Dynamic Balancing

ES2A7

0919739

School of Engineering, University of

Warwick

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Summary

The main aims of the laboratory were to appreciate the problem of rotating masses, and produce a statically and dynamically balanced rotating system.

To produce a statically balanced system I arranged four masses at different angles on the rotating shaft. The angle at which each mass was placed was calculated using a vector diagram, following the theory of static balancing. To produce a dynamically balanced system I arranged the masses at the angles for static balance and also positioned each mass at a certain distance along the shaft. Again I used a vector diagram and followed the theory for dynamic balance.

The results gained showed that applying theory to balance rotating systems can considerably reduce vibration and noise.

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Table of Contents1. Introduction 4

2. Theory5

i. Definitions 5ii. Static balancing 5iii. Dynamic balancing…………………………………………………………………………………………..…5

3. Apparatus and Methods………………………………………………………………………………………………….6i. Static balancing procedure……………………………………………………………………………..7ii. Dynamic balancing procedure………………………………………………………………………...7

4. Observations and Results………………………………………………………………………………………………..7

5. Analysis of Results………………………………………………………………………………………………………….10

6. Discussion………………………………………………………………………………………………………………………11

7. Appendix……………………………………………………………………………………………………………………….11

i. Figures……………………………………………………………………………………………………………..…..11

ii. Images……………………………………………………………………………………………………………….…13

iii. Tables…………………………………………………………………………………………………………….…….14

8. References……………………………………………………………………………………………………………….…..15

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1. Introduction

The aim of the laboratory was to find out how to produce a static and dynamic balance for a simple rotating system. The laboratory briefing sheet provided an outline of the fundamental theoretical principles relating to static and dynamic balancing. These principles were then used to find the arrangement of masses required to produce an optimally balanced system.

Rotating machinery is integral in almost all mechanical engineering machinery, for example car wheels and turbocharger spools. Excessive vibration in rotating machinery can cause unacceptable levels of noise and more importantly, substantially reduce the life of shaft bearings. Therefore it is important to minimise vibration in order to preserve rotating machinery and ensure that it is operating most efficiently. An ideal scenario would be to remove all causes of vibration, resulting in the unit running totally “smoothly”. Unfortunately in practice this ideal scenario cannot be achieved and some inherent cause of vibration or unbalance will always remain.

This laboratory will show what theory can be used to reduce unbalanced forces that cause vibration in rotating machinery. The unbalance is caused by an effective displacement of the mass centre line from the true axis caused by some mass eccentricity in the unit. This process of balancing is the removal or addition of weight to the unit so that the effective mass centre line approaches the theoretically true axis. An important example of dynamic balancing is engine design. Increased balancing within an engine leads to improved performance, efficiency, cost of ownership, and reliability.

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2. Theory

2.i. Definitions

For this experiment it is particularly important to understand and know when to use the definitions of some key terms related to the balancing of rotating masses.

Static Balance: When a shaft carrying several eccentric masses is in static balance, the summation of all inertia forces is equal to zero, so that the shaft and attached masses remain in any position in which it is placed.

Dynamic Balance: When the shaft rotates centripetal forces act upon the masses, however there are also moments created by each mass-radii with respect to another point on the shaft. For dynamic balance the sum of these inertia moments must also equal zero.

2.ii. Static Balancing

For static balancing the masses on the shaft can be expressed as force vectors:

F⃗=mr

The object of balancing is ideally to provide zero resultant force on the bearings of the shaft.

We shall consider a rigid shaft supported by bearings A and B at each end as in figure 1 (Appendix). The shaft is carrying 4 eccentric massesm1,m2,m3 and m4 with radii of rotation r1,r2,r3 and r 4 then the sum of the forces is given by:

∑n=1

4

Fn=∑n=1

4

mnrnω2

∑n=1

4

Fn=(m1 r1+m2r2+m3 r3+m4 r4 )ω2=0

Since ω2 remains constant it can be removed from the equation meaning that the static balance of the system is only affected by the mass-radii configuration of the system. Vector diagrams can then be used to find the angular arrangement of the masses that is necessary to provide static balance. An example of a vector diagram for a statically balanced system is shown in figure 2 (Appendix). For a statically balanced system the vector diagram must be closed.

2.iii. Dynamic Balance

The case for dynamic balancing is very similar to that of static balancing. An arrangement of masses will produce centripetal forces which are equal to:

∑n=1

4

Fn=∑n=1

4

mnrnω2

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This is clearly equal to zero as at this point the system will already be statically balanced having already followed the procedure above.

However for the system to achieve complete dynamic balance:

(a) The resultant centripetal force acting upon the shaft must be zero (b) The resultant moment acting upon the shaft must also be zero

∑ F=0 and ∑M=0

Otherwise the rotation of the shaft would cause large vibrations; this could result in damage of the assembly and cause adverse effects on performance of the machine.

Having already achieved part (a) through static balance, we must now consider part (b). Each eccentric mass will also produce a moment with respect to a point on another axis – the shaft itself. Consider the diagram shown in figure 1 (Appendix). Each mass will create a moment relative to the front bearing A due to the distance L between the mass and the bearing. These moments act in a different plane to the centripetal forces acting on the shaft.

The equation for the sum of moments acting on the shaft is therefore given by:

∑n=1

4

M=∑n=1

4

mnr nLnω2

Thus to achieve complete dynamic balance the sum of the moments in the reference plane must be equal to zero:

∑n=1

4

mn rn ln=0

m1r 1L1+m2r2 L2+m2 r2L2+m3 r3L3+m4 r 4L4=0

This can once again be solved by using a closed vector diagram to find the critical arrangement of the masses on the shaft to achieve dynamic balance.

3. Apparatus and Methods

The experiment was conducted using a static and dynamic balancing rig. In this laboratory each group used a different rig; the assembly that will be used for this experiment is ‘rig 7’. The rig has a motor attached to drive the shaft during the test for dynamic balancing. The arrangement of the apparatus can be seen in Images 1,2,3,4 (Appendix).

Mounted on the rig are four eccentric bodies, which each had different mass-radii products to provide different levels of unbalance to the system. The values for each mass-radii product can be found in the table 1 (Appendix), the method for calculation of the mass radii product can also be found.

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The axial and angular positions of each mass could be changed by loosening the fastening mechanism using an Allen key. On the rig there is a scale for measuring the position of each mass. There is a ruler to measure the axial position along the shaft and there is also a dial to measure the angular orientation of the shaft relative to a fixed point. The angular position of each mass could also be fixed using this dial.

3.i Static Balancing Procedure

1. Fix two bodies at arbitrary positions on the shaft, preferably at 0° and 90° relative to each other using the dial on the rig assembly.

2. Use vector analysis to determine the angular settings for the other two bodies. Use the mass-radii values for the length of each vector.

3. Use compasses set at the length for the mass-radii of weights 3 and 4 to draw two arcs, one for each mass.

4. The point of intersection of these arcs is the final node in the vector diagram5. Draw lines to show the vectors for the two remaining masses6. Measure the angles of these vectors relative to the vector for mass 1 which we know is at 0°7. Adjust the positions of the remaining two masses on the shaft to the angle found using the

vector diagram

3.ii. Dynamic Balancing Procedure

1. Using the vector diagram used to find static balancing set weight 1 at an arbitrary length value. The formula m3 r3L3a gives the length for the vector. ‘a’ is a scale factor to ensure the L values found are appropriate for the scale on the rig.

2. Weight 2 is also set at an arbitrary length using the same scale factor ‘a’ 3. The two remaining lines can be extended until they intersect and the lengths can then be

measured 4. The actual L value for these two masses can then be found by working backwards using the

equation m3 r3L3a

5. Each mass can then be moved along the shaft to the L value found from the vector diagram to give a dynamically balanced system

4. Observations and Results

Table 2 below shows the results for the static balancing of the system. The angles for each mass were found using the graphical analysis method detailed in the theory section. The vector diagrams produced are also shown below.

Mass Number Mass-radii product Angular Position (degrees)1 105.57 0

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2 108.12 903 98.77 201.14 76.85 280

Table 2. Results from vector diagram for static balancing

Figure 7. Vector Diagram of mass arrangement

Drawing 1.

Following the rearrangement of the masses on the rig so that the shaft was statically balanced, there was a noticeable difference in the behaviour of the system. The rig was tested to determine the

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degree improvement of balance by eye and applying a force and moving the position of the rig by hand. Before the eccentric masses were moved the system was also tested using this method.

The unbalanced system would not remain in any position placed and after being moved to a new position rotated quickly following release, eventually returning to the same position of rest. Only a slight displacement was required to cause the system to rotate. The force required to rotate the shaft was relatively large meaning that the shaft felt ‘heavy’.

Once the masses had been rearranged into a statically balanced orientation, the shaft rotated slowly when a force was applied by hand and soon came to rest. The system would also remain at any angle to which it was placed, the system was generally easier to move and seemed ‘lighter’.

Table 3 below shows the results calculated for the dynamic balancing of the system using graphical vector analysis. The vector diagrams used are also shown below.

Mass Number Mass-radii productAngular Position (degrees)

Axial Position (cm)

1 105.57 0 `102 108.12 90 153 98.77 201.1 8.94 76.85 280 17.65

Table 3. Results from vector diagram from dynamic balancing

Drawing 2. Vector diagram for dynamic balancing

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To achieve dynamic balance the angular positions of the masses were kept constant, however the axial positions (distance along the shaft) of each mass was altered. The system was tested by driving the rig with a motor to provide an angular velocity. Once again the system was tested before and after the position of the eccentric masses had been optimised.

Preceding the rearrangement of the masses the system suffered large vibrations while being driven, the rig seemed to shake and seem generally unstable and unsecure. There was also a large amount of noise due to the vibrations. The second test, after balancing showed noticeable changes, the shaft ran much for smoothly with less shaking and vibration. There was also a difference in the volume of sound created, it was much quieter than the previous test.

5. Analysis of Results

Due to the nature of the results it is hard to compare and analyse them. The results for static balancing are correct if the shaft can be put to any position and it will remain static and not rotate. This indicates that the axis is in line with the centre of balance, and therefore statically balanced. The masses on the shaft are at the correct angles, relevant to their mass-radii values. This means that the total reactive force acting on the shaft is zero. The rig will dynamically balance if there is no vibration when the shaft is spinning. This will indicate that the masses are at the correct angles, relevant to their mass-radii values (statically balanced), as well as being the correct axial distance apart from each other. Each mass-radii will produce a moment with respect to another point on the axis. For the system to be dynamically balanced this value must equal zero.

The results that I collected are reasonably accurate. After statically balancing the arrangement, the shaft and attached masses remained in any position in which they were placed. Once dynamically balancing the rig the masses and shaft rotated “smoothly”. However the results of both the static and dynamic balancing were not perfectly accurate by any means. There must have been some error during the experimenting. The most likely source of error is human error. The whole balancing rig is operated manually using analogue dials and measuring equipment. Therefore setting the angles and axial distances of the masses, reading off the angles on the vector diagrams and setting the compass to mark out the length of the sides on the vector diagrams all contributed towards the degree of inaccuracy in the results. Improving the accuracy of the results is something that I will research when looking to make further improvements to the laboratory.

Overall, there is much to learn from this laboratory. Seeing the behaviour of the shaft and masses before making the calculated changes is very noticeable. Although not being entirely accurate, it gives enough indication to prove the theory that is used.

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6. Discussion

An interesting point that has arisen from the analysis is that the human error is the biggest error factor in the results. I have researched various methods that can eliminate this inaccuracy and make the experiment more precise. The most common way to overcome this, and a method often used in industry is a balancing machine. A balancing machine has its bearings connected to sensors (displacement or acceleration type depending on the design of the machine) which detect the "heavy" point, in relation to a datum on the unit, whilst it is being rotated. This increases the sensitivity and, hence, the accuracy of the balance.

7. Appendix

7.i. Figures

Figure 1. Space diagram showing a rigid shaft freely supported by A and B. (Dynamic Balancing briefing sheet, University of Warwick, department of engineering)

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Figure 2. Vector Diagram. (Dynamic Balancing briefing sheet, University of Warwick, department of engineering)

Figure 7. Vector Diagram of mass arrangement

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7.ii. Images

Image 1. Set up of balancing rig.

Image 2. Balancing rig in motion – testing dynamic balancing

Image 3. Arrangement of weights on balancing rig

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Image 4. End view of balancing rig.

7.iii. Tables

Table 1. Table showing mass-radii values for rig 7.

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Mass Number Mass-radii product Angular Position (degrees)1 105.57 02 108.12 903 98.77 200.14 76.85 260.12

Table 2. Results from vector diagram for static balancing

Mass Number Mass-radii productAngular Position (degrees)

Axial Position (cm)

1 105.57 0 `102 108.12 90 153 98.77 200.1 8.94 76.85 260.12 17.65

Table 3. Results from vector diagram for dynamic balance

8. References

Uicker, J. J., Pennock, G. R., & Shigley, J. E. (2003). Theory of Machines and Mechanisms; Third Edition . New York: Oxford University Press, Inc.

Hannah, J., & Stephens, R. C. (1984). Mechanics of Machines: Elementary Theory and Examples 4th Edition. London: Edward Arnold.

Wahab, M.A., (2008). Dynamics and Vibration; Revised First Edition. London: John Wiley & Sons.

http://www.dynamicbalancing.co.uk/whatis.html

http://dictionary.babylon.com/static%20balance/

http://www.torquecars.com/tuning/engine-balancing.php

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