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13 Finite Difference Method for the Vibration Analysis …
Fig. 13.1 Representation of
a curve
The first derivative is given by
dw
dx
i
=
1
2h
(w i+1 − w i−1 )
(13.1)
Equation (13.1) suggests that the slope of the curve between stations i + 1 and i
− 1 has been considered to be constant.
The second derivative can be approximated by
d
2
w
dx 2
i
=
1
h
dw
dx
i+1/2
−
dw
dx
i−1/2
(13.2)
It can be shown that
dw
dx
i+1/2
=
1
h
(w i+1 − w i )
and
dw
dx
i−1/2
=
1
h
(w i − w i−1 )
(13.3)
Combining Eqs. (13.2) and (13.3), we get
d
2
w
dx 2
i
=
1
h 2 (w i+1 − 2w i + w i−1 )
(13.4)
Likewise, the third and fourth derivatives can be shown to be
d
3
w
dx 3
i
=
1
2h 3 (w i+2 − 2w i+1 + 2w i−1 − w i−2 )
(13.5)