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Актуальные проблемы авиации и космонавтики - 2015. Том 2
UDK 539.3
NONLINEAR BENDING OF THIN ELASTIC ROD LOADED BY A TRANSVERSAL FORCE - COMPARISON OF ANALITYC SOLUTIONS
D. (M.) Zuev Scientific supervisor - Yu. (V.) Zakharov Foreign language supervisor - I. (N.) Shostak
Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation
In this paper nonlinear bend analytic solution of thin elastic rod loaded by a transversal force proposed at Ritelli and Scarpello was compared with approximate stength of materials theory solution and analytic solution proposed at Zakharov and Zakharenko. Necessary graphics were made and the comparison of solution was made.
Keywords: thin elastic rod, bend, transversal load.
ИЗГИБ КОНСОЛЬНОГО СТЕРЖНЯ ПОПЕРЕЧНОЙ НАГРУЗКОЙ -СРАВНЕНИЕ АНАЛИТИЧЕСКИХ РЕШЕНИЙ
Д.М. Зуев Научный руководитель - Ю. В. Захаров Консультант по иностранному языку - И. Н. Шостак
Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнева
Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31
Проведено сравнение точного решения для изгиба нагруженного поперечной нагрузкой консольного стержня, предложенного в работе Рителли и Скарпелло с приближенным решением сопротивления материалов и точным решением, предложенным Захаровым и Захаренко. В процессе работы были построены необходимые графические зависимости и проведено сравнение решений.
Ключевые слова: консоль, изгиб, поперечная нагрузка.
Microelectromechanical system devices and bar systems of satellites design need exact and approximate solutions of nonlinear bend of thin elastic rod for different cases. There is an actual task of comparison of exact and approximate solutions for finding the range of parameters, where it is appropriate to apply exact or approximate solution.
The aim of this paper is the comparison of nonlinear bend solutions of thin elastic rod loaded by transversal force (solutions proposed at Ritelli and Scarpello (RS) [1], Zakharov and Zakharenko (further exact solution) [2]) and approximate strength of materials (SoM) solution [3].
Solutions were compared in region of application of RS solution: 0 < и < 1, where и = PL / EJ is a dimensionless parameter of force intensity. This range of и is below the first stability threshold, in units of
dimensionless parameter of a load: 0 < X < 8 / n2, where и = n2X / 8. RS solution doesn't exist outside the range.
Graphs of forms of bend, maximum deflection and deviation of the solutions from each other were
made.
Rod bending forms are shown in Figure 1. We can see that RS solution has increasing deviation from bend forms of other solutions, when load intensity и increases. Bend forms of exact and SoM solution have the same form with insignificant deviation of SoM solution from exact solution.
Figure 2 provides maximum sagging deflections of rod depending from intensity parameter и . Maximum sagging deflection of all solutions coincides in range: 0 < и < 0.15, that fits deflection 0 < f < 0.1. Interval of application of SoM solution, which is f < 0.05, is located at this range. In addition, we can see sagging deflection of RS solution is close to SoM solution on the entire range, which is interpreted by Ritelli
Секция «Актуальнее на учнье проблемы в мире (глазами молодьш исследователей)»
and Scarpello as good accuracy rate. However, this interpretation is not correct because SoM solution was compared with RS solution outside the interval of application.
Figure 1. Forms of rods bending under different ^ values
Figure 2. Maximum sagging deflection of rod bendings
RS solution has large deviation from exact solution, as we see in Figure 3. Maximal deviation is 25% in the limiting case j ^ 1. This discrepancy and low applicability range (0 < A < 8 / n2 ) are the reasons to make a conclusion about low practical applicability.
Figure 3. Maximum deflection deviations of solutions from each other
This comparison is an example of low practical applicability of RS solution for several reasons. RS solution has large discrepancy with exact solution and low applicability range. There is some range where suggested RS solution competes with approximate solution of strength of materials. Ritelli and Scarpello solution shouldn't be used in such cases because of cumbersome formulae of solution equations.
References
1. Scarpello G. M., Ritelli D. (2011) Exact solutions of nonlinear equation of rod deflections involving the Lauricella hypergeometric functions. International Journal of Mathematics and Mathematical Sciences. Vol. 1.
2. Zakharov, Yu V., Zakharenko A. A. [Dynamic instability in the nonlinear problem of a cantilever]. Vychislitel'nye tekhnologii. 1999, vol. 4, no.1, p.48-54. (In Russ)
3. O. Belluzzi, (1970) Scienza delle Costruzioni, Zanichelli Italy: Bologna, vol. 1, p. 680.
© Zuev D. (M.), 2015
