| abstract.
In this lab we tried to model the behavior of two coupled pendulums. We first experimented with an analytical system and then modeled the system in Simulink to verify our results from the first part. We showed that all possible motions of the two pendulums can be expressed by the combination of two functions which are the modes of the system.
introduction.
We demonstrated that pendulums acted like two masses connected to walls by two springs and to each other by a spring and that the motions of the two pendulums can be expressed by a combination of the mode functions of the system.
In order to find the modes of the system, we used two different initial conditions and found the behavior of the system for those ICs (we made sure that the two setups would not yield the same mode). Using the functions obtained by the two initial conditions we observed that the functions for other initial conditions could be expressed by the sum of scalar multiples of the first two functions.
In Simulink, we simulated the motion of the pendulums using an analogous electrical system and verified our results from the analytical system analysis.
procedure.
The procedure followed in the laboratory did not deviate significantly from the procedure described in the experiment description, which can be found at: http://palantir.swarthmore.edu/maxwell/classes/e12/S04/labs/lab03/
When processing the data acquired from the pendulum potentiometer, we did not use a Hamming window. The lab instructor present instructed us to avoid this step, since it seemed to blur the relatively close frequency peaks together.
calculations, and matlab methods.
Measured Values and Calculated K Value (Spring Constant)
M1 = M2 = 1.5 kg
L = 0.6m
Ls=0.1m
From time data
Wp = 0.6452
Wc = 0.6897
From Fourier transform (frequency domain)
Wp = 0.63
Wc = 0.69
From equation (16) in the procedure
Wa = (Wc + Wp)/2 = 0.66
Wb = (Wc - Wp)/2 = 0.03
K = 67.5 N/m
Matlab Sourcefiles
- god.m - Generate Plots for the theta1 and theta2 Simulink
- do_lab3.m - Compute Plots from Simulink Output
- angel.m - Process the NiDAQ output and generate FFT and signal vs. time plots
- lab3.mdl - Simulink simulation file
NiDAQ pendulum experiment results.
simulink simulation results.
discussion.
The main significance of our results are that any motion of the pendulum system can be described using a combination of the coupled and parallel motions. This can be seen in our offset graph in the frequency domain where the peaks at two different frequencies. The peak at about 0.6 Hz is the result of the parallel motion in the system and the peak at about 0.7 Hz is the result of the coupled motion in the system. We know this because the frequency of the oscillation of the pendulum with the masses offset in the same direction was about 0.6 Hz and with the masses offset in opposite directions was about 0.7 Hz. Therefore, the peaks at 0.7 Hz and 0.6 Hz show that when only one of the masses is offset, the motion of the system is a combination of the parallel and coupled motions.
Overall, our actual results from the pendulum matched the theoretical results obtained using Simulink relatively well. For the parallel offset of the two masses, both the theoretical and actual results showed that the masses would move in a periodic motion similar to each mass moving on its own. This makes sense because when the two masses are offset by the same amount in the same direction, the spring will have no effect on the motion of the system. For the coupled offset of the two masses, both the theoretical and actual results showed that the pendulums would again move in a periodic manor, but that the frequency of oscillation would be higher than for the parallel offset. This again makes sense because as the two masses move away from each other the spring stretches and pulls them back towards each other. However, when they move towards each other the spring contracts and pushes both the masses away. This additional force on the masses causes the frequency of oscillation to increase which is shown in both the theoretical and actual results. Again both the theoretical and actual results show that when only one of the masses is offset, the resulting motion is a combination the parallel and coupled motions. However, there was one major difference between the theoretical and actual results. The theoretical results show the amplitudes of the oscillations as constant while the amplitudes of the actual result decrease over time. This discrepancy is the result of damping in the real pendulum system that is not accounted for in the theoretical model.
For our extinction, we tested what would happen if the one pendulum was offset by more than just a small angle so that the small angle approximation on longer held and the other pendulum was not offset at all. This setup was the same as the setup where the motion was a combination of the parallel and coupled motion except that the weight was offset by much more. However, the result from this setup was much more similar to the parallel setup than the offset setup. This is because the spring was initially stretched by a large amount, but when the masses were let go the spring was not able to contract by as much as it had been stretched and slack appeared in the spring. Therefore, the spring ended up acting much more like a string because it applied a force only when the masses were far apart. That means that the motion of the masses no longer acted like a combination of the coupled and parallel offsets but instead that their motion was much like a parallel offset.
conclusion and future work. The laboratory experiments performed confirmed our theories and equations. The pendulums do move as a linear combination of the two modal frequencies no matter the initial conditions.
Future work will involve more labs in the future.
acknowledgments.
Much of the text of this writeup is more or less directly taken from the original lab description, which is available at:
http://palantir.swarthmore.edu/maxwell/classes/e12/S04/labs/lab03/ |