Boundary conditions are (i) y and φ_x at station 0 is zero, and (ii) at the free end shear force, S_y , and bending moment, M_yz are zero. On substituting these boundary conditions in equation (j), we get

(k)

From the last two expressions of equation (k), we have

(l)

Hence, on taking the determinant of equation (k) equal to zero, the frequency equation is

(m)

which is,

From which transverse natural frequencies can be obtained by the by root searching numerical method described earlier by expressing the frequency equation as a function of the following form

(n)

Alternatively, through commercial software directly roots of equations can be obtained. Roots of this function are natural frequencies and are obtained as

For obtaining the mode shape, from the first two equations of (k), we have

(o)

For a given natural frequency, the first equation of equation (l) gives

(p)

By choosing _Ry₂ =1 as a reference value of the displacement, the first equation of equation (m) gives

(q)

On substituting equation (p) into equation (q), the shear force at station 0 can be obtained as

(r)

The bending moment at station 0 can be obtained now from equation (p). Now, we have obtained the state vector at station 0 completely, and using the transformation matrices the state vectors at all other station can be obtained and are given as

(i) For first natural frequency : State vectors are given as

So that from above state vectors, we have

Fig. 8.28 Mode shapes corresponding to the first natural frequency (top) for the translational displacement and (bottom) for the rotational displacement