-domain spectral finite element method, J. Sound Vib. 409 (2017) 318–335.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019)