Optimal configurations of circular bars under free torsional and longitudinal vibration based on Pontryagin’s maximum principle.

*(English)*Zbl 1382.74098Summary: Optimal configurations of circular bars under free torsional and longitudinal vibration are investigated using the Pontryagin’s maximum principle (PMP). The necessary optimality condition of PMP is analysed by considering the cross-sectional area of bars as control variable and by using Maier objective functional to control the final state of the objective functional. Optimal configurations associated to different orders of eigen frequencies, eigen vectors, and specific boundary conditions are demonstrated. Their relations are also explained qualitatively. Numerical results show the equivalent about optimal configurations and eigen frequencies under specific boundary conditions.

##### MSC:

74P10 | Optimization of other properties in solid mechanics |

74H45 | Vibrations in dynamical problems in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

PDF
BibTeX
XML
Cite

\textit{H.-L. Bui} et al., Meccanica 51, No. 6, 1491--1502 (2016; Zbl 1382.74098)

Full Text:
DOI

##### References:

[1] | Hagedorn P, DasGupta A (2007) Vibrations and waves in continuous mechanical systems. Wiley, England · Zbl 1156.74002 |

[2] | Rao SS (2007) Vibration of continuous systems. Wiley, USA |

[3] | Ma, H; Lu, Y; Wu, Z; etal., A new dynamic model of rotor-blade systems, J Sound Vib, (2015) |

[4] | Sihler, C, A novel torsional exciter for modal vibration testing of large rotating machinery, Mech Syst Signal Process, 20, 1725-1740, (2006) |

[5] | Murawski, L; Charchalis, A, Simplified method of torsional vibration calculation of marine power transmission system, Mar Struct, 39, 335-349, (2014) |

[6] | Astashev VK, Babitsky VI (2007) Ultrasonic processes and machines—dynamics, control and applications. Springer, USA · Zbl 1182.74068 |

[7] | Al-Budairi, H; Lucas, M; Harkness, P, A design approach for longitudinal-torsional ultrasonic transducers, Sens Actuators A, 198, 99-106, (2013) |

[8] | Yi, Y; Seemann, W; Gausmann, R; Zhong, J, Development and analysis of a longitudinal and torsional type ultrasonic motor with two stators, Ultrasonics, 43, 629-634, (2005) |

[9] | Kubojima, Y; Sonoda, S, Measuring young’s modulus of a wooden bar using longitudinal vibration without measuring its weight, Eur J Wood Prod, (2015) |

[10] | Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, England |

[11] | 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 |

[12] | 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 |

[13] | Atanackovic, TM; Djukic, DS, The influence of shear on the stability of a pflüger column, J Sound Vib, 144, 531-535, (1991) |

[14] | Atanackovic, TM; Simic, SS, On the optimal shape of a pflüger column, Eur J Mech A Solids, 18, 903-913, (1999) · Zbl 0978.74057 |

[15] | Glavardanov, VB; Atanackovic, TM, Optimal shape of a twisted and compressed rod, Eur J Mech A Solids, 20, 795-809, (2001) · Zbl 0998.74058 |

[16] | Atanackovic, TM; Braun, DJ, The strongest rotating rod, Int J Non-Linear Mech, 40, 747-754, (2005) · Zbl 1349.74217 |

[17] | 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 |

[18] | Atanackovic, TM, Optimal shape of a strongest inverted column, J Comput Appl Math, 203, 209-218, (2007) · Zbl 1113.49021 |

[19] | Atanackovic, TM, Optimal shape of a rotating rod with unsymmetrical boundary conditions, J Appl Mech, 74, 1234, (2007) |

[20] | Jelicic, ZD; Atanackovic, TM, Optimal shape of a vertical rotating column, Int J Non-Linear Mech, 42, 172-179, (2007) |

[21] | 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 |

[22] | 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) |

[23] | Glavardanov, VB; Spasic, DT; Atanackovic, TM, Stability and optimal shape of pflu¨ger micro/nano beam, Int J Solids Struct, 49, 2559-2567, (2012) |

[24] | 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 |

[25] | 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 |

[26] | Bui, HL; Tran, DT; Le, MQ; Tran, MT, Multi-objective optimal control for eigen-frequencies of a torsional shaft using pontryagin’s maximum principle, Meccanica, (2015) · Zbl 1325.74081 |

[27] | Thorby D (2008) Structural dynamics and vibration in practice. Elsevier, USA |

[28] | Geering HP (2007) Optimal control with engineering applications. Springer, Berlin · Zbl 1121.49001 |

[29] | Lebedev LP, Cloud MJ (2003) The calculus of variations and functional analysis with optimal control and applications in mechanics. World Scientific, Singapore · Zbl 1042.49001 |

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.