Non-Coplanar Neutralize of a Spindle - Report Example

SUMMARY This report presents the outcomes of study carried out in laboratory to determine the balancing of non-coplanar rotating masses. The study involved the use of tecquipment TM-1002 in order to apply the analysis of experimental date to create a balance. This report includes the methods of finding these important unknowns using force balance and couple balance. Masses connected to a shaft can cause rotating reactions and a bending stress effect to the shaft if not well balanced. The process of balancing these masses require that both the unbalancing forces and moments acting on the shaft are made to diminish. As a conclusion, this method of balancing is proved to be accurate and applicable in real life problems such as balancing of car shafts, turbines and generators. A detailed background theory followed by the experimental procedure, equipment and results is included along with the discussion on results. Valuable conclusions are drawn in the end with recommendations. INTRODUCTION Quickly after the invention of wheel, man realized that the uneven, round surface of wheel is causing problems. The problem was nothing but the vibrations due to uneven mass distribution which can also damage the mechanism supporting it along with the wheel itself. Today, high speed engines and high speed rotating machines has become a common phenomenon. Therefore, it becomes very essential to ensure that all revolving parts become completely balanced the best possible. Improper balancing of these parts would mean that dynamic forces are set-up in the rotating system, which in turn produces vibrations on the machine. In a system of rotating masses, there is likelihood of the masses to have eccentricity as a result of inaccuracy in manufacturing, variation in density of shaft material or fitting tolerances. Masses attached on a shaft will rotate with the shaft, and therefore, if the center of gravity of the masses are not aligned with the axis of the shaft, then the masses will rotate about an axis at a radius equivalent to the eccentricity. The masses will have to maintain their positions relative to the shaft, meaning that the shaft will be forced to follow the direction of the masses by a centrifugal force resulting from the inertia of the rotating masses. The centrifugal force produces harmonic excitation to the rotating system, and the resulting effect is forced vibrations. To avoid vibrations, the balancing of rotating body is of immense importance. Vibration is mostly undesirable due to many reasons. Vibration is uncomfortable and in severe cases, it can cause disastrous failure in rotation machines used in heavy industries. Similarly an unbalanced wheel of car will make your ride unpleasant and tiring one. Some method must be found to eliminate or minimize this problem. In a rotating system, all the masses attached to it must be evenly distributed to avoid vibrations. In this lab, we did an experiment involving balancing of forces resulting from rotating masses, so as to nullify unbalanced forces when the shaft is running. The primary objective of this experiment was to dynamically balance a rotating shaft. Other specific objectives of this experiment include: Develop an understanding of methods to reduce vibrations and severe stress. Application of those methods practically. To find the masses to be attached for balancing. To find the location of these masses and angels in rotating body. Observing the accuracy of method. Presenting recommendations for improvement of procedure. BACKGROUND/THEORY In rotor machines, unbalanced forces are a times produced as a result of inertia forces, which are largely associated with rotating masses. A rotating mass experiences a centripetal acceleration produced by a force. Another opposite and equal force, the centrifugal force, acts radially outwards the rotating mass and is an interfering force on the rotation axis. The magnitude of the centrifugal force remains the same, but its direction changes with the rotating mass. Balancing of forces is the technique of eliminating unwanted moments or forces in rotating masses, achieved by varying of the location of the centers of the masses. It involves redistribution of the masses, either by removal or addition of masses from various members. Improper balancing of these masses leads to setting up of dynamic forces, which increase loads and stresses in various components, as well as aforementioned vibrations. A rotating body needs two things to be in balance: static balance and dynamic balance. Static balance occurs when the resultant centrifugal force is zero, and the gravitational center is on the rotation axis. If a shaft is rotating at an angular velocity  rad/s, the dynamic force F will pull the shaft in a direction towards the mass M connected to it at a radius r from the axis of the revolving shaft. The magnitude of the force F is given by: F = M2r The magnitude and positions of the masses in the rotating system can be obtained using two methods: (i) analytical method and (ii) graphical method. i. Analytical method This method involves finding and resolving the centrifugal forces due to the inertia of the attached masses vertically and horizontally, and obtaining their sums. The balancing force is treated as equal to the magnitude of the resultant force in an opposite direction i.e. V = m1r1 sin1 + m2r2 sin2 + …… And, H = m1r1 cos1 + m2r2 cos2 + …… The resultant centrifugal force, Fc = V + (H If the angle between the resultant force and the horizontal is, say, then, tan = V/H. The balancing force, Fc is given by: Fc = mr Where: m – balancing mass r – radius of rotation ii. Graphical method This method involves drawing space diagrams showing the relative positions of the attached masses and finding the centrifugal force exerted by each of the masses on the shaft. The centrifugal forces obtained are then used to draw a vector diagram with a scale representative of the magnitude and direction of the forces. As per the law of polygon of forces, the side that completes a closed polygon represents the magnitude and direction of the resultant force. The balancing force is then equal and opposite of the resulting force. The magnitude of the balancing mass is obtained from: m. = Resultant centrifugal force m.r = Resultant of m1r1, m2r2…… It is worth noting that the masses in this experiment are in balance, hence, there is no resultant force or unbalanced mass. EXPERIMENTAL PROCEDURE AND EQUIPMENT Equipment: The equipment used in this experiment was a static and dynamic balancing equipment, known as the TecQuipment TM-1002, with four masses. A picture of this equipment was taken during the experiment and is shown in figure 1. Figure 1: The TecQuipment TM-1002 static and dynamic balancing equipment Procedure: Four masses were correctly weighed and then placed in their correct positions and angles on the shaft of the equipment and relative to each other to achieve a dynamically balanced system. The table shown below was provided for filling in the missing data. After all the required angles and forces were obtained, the polygon of forces was drawn. The relative linear positions of the masses on planes B and C along the shaft were determined by drawing the polygon of moments. Figure 2: Masses being weighed on a scale during the experiment The data obtained after the experiment was used to fill the missing values in the table provided. Table 1: Provided values Plane Mass (kg) r (mm) mr (kg.mm) l (mm) mrl θ A 0.128 50 47 B 50 9.9 120 C 0.128 50 240 D 0.128 50 RESULTS AND CALCULATIONS Plane A: = (0.128 x 50) = 6.4 kg.mm = (6.4 x 47) = 300.8 kg.mm2 θ = 30o Plane B: = (0.198 x 50) = 9.9 kg.mm = (9.9 x 95.95) = 950 kg.mm2 θ = 120o Plane C: = (0.128 x 50) = 6.4 kg.mm = (6.4 x 56.25) = 360 kg.mm2 θ = 240o Plane D: = (0.128 x 50) = 6.4 kg.mm = 768 kg.mm2 θ = 298o After establishing the missing values of and their correct angles, the table 1 was filled with those values as shown in table 2 below. These values were then applied in the experiment. A force polygon and polygon were drawn with the help of data obtained in column four and six of table 2 respectively. Table 2: Obtained values Plane Mass (kg) r (mm) mr (kg.mm) l (mm) mrl θ A 0.128 50 6.4 47 300.8 30 B 0.198 50 9.9 95.95 950 120 C 0.128 50 6.4 56.25 360 240 D 0.128 50 6.4 120 768 298 The figure below shows a space diagram of the four forces relative to the axis of rotating shaft and the plane OX. Figure1: Space diagram Figure 2: Force Polygon In figure 2 above, a force polygon of the four balanced masses has been drawn to scale. From this vector diagram, it is clear that the masses on the shaft are in complete balance, i.e. a balanced system, hence, a closed polygon. Figure 4 below shows a polygon of moments that has been drawn to a scale. Figure 3: mrl polygon of moments The relative linear positions of B and C along the shaft are 95.9mm and 56.2mm respectively. DISCUSSION When the dynamic balancing equipment was set with masses at values of radii and angle positions obtained in table 2 above, and the motor switched on, it was noticed that the machine revolved without any noise or vibrations. This was an indication that all the masses attached to the shaft were dynamically balanced. A force polygon dawn to scale for the system is completely closed with sides representing the magnitude of the respective forces. This closed loop verifies the point that the masses attached to the shaft are in balance, hence the main objective of the experiment was achieved. The centrifugal force produced by each of the four masses is proportional to the product of its mass (m) and the radius of rotation (r). All the four forces act radially outwards, forming a concurrent coplanar system of forces at the point O, which is the axis of rotation. The forces produce a bending moment on the rotating shaft that has to be counteracted by a balancing mass attached on the same plane. Since the masses are in balance, the resultant force is zero, i.e. the centrifugal forces from the four masses are in balance. A complete balance of the several rotating masses in separate planes require that the forces falling in the reference plane (OX) become balanced as well as the couples about the same plane. Dynamic balance in a rotating system occurs when the resulting turning moment along the rotation axis is zero. In a rotating machine, the centrifugal force will always remain in balance provided the mass of the rotor is on the rotation axis of the shaft. Otherwise, eccentricity is observed, producing an unbalanced force. This kind of unbalancing is experienced in engine crankshafts, steam turbine rotors, centrifugal pumps, rotors of compressors etc. CONCLUSION The four masses are dynamically balanced on the shaft. Forces and couples have to be always in balance about the reference plane for the rotating system to be said that it is balanced. Balancing of the shaft in rotating machines reduces vibrational effect and noise as was observed in this experiment when the motor was turned on. Errors in this experiment could result from inaccuracy in weighing the masses and measuring the angles and radii of rotation. RECOMMENDATIONS The experiment can be improved by following the steps given below, i. Shaft masses must be perfectly rotating on the shaft axis. ii. Shaft must be clean to avoid any significant mass attached which is not incorporated in calculations. iii. Masses used must be checked on weight balance before experiment to make sure their mass is exactly as written on them. iv. Make sure that the masses are perfectly attached to shaft i.e. there is no clearance between them. REFERENCES Read More