Mechanical Resonance

Equipment

• Computer with Science Workshop(TM)
• Pasco Interface
• Pasco Power Amplifier
• Pasco Track with carts (with sound reflector) and springs
• Pasco Mechanical Oscillator Driver: 0-12V = 0.3 - 3 Hz
• Pasco Sonic Ranger / Motion Sensor
• Pasco Smart Pulley
• Thread/string
  
• 2 Rod stands
• Meter stick
  
• DataStudio Software
• DataStudio file ResonanceLab.ds
  

Proceedure

• A sample software set-up is available
• Make sure the Interface power is on before booting the computer.
• Connect the 2 phono jacks of the sonic ranger to the Interface (notice the order matters)
• Connect the power amplifier multi-pin connector to an Analog conector of the Interface
• Connect the power amplifer to the motor

1. Setup: Match the physical connections to the virtual connections in the image above:
1. Plug the Motion Sensor into Channels 1 & 2 of the Interface (yellow into Channel 1)
1. Carefully set the
   
2. Plug the Smart Pulley into Channel 3 (optional).
3. Plug the Power Amplifier into the Interface Analog A channel
4. Plug the Mechanical Oscillator into the the Power Amplifier (Banana Jacks)
5. Set the WaveForm of the Signal Generator to Positive Up Ramp Wave
1. Amplitude 5V, Frequency 0.003 Hz
2. Set the Signal Generator Control to OFF
6. Setup the oscillator
1. Connect the spring to the tab on the Mechanical Driver.
2. Adjust the radial arm to 1-2 cm
3. Cut a piece of thread in length twice your outstretched arms. Tie it in one big loop.
4. Connect the loop to the spring, pass it over the pulley, and hang the mass hanger, with 150g total.
7. Setup the Graph and Table Windows (mayber already done in setup)
   
1. Drag the Ch 1&2 Position icon to the Graph icon
2. Now drag the Ch 1&2 Velocity icon to the Graph 1 icon (this will make both graphs with a common time axis)
3. Drag the Ch 3 Velocity icon to the Graph 1 icon (this will superimpose the Smart Pulley and Motion sensor signals)
4. Drag the Output Volts icon to the Graph 1 icon
5. To the sme for the Table icon
   
2. Static String Measurements:
1. Measure the unstretched length of the spring.
2. Measure the stretched length with just the 50 g mass hanger
3. Measure the stretched length with a total of 150 g on mass hanger.  Compute the spring constant value k. 
   

4. Mass: m	Spring Length: L	Net Stretch: L(m) - L(0)	Spring constant k = mg/(L-L0), g = 980 cm/s2
   0 g		0
   50 g
   150

   Predict the resonant frequency for the spring with mass 150 g:
1. f = 2(3.14159) sqrt[k/m] = _________________
   
3. Damped oscillations (undriven)
1. Before taking data, click on the sampling options, and make sure you are running at at least 20 Hz.
2. Make sure the signal generator (in software window) driving the power amplifier is off.
   
3. Click on the GRAPH window to follow the motion of the mass on the computer.
   
4. Click on the START button  to start recording data.
5. Gently pull on the spring (Pulling on spring rather than mass will ensure the motion stays vertical), to move the hanging mass 15-20 cm vertical and release.
6. After about 10 oscillations, click STOP to stop data acquisition.
7. You can export your data table to a format readable by a spreadsheet.
   
1. From the Window Menu, select Table, then select Export Data
2.  From the Display Menu, select Export Picture to capture a bitmap image.  You can print your graph to the BW printer on the 2nd floor (maybe).
8. In your write up, explain the junk at the beginning of the data, before the simple oscillations
9. From either your graph or your table, find the natural frequency of oscillation. 
   
1. This is best done by finding the time it takes for e.g. 5 full oscillations, or averaging several measurements of the period of oscillation (see table below-10). Report this value in your write-up.
2. In your write-up, compare the measured value of the period T, with the predicted value T=1/f from your static measurements of the spring/mass combination.
   
10. From the data table, compute the ratio (A_n+1/A_n ) of successive amplitude maxima of the oscillations.  Record at least 4 or 5 values, compute the mean value.  Notice than amplitude is the mean of the maximum and minimum positions of the oscillator, since the motion is not centered around zero position.
    

11. Oscillation maximum: n	Time  of nth Maximum:  tn	Period tn+1-tn	Minimum Position ymin	Maximum  Position ymax	Amplitude An = (ymax-ymin)/2	Amplitude Ratio: An+1/An	Q-Value Qn=1/[1-(A n+1/An)2]
    1
    2
    3
    4
    5		XXX				XX	XX
    Averages	XXX		XXX	XXX	XXX

    
    
12. The energy of the oscillator is proportional to the amplitude squared. 
    
1. The Q value of an oscillator measure how "good" an oscillator it is.
2. 1/Q = Energy stored in oscillator divided by energy lost in each full oscillation. 
   
3. If Q is much larger than one, then, [1/Q] = 1- [A_n+1/A_n]², or Q=1/[1-(A _n+1/A_n)²]. 
   
4. Compute the Q value for your oscillator in the table above. 
   
5. How constant are the values of Q as the motion decays?
4. Resonance Scan
1. Set the Signal Generator to the triangle wave, with amplitude 7 V and frequency 0.003 Hz.  This will slowly sweep through all frequencies for the drive.
2. Start data acquisition.  It may be a minute before the driver starts to rotate.  This is normal, watch the graph display of voltage.  Make sure you start with the frequency very different from the approximately 1 Hz resonance frequency.  You should see the graph gradually build up a resonance scan like the figure above.  It does not matter whether the frequency is increasing or decreasing (remember, the DC voltage out controls the frequency of the motor).
3. After the hanging mass goes through resonance, the mass oscillates at both its natural frequency with decaying amplitude and at the drive frequency.  This creates very strong beats, that bring the mass almost to rest.
4. Stop the data acquisition after the hanging mass has resumed a simple motion.  Here is a sample image at 0.001 Hz.  Notice the "glitch" data point near resonance, where the echo missed the hanging mass.
   
5. 
   
6. Find the Resonance Frequency (frequency for maximum amplitude).
1. Use the controls to the graph window to zoom in on the position graph.
2. Find the oscillations with maximum amplitude.  Use the Sine-Fit or the graph itself to record the amplitude and period of the maximum oscillations.  In your write-up compare this frequency with the natural frequency of the free oscillation (measured above).  Use the table below to record your measurements.  Make two or these measurements of consecutive oscillations.

   Time of resonance maximum	Period of oscillation at resonance maximum	Frequency at resonance maximum	Amplitude of oscillation at resonance maximum	Motor Driving Voltage at Resonance

7. Find the the Frequency at which the amplitude is 1/2 its  maximum value.
1. For example, in the sample resonance scan above, the resonance maximum is approximately at time 860 sec, and the amplitude is 1/2 the maximum at about 840 sec.
2. Measure the Period of oscillation and amplitude of oscillation at several positions near the driving voltage at one-half maximum amplitude oscillation.  Interpolate to the 1/2 max value.
   

   Time	Period of oscillation	Freqeuncy of oscillation	Amplitude of oscillation	Motor driving voltage
   Interpolated value for 1/2 maximum oscillation

3. The Q-value of the oscillator is also related to the width of the resonance peak.  If f₀ is the frequence at resonance, and f_1/2 is the frequency of 1/2 maximum oscillation, then compute Q = f₀/(f₀-f_1/2).  compare this value with the Q-value you obtained above.