Optimal design for eigen-frequencies of a longitudinal bar using Pontryagin's maximum principle considering the influence of concentrated mass

Bui Hai Le, Tran Minh Thuy

Abstract


In this paper, the problem of optimal design for eigen-frequencies of a longitudinal bar using Pontryagin's maximum principle (PMP) considering the influence of concentrated mass is presented. The necessary optimality condition when simultaneously maximizing system's eigen frequencies and minimizing system's weight considering the influence of concentrated mass is established by using Maier objective functional in order to control the final state of the objective functional. By considering eigen frequencies as state variables, the analogy coefficient k in the necessary optimality condition is explicitly determined. Numerical results obtained in this paper include: (1) the bar's optimal configurations as well as frequency responses in different cases of objective functions; (2) the Pareto front for the system's first eigen frequency and weight; (3) the influence of concentrated mass on the bar's optimal configuration.

Keywords


Eigen frequencie; optimal design; longitudinal bar; concentrated mass; Pareto front; Pontryagin's maximum principle

Full Text:

PDF

References


A. DasGupta and P. Hagedorn. Vibrations and waves in continuous mechanical systems. Wiley, New York, (2007). doi:10.1002/9780470518434.

Y. Yi, W. Seemann, R. Gausmann, and J. Zhong. Development and analysis of a longitudinal and torsional type ultrasonic motor with two stators. Ultrasonics, 43, (8), (2005), pp. 629–634. doi:10.1016/j.ultras.2005.03.007.

V. K. Astashev and V. I. Babitsky. Ultrasonic processes and machines: dynamics, control and applications. Springer Science & Business Media, (2007). doi:10.1007/978-3-540-72061-4.

H. Al-Budairi, M. Lucas, and P. Harkness. A design approach for longitudinaltorsional ultrasonic transducers. Sensors and Actuators A: Physical, 198, (2013), pp. 99–106. doi:10.1016/j.sna.2013.04.024.

Y. Kubojima and S. Sonoda. Measuring Youngs modulus of a wooden bar using longitudinal vibration without measuring its weight. European Journal of Wood and Wood Products, 73, (3), (2015), pp. 399–401. doi:10.1007/s00107-015-0884-2.

C. Szymczak. Optimal design of thin walled I beams for extreme natural frequency of torsional vibrations. Journal of Sound and Vibration, 86, (2), (1983), pp. 235–241. doi:10.1016/0022-460x(83)90751-4.

C. Szymczak. Optimal design of thin-walled I beams for a given natural frequency of torsional vibrations. Journal of Sound and Vibration, 97, (1), (1984), pp. 137–144. doi:10.1016/0022-460x(84)90474-7.

T. M. Atanackovic and S. S. Simic. On the optimal shape of a Pfl¨ uger column. European Journal of Mechanics-A/Solids, 18, (5), (1999), pp. 903–913. doi:10.1016/s0997-7538(99)00128-x.

V. B. Glavardanov and T. M. Atanackovic. Optimal shape of a twisted and compressed rod. European Journal of Mechanics-A/Solids, 20, (5), (2001), pp. 795–809. doi:10.1016/s0997-7538(01)01165-2.

T. M. Atanackovic and D. J. Braun. The strongest rotating rod. International Journal of Non-Linear Mechanics, 40, (5), (2005), pp. 747–754. doi:10.1016/j.ijnonlinmec.2004.09.002.

T. M. Atanackovic and B. N. Novakovic. Optimal shape of an elastic column on elastic foundation. European Journal of Mechanics-A/Solids, 25, (1), (2006), pp. 154–165. doi:10.1016/j.euromechsol.2005.06.008.

T. M. Atanackovic. Optimal shape of a strongest inverted column. Journal of Computational and Applied Mathematics, 203, (1), (2007), pp. 209–218. doi:10.1016/j.cam.2006.03.019.

T. M. Atanackovic. Optimal shape of a rotating rod with unsymmetrical boundary conditions. Journal of Applied Mechanics, 74, (6), (2007), pp. 1234–1238. doi:10.1115/1.2744041.

Z. D. Jelicic and T. M. Atanackovic. Optimal shape of a vertical rotating column. International Journal of Non-Linear Mechanics, 42, (1), (2007), pp. 172–179. doi:10.1016/j.ijnonlinmec.2006.10.020.

D. J. Braun. On the optimal shape of compressed rotating rod with shear and extensibility. International Journal of Non-Linear Mechanics, 43, (2), (2008), pp. 131–139. doi:10.1016/j.ijnonlinmec.2007.11.001.

T. M. Atanackovic, B. B. Jakovljevic, and M. R. Petkovic. On the optimal shape of a column with partial elastic foundation. European Journal of Mechanics-A/Solids, 29, (2), (2010), pp. 283–289. doi:10.1016/j.euromechsol.2009.08.003.

V. B. Glavardanov, D. T. Spasic, and T. M. Atanackovic. Stability and optimal shape of Pfluger micro/nano beam. International Journal of Solids and Structures, 49, (18), (2012), pp. 2559–2567. doi:10.1016/j.ijsolstr.2012.05.016.

T. M. Atanackovic, B. N. Novakovic, and Z. Vrcelj. Shape optimization against buckling of micro-and nano-rods. Archive of Applied Mechanics, 82, (10-11), (2012), pp. 1303–1311. doi:10.1007/s00419-012-0661-1.

M.-Q. Le, D.-T. Tran, and H.-L. Bui. Optimal design of a torsional shaft system using Pontryagins maximum principle. Meccanica, 47, (5), (2012), pp. 1197–1207. doi:10.1007/s11012-011-9504-3.




DOI: https://doi.org/10.15625/0866-7136/6058

Refbacks

  • There are currently no refbacks.


 

                      

Editorial Office of Vietnam Journal of Mechanics

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam

Tel: (+84) 24 3791 7103

Email: vjmech@vjs.ac.vn