Accelaration of Geared System
November 18, 2022 | Author: Anonymous | Category: N/A
Short Description
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Description
Acceleration of Geared System Objective
To conduct an experiment to measure the acceleration of a geared system and compare it with calculated theoretical values. Theory
As given in the lab sheet
Apparatus
Geared system rig “Orbit” counter timer plus inductive probe Selection of masses and wire
Figure 2: Selection of masses used in experiment and Orbit counter
Figure 1: Geared system rig
© Nizam Inc. 2017
ENGD2005- Theory of Machines
Data Given in lab sheet
The torque drum diameter of shaft 1 is 76.2mm, and the torque drum diameters of shafts 2, 3 and 4 are 50.8mm.
t1 = 90 teeth t2 = 30 teeth and 96 teeth
3 t = 24 teeth and 100 teeth t4 = 20 teeth
The following data is determined from an experiment I1 = 22.6x10-3kgm2
T FF11 = 2.19x10-3Nm
η12 =
90.4%
I2 = 23.8x10-3 kgm2
T FF22 = 3.63x10-3Nm
η23 =
94.0%
I3 = 26.1x10-3kgm2
T FF33 = 3.12x10-3Nm
η34 =
97.0%
I4 = 14.0x10-3kgm2
T FF44 = 3.11x10-3Nm
Procedure As given in lab sheet
1. Mesh all gears to produce a four-shaft train. 2. Mount the probe to monitor 60-hole circle on the inertia disc attached to shaft 4. 3. Apply accelerating mass via a wire to the torque drum on shaft 1. This mass will produce accelerating torque T1. Use masses: 6, 8, 10, 12 kg. 4. Using the ratchet handle raise the mass m ass and stop any rotation in the mechanism. 5. Release the mass by removing the ratchet handle and allow it to accelerate freely under the action of gravity. 6. Note the readings displayed on the counter co unter timer sequentially as they appear until mass hits the floor. Since there are 60 holes, each frequency reading will represent the average angular velocity of shaft 4 in revolutions per minute displayed at two seconds’ intervals. It is recommended recommended to use a video recorder (mobile phone camera) to record these readings for po post-processing. st-processing. 7. Repeat steps 3 - 6 for other accelerating masses. 8. For each run (accelerating mass): a. plot angular velocity (in rad/s) against time (s) for shaft 4. b. from the plots deduce the angular accelerations of shafts 4 (slope of the curve in 8a). 8a) . c. calculate transmission ratio of the gear box d. compute angular acceleration of shaft 1using results obtained in 8b for shaft 4. 9. Calculate values of angular accelerations of shaft 1 for the same accelerating masses theoretically. 10. Compare the results with those obtained experimentally in section 8.
Nizamuddin Patel
P15219444
ENGD2005- Theory of Machines
Results
6 kg Time (seconds) 0
Velocity (RPM) 0
Velocity (rads/s) 0
90
2
59
6.178466
80
4 6
101 143
10.5767 14.97492
8
183
19.16372
10
223
23.35251
12
261
27.33186
14
300
31.41593
) s 70 / s 60 d a r 50 ( y t i 40 c o l e 30 V
16
337
35.29056
10
18
374
39.16519
0
20
410
42.9351
22
446
46.70501
24
481
50.3702
26
515
53.93067
28
548
57.38643
30
581
60.84218
32
613
64.19321
34
644
67.43952
36
675
70.68583
38
682
71.41887
40
665
69.63864
42
650
68.06784
44
634
66.39232
46
620
64.92625
48
605
63.35545
50
590
61.78466
52
576
60.31858
54 56
563 550
58.95722 57.59587
58
536
56.12979
60
524
54.87315
Velocity/Time for shaft 4 (6 kg) y = 2.0348x
20
0
5
10
15
20
25
30
35
40
45
50
Time (seconds) Figure 3: Graph of results from 6kg load
Computational methods show an angular acceleration of 2.035rad/s2 Acceleration =
2
.− −. = 1.781 rad/s
Table 1: Experimental results for 6 kg
Nizamuddin Patel
55
P15219444
60
65
ENGD2005- Theory of Machines
8 kg Time (seconds) 0
Velocity (RPM) 0
Velocity (rads/s) 0
100
2
50
5.235988
90
4
108
11.30973
80
6
165
17.27876
8
220
23.03835
10
275
28.79793
12
328
34.34808
14
381
39.89823
16
432
45.23893
10
18
482
50.47492
0
20
532
55.71091
22
580
60.73746
24
628
65.76401
26
675
70.68583
28
720
75.39822
30
766
80.21533
Velocity/Time for shaft 4 (8 kg)
) s / s d a r ( y t i c o l e v
y = 2.7295x
70 60 50 40 30 20
0
5
10
15
20
25
30
35
40
45
50
time (seconds) Figure 4: Graph of results from 8kg load
Computational methods show an angular acceleration of 2.7295rad/s 2 2
32 34
810 800
84.823 83.7758
36
783
81.99557
38
766
80.21533
40
749
78.4351
42
733
76.75958
44
716
74.97934
46
700
73.30383
48
685
71.73303
50
669
70.05752
52
655
68.59144
54
640
67.02064
56
626
65.55457
58
612
64.08849
60
548
57.38643
Acceleration =
.− −. = 2.461 rad/s
Table 2: Experimental results for 8 kg
Nizamuddin Patel
P15219444
55
60
65
ENGD2005- Theory of Machines
10 kg Time (seconds) 0
Velocity (RPM) 0
Velocity (rads/s) 0
2
56
5.864306
4
134
14.03245
6
207
21.67699
8
278
29.11209
10
347
36.33776
12
414
43.35398
14
480
50.26548
16
545
57.07227
18
609
63.77433
20
672
70.37168
22
732
76.65486
24
793
83.04277
26
854
89.43067
28
912
95.50442
30
940
98.43657
Velocity/Time for shaft (10 kg) 120 y = 3.4513x
100 ) s / s d a r ( y t i c o l e v
80 60 40 20 0 0
5
10
15
20
25
30
35
40
45
50
time (seconds) Figure 5: Graph of results from 10kg load
Computational methods show an angular acceleration of 3.4513rad/s 2 2
32 34
921 902
96.44689 94.45722
36
884
92.57226
38
866
90.68731
40
848
88.80235
42
830
86.9174
44
813
85.13716
46
796
83.35693
48
780
81.68141
50
763
79.90117
52
747
78.22566
54
731
76.55014
56
716
74.97934
58
701
73.40855
60
686
71.83775
Acceleration =
.− −. = 3.194 rad/s
Table 3: Experimental results for 10 kg
Nizamuddin Patel
P15219444
55
60
65
ENGD2005- Theory of Machines
12 kg Time (seconds) 0
Velocity (RPM) 0
Velocity (rads/s) 0
2
10
1.047198
4
119
12.46165
6
210
21.99115
8
297
31.10177
10
380
39.79351
12
462
48.38053
14
543
56.86283
16
621
65.03097
18
700
73.30383
20
776
81.26253
22
852
89.22123
24
927
97.07521
26
1000
104.7198
28
1057
110.6888
30
1037
108.5944
Velocity/Time for shaft 4 (12 kg) 120 y = 4.0165x 100 ) s / s d a r ( y t i c o l e v
80 60 40 20 0 0
5
10
15
20
25
30
35
40
45
50
time (seconds) Figure 6: Graph of results from 12kg load
Computational methods show an angular acceleration of 4.0165rad/s 2 2
32 34
1017 996
106.5 104.3009
36
977
102.3112
38
958
100.3215
40
939
98.33185
42
922
96.55161
44
903
94.56194
46
885
92.67698
48
868
90.89675
50
851
89.11651
52
834
87.33628
54
817
85.55604
56
801
83.88052
58
785
82.20501
60
769
80.52949
Acceleration =
.− −. = 3.8223 rad/s
Table 4: Experimental results for 12 kg
Nizamuddin Patel
P15219444
55
60
65
ENGD2005- Theory of Machines
Calculations
Theoretical
– angular acceleration of shaft 1
– resistance torque
=
– applied torque at shaft 1
– equivalent moment of inertia
I1 = 22.6x10-3kgm2 I2 = 23.8x10-3 kgm2 I3 = 26.1x10-3kgm2
T FF11 = 2.19x10-3Nm T FF22 = 3.63x10-3Nm T FF33 = 3.12x10-3Nm
I4 = 14.0x10-3kgm2
T FF44 = 3.11x10-3Nm
η12 =
90.4% η23 = 94.0% η34 = 97.3%
= (ƞ) (ƞnƞn) (Tƞnƞnƞn) ) (.x¯³) .x¯³ ( . x¯ ³ . .. ...
=
2.19x10-3 +
TR(1) = 0.284 Nm
= .× × . × =22.6×10− . ... .....
IEQ(1) = 65.639 kgm2
: =× == × = 2 = = 6×9.81 6×9.81 × (76.2 2) × 10− = 2.224343 8×9.81 × (76.2 2) × 10− = 2.9999 = 8×9.81 = 10×9.811 × (76.2 2) × 10− = 3.773838 = 12×9.811 × (76.2 2) × 10− = 4.448585
Nizamuddin Patel
P15219444
ENGD2005- Theory of Machines
= 2.243 2465.3 6390.228484 = =0.0298 / 2.99 9965.6390.228484 = = 0.0412 412 / / 3.738 7365.8 6390.228484 = = 0.0552626 // 4.485 4865.5 6390.228484 = = 0.0664040 //
Experimental Figures given from lab
= = = 9030 = 3
== == == 9624100 24===45 20 = =3×4×5=60 Therefore, the transmission ratio for this geared system is 60.
Angular acceleration for shaft 1
ℎ 1 = 60 ℎ 4 Load (kg) 6 8 10 12
Angular acceleration for shaft 4 2.0348 2.7295 3.4513 4.0165
Angular acceleration for shaft 1 0.0339 0.0455 0.0575 0.0669
Table 5: Experimental results for angular acceleration of shaft 1 and shaft 4
Load (kg)
6 8 10 12
Experimental angular acceleration for shaft 1 (rads/s2) 0.0339 0.0455 0.0575 0.0669
Theoretical angular acceleration for shaft 1 (rads/s2) 0.0298 0.0412 0.0526 0.0640
Percentage error (%)
13.76 10.44 9.36 4.53 Average: 9.52
Table 2: Experimental and theoretical angular acceleration of shaft 1 and percentage error
Nizamuddin Patel
P15219444
ENGD2005- Theory of Machines
Discussion
1. Derive the formula for calculation of the angular acceleration of shaft 1 theoretically and present all computations.
=
ma = mg – F1
I α = F 2 = = = = = ² = = +² = = [ +] = [ [1 +] = [ [1 + ] − + = + = + = + g = + = (+ )
2. Comment on the agreement between measured and calculated values of angular acceleration. Table 6 shows a correlation c orrelation between the load and the percentage error, as the load increases the percentage error decreases. The largest percentage error in this experiment is 13.76%, this was using the 6 kg load. This is quite significant compared to 4.53% error from the 12 kg load. Theoretically the acceleration of the weights should be linear until the weight drops to the ground and then immediately starts decelerating. However, as this experiment was conducted in an uncontrolled environment it was hard to minimise all of the factors that could impact the experiment (The errors in the experiment are mentioned in more detail in answer 3).
Nizamuddin Patel
P15219444
ENGD2005- Theory of Machines 3. Is the angular acceleration constant? If not, to what do you attribute any discrepancy and is the assumption of constant acceleration justified? From figures 3,4,5,6 we can see that the acceleration is mostly linear, with a few anomalies. These anomalies could have been caused by many different factors; such as air resistance, parallax error, apparatus malfunction, percent error and random error. Air resistance could have occurred whilst the weight was being dropped, and whilst the gears were spinning. The increase in air resistance would have decreased the acceleration of the acceleration of the weight whilst it was dropping to the ground. Parallax error could have occurred whilst a student was measuring one metre from the ground to the weight. This could have caused the height to be more/less than one metre; this will have an impact on the acceleration value. The apparatus used (figures 1,2) could have been damaged or not calibrated; This would have been caused by continuous use or lack of maintenance. T Random error could have been caused when the readings from the timer were taken. As the readings were given every 2 seconds, there is a possibility that the results reader did not capture the result at the specific time, which may have resulted in a value being m missed issed out. Although, there seems to be an abundance of errors that could’ve occurred during the experiment, the percentage errors seem to be relatively small. 4. This is one method of angular acceleration measuring. Is it satisfactory? Can you offer an alternative and/or better method? There are many ways in which this experiment could have been more accurate and minimise the risk or errors. However, there is not one method that could eliminate all errors and give the perfect reading. Below you will see a brief description on how the errors mentioned in question 3 could have been minimised. The likelihood of Air resistance being prevented is very minimal, as the experiment will most likely have to be conducted in a controlled environment where it is placed in a vacuum. Parallax error could be minimised if an electronic tool was used to measure the distance between the ground and the weight. Apparatus error could be minimised mi nimised by either getting more accurate measurement devices, or having the equipment maintained on a regular basis. Random error could have been prevented if an electronic device recorded the results In conclusion, the experiment held was satisfactory as the acceleration calculated and obtained was very close to the theoretical value even though there could have been a few errors. However, there could have been other ways to carry out this experiment such as connecting a motor instead of weights and calculating the rate of acceleration through this way. This would ensure less errors as parallax, air resistance should be less than the method used here. The load will be connected to one side of the gears with the motor on the other side. The load will be varied to calculate the rate of acceleration; this would result in different acceleration rates like this experiment.
Nizamuddin Patel
P15219444
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