13.2 Central Difference Method
493
Fig. 13.2 Schematic representation of derivatives
and
d
4
w
dx 4
i
=
1
h 4 (w i+2 − 4w i+1 + 6w i − 4w i−1 + w i−2 )
(13.6)
The central finite difference schemes have been adopted in the above derivations.
It may be noted that the nodes are located symmetrically about the point, at which the
derivatives are of interest to us. Though there are two other schemes, viz. the backward
differences and the forward differences, it is the central difference schemes that are
adopted for the solution of vibration problems, vis-à-vis the structural mechanics
problems. The above four derivatives are shown schematically in Fig. 13.2.
13.3 Free Vibration of Beams
The differential equation for the free vibration of a beam is given by
d
2
dx 2
E I
d
2
w
dx 2
= p
2 mw
(13.7)
where m is the mass per unit length.
For a uniform beam, Eq. (13.7) becomes
EI
d
4
w
dx 4 = p
2 mw
(13.8)
When expressed in finite difference form, Eq. (13.8) becomes
493
Fig. 13.2 Schematic representation of derivatives
and
d
4
w
dx 4
i
=
1
h 4 (w i+2 − 4w i+1 + 6w i − 4w i−1 + w i−2 )
(13.6)
The central finite difference schemes have been adopted in the above derivations.
It may be noted that the nodes are located symmetrically about the point, at which the
derivatives are of interest to us. Though there are two other schemes, viz. the backward
differences and the forward differences, it is the central difference schemes that are
adopted for the solution of vibration problems, vis-à-vis the structural mechanics
problems. The above four derivatives are shown schematically in Fig. 13.2.
13.3 Free Vibration of Beams
The differential equation for the free vibration of a beam is given by
d
2
dx 2
E I
d
2
w
dx 2
= p
2 mw
(13.7)
where m is the mass per unit length.
For a uniform beam, Eq. (13.7) becomes
EI
d
4
w
dx 4 = p
2 mw
(13.8)
When expressed in finite difference form, Eq. (13.8) becomes