School of Mechanical and Manufacturing Engineering
Static & Dynamic Balance Experiment
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Shafts which revolve at high speeds must be carefully balanced if they are not to be a source of vibration. If the shaft is only just out of balance and. revolves slowly .the vibration may merely be a nuisance but catastrophic failure can occur at high speeds even if the imbalance is small.
For example if the front wheel of a car is slightly out of balance this may be felt as a vibration of the steering wheel. However if the wheel is seriously out of balance, control of the car may be difficult and the wheel bearings and suspension will wear rapidly, especially if the frequency of vibration coincides with any of the natural frequencies of the system. These problems can be avoided if a small mass is placed at a carefully determined point on the wheel rim.
It is even more important to ensure that the shaft and rotors of gas turbine engines are very accurately balanced, since they may rotate at speeds between 15,000 and 50,000 rev/min. At such speeds even slight imbalance can cause vibration and rapid deterioration of the bearings leading to catastrophic failure of the engine.
It is not enough to place the balancing mass such that the shaft will remain in any stationary position, i.e. static balance. When the shaft rotates, periodic centrifugal forces may be developed which give rise to vibration. The shaft has to be balanced both statically and dynamically.
Usually, shafts are balanced on a machine which tells the operator exactly where he should either place a balancing mass or remove material. The apparatus requires the student to balance a shaft by calculation or by using a graphical technique, and then to assess the accuracy of his results by setting up and running a motor driven shaft. The shaft is deliberately made out of balance by clamping four blocks to it, the student being required to find the positions of the third and/or fourth blocks necessary to statically and dynamically balance the shaft.
The goals of this experiment are:
A shaft with masses mounted on it can be both statically and dynamically balanced. If it is statically balanced, it will stay in any angular position without rotating. If it is dynamically balanced, it can be rotated at any speed without vibration. It will be shown that if a shaft is dynamically balanced it is automatically in static balance, but the reverse is not necessarily true.
Figure SDB1 shows a simple situation where two masses are mounted on a shaft. If the shaft is to be statically balanced, the moment due to weight of mass (1) tending to rotate the shaft anti-clockwise must equal that of mass (2) trying to turn the shaft in the opposite direction.
Figure SDB 1 - Static Balance
Hence for static balance,
m1r1cosa1 = m2r2cosa2
The same principle holds if there are more than two masses mounted on the shaft, as shown in figure SDB2.
Figure SDB 2 - Static Balance of Three masses
The moments tending to turn the shaft due to the out of balance masses are:-
For static balance,
In general the values of m, r and a have to be chosen such that the shaft is in balance. However, for this experiment the product W.r can be measured directly for each mass and only the angular positions have to be determined for static balance.
If the angular positions of two of the masses are fixed, the position of the third can be found either by trigonometry or by drawing. The latter technique uses the idea that moments can be represented by vectors as shown in figure SDB3(a). The moment vector has a length proportional to the product mr and is drawn parallel to the direction of the mass from the centre of rotation.
Figure SDB 3 - (a) MR Vectors from Centre to Mass (b) Rearrangement to show Closed Polygon
For static balance the triangle of moments must close and the direction of the unknown moment is chosen accordingly. If there are more than three masses, the moment figure is a closed polygon as shown in Figure SDB3(b). The order in which the vectors are drawn does not matter, as indicated by the two examples on the figure.
If on drawing the closing vector, its direction is opposite to the assumed position of that mass, the position of the mass must be reversed for balance.
The masses are subjected to centrifugal forces when the shaft is rotating. Two conditions must be satisfied if the shaft is not to vibrate as it rotates:
a) There must be no out of balance centrifugal force trying to deflect the shaft.
b) There must be no out of balance moment or couple trying to twist the axis of the shaft.
If these conditions are not fulfilled, the shaft is not dynamically balanced.
Figure SDB 4 - Dynamic Out-of-Balance for a Two Mass System
Applying condition a) to the shaft shown in Figure SDB.4 gives:
F1 = F2
The centrifugal force is mrw2
the angular velocity, w, is common to both sides then for dynamic balance
is the same result for the static balance of the shaft. Therefore if a shaft is dynamically balanced it will also be
second condition is satisfied by taking moments about some convenient datum such
as one of the bearings.
this simple case where m1 and m2 are diametrically
opposite and F1 = F2 (condition a) then dynamic balance
can only be achieved by having a1 = a2 which means that
the two masses must be mounted at the same point on the shaft.
static balancing where the position of the masses along the shaft is not
important, the dynamic twisting moments on the shaft have to be eliminated by
placing the masses in carefully calculated positions. If the shaft is statically balanced it does not follow that
it is also dynamically balanced.
order for static balance to be achieved the sum of the vectors representing the
couple due to each rotor must form a closed polygon.
In the case where there are three rotors, the simplest arrangement to
give balance is shown in
- Dynamically Balanced Shaft with 3 Eccentric Masses
this case, it is clear from the first requirement that:
= 2 x F1
second criterion then says that:
= a1F1 + a3F1
= a1F1 + a3F1
= (a1 + a3)/2
that the eccentric mass in the middle has twice the m.r value of the two masses
on either side and is equidistant from both masses.
general case, where the eccentric masses differ on each rotor and the directions
are not exactly opposite is shown in Figure SDB6.
- General Case for Three Out-of-Balance Masses
The method for balancing such shafts requires the addition of two extra eccentric masses to the system at locations chosen by the engineer. These masses are determined in many ways. One method will be outlined in the lectures, while this experiment shows a method that uses preset out of balance forces and determines the orientation and position of the masses relative to the eccentric masses already on the shaft.
The Static and Dynamic Balance Equipment consists of a shaft mounted on a plate isolated from the base by rubber bushes. A motor, also attached to the plate, may be used to drive the shaft using a belt. Four identical eccentric mass blocks are provided, together with four individual inserts which may be used to alter the imbalance on each block. An extension shaft and pulley are stored on the base and used in conjunction with the string/buckets and ball bearings to determine the imbalance associated with each block. Two hexagonal keys (Allen keys) are provided to clamp the blocks on the shaft and the inserts into the blocks. Two guides, one on the pulley at the end of the shaft and one on the plate, are used to measure the relative angle and position along the shaft of each block. At either side of the plate two clamps allow the plate to be locked to the base. The plate should be clamped for the static parts of the experiment. If the motor is in use then the clamps should be released and secured away from the plate by tightening the screws.
Figure SDB 7 - Eccentric Masses and Axial Positioning
The experiment is divided into three parts:
Demonstration of Static and Dynamic Balance
For this part of the experiment the four blocks will be used without the inserts. This provides four identical eccentric masses for use in demonstration of static and dynamic balance.
Step 1 - Static Imbalance
Lock the plate to the base. Attach one of the blocks securely to the shaft near the pulley. The mark on the protractor should align with zero when the block is positioned against the guide. Slide the guide clear of the block. Rotate the shaft and observe the behaviour. Record your observations.
Step 2 - Static Balance
Attach a second block near the opposite end of the shaft. Align this block such that the protractor reads 180o. Rotate the shaft and observe the behaviour. Record this and compare with step 1.
Step 3 - Dynamic Imbalance
Release the plate and secure the clamps clear of the plate. Attach the belt between the motor and the pulley on the end of the shaft. Ensure that all loose components are removed from the equipment and then place the safety cover over the motor and shaft. Switch on the motor controller and the motor. Slowly increase the speed of the motor and observe the behaviour of the shaft and plate. Record your observations. Switch off both the motor & controller and allow the shaft to come to rest before removing cover.
Step 3 - Dynamic Balance
There are three steps to this part of the experiment
The axial position of the two balancing masses needs to be calculated next. In this case, it is simplest to take the largest eccentricity as the reference axial location, eliminating it from this part of the calculation.
Assemble all the information into a table indicating the eccentricity (mr) of each block, its axial location (l) and its angular orientation (q).
Figure SDB 8 - Graphical Solution to Dynamic Balance
Your report should give a detailed description of your observations and include all rough work and calculations. The graphs should be clearly labelled and neat.
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