Abstract
Free axial vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the pro-posed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the proposed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.
Highlights
The vibration of non-uniform rods is a subject of considerable scientific and practical interest that has been studied extensively, and is still receiving attention in literature
In this case one could obtain its exact solution of free axial vibrations with Kummer function, which is a special case of (Guo, 2010; Ray, 1997)
The problem could be investigated by the WKB method, which leads to a series solution
Summary
The vibration of non-uniform rods is a subject of considerable scientific and practical interest that has been studied extensively, and is still receiving attention in literature. In a study using an approach Abrate [3] sought exact solutions for the problem He obtained a closed form solution for rods whose cross-sectional area varies as. In a recent study Anil and Sujith (2005) solved the problem with cross-sectional area A( x) = kxnebx and A( x) = kxnebx. A method is proposed to solve the free axial vibration of non-uniform rods, which results in a series solution. The concerned non-uniform rod is shown as Fig. 1, in which Young’s modulus E ( x) , density ρ ( x) and cross-sectional area A( x) vary with the axial coordinate x.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
