Page 91 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

88 Chapter 1 Fundamentals oF Vibration
>
d >
ivt
dX = 1Ae 2 = ivAe ivt (1.52)
dt dt = ivX
>
2 d >
ivt
2
2
d X = 1ivAe 2 = -v Ae ivt = -v X (1.53)
dt 2 dt
Thus the displacement, velocity, and acceleration can be expressed as 4
ivt
displacement = Re[Ae ] = A cos vt (1.54)
velocity = Re[ivAe ivt ] = -vA sin vt
= vA cos 1vt + 90 2 (1.55)
2
2
acceleration = Re[-v Ae ivt ] = -v A cos vt
2
= v A cos 1vt + 180 2 (1.56)
where Re denotes the real part. These quantities are shown as rotating vectors in Fig. 1.49.
It can be seen that the acceleration vector leads the velocity vector by 90°, and the latter
leads the displacement vector by 90°.
Harmonic functions can be added vectorially, as shown in Fig. 1.50. If
>
>
Re1X 2 = A cos vt and Re1X 2 = A cos1vt + u2, then the magnitude of the resultant
2
2
1 >
1
vector X is given by
2
A = 21A + A cos u2 + 1A sin u2 2 (1.57)
2
1
2
and the angle a by
2
-1
a = tan ¢ A sin u ≤ (1.58)
A + A cos u
1
2
Im
v x(t)
X ivX
p/2 X x(t)
p/2
vt
Re O vt
p 2p
x(t)
2
X X
v
FiGure 1.49 Displacement, velocity, and accelerations as rotating vectors.
4 If the harmonic displacement is originally given as x1t2 = A sin vt, then we have
ivt
displacement = Im[Ae ] = A sin vt
ivt
velocity = Im[ivAe ] = vA sin1vt + 90 2
2 ivt 2
acceleration = Im[-v Ae ] = v A sin1vt + 180 2
where Im denotes the imaginary part.