Natural modes for finite elements

Abstract

In the natural mode method we express the deformation u(x, y, z) at any point in a finite element as a linear combination of the nodal cartesian displacements r _e

$$ \mathop u\limits_{(3x1)} (x,y,z) = \mathop C\limits_{(3xn)} (x,y,z)\mathop {{r_e}}\limits_{(nx1)} $$

((3.1))

where n represents the nodal degrees of freedom. A finite element e can deform in n different modes. Then, the total deformation of the element may be expressed as a linear combination of the imposed n modes so that

$$ u = {u_1} + {u_2} + ... + {u_n} $$

((3.2))