Grand Valley State University

PERFORMED: 14 OCT 99

Mike Cool

Table of Contents *

List of Figures and Tables 3

Table 1 - Calculated and Measured Data for both setups 11

Table 2 - Percent Errors between Theoretical and Actual Frequencies and Spring Constants 11

Table 3 - List of equipment *

In this lab two experiments were performed. In both experiments, a rod was used as a torsional spring. One rod was an aluminum round, and the other, a steel rectangle rod. A mass was then mounted at the bottom end of the rods. The aluminum rod had a circular plate attached to it. The steel rod had two tubular aluminum pieces attached to the bottom. The masses were deflected at a certain angle. This was done by applying a given force to the mass. The spring constant was then calculated. A potentiometer was mounted on the bottom of each mass. This allowed the output voltage to be converted into frequency. This frequency was then compared to the theoretical frequency calculated earlier. Results from the experiment showed that as the degrees of displacement were increased, the frequency of oscillation did not increase. Another important concept that was discovered is that the calculated and measured spring constants for both rods were reasonable. Some future recommendations for this lab include some of the following ideas. It would be helpful if a larger variety of materials in the shop were readily available to the student. This would allow the student to derive a better, more creative setup. Also, the time allowed to perform the lab was not sufficient. A lab of this complexity should be given a longer time frame to complete.

This lab was created in order to give a student a better understanding of a basic torsional spring setup. The purpose is to instruct a student on how to setup a correct system using common steel and aluminum rods. This shows that any piece of steel or aluminum with the a correct cross-section can be used to create torsion. The student was instructed to go find and modify a piece of steel or aluminum into a configuration that will demonstrate torsion.

The procedure consisted of finding a specimen, setting up the LabVIEW program to read in the data. LabVIEW is a powerful program that enables a student to read in mechanical data via a DAQ board and computer. A DAQ board is a Data Acquisition Board. This is used to connect the potentiometer to the computer. The potentiometer enables the data to be read in by the difference in voltage read in on the computer.

Given that a large symmetric rotating mass with rotational inertia J, is attached to a twisting rod with a torsional spring coefficient K, the following basic relationships can be used to determine the frequency of the system. J is the rotational inertia of the rotating mass. G is the moment of inertia of the mass. A is the area, and L is the length of the part to be rotated. The greek symbol Theta is the angle of rotation and the greek symbol Omega is the angular velocity.

Applying the sum of torques, we get the following:

Now by using these basic relationships, we can rearrange them to find the frequency of the system:

or

Given the rotational element of the part,

And finally the frequency is derived to be:

The following is a list of all the equipment used in the lab:

Along with these items which need serial number designation, the following items were used. An interface cable was used to get the data from the potentiometer to the computer. A multimeter was used to verify voltages throughout the experiment. A Sargent-Welch Scale was used to pull on the apparatus in order to give the system an initial force. A meter stick was used for height measurements. Various clamps were used to hold the apparatus in place throughout the experiment. And lastly, a protractor was used to measure the rotational displacement while the initial force was introduced.

In this lab two experiments were done simultaneously. In both experiments a torsional spring (aluminum rod) was clamped firmly to a support and hung perpendicular to the floor. In experiment 1 a round circular mass of constant thickness and radius of 8.5 in. was firmly attached to the bottom of the spring. For experiment 2, two 48 in. long two inch square aluminum tubing were clamped firmly together with the spring between them so that the spring is unable to rotate between them. A model of the experiment is shown below in figure 1.

The spring constant can now be found from this experiment by applying a measured force to the outside of the mass and rotating the mass a measured amount of degrees. These values are put into the equation (2) below to find the experimental value of K (Table 1). Where F is the force applied over a radius, R and s is the angle of rotation.

(2)

This value is compared to the theoretical value of K (Table 1,Table 2), which is found using the basic mechanics of materials. This is shown in equation (3) below.

In this part of the experiment a potentiometer is attached to the bottom of the mass as seen in figure 1 above. The potentiometer is given a constant voltage of 5V. The output voltage from the potentiometer is fed into a LabVIEW program, which plots voltage vs. time. The bottom of the potentiometer is held still with visegrips. As the mass is rotated the output voltage changes. When an unequal force is applied to the mass it begins to oscillate which displays a graph that looks like a sine wave. The potentiometers used in these experiments were found to be linear by graphing several values of voltage vs. angular position. From this a ratio was developed to convert voltage to angular position. When the system is oscillating the frequency of oscillation can be found from the graph in LabVIEW (Table 1). This value will then be compared to the theoretical value found by using equation (4) below (Table 1,Table 2).

(4)

The data was collected and analyzed using LabVIEW. The frequency was then calculated using Mathcad and compared to the experimental frequency. The results of the lab are below:

The values calculated in Table (1) were done using equations (2) & (4). They are the equations needed to find the spring constant of the rod, K, and the frequency of the system, f. The forces were measured using the Sargent-Welch Scale. The apparatus was initially rotated using the scale. The protractor was used to measure the rotational displacement. The following are the %-errors found from the data:

The error that was found between the calculated and experimental data could be due to several reasons. For one, the potentiometer might have had some error to it. Another reason could be that the setup was not completely flawless. Small movements when the oscillations occurred might have contributed to the error. The data shows that in fact the deflection did not change the frequency of oscillation for the rods. It also shows how the oscillation is in fact the shape of a sine wave. This was as expected.

The experiment went well. The data could’ve been a bit closer to the expected values, but the general idea of the lab was achieved. Future recommendations include the following. For one, I believe that a larger selection of materials would be nice. This would allow the student to be more creative when attempting to choose a suitable setup. During lab time, some direction would be nice when trying to distribute jobs amongst the group. The idea that any aluminum or steel bar with a relatively small cross-section can be used to model torsion, is a valid one. This idea can be used when making a rough design or trying to prototype a design.

Appendix A-Mathcad calculations for experiment 1

Appendix B-Mathcad calculations for experiment 2