Theory of Unbalanced Vibration, Theory of Balancing, Determination of Unbalanced Mass - Lab Report Example

Name Institution Lecturer Course Date Introduction Balancing is a crucial aspect of all mechanical systems particularly rotating mechanical elements, such as wheels, flywheels, shafts, pulleys and gears among others. Unbalance in rotating elements of machines is usually a primary source of unbalanced vibrations, due to the development of unbalanced forces in rotating systems. Unbalanced forces create excessive vibrations that ultimately lead to premature failure of machine components and the entire machine system especially when unbalanced vibrations from rotating elements are transmitted to other parts of the machine system. For example, vibrations from an unbalanced propeller shaft of an automotive can be transmitted to the body of the automotive, which will results to wearing out of the entire automotive. The magnitude of unbalanced forces and hence the intensity of unbalanced vibrations, increases with increase in rotation speeds. This project seeks to study the impact of unbalanced mass, eccentricity and speed on unbalanced vibrations. Theory of Unbalanced Vibration There are various sources of unbalanced vibrations in machine components including non uniform machine wear, repairs and production defects. Non uniform machine wear: the occurrence of non uniform wearing of machine components result to unbalanced masses within the respective component, which leads to unbalanced forces when the component is rotating. This is usually prominent in automotive wheel rims where non uniform wear, which leads to unbalanced vibrations, result from impact with hard objects. Repairs: repairs should be properly carried out to avoid unbalanced masses after the repair work. During several instances, vibrations from unbalanced forces usually begin after the first repair work on a machine component, such as a shaft. These vibrations intensify with subsequent repair works wherein the rate of repair work on the respective component increases due to unbalanced vibrations. For example, repairing a broken shaft through welding can result to unbalanced vibrations especially if the shaft is not checked for balancing. Production flaws: especially when a production unit does not have sufficient equipments for checking and ensuring that machine components are balanced. Unbalanced vibrations are usually eliminated through balancing, which is usually done on rotating machine elements, such as wheels. During balancing, the existence of unbalanced vibrations is established and the masses causing unbalanced vibrations, alongside their relative position from the component’s centre of rotation. Technically, the distance between the unbalanced mass and the centre of rotation of the component is known as eccentricity (e). The determination of unbalanced mass and eccentricity is followed by the addition of well determined masses at strategic positions relative to the centre of rotation of the component. The added mass (balancing mass) helps to distribute the mass of the component evenly to eliminate or reduce unbalanced vibrations. The intensity of unbalanced vibrations is not only dependent on the unbalanced mass and its distance from the center of rotation of the component, but also on the angular speed of rotation of the rotating component (Haddow 28). Research experiments have shown that steady state vibrations arising from unbalanced masses are proportional to the amount of the unbalanced mass (m) and its eccentricity (e) (Srinivas and Srinivas 95). Research experiments also show that steady state unbalanced vibrations arising from unbalanced masses are proportional to the square of the rotation speed of the component under consideration (Haddow 28). This means, therefore, that unbalanced vibrations increase with increase in rotation speed. This is perhaps the concept of critical speed of rotation beyond which machine operation can be dangerous because it can lead to mechanical breakdown due to excessive vibrations. This is especially the case when unbalanced vibrations are beyond the bearing capacity of the rotating component, which depends on material properties and component geometry. It is worth noting that it is not possible to eliminate unbalanced vibrations completely from rotating machine components. This is especially because it is practically impossible to eliminate all design and production flaws during the design and production of machine components. Additionally, it is not usually possible to ensure uniform wear during machine operation as a seemingly negligible non uniform wear, such as a small dent on a rotating shaft, can lead to significant unbalanced vibrations at super high speeds of operation. Therefore, design procedure for unbalanced vibrations involves determining the maximum permissible vibrations, through the determination of maximum permissible unbalance in order to limit the steady state deflection to a given value, such as amplitude of a given value. This is, of course, dependent on the safe (or maximum) speed of operation of the component in consideration, such as the maximum rotation speed of a car wheel. Such an analysis facilitates the determination of the appropriate mass (m) and eccentricity (e) to bring down unbalanced vibrations to below the maximum permissible vibrations. Consider the system below that comprises of an unbalanced system of mass (m) and eccentricity (e). The system also comprises of a spring of spring constant (k) and damper of damping constant (c). Figure 1: An unbalanced system in rotation As shown in figure 1, an unbalanced mass (m) located at a radius (e) is rotating at an angular velocity w. the entire system is of mass M. The motion of the entire system can be represented by the equation; (1) Where represents the angular acceleration of the unbalanced mass rotating in the x direction Equation 1 can be re-written as (2) However,, which is defined as the critical damping coefficient that can be defined as, Also, the natural frequency of the system is defined as, If equation 1 is compared to equation 2, it is seen that the two components are of the same form. Therefore, it is possible to re-write the steady state component as; (3) Where, The phase angle is fixed and it is given by the equation, Therefore, This can be re-written into a more compact form as, In which, the element, Br is a magnifying factor defined by the equation, Therefore, x (t), a steady state response is given by the equation, Equation 9 can be re-written as, Theory of Balancing Unbalanced vibrations can be eliminated through mass balancing. This involves the detection and determination of unbalanced masses and consequent application of masses at strategic location to counter the unbalanced masses. There are two types of balancing, static balancing and dynamic balancing. Static balancing Static balancing is used to distribute masses about the rotation axis of the mechanical component under consideration. Static balancing focuses mass distribution on the radial direction of rotation but does not consider the longitudinal/axial direction of rotation (Camhaoil 6). To balance a mechanical component, the component is rotated with its axis of rotation held horizontally. After every complete run of rotation of the component, such as a disk, a mark is made on one of the disk faces and at the lower part of the disk. After making several runs and consequent marks, the marks are studied for distribution, and if they are evenly distributed over the face of the disk, the disk is considered to be balanced (its masses are evenly distributed). However, if it is found that the marks are concentrated on one side of the disk’s face, the masses of the disk are said to be concentrated on that side of the disk (Vaughan 4). Balancing is then done by subtracting masses from the region of mass concentration and/or adding masses on the opposite of the region of mass concentration. Alternatively, balancing can be done by placing the disk with its axis of rotation along the y axis as shown in figure 2, which is the concept of the static balancing machine. Before loading, the system is like a pendulum that is free to swing around the ball joint with negligible friction for accuracy. The right side of figure 2 shows an unbalanced mass having been loaded. As shown, mass imbalance in the disk causes the system to tilt with the side having more mass being on the lower side while the side having less mass being on the upper side. Mass is added on the lighter side and/or removed from the heavier side for balancing. This is repeated until the desired balance level is achieved. Figure 2: determination of unbalanced mass Static balancing is done based on the following equations: For static balancing to be achieved, the resultant force of all forces arising from the rotation of the unbalanced mass should be zero (Rao and Karse 1-6). Mathematically, Where, In which mi is the unbalanced mass (eccentric mass) at ei eccentricity (distance from the rotation axis) in a part rotating at radians per second. Dynamic balancing In dynamic balancing, the presence of unbalanced mass is detected in all directions unlike in static balancing where unbalanced mass is detected on the central axis only. Therefore, dynamic balancing can be considered a 3D way of balancing unbalanced bodies while static balancing is a 2D version (Stadelbauer 12). Therefore, in dynamic balancing, the unbalanced mass is detected with respect to its magnitude, eccentricity and its position in the rotating body’s longitudinal (axial) direction. This is usually done by selecting one plane as the reference plane that is used to analyse the other planes that contain eccentric masses as shown in figure 3 (Grim, Haidler and Mitchell 13). Accordingly, each of the planes is located at a distance ai from the plane of reference. Figure 3: Dynamic (3D) balancing Dynamic balancing, like static balancing, is based on the condition that for a balanced system, the resultant force of all forces arising from the rotation of the unbalanced mass should be zero. Additionally, the summation of the moments of all the unbalanced masses about any point (this case the reference point) should be zero (Rao and Karse 1-6). Mathematically, a dynamically balanced system must fulfill the following two equations. The forces are given by; And the moments are given by; Therefore, all the centrifugal forces in all planes are translated to the reference plane as forces and moments. Afterwards, the application of vector summation of all the forces and moments, separately, helps in the determination of the unbalanced mass value and its 3D distance from the axis of rotation (eccentricity). This project is based on dynamic balancing. Experimental Setup The system is comprised of 4 blocks having the same dimensions and geometry. However, each of the 4 blocks has different hole of different size from the rest, which makes the four blocks to have different eccentric masses. The four blocks are placed onto a shaft and given some spacing so that each block has some distance (Si) from the end of the shaft. Angle i is measured with respect to the horizontal direction. The shaft carrying the blocks is then attached to an electric motor, which rotates the shaft and the blocks at different angular speeds. Vibrations on the rotating system are sensed by a rigid frame that is attached to the rotating shaft by flexible mountings. Before operating the system, a balanced state is achieved using the principles of balancing outlined. Si and i for two of the blocks are given and the same parameters for the remaining two blocks are to be determined analytically such that the system achieves a balanced state. Figure 4: Blocks nomenclature Experimental procedure Take all the necessary dimensions and parameters and determine the missing variables to ensure balanced state. Then, fix the four blocks to the shaft with their corresponding ai and  based on the balancing calculations. Run the shaft at different rotation speeds and observe unbalanced vibrations in the system. The parameters that will be changed include mass, eccentricity and rotation speed. The system will be tested for three different speeds, 1000 rpm, 1250 rpm and 1500 rpm. Mass and eccentricity values will be altered by using different sets of blocks such that the experimental matrix will be as follows. The variable Constant parameters Constant parameters Speed 1 = 1000 rpm e1 = 0.0558 e1 = Speed 2 = 1250 rpm e1 = 0.0558 e1 = Speed 3 = 1500 rpm e1 = 0.0558 e1 = m1 = 0.0558 Speed 1 = 1000 rpm e1 = m2 = 0.0565 Speed 1 = 1000 rpm e1 = e1 = Speed 1 = 1000 rpm m1 = 0.0558 e2 = Speed 1 = 1000 rpm m1 = 0.0558 For this experiment, To find the various missing parameters, take the necessary dimensions, as shown in figure 5, of all the four blocks. Determine the values of m and e for the four blocks from the following two governing equations. Where L1, L2, D1, D1, d and b are as shown in the nomenclature, and , where t is the thickness of the blocks (same for all the four blocks) Then, find the product of m and e (me) for the four blocks and quantify ame for the two blocks whose parameters are given. Draw a scale on a graph paper, from the point (0,0), vector m1e1 at and continue with vector m2e2 and 2 from the tip of the previous vector. Then, from the end of the latter vector, draw a circle whose radius is equal to m3e3. Then, draw a circle of radius m4e4 from the origin of the same vector. Join the point of intersection of the two circles with the end of the second vector to get vector 3, which is joined with the origin to get vector 4. Get 3 and 4 by measuring the angles for vectors 3 and 4 respectively. Get another graph paper and draw, from (0,0), vector a1m1e1 at angle 1 and continue the vector with a2m2e2 at 2. Draw a line at 3 from the end of the previous vector. From the origin of the vector, draw a line at 4. Vectors 3 and 4, with lengths, a3m3e3 and a4m4e4 respectively are identified by the intersection of these two lines (Dado and Abu-Farha 63). Then, it is possible to find a3 and a4 and then S3 and S4 depending on the scale chosen. Cited Works Dado, M & Abu-Farha, F. Mechanical Vibrations Lab, Manual. N.d. Web. January 04, 2013. Grim, Gary K., Haidler, John W. & Mitchell, Bruce J. The Basics of Balancing. Balance Technology, Inc. n.d. Haddow, A. Mechanical Vibrations Laboratory Manual. 2009. Web. January 04, 2013. MacCamhaoil, Macdara. Static and Dynamic Balancing of Rigid Rotors. Bruel & Kjaer. N.d. Rao, Mohan & Karsen, Chuck. Mechanical Vibrations. MichiganTech. 2003. Srinivas, Dukkipati & Srinivas Rao. Textbook of Mechanical Vibrations. PHI Learning Pvt. Ltd. 2004. Stadelbauer, Douglas G. Balancing of Rotating Machinery. N.d. Web. January 04, 2013. Vaughan, John. Static and Dynamic Balancing using Portable Measuring Equipment, 2nd edition. Bruel & Kjaer. N.d. Read More