Coupled Pendulums Best Results

First created: Tue Jul 10 2001
Last updated: Thu Jul 12 2001

Five Days Of Data: 2001-07-[04-08]

The five days from July 4th to July 8th, 2001 are specially interesting because the house was empty and the air conditioner was off. Analysis of the data collected during this time shows two strong periodic components, one with a period of 24 hour, the other with a period of 12 hours.

Original Data

Pendulum and temperature data, after mean subtraction. The pendulum is in red, the temperature in black. A rising black curve indicates a falling temperature, and vice versa.

Residuals After Subtracting Temperature Effect

Data, temperature and residuals. The residuals (green) after a best fit of the pendulum data (red) to the temperature (black), after mean subtraction.

    Pendulum(t) = 0.1186 - 1.78*Temp(t) - 2.98*Temp(t)2

The statistics for this fit shows a low R-Square, indicating that the temperature change alone cannot explain the variation in the pendulum data.

Residual Standard Error=0.5379
R-Square=0.2938

          Estimate Std.Err  
Intercept   0.1186  0.0084 
X1         -1.7824  0.0329
X2         -2.9828  0.1374

Frequency Analysis Of Residuals

The Discrete Fourier Transform (DFT) of the residuals for the 5 day sequence shows pronounced peaks at 5+1 and 10+1, corresponding to effects with periods of 24 h and 12 h, resp. This graph is the modulus of the DFT of the residuals, after mean subtraction.

The presence of a 12 hour peak is of course what makes this data set interesting. Whether this is a tidal effect or not, only better observations and more refined analysis will tell, but no doubt this is encouraging. Score = 1!

If I isolate the 24 and 12 h periodic components, it is easier to visualize exactly how convincing (or unconvincing!) the two period fit actually is. The residuals are in black, the 2 interfering periodic components are in red. If I also include the frequencies immediately adjacent to the 2 peak periods the fit is much improved. This means more data is needed to sharpen the peaks, and/or the actual peaks are not exactly at 24 and 12 h.

It is interesting to compare the projections of the theoretical tidal vector for July 5th to the curve for any of day in this sequence. The fit best matches the east-west projection, in relative phase and relative magnitude, and indeed the pendulum is oriented east-west. Score + 1!

There is also the overall sign of the effect to consider. When I press down on the table to the west of the pendulum, the pendulum tilts west and the pendulum curve rises. So a positive east-west projection corresponds to a rising pendulum curve. This is indeed what is observed. Score + 1!

Conclusion

Before I can claim that tidal forces are being detected, I need to show at least approximate agreement between data and theory in periodicity, relative magnitude, relative and absolute phases. On all these counts this data set is promising.

To be really convincing, I need a more consistent fit needed over a longer stretch of time. These are things I hope to do in the future: stabilize the temperature, increase the temperature measurement sensitivity 10 fold and increase the pendulum measurement sensitivity 100 fold.