Short Answer:
Balancing of several rotating masses means arranging two or more masses rotating in different planes or at different positions on a shaft so that their resultant centrifugal force and resultant couple become zero. It ensures that the shaft remains stable during rotation without vibration or bending. To achieve this, balancing masses are added in suitable positions and planes to counteract the unbalanced forces produced by all the rotating masses.
This process is important in engines, turbines, and other machines where multiple rotating parts exist. Proper balancing prevents vibrations, bearing damage, and mechanical failure, ensuring smooth and efficient machine operation.
Detailed Explanation:
Balancing of Several Rotating Masses
When several masses are attached to a shaft at different angular positions and planes, each rotating mass produces its own centrifugal force. These forces differ in magnitude, direction, and position. The combination of these forces causes the shaft to experience vibrations and bending moments if they are not properly balanced. To achieve smooth and stable operation, both the resultant centrifugal force and the resultant couple must be made zero. This process is known as the balancing of several rotating masses.
Each mass produces a centrifugal force given by:
where,
= centrifugal force,
= mass of the rotating body,
= angular velocity, and
= radius of rotation.
If these forces are not balanced, the machine will vibrate, causing wear and possible failure of components like bearings and shafts. Therefore, proper balancing of all rotating masses is essential in mechanical systems such as engines, turbines, compressors, and rotors.
Conditions for Complete Balancing
To completely balance several rotating masses, two important conditions must be satisfied:
- The vector sum of all centrifugal forces must be zero:
This condition ensures there is no resultant force acting on the shaft, eliminating linear vibration.
- The vector sum of all centrifugal couples must be zero:
where is the distance between the planes of rotation. This ensures that there is no resultant moment or torque causing bending or twisting of the shaft.
When both these conditions are satisfied, the system of several rotating masses is said to be completely balanced.
Procedure of Balancing
- Representation of Centrifugal Forces:
Each rotating mass is represented by a vector in magnitude and direction corresponding to its centrifugal force. These vectors are drawn from a common point at angles representing their angular positions. - Graphical Method (Polygon of Forces):
The polygon of forces is constructed by drawing the centrifugal force vectors one after another in sequence. If the polygon closes (forms a closed shape), the system is already balanced. If not, the closing side of the polygon gives the magnitude and direction of the resultant unbalanced force. - Balancing in Different Planes:
If the masses are in different planes, then in addition to forces, their moments or couples must be considered. By drawing a couple polygon, the magnitude and position of the balancing mass in a specific plane can be determined to eliminate both the resultant force and the couple. - Balancing by a Single Mass:
In some cases, one properly chosen mass placed in a particular plane and at a suitable angular position can balance the entire system. This mass produces both an equal and opposite resultant force and an equal and opposite resultant couple.
Analytical Method of Balancing
In the analytical method, the balancing problem is solved using equations instead of diagrams. The centrifugal forces of all the masses are resolved into horizontal and vertical components.
If the shaft rotates in the plane, then for perfect balancing:
These four equations ensure that the resultant force and couple are both zero, giving complete balance.
Applications
Balancing of several rotating masses is applied in:
- Crankshafts of multi-cylinder engines,
- Turbines and rotors,
- Compressors and blowers,
- Centrifuges and high-speed machinery,
- Electric motors and generators.
In all these cases, unbalanced forces due to multiple rotating parts can cause heavy vibrations and damage if not properly balanced.
Practical Example
Consider three rotating masses , , and mounted on a shaft at different angular positions and planes. Each mass produces centrifugal forces , , and . By drawing a force polygon representing these forces and a couple polygon representing the moments (force × distance between planes), we can determine the magnitude and position of the balancing mass that will completely balance the system.
Conclusion
Balancing of several rotating masses is the process of arranging multiple rotating components so that their combined centrifugal forces and couples cancel each other. It ensures smooth rotation of shafts without vibration, reduces bearing loads, and prevents mechanical failure. This process is essential for reliable and long-lasting operation of machines with multiple rotating parts such as turbines, engines, and compressors. Complete balancing improves performance, safety, and efficiency in mechanical systems.