Page 123 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 123
110 Transient Vibration Chap. 4
where the time interval is h = At. Subtracting and ignoring higher-order terms, we
obtain
1
2h (^,+1 - Ji,-,) (4.7-5)
Adding, we find
X,- = 1 - 2x,- +x,_,i) (4.7-6)
In both Eqs. (4.7-5) and (4.7-6), the ignored terms are of order h^. By substituting
from the differential equation, Eq. (4.7-3), Eq. (4.7-6) can be rearranged to
^, + 1 = 2x, - x ,_ i -I- /zV(x,,i,) i > 2 (4.7-7)
which is known as the recurrence formula.
(Starting the computation.) If we let / = 2 in the recurrence equation, we
note that it is not self-starting, i.e., jCj is known, but we need X2 to. find JC3. Thus,
to start the computation, we need another equation for ^2. This is supplied by the
first of Taylor’s series, Eq. (4.7-4), ignoring higher-order terms, which gives
X2 = X\ -\- hx^ + “y-^1 = -^1 + 0 (4.7-8)
Figure 4.7-1. Flow diagram (undamped system).
