Question

Which of the following statements are associated with the complete dynamic balancing of rotating systems?
I. Resultant couple due to all inertia forces is zero
II. Support reactions due to forces are zero but not due to couples
III. The system is automatically statically balanced
IV. Centre of masses of the system lies on the axis of rotation

• A. I, II, III and IV
• B. I, II and III only
• C. II, III and IV only
• D. I, III and IV only

Hint:

Remember the absolute hierarchy of balancing rules: - Dynamic Balancing $\implies$ Static Balancing (Always). - Static Balancing $\centernot\implies$ Dynamic Balancing. - Complete dynamic balance means Net Force = 0 AND Net Couple = 0, resulting in zero dynamic bearing reactions.

Answer 1

The Correct Option is D

Concept: Balancing of rotating masses is categorized into two types: static balancing and dynamic balancing.
• Static Balancing: Occurs when the center of gravity of the system of masses lies precisely on the axis of rotation. Under this condition, the net eccentric force or the sum of all centrifugal forces acting on the shaft is zero: \[ \sum \vec{F}_c = \sum m_i \cdot r_i \cdot \omega^2 = 0 \quad \Rightarrow \quad \sum m_i \cdot r_i = 0 \]
• Dynamic Balancing: Occurs when the shaft is rotating and the system experiences no net centrifugal forces as well as no net centrifugal couples along its length. For complete dynamic balancing, two separate conditions must be simultaneously satisfied:
• The net dynamic force must be zero (\(\sum \vec{F} = 0\)). This condition automatically ensures that the system satisfies static balancing and that the center of mass lies exactly on the axis of rotation.
• The net dynamic couple about any reference plane along the shaft axis must be zero (\(\sum \vec{C} = 0\)). Let us evaluate each of the given analytical statements step-by-step:

Step 1: Analyzing Statement I (Resultant couple due to all inertia forces is zero).
In dynamic operation, as different masses rotate in different planes, they set up centrifugal couples about any arbitrary point along the shaft axis. If the net dynamic couple is not zero, it creates a turning moment that tends to rock the shaft in its bearings, inducing varying support reactions. For complete dynamic balancing, the net dynamic couple must be zero. Therefore, Statement I is correct.

Step 2: Analyzing Statement II (Support reactions due to forces are zero but not due to couples).
When a system is completely dynamically balanced, both the net centrifugal force and the net centrifugal couple are identically zero. Because both are zero, the dynamic reactions at the supporting bearings are completely eliminated under ideal operating speeds. Statement II claims that reactions due to couples are not zero, which directly contradicts the requirement for complete dynamic balancing. Therefore, Statement II is incorrect.

Step 3: Analyzing Statement III (The system is automatically statically balanced).
Dynamic balancing requires that the condition \(\sum m_i \cdot r_i = 0\) is satisfied along with \(\sum m_i \cdot r_i \cdot z_i = 0\). Since satisfying the force equilibrium condition (\(\sum m_i \cdot r_i = 0\)) is a sub-requirement of dynamic balancing, any system that is dynamically balanced must also be statically balanced. Therefore, Statement III is correct.

Step 4: Analyzing Statement IV (Centre of masses of the system lies on the axis of rotation).
Static balancing physically means that the net static mass moment about the axis of rotation vanishes. Mathematically, this shifts the combined center of mass of all rotating components so that it aligns perfectly with the geometric axis of rotation. Since dynamic balancing guarantees static balancing, it guarantees that the center of mass lies on the axis of rotation. Therefore, Statement IV is correct. Combining our findings, Statements I, III, and IV are correct, which matches Option (4).