Page 93 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

90 Chapter 1 Fundamentals oF Vibration
Im
x(t)

x (t)
2

14.1477
15
74.6
(1.30 rad)
10
vt 1 114.6 vt x (t)
1
(vt 1 2 rad) O Re
FiGure 1.51 Addition of harmonic motions.

Method 2: By using vectors: For an arbitrary value of vt, the harmonic motions x 1 1t2 and x 2 1t2 can
be denoted graphically as shown in Fig. 1.51. By adding them vectorially, the resultant vector x(t)
can be found to be

x1t2 = 14.1477 cos1vt + 1.30 rad2 (E.6)
Method 3: By using complex-number representation: The two harmonic motions can be denoted in
terms of complex numbers:

ivt
ivt
x 1 1t2 = Re[A 1 e ] K Re[10e ]
x 2 1t2 = Re[A 2 e i1vt + 22 ] K Re[15e i1vt + 22 ] (E.7)
The sum of x 1 1t2 and x 2 1t2 can be expressed as

x1t2 = Re[Ae i1vt + a2 ] (E.8)
where A and a can be determined using Eqs. (1.57) and (1.58) as A = 14.1477 and a = 74.5963
(or 1.30 radians).

■

1.10.5 The following definitions and terminology are useful in dealing with harmonic motion and
definitions and other periodic functions.
terminology
Cycle. The movement of a vibrating body from its undisturbed or equilibrium position to
its extreme position in one direction, then to the equilibrium position, then to its extreme
position in the other direction, and back to equilibrium position is called a cycle of vibra-
tion. One revolution (i.e., angular displacement of 2p radians) of the pin P in Fig. 1.46 or
>
one revolution of the vector OP in Fig. 1.47 constitutes a cycle.

Amplitude. The maximum displacement of a vibrating body from its equilibrium position
is called the amplitude of vibration. In Figs. 1.46 and 1.47 the amplitude of vibration is
equal to A.