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MECH3316SampleLabWriteup1092819038209382048782074.pdf

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February 28, 2016 Dr. Samuel Daniels Associate Professor of Mechanical, Civil and Environmental Engineering University of New Haven Tagliatela College of Engineering 1124 Campbell Ave., Room 102 West Haven, CT 06516

Dear Dr. Daniels,

This report outlines the procedures and results of the MECH 3315 Impact Force Lab. The lab includes a

concise description of the lab setup, the results and analysis of the results, and description of anything

that went wrong as well as future recommendations. With the numerical values of the data, error

propagation and uncertainties are included so that the error of the equipment and the calculations are

already taken into account. The purpose behind this information is to determine how different distances

impact the force on a beam through the deformation of the beam.

Although the calculations were the majority of the report there is a strong emphasis on the theory behind

the calculations and the data is explained as well so that it can be converted into knowledge for everyday

use. Data tables with paired graphs were created as the visual portions of the analysis. The raw data will

be included in the appendix as well so that there is proof of where the data originated from and with that,

pictures of the lab setup to create an actual picture of what the lab consisted of. It is our hope that from

this document, the theory of impact force will be clarified, as well as make recommendations to improve

the project for future students performing the same lab.

Kind Regards,

xxxxxxxxxxxxxxxx University of New Haven Tagliatela College of Engineering Mechanical Engineering

University of New Haven

Tagliatela College of Engineering

The Effect of Impact on a Beam

xxxxxxxxxxxxxx

Performed February 24, 2016

Due March 27, 2016

Abstract The objective of this experiment was to describe how a small rotating steel hammer impacts an

aluminum cantilever beam at different angles. When the hammer hits the beam, it creates a dynamic

point load which then is converted into a strain. An accelerometer was used measure the different

angels, while a strain gauge measured the beam deflection. From the strain, the force which was

imposed on the beam was calculated using the equation for strain and then plotted against the angles.

From that plot the conclusion was made that as the angle increases so does the force on the beam and

with analysis grouped into a specific function (e.g linear, exponential, logarithmic, and sinusoidal). The

conclusion was drawn that values of the theoretical were very similar to the experimental.

Contents Abstract ......................................................................................................................................................... 3

Executive Summary ....................................................................................................................................... 5

Introduction .................................................................................................................................................. 6

Methods and Materials ................................................................................................................................. 9

Data and Results ......................................................................................................................................... 11

Discussion.................................................................................................................................................... 13

Conclusion and Recommendations ............................................................................................................ 14

Works Cited ................................................................................................................................................. 15

Appendices .................................................................................................................................................. 16

Appendix A – Picture of Lab Setup .......................................................................................................... 16

Appendix B - Screenshot of LabView Code for Strain Measurements .................................................... 17

Appendix C – Screenshot of LabView code for the Angle Encoder......................................................... 18

Appendix D- Dimension of Beam and Tup .............................................................................................. 19

Appendix F – Raw Waveform graphs of 30 and 45 degrees ................................................................... 20

Appendix G – List of Symbols .................................................................................................................. 21

Executive Summary To test the impact on a beam, there were several components that needed to be set up so that

everything was converted and calculated. The first was the actual set up. A half- bridge was used for the

wiring of the strain gauge. A LabView code was also created to program the sensor and equipment so

that data could be collected. Lastly the data needed to be converted from strain into force so that the

relationship between angle and impact could be compared. In addition, the experimental force was

compared to a theoretical force to decide whether or not the experimental data followed the trend of

the theoretical. The comparison of the two is seen in the data and results. The impact created “noise”

or resonance in the beam, which can be interchanged with the term natural frequency. The calculation

of natural frequency was the main portion of the lab to determine how well it compared to the period of

oscillation, which was experimental. Below are the results

Period of Oscillation

(theoretical) 0.0174 sec

Period of Oscillation

(experimental) .0166 sec

Natural Frequency (theoretical) 57.47 Hz

Natural Frequency (experimental) 60 Hz

Table 1 Comparison of Frequency and period of oscillation for experimental and theoretical data

Introduction Impact can be described as a high force or shock over a short period of time. Most of the time, impact

occurs when two bodies collide with each other, such as two cars or a nail and a hammer. As a result the

bodies or objects will deform a certain way so that the shock is absorbed, creating a sort of cushion. The

deformation also causes vibration in certain objects resulting in a so called “noise”. The magnitude of

the impact relies on not only the material, but also the distance between as well as the mass. Below is a

schematic of the general theory of the impact.

Figure 1 Schematic for Theory of Impact of a hammer on a cantilever beam (University of New Haven Blackboard SHOCK TEST EQUIPMENT THEORY (“Mechanical Measurements”, 5th ed., Beckwith, pg.770-772)

The theoretical force can be calculated through the help of the schematic. Since all the components can

be solved for, the Fth can be used when comparing the trend of the actual experimental data. The

equation below is derived from the schematic and can be seen below. (note- a list of symbols can be

found in appendix F).

𝑭𝒕𝒉 = √ 𝟔𝑬𝑰𝑼𝟐

𝑳𝟑 (1)

The majorities of the values are either given or can be solved for the Young’s modulus E is specific to the

material and the moment of inertia I is specific to the shape of the piece involved. The U term is the

energy portion which can be broken down to

𝑼𝟐 = 𝟏

𝟐

(𝒎𝟏+𝒎𝟐)𝑽𝟐 𝟐

𝒈𝒄 (2)

And the second velocity (V2) can also be broken down further into

𝑽𝟐 = ( 𝒎𝟏

𝒎𝟏+𝒎𝟐 )√𝟐𝒈𝒉 (3)

where the height can be solved in terms of the length of the hammer and the angle of the hammer. The

units are Newtons (N) since the force is being calculated.

In many engineering applications impact is measured through the use of strain gauges. These sensors

don’t directly take the force into account, but rather the strain or deformation of the object. These two

values are related through the following equation:

𝜺 = 𝟔𝑭𝑳

𝑬𝒘𝒕𝟐 (4)

Equation (4) can be manipulated so that the force is the variable to solve for rather than the strain since

it is the only unknown. There will be multiple forces since there are multiple measurements for each

angle as well as angles from 5° to 120°. Although not directly stated, the angle/distance that the

hammer travels is taken into account through the different values in the strain. An accelerometer was

used to measure the different angles. It uses the acceleration forces such as gravity or even slight

vibrations and converts these values to angles.

As previously mentioned, whenever there is any sort of impact or force on an object, vibrations run

through the object causing a disturbance within. These disturbances are often referred to as natural

frequency. The natural frequency also causes sound waves which follow the pattern of a sinusoidal

wave. When calculating for the natural frequency of an object, the Moment of Inertia (I) needs to be

determined. Inertia is often used when describing rotational inertia, and describes how a body resists

the angular acceleration it undergoes. Every 3D object has a different Moment of Inertia based on its

dimension and shape. The equation for the natural frequency is stated below:

(5)

Since a cantilever beam was used for the lab, the shape of it is a rectangular beam for which the

moment of inertia can be calculated from the equation

𝑰 (𝒐𝒇 𝒃𝒆𝒂𝒎) = 𝟏

𝟏𝟐 𝒃𝒉𝟑 (6)

Since every piece in the setup was connected, the actual mass of the beam and tup could not be

weighed. Since all dimensions were taken at the beginning of the lab, the mass could be calculated with

the density through the equation

𝒎 = 𝝆𝑽 (7)

Since the material of the beam and the tup was known, the specific densities were used. The natural

frequency that was calculated can technically be seen as “theoretical” since from the data, this is what it

is supposed to do.

𝒇 = 𝟏

𝟐𝝅 √ 𝟑𝑬𝑰

(𝒎𝒆 + 𝟑𝟑 𝟒𝟎

𝒎) ∗ 𝒍𝟑

The inverse of the natural frequency is the period of oscillation (T). The period can be found both

experimentally through a graph and theoretically through calculations. The relationship between the

frequency and period of oscillation is

𝒇 = 𝟏

𝑻 𝒐𝒓 𝑻 =

𝟏

𝒇 (8)

Since the natural frequency was calculated, it could be compared to the period of oscillation by taking

the inverse of that value.

Methods and Materials Before the actual calculations were done, the lab had to be set up so that the data could be read. Since

most of the equipment is already connected, the only critical components to connect were the wires of

the strain gauges. A picture of the setup can be seen in Appendix A, along with the written LabView

codes. The procedure of the lab is outlined below

1) Record all information of all sensors and equipment

a. Included are dimensions of the beam and the tup (note the mass of the beam and tup

cannot be determined, however it will be later calculated from the volume and density)

2) Examine equipment to ensure safety as well as leveling of the machine.

3) Wire the strain gauge to the DAQ assistant. The wiring diagram can be found in the LabView

program under gage completion block.

a. Note – the configuration should be half-bridge

4) Write the code using Labview. Included should be maximum strain as well as a waveform graph

and chart to record the raw waveform.

a. When calibrating, un-check the box labeled “shunt calibration”

5) Open the pre-written LabView code for the accelerometer. Turn on the NI Elvis board which is

the power source for the accelerometer

a. Check to make sure the angle encoder works properly and reads the angles correctly

b. The uncertainty for the strain gauges can be found on the sheet posted on the

backboard of the machine.

6) Measure in increments of 5° from 5-120° making sure to record all trials.

a. Include a screenshot of the running code.

7) Take 10 measurements at a low angle for later calculations of the precision uncertainty.

8) Take data for the raw waveform by changing the LabView code to record a specific time range

rather than one point at impact.

a. This raw data will be later used to relate the period of oscillation (experimental) to the

natural frequency (theoretical).

9) After the completion of the lab, use the data to convert the captured strain to force using

equation 1

10) Plot in excel the force vs. the angle

a. Theoretically the graph should follow a sinusoidal trend

11) Calculate the natural frequency from equation 2

12) Calculate the period of oscillation from the raw waveform graph.

As previously stated, the screenshot of the LabView codes can be seen in the Appendix along with some

of the raw waveform graphs.

Even though there were not many components to connect for the setup, the machine contained many

little pieces which all played a role in the measurements. A table of materials is presented below

Table 2 Equipment List

Common Name

Manufacturer Model # Serial # Bias Uncertainty

Comments

Stainless Steel Ruler

Helix n/a n/a +/- 1/64 in Flexible, made in China

Caliper Lufkin W-24796 No. 701 +/- .0005 Stainless steel

Strain Gage Conditioner

National Instrument

NI-9237 0626664X +/- .0025 V Inside a NI-9162

Completion Block

National Instrument

NI-9949 196773B-01 +/- .2 A Half Bridge

Strain Gage Micro- Measurements

6835 CF550978 +/- .003 Ω One strain gage

The ruler was used for the more broad measurements such as length of the beam and of the tup. The

caliber was used for smaller measurements such as the cantilever beam width or the diameter of the

tup. As stated in the procedure, the mass of them beam and the tup could not be measured directly

since they were fastened and could not be taken off. Instead the density of the material was used along

with the dimensions.

Data and Results The lab was successfully performed with data for the impact. The strain was converted into the force

with equation 4. This force was then plotted with respect to the angle. Theoretically the data is

supposed to follow the general trend of a sinusoid. The data ranges from 5-120° +/- 1° with several

points at each angle. Figure 2 is the graph so the general trend can be seen along with a graph of the

theoretically calculated Force for every angle.

Figure 2 Graph of calculated impact force versus the angle

Figure 3 Graoh of theoretical force versus the angle

The natural frequency was calculated as a second step from equation 5 with dimensions in Hertz (Hz).

The dimensions taken from the procedure were used to calculate this value, and the table can be seen in

the Appendix. The expectation for this value was that it would be fairly small since it is the inverse of the

period of oscillation which is generally higher. To determine the period of oscillation, the raw waveforms

had to be used, one of which can be seen below.

From this graph the period of oscillation was calculated by taking the freqnucy of the waves and

dividing it by the time. The period of oscillation is the inverse of the natural frequency and since the

natural frequency is generally smaller, the period is going to be larger. The comparison of the two can be

seen below

Period of Oscillation

(theoretical) 0.0174 Sec

Period of Oscillation

(experimental) .0166 sec

Natural Frequency (theoretical) 57.47 Hz

Natural Frequency (experimental) 60 Hz

Table 2 Comparison of Frequency and period of oscillation for experimental and theoretical data

Figure 4 Raw waveform of strain at 15 degrees

Discussion From the comparison of the natural frequency to the period of oscillation, it can be seen that there was

not a big difference between the two. This analysis is very positive since it means the data can be used,

and is a good representation of what is supposed to happen. From the graph, it was difficult to see

whether or not the data followed a sine function since the range was only to 120° rather than the full

180° that were originally stated. As a result, no real conclusion can be made. Along with the theoretical

frequency and period, the theoretical force was calculated and then plotted, just as with the actual force

calculated from the strain. When the trends were compared, there was not much difference between

the values. The only noticeable difference is that the theoretical force did not range as high as the

actual, with the actual ranging all the way to 1000 N +/- 8.74 N while the theoretical only reaching about

600N +/- 8.74 N.

Although the data was not off by very much for the natural frequency, there are some errors that did

occur during the lab which may have had an impact on the data. The first was with the raw waveform,

which affected the period calculations. The main focus of the graph was the very beginning, which was

zoomed in and manipulated so the general trend could be actually seen. There was a lot of extra noise

within the points, so it was a little more difficult to determine how many times the graph actually

completed a cycle. There is no clear explanation to this error, however one possible reason is that there

were too many times the tup hit the beam causing vibrations that collided with each other. As a result,

there was a sort of lack of certainty in the calculations. In this case, the error acted favorably in the

sense that the data was fairly close to the expected value.

As stated in the first paragraph, another error resulted from not being able to use the full range the data

would fall under. What this means is the angles were shorted, and could not fully complete the

revolution. Initially the lab called for impact from 5-180°, and then was modified to 120° to protect the

machine. The beam which held the tup was already very frail and had snapped previously, which caused

the limitation within the range of angles. Similarly, the cantilever beam that took the impact was not

firmly connected to the machine. As a result, there was no guarantee that the tup struck the beam

evenly and at the same spot every time. This inaccuracy could cause data values that are different from

those if there was an even distribution of impact on the beam.

There was one last malfunction which occurred during the data, caused by the premade LabView code

for the angle encoder. The first was that the angles measured where converted backwards. What this

means is that rather than increasing from 0-360°, the indicator in the code would go from 360-0°. In

addition, it had to be zeroed multiple times during the data. When the tup hit the beam at higher

angles, the angle encoder sensor became uncalibrated. To fix this, the code had to be stopped and then

restarted again.

Although these problems did occur, the data can still be deemed as successful since the values that were

compared a very similar.

Conclusion and Recommendations Even though there were some limitations during the lab, the overall outcome was a success due to the

similarity in natural frequency data. The theoretical was determined to be .0174 Hz and the

experimental at .0166 Hz. It was determined that at higher angles, there was a greater impact.

Theoretically, the impact should follow a sinusoidal trend however due to the limitations in equipment

this conclusion could not be drawn since there was no evidence to back up this conclusion. On the other

hand, the comparison of the natural frequency to the period of oscillation did successfully prove the

prediction that the two are the inverse of each other.

For the lab to be successful in the future there are some recommendations. The first is to make the

actual impact machine studier. Although fastened to the table, the machine was not tightly bound which

allowed it to move and shift during impact, especially the impact caused at greater angles. There were

also a couple screws that were hanging loosely that should be fastened. Similarly, it is recommended

that the cantilever beam that took the impact be fastened so that a more accurate data points could be

acquired. In addition, a more sturdy material could be used to support the swinging tup. Not only had

the beam snapped before, the fixed beam also had a slight bend to it. To be able to fulfill the full range,

the beam needs to be able to withstand the full impact associated with the greater angles.

Another recommendation is to possibly fix the accelerometer. The device had to be recalibrated many

times during the lab. Since the goal is to have the most accurate readings, the recalibration hinders this

goal and should be fixed so that it is not a factor.

Overall, the lab was a success, and with the following recommendations can be made even more

successful. Data from this lab can be used for further experimentations and converted in real life

situations.

Works Cited 1) "A Beginner's Guide to Accelerometers." Dimension Engineering: Robotics, Radio

Control, Power Electronics. Dimension Engineering Inc. Web. 2 Mar. 2016.

<http://www.dimensionengineering.com/info/accelerometers>.

2) Henderson, Tom. "Natural Frequency." The Physics Classroom. The Physics Classroom, 2016. Web. 2 Mar. 2016. <http://www.physicsclassroom.com/class/sound/Lesson-

4/Natural-Frequency>.

3) Beckwith. Mechanical Measurements. 5th ed. University of New Haven Blackboard. Web. 5 Mar. 2016. <https://blackboard.newhaven.edu/bbcswebdav/pid-728699-dt-

content-rid-3438857_1/courses/MECH-3315-02-S16/SHOCK TEST EQUIPMENT

THEORY(1).pdf>.

4) University of New Haven. "Impact Force." Blackboard. University of New Haven, 24 Sept. 2015. Web. 3 Mar. 2016.

Appendices

Appendix A – Picture of Lab Setup

Appendix B - Screenshot of LabView Code for Strain Measurements

Appendix C – Screenshot of LabView code for the Angle Encoder

Appendix D- Dimension of Beam and Tup

For Tup

Diameter (in) +/- .016

D (m) +/- .016

Length (in) +/- .017

Length (m) +/- .016

0.888 0.0225552 0.94 0.0238125

0.891 0.0226314 0.91 0.02301875

0.89 0.022606 0.94 0.0238125

0.89 0.022606 0.91 0.02301875

0.886 0.0225044 0.91 0.02301875

Average 0.0225806 Average 0.02333625

thickness (cm)

+/- .033

thickness (m)

+/- .0005

width (cm)

+/- .037

width (m)

+/- .0005

length (to SG)

+/- .0471

length (m)

+/- .0005

total length (cm)

+/- .037

total length (m)

+/- .0005

10.53 0.01053 19.46 0.01946 47.77 0.04777 109.51 0.10951

10.43 0.01043 19.45 0.01945 47.51 0.04751 109.68 0.10968

10.49 0.01049 19.22 0.01922 47.51 0.04751 109.49 0.10949

10.31 0.01031 19.39 0.01939 47.53 0.04753 109.44 0.10944

10.48 0.01048 19.39 0.01939 47.46 0.04746 109.46 0.10946

Average 0.010448 Average 0.019382 Average 0.047556 Average 0.109516

For aluminum beam

E 69000000000 N/m^2

w 0.019382 m

t 0.010448 m

L 0.047556 m

L total 0.109516 m

Volume beam 2.21773E-05 m3

mass beam 0.0598788 kg

Volume Tup 9.34527E-06 m3

Mass tup 0.075229432 kg

Moment of

Inertia 6.33939E-09 m4

Natural

Frequency 0.017440887 Hz

17.44088693 MHz

Natural Frequency Calculations

Appendix F – Raw Waveform graphs of 30 and 45 degrees

Appendix G – List of Symbols

F = Force (N) L = Length of beam (m) E = Young’s Modulus of material (Pa) w = width of beam (m) t = thickness of the beam (m) ɛ = strain (unit less) I = moment of inertia (m4) me = mass of the tub at the end of the beam (kg) m = mass of beam (kg) U2 = energy H = height V = velocity g = gravitational acceleration