Page 39 - Rotating Machinery Pratical Solutions to Unbalance and Misalignment

Developing the Vibration Equations

DERIVING VIBRATION EQUATIONS

Referring to the Figure 2-6, the vertical displacement X at any
time t can be expressed as:

X = Xo Sin ωt (2.5)

Where:
Sin = the sine of the angle ω at time t
ω = 2πf (2.6)
f = 1/t [frequency in cycles per second]

Note: Cycles per second will be expressed in Hertz (Hz)
X occurs @ Sin ωt) = 1; Sin ωt =1 occurs at, π/2, 3π/2, 5π/2, 7π/
max
2… (2n–1) π/2

Figure 2-6. Basic Sine Wave

The velocity at any time t is expressed by:

V = dx/dt (2.7)

And:
V = X ωCos (ωt) (2.8)
o
V occurs @ Cos (ωt) = 1; Cos ωt =1 occurs at, π, 2 π, 3 π,
max
4 π…nπ

Where:
V = Maximum Velocity, See Figure 2-7.
max
Cos = cosine of the angle ω @ time t