Balancing of Rotating Masses free study notes

1. Introduction

Balancing of rotating masses is the process of arranging masses in a rotating system so that the resultant centrifugal force and couple become zero, thereby eliminating vibration.

Why Balancing is Needed

• To reduce vibration
• To avoid excessive bearing loads
• To ensure smooth operation
• To increase machine life

2. Unbalanced Force in Rotating Mass

When a mass rotates, it produces centrifugal force:

$F = m \omega^2 r$F=mω2r

Where:

• $m$m = mass
• $\omega$ω = angular velocity
• $r$r = radius of rotation

Effect of Unbalance

• Vibration
• Noise
• Shaft bending
• Wear and tear

3. Condition for Balancing

(a) Static Balance

• Resultant force = 0

$$\sum m r = 0$$∑mr=0

(b) Dynamic Balance

• Resultant force = 0
• Resultant moment (couple) = 0

$$\sum m r = 0 \quad \text{and} \quad \sum m r l = 0$$∑mr=0and∑mrl=0

Where:

• $l$l = distance between planes

4. Types of Balancing

(a) Single Plane Balancing

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• All masses lie in the same plane
• Only static balance required
• Example: thin disc

(b) Multi-Plane Balancing

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• Masses lie in different planes
• Both force and couple must be balanced
• Example: crankshaft

5. Methods of Balancing

(a) Analytical Method

• Uses vector equations
• Suitable for simple systems

(b) Graphical Method (Polygon Method)

• Forces represented as vectors
• Closed polygon → balanced system

6. Balancing of Several Masses in Same Plane

Condition:$$\sum (m r \cos \theta) = 0$$$$\sum (m r \sin \theta) = 0$$

• Forces resolved in horizontal and vertical directions
• Resultant must be zero

7. Balancing of Several Masses in Different Planes

Conditions:

• Force balance: $\sum m r = 0$∑mr=0
• Couple balance: $\sum m r l = 0$∑mrl=0

Procedure

1. Choose reference plane
2. Resolve forces
3. Calculate moments
4. Determine balancing masses

8. Balancing of Reciprocating Masses

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• More complex due to linear motion
• Produces:
  • Primary forces
  • Secondary forces

Primary Force

$$F_p = m r \\omega^2 \\cos \\theta$$Fp​=mromega2costheta

Secondary Force

$$F_s = m r \\omega^2 \\frac{r}{l} \\cos 2\\theta$$Fs​=mromega2fracrlcos2theta

9. Applications of Balancing

• Rotating shafts
• Turbines
• Electric motors
• Crankshafts in engines
• Fans and rotors

10. Advantages of Proper Balancing

• Smooth running
• Reduced vibration
• Less noise
• Increased machine life
• Improved efficiency

11. Consequences of Poor Balancing

• Excessive vibration
• Bearing failure
• Shaft damage
• Noise pollution
• Reduced efficiency

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