Vibration Of Non-uniform Rods Research Articles

Erratum for “Numerical Analysis of Free Longitudinal Vibration of Nonuniform Rods: Discrete Singular Convolution Approach” by Mohammad Shokrollahi and Amir Zayeri Baghlani Nejad

C-F −2.5 4.884 2.96 × 10−02 3.455 8.28 × 10−02 2.904 1.17 × 10−01 1.674 2.59 × 10−01 −0.5 3.233 −4.78 × 10−03 2.109 3.06 × 10−02 1.684 4.98 × 10−02 0.812 −5.66 × 10−03 0.5 2.350 −1.65 × 10−02 1.432 −5.79 × 10−02 1.100 1.42 × 10−02 0.470 1.39 × 10−03 2.5 0.864 2.99 × 10−03 0.424 2.87 × 10−02 0.295 5.03 × 10−03 0.101 4.05 × 10−02 C-C −2.5 5.286 5.77 × 10−02 4.486 9.52 × 10−02 4.316 1.02 × 10−01 4.353 8.18 × 10−02 −0.5 3.622 5.47 × 10−03 3.309 4.21 × 10−02 3.384 3.95 × 10−02 4.015 1.88 × 10−03 0.5 2.817 −3.78 × 10−03 2.906 −2.94 × 10−03 3.154 2.59 × 10−03 4.115 −2.03 × 10−03 2.5 2.030 3.89 × 10−02 3.074 5.56 × 10−02 3.598 3.99 × 10−02 5.012 2.38 × 10−03

Numerical Analysis of Free Longitudinal Vibration of Nonuniform Rods: Discrete Singular Convolution Approach

AbstractIn this paper, the problem of longitudinal free vibration of nonuniform rods is investigated for the first time by using the discrete singular convolution (DSC). This method is a relatively novel and powerful numerical approach especially in analyzing vibration problems. The governing equation of this problem is nonlinear and has analytical solutions only in a few cases, so the analysis uses DSC as a numerical method. The regularized Shannon delta kernel is used in the present algorithm. The nondimensional natural frequencies of rods with various-area cross sections have been studied in this paper. Results show good agreement with the previous analytical solutions. This study indicates the efficiency of the DSC in solving nonuniform rod problems.

Free Axial Vibrations of Non-Uniform Rods

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.

Exact solutions for the longitudinal vibration of non-uniform rods: Author's reply

Exact solutions for the longitudinal vibration of non-uniform rods: Author's reply

Exact solutions for forced vibration of non-uniform rods by Laplace transformation

Exact solutions for forced vibration of non-uniform rods by Laplace transformation

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.

Vibration of non-uniform rods and beams

We show that for some non-uniform rods and beams the equation of motion can be transformed into the equation of motion for a uniform rod or beam. Then, when the ends are completely fixed, the eigenvalues of the non-uniform continuum are the same as those of uniform rods or beams. For other end support conditions, exact solutions are obtained. An efficient procedure is used to analyze the free vibration of non-uniform beams with general shape and arbitrary boundary conditions. Simple formulas are presented for predicting the fundamental natural frequency of non-uniform beams with various end support conditions.