Eigenvalues of Fully-Clamped Skew Beams — An Edge-Function Method

Abstract

The plane isotropic elastodynamic eigenvalue problem has the form:

$$\left( {\lambda + \mu } \right)u_{j,ji} + \mu u_{i,jj} + \omega ^2 \rho u_i = 0\;in\;\Omega$$

(1)

$$u_i = 0\;in\;\Gamma _1$$

(2)

$$\lambda {u_{j,}}_j{v_i} + \mu \left( {{u_{i,j}} + {u_{j,i}}} \right){v_j} = 0\;in\;{\Gamma _2}$$

(3)

where λ and μ are Lamé’s constants, ρ is the density, ω is the angular frequency, Ω denotes the cross-section of the body with boundary $\overline \Gamma _1 \cup \overline \Gamma _2$, and the i’th components of displacement and exterior boundary normal are given by u_i and v _i, respectively. An Edge-function approach is developed for the numerical solution of the problem. The analysis is confined, in this paper, to rectilinar polygonal cross -sections.