Page 35 - Rotating Machinery Pratical Solutions to Unbalance and Misalignment

Developing the Vibration Equations

radius line will start at zero, increase to plus one, decrease to zero,
continue to decrease to a minus one, and finally increase back to
zero at 360 degrees. Note that 0 degrees and 360 degrees are the
same point [3 o’clock].
This is the trigonometric sine function. The sine of any angle
is defined as the Y value divided by the radius R or [Y/R]. In this
special case where R =1, the sine of the angle is equal to the Y
displacement. Note that the sine is zero for both angles 0 and 180,
while the sine of angle 90 is plus one, and the sine of angle 270 is
minus one.
In mathematical calculations, the sine of an angle is written
as Sin(a), where a is the angle expressed in degrees or radians.
Since this book uses degrees, all angles and their trigonometric
functions are presented in degrees.

DEGREES OR RADIANS

There are 2π radians in a circle, and therefore one radian is
equal to 180° degrees.

Radians = Degrees × pi/180 (2.1)
Or:
Degrees = Radians × 180/pi (2.2)

Thus the angle 27 degrees expressed in radians would be:

27 degrees = 27 × pi/180 radians, or .4712388980 radians.

When using calculators to obtain the sine or cosine of an
angle, be sure to note whether the calculator calculates the func-
tions in degrees or radians. If the sine function of 27 degrees was
taken but the calculator was in radians, the answer would be .9564
instead of the correct answer of .4540. When 27 degrees is ex-
pressed in radians, it is .4712389. Calculating the sine of that num-
ber would yield the correct answer, namely .4540.