If a body is moving linearly through a curved path then a centripetal force is necessary to apply on the body. This centripetal force is also known as active force. But if a body at its own, moves through a circular path, a radially outward force also known as reactive force acts on it. The term “gyroscope”, conventionally referred to the mechanical class of gyroscopes. Gyroscope applications are mainly for navigation and control systems in aviation as well as space engineering. In
A number of researchers have studied gyroscopic systems along with its properties. But, most of the work that has been carried out is theoretical and difficult to apply experimentally. Therefore, author, performed the experiment in the machine dynamics laboratory to relate both the theory and practical.
Motorized Gyroscope Apparatus available in machine dynamics laboratory is shown in
The mass is kept in the pan at a fixed distance from rotor and the applied couple is calculated. The Gyroscopic couple is also found with the help of mass moment of inertia, rotor speed and precession speed. We keep rotor at zero position. The variable motor is started with the help of rotary switch. We increased the speed of variable motor gradually and let to stable and measured its speed as 881 rpm with the help of tachometer. Now, we put the 1 kg weight in the weight pan and yoke rotate in counter clockwise direction recorded the time taken for the rotation of angle 45°, 90°, 135° and 180° using stop watch. Now, at the same speed (881 rpm), we increased the external weight from 1 kg to 2 kg and recorded the time taken again in rotation of angle 45°, 90°, 135° and 180°. Similarly, at the same speed (i.e. 881 rpm), we recorded the third and fourth observations at 3 kg and 4 kg respectively for the rotation of 45°, 90°, 135° and 180° each time. The author recorded all these four observations in
Now, we increase the rotor rpm up to 1608 rpm and at this constant speed, the time taken for the second set of observations for external weight of 1kg, 2kg, 3kg and 4 kg is recorded for rotation of angle 45°, 90°, 135° and 180° for each mass in
I = Mass moment of inertia of disc or rotor about Z axis in kg.m2.
N = Speed of motor in RPM.
ω = Angular velocity of spinning rotor in rad/sec.
ωp = Angular velocity of precession of yoke about vertical axis in rad/sec.
dθ = Angle of precession in degree.
dt = Time required for this precession angle in sec.
M = Mass of the disc or rotor in kg.
m= External mass applied in the pan in kg.
L = Distance of bolt of Weight pan from disc Centre in m.
R = Radius of rotor, in m.
C = Reactive or Gyroscopic couple = I. ω. ωp in N. m.
T = Active or Applied couple = W.L = m.g.L in N. m.
D = Diameter of the disc or rotor in m.
R = Radius of the disc or rotor in m.
D= Rotor Diameter = 300 mm = 0.3m, L = Distance of bolt of Weight pan from disc Centre =197 mm, M = Mass of the disc or rotor = 6.5 kg, I = M.D2/8 = 0.073 kg.m2, m1 =1 kg, m2 =2 kg,m3 =3 kg, m4 =4 kg, T1=1.933 N.m, T2=3.865 N.m, T3=5.799 N.m, T4=7.732 N.m.
S. N. |
θ (deg) |
dt1 (at m=1 kg) |
dt2 (at m=2 kg) |
dt3 (at m=3 kg) |
dt4 |
1 |
45 |
2.15 |
1.38 |
1.09 |
0.67 |
2 |
90 |
4.4 |
2.68 |
1.93 |
1.35 |
3 |
135 |
6.79 |
4.08 |
2.73 |
2.12 |
4 |
180 |
10.02 |
5.42 |
3.6 |
2.84 |
S. N. |
θ (deg) |
dt1 (at m=1 kg) |
dt2 (at m=2 kg) |
dt3 (at m=3 kg) |
dt4 |
1 |
45 |
3.94 |
2.25 |
1.55 |
1.09 |
2 |
90 |
8 |
4.69 |
2.85 |
2.41 |
3 |
135 |
12.94 |
6.68 |
4.17 |
3.59 |
4 |
180 |
17.42 |
9.48 |
6.12 |
4.59 |
S. N. |
θ (deg) |
dt1 (at m=1 kg) |
dt2 (at m=2 kg) |
dt3 (at m=3 kg) |
dt4 |
1 |
45 |
6.4 |
2.78 |
2.12 |
1.53 |
2 |
90 |
12.76 |
5.54 |
4 |
2.77 |
3 |
135 |
19.45 |
8.29 |
6.07 |
4.44 |
4 |
180 |
27.27 |
11.15 |
8.01 |
5.97 |
S. N. |
θ (deg) |
dt1 (at m=1 kg) |
dt2 (at m=2 kg) |
dt3 (at m=3 kg) |
dt4 |
1 |
45 |
7.41 |
3.7 |
2.11 |
1.99 |
2 |
90 |
14.66 |
7.08 |
4.07 |
3.59 |
3 |
135 |
22.98 |
10.45 |
6.46 |
5.23 |
4 |
180 |
31.9 |
14.11 |
8.8 |
6.82 |
The results and various calculations at spinning speed (N) 881 rpm are recorded in
We observe from
The author observes from
It is clear from
SN |
1 |
2 |
3 |
4 |
Avg |
θ (deg |
45 |
90 |
135 |
180 |
|
m in kg |
1 |
2 |
3 |
4 |
|
ωp1(rad/s) |
0.365 |
0.357 |
0.347 |
0.314 |
|
ωp2(rad/s) |
0.569 |
0.586 |
0.578 |
0.58 |
|
ωp3(rad/s) |
0.721 |
0.814 |
0.863 |
0.873 |
|
ωp4(rad/s) |
1.172 |
1.164 |
1.111 |
1.106 |
|
C1(N-m) |
2.46 |
2.41 |
2.34 |
2.11 |
2.33 |
C2(N-m) |
3.84 |
3.95 |
3.89 |
3.91 |
3.896 |
C3(N-m) |
4.86 |
5.489 |
5.821 |
5.885 |
5.514 |
C4(N-m) |
7.91 |
7.85 |
7.5 |
7.46 |
7.68 |
%ErrorC1 |
27.26 |
24.67 |
21.06 |
9.16 |
20.54 |
%ErrorC2 |
0.65 |
2.2 |
0.65 |
1.16 |
1.16 |
%ErrorC3 |
16.19 |
5.35 |
0.400 |
1.48 |
5.86 |
%ErrorC4 |
2.30 |
1.53 |
3.00 |
3.51 |
2.58 |
S.N. |
1 |
2 |
3 |
4 |
Avg |
θ (deg) |
45 |
90 |
135 |
180 |
|
m in kg |
1 |
2 |
3 |
4 |
|
ωp1(rad/s) |
0.199 |
0.196 |
0.182 |
0.18 |
|
ωp2(rad/s) |
0.349 |
0.335 |
0.353 |
0.331 |
|
ωp3(rad/s) |
0.507 |
0.551 |
0.565 |
0.513 |
|
ωp4(rad/s) |
0.721 |
0.652 |
0.656 |
0.684 |
|
C1(N-m) |
2.45 |
2.42 |
2.24 |
2.22 |
2.333 |
C2(N-m) |
4.3 |
4.12 |
4.34 |
4.08 |
4.21 |
C3(N-m) |
6.237 |
6.784 |
6.955 |
6.319 |
6.574 |
C4(N-m) |
8.87 |
8.02 |
8.08 |
8.43 |
8.35 |
%ErrorC1 |
26.74 |
25.20 |
15.88 |
14.85 |
20.69 |
%ErrorC2 |
11.25 |
6.98 |
12.29 |
5.56 |
9.02 |
%ErrorC3 |
7.55 |
16.98 |
19.93 |
8.96 |
13.37 |
%ErrorC4 |
14.71 |
3.72 |
4.50 |
9.03 |
7.99 |
S.N. |
1 |
2 |
3 |
4 |
Avg |
θ (deg) |
45 |
90 |
135 |
180 |
|
m in kg |
1 |
2 |
3 |
4 |
|
ωp1(rad/s) |
0.123 |
0.123 |
0.121 |
0.115 |
|
ωp2(rad/s) |
0.283 |
0.284 |
0.284 |
0.282 |
|
ωp3(rad/s) |
0.37 |
0.393 |
0.388 |
0.392 |
|
ωp4(rad/s) |
0.513 |
0.567 |
0.531 |
0.526 |
|
C1(N-m) |
1.88 |
1.89 |
1.86 |
1.77 |
1.85 |
C2(N-m) |
4.33 |
4.35 |
4.36 |
4.32 |
4.34 |
C3(N-m) |
5.683 |
6.024 |
5.955 |
6.017 |
5.919 |
C4(N-m) |
7.87 |
8.7 |
8.14 |
8.07 |
8.195 |
%ErrorC1 |
2.74 |
2.22 |
3.77 |
8.43 |
4.29 |
%ErrorC2 |
12.03 |
12.55 |
12.81 |
11.77 |
12.29 |
%ErrorC3 |
2.00 |
3.88 |
2.69 |
3.76 |
3.08 |
%ErrorC4 |
1.78 |
12.52 |
5.28 |
4.37 |
5.98 |
S.N. |
1 |
2 |
3 |
4 |
Avg |
θ (deg) |
45 |
90 |
135 |
180 |
|
m in kg |
1 |
2 |
3 |
4 |
|
ωp1(rad/s) |
0.106 |
0.107 |
0.103 |
0.098 |
|
ωp2(rad/s) |
0.212 |
0.222 |
0.225 |
0.223 |
|
ωp3(rad/s) |
0.372 |
0.386 |
0.365 |
0.357 |
|
ωp4(rad/s) |
0.517 |
0.438 |
0.451 |
0.461 |
|
C1(N-m) |
2.07 |
2.09 |
2.00 |
1.92 |
2.02 |
C2(N-m) |
4.14 |
4.32 |
4.39 |
4.34 |
4.296 |
C3(N-m) |
7.255 |
7.522 |
7.109 |
6.958 |
7.211 |
C4(N-m) |
7.68 |
8.53 |
8.78 |
8.98 |
8.4925 |
%ErrorC1 |
7.09 |
8.12 |
3.46 |
0.67 |
4.5 |
%ErrorC2 |
7.11 |
11.77 |
13.58 |
12.29 |
11.26 |
%ErrorC3 |
25.11 |
29.71 |
22.59 |
19.98 |
24.34 |
%ErrorC4 |
6.23 |
10.32 |
13.55 |
16.14 |
11.56 |
It is clear from Table 5 that at precession angular velocity of 92.26 rad/s, minimum error is 0.65 % at an externally applied load of 2kg. Also, the average minimum error is 1.16 % at a load of 2kg for angular precession from 45° to 180°. It is clear from Table 6 that at precession angular velocity of 168.39 rad/s, the average minimum error is 7.99 % at a load of 4kg for angular precession from 45° to 180°. It is clear from Table 7 that at precession angular velocity of 209.86 rad/s, the average maximum error is 12.29 % at a load of 2 kg for angular precession from 45° to 180°. It is clear from Table 8 that at precession angular velocity of 266.62 rad/s, the average minimum error is 4.5 % at a load of 1 kg for angular precession from 45° to 180°. Finally, we conclude that at lower precession angular speed, there is maximum error using the lower externally applied load. We should use higher external load at lower precession speed. But, as we increase the precession speed, we obtained minimum error at lower external loads also. With the increase of angular precession speeds, we get the average error within permissible limits. In short, we conclude from
The author is thankful to Prof. Emran Khan, Head, Department of Mechanical Engineering, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi for granting all the facilities along with the permission to work in the Laboratory and all those who participated in the study and helped me to facilitate the research process successfully.