Designing cylindrical springs of minimum weight with restriction on the own frequency of longitudinal oscillations under condition of full and fuzzy information

V. O. Baranenko, D. L. Volchok, M. S. Hryhorovych


Formulation of the problem. Elastic elements such as springs are common іn many products of the machine-building, railway, construction industry. They are designed to accumulate or absorb mechanical energy. These elements of various designs and devices have been and are being given enough attention. There is a large number of publications in this field, where questions of calculation and manufacturing are considered. At the same time, a small number of scientific papers are devoted to optimal design, in which the optimization of the characteristics of springs by various criteria is carried out using nonlinear programming methods.

This paper is devoted to the problems of optimal design of a cylindrical tension spring by the weight criterion, while its own frequency of longitudinal oscillations is limited. Direct and dual optimization problems are formulated. Realization is provided with  the method of Lagrange multipliers and necessary conditions for the existence of an extremum. Purpose of the article. To consider optimization dynamic problems of design of helical cylindrical springs in conditions of complete and incomplete information about the output data. To analyze the influence of the number of active spring turns on the optimal design parameters, as well as the influence of the fuzzy information about weight setting and the value of the natural frequency of the longitudinal oscillations. Conclusion. As a result of solving the direct and dual optimizing dynamic problem for a helical spring it is found that the dependence of the optimal weight from the natural frequency of the intrinsic longitudinal oscillations and the number of active turns is a nonlinear function. The dependence of the optimum frequency of natural oscillations and the diameter of the wire from the given weight is also nonlinear. The estimation of the influence of the fuzzy initial data on the result of the project (the diameter of the wire of the spring) is obtained. The transformation of fuzzy numbers into deterministic ones is performed by the center method (defuzzification operation). Accounting of fuzzy information leads to increasing in the weight parameter and in the wire diameter parameter.


cylindrical spring; optimal design; fuzzy sets


Anur'ev V.I. Spravochnik konstruktora-mashinostroitelya: v 3 tomax [Reference book of the machine-builder designer: in 3 volumes]. Ed 8, Moskva: Mashinostroenie, 2001, no. 3, 864 p. (in Russian)

Baranenko V.A., Ivanec M.V. and Chaplygina S.N. Optimal'noe proektirovanie cilindricheskix pruzhin v usloviyax nechetkoj informacii [Optimal design of cylindrical springs under conditions of fuzzy information] Zbіrnyk naukovykh prats fіzyko-matematychnі nauky №3 [Collection of scientific Physics and Mathematics works no. 3]. Zaporіzhzhya, 2015, pp. 23 – 27 (in Russian)

Markina M.V. Naxozhdenie paretovskogo mnozhestva mnogoekstremal'nyx zadach [Finding a Paretov set of multi-extremum problems] Prikladnye problemy prochnosti i plastichnosti: vsesojuz. mezhvuz. sb. [Applied problems of durability and plasticity: All-Union Interuniversity Collection]. Issledovanie i optimizaciya konstrukcij [Investigation and optimization of structures]. Gor'kij, 1982, pp. 137 – 143 (in Russian)

Ponomarev S.D. and Andreeva L.E. Raschet uprugix elementov mashin i priborov [Calculation of elastic elements of machines and instruments]. Moskva: Mir, 1980, 326 p. (in Russian)

Rutkovskaya D., Pilinskij M. and Rutkovskij L. Nejronnye seti, geneticheskie algoritmy i nechetkie sistemy [Neural networks, genetic algorithms and fuzzy systems]. Moskva: Goryachaya liniya-Telekom. 2008, 383 p. (in Russian)

Belegundu Ashok D. and Jasbir S. Arora A study of mathematical programming methods for structural optimization. Part II: Numerical Results. International journal for numerical methods in engineering. 1995, vol. 21, iss. 9, pp. 1601–1623.

Haug Edward J. and Jasbir. S. Arora Applied Optimal Design: Mechanical and Structural Systems. New York: Wiley-Interscience, 1979, 520 p.

Shigley J. E. Mechanical engineering design. 3rd Ed. New York: McGraw-Hill, 1977, 695 p.

GOST Style Citations

  1. Анурьев В. И. Справочник конструктора-машиностроителя : в 3 т. Т. 3 / В. И. Анурьев. – 8-е изд., перераб. и доп. – Москва : Машиностроение, 2001. – 864 с.
  2. Бараненко В. А Оптимальное проектирование цилиндрических пружин в условиях нечёткой информации / В. А. Бараненко, М. В. Иванец, С. Н. Чаплыгина // Вісник Запорізького національного університету. Фізико-математичні науки : зб. наук. ст. – Запоріжжя, 2015. – № 3. – С. 23–27.
  3. Маркина М. В. Нахождение паретовского множества многоэкстремальных задач / М. В. Маркина // Прикладные проблемы прочности и пластичности : всесоюз. межвуз. сб. / Горьков. гос. ун-т. – Горький, 1982. – С. 137–143. – (Исследование и оптимизация конструкций).
  4. Пономарев С. Д. Расчет упругих элементов машин и приборов / С. Д. Пономарев, Л. Е. Андреева. – Москва : Машиностроение, 1980. – 326 с.
  5. Рутковская Д. Нейронные сети, генетические алгоритмы и нечёткие системы / Д. Рутковская, М. Пилиньский, Л. Рутковский ; пер. с польск. И. Д. Рудинского. – Москва : Горячая линия-Телеком, 2008. – 383 с.
  6. Belegundu Ashok D. A study of mathematical programming methods for structural optimization. Part II: Numerical Results / Ashok D. Belegundu, Jasbir S. Arora // International journal for numerical methods in engineering. – 1995. – Vol. 21, iss. 9. – P. 1601–1623.
  7. Haug Edward J. Applied Optimal Design: Mechanical and Structural Systems / Edward J. Haug, Jasbir. S. Arora – New York : Wiley-Interscience, 1979. – 520 p.
  8. Shigley J. E. Mechanical engineering design / J. E. Shigley – 3rd Ed. – New York : McGraw-Hill, 1977. – 695 p.