C.1.2 Powers of Two-Term Sums
ða þ bÞ
3 ¼ a
3
þ b
3
þ 3a
2 b þ 3ab
2
;
ða þ bÞ
4 ¼ a
4
þ b
4
þ 4a
3 b þ 4ab
3
þ 6a
2 b
2
;
ða þ bÞ
5 ¼ a
5
þ b
5
þ 5a
4 b þ 5ab
4
þ 10a
3 b
2
þ 10a
2 b
3
:
ðC:2Þ
C.1.3 Averaging Integrals
These integrals often occur with averaging or ‘slow/fast’ analysis with harmonic
excitation. Let T = 2p/X be the averaging period and X the frequency, then:
1
T
Z T
0
ðsin XtÞ
2iÀ1 ðcos XtÞ
jÀ1 dt
¼
1
T
Z T
0
ðcos XtÞ
2iÀ1 ðsin XtÞ
jÀ1 dt ¼ 0; for i; j ¼ 1; 2; Á Á Á ;
1
T
Z T
0
sin
2
ðXtÞdt ¼
1
T
Z T
0
cos
2
ðXtÞdt ¼
1
2
;
1
T
Z T
0
sin
2
ðXtÞ cos
2
ðXtÞdt ¼
1
8
;
1
T
Z T
0
sin
4
ðXtÞdt ¼
1
T
Z T
0
cos
4
ðXtÞdt ¼
3
8
;
1
T
Z T
0
cosðXtÞ cosðXt À uÞdt ¼
1
T
Z T
0
sinðXtÞ sinðXt À uÞdt ¼
1
2
cos u;
1
T
Z T
0
sinðXtÞ cosðXt À uÞdt ¼ À
1
T
Z T
0
cosðXtÞ sinðXt À uÞdt ¼
1
2
sin u:
ðC:3Þ
In terms of a phase variable w = Xt the same integrals becomes:
478
Appendix C: Mathematical Formulas