Topology optimization for elastic base under rectangular plate subjected to moving load

Open access

Abstract

Distribution optimization of elastic material under elastic isotropic rectangular thin plate subjected to concentrated moving load is investigated in the present paper. The aim of optimization is to damp its vibrations in finite (fixed) time. Accepting Kirchhoff hypothesis with respect to the plate and Winkler hypothesis with respect to the base, the mathematical model of the problem is constructed as two-dimensional bilinear equation, i.e. linear in state and control function. The maximal quantity of the base material is taken as optimality criterion to be minimized. The Fourier distributional transform and the Bubnov-Galerkin procedures are used to reduce the problem to integral equality type constraints. The explicit solution in terms of two- dimensional Heaviside‘s function is obtained, describing piecewise-continuous distribution of the material. The determination of the switching points is reduced to a problem of nonlinear programming. Data from numerical analysis are presented.

[1] P.W. CHRISTENSEN and A. KLARBRING: An Introduction to Structural Optimization. Solid Mechanics and its Applications. Springer, Berlin, 2009.

[2] M.P. BENDSO/E and O. SIGMUND: Topology Pptimization. Theory, Methods and Applications. Springer, Berlin, 2003.

[3] P.A. BROWNE: Topology optimization of linear elastic structures. Thesis submitted for the PhD degree, Bath, 2013.

[4] H.A. ESCHENAUER and N. OLHOFF: Topology optimization of continuum structures: A review. ASME Applied Mechanics Reviews, 54(4), (2001), 331-390.

[5] J. HASLINGER and P. NEITTAANMA¨KI: Finite Element Approximation for Optimal Shape, Material and Topology Design. 2nd edition. Wiley, New York, 1996.

[6] J. HASLINGER and R.A.E. MA¨KINEN: Introduction to Shape Optimization: Theory, Approximation, and Computation. Advances in Design and Control. SIAM, Philadelphia, 2003.

[7] J. HASLINGER, J. M´ALEK and J. STEBEL: A new approach for simultaneous shape and topology optimization based on dynamic implicit surface function. Control and Cybernetics, 34(1), (2005), 283-303.

[8] S.H. JILAVYAN, AS.ZH. KHURSHUDYAN and A.S. SARKISYAN: On adhesive binding optimization of elastic homogeneous rod to a fixed rigid base as a control problem by coefficient. Archives of Control Sciences, 23(4), (2013), 413-425.

[9] P.M. PRZYBYLOWICZ: Active reduction of resonant vibration in rotating shafts made of piezoelectric composites. Archives of Control Sciences, 13(3), (2003), 327-337.

[10] Z. GOSIEWSKI and A. SOCHACKI: Control system of beam vibration using piezo elements. Archives of Control Sciences, 13(3), (2003), 375-385.

[11] L. LENIOWSKA and R. LENIOWSKI: Active control of circular plate vibration by using piezoceramic actuators. Archives of Control Sciences, 13(4), (2003), 445-457.

[12] A. BRANSKI and S. SZELA: On the quasi optimal distribution of PZTs in active reduction of the triangular plate vibration. Archives of Control Sciences, 17(4), (2007), 427-437.

[13] Z. GOSIEWSKI and A. SOCHACKI: Optimal control of active rotor suspension system. Archives of Control Sciences, 17(4), (2007), 459-468.

[14] L. STAREK, D. STAREK, P. SOLEK and A. STAREKOVA: Suppression of vibration with optimal actuators and sensors placement. Archives of Control Sciences, 20(1), (2010), 99-120.

[15] AS.ZH. KHURSHUDYAN: The Bubnov-Galerkin procedure in bilinear control problems. Automation and Remote Control, 76(8), (2015), 1361-1368.

[16] AM.ZH. KHURSHUDYAN and AS.ZH. KHURSHUDYAN: Optimal distribution of viscoelastic dampers under elastic finite beam under moving load. Proc. of NAS of Armenia, 67(3), (2014), 56-67 (in Russian).

[17] S.V. SARKISYAN, S.H. JILAVYAN and AS.ZH. KHURSHUDYAN: Structural optimization for infinite non homogeneous layer in periodic wave propagation problems. Composite Mechanics, 51(3), (2015), 277-284.

[18] L.C. NECHES and A.P. CISILINO: Topology optimization of 2D elastic structures using boundary elements. Engineering Analysis with Boundary Elements, 32(7), (2008), 533-544.

[19] P.M. PARDALOS and V. YATSENKO: Optimization and Control of Bilinear Systems. Springer, Berlin, 2008.

[20] K. BEAUCHARD and P. ROUCHON: Bilinear control of Schrodinger PDEs. In Encyclopedia of Systems and Control, 24 (to appear in 2015).

[21] M.E. BRADLEY and S. LENHART: Bilinear optimal control of a Kirchhoff plate. Systems & Control Letters, 22(1), (1994), 27-38.

[22] V.F. KROTOV, A.V. BULATOV and O.V. BATURINA: Optimization of linear systems with controllable coefficients. Automation and Remote Control, 72(6), (2011), 1199-1212.

[23] I.V. RASINA and O.V. BATURINA: Control optimization in bilinear systems. Automation and Remote Control, 74(5), (2013), 802-810.

[24] M. OUZAHRA: Controllability of the wave equation with bilinear controls. European J. of Control, 20(2), (2014), 57-63.

[25] AS.ZH. KHURSHUDYAN: Generalized control with compact support of wave equation with variable coefficients. International J. of Dynamics and Control, (2015), DOI: 10.1007/s40435-015-0148-3.

[26] V.S. VLADIMIROV: Methods of the Theory of Generalized Functions. Analytical Methods and Special Functions. CRC Press, London-NY, 2002.

[27] A.H. ZEMANIAN: Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications. Dover Publications, New York, 2010.

[28] AS.ZH. KHURSHUDYAN: Generalized control with compact support for systems with distributed parameters. Archives of Control Sciences, 25(1), (2015), 5-20.

[29] S.G. MIKHLIN: Error Analysis in Numerical Processes. John Wiley & Sons Ltd, New York, 1991.

Archives of Control Sciences

The Journal of Polish Academy of Sciences

Journal Information


IMPACT FACTOR 2016: 0.705

CiteScore 2016: 3.11

SCImago Journal Rank (SJR) 2016: 0.231
Source Normalized Impact per Paper (SNIP) 2016: 0.565

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 192 129 9
PDF Downloads 75 62 4