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: 

L

1 m

E

210E+09 N/m2

I

5E-05 m4

rA

200 kg/m

k

2E+06 N/m

      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
([MASS]-1[STIFF] - w2I) must vanish, which leads to the eigenvalue determination

             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)}