Free Vibrations of Arbitrary Tapered Beams with Static Deflections due to Arbitrary Distributed Dead Loads

임의분포 사하중에 정적변위를 갖는 변단면 보의 자유진동

  • 이병구 (원광대학교 공과대학) ;
  • 이용 (원광대학교 대학원)
  • Published : 1996.09.01

Abstract

A numerical method is presented to obtain the natural frequencies and mode shapes of the arbitrary tapered beams with static deflection due to arbitrary distributed dead loads. The differential equation governing free vibration of such beams is derived and solved numerically. The double integration method using the trapezoidal rule is used to solve the static behaviour of beams loaded arbitrary distributed dead load. Also, the Improved Euler method and the determinant search method are used to integrate the differential equation subjected to the boundary conditions and to determine the natural frequencies of the beams, respectively. In the numerical examples, the various geometries of the beams are considered : (1) linearly tapered beams as the arbitrary variable cross-section, (2) the triangular, sinusoidal and uniform loads as the arbitrary distributed dead loads and (3) the hinged-hinged, clamped-clamped and hinged-clamped ends as the end constraints. All numerical results are shown as the non-dimensional forms of the system parameters. The lowest three natural frequencies versus load parameter, slenderness ratio and section ratio are reported in figures. And for the comparison purpose, the typical mode shapes with and without the effects of static deflection are presented in the figure. According to the numerical results obtained in this analysis, the following conclusions may be drawn : (1) the natural frequencies increase when the effects of static deflections are included, (2) the effects are larger at the lower modes than the higher ones and (3) it should be betteF to include the effect of static deflection for calculating the frequencies when the beams are supported by both hinged ends or one hinged end.

Keywords

References

  1. 한국소음진동공학회 논문집 v.4 no.4 사하중에 의한 정적변위를 고려한 보의 자유진동 이병구;박광규;오상진;모정만
  2. 대한토목학회논문집 v.11 no.4 변단면 보의 자유진동 해석 이병구;오상진;최규도
  3. Ph.D Thesis,Utah State University Vibration Analysis of Linearly Tapered Beams using Frequency-Dependent Stiffness and Mass Matrix A.K.Gupta
  4. Journal of the Structural Division v.11 no.1 Vibration of Tapered Beams A.K.Gupta
  5. Journal of the Engineering Mechanics Division v.93 no.EM 2 Vibration of Non-Uniform Simply-Supported Beams A.C.Heidebrecht
  6. Proceeding of the International Conference on Vibration Engineering, ICVE'94 Free Vibrations and Buckling Loads of Beam-Columns on Elastic Foundations B.K.Lee;S.J.Oh
  7. Proceeding of Sixth International Conference on Conputing in Civil and Building Engineering Free Vibrations and Buckling Loads of Beam-Columns with Intermediate Multiple Elastci Springs B.K.Lee;S.J.Oh;J.M.Mo
  8. Journal of the Structural Engineering v.117 no.4 Effect of Dead Loads on Natural Frequencies of Beams H.Takabatake
  9. Ph. D Thesis,M.I.T. Nonlinear Vibration of Straight and Buckled Beams under Harmonic Excitation W.Y.Tseng
  10. Journal of Applied Mechanics v.136 Generalized Hypergeometric Function Solutions on the Transverse Vibration of a Class of Non-Uniform Beams H.C.Wang