Computable upper and lower bounds on eigenfrequencies

Improved eigenfrequencies of mechanical systems by combining complementary models

“…Assuming that this is effectively a bound, we note that it is not sharp, and we believe that it may be possible to improve it. Nevertheless, as shown in the next section for the plane stress examples, it is of the same order of magnitude as the bound presented in the work of Wang et al…”

“…The planar beam elements are used to analyse a clamped‐hinged beam (a propped cantilever), without longitudinal vibration, and a rectangular frame, hinged at both supports. The plane stress elements are used to analyse the two examples in the work of Wang et al: a rectangular plate with sliding supports and an L‐shaped plate clamped on two sides. For all these models, the material properties assumed and the initial meshes are presented in Figures and .…”

“…Problems, the same as in the work of Wang et al, considered for the 2‐dimensional continua model. The material properties are E =1, ν =0.3, and ρ =1.…”

“…More recently, these authors proposed an alternative approach for the computation of bounds, based on the Rayleigh (kinematic) quotient , using a static approach to compute an appropriate value of the constants involved.…”

“…We present in Table the relative errors obtained from the combined model and those presented in that paper, computed respectively as Keeping in mind that a fair comparison for meshes of elements of such different types is difficult to obtain, we just note that the order of magnitude of these errors is similar. In terms of degrees of freedom of the compatible model, the meshes with 128 and 2048 quadrilateral elements in the work of Wang et al have 254 and 4094 degrees of freedom. The meshes with 22 and 86 triangular elements have 20 and 84 or 196 and 772 degrees of freedom when a linear or a cubic approximation of the displacements is used, respectively.…”