Static and Dynamic Balancing PDF

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University of Birmingham Achieving Static and Dynamic Balance and Why it is Important William Lubiantoro Student ID: 1891452 Mechatronics and Robotics Engineering Personal Tutor: Peter Jankovic Mechanics Laboratory Session – Group 1C AM 2.3 Static and Dynamic Balancing Lab Date: December 11, 2019 La...

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Static and Dynamic Balancing William Lubiantoro

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University of Birmingham

Achieving Static and Dynamic Balance and Why it is Important

William Lubiantoro Student ID: 1891452 Mechatronics and Robotics Engineering Personal Tutor: Peter Jankovic

Mechanics Laboratory Session – Group 1C AM 2.3 Static and Dynamic Balancing Lab Date: December 11, 2019 Lab Coordinator: Carol Kong

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Assessment and Feedback Previous Feedback from previous assignment, areas needed for improvement: • • •

Don’t get distracted and go off topic, write only what is important and don’t be too wordy. Provide strong conclusions and applications related to the experiment to back-up your results. Address figures and tables in the report as evidence and back-up your discussion. Provide reference if taken from somewhere else.

How I attempted to act on previous feedback: • • •

Kept looking back at the aim and objectives to see what goal is and based conclusion on it. Searched for similar reports to provide useful applications and summarized the aims and objectives in the conclusion. All figures and tables are addressed in the main body of the report as well as the methods.

Feedback on this assignment that would be helpful: • • •

Structure, presentation of results as there not many to provide from the actual experiment. Issues, improvements and suggestions as discussed in the discussion. Resources are limited and unreliable. How concise and important is the theory and methodology of getting the results in the experiments using the machine (TM1002).

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Abstract This report was written to examine the methods of statically and dynamically balancing a rotating system. This experiment focuses on a more theoretical approach of balancing using the machine provided by TecQuipment’s Static and Dynamic Balancing (TM1002). Unbalance is the problem for many current and past operating machines as it produces unwanted vibrations that decrease the life of machines and parts. Vibrations can even lead to catastrophic failures and dangerous situations. In fact, the faster something rotates, the more force it creates (centripetal force equation), so it is even more important to be balanced on higher speeds. Different experiments along with theories using the machine was conducted to prove that a statically balanced system does not mean it is automatically dynamically balanced. There are two types of moments in a rotating system and in this experiment, they are stated as (i) Block moment (rotation on horizontal axis) and the (ii) Twisting Moment (centrifugal force or couple). To achieve dynamic balance, both (i) and (ii) need to equal to zero, which is also referred to as ‘balance’. Rotating systems with two, three and four masses were used as examples in this experiment to show how a system is statically and dynamically balanced just by using the moments equation and looking at which direction the force is heading towards. Blocks attached to shafts with angles on them require more calculation than symmetrical configurations as the block moment and twisting moments becomes a lot more complex. Therefore, drawing vector diagrams and moment triangles/ polygons can help visualize the angles more easily. This report concluded that static and dynamic balancing are important factors to consider for engineers building vehicles or machines with any rotating parts as balancing them can help reduce risks of damage and improve its functionality. Static and dynamic balancing can be calculated using moments equation and understanding the difference between both can help engineers create a better and safer machine.

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Introduction Machines and vehicles use a lot of rotating parts and ‘unbalance is the most common source of vibration in machines with rotating parts’ [1]. Unbalance happens when the mass is distributed unevenly on a rotating object that produces unwanted vibration. Vibrations are naturally bad for rotating objects as it decreases their useful life a lot faster than normal and at excessive vibrations, the damage can be detrimental. To avoid creating vibrations, the rotating object must have balance. Balance is created by improving the mass distribution by calculating the angles and ‘adding or removing weight from the rotating element’ [2]. There are two types of balancing: static and dynamic balancing. Static balancing is when an object stays in any angular position without rotating. Dynamic balancing is when an object rotates at any speed without vibrating or any resultant centrifugal force and couple. Knowing the difference between the two allows engineers to position correctly different points of mass in a rotating object to negate unbalance. The equation for centripetal force is expressed as: 𝐹 = 𝑚𝑟𝜔2 (Equation 1)

This shows that ‘the force caused by unbalancing increases by the square of speed’ [3]. Doubling the speed quadruples the force. Therefore, balancing is a very important factor to consider when building machines and vehicles with rotating parts, ‘especially where high speed and reliability are significant considerations’, as it increases a product's lifetime and safety [1].

Aim The Experiment aims to further understand the methods to achieve static and dynamic balancing with different amounts of masses and prove that a dynamically balanced system is automatically statically balanced, but a statically balanced system is not always dynamically balanced. It is also crucial to understand the importance of balance in a rotating object practically using real-world examples.

Objectives • • • •

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Further understand static balancing using moment equations and derivations Compare dynamic and static balancing and how it differs by calculating its forces Test balancing using the TecQuipment’s Static and Dynamic Balancing (TM1002) machine and record the results. Using previous results and knowledge of moments and forces for dynamic balancing, find angular and horizontal positions (using vector drawings) for a four-mass system and test results to see if static and dynamic balancing is achieved. Discuss real-life applications where achieving static and dynamic balance is of the utmost importance.

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Methods and Apparatus Removable Protective Dome

Extension Pulley

Figure 2: Method of attaching the balance blocks onto the horizontal shaft

Weight Hanger and Masses Figure 1: Static and Dynamic Balancing (TM1002) by TecQuipment. All images are taken from the TM1002 User Guide: http://www.robotgrubu.com/robo/wpcontent/uploads/2017/03/TM1002-User-Guide_0215.pdf

Figure 1 shows the equipment that was used in the experiment. Motors turn the horizontal shaft in the center using a pulley system that can hold up to four balance blocks in any position along the shaft longitudinally and angularly. Figure 2 shows how the balanced blocks are attached to the shaft. To find the static balance for the horizontal shaft, the total moment (anticlockwise + clockwise) must be equal to zero. The moment is expressed as 𝑀 = 𝐹∙𝑑 = 𝑊∙𝑟

(Equation 2)

Where 𝑊 is the weight of the balancing block in Newtons,

and 𝑟 is the distance to the center of the mass of the block in meters.

To calculate the moment of one balancing block, the extension pulley is attached to the end of the shaft so that the weight hanger hanged to a cord can be used to lift the block 90 degrees (to equilibrium). Figure 3 shows how this looks like from the side. Weights of 10g are added to the weight hanger until the block becomes perpendicular to the hanging weight, which means the total moment is zero, and the moment of one balancing block is calculated.

Figure 3: method of using the extension pulley to find Wr of one balancing block

Experiment 1 uses a two-mass system and it examines how statically balanced systems are not necessarily a dynamically balanced system. Using equation 2 to get a total momentum of zero in the system, Figure 4 shows how 𝑊1 would turn the shaft clockwise while 𝑊2 would balance it by turning the shaft anticlockwise. Figure 5 is the result of attaching balanced blocks following Figure 4 to get a statically balanced system because the moments would equal each other since moments 1 and 2 are known due to the previous method of calculating the moment of one balancing block.

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𝑊1 𝑟1 = 𝑊2 𝑟2

Twisting Moment

1

2

Block Moment

Figure 4: Static Balancing of a twomass system. Shows block moments for both blocks for a statically balanced system.

Figure 5: Front view of the shaft after attaching the balancing blocks. Blocks 1 and 2 are labeled.

Figure 6: Twisting Moment about the center of the shaft for a two-mass system (experiment 1).

Experiment 2 examines the dynamic balancing of a three-mass system using four balancing blocks, with the two middle blocks simulated as one big block for twice the mass. The configuration is as seen in Figure 7. Similarly, experiments 3 and 4 examines and tests a four-mass system for static and dynamic balancing using two different configurations as seen in Figures 8 and 9 respectively.

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Figure 7: Block configuration for experiment 2 of a three-mass system

1

2

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1

3

4

Figure 8: Block configuration for experiment 3 of a four-mass system

Figure 9: Block configuration for experiment 4 of a four-mass system after calculation of blocks 3 and 4’s angles and position.

Experiment 4 uses four blocks with different block moments as shown in Table 1.

Table 1: Block moment for blocks 1 – 4 in experiment 4

Results 84g

Weight of Hanger and Masses for block to become Equilibrium Radius of block Block Moment (Wr) for 1 balancing block

0.04𝑚 3.30 × 10−3 𝑁𝑚

Table 2: Result of block moment using the extension pulley in experiments 1,2 and 3 as in Figure 3. Experiment 4 uses different blocks.

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Experiment 1 – Two-mass system

Block 1 Block 2

Type of Moment Block Moment Twisting Moment Block Moment Twisting Moment

Direction Clockwise Clockwise Anti-clockwise Clockwise

Table 3: Direction of the moment for blocks 1 and 2 from Figures 4, 5 and 6 with the same weight for experiment 1 – two-mass system. Figure 4 show the block moment while Figure 6 shows the twisting moment.

Experiment 2 – Three-mass system

Block 1 Block 2 Block 3

Type of Moment Block Moment Twisting Moment Block Moment Twisting Moment Block Moment Twisting Moment

Direction Clockwise Clockwise Anti-clockwise Clockwise Anti-clockwise

Table 4: Direction of the moment for experiment 2. It is assumed that block 1 is turned clockwise. Taken from Figure 7.

Experiment 3 – Simple four-mass system

Block 1 Block 2 Block 3 Block 4

Type of Moment Block Moment Twisting Moment Block Moment Twisting Moment Block Moment Twisting Moment Block Moment Twisting Moment

Direction Clockwise Clockwise Anti-clockwise Anti-clockwise Anti-clockwise Clockwise Clockwise Anti-clockwise

Table 5: Direction of the moment for experiment 3. It is assumed that block 1 is turned clockwise. Taken from Figure 8.

Experiment 4 – Four-mass system with two initial block placements Block 1 Block 2 Configuration 𝜽°𝟏 𝒙𝟏 mm 𝜽°𝟐 𝒙𝟐 mm 1 0 20 100 120 2 0 5 150 105 3 0 5 160 85

Block 3 𝜽°𝟑 𝒙𝟑 mm 192 5 190 24 195 56

Block 4 𝜽°𝟒 𝒙𝟒 mm 272 147 337 148 353 159

Block Order (Left to Right) 3124 1324 1324

Table 6: Theoretical angles and positions of the balancing block given the positions and angles of blocks 1 and 2. These are calculated to achieve a statically and dynamically balanced system. See Figure 9. 𝑥 represents the distance from the left end of the shaft while 𝜃 ° represents the angular position of the balancing block. All the working out is in the appendix.

Balancing methods and theories were used to calculate the angles of the balancing blocks and the horizontal positions.

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Discussion To achieve dynamic balancing, two forces need to be balanced; the block moment or net dynamic force on the shaft must be equal to zero’ (condition for static balancing) and the twisting moment (from centrifugal force) of the shaft must be equal to zero as seen in Figure 6. This means, from Figure 6, the left-hand moment and the right-hand moment must be equal but opposite to each other. Figure 4, 5 and 6 are essentially the same system, with the force going in the same direction resulting in unbalance because the total twisting moment would not be zero and is expressed by: 𝑀 = 𝐿1 𝐹𝑐1 + 𝐿2 𝐹𝑐2 (Equation 3) Where 𝐹𝑐 = 𝑚𝑟𝜔2

Experiment 1 with two masses is statically balanced but dynamically unbalanced. From Table 2, the block moment when added equals to zero because they are opposite to each other. On the other hand, it is dynamically unbalanced because the twisting moments are additive and are heading towards the same direction, proving that a statically balanced system does not mean it is automatically a dynamically balanced system. When the configuration of Figure 5 was tested on the machine, small vibrations could be seen when the balancing blocks are rotating. Experiment 1 was reliable because only static balancing was tested. So, as long as the shaft was not swaying left and right when the balancing blocks were attached, it was already statically balanced. From Table 3 and Figure 7 the total block moment is zero because block two has double the mass of blocks 1 and 3. Clockwise, in this case, would be positive and anti-clockwise is negative. The total block moment is 𝑀 = 𝑊1 𝑟1 − 2𝑊2 𝑟2 + 𝑊3 𝑟3 = 0, Assuming all the weights are the same. However, the dynamic balancing for this configuration would be considered balanced because the net twisting moment is zero because blocks 1 and 3 cancel each other as seen in Figure 7. From Table 3, the twisting moment of block 2 is negligible because block 2 is in the center and is symmetric, that part is already balanced. Therefore, the twisting moment is 𝑀 = 𝐿1 𝐹𝑐1 − 𝐿3 𝐹𝑐3 = 0 (Clockwise is positive). Experiment 2 with 3 balancing blocks is both statically and dynamically balanced. Attaching this configuration in the machine would produce rotations with very little to no vibration if attached correctly and accurately. Comparing experiments 1 and 2, it can be seen clearly that there are fewer vibrations in experiment 2. These balancing theories are helpful and accurate but actual experiments can be a little different every time as there can be human error when placing and attaching the balancing blocks to the shaft as the slightest error in placement can produce unwanted vibrations. The human eye can also be unreliable, so to further increase the accuracy of this experiment, a third-party device may be needed to record the vibrations digitally and record the results in table form over a period of time, but it was not provided. Using a device to measure the vibration instead of just using human eyes can improve the accuracy and reliability of the experiments drastically. This issue applies to all the experiments done using this TM1002 machine. Experiment 3 is similar to experiment 2 with the only difference being block 2 is split up into 2 different blocks. Using the center of the shaft as a reference, it can be seen from Figure 8 and Table 4 that the system is statically and dynamically balanced. The clockwise and anti-clockwise directions cancel each other to give a moment of zero for both the block and twisting moment. This further proves that a dynamically balanced system will automatically be a statically balanced system. So far, the distance and the angle between each block have been negligible because they are symmetrical from the center of the shaft. But in the case where the distance and the angles are asymmetric and random, other actions such as 8

adding or removing weights are necessary to make the rotating system statically and dynamically balanced. The moment polygon of experiment 4, configuration 1 is shown in Figure 10.

Figure 10: moment polygon and angular positions of the angles of experiment 4, configuration 1.

Figure 11: Example of rearranging a complex moment polygon diagram into one big loop.

As seen from Figure 10, when trying to achieve static balance, adding the angles together will result in the angles going back to where it started. A statically balanced system will not have any net moment therefore the angles will cancel each other out and therefore return to where it started, producing a net moment of zero. This also applies to three and above mass systems that have angles. Experiments 1, 2 and 3 do not have diagrams for their angles because their directions are 180 degrees apart. Vector diagrams are only applicable when the masses have an angle to them so that can loop back to the beginning. The order does not have to be specifically one to four, as seen in Figure 11. When their directions are 180 degrees apart, the clockwise and anti-clockwise direction needs to be equal. To achieve a dynamic balance, the twisting moment of the entire system must also be equal to zero. This can be calculated using equation 3 to calculate the length between all the blocks and a chosen reference block, which in this case is block 1 for experiment 4. Figure 9 shows the block configuration of Table 6 clearly. This time there is no centerline since the configuration is not symmetric around the center, so block 1 was used as a reference since its angular position was at 0˚. Therefore, the twisting moment on the right side of block 1 needs to be equal to that of the left-side. Given the angles and positions of blocks 1 and 2 allows the angles and positions of blocks 3 and 4 to be found theoretically. The methods are presented in the appendix. There are of course many possibilities and configurations using the same starting 2 blocks that can achieve a statically and dynamically balanced system, but not all of them have realistic distances between the blocks or not fitting into the shaft. Some calculations may require trial and error using different correct possibilities and this is the limit to using this machine. Many vehicles and machines require dynamic and static balancing to properly work at its best and be safely used. Unbalancing produces unwanted vibration that can cause fatal damage and decrease in life to the object that is rotating and its surroundings. One example is a turbopump that runs at 40,000 rpm. ‘Vibrations can cause a rub to occur’ which results in ‘catastrophic failure’, as vibration forces can ‘be many times the gravitational load’ that bearings or any other small parts cannot handle, which causes ‘early bearing failure’ [4]. Parts of machinery and vehicles have a certain lifetime and wears the more it is used. Vibrations due to unbalance speeds this up and builds up contamination in the parts, making it break faster. Proper control of balance can help extend the life of machines and parts, it can also improve the functionality of many machines by making it more efficient and safer. 9

Conclusion Static and dynamic balancing are both an important factor in many current and upcoming machines and vehicles with rotary parts. Balancing reduces the risk of dam...