Introduction
When objects are accelerated, inertia-forces are generated. This is the case with rotating machine elements. The design of such parts must ensure that they perform their intended work under all the existing combinations of service loads and rotational forces or inertia forces. The rotation of a machine element results in an external vibrating force. The shaking force disturbs the supporting frame of the machine and the adjacent parts. It is also worth to note that the vibration generated by rotating machine elements and consequently the associated noise is also a serious problem to the machine operators. Therefore, it is very important for the designer to devise ways to minimize the effect of vibrations on the machine. To achieve this, engineers introduce another shaking force equal in magnitude and position it to act in the opposite direction to that of the first shaking force. The main objective of this report is to explicate the character of static and dynamic balancing and to determine the arrangement required to balance rotating shaft, both statically and dynamically.
Objectives
Theory
Static unbalance refers to the concern of objects at rest, while, dynamic unbalance refers to a rotating body. For a statically balanced case, the body under consideration does not rotate about its centroidal axis when at rest regardless of the orientation of the body . This implies that, in a statically balanced body the axis or rotation passes through the centroid or center of gravity of the body. Figure 1 below demonstrates masses at an eccentric distance r from the axis of the shaft. The condition of static balancing of the mass system in Fig. 1 is F=0. We can expresses this condition as :
i=14miri=0 (1)
In Eq.1, r is the perpendicular distance from the mass of the eccentric element to the axis of rotation of the shaft. It is very important to note that an object may be statically balanced and dynamically unbalanced. This is because dynamic balancing involves an additional condition due to the rotation of the masses. For engineers to achieve dynamic balancing, they must achieve the following two conditions. First, the rotation axis must pass through the center of gravity and secondly, the rotation axis must be the principal axis of inertia. We can express this as follows:
F=0 (2)
M=mrd=0. (3)
Where, d is the axial distance of the mass along the axis of the shaft. It is also important to note that in dynamic balancing corrections in two separate must be taken into consideration when balancing the mass. Inspection of equation e and 3 shows that in dynamic balancing the shaft must be balanced when stationary and when rotating.
Figure 1: Rotating mass system
Methodology
Experimental apparatus
- Static and dynamic balancing equipment
- Assorted weight/mass
- Pulley
Experimental set-up
The Static and dynamic balancing equipment used in this work is shown in Fig.2.
Figure 2: Static and dynamic balancing equipment
Procedure
The experiment was carried in two parts.
Part 1
This part involved the observation of static and dynamic balancing on the various configurations of the shaft. The four shaft configurations considered in this case are shown in the Figure 3. The disks were removed from the blocks. Extreme care was ensured in order to match the number of the disk with the corresponding block. The configuration was set-up as shown in Fig.3 (a). The behavior of this configuration was noted for non-rotating (static) and rotating (dynamic) case. The same approach was adopted for all the three remaining configurations. The observation made in this case as presented in Table 1.
Figure 3: Shaft configurations
Experimental observation (part 1)
The experimental observation for part 1 is shown in Table 1.
Part 2
This part involved determination of the various unknown parameters needed in order to balance the shaft dynamically. The disk was placed in the block as described in the first part. By using pulley and weights, the appropriate 'mr' values were obtained. The number of metal balls gave the 'mr' values. Table 2 shows the various parameters obtained in this case. In this section, the missing values required in order to balance the shaft were determined from the vector diagrams. The missing parameters were d4, d3, θ3 and θ4. The determination of these parameters is presented in the following section.
Vector diagrams
In order to determine the missing parameters in Table 2, two vector diagrams were plotted to a scale. The first vector diagram was for 'mr' values and the second one was for 'mrd'. The two vector diagram are as shown below. Fig 4 shows a vector diagram of 'mr' values drawn to a scale 1:5. This was done in AutoCAD 2012.
Figure 4: Vector diagram for mr Values
θ3 = 360 -76-90= 1940
θ4= 360-101= 2590
Figure 5: Vector diagram for mrd values
The remaining parameters were obtained from Fig. 5 as shown below:
m3 r3 d 3= 2.5264 mm
but the scale adopted in this case was 1:200. Thus
m3 r3 d 3= 2.5264*200=505.28 mm
also
m3 r3=22
22 d 3=505.28 mm
d 3 = 505.2822=22.967 mm
Also
m4 r4 d 4= 12.8472 *200 =2569.44
also m4 r4 =19
19d 4 = 2569
d 4 = 2569/19=135.233 mm
Discussion
The character of static and dynamic balancing was demonstrated in the first part. The results obtained in Table 1 were accurate and corresponded to our expectations. From the first case, it was very evident that static balancing required only the condition F=0 to be met while dynamic balancing also required M=mrd=0 in addition to F=0. For dynamic balancing, the two conditions must be satisfied otherwise the shaft will not be balanced. This was the case with the first configuration where the shaft was statically balanced and dynamically unbalanced. From this part, it was very evident that the rotation of unbalanced masses always results to distracting levels of noise and vibrations that engineers must contain.
The second section involved determination of missing parameters that were required in order to balance the shaft statically. This was achieved by construction of vector diagrams. The method applied was vector polygon method. For a dynamically balanced case, the vector polygons drawn must close. The values of θ3 and θ4 were obtained from the first vector diagram of 'mr' values. The values of θ3 and θ4 obtained were then used plot the second vector diagram of 'mrd' values from which the axial locations d3 and d4 were obtained. From the plotted vector polygon, the unknown parameters were obtained as follows: d3= 23 mm and d4=135.21 mm, and angular orientations were obtained as θ3=1940 and θ4= 2590. Static and dynamic balancing is very important in daily life situations and engineers must address these issues in the design stage.
Conclusion
Works Cited
Ashby, Michael and David Hugh. Materials: Engineering, Science, Processing and Design. Butterworth-Heinemann, 2007.
Askeland, David and Philemon Phulé. The science and engineering of materials. 5. Cengage Learning., 2006.
Higdon, Askeland. Mechanics of Materials. 2. John Wiley & Son., 2009.
