Natural Frequency: Masses on Rods
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Abstract
The mass and stiffness of a building affect how much an earthquake damages the building. The natural frequency, related to mass and stiffness, is the frequency that the structure will shake at. Since structures are different weights, different heights, and made of different materials, each structure has its own natural frequencies. If you apply a forcing motion to a structure that is at one of the structure's natural frequencies, you can cause the structure to resonate. Resonance can cause a lot of damage! When engineers design a building in an area where there are a lot of earthquakes, they want to calculate the natural frequencies of the structure. This activity uses lumped masses on a single rod to understand the concepts associated with structural frequencies and modes of vibration.
Authors: Leslie Bucar, 712 Science Teacher, Fond du Lac Ojibwe School, B.S. Biology, B.A.S. Teaching Biology, B.A.S. equiv Teaching Chemistry
Dan Johnson, 712 Math Teacher, Fond du Lac Ojibwe School, B.A.S. Teaching Math
Introduction
The movement of rocks along a fault releases stored energy as vibrations. This sudden movement causes the earthquake. A vibration produced by an earthquake is called a seismic wave. The frequency of a wave is the number of waves per unit of time, and the amplitude is how tall the wave is. We measure frequency in hertz.
The mass and stiffness of a building affect how much damage an earthquake has on the building. Mass is related to weight. Weight is mass affected by gravity. If you were to go to the moon, your mass would be the same but your weight would be different! Stiffness is related to the "column" of the building, how long the column is, and the shape of the column.
The natural frequency, related to mass and stiffness, is the frequency that will disturb the structure. It is the frequency that the structure will shake at. Since many structures are different weights, different heights, and made of different things, each structure has its own natural frequencies.
If you apply a forcing motion to a structure that is at one of the structure's natural frequencies, you can cause the structure to resonate. Resonance can cause a lot of damage!
A point of natural frequency is called a mode. Some structures have more than one natural frequency, or mode. It all depends on how many levels are on the structure. A twostory building has two modes, a threestory building has three and so on. We call these higher modes.You can find the second mode of a twostory structure by increasing the frequency until you hit the resonance of that mode. Of course, by increasing the frequency, you've moved past the resonance point of the first mode, so the first mode will stop vibrating.
If an engineer were going to build a building in an area where there are a lot of earthquakes, she would want to know what the frequency of each mode is. When engineers are trying to analyze structures, it is easier to think of them as a lumped mass on a single rod. So, a single story building would be seen as a rod with all of the weight located at the top in a lump. They use an equation to figure out the natural frequency (f):
f = 0.159 * sqrt(K/M)
The K stands for the stiffness of the building. The M stands for the mass of the building. The constant 0.159 = 1/(2pi)
Earthquake Engineering Component
Earthquake engineers calculate the natural frquencies of a building as important parameters in determining the required stiffness and strength of the structure.
Learning Objectives and Standards
Links to the National Science Standards and to individual State Science Standards are available by using this link:
http://nees.org/education/forteachers/k12teachers#standards
Material List
 Shake Table
 Shake Plate
 Four Rods
 Various Masses
 Screws for the shake table
Procedure
Procedure:
This experiment will attempt to show what happens when a building is shaken at its natural frequency. To show this, several small rods will be fixed to the shake table with different weights placed at different heights to show the effects of more weight and greater height.
1. Attach the masses on rods to the shake table using the two screws to securely fasten the plate to the table.
2. Test to see if the plate is fastened to the table by shaking it with your hands.
3. Start the shake table and allow it to calibrate using the procedure outlined in the shake table operations manual.
4. Navigate to the sine mode (option 2 at main menu) ad select displacement mode (option 1). Press the # key to start the experiment and then slowly increase the frequency.
5. Write down the frequency that each rod vibrates in the table below.
Links and Resources
Brecht, W., Ludwig, F., Mann, B., Snyder, R. and Stasik, J. 1991. Earth Science. D.C.
Heath and Company, Massachussetts.
Chopra, A. 2001. Dynamics of Structures: Theory and Applications to Earthquake
Engineering. 2^{nd} Edition, Prentice Hall, New Jersey.
Feather, R., Snyder, S., and Zike, D. 2002. Earth Science. Glencoe McGrawHill, New York, NY.
Levy, M. and Salvadori, M. 1995. Why the Earth Quakes. W.W. Norton and Company, New York.
Assessment
Collecting and Analyzing Data:
1. Write down the frequency that each rod vibrates (natural frequency) below.(Remember: the twostory will have 2 modes!)
Natural Frequency (Hz)

Mode 1 
Mode 2 
Short Rod/One Weight 


Medium Rod/One Weight 


Medium Rod/Two Weights 


Tall Rod/ 2 Weights 


Conclusion:
 As the frequency increases, will the rods swing back and forth more?
 What effect will increasing the weight have on the natural frequency?
 What effect will increasing the height of the weight have on natural frequency?
Teacher Lesson Guide and Key: Masses on Rods
Collecting and Analyzing Data Key:
Natural Frequency (Hz)

Mode 1 
Mode 2 
Short Rod/One Weight 
7.2 Hz 
N/A 
Medium Rod/One Weight 
3.2 Hz 
N/A 
Medium Rod/Two Weights 
2.4 Hz 
N/A 
Tall Rod/ 2 Weights 
1.4 Hz 
10.4 Hz 
Conclusion Key:
 As the frequency increases, will the rods swing back and forth more? No, the biggest movement back and forth will happen when the rod is being shaken at its natural frequency. Once the frequency increases past this point the rods will shop shaking as hard.
 What effect will increasing the weight have on the natural frequency? As the mass increases, the natural frequency will go down.
 What effect will increasing the height of the weight have on natural frequency? As the length increases, the natural frequency will go down.
Extensions
The 1985 M8.1 Mexico earthquake caused significant damage (resulting in collapse) to buildings of 9 to 14 stories, and much less damage to low rise and very tall buildlngs. It is generally felt that the dominant frequencies of the ground motion in Mexico City were in the same range as the natural frequencies of the 9 to 14 story buildings, causing resonance. More detail about the damage to buildings in the 1985 earthquake is found at http://failures.wikispaces.com/1985+Mexico+City+Summary+%26+Lessons+Learned
A video of a simple demonstration of resonance is found at http://www.iris.edu/hq/files/programs/education_and_outreach/seismographs_in_schools/docs/14A.Resonance1_Spaghetti.mov
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