Modelling 334 Experiment
2017
Modelling 334 Experiment
Jeandré Rossow (19301146) 10 March 2017
University of Stellenbosch
Abstract The purpose of the report is to determine the parameters of a model, experimentally. The experiment is a mixed-system, as it consists of an electrical and a mechanical system. The system is a DC motor that is connected to a gearbox that drives a disk. This is a second-order system, but is simplified to a first-order system. The system can be defined as a1Ω ̇ + a0Ω = F. The parameters a1 and a0 is going to be determined. The experiment is done for the input voltages of 3V, 6V and 9V. A Video is taken of the rotating disk and is analyzed on Tracker . Tracker gives the displacement and angular velocity of the disk. Matlab is used to plot the graph of the disk’s angular velocity respect to time. The parameter a0 is determined at steady state, where the angular acceleration (Ω ̇) is equal to zero. The force F is your input value (ei). To find a1, the angular velocity and angular acceleration values you choose must be in the non-steady state area, because then Ω ̇ is not equal to zero. The averages are taken of the parameters for the three different voltages.
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Table of contents Abstract…………………………………………………………………………………..……… i Plagiarism declaration ……………………………………………………….…………………..ii List of Figures…………………………………………………………………...………..…….. iv List of Symbols………………………………………………………………….………..….….. v 1. 2. 3. 4.
Introduction………………………………………………………………….......….…… 1 Methodology of data collection……………………………………………...……….….. 1 Processing of the data………………………………………………………………...….. 2 Data analysis……………………………………………………………………… ...…… 4 4.1 Input voltage of 3V……………………………………………………..……...…….. 4 4.2 Input voltage of 6V……………………………………………………..…………..... 5 4.3 Input voltage of 9V………………………………………………. ..........…………… 6 4.4 Results……………………………………………………………………..…….…… 7 5. Conclusions………………………………………………………………………………. 7 6. References…………………………………………………………………...…………… 8 Appendix A: Calculation of a1, a0 and response ………..……………………………...……….. 9 A.1. Input voltage of 3V…………………………………………………………………. 9 A.2. Input voltage of 6V…………………………………..………...………………….. 10 A.3. Input voltage of 9V…………………………………………………………….….. 11 Appendix B: Plot of graphs ……………………………………………………………...………13
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List of Figures
Figure 1: DC motor mixed-system ……………………………………………..……….. 2
Figure 2: Angular velocity vs. time for 3V ………………………………..…………….. 4
Figure 3: Angular velocity vs. time for 6V ……………………………………..……….. 5
Figure 4: Angular velocity vs. time for 9V ……………………………………………… 6
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List of Symbols - input voltage 12 - induced voltage in winding L - Rotor inductance
- armature current - coupling coefficient Ω - angular velocity Jm - rotor inertia J1 - disk inertia Bm - damping of motor B1 -damping of disk Ke - Back-EMF constant of the motor
Ω ̇ - angular acceleration Ω - angular velocity T – torque F – forcing function
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1.Introduction The purpose of the report is to analyze, interpret and derive information from the data of the experiment. The experiment is a mixed-system that consists of a DC motor that is connected to a gearbox that drives a disk. The system is defined as a differential equation, a1Ω ̇ + a0Ω = F. The parameters (a1 and a0) of the model are going to be determined experimentally. The experiment is going to be conducted over three different input voltages. This will secure that the results obtained from the experiment is more accurate. A video is taken of the rotating disk. The data of the video is then analyzed on Tracker and Matlab. The accuracy of the data is going to decrease when the input voltage is increased, because when the disk is rotating at high angular velocity the video will cover fewer pixels and results that the dot will blur. Tracker will struggle to track the dot.
2.Methodology of data collection 1. Connect a gearbox on a DC motor. 2. A disk is then connected on the gearbox that will cause the disk to rotate when there is an input voltage over the DC motor. The input voltage of the DC motor can be changed to regulate the angular velocity of the disk. 3. Paste two dots on the disk with one dot at the outer radius and the other dot at the inner radius. 4. Set the input voltage to 3V. 5. Place your camera close to the disk with the camera pointing to the middle of the disk. 6. Start recording the rotation of the disk just before you put the output of your source on. 7. Stop recording after the disk has reached its maximum angular velocity. 8. Put the output of the source off. 9. Repeat this procedure for also the input voltages of 6V and 9V. 10. Open your videos on Tracker. 11. Calibrate your videos by using the diameter of the disk as 15.9 cm. 12. Select the center point of your axes in the middle of the disk. 13. Set the video on auto tracker. Tracker will then track the dots. 14. Select on Tracker to track the outer radius dot. 15. Tracker will then display the displacement of the dot from the origin in the y- and xdirections. 16. Insert the angular velocity and angular acceleration to the data tables. 17. Export the data from Tracker and import it into Matlab.
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3. Processing of the data Figure 2, 3 and 4 displays the graph for the angular velocity of the disk respect to time. When a system reached his steady state, then all the state variables, like the angular velocity stays constant over time. When a system is not in his steady state then all the state variables can change over time. There are oscillation occurring at the steady state of Figure 2, 3 and 4.
Figure 1: DC motor mixed-system (Dynamic Modeling and Control of Engineering Systems, page 258).
In Figure 1 there is a DC motor that is connected to a gearbox and the gearbox drives a disk. KVL of the DC motor from Figure 1:
= × () + × 12 =
() + 12
×Ω
∴ = × () + ×
()
+
× Ω
(1)
Ω = ̇ ̈ Ω ̇ = Damping torque of the disk:
1 = 1 × Ω Damping torque of the motor:
= × Ω
2
̈ Ʃ = ̈ () 1 = ( + 1) () = ( + 1)Ω ̇ + ( + 1)Ω Where () =
=
(2)
() (Dynamic Modeling and Control of Engineering Systems, page 258).
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∴ () = ∝ ( + 1)Ω ̇+ ∝ ( + 1)Ω
(3)
Differentiating both sides: ()
̈ + ∝ ( + 1)Ω ̇ = ∝ ( + 1)Ω
(4)
Substituting (3) and (4) in (1): ̈ + ∝ ( + 1)Ω ̇ + = ∝ ( + 1)Ω ̇ + ∝ ( + 1)Ω + ∝ ( + 1)Ω
∝
Ω (5)
The inductance of the motor is so small that it can be chosen as zero, therefore the new differential equation for the system will then be = ∝ ( + 1)Ω ̇ + ( ∝ ( + 1) +
∝
)Ω . This is a
first-order system. To simplify the experiment the system can be de fined as a1Ω ̇ + a0Ω = F.
a1Ω ̇ + a0Ω = F
(6)
From equation (6) and (5), it can be seen that 0 = (( + 1) +
)and1 = ( +
1). The parameter a0 is determined at steady state, when there is no change in angular velocity over time. The angular acceleration (Ω ̇) is equal to zero. The angular velocity is read of Figure 2, 3 and 4 for all three of the different input voltages. The parameter a1 is determined at non-steady state. In this time difference the an gular velocity is not constant over time. The average angular acceleration is taken as the difference in angular velocity at steady state and when time is equal to zero over the time it took for the system to reach steady state. The average angular velocity is the difference in the angular velocity at steady state and when the time is equal to zero over two. To determine a1 the equation, a1Ω ̇ + a0Ω = F is going to be used. The forcing function is equal to the input voltage and a0 is already determined in Appendix A. The parameter a1 can then be determined.
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A response for the differential equation can be determined. The response will give a superior solution for the data points. There is a homogeneous – and a particular solution. The homogeneous solution is,Ωh(t ) = is,Ωp(t ) =
−
. C is a constant and =
. The particular solution
− . The complete solution is then,Ω(t ) = Ωh(t ) + Ωp(t ) = +
. The response
is plotted with the data points in Figure 2, 3 and 4.
4.Data analysis 4.1 Input voltage of 3V:
Figure 2: Angular velocity vs. time for 3V
The experimental data is very accurate, because the oscillations are not that big. The oscillations are between 9.5- and 11 rad/s. The angular velocity reached its steady state value of 10.26
in
a time of ±0.4 seconds. The non-steady state area is in the time interval of 0 to ±0.4 seconds. In 4
the non-steady state the angular velocity increased exponentially. The angular acceleration will then also change at non-steady state over time, because when you take two points on the curve of Figure 2 at non-steady state, the slopes of the tangential lines on both of the points won’t be the same. The average angular acceleration is45.6899 rad/s . The parameter a0 can be determined at steady state, because then the angular acceleration is zero. a0 is equal to 0.292. The parameter a1 is 0.0328 at the non-steady state. The equation for the response is, Ω(t ) = 10.268
10.268
.
−
. The response is a good estimate of the data.
4.2 Input voltage of 6V:
Figure 3: Angular velocity vs. time for 6V The experimental data is not that accurate, because there are big oscillations in the data. The oscillations are between 18- and 25 rad/s. The oscillation does also not look like sinus waves. If the oscillations for this experiment looked like sinus waves, then it would be valuable data. The angular velocity at steady state is, Ω = 21.53
. Steady state is reached at ±0.4 seconds. The
average angular acceleration for the non-steady sate is, Ω ̇ = 84.3122 /. The parameters a1 and a2 for the first-order model are 0.0356 and 0.2 7868 respectively. The response is an accurate solution for the experimental data. At the non -steady state it increases exponentially and then 5
stays horizontal for the steady state. The response for the 6V input voltage is, Ω(t ) = 21.53
21.53
.
−
.
4.3 Input voltage of 9V:
Figure 4: Angular velocity vs. time for 9V The experimental data is not accurate; therefore the solutions for the model will also not be accurate. There are very big oscillations occurring. The oscillations are between 23- and 40 rad/s. The steady state of the 9V experimental data begins at ±0.4 seconds. The steady state angular velocity is equal to 32.16
. The non-steady state area is between zero- and 0.4 seconds. The
angular velocity is exponentially increased in the non-steady state, therefore the average angular acceleration is 112.381
. The parameter a0 is obtained at steady state and is equal to 0.27985.
The parameter a1 is equal to 0.0399 at non-steady state. The response for the 9V input voltage can be written as Ω(t ) = 32.16 32.16
.
−
.
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4.4 Results
The average for the parameters is taken between the three different input voltages.
0 = 0.28351 1 = 0.0361 The first-order differential equation can then be written as:
0.0361Ω ̇ + 0.28351Ω =
5 Conclusion The first-order differential equation is obtained as 0.0361Ω ̇ + 0.28351Ω = . The parameters of the model that is calculated is not entirely correct, but is accurate enough. There are a few factors that caused the measurements to be a little inaccurate. The video that was taken of the rotating disk did not cover enough pixels. The disk was spinning too fast for the video and caused the dot to blur. There were oscillations in the experimental data. The oscillations were caused by: the input voltage you put over the power supply, could not keep the voltage steady, there were a hole in the disk. The weight of the disk is then not evenly distributed when the disk rotates, there can be a big tolerance on the disk. The disk will then not be perfectly circular. This w ill also influence the weight that is not evenly distributed around the middle axis, when rotating. The higher the input voltage, the faster the disk will rotate and this will lead to bigger oscillations. The experimental data will then be less accurate.
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References: “ Dynamic Modeling and Control of Engineering Systems”.3rd ed.Cambridge University: John F. Gardner, 2007. “Science Fair Project Final Report ”. Science Buddies.N.p, 2017. Web. 8 Mar. 2017. “Guide for Writing Technical Reports”.4rd ed.Stellenbosch University: AH Basson, TW von Backström, 2012.
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Appendix A: Calculation of a1, a0 and response A.1. Input voltage of 3V: Determining a0:
= 3 Ω ̇ = 0 The angular velocity at steady state is: Ω = 10.2677
∴ = (( + 1) +
1
)Ω
= 0Ω
3 = 0(10.2677) ∴ 0 = 0.29217 Determining a1:
= Ω ̇ = = Ω =
Ω@ Ω@ @
Ω@ Ω@ 2
= = 3 a1Ω ̇ + a0Ω = F
1(45.6899) + (0.29217)(5.132) = 3 ∴ 1 =
3 (0.29217)(5.132) 45.6899
= 0.0328
Response:
=
1 0
=
0.0328 0.29217
= 0.1123
− − Ωh(t ) = = .
9
= 45.6899 /
= 5.132
3
Ωp(t ) =
0.29217
= 10.268
− Ω(t ) = Ωh(t ) + Ωp(t ) = . + 10.268
At Ω(0) = 0
∴ = 10.268
− Ω(t ) = 10.268 10.268 .
A.2. Input voltage of 6V: Determining a0:
= 6 Ω ̇ = 0 The angular velocity at steady state is: Ω = 21.53
∴ = (( + 1) +
1
)Ω
= 0Ω
6 = 0(21.53) ∴ 0 = 0.27868 Determining a1:
= Ω ̇ = = Ω =
Ω@ Ω@ @
Ω@ Ω@ 2
= = 6 a1Ω ̇ + a0Ω = F
1(84.3122) + (0.27868)(10.759) = 6 ∴ 1 =
6 (0.27868)(10.759) 84.3122
= 0.0356 10
= 84.3122 /
= 10.759
Response:
=
1 0
=
0.0356 0.27868
Ωh(t ) = Ωp(t ) =
−
= 0.1277
=
6 0.27868
.
−
= 21.53
− Ω(t ) = Ωh(t ) + Ωp(t ) = . + 21.53
At Ω(0) = 0
∴ = 21.53
− Ω(t ) = 21.53 21.53 .
A.3. Input voltage of 9V: Determining a0:
= 9 Ω ̇ = 0 The angular velocity at steady state is: Ω = 32.16
∴ = (( + 1) +
1
)Ω
= 0Ω
9 = 0(32.16) ∴ 0 = 0.27985 Determining a1:
= Ω ̇ = = Ω =
Ω@ Ω@ @
Ω@ Ω@ 2
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= 112.381 /
= 16.11
= = 9 a1Ω ̇ + a0Ω = F
1(112.381) + (0.27985)(16.11) = 9 ∴ 1 =
9 (0.27985)(16.11) 112.381
= 0.0399
Response:
=
1 0
=
0.0399 0.27985
= 0.1426
− − Ωh(t ) = = .
Ωp(t ) =
9 0.27985
= 32.16
− Ω(t ) = Ωh(t ) + Ωp(t ) = . + 32.16
At Ω(0) = 0
∴ = 32.16
− Ω(t ) = 32.16 32.16 .
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Appendix B: Plot of graphs %experimental data at 3V t3 = nuwedatadrievolt(15:108,1)-0.5; x3 = nuwedatadrievolt(15:108,2)*1e-2; y3 = nuwedatadrievolt(15:108,3)*1e-2; w3 = -nuwedatadrievolt(15:108,5)*(pi/180); %experimental data at 6V t6 = datasesvolt(22:61,1)-0.7333; w6 = -datasesvolt(22:61,5)*(pi/180); %experimental data at 9V t9 = datanegevolt(17:41,1)-0.56666; w9 = -datanegevolt(17:41,5)*(pi/180); %% %Plot of 3V input voltage wave3 = sum(w3(9:94))/numel(w3(9:94)); a03 = 0.29217; ei3 = 3; acc3 = (w3(8) - w3(1))/(t3(8)-t3(1)); a13 = (ei3 - a03*(wave3/2))/acc3; tau3 = a13/a03; yp3 = ei3/a03; c3 = -yp3; omega3 = c3*exp(-t3/tau3) + yp3; plot(t3,omega3) hold on grid on plot(t3,w3,'-x') xlabel('time[s]') ylabel('Angular velocity[rad/s]') title('Angular velocity vs time for 3V input voltage') legend('Line of best fit for data','Experimental data') hold off %% %% %% %Plot of 6V input voltage wave6 = sum(w6(16:39))/numel(w6(16:39)); a06 = 0.29217; ei6 = 6; acc6 = (w6(9) - w6(1))/(t6(9)-t6(1)); a16 = (ei6 - a06*(wave6/2))/acc6; tau6 = a16/a06; yp6 = ei6/a06; c6 = -yp6; omega6 = c6*exp(-t6/tau6) + yp6; plot(t6,omega6) hold on 13
grid on plot(t6,w6,'-x') xlabel('time[s]') ylabel('Angular velocity[rad/s]') title('Angular velocity vs time for 6V input voltage') legend('Line of best fit for data','Experimental data') hold off %% %% %% %Plot of 9V input voltage wave9 = sum(w9(5:24))/numel(w9(5:24)); a09 = 0.29217; ei9 = 9; acc9 = (w9(9) - w9(1))/(t9(9)-t9(1)); a19 = (ei9 - a09*(wave9/2))/acc9; tau9 = a19/a09; yp9 = ei9/a09; c9 = -yp9; omega9 = c9*exp(-t9/tau9) + yp9; plot(t9,omega9) hold on grid on plot(t9,w9,'-x') xlabel('time[s]') ylabel('Angular velocity[rad/s]') title('Angular velocity vs time for 9V input voltage') legend('Line of best fit for data','Experimental data') hold off
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