Problem Statement
The form of DP for the vibrating Euler-Bernoulli beam was developed in the lecture. The boundary conditions for the cantilever beam shown above are y(0,t)
= 0
The data defining the
problem include L, length of the beam, E, Young’s modulus, I, centroidal
moment of inertia, r,
mass density, A, cross-sectional area and k, the spring constant. The
selected data are:
The Galerkin weak statement FE implementation produces
The normal mode solution process assumes
yh(x,t) = Y(x)eiwt hence
Since this GWSh matrix statement is homogeneous,
the determinant of w2 Þ wi2, i = 1,2,3,….,n,… Choosing the positive root, the natural vibration frequencies are w Þ wih = Ö(eigenvalues of [MASS]-1[STIFF]) The corresponding normal mode are characterized by the nodal DOF arrays {Q} Þ {Qi (wi)} |