Generalized formula for estimating the oscillation frequency response of a cantilever bar with point masses
DOI:
https://doi.org/10.26577/JMMCS.2022.v116.i4.03Keywords:
cantilever bar, variable bending stiffness, main unit coefficients, oscillations frequency response for natural oscillations, graphical dependence of the oscillation frequency response,, reduced mass, calculation reliability, calculation nomogramAbstract
This paper presents a study of natural oscillations of a cantilever bar with five point masses with variable geometric and stiffness parameters (distances between locations of the masses, coefficients of variability of the bending stiffness of the bar sections). Using the exact method of forces based on the Mohr formula, there have been obtained expressions in general form for calculating the main unit coefficients of the secular equation, which makes it possible to perform calculations and to determine the oscillation frequency response of natural oscillations with a wide range of changes in the initial parameters of the physical and geometric state of cantilever bars. A numerical example has been given to illustrate the proposed theoretical approaches. The results have been compared with the results based on calculating a similar can tilever bar with one( reduced by masses) degree of freedom. A graphical dependence of the oscillation frequency response value on changing the value of the bending stiffness along the length of the cantilever bar gas been obtained. The theoretical provisions and applied results presented in the work will be widely used both in the practical design of bar systems and in scientific research in the field of mechanics of a deformable solid body.
References
[2] Numerical methods for solving differential and integral equations and quadrature formulas. Digest of articles. M.: Science.
1964.
[3] Doyev V.S., Doronin F.A., Indeikin A.V. Theory of vibrations in transport mechanics (under the editorship of A. V.
Indeikin). M.: FGOU "Training and Methodological Center for Education in Railway Transport". 2011. 352 p.
[4] Gorelov Yu.N. Numerical methods for solving ordinary differential equations (the Runge-Kutta method): textbook.
allowance. Samara: Samara University. 2006. 48 p.
[5] Konstantinov I.A., Lalina I.I. Structural mechanics. Calculation of rod systems. St. Petersburg: Publishing house of
Politekhn. university 2005. 155 p.
[6] Mrdak Ivan, Rakocevic Marina, Zugic Ljiljana, Usmanov Rustam, Murgul Vera, Vatin Nikolay. Analysis of the influence
of dynamic properties of structures on seismic response according to Montenegrin and European regulations. 2014. Vol.
633-634. pp. 1069-1076.
[7] Timoshenko S.P. Fluctuations in engineering. M.: Nauka, 1967. 444 p.
[8] Vatin N.I., Sinelnikov A.S. Long-span elevated pedestrian crossings made of light cold-formed steel profile // Construction
of unique buildings and structures. 2012. No. 1. P. 47-52.
[9] Lyakhovich, Leonid Semyonovich. Special Characteristics of Forms of Bending of The Bars With Equal Resistance and
Forms of Natural Oscillation Under Fundamental Tone Value Restriction for Linear Dependence of Moments of Cross-
Section Inertia and Varied Parameter // Vestnik TGASU No 1, 2010, C. 102-109.
[10] Ufimtsev E.M., Voronin M.D. Dynamic Calculation of a Roof Truss during the Process of Free Structurally Nonlinear
Oscillations // Bulletin of the South Ural State University. Ser. Construction Engineering and Architecture. 2016 Vol. 16,
No. 3, PP. 18–25.
[11] IGNATIEV V.A., IGNATYEV A.V. Application of the mixed form of the finite element method to the calculation of bar
systems for free oscillations. 2012. No. 16. pp. 101-110.
[12] Chernov S. A. Analysis of free oscillations of an arbitrary planar bar system on a computer // Bulletin of the Ulyanovsk
State Technical University. 2015. №2. pp. 17-21.
[13] Postnoe V.A. Dynamic Stiffness Matrices of Beam Elements and their Use in the Finite Element Method in Calculating
Forced Vibrations of Rod Systems // Bulletin of Civil Engineers. 2005. S. 42-45.
[14] Klein G. K., Rekach V. G., Rosenblat G. I. Guide to practical exercises in the course of structural mechanics. M.: Higher
school, 1980. 320 p.
[15] Sovremennye metody proektirovaniya mostov [Modern methods of bridge design] // M.: Transport. – 2006. – 664.
