Harmonious Spring Pendulum (xyz)MechanicsAnalyze the frequency and period of a spring oscillator and display the elogation of the pendulum.
This experiment uses the accelerometer to measure the oscillator movement and calculates the oscillation period T. Using the circular frequency it transfers the data of the accelerometer into the elogation.
Further details:
The oscillation period is obtained through the autocorrelation of each accelerometer component. The sum of three autocorrelations is then analyzed for its first maximum. As an autocorrelation shows a maximum at dt = 0, we look for the first time t0 it crosses zero. From there we expect the autocorrelation to oscillate as the pendulum gives a sine function. Through this assumption we extract the position of the first full maximum to be in the range 3 t0 to 5 t0 in which we look for the maximum value.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Harmonisches Federpendel (xyz)Mechanik
Misst die Frequenz und Periode eines Federpendels und zeigt die Auslenkung des Pendels an.
Dieses Experiment nutzt den Beschleunigungssensor um die Pendelbewegung zu erfassen und berechnet hieraus die Schwingungsperiode T. Mit Hilfe der Kreisfrequenz wird aus den Beschleunigungsdaten die Elogation bestimmt.
Weitere Details:
Die Schwingungsperiode wird durch eine Autokorrelation jeder Komponente des Beschleunigungs-Vektors ermittelt. In der Summe aller drei Autokorrelationen wird dann nach dem ersten Maximum nach dem für Autokorrelationen üblichen Maximum bei dt = 0 gesucht. Hierzu wird der erste Zeitpunkt t0 ermittelt, bei welchem das Signal unter Null fällt. Unter der Annahme, dass die Autokorrelation periodisch um Null schwingt, wird das Maximum dann im Intervall 3x t0 bis 5x t0 gesucht.
Elongation (x-Achse des Smartphones)Elongation (y-Achse des Smartphones)Elongation (z-Achse des Smartphones)ErgebnissePeriodeFrequenzKreisfrequenzFederkonstanteMasseFederkonstanteRohdatenBeschleunigung XBeschleunigung YBeschleunigung ZaccXaccYaccZacc_timeautocorrelation_xautocorrelation_yautocorrelation_zautocorrelation_tautocorrelationdtt0t1t2search_tsearch_yperiodfrequencyautocorrelation_t (1)autocorrelation_t (2)circular frequencyomega2eloXampXampX (1)eloYeloZmassproductpowerspring constanteloXcmeloYcmeloZcm
acc_time
accX
0
2.5
acc_time
accY
0
2.5
acc_time
accZ
0
2.5
autocorrelation_x
autocorrelation_y
autocorrelation_z
autocorrelation_t (2)
autocorrelation
t0
2
t0
dt
t1
dt
autocorrelation_t (2)
t1
t2
autocorrelation
search_t
search_y
1
period
frequency
6.2832
circular frequency
2
accX
omega2
-1
100
eloX
accY
omega2
-1
100
eloY
accZ
omega2
-1
100
eloZ
mass
39.4784
period
2
product
power
eloXcm
acc_time
eloYcm
acc_time
eloZcm
acc_time
period
frequency
circular frequency
spring constant
accX
acc_time
accY
acc_time
accZ
acc_time
acc_time
eloXcm
eloYcm
eloZcm
period
frequency
spring constant
acc_time
accX
accY
accZ