Dynamic Higher-Order Equations for Finite Rods

Abstract

This work considers longitudinal wave propagation in circular cylindrical rods adopting Boström's power series expansion method in the radial coordinate. Equations of motion together with consistent sets of general lateral and end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton's principle. Analytical comparisons are made between the present theory to low order and several classic theories. Numerical examples for eigenfrequencies, displacement and stress distributions are given for a number of finite rod structures. The results are presented for series expansion theories of different order and various classical theories, from which one may conclude that the present method generally models the rod accurately.

Higher Order Beam Equations

“…Unlike the lateral conditions the entire end surface must be prescribed with one condition for each direction (r, θ, z). If different parts of the end surface have different conditions it is not possible to directly use the method presented here, instead one should use the theory discussed in [3]. However the method as presented here can be generalized to cases where conditions are dependent on r, θ and t. The combinations of end conditions are…”

“…The second constraint (3.7) can be obtained by setting k = −1 in (3.3). For the case of m = 0 the constraint equations vanish in accordance with [2] and [3]. The recursion formulas are used to write all coefficients u i , v j and w k as functions of the coefficient with lowest index.…”

“…The discussed truncation scheme is only to be used when m ≥ 1, while for m = 0 the truncation scheme discussed in [3] should be used.…”

“…This uncoupling makes it possible to derive dispersion relations for the torsional modes analytically. The case of longitudinal waves is treated in [2], [3]. For the analysis of the torsional dispersion relation the displacement fields u and w are considered to be nonexistent.…”

“…This method was used to investigate the governing equation and dispersion relations for rods [2]. The method was also used to investigate stresses and eigenfrequencies for finite rods [3]. Furthermore it was used also for cylindrical shells [4] and plates [5], [6].…”