Natural Frequency
Masses on Rods Student Module
Introduction:
A common technique for analyzing structures is to treat them as lumped masses on a single rod. The vertical
elements in the building (i.e., columns) provide resistance to lateral movement. Most of the weight on the building
(i.e., furniture, occupants, floor system) is located at each story level. The rod can be thought of as the columns in the
building, and the lumped masses represent the weights applied at each story level. A single story building would be
shown with a rod representing the height of the structure, with the entire weight of the structure located at the top. A
five story building, as shown below, would have a rod extending for five stories (columns idealized as a single
column) with the weights attached at each of the five floor levels. This analysis allows researchers to scale the
structure for testing and make calculations as to how it would respond in an earthquake.
Two important considerations in this model is the mass of the structure (note: weight = mass x gravity) and the
stiffness. The stiffness of a structure corresponds to how well it will resist a force being applied to it. In earthquake
design, the outside force is the ground motion which ends up forcing the structure to move side to side (horizontal
force). Therefore, the walls of the building end up being the contributing factor for the stiffness.
The stiffness is determined by four parameters, the length, material properties, moment of inertia (I), and boundary
conditions (how the member is attached at its ends). The material property is called the modulus of elasticity (E) and
can be thought of as how well the building material can resist force without bending. The two most common
materials are steel and concrete. Steel is about eight times stiffer than concrete, so its modulus of elasticity is eight
times larger than concrete. The moment of inertia can be thought of as how much material there is. This makes sense
because a larger wall will end up being stiffer than a smaller wall of the same material.
The stiffness and the mass of the building, therefore, control how the building will respond to an earthquake. An
interesting response to an earthquake is when a structure vibrates at its natural frequency. Frequency is how many
times per second something will move back and forth. For example, if you stretch a spring it will not only bounce
back to its original position, but it will also overshoot this position and end up stretching back the other way. When it
gets to the end of that side it will repeat the process bouncing back in the other direction. This is similar to what an
earthquake does to a building because during an earthquake the ground moves back and forth, which is like stretching
the spring in one direction and then moving it back to the other repeatedly.
The natural frequency is a special rate of moving back and forth. At this frequency the building will shake
uncontrollably. This is especially damaging during an earthquake because the building cannot oppose the force of the
earthquake and will increasingly move further away on each back and forth movement of the earthquake.
Calculations:
Engineers use a simple equation to determine the natural frequency of a structure when it is modeled as a lumped
mass. However, for this experiment, we will be looking at exactly what the model says a building is like.
Therefore, we can accurately calculate the natural frequency and then test to see if our calculations are correct.
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The equation is: f 
1
2
K
M
In this equation, f is the natural frequency which we are trying to calculate and M is the
mass of the weight at the top of the rod, which is found by dividing the weight by
gravity. K is the stiffness of the rod and is dependent upon the length of the rod,
modulus of elasticity (material properties), and the moment of inertia (how much
material). In this case, we can measure the length and the moment of inertia. We also
know that we are using steel so we know this quantity as well. We can use the
following equation to determine the stiffness (K).
K
3EI 3 * (29,000,000 psi ) * (1.6 * 10 6 in 4 )

 0.64 lbs in
L3
(6.5in ) 3
This means that to move the weight an inch, you need to push on it with a force of ~0.6
pounds. This is a small force and will show that our rods will move back and forth quite a bit.
Now that the stiffness has been calculated and because we know how much the metal weight weighs, we can calculate
the natural frequency using the equation discussed earlier.
f 
1
2
K
1

W
2
g
0.64 lbs in
0.133lbs
386.1in
 6.9Hz
sec 2
The unit of the frequency is Hertz (Hz), which is the number of cycles (movements back and forth) per second. This
calculation is for the tall single mass rod. This calculation of natural frequency holds true for a single story building.
Unfortunately, most buildings at risk of earthquake damage are taller than one story, and thus the engineer needs to
find another method to analyze these buildings.
Using the simplified model of a lumped mass, the engineer can imagine a multi-story building as a rod with multiple
weights placed at the level of each floor. This arrangement allows the engineer to make simplified calculations of the
natural frequencies. However, because there is more than one weight, there is more than one natural frequency.
These new natural frequencies are related to the mode shapes of the structure. A mode shape is the way a structure
will shake back and forth. With a single weight there is only one mode, however, with two stories there are two
modes. These are shown below.
First Mode
Second Mode
As a structure has more and more floors the number of mode shapes increase. However, the first shapes are usually
the most important because as the mode number goes up, so does the corresponding natural frequency. This means
that the earthquake needs to shake back and forth faster and faster to get the structure to move in these ways.
Experimental Procedure:
An experiment can demonstrate what happens when a building is shook at its natural frequency.
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One way to measure the natural frequency of a structure is to determine how many times the structure oscillates back
and forth within a specified period of time (e.g., one minute). The frequency would then be the number of cycles that
occurred during the interval divided by the time in seconds. Inversely, the period of the structure, which is another
term commonly used, is T=1/f. This represents the time it takes for the structure to oscillate back and forth once.
In this experiment, a plate with four rods and various weights will be placed on the shake table and shook to
determine the natural frequencies of the various rods. This is analogous to several different sized buildings being
shaken by an earthquake.
Discussion Questions:



Which building will have the highest frequency? Which one will have the lowest frequency?
Make a guess, displace each of the structures one at a time and then release. Did you guess correctly?
How do mass and stiffness affect the natural frequency of each structure?
Resonance:
Now we will investigate a property called “resonance.” Resonance occurs when the forcing (excitation) frequency
equals the natural frequency, which can lead to catastrophic results. If you can excite a structure at its natural
frequency, it will resonate. We will use the sine wave feature of the shake table to investigate resonance. A sine wave
feature of the shake table will be used to investigate resonance.
As the frequency is increased, note how the different rods respond to the shaking, especially when you reach a natural
frequency of one of the rods. When this happens, stop increasing the frequency to observe the increased movement of
the rods. Then keep increasing the frequency of the sine waves until you have hit all of the natural frequencies of the
rods. The natural frequencies are labeled in the picture on the next page.
3.7 Hz
2.4 Hz
First Mode: 1.3 Hz
Second Mode: 10.5 Hz
6.9 Hz
Discussion Questions:



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 the natural frequency?
Advanced Topics:
Now that we have seen the effect that natural frequency and resonance can have on a system, it is important to find a
way to use this information in designing safe structures. An important part of an earthquake record is the frequency at
which the ground moves. If this frequency is the natural frequency of a structure, the building will resonate, move
back and forth uncontrollably, and cause large forces on the building.
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A unique trait of earthquakes is that the ground motion moves in repeating frequencies. These frequencies are
determined by the local geology of the area and can be determined from historical records. Therefore, engineers try to
figure out what the frequencies for a particular fault and region are so they can make sure that buildings in those areas
do not have natural frequencies that are the same as the earthquakes. If this can be avoided, buildings generally
survive the earthquake better and will not have the extreme movement caused by resonance.
One tool that engineers use to accomplish this goal is to develop response spectra. A response spectra can be thought
of as a graph that has the natural frequency on the x-axis and the maximum displacement or acceleration on the yaxis. This corresponds to the maximum displacement or acceleration for a given natural frequency. This spectra is
created by testing an earthquake record with different masses, on rods which have varying natural frequencies, and
then determining the maximum acceleration and displacement. Then the information is plotted on a response spectra,
which gives a visual representation of the worst case scenario for each natural frequency. An example is shown
below:
http://www.isr.umd.edu/~austin/aladdin.d/matrix-eq-spectra.html
On this plot, the maximum acceleration is plotted on the y-axis and the natural period is plotted on the x-axis. The
natural period is the time it takes for the mass to move back and forth once, and is directly related to the natural
frequency. It can be determined by the following equation:
Tn 
2
fn
Where Tn is the natural period and fn is the natural frequency. Therefore, it is easy to
switch back and forth between the natural period and the natural frequency.
From this plot we see that the worst acceleration happens when the natural period is 0.5 seconds, which corresponds
to a natural frequency of 12.6 Hz. Therefore, if an engineer is designing a building in the area, it would be prudent to
make sure that it does not have a natural frequency close to this value.
Discussion Question:

Which rods would be affected the most by this recorded earthquake, based on the response spectra?
Some things to take away:
1. Structures vibrate at a natural frequency that depends on E, I, L, boundary conditions of the structure and the
distribution of masses.
2. The response of a structure is greatly amplified when it is excited (vibrated) at its natural frequency.
References:
Chopra, Anil. Dynamics of Structures: Theory and Applications to Earthquake Engineering. 2nd Edition, Prentice
Hall, New Jersey, 2001.
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