Version 08 June 16 2017 Page 1 School of Engineering and Technology Higher Education Division ENEM14015 – Dynamic System Modelling and Control Laboratory Instruction Sheet Mass Balance ___________________________________________________________________________ Compulsory Personal Protection Equipment (PPE) required to access laboratories. Closed footwear (leather uppers) Long or short sleeve shirts. No tank tops allowed. Lecturer: Mitchell McClanachan Lab OIC: Anwar Mohammad Learning outcomes (LOs): LO1: Design mathematical models that analyse and evaluate mechanical systems LO4: Relate theory to problems of introducing, operating and maintaining mechanical systems in the industrial context Introduction When any eccentric mass exists in a rotation system, the centrifugal force will be acting and rotating together with the shaft. These unbalance forces can be harmful, causing extra loading of bearings and vibration of the system. That is why the balancing process is of great importance for many practical cases. Multiple unbalances can exist along a single shaft due to unbalance in individual components fitted to the shaft. Objectives – Observe the difference between static balance and dynamic balance. – Observe multiple plane unbalance in rotating machinery. – Statically and dynamic balance a shaft by using additional masses. Equipment The Mass Balance test unit TecQuipment TM1002 and associated accessories are used in this lab. The lab instructor will introduce the equipment at the beginning of the Laboratory. NOTE: Large vibrations occurring on the equipment for extended period of time can cause permanent damage; the lab technician will guide you on the safe operation of the equipment. If in doubt, please ask.Version 08 June 16 2017 Page 2 Theoretical Background Rotating Unbalance The centrifugal force value F due to a mass unbalance can be expressed as shown in Equation 1. This centrifugal force is acting radially outwards and in the plane where the centre of mass m rotates, as shown in Figure 1. ð = ð âˆ™ ð âˆ™ ð² (Eqn. 1) where m = unbalance mass (kg) r = unbalance radius (m) Ï‰ = angular velocity of shaft (rad/s) Figure 1: Rotating unbalance Static and Dynamic Balance When an unbalanced system is in a static state, gravity forces will rotate the shaft to one position. This is static unbalance. A system in static balance is not necessarily in dynamic balance, but if a system is in dynamic balance it will also be in static balance. When unbalances lie in different planes down a shaft, each unbalance can be expressed as a force vector and a moment vector based on the unbalance mass, angle and length down the shaft, Figure 2. The static and dynamic balance of the shaft will take place only if the total unbalance force vectors is equal to zero, and the sum of the unbalance moment vectors is equal to zero. Figure 2: Shaft Unbalance Forces LB FA FB FC FD LC LD A B C D Î¸Î‘ Î¸Î’ Î¸C Î¸D r F Ï‰ mVersion 08 June 16 2017 Page 3 Determining Static balance for four masses The static balancing of the shaft is based on the unbalance masses, radius and angles. The unknowns can be determined either by use of static equilibrium equations or graphically by the use of vectors. Figure 3: Static Balancing of a four mass shaft The sum of the moments due to the weights, radius and angle of the masses have to equal to zero. For the Figure 3 the anticlockwise forces have to equal the clockwise forces as per equation 2. ð±ð±ðµð âˆ1+ ð²ð²ðµð âˆ2= ð³ð³ðµð âˆ3+ ð´ð´ðµð âˆ4 (Eqn 2.) If all the blocks â€˜Wrâ€™ values, and blocks Î±1 and Î±2 angles are known, then the other two angles can be determined by using a vector diagram method: 1) Choose the first block for which the angle and â€˜Wrâ€™ value is known. Draw its vector of with the length â€˜W1r1â€™ at the same angle as Î±1 shown in Figure 3. 2) Choose the second block for which the angle and â€˜Wrâ€™ value is known. Starting at the end of the vector drawn in step 1), draw the corresponding vector of length â€˜W2r2â€™ at its angle Î±2 shown in Figure 3. Step 1 and 2 are shown in Figure 4. 3) Using a compass draw an arcs of length â€˜W3r3â€™ and â€˜W4r4â€™ and find the intersection, Figure 4. 4) Draw the vectors â€˜W3r3â€™ and â€˜W4r4â€™ head to tail to complete the vector polygon and determine the angles Î±3 and Î±4. Figure 4: Vector Polygon Method r4 Î±4=? Î±3=? W2r2 W1r1 Step (1) and (2) W2r2 W1r1 Step (3) W3r3 W4r4Version 08 June 16 2017 Page 4 Determining Dynamic balance for four masses For dynamic balance the sum of the unbalance moment vectors has to equal zero. Unbalance moment vectors occur since the unbalance masses are placed in different positions along the length of the shaft as shown in Figure 5. If the distance of any mass from the reference point is â€˜Lâ€™ then the value of the moment vector will be: Unbalance moment vector for a single mass = F â‹… L Figure 5: Dynamic forces on shaft The rotational speed can be removed from the unbalance calculations as it is the same for all moment vectors. Moments are taken about a single reference point at one of the unbalance masses for the example in Figure 5 and Figure 6 the reference point is at the first block. If lengths L3 and L4 are unknown then these can be determined from equilibrium equations or from accurately drawing the moment polygon as shown in Figure 6. The angles are the same as those determined by the previous static balancing method. Summing moment vectors, Horizontal direction (Fig 6): -W2r2L2 cos Î±2 + W3r3L3 cos Î±3 + W4r4L4 cos Î±4 = 0 Summing moment vectors, Vertical direction (Fig 6): W2r2L2 sin Î±2 + W3r3L3 sin Î±3 – W4r4L4 sin Î±4 = 0 The unknowns above (L3 and L4) can be determined by solving the two simultaneous equations. When drawing the vector diagram first draw the W2r2L2 vector at angle Î±2. Now draw light/dotted lines at angles Î±3 and Î±4 respectively from both ends of the W2r2L2 vector and find the intersection. The lengths of the dotted lines are equal to the W3r3L3 and W4r4L4 vectors. If the W3r3L3 and W4r4L4 vectors in Figure 6 go in the opposite direction to the force vectors in Figure 5 this indicates that the â€˜Lâ€™ value is negative, i.e. it is on the other side of block 1. As W3r3 and W4r4 are known the L3 and L4 values can be calculated using: L3 = W3r3L3 / W3r3 and L4 = W4r4L4 / W4r4. Figure 6: Moment Polygon W1r1L1 = 0 W2r2L2 W3r3L3 W4r4L 4 Î±2 Î±3 Î±4Version 08 June 16 2017 Page 5 Lab Procedure: Experimentally determining the Wr values of the balance blocks The unbalance of the blocks can vary from the nominal values as shown below as the mass insert alignment may not be exact. The Wr values (in Nm) of the balance blocks can be determined experimentally as detailed in this section. The base should be level and clamped down with the two locking clamps during the procedure. â€˜TM1002 Static and Dynamic Balancing User Guideâ€™, TecQuipment 2011Version 08 June 16 2017 Page 6 Balancing exercise: 1) Using the values given for the positions of two of the balance blocks for your group in Table 3, fit these to the shaft. 2) Observe static unbalance: a. Engage the two locking clamps to lock the base in place, and remove the drive belt. b. Observe the static unbalance of the system by rotating the shaft 180 degrees by hand and releasing it. 3) Observe dynamic unbalance: a. Release the two locking clamps that hold the base in place. b. Ensure the shaft can rotate without hitting anything. c. Fit the drive belt and fit the safety dome. d. Start the motor and observe the unbalance and vibration. Only run the motor for a few seconds to avoid damage. 4) Determine the positions of the remaining two balance blocks to statically and dynamically balance the system. Calculate the unknown angles (Î±3 and Î±4) using static balance procedure described in the theory section. Calculat
e the lengths (L3 and L4) along the shaft using dynamic balance procedure described in the theory section. 5) Fit the two other balance blocks at your calculated positions. Again, check the static balance of the system as per step (1). If it is not statically balanced check the positions of the blocks and your calculations. Record any changes made to the system along with any observations. 6) Observe the dynamic balance of the system as per step (3). 7) If the system is still unbalanced check the positioning of the balance blocks and your calculations and go back to step (5). Record any changes made to the system along with any observations. 8) Make notes of any other observations. Observe the difference between static and dynamic balance 1) If you managed to balance the shaft both statically and dynamically, remove the drive belt and secure the base clamps and again observe the static balance. 2) Choose one of the blocks and keeping the angle the same, change the position along the shaft. 3) Again observe the static balance (it should still be statically balanced) 4) Refit the drive belt, remove the base clamps and start the motor and observe any dynamic imbalance. WARNING: â€¢ Always fit the safety dome when operating the machine. â€¢ Ensure the guide block is moved out of the path of the rotating blocks â€¢ Only run the motor for a few seconds if the system is unbalanced.Version 08 June 16 2017 Page 7 Results 1) Record the determined Wr values for the blocks. See Table 1 for a possible format. Table 1: Wr Values Block Applied mass Hanger mass Total Mass (kg) Total Weight (N) Pulley radius (m) Wr (N.m) 1234 2) Record the balance block positions for every trial run and the related observations. See Table 2 for a possible format. Table 2: Trial Block Positions Block 1 Block 2 Block 3 Block 4 Trial Angle (degrees) Distance from zero (mm) Angle (degrees) Distance from zero (mm) Angle (degrees) Distance from zero (mm) Angle (degrees) Distance from zero (mm) A Comments: B Comments: Etcâ€¦ Comments: 3) Observations on the difference between static and dynamic balance: Discussion â€¢ Discuss the results of the balancing exercises including any sources of inaccuracies. â€¢ Provide a discussion on the listed laboratory objectives. â€¢ Discuss how the concepts in this lab could be applied to engineering practice. Table 3: Initial Positions of Blocks for Each Lab Group Use one set of initial positons for your group or as directed by your instructor. Group Block 1 Block 2 Block 3 Block 4 Angle (degrees) Distance from zero (mm) Angle (degrees) Distance from zero (mm) Angle (degrees) Distance from zero (mm) Angle (degrees) Distance from zero (mm) 1 0 5 145 105 ? ? ? ? 2 65 20 170 105 ? ? ? ? 3 0 20 100 115 ? ? ? ? 4 90 10 180 120 ? ? ? ? 5 45 60 145 125 ? ? ? ?