N otes on Chapter 9B
pp 96 - 103 of your lab book

We have discussed a simple spring system previously in the natural oscillations lab. We have even shown that this simple spring model can work for other physical objects such as compressions in a jar of air. Now it's time to "kick it up a notch"! In real life, the simple spring oscillating at its natural frequency almost never happens. Why? Because we are almost always going to have a damping force such as friction working on a real system.

Such oscillations, where you have only the spring force and the damping force, are called damped oscillations.

The frequency at which these oscillations vibrate at is not equal to the natural frequency, but if the damping force is small, then the frequency is nearly the natural frequency. We have depended on this fact for all of the labs in which we obtained the natural frequency (pendulum, natural oscillations, spring of air). In this lab, we will not make the damping force small. We want to see damping occur, so we will place part of our apparatus in water.

Our goal in this lab is not to observe damped oscillations, however. Our goal is to observe driven and damped oscillations! Another words, we are interested in a spring system which has a large damping force (provided by the water) and has a driving force. A driving force is provided by a motor which can drive the entire system at varying frequencies.

What will you see when you let this entire system start oscillating? The spring will still have a sine wave oscillation. The frequency it oscillates at will be the frequency you are driving it at (the driving frequency). That sounds just like what we saw in the natural oscillations lab except that the frequency is different. But wait! There is one important difference! When you change the driving frequency the amplitude of the oscillations changes! Without changing the driving amplitude, the amplitude of the spring oscillations increases or decreases as you change the driving frequency. You will find that there is a frequency at which the amplitude is maximum.

Apparently there are some driving frequencies which yield a large amplitude response. This is called resonance. The frequencies where you observe the maximum responses are called resonant frequencies. This is a very important concept in applied physics. This concept is used to make radio tuners. Resonance was to blame for the amazing Tacoma Narrows bridge collapse. It's a fancy word, but you will find that you already understand resonance in every day life.

Example: Pushing a child on a swing (a pendulum).
Class Participation: What frequency should you push (drive) the swing at?





What happens to the swing if you push it at some other frequency?





We can make a plot of the amplitude versus the driving frequency. This plot is so important, it has a name: it is called the response curve. Theoretically, this curve is given by the equation:

where ₀ is the natural frequency, A₀ is the driving amplitude, d is a damping term, and  is the driving frequency.

Procedure:
Important: This lab is very long. Please make sure you know what to do before you come to class so that you can do this procedure quickly.

1. First, we want to find the natural frequency and driving amplitude of the spring. Record the distance from the center of the motor wheel to the pin. This is the driving amplitude A₀.
2. To find the natural frequency, you need to take the spring and mass out of the water. Stretch the spring a little and let it go without any driving force (the motor should be off). Find the period of the oscillations in the same way we did in part A of this chapter.
3. Since the natural frequency of a spring system is dependent on mass, you will need to find the natural frequency with and without the second damper (which adds a small mass to the system).
4. Next, we want to find the response curve, A( ) versus . Place the spring back in the water with only one damper attached. Turn on the motor.
5. Try to find the frequency at which you see the maximum possible amplitude. Change the "speed" on the motor dial until you see the spring has the most amplitude you can get. Notes:  1) Atno time should the mass hit the water. Only the damper should be in the water. 2) The mass should not collide with the side of the glass nor the bottom of the container. 3) Transient oscillations should be ignored. 4) The "speed" of the motor is not the frequency. Do not record the "speed".
6. Record the amplitude for this frequency by measuring the peak to peak distance and dividing by two.
7. Measure the driving frequency by measuring the period of the oscillation.
8. Repeat steps 6-7 six more times for "speeds" close to where you found the maximum amplitude. (You must have at least five total data points on your graph).
9. Increase the damping force by adding a damper. Repeat steps 5-8 for this new damping force.
10. Plot A( ) versus for the two cases. Fit the curves to the theoretical response curve by doing an "Automated Curve Fit" in GA

The fit which you should type in is (copy this exactly) y = A*w^2/((w^2-x^2)^2 + d^2*x^2)^0.5 , but instead of A, put your driving amplitude in whatever units you measured A( ) in. Example: I measured A₀ = 1.35 cm, and I measured the amplitude in cm, so I would type y = 1.35*w^2/((w^2-x^2)^2 + d^2*x^2)^0.5

Data:
Driving Amplitude = _______________+/-________ (Distance from the center of the wheel on the motor to the pin)
Once again you can use my SDM Calculator .

Table 1: Finding the natural frequency. (Remove the spring and mass from the water)

1 damper	Trial	Time (5 osc.) (         )	Period (    )
	1
	2
	3
	4
	5
Tave=		SD =	SDM =

Table 2: Finding the natural frequency for 2 dampers attached:

2 dampers	Trial	Time (5 osc.) (         )	Period (    )
	1
	2
	3
	4
	5
Tave=		SD =	SDM =

Table 3: Response Curve, one damper attached

A1= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =	A2= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =
A3= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =	A4= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =
A5= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =	A6= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =
A7= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =

Table 4: Response Curve, two dampers attached

A1= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =	A2= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =
A3= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =	A4= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =
A5= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =	A6= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =
A7= ____ Trial Time (5 osc.) (         ) Period (    ) 1 2 3 4 5 Tave= SD = SDM =

A ssignments: I am going to give you an Extra Credit assignment which is similar to the Spring Test , in which you can change the damping force and driving frequency at will on the computer. You will be getting some simple plots from this program. The program calculates the numerical solution for the position of the mass based upon the three forces involved, and each force can be changed in any way. Presently there may be a bug in the appearance of the plot on some computers. Please give me feedback. Don't tell me "it doesn't work'. That is not helpful. If you have problems with the program, you should tell me: 1) What OS you have. 2) Does it run at all, or do you get an error message? If you get an error message report to me the error message. 3) What browser do you have? 4) Did you check to see if Java was enabled in your browser? (Look under preferences). 5) If the graph is plotting off center or strangely, please describe the problem, and tell me the resolution of your monitor.  If you cannot get the program to work, then give me the above feedback and then go to Wat 421 to use that computer. It is set up for your use, and it does work correctly on this computer.

The extra credit assignment instructions and program are at http://www2.hawaii.edu/~jmcfatri/labs/SpringAssignment.html The program itself is located at http://www2.hawaii.edu/~jmcfatri/Java/SpringTest2.html