Page 89 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

86 Chapter 1 Fundamentals oF Vibration
1.10.2 As seen above, the vectorial method of representing harmonic motion requires the descrip-
Complex- tion of both the horizontal and vertical components. It is more convenient to represent
>
number harmonic motion using a complex-number representation. Any vector X in the xy-plane
can be represented as a complex number:
representation >
of harmonic X = a + ib (1.35)
motion >
where i = 1-1 and a and b denote the x and y components of X , respectively (see
Fig. 1.48). Components a and b are also called the real and imaginary parts of the vector
>
>
X . If A denotes the modulus or absolute value of the vector X , and u represents the argu-
>
ment or the angle between the vector and the x-axis, then X can also be expressed as
>
X = A cos u + iA sin u (1.36)
with

2 1>2
2
A = 1a + b 2 (1.37)
and
b
-1
u = tan (1.38)
a
4
3
2
Noting that i = -1, i = -i, i = 1, c, cos u and i sin u can be expanded in a series as
2 4 2 4
u u 1iu2 1iu2
cos u = 1 - + - g = 1 + + + g (1.39)
2! 4! 2! 4!
3 5 3 5
u u 1iu2 1iu2
i sin u = iJu - + - g R = iu + + + g (1.40)
3! 5! 3! 5!

y (Imaginary)

b X a ib Ae iu

u
x (Real)
O a
FiGure 1.48 Representation of a complex
number.