Acceleration of Gears (Exp)

 

 

Experiment # 01

Acceleration of geared system

Submitted by: Submitted to: Lab code:

Table of contents

 

Measurement of acceleration of geared system......................................................................................2 1.1.

Objective Objec tive............... ............................. ............................. .............................. ............................. ............................. .............................. ............................................... ................................ 2

1.2.

Introductio Intro duction........... n......................... ............................. .............................. ............................. ............................. ............................. ............................. ................................ ................. 2

1.3.

Theory.......... Theor y........................ ............................. .............................. ............................. ............................. .............................. ............................. ......................................... ........................... 2

Types of gears............. gears........................... ............................. .............................. ............................. ............................. .............................. ............................. .................................. ....................2 2 Spur gears............. gears............................ .............................. ............................. ............................. .............................. ............................. ................................................... ..................................... 2 ............................. ............................. ............................. ............................. .............................. .................................................. ................................... 2 Helical gear............. gear............................ Worm gears............. gears............................ ............................. ............................. ............................. ............................. .............................. ............................. ...................................3 .....................3 Bevel gears............... gears.............................. ............................. ............................. .............................. ............................. ............................. ............................. ............................. .....................3 ......3 ............................. ............................. .............................. ............................. ............................. .............................. ............................. ..........................4 ............4 Gear ratio............. ratio............................ Acceleration of geared system.............. system............................. ............................. ............................. .............................. ............................. .................................... ...................... 4 1.4.. 1.4

Appara App aratus tus..... .......... ......... ......... .......... .......... .......... .......... .......... .......... ......... ......... .......... .......... .......... .......... .......... .......... ......... ......... .......... .......... .............. .................. .................5 ........5

1.5.

Procedure... Proce dure.................. .............................. ............................. ............................. ............................. ............................. .............................. ............................. ............................ .............. 5

1.6.

Calculati Calcu lations ons & Resul Results.......... ts........................ ............................. ............................. ............................. .............................. ............................. .............................. ................ 6

1.7. 1.8.

Discussion............. Discussio n............................ .............................. ............................. ............................. .............................. ............................. ............................. ............................. .................. .... 7 Conclusio Concl usion........... n.......................... ............................. ............................. .............................. ............................. ............................. .............................. ............................. ...................8 .....8

References Refer ences.............. ............................. ............................. ............................. .............................. ............................. ............................. .............................. ............................. ............................ .............. 9

List of Figures Figure Figure Figure Figure Figure

1 Spur gear...........................................................................................................................2 2 Helical gear.......................................................................................................................3 3 Worm gear.........................................................................................................................3 4 bevel gear..........................................................................................................................4 5 Acceleration Vs mass........................................................................................................7

List of Tables Table 1 Acceleration against different masses................................................................................7

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Measurement nt of acceleration of geared system Measureme 1.1. 1. 1. Ob Obje ject ctiv ivee The objective of performing this experiment is to determine the acceleration of geared system and compare it with the theoretical values to confirm the accuracy level.

1.2. 1. 2. In Intr trod oduc ucti tion on Different types of rotating members i.e. gears and belt drives are used in different application for  transfer of rotary motion from one place or member to another. Some of its applications are conveyer belts, vehicles, flywheels and majority of the industrial machineries. Due to their vast use, analysis of performance of gear systems is also of great importance. As major purpose of  gears is to increase or decrease the rotational acceleration of the system, in this experiment we are going to measure the acceleration of geared system. For this we will consider the gear system of four shafts with each one having its own gear of specific teeth amount, and measure the acceleration of this system.

1.3. Theory Gears are one of the machine elements having teeth cut on a cylindrical surface. They are used to transfer the rotational motion from driving shaft to the driven shaft, using the meshing of gear  elements on both shafts. Through shafts increase, decrease in motion, even change in direction of  motion can also achieved. Based on the application, there are different types of gears. Major gear  types are explained below:

Types of gears Spur gears They are also called as spur gears. In this gears, teeth are parallel to direction of shaft, as shown in the figur figuree below. In meshi meshing ng of spur gears, large one is call called ed gear and smaller one is calle called d  pinion. Such gears are easy to produce and have good efficiency for energy transmission[ CITATION Typ212 \l 1033 ] .

Figure 1 Spur gear 

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Helical gear These gears are similar to spur gear, one difference is that its teeth are not parallel to shaft. Instead they are twisted in left or right direction. Due to this twist they are able to resist axial loads and able to withstand high amount of loads. Moreover, their meshing is also smoother and their operation is also quite. Below figure shows the helical gears [ CITATION Rob21 \l 1033 ] .

Figure 2 Helical gear 

Worm gears These are the type of gear that contains shaft with spiral threads, along with the toothed wheel.

There is no point of intersection of both of the meshed shafts. Teeth of shaft are twisted while the other one is spur gear as shown in figure below. This gear also results in change in transmission direction by 90o.

Figure 3 Worm gear 

Bevel gears These are the gears with conical pitch as can be seen in figure. Number of teeth of both of its gears are not similar. It means there is some gear ratio. In order for perfect meshing of gears,  pressure angle, pitch and face width of both gears must need to be same. Speed reduction or  increase in these gears us is also right angled i.e. 90 degree. In this type of gears, large one is called gear and smaller one is called pinion.

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Figure 4 bevel gear 

Density is one the most important Density important parameter of flui fluid d especi especially ally when discus discussing sing their flows. It 3 tells us about the mass unit volume of the substance. In SI its unit is kg/m .Another term used in flows is specific gravity. Gear ratio It is basically the ratio of number of turns that driven shaft will take when one turn of driver shaft will be completed. In other words it is the ration of number of teeth of driver and driven shaft. When two gears of different diameters and meshed together, are rotated [ CITATION Sai21 \l 1033 ] . Pinion will rotate more quickly than the driver shaft. Gear ratio is calculated using the below equation:

Gear ratio (from 1 to 2) =

 Number of teeth of gear 1  Number of teeth of gear 2

Suppose gear has 120 teeth and meshed pinion has 30 teeth. When rotated together, gear ratio comes out to be: Gear ratio =

 Number of teeth of gear  =  Number of teeth of pinion

120 30

 = 4

For this case, gear ratio comes out to be 4 Acceleration of geared system As pur purpos posee of the gears in inc increa rease se or dec decrea rease se in rot rotati ation on dur during ing its tra transm nsmiss ission ion,, so it itss acceleration is also one of the most important concerning factor. Acceleration of gear system can  be calculated after determining the torque and moment of inertia of all shafts of the system[ CITATION Acc21 \l 1033 ] . Applying the Newton’s law, it comes out to be:

T1 – TR  =  = I EQ α1 Another equation for the determination of acceleration of this system is: α1 =

/ + T  R 1 / 4 m d + I  1 2 mgd 2

 EQ

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Where, T1 is the torque applied at first or driving shaft TR is the resisting torque IEQ = Total moment of inertia α1 is the angular acceleration of first or driving shaft Total moment of inertial means the total amount of inertia generated by all shafts in the system. Mathematically it is calculated using the equation below: IEQ (1) = I1 +

 I 2 n21 ƞ21

2

2

 +

2

2

 I 3 n21 n 32 ƞ21 ƞ32

 +

2

2

 I 4 n 21 n32 n 43 ƞ 21 ƞ32 ƞ43

Moreover, resistance torque is calculated using the equation below: TR (1) = TF1 +

T  F 2 n21 ƞ21

 +

T  F 3 n21 n 32 ƞ21 ƞ32

 +

T  F 4 n21 n32 n 43 ƞ21 ƞ32 ƞ43

Where, n ij is gear ration between i and j gears. It is calculated using the equation: n ij =

t i t  j

Where t i and t  j are number of teeth of i and j gears. Moreover, ƞ ij is the efficiency of torque transmission between i and j gears.

1.4. 1. 4. Ap Appa para ratu tuss Apparatus used in performing this experiment are:     

1.5.

Equipment of geared system Inductive probe Stopwatch Selected masses (6,8,10,12 kg) Wire for connection with the drum

Proc Pr oced edur uree

1. First First accele accelerat rating ing mass mass is applied applied using using the wire on the torque torque drum of shaft shaft 1, due to which torque T1 is produced. 2. Using the the ratchet, ratchet, rotati rotation on can be stopped stopped at at any moment moment by raisi raising ng the mass. mass. 3. Rel Releas easing ing it, will will the resume resume the accelera acceleratio tion n as the mass will will fall fall under the action action of  gravity. 4. Take the stop stop watch watch and measure measure the the readings readings of time time when when mass will will touch touch the ground ground and also after 20 seconds, as there are 60 holes present.

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Each fre frequen quency cy will represen representt the angu angular lar velocit velocity y (av (avera erage) ge) of sha shaft ft 4. Vid Video eo recorder  recorder  5. Each can also be used for post p ost processing and better data collection. 6. After that, this procedure procedure is repeated repeated for for other other masses masses present. present.

1.6.. Cal 1.6 Calcul culati ations ons & Re Resul sults ts To start the calculation of theoretical acceleration of the geared system. We need the inertia and torque (due to friction) values of all shafts. Such values va lues are given below: I1 = 22.6 (10-3) kgm2 & T F1 = 2.19 (10-3) Nm I2 = 23.8 (10-3) kgm2 & T F2 = 3.63 (10-3) Nm I3 = 26.1 (10-3) kgm2 & T F3 = 3.12 (10-3) Nm I4 = 14.0 (10-3) kgm2 & T F4 = 3.11 (10-3) Nm Moreover, efficiency of all meshed gears is as: Ƞ12 = 90.4 % Ƞ23 = 94.1 % Ƞ34 = 97.9 % Teeth of all these gears are: t1 = 90 teeth t1 = 30 & 96 teeth t3 = 24 & 100 teeth t4 = 20 teeth   From the given data, equivalent moment of inertia comes out to be: -3

−3 23.8 10

(

IEQ (1) = 22.6 (10 ) +

−3 26.1 10

2

0.904

− 14 ( 10 )( 3 ) (4 ) ( 5 ) )( 3 ) ( 4 ) (0.909 )( 0.941)  + ( 0.909 )( 0.941 )( 0.979 )

(

)( 3 )

 +

2

2

3

2

2

2

Solving the equation, total moment of inertia comes out to be: IEQ (1) = 65.19647 kgm2

Similarly, resistance torque will be: -3

TR (1) = 2.19 (10 ) +

−3 23.8 10

(

0.909

)( 2 )

−3 23.8 10

− )( 3 )( 4 ) 23.8 ( 10 )( 3 )( 4 )( 5 )  +  + (0.909 )( 0.941 ) (0.909 )( 0.941 )( 0.979 )

(

3

From that, it comes out to be: TR = 0.285907 Nm

 Now for value of mass m = 6 kg, acceleration value will be: 6

 

/ ( ∗9.81∗0.0762 )+ 0.285907  = 0.02995 m/s2 ( ) 1 / 4 ( 6 ) 0.762 + 65.19647

1 2 6

α1 =

2

For other values of masses, calculations are performed in a similar way and results obtained are summarized in the table below: Mass

Acceleration α1

(kg) 6 8 10 12

(m/s2) 0.029952514 0.041414821 0.052876107 0.064336372  

Table 1 Acceleraton agains diferen masses

Based on the calculated values of acceleration, graph is plotted to check its trend. For this values are inserted in Excel and graph is plotted, shown in figure 5. 0.07 0.06     )    2    s     /    m     (    n    o    t    a    r    e     l    e    c    c    A

0.05 0.04 0.03 0.02 0.01 0

6

8

10

12

Mass (kg)

Figure 5 Acceleraton Vs mass

1.7. 1. 7. Di Disc scus ussi sion on It can be seen that with the increase in mass applied, there is increase in resulting acceleration of  the geared system. It is because more mass will create more force, which result in more torque generation. This torque will lead to acceleration of the gears. For this reason, it is also termed as the accelerating mass. Moreover, there is efficiency of the gears considered in calculations. In ideal cases, efficiency of transmission of rotation between two meshed gears is not considered or  taken equal to 100%. But in our case, based on the type of gears, their meshing smoothness, we have defined the efficiency value for each meshed gears. This will lead us to the more actual results.

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Resisting torque is also included in the calculations. It is majorly due to friction between all these meshed components. In actual, friction is one of the major concerning factor for any moving  body, so this factor, include all such kind of losses in it. It can be seen that its value is subtracted in the equation of acceleration determination. In our case, its value comes out to be 0.285 Nm. The final value of acceleration comes out to be 0.02995 m/s2. In last, we havemass, also made graph of acceleration Vs mass.ofItthe hasgeared been seen that with the increase in accelerating there is increase in the acceleration system.

1.8. 1. 8. Co Conc nclu lusi sion on In this experiment, acceleration of the geared system is determined. For this geared system equipment is taken and different masses are applied on it. In this system, there are four shafts, each one have their specific gears and its teeth. Based on this total moment of inertia and resisting torque is determined. For our case, it comes out to be 65.19647 kgm2  and 0.2854 Nm. Using both of these values and Newton’s second law, acceleration of the system is calculated, which is 0.02995 m/s2 for mass of 6 kg. Its value is also calculated for value of masses i.e. 8, 10 and 12 kg and graph is plotted for all given mass values. It has been be en seen that with the increase in accelerating mass, there is also increase in resulting mass.

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References [1]] "Typ [1 "Types es of gear ge ar (Spu (S purr gear ge ar), ),"" Soc S ock k gears gea rs,, 2021 20 21.. [Onl [O nlin ine] e].. Avai Av aila labl ble: e: hps://khkgears.ne/new/gear_knowledge/inroducton_o_gears/ypes_of_gears.hml. [2] R. Olson Olson,, "The diere dieren n ypes of gears, gears,"" REXNO REXNORD, RD, hps://www.rexnord.com/blog/artcles/gear/ypes-of-gears hps://www.rexnord.com/blog/artcles/gear/ypes-of -gears .

2021.

[Online]. [Onl ine].

Available: Available:

[3] S. M4, "WHAT IS GEAR GEAR RATIO? IT’S FORMULA FORMULA AND CALCULAT CALCULATION ION ON GEAR RATIO," RATIO," he Engineer Engineer Pos, Pos , Novem No vember ber 2021. 202 1. [Onlin [On line]. e]. Availa Ava ilable ble:: hps:/ hp s://se /searc arch.y h.yaho ahoo.c o.com om/sea /search rch? ? fr=mcafee&ype=E210US91213G0&p=gear+rato. [4] "Acceleraton "Acceleraton of a Geared Sysem," 2021. [Onli [Online]. ne]. Available: Available: hps hps://www ://www.sud .sudocu. ocu.com/en com/en-au/documen/universiy-of-queensland/inermediae-mechanical-space-dynamics/mech2210experimen-1-acceleraton-of-a-geared-sysem/17143986.

 

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