Dynamic Response of Non-linear Beam Structures in Deterministic and Chaos Perspective
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Abstract
The behavior of large deformation beam structures can be modeled based on non-linear geometry due to geometric
nonlinearity mid-plane stretching in the presence of axial forces, which is a form a nonlinear beam differential equation
of Duffing equation type. Identification of dynamic systems from nonlinear beam differential equations for
deterministic and chaotic responses based on time history, phase plane and Poincare mapping. Chaotic response based
on time history is very sensitive to initial conditions, where small changes to initial terms leads to significant change in
the system, which in this case are displacement x (t) and velocity x’(t) as time increases (t). Based on the phase plane, it
shows irregular and non-stationary trajectories, this can also be seen in Poincare mapping which shows strange attractor
and produces a fractal pattern. The solution to this Duffing type equation uses the Runge-Kutta numerical method with
MAPLE software application.
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How to Cite
Dolu, A., & Nasution, A. (2019). Dynamic Response of Non-linear Beam Structures in Deterministic and Chaos Perspective. Indonesian Journal of Physics, 30(2), 14 - 19. https://doi.org/10.5614/itb.ijp.2019.30.2.3
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References
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[4] Bayat, M., Pakar, I., Shahidi, M. (2011). Analysis of Nonlinear Vibration of Coupled Systems with Cubic Nonliearity. Journal Mechanica.17(6):620– 629.
[5] Belendez, A., Bernabeu, G., Frances, J., Mendez, DI., Marini, S. (2010). An Accurate Closed-form Approximate Solution for the Quintic Duffing Oscillator Equation. Journal Mathematical and Computing Modelling, Vol. 52, Issues 3-4, August 2010 : pp. 637 – 641.
[6] Clough, R.W., Penzien, J. (2003). Dynamics of Structure. Third Edition. Computers & Structures, Inc. University Berkeley. USA.
[7] Chopra, A.K., (2001). Dynamics Structure ; Theory and Application to Earthquake Engineering, Second Edition, byPrentice Hall.
[8] Ens, R.H., McGuire, G.C. (2001). Nonlinear Phisics with Mathematica for Scientiest and Engineers. Birkhauser, Boston. USA.
[9] Fahzi, A., Belhaq, M., Lakrad, F. (2009). Suppression of Hysteresis in a Forced Van der Pol – Duffing Oscillator. Journal Commun Nonlinear Sci Numer Simulat 14 (2009) : 1609 – 1616.
[10] Kovacic, I,. Brennan, MJ. (2011). The Duffing Equation – Nonlinear Oscillator and their Behaviour. London, John Wiley & Sons.
[11] Lai, S.K., Lim, C.W., Wu, B.S., Wang, C., Zeng, Q.C., He, X.F. (2009). Newton–Harmonic Balancing Approach for Accurate Solutions to Nonlinear Cubic–Quintic Duffing Oscillators. Applied Mathematical Modelling 33, 852–866.
[12] Lorenz, E.N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, Volume 20. March 1963. pp 130 -141
[13] Lynch, S. (2010). Dynamical Systems with Applications using Maple, second edition. Birkhauser Boston - Springer Science Bussines Media. USA.
[14] Moon, F.C. (1983). Chaotic Vibrations of a Beam with Non-LinearBoundary Conditions. Int Journal Non-Linear Mechanics, Vol. 18, No. 6, pp. 465-477. Great Britain.
[15] Moon, F.C. (2004). Chaotic and Fractal Dynamics, An Introduction for Applied Scientist and Engineers.Wiley-VCH Verlag GmbH&Co.KgaA. Weinheim.
[16] Nayfeh, A.H., Pai, P.F. (2004). Linear dan Nonlinear Structural Mechanics. John Wiley & Sons. USA.
[17] Paz, M. (1996). Dinamika Struktur ; Teori dan Perhitungan. Penerbit Erlangga. Jakarta.
[18] Pai, P.F. (2006). Highly Flexible Structures ; Modeling, Computation, and Experimentation. AIAA Inc, Reston, Virginia
[19] Portela, A., Charafi, A. (2002). Finite Elements using MAPLE, A Symbolic Programming Approach. Springer-Verlag, Berlin. Germany
[20] Rand, Richard H. (2012). Lecture Notes on Nonlinear Vibrations.
https://ecommons.cornell.edu/bitstream/handle/1813/28989/NonlinearVibrations_ver53.pdf
[21] Sathyamoorthy, M. (1998). Nonlinear Analysis of Structures. CRC Press LLC. USA
[22] Starosvetsky, Y., Gendelman, O.V. (2010). Bifurcations of Attractors in Forced System with Nonlinear Energy Sink: the Effect of Mass Asymmetry. Journal Nonlinear Dynamic 59 : 711–731.
[23] Thompson, J.M.T. and Stewart, H.B. (1986), Nonlinear Dynamics and Chaos Geometrical Methods for Engineers and Scientists. John Wiley & Sons.New York
[24] Thompson, W.T. (1982). Theory of Vibration with Applications. Second Edition. Prentice – Hall of India.
[25] Ueda, Y. (1991). The Road to Chaos. Aerial Press, Inc. Santa Cruz.
[26] Urabe, M. (1969). Numerical Investigation of Subharmonic Solutions to Duffing's Equation. Publ. RIMS, Kyoto Univ. Vol. 5, pp. 79-112.
[27] Zardar, Ziauddin,. Abrams, Iwona, (1998), Chaos for Beginners, Cambridge, Inggris, Icon Brooks
[28] Zienkiewicz, O.C., Morgan, K. (1983). Finite Element and Approximation. John Wiley & Sons