eISSN 2658–5782

DOI 10.21662/mfs

☰
2019. Vol. 14. Issue 3, Pp. 214–217
URL: https://multiphasesystems.online/mfs2019.3.028,en
DOI: https://doi.org/10.21662/mfs2019.3.028
Determination of the dimensions of the cylindrical weight at the end of the rod
Aitbaeva A.A.
Mavlyutov Institute of Mechanics UFRC RAS, Ufa

Abstract

This article discusses free transverse vibrations of a homogeneous rod. The left end of the rod is clamped, and a cylindrical weight is concentrated at the right end. The eigenfrequencies of the rod vibration are known. The purpose of this work is to determine the parameters of the end cylindrical weight of the rod (mass, moment of inertia, length and radius) by the natural frequencies of the rod vibrations. We use a partial differential equation derivative of the fourth order to solve this problem. This equation and boundary conditions are reduced to a spectral problem. To find the mass and moment of inertia of the weight, the «Method of an additional unknown» was applied. In the characteristic determinant of the spectral problem, there are terms that contain products of unknown coefficients. The essence of the «Method of an additional unknown» is that some of these products are proposed to be considered new additional unknowns, through which the rest can be expressed. It is shown that the mass and moment of inertia of the weight can be found using the three natural frequencies of the rod vibrations. Formulas for finding the length and radius of a cylindrical weight are obtained, and corresponding examples of finding unknown parameters are considered.

Citation

Aitbaeva A.A. Determination of the dimensions of the cylindrical weight at the end of the rod. Multiphase Systems. 14 (2019) 3. 214–217 (in Russian).

Keywords

eigenvalues,
natural frequencies,
cylindrical weight,
rod

Article outline

Problem: To determine the parameters of the end cylindrical weight using the natural frequencies of the rod’s vibrations. The parameters are the mass and moment of inertia, length and radius of the cylindrical end weight.

Methods: This article discusses the free bending vibrations of a homogeneous rod. The left end of the rod is sealed, and a cylindrical weight of mass m1 and moment of inertia I1 is concentrated on the right end. Length h and radius r2 are the dimensions of the cylindrical weight. These parameters are considered unknown. The eigenfrequencies of vibrations of the rod is used as known acoustic data.

To solve this problem, we use a fourth-order partial differential equation that describes the bending vibrations of the rod that has a constant bending stiffness. The equation and boundary conditions are reduced to the problem with the spectral parameter λ by replacement and variable separation method. Next, we find the characteristic determinant of the spectral problem. In the characteristic determinant containing unknown coefficients of the boundary conditions, there are also products of unknown coefficients. To find the mass and moment of inertia of the weight, the method of additional unknown quantity is used. The essence of the method is that some of these products of unknown coefficients are proposed to be considered new additional unknowns, through which the rest can be expressed. Thus, new coefficients are introduced: a1 = m/(ρFL), a2 = I1/(ρFL), a3 = a1a2. Substituting three known eigenvalues into the characteristic determinant we obtain a system of three equations from three unknowns. To solve the system, we use Kramer's method and find the dyenhrtcoefficients a1 and a2. Using the known formulas for determining the mass and moment of inertia of the cylinder, as well as knowing the coefficients a1 and a2, we find the length and radius of the weight:
h = a12r12ρL/(2a2ρ1L2 - 2r12a1ρ1), r2 = √((2a2L2 - r12a1)/a1),
where ρ1 is the density of material of weight, ρ is the density of the material of the rod, Lis the length of the rod, r1is the radius of the rod (inner radius of the cylinder).

In a study was determined:

The mass and moment of inertia of the cylindrical weight attached to the right end of the rod are found by three natural frequencies of bending vibrations of the rod.

Formulas for finding the length and radius of the end cylindrical weight are obtained.

References

  1. Strutt J.W. Teoriya zvuka. V. 1. M.: Gostekhizdat, 1955. P. 504 (in Russian).
  2. Kolatts L. Zadachi na sobstvennyye znacheniya (s tekhnicheskimi prilozheniyami). M.: Nauka, 1968. P. 504 (in Russian).
  3. Vibratsii v tekhnike: spravochnik v 6 tomakh. T. 1. Kolebaniya nelineynykh sistem / Pod redaktsiyey V.V. Bolotina. M.: Mashinostroyeniye, 1978. P. 352 (in Russian).
  4. Akulenko L.D., Nesterov S.V. A frequency-parametric analysis of natural oscillations of non-uniform rods // Journal of Applied Mathematics and Mechanics. 2003. V. 67, No. 3. Pp. 525–537.
    https://elibrary.ru/item.asp?id=28927800
  5. Gladvell G.M.L. Obratnyye zadachi teorii kolebaniy. M.-Izhevsk: NITS «Regulyarnaya i khaoticheskaya dinamika», Institut komp’yuternykh issledovaniy, 2008. P. 608 (in Russian).
  6. Yurko V.A. Vvedeniye v teoriyu obratnykh spektral’nykh zadach. M.: Fizmatlit, 2007. P. 384 (in Russian).
  7. Morassi A., Dilena M. On point mass identification in rods and beams from minimal frequency measurements // Inverse probl. eng. 2002. V. 10, No. 3. P. 183–201.
    DOI: 10.1080/10682760290010378
  8. Akhtyamov A.M. Teoriya identifikatsii krayevykh usloviy i yeye prilozheniya. M.: Fizmatlit, 2009. P. 272 (in Russian).
  9. Akhtyamov A.M., Urmancheev С.F. Determination of the parameters of a rigid body clamped at an end of a beam from the natural frequencies of vibrations // Journal of Applied and Industrial Mathematics. 2010. V. 4.
    DOI: 10.1134/S1990478910010011
  10. Vatul’yan A.O. Obratnyye zadachi v mekhanike deformiruyemogo tela. M.: Fizmatlit, 2007. P. 224 (in Russian).
  11. Aitbayeva A.A. Matematicheskoye modelirovaniye i identifikatsiya vida i parametrov zakrepleniya kontsa sterzhnya po sobstvennym chastotam yego kolebaniy: diss. ... kand. fiz.-mat. nauk: 05.13.18. Ufim. gos. aviats. universitet, Ufa, 2018. P. 95 (in Russian).
    https://www.ugatu.su/assets/files/documents/dissov/06/2018/AitbaevaAA/Autoref_AitbaevaAA.pdf