Page 74 - Rotating Machinery Pratical Solutions to Unbalance and Misalignment

Rotating Machinery: Practical Solutions

LAWS OF SIN & COS

β
c

α a

b
γ

Figure 5-4. Relationships Between Angles and Sides of a Triangle

sin α/sin β = a/b (5.2)
sin α/sin γ = a/c (5.3)
sin β/sin α = b/a (5.4)
sin β/sin γ = b/c (5.5)
sin γ/sin α = c/a (5.6)
sin γ/sin β = c/b (5.7)

LAW OF COSINE

2
2
2
a = b + c – 2bc cos α (5.8)
2
2
2
b = a + c – 2ac cos β (5.9)
2
2
2
c = a + b – 2ab cos γ (5.10)
In our example, the angle between the O vector and the O +
T vector is β, and the opposite side is the T vector. The angle
between the O vector and the T vector is represented by α and the
opposite side is the O + T vector. Finally, the angle between the O
+ T vector and the T vector is represented by γ and the opposite
side, the O vector. This is illustrated in Figure 5-5.
We know that β = 60 degrees, but we don’t know anything
about the other two angles. Also, we know a = 5 mils and c = 3.5
mils. We need to know the angle α and the length b. The law of