Page 90 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.10 harmoniC motion 87
Equations (1.39) and (1.40) yield
2 3
1iu2 1iu2
1cos u + i sin u2 = 1 + iu + + + g = e iu (1.41)
2! 3!
and
2 3
1iu2 1iu2
1cos u - i sin u2 = 1 - iu + - + g = e -iu (1.42)
2! 3!
Thus Eq. (1.36) can be expressed as
>
X = A1cos u + i sin u2 = Ae iu (1.43)

1.10.3 Complex numbers are often represented without using a vector notation as
Complex z = a + ib (1.44)
algebra
where a and b denote the real and imaginary parts of z. The addition, subtraction, multi-
plication, and division of complex numbers can be achieved by using the usual rules of
algebra. Let
z = a + ib = A e iu 1 (1.45)
1
1
1
1
z = a + ib = A e iu 2 (1.46)
2
2
2
2
where
2
2
A = 2a + b ; j = 1, 2 (1.47)
j
j
j
and
b j
-1
u = tan ¢ a j ≤; j = 1, 2 (1.48)
j
The sum and difference of z and z can be found as
2
1
z + z = A e iu 1 + A e iu 2 = 1a + ib 2 + 1a + ib 2
2
1
2
2
2
1
1
1
= 1a + a 2 + i1b + b 2 (1.49)
1
2
2
1
z - z = A e iu 1 - A e iu 2 = 1a + ib 2 - 1a + ib 2
1
1
2
2
1
2
2
1
= 1a - a 2 + i1b - b 2 (1.50)
2
1
2
1
>
1.10.4 Using complex-number representation, the rotating vector X of Fig. 1.47 can be written as
>
operations on X = Ae ivt (1.51)
harmonic >
Functions where v denotes the circular frequency (rad/s) of rotation of the vector X in counterclock-
wise direction. The differentiation of the harmonic motion given by Eq. (1.51) with respect
to time gives