zbMATH — the first resource for mathematics

Multi-objective optimal control for eigen-frequencies of a torsional shaft using Pontryagin’s maximum principle. (English) Zbl 1325.74081
Summary: The present work investigates through Pontryagin’s maximum principle the multi-objective optimal control for eigen frequencies of a torsional shaft. Control variables are diameters of the shaft’s segments. Maier objective functional is used to control the final state of the objective functional for solving multi-objective optimal problem, in which maximizing eigen frequencies and minimizing system’s weight are simultaneously involved. The analogy coefficient \(k\) in the necessary optimality condition is explicitly determined by considering eigen frequencies as state variables. Numerical simulations demonstrate the relationship between the optimal configuration of the shaft and their eigen modes. The Pareto fronts and the boundary of the feasible region are constructed for the objectives. The Pareto fronts and the feasible region facilitate the estimation of the level of the trade-off between objectives, and the selection of a suitable solution among a set of competitive objectives.

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
90C29 Multi-objective and goal programming
49K15 Optimality conditions for problems involving ordinary differential equations
49N90 Applications of optimal control and differential games
Full Text: DOI
[1] Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, England
[2] Szymczak, C, Optimal design of thin walled I beams for extreme natural frequency of torsional vibrations, J Sound Vib, 86, 235-241, (1983) · Zbl 0508.73079
[3] Szymczak, C, Optimal design of thin walled I beams for a given natural frequency of torsional vibrations, J Sound Vib, 97, 137-144, (1984) · Zbl 0562.73085
[4] Atanackovic, TM; Djukic, DS, The influence of shear on the stability of a pflüger column, J Sound Vib, 144, 531-535, (1991)
[5] Atanackovic, TM; Simic, SS, On the optimal shape of a pflüger column, Eur J Mech A Solids, 18, 903-913, (1999) · Zbl 0978.74057
[6] Glavardanov, VB; Atanackovic, TM, Optimal shape of a twisted and compressed rod, Eur J Mech A Solids, 20, 795-809, (2001) · Zbl 0998.74058
[7] Atanackovic, TM; Braun, DJ, The strongest rotating rod, Int J Non-Linear Mech, 40, 747-754, (2005) · Zbl 1349.74217
[8] Atanackovic, TM; Novakovic, BN, Optimal shape of an elastic column on elastic foundation, Eur J Mech A Solids, 25, 154-165, (2006) · Zbl 1083.74040
[9] Atanackovic, TM, Optimal shape of a strongest inverted column, J Comput Appl Math, 203, 209-218, (2007) · Zbl 1113.49021
[10] Atanackovic, TM, Optimal shape of a rotating rod with unsymmetrical boundary conditions, J Appl Mech, 74, 1234, (2007)
[11] Jelicic, ZD; Atanackovic, TM, Optimal shape of a vertical rotating column, Int J Non-Linear Mech, 42, 172-179, (2007)
[12] Braun, DJ, On the optimal shape of compressed rotating rod with shear and extensibility, Int J Non-Linear Mech, 43, 131-139, (2008) · Zbl 1203.74077
[13] Atanackovic, TM; Jakovljevic, BB; Petkovic, MR, On the optimal shape of a column with partial elastic foundation, Eur J Mech A Solids, 29, 283-289, (2010)
[14] Glavardanov, VB; Spasic, DT; Atanackovic, TM, Stability and optimal shape of pflüger micro/nano beam, Int J Solids Struct, 49, 2559-2567, (2012)
[15] Atanackovic, TM; Novakovic, BN; Vrcelj, Z, Shape optimization against buckling of micro- and nano-rods, Arch Appl Mech, 82, 1303-1311, (2012) · Zbl 1293.74342
[16] Le, MQ; Tran, DT; Bui, HL, Optimal design of a torsional shaft system using pontryagin’s maximum principle, Meccanica, 47, 1197-1207, (2012) · Zbl 1293.74349
[17] Hagedorn P, DasGupta A (2007) Vibrations and waves in continuous mechanical systems. Wiley, England · Zbl 1156.74002
[18] Thorby D (2008) Structural dynamics and vibration in Practice. Elsevier, USA
[19] Geering HP (2007) Optimal control with engineering applications. Springer, Berlin
[20] Lebedev LP, Cloud MJ (2003) The calculus of variations and functional analysis with optimal control and applications in mechanics. World Scientific, Singapore
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.