Introduction

In response to a memo distributed by the principal engineer of Vulture Capital Associates, a force transducer was designed to be marketed to high schools as a teaching aid. The memo described the need for a device that would help high school students learn about the fundamental laws of physics. With the use of this equipment, students will be provided with hands-on experience when learning about Newton’s Laws. Potentially, there is a very large market for this type of device so production of this transducer would give Vulture Capital an edge over other companies in this particular market.

Since the force transducer will undergo repeated use, it is designed to be reliable, easy to use, and inexpensive to manufacture. It will be used along with the Estes B6-4 model rocket engines manufactured by Estes Corporation since this particular model would be a good input source for the transducer. Force applied by the rocket during firing changes continuously, but the maximum thrust produced is just three pounds over one second. When fired, the force transducer output will produce a thrust versus time curve which will have a resolution of at least 100 points for both force and time. This data, which is displayed and recorded by a software program, can be stored and analyzed. The force transducer operates with an accuracy of 9.5% using an unfired rocket and 5.6% using a fired rocket at 20Hz.

The following report explains the steps taken during the design, construction, testing, and refining stages of building the force transducer. Also, the methods used to perform data corrections are provided along with the results.

Methods

Design

Careful analysis was conducted prior to construction of the force transducer to determine which materials should be used. This device consists of two parts, the cantilever beam and the base. Optimally, the cantilever beam should be made from a material in which the yield strength is maximized and the modulus of elasticity minimized. A low modulus of elasticity would cause the material to have higher strain when compared with the amount of stress. These properties would allow the beam to withstand greater forces without permanent deformation and the product could be used repeatedly. Since 6061 T-6 aluminum has these desired physical properties and is readily available, it was chosen as the material for the cantilever beam.

To determine the physical dimensions of the beam, a factor of safety of 1.6 for the 3lb force was taken into account to ensure the beam would not fracture during loading. The length of the beam was calculated using the following equations³:

σ = stress [1]

M = bending moment

I = moment of inertia

E = modulus of elasticity [2]

ε = strain

F = applied force

L = beam length [3]

b = beam width

h = beam thickness [4]

C = ½ beam thickness

Combining equations [1] through [4] and rearranging yields the equation used to calculate the beam length.

[5]

The modulus of elasticity used during the calculations was 10⁷psi.⁵ Using a beam width of 1.0” and thickness of 1/16” the calculated length is 5.08”. Since it would be difficult to cut a beam to this precise length, for the subsequent equations, the beam length was rounded to 5”. Next, in order to ensure there is enough clearance to allow for the beam deflection during firing, the magnitude of the deflection needed to be calculated. This was determined using the following equation³:

[6]

The maximum deflection, δ, was calculated to be 0.0384”. Since the vertical deflection is very small, an L-shaped wood base was chosen for the transducer. This would be easy to manufacture, inexpensive and simple in design. To ensure maximum shear stress of 37,500psi ¹ is not exceeded, the following equation was used to calculate the stress of the rocket on the beam with a length of 5”.

[7]

From this equation, the shear stress was calculated to be 23,040psi, which is less than the maximum shear stress. Thus, the beam will be able to withstand the force produced by the rocket during firing.

Data Acquisition System

A data acquisition system was set up to transmit and record the output signal from the transducer so it may be analyzed. Strain gauges were attached (one on top and one on the bottom) of the beam to read the degree of beam deflection according to tension and compression, respectively. The strain gauges output signals to a Wheatstone bridge circuit which is attached to the base of the force transducer.

Figure 1.1. A diagram of the Wheatstone bridge circuit used to transmit input and output signals. The green pins are connected to the input voltage from the DC power supply and the gray pins are connected to the A/D board to transmit the output voltage readings from the strain gauges.

Output signals from the circuit are transmitted to the A/D board connected to the computer, then recorded by the software program, LabVIEW. LabVIEW is used to create a virtual instrument (VI) which can record data, display the time and frequency domain data, and save the data for future use. The program is shown in the following diagram.

Figure 1.2. Block diagram of the LabVIEW program used to record the force transducer output data.

The sampling rate at which data is recorded in LabVIEW is 1024 points per second. This is a convenient sampling rate to use since the Nyquist frequency will be 512Hz and the spacing for each data point will be 1Hz. Also, this enables the use of the Fast Fourier Transforms function in Microsoft Excel to perform calculations since a number of 2ⁿ data points are needed for the function to operate. The LabVIEW program produces two graphs to display the data recorded by the transducer--amplitude (V) vs. time (s) and power spectrum (dB) vs. frequency (Hz).

Calibration

Before firing the rocket, calibration tests were conducted to ensure there is a positive correlation between the voltage reading and the applied force. If so, an accurate curve will be produced in response to the constantly changing applied force. Three weights were hung, added one at a time, from the free end of the beam. Voltage output readings by the transducer were taken with input voltages of 1V, 2V, 3V, and 4V. A graph of the output voltage versus the applied force by the weights is graphed for each input voltage. The slopes of these graphs were then graphed against the applied voltage to confirm a linear voltage relationship. This ensures the voltage readings taken by the transducer increase in response to an increase in applied force. Since the force applied by a rocket during firing is constantly changing, it is important that the voltage output correspond to change in applied force correctly.

Initial Firing

After setting up the data acquisition program, the first test firing was conducted. A steel cup was attached to the free end of the beam to support the rocket during firing. Using the power spectrum vs. frequency graph produced by LabVIEW, the ringing frequency was determined. The ringing frequency is the frequency at which the first significant power spectrum peak occurs.

Refining the Design

Results from the initial firing were used to determine the value of the natural frequency (f_n) and damping ratio (ξ) of the beam using the following equations⁴.

[8]

[9]

The accuracy of the transducer was then determined using the following equation.

[10]

An increase in natural frequency, and an increase in accuracy, is obtained by decreasing the mass associated with the beam. Therefore, the steel cup was removed and for future testing, the rocket was glued directly onto the beam. Also, the length of the beam was shortened to further decrease the mass of the system. Segments ranging from ¼” to 1” were cut off from the beam and a series of step input tests were conducted each time. With the unfired rocket glued to the free end of the beam, a step input was applied to the beam by pulling down and letting go. Output from the transducer was recorded and analyzed using LabVIEW. The ringing frequency (f_d) was determined using the graph of the power spectrum vs. frequency. Decay in the peaks from the voltage vs. time graph was used to produce a graph of the natural log of the peak amplitude versus time. This data was then used to calculate the corresponding natural frequency, damping ratio, and accuracy (using equations [8], [9], [10]) for each beam length. These tests were repeated until the beam was short enough to operate with the desired accuracy of 10% at 20Hz. When the optimum beam length was obtained, the step input test was repeated once more using a fired rocket instead. Again, the natural frequency, damping ratio, and accuracy were calculated.

Final Design

The final design of the BML Force Transducer consists of a 6061 T-6 aluminum cantilever beam mounted to a wood base. The circuit board containing the Wheatstone bridge is attached to the back of the transducer.

Figure 1.3. A 3-D frontal view of the final force transducer design is shown. The final beam length is 2.5”.

Final Firing

To prepare the rocket for the final firing, all the ejection charge was removed from the rocket since it does not contribute to the force produced during firing. The majority of the thrust produced during firing is due to the solid propellant.⁶ This second firing was conducted using the modified beam length of 2.5” with the rocket glued directly to the beam. From the final firing, a second set of data was recorded and later corrected to produce the thrust versus time curve.

Producing the Thrust vs. Time Curve

After the second firing, a sensitivity test was conducted to determine the constant used to convert the voltage readings (which are in the frequency domain) to force (which is measured in the time domain). In a pattern similar to the calibration test, three weights were hung successively one at a time from the free end of the beam. Using the same input voltage as during the firing (5V), the output voltage versus applied force from the weights was graphed. The slope of this graph was used as the conversion factor. All of the corrected voltage values were divided by the value of the slope to calculate the corresponding force.

The raw data recorded by LabVIEW during the firing was transformed using Microsoft Excel to perform Fourier analysis calculations using a series of Fast Fourier Transforms (FFT). Ideally, the final curve would model the following curve provided by the manufacturer.

Figure 1.4. Thrust vs. time curve for the Estes B6-4 model rocket engine provided by Estes Corporation is shown.² The maximum thrust is about 3lbs.

To obtain a smooth curve similar to Figure 1.4, the data was manipulated using several transformations. First, a logarithmic plot of the magnitude vs. frequency was used to determine at which frequencies a significant amount of mechanical distortion and electrical noise occurs. Distortion due to the beam was removed by dividing the FFT by the gain to correct the output data. Since the beam amplifies the output signal of the transducer, the amplification was removed at the frequency at which it occurs. This obtains a more accurate value for the actual input signal. Electrical noise was corrected by creating low pass filters to allow only low frequencies to be present in the output signal. This series of corrections obtain a smooth force versus time curve which models the manufacturer’s curve seen in Figure 1.4.

Results

Preliminary Testing

The following are results from the calibration test condcuted prior to firing.

Figure 2.1. Voltage output of the transducer is graphed versus the weight applied. Readings for each weight were taken with an input voltage of 1V, 2V, 3V, and 4V. The equation of the line for each voltage graph shows its slope. The linear relationship between the force applied and output voltage is shown.

Figure 2.2. The slope of each of the curves from Figure 2.1 are graphed versus the input voltage. This shows the relationship between the output voltage of the transducer and applied force is linear. With a linear relationship, it can be assured the voltage readings will accurately correspond to a varying applied force.

Initial Firing

The first test firing was conducted using the force transducer with a beam length of 5”. A steel cup was attached to the end of the beam to support the rocket.

Figure 2.3. Data recorded during the first firing is shown in the graphs produced by LabVIEW. The top graph shows the output voltage versus time, which is significantly different from the thrust vs. time curve provided by Estes (see Methods, Figure 1.4). From the lower graph, it can be seen that the ringing frequency is 10.5Hz since this is the frequency at which the first significant peak occurs. *Note the y-axis for the lower graph should read Power Spectrum (dB), not Amplitude and the x-axis should read Frequency (Hz), not Time.

Design Modifications

Results from the first firing show the transducer has a low ringing frequency which is largely due to the mass of the steel cup attached to the beam. To improve the transducer, the steel cup was removed and a series of step input tests were conducted with the rocket glued to the free end of the beam. The proceeding figures give an example of how the natural frequency, damping ratio, and accuracy were obtained from the step input test.

Figure 2.4. The natural log of the voltage versus time from the step input response of the beam with a fired rocket attached is shown. The voltage measurements were taken from the peak amplitude of the voltage versus time graph produced by LabVIEW. The slope of the graph shown was used (along with equation [8] from Methods) to calculate the natural frequency, damping ratio and accuracy corresponding to the beam length.

Damping Ratio (ξ) and Natural Frequency for Fired Rocket
beam length	fd (Hz)	slope (m)	ξ	fn (Hz)	Accuracy (%)
2.5"	86	-6.7803	0.01255	86	5.6

Figure 2.5. This chart shows the results from the step input test for a 2.5” beam using a fired rocket.

As the length of the beam decreased, its stiffness, natural frequency, damping ratio, and accuracy increased. After repeating the step input test several times using an unfired rocket, the desired accuracy was obtained when the beam length was 2.5”. At this length, the beam has a natural frequency of 68Hz, damping ratio of 0.001223, and accuracy of 9.5% at 20Hz using an unfired rocket. Using a fired rocket, the natural frequency of the beam is 86Hz, damping ratio is 0.01259, and accuracy is 5.6%. The accuracy using a fired rocket is slightly less than double the accuracy using an unfired rocket. The natural frequency and accuracy is greater since the mass of the fired rocket is less than when it is unfired. With a 2.5” beam, accuracy of the transducer using either an unfired or fired rocket exceeds the initial goal of 10%.

Final Firing

After shortening the beam to obtain the desired accuracy, a second firing was conducted. The rocket was then glued to the free end of the 2.5” beam.

Figure 2.6. This is a screen shot of the final rocket firing using a beam 2.5” in length. The top graph shows a curve which is similar to the thrust vs. time curve provided by Estes. Although there is significantly less noise and distortion compared to the initial firing, there is still some noise present in the output signal. The lower graph shows the ringing frequency is measured as 68Hz, which corresponds to the calculated value.

A sensitivity test was conducted in order to convert the output voltage into force to form a thrust versus time graph. The result is shown below.

Figure 2.7. A constant used to convert the voltage readings to force was determined from the slope of the graph produced during the sensitivity test. Using an input voltage of 5V (equal to the voltage used during firing), the output voltage readings were recorded and graphed vs. the applied force. The corrected output voltage values will be divided by 0.001 to calculate the corresponding force.

A series of data manipulations were performed to obtain a curve that models the thrust (or force) versus time curve for the rocket provided by Estes. First, the original, uncorrected data is shown.

Figure 2.8. The force vs. time graph using the raw, uncorrected data is shown. This curve models that produced by Estes, but has a significant amount of noise present.

Data Manipulations and Corrections Using Microsoft Excel

A series of several data manipulations and corrections were done using various functions in Microsoft Excel to remove mechanical distortions and electrical noise. The 68Hz distortion seen in Figure 2.9, characterized by the large peak, is most likely to be caused by the amplification of the output signal due to the natural frequency of the beam, which is 68Hz for an unfired rocket. Since the beam amplifies the output at this frequency, the 68Hz frequency was divided out to obtain the actual input data. The removal of the 68Hz distortion is shown in Fig. 2.10. As mentioned in Figure 2.9, a significant amount of distortion from the 86Hz natural frequency of the fired rocket is not present nor is there distortion from 76Hz till 86Hz. Distortion in this range would be due to the changing natural frequency of the beam during firing, as a result of the changing mass of the rocket. Since there is no distortion present in Figure 2.9, no correction was made to suppress these frequencies. However, corrections to the data were made to remove the electrical noise caused by frequencies greater 170Hz by simulating the use of a low pass filter.

Figure 2.9. The uncorrected magnitude vs. frequency graph is shown. It can be seen there is a significant amount of distortion around 68Hz and noise around 176Hz and 186Hz due to the significant peaks on the graph. Distortion at 86Hz from the fired rocket is not present in the graph, so corrections to suppress this frequency are unnecessary.

Figure 2.10. The corrected magnitude vs. frequency graph is shown. It can be seen that the 68Hz distortion has been removed since there is no longer a significant peak at this frequency. Also, the effect of the low pass filter is evident as all the frequencies above 170Hz have been removed.

Figure 2.11. Magnitude ratio versus frequency is graphed on a logarithmic scale using the data from the unfired and fired rockets. The unfired rocket has a natural frequency of 68Hz and the fired rocket 86Hz, which is indicated by the peaks in the graphs. The two curves intersect at a frequency of 76Hz. As shown graphically, the magnitude ratio for the unfired rocket has values less than 1 for frequencies greater than 97Hz.

Since the graph from the unfired rocket in Figure 2.11 crosses the horizontal axis of a magnitude ratio equal to 1 at a frequency of 97Hz, this frequency is used as the cutoff for corrected data. Only data ranging from frequencies between 0 and 96Hz were corrected ¹ using successive data manipulations. Whatever corrections were made to the data to suppress the distortion and noise of the output signal, the uncorrected values were used from 97Hz to 512Hz. The complex conjugate of the uncorrected FFT of the output voltages for frequencies from 97Hz to 512Hz was equated. These uncorrected values were mirrored to the 97Hz frequency occurring on the other side of the Nyquist frequency. The following curves show the force versus time curves using the various corrections.

Figure 2.12. The corrected force vs. time curve of the final firing with only the simulation of a 170Hz low pass filter is shown. A significant amount of noise was removed from the original uncorrected curve. The maximum thrust of the rocket is approximately 3.17 lb which is larger than the estimated 3 lb thrust from the manufacturers curve. This is reasonable due to the natural frequency of the beam amplifying the output signal from the transducer.

Figure 2.13. Corrections made to obtain this force vs. time curve include the 170Hz low pass filter and removal of the 68Hz distortion. This curve is an even closer model of the Estes curve. The maximum thrust is approximately 3.07Hz.

The successive force versus time curves are compared in the following figures to show the effectiveness of the series of data corrections.

Figure 2.14. A comparison of the uncorrected (raw) and corrected (170Hz low pass filter and 68Hz distortion removal) force vs. time curves is shown. Through these corrections, a significant amount of noise and distortion was removed from the original data to provide a smoother curve.

Using only the data from the input and output signals, the curve displaying the low pass filter and removal of the 68Hz distortion is the smoothest curve attainable. Removing the mechanical distortion and electrical noise in the output signal, a close model of the actual input from the rocket signal was obtained. To further correct the data, all frequencies greater than 20Hz can be removed since they contribute less than 1% of the input signal and are deemed insignificant.¹ This curve most closely models the thrust versus time curve provided by the manufacturer. However, all the data from the input signal is removed, so again, the best model based on the experimental data alone is that shown in Figure 2.13.

Figure 2.15. The force vs. time curve is shown using the simulation of a 20Hz low pass filter and the removal of 68Hz distortion. Using these two data corrections, most of the noise and distortion was removed from the input signal to obtain a smooth curve. Thus, this curve most closely resembles the thrust vs. time curve provided by Estes.

Figure 2.16. The force vs. time curve with frequencies greater than 20Hz removed compared to the uncorrected (raw) data.

Conclusion

After the initial firing, the output voltage curve produced by the force transducer was distorted and did not resemble the thrust vs. time curve produced by Estes. However, after several modifications were made to the cantilever beam, the natural frequency and accuracy of the transducer was greatly increased. The final design of the force transducer consists of a beam 2.5” in length. It operates with an accuracy of 9.5% for an unfired rocket and 5.6% for a fired rocket at 20Hz. In order to obtain a curve that closely resembles that from Estes, a series of corrections using Fourier transformations were made to remove the distortions and noise present in the output signal. The simulation of a low pass filter and removal of the 68Hz distortion provided a clearer, smoother graph of the actual input from the rocket signal as shown in Figure 2.13. Overall, a successful model of the thrust versus time curve for the Estes B6-4 rocket was obtained with the use of the force transducer.

In general, the force transducer worked well as there were no problems with the cantilever beam. However, there are some potential problems with the base. In the future, a couple modifications should be made to improve the quality of the force transducer. First, the material of the base should be changed. Wood was originally chosen since it is lightweight (which is good for mobility), readily available, and inexpensive. However, since the majority of its use will be from high school students using rockets, a material that is fire-resistant would be preferable. Perhaps the base could be built from an L-shaped piece of aluminum instead to match the beam material. Also, extending the length of the base and changing its material to a type of metal would provide greater stability. These modifications would enhance the quality of the transducer, making it an even more valuable tool for high school students.

References

1. B.E. Liebert, personal communication (2005)

2. B.E. Liebert. ME 402 Website, http://www.eng.hawaii.edu/~liebert/ME401/

3. James M. Gere, Mechanics of Materials, 5^th Ed., pp. 899, Brooks/Cole (2001)

4. Richard S. Figliola and Donald E. Beasley, Theory and Design for Mechanical Measurements, 3^rd Ed., pp. 88-93, Wiley (2000)

5. William D. Callister, Jr, Materials Science and Engineering An Introduction by: William D. Callister, Jr., 5^th Ed., pg. 793, John Wiley & Sons, Inc

6. William Simon, Model Rocketry Technical Manual, http://courses.washington.edu/engr100/All_Sections/Rocket/HTML%20Handouts/02_hnd_building_rocket.pdf

BML Force Transducer Design

Meredith Chee

ME 402

May 2005

Abstract

A prototype force transducer, the BML Force Transducer, was designed, built, and tested to provide a product that could be marketed for use by high school students to help demonstrate the basic principles of physics. The transducer consists of a cantilever beam mounted on a wood base to provide a lightweight, inexpensive, and user-friendly design. A data acquisition system was set up to record the output signals from the transducer using a Wheatstone bridge circuit, an A/D board, and a virtual instrument software program. The recorded data is then manipulated to correct any noise or distortion occurring in the output signal. The force transducer operates with an accuracy of 9.5% using an unfired rocket and 5.6% using a fired rocket at 20Hz. Using the BML Force Transducer, the thrust of an Estes B6-4 model rocket engine was measured and a graph modeling the thrust versus time diagram provided by the manufacturer was successfully obtained.

Table of Contents

Abstract ............................................................................. i

I. Introduction 1

II. Methods ............................................................................. 2

Design............................................................................ 2

Data Acquisition System................................................. 3

Calibration...................................................................... 5

Initial Firing..................................................................... 6

Refining the Design......................................................... 6

Final Design.................................................................... 7

Final Firing..................................................................... 8

Producing the Thrust vs. Time Curve............................... 8

III. Results ............................................................................. 10

Preliminary Testing.......................................................... 10

Initial Firing..................................................................... 11

Design Modifications...................................................... 12

Final Firing..................................................................... 14

Data Manipulations and Corrections

Using Microsoft Excel............................................ 17

IV. Conclusions ............................................................................. 26

V. References ............................................................................. 27