Page 102 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.11 harmoniC analysis 99
x(t)
(a) (b) (c)

A
(d) A t
(e)
O t t
2
(i)
x (t)
1
A
t
O t t
A 2
(ii) Odd function

x 2 (t)
A
t
O t
A
t
2
(iii) Even function
FiGure 1.58 Even and odd functions.

we find from Eq. (1.91),

t 4A 1 2p12n - 12 t
x ¢t + ≤ = sin ¢t + ≤
a
1
4 p n = 1 12n - 12 t 4
4A 1 2p12n - 12t 2p12n - 12
= a sin b + r (1.94)
p n = 1 12n - 12 t 4

Using the relation sin1A + B2 = sin A cos B + cos A sin B, Eq. (1.94) can be expressed
as

t 4A 1 2p12n - 12t 2p12n - 12
x ¢t + ≤ = b sin cos
a
1
4 p n = 1 12n - 12 t 4
2p12n - 12t 2p12n - 12
+ cos sin r (1.95)
t 4