## Understanding Natural Frequency: Masses on Rods

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#### Abstract

A common technique for analyzing structures is to treat them as lumped masses on a single rod, which simplifies the analysis. A single story building would then be modeled as a rod of the height of the structure with all of the weight of the structure located at the top of the rod. A multi-story building would be modeled by a rod with a weight located at each floor. This simplified **lumped-mass model** allows researchers to scale the structure for testing and make simple calculations as to how it will respond to an earthquake.

#### Learning Objectives and Standards

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#### Procedure

**Understanding Natural Frequency: Masses on Rods**

**Introduction:**

A common technique for analyzing structures is to treat them as lumped masses on a single rod, which simplifies the analysis. A single story building would then be modeled as a rod of the height of the structure with all of the weight of the structure located at the top of the rod. A multi-story building would be modeled by a rod with a weight located at each floor. This simplified **lumped-mass model** allows researchers to scale the structure for testing and make simple calculations as to how it will respond to an earthquake.

*Figure 1: Actual structure (left), five-story structural support system (middle), simplified lumped-mass model (right).*

The two important considerations in this model are the **mass** (weight) of the structure and the **stiffness**. The stiffness and the mass of the building control how the building will respond to an earthquake. The stiffness of a structure corresponds to how much it will deform when a force is applied to it. In the instance of earthquake design, the outside force is the ground motion that forces the structure to move side to side (horizontal force). The building walls or the structural frame are the main contributor to the stiffness.

The stiffness of a structural member is determined by three parameters:

- the length,
- material property, and
- the moment of inertia (I).

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. Two common building materials are steel and concrete. Steel is about eight times stiffer than concrete, meaning its modulus of elasticity is eight times larger than concrete. The moment of inertia can be thought of as an indicator of how much material there is. This makes sense because a larger wall will be stiffer than a smaller wall of the same material.

An earthquake causes a building to vibrate, and engineers are interesting in studying the **frequency** of these vibrations. Frequency is how many times per second something will move back and forth. For example, if you stretch a spring it will contract back to its original position but will also overshoot this position and end up stretching back the other way and when it gets to the end of that side it will repeat the process stretching 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.

Engineers are particularly interested in when the earthquake causes the structure to vibrate at its **natural frequency**. The natural frequency is a special rate of moving back and forth because at this frequency the building will shake uncontrollably. This uncontrollable vibration causes damage because the building bends more than it was designed for. Once damage starts, each back and forth movement of the building will be farther away from its straight shape.

This experiment attempts to show what happens when a building is shaken at its natural frequency. To illustrate this concept, several small rods will be fixed to the shake table with different weights placed at different heights to show the effects of increasing weight and height.

*Figure 2: Experiment set up.*

**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 observing exactly how the model behaves. Therefore, we can accurately calculate the natural frequency and then test to see if our calculations are correct.

The equation for natural frequency of lumped-mass structure is:

In this equation, f is the natural frequency and M is the mass of the weight at the top of the rod, which is found by dividing the weight (W) by gravity. M=W/g

K is the stiffness of the rod and depends upon the length of the rod, Modulus of Elasticity (material property), and the moment of inertia (how much material).

In this case, we can measure the length and the moment of inertia. Also, we know that we are using steel so we know the Modulus of Elasticity as well. Therefore, we can use the following equation to determine the stiffness (K).

This means that to move the weight an inch to the side, you need to push on it horizontally with a force of 0.51 pounds. This is a small force and will be seen that our rods will move quite a bit back and forth.

Because we know how much the metal weight weighs, we can calculate the natural frequency using the equation discussed earlier.

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 the natural frequency holds true for a single story building. Unfortunately, most buildings at risk of damage due to earthquakes 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 simply shape of a structure as it shakes back and forth. With a single weight there is only one mode, however, with two stories there are two modes. These are shown in Figure 3.

*Figure 3: Modes of vibration of a two-story building*

The number of mode shapes increases with the number of floors. However, the first mode shape is usually the most important because as the mode number goes up, so does the corresponding frequency of vibration. This means that the earthquake needs to shake back and forth faster and faster to get the structure to move in these ways. This is shown in the demonstration for the two-mass rod.

**Experimental Procedure:**

The lumped mass model of buildings simplifies not only the analysis of a building but the experimental testing as well. With minimal setup, the behavior of a building can be experimentally verified.

In this experiment, a plate with four rods and various weights will be placed on the shake table and shaken to determine the natural frequencies of the various rods. This is analogous to several different sized buildings being shaken by an earthquake.

The first step is to attach the masses on rods plate to the shake table using the two screws to securely fasten the plate to the table. Test to see if the plate is fastened to the table by shaking it with your hands. Next, start the shake table and allow it to calibrate using the procedure outlined in the shake table operations manual.

Navigate to the sine mode (option 2 at main menu) and select displacement mode (option 1). Then before selecting any other option, move the displacement and frequency models and establish how quickly they change the values of either the displacement or the frequency. Then set the displacement to 12% and set the frequency to 0.1Hz. Press the # key to start the experiment and then slowly increase the frequency.

While you are increasing the frequency, point out to the students 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 and allow the students to see the increased movement of the rods. Keep going until you have hit all of the natural frequencies of the rods. The natural frequencies are labeled below on the picture.

As a check, calculate the frequencies by counting the number of cycles that the systems undergo over a one minute interval after the structures are each individually deflected and released. The frequency would be the number of cycles that occurred during the interval divided by the time in seconds.

To exit the displacement mode simply press the 0 key and you will abort to the main menu. Simply remove the rods from the table by unscrewing the plate from the table.

**Discussion Questions:**

Below are some discussion questions to ask the students and encourage discussion.

- As the frequency increases, will the rods swing back and forth more?
- No, the biggest deflection (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 stop shaking nearly as hard.
- What effect will increasing the weight have on the natural frequency?
- As the mass increases, the natural frequency will go down. This is an actual technique used in practice, where mass is added to a structure to lower the natural frequency out of the range of frequency in the earthquake.
- What effect will increasing the height of the weight have on the natural frequency?
- As the length increases, the natural frequency will go down. This can be seen by comparing the short single weight rod to the long single weight rod.

**Advanced Topics:**

Now that we have seen the effect that natural frequency and resonance can have on system, it is important to find a way to use this information to design safe structures. An important part of an earthquake record is the frequency that 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.

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 these frequencies for a particular fault and region are so that 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 will generally survive the earthquake better because it will not have the extreme movement caused by resonance, due to shaking the building near its natural frequency.

One tool that engineers use to accomplish this goal is to develop response spectra (plural of spectrum). A response spectrum can be thought of as a graph that has the natural frequency on the x-axis and the maximum displacement or acceleration on the y-axis. This corresponds to the maximum displacement or acceleration for a given natural frequency. This spectrum is created by testing an earthquake record with different masses on rods which have varying natural frequencies and determining the maximum acceleration and displacement. Then the information is plotted on a response spectrum 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 and is given by the following equation:

Where T_{n} is the natural period and f_{n} 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.

- Which rods would be affected the most by this earthquake record based on the response spectrum?
- All of the rods except the two story rod would fair well because their natural frequencies are significantly lower than the worst case. However, the two story mass-rod system has a second mode natural frequency of 10.4 Hz which is relatively close to the worst case scenario.

**References:**

Chopra, Anil. Dynamics of Structures: Theory and Applications to Earthquake Engineering. 2nd Edition, Prentice Hall, New Jersey, 2001.

**Some things to take away**

- Structures vibrate at a natural frequency that depends on E, I, L, boundary conditions of the structure and the distribution of masses.
- The response of a structure is greatly amplified when it is excited (vibrated) at its natural frequency.