Free vibration analysis of non-cylindrical helices with, variable cross-section by using mixed FEM

The objective of the study is to present a mixed finite element (FE) formulation for the free vibration analysis of non-cylindrical helices with variable cross-section. Non-cylindrical helix geometry was well approximated through the variable arc lengths and linear variations in curvatures via shape functions. The mixed FE model for non-cylindrical helices was developed by using the exact cylindrical helix geometry and corresponding field equations. The element and consistent mass matrices are attained under these assumptions. The element matrix is derived based on the Timoshenko beam theory and the effects of rotary inertia are involved into the mass matrix. Although the formulation is quite simple, the generated element is capable to provide highly accurate solutions for the conical, barrel and hyperboloidal geometries of non-cylindrical helices. The results of presented FE model based on an approximated geometry approach are in a good agreement with the other studies in the actual literature. Some original examples were generated and solved for the literature as well.

[1]  V. Yıldırım.  EXPRESSIONS FOR PREDICTING FUNDAMENTAL NATURAL FREQUENCIES OF NON-CYLINDRICAL HELICAL SPRINGS , 2002 .

[2]  Wisam Busool,et al.  Free Vibration of Helicoidal Beams of Arbitrary Shape and Variable Cross Section , 2002 .

[3]  David Thompson,et al.  Dynamic stiffness formulation, free vibration and wave motion of helical springs , 2001 .

[4]  V. Yıldırım.  FREE VIBRATION ANALYSIS OF NON‐CYLINDRICAL COIL SPRINGS BY COMBINED USE OF THE TRANSFER MATRIX AND THE COMPLEMENTARY FUNCTIONS METHODS , 1997 .

[5]  A. C. Scordelis.  Internal Forces in Uniformly Loaded Helicoidal Girders , 1960 .

[6]  W. H. Wittrick,et al.  An exact solution for the vibration of helical springs using a Bernoulli-Euler model , 1986 .

[7]  J. Mottershead.  The large displacements and dynamic stability of springs using helical finite elements , 1982 .

[8]  Y. Xiong,et al.  A finite element model for the vibration of spatial rods under various applied loads , 1992 .

[9]  Sadahiko Takeda,et al.  Free vibration of coil springs of arbitrary shape , 1986 .

[10]  J. Mottershead.  FINITE ELEMENTS FOR DYNAMICAL ANALYSIS OF HELICAL RODS , 1980 .

[11]  S. Timoshenko,et al.  LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars , 1921 .

[12]  Mehmet H. Omurtag,et al.  The mixed finite element formulation for three-dimensional bars , 1991 .

[13]  W. H. Wittrick.  On elastic wave propagation in helical springs , 1966 .

[14]  I. Epstein.  The Motion of a Conical Coil Spring , 1947 .

[15]  Vebil Yıldırım,et al.  NATURAL FREQUENCIES OF HELICAL SPRINGS OF ARBITRARY SHAPE , 1997 .

[16]  Vebil Yıldırım,et al.  INVESTIGATION OF PARAMETERS AFFECTING FREE VIBRATION FREQUENCY OF HELICAL SPRINGS , 1996 .

[17]  A. C. Scordelis,et al.  An Analytical and Experimental Study of Helicoidal Girders , 1958 .

[18]  W. L. Cleghorn,et al.  On the natural frequencies of helical compression springs , 2002 .

[19]  Mehmet H. Omurtag,et al.  The mixed finite element solution of helical beams with variable cross-section under arbitrary loading , 1992 .