Vibrations of a Cantilever Beam PDF

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This is lab 11...

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Vibrations of a Cantilever Beam Author: Austin Cole Lab Partner: Steffano Sanchez MECE 3360 Conducted: 28 May 2019 Due: 30 May 2019 Submitted: 29 May 2019

Abstract The purpose of this lab is knowing how to calculate the natural frequency and damping ratio. This is done by plotting the vibration readings against time for each weight once it is added to the beam. Plotting the natural frequency vs the mass you can then determine the effective mass and Young’s Modulus. In the end I calculated a Young’s Modulus of 61.4 GPa which is slightly below the true value of 69 GPa. Objectives •

To calculate the natural frequency and damping ratio of a beam using weights and a proximity probe.

•

To determine Youngs Modulus and the density of aluminum using this method.

Introduction Understanding how vibrations affect a material is a major part in what materials should be used for what projects. This stems from understanding Young’s Modulus for the different metals. Using certain computer programs, you can measure the vibrations these metals are giving off and design around it, that way the dynamic system works together not against its self. To be able to understand the math behind a cantilever beam you need to make a lot of assumptions as the math is very complex and a solver would be needed. To understand it in this lab we split the beam up into 2 parts: one was a beam with no mass and a weight on the end, and the second part was a beam with distributed mass and no mass on the end. Formulas being used: 𝑦 = # 𝐶% + 𝐶' 𝑒 )*+ , 𝑆𝑖𝑛(𝐶1 𝑥 + 𝐶3)

𝐸=

67

1*8

𝑚:;; = .2427𝑚@

𝑘 𝜔B = 𝜔C = D = 2𝜋𝑓 𝑚

Procedures 1. Measure the weights and the weight hanger 2. Prepare the beam by mounting it using the c-clamps to hold it steady along with the proximity probe 3. Connect the probe to the power, oscilloscope, and the computer using the cables 4. Use Scope VI to capture the signal 5. Set sample rate to 10 points per cycle and time to get the tail at 10% of the initial amplitude 6. Starting with just the beam record the data from the computer and the frequency from the oscilloscope 7. Repeat step 6 adding the hanger initially and then each mass thereafter until all have been added 8. Using excel solver determine the best fit coefficients Results RUN

WEIGHT

FREQUENCY (Hz)

0 1 2 3 4 5 6

N/A HANGER W1 W2 W3 W4 W5

73.9 25 21 18.65 16.3 14.74

Table 1 above shows the initial data collected using the oscilloscope. There were no calculations needed as it showed the Frequency directly in Hz. I will talk more about it but due to our Probe not working we were not able to collect data for the run using just the hanger. Weight Name N/A HANGER W1 W2 W3 W4 W5

Mass(g) 0 49.8 126.3 118.3 99.8 116.7 127

Mass(kg) 0 0.0498 0.1263 0.1183 0.0998 0.1167 0.127

Table 2 above shows the conversion from the mass in grams to kilograms after I measured the weights using a triple beam balance. This was done for each weight

Measurement Length T1 T2 T3 T4 T5 xp xg width

length (cm) 25.5 0.6 0.6 0.6 0.5 0.7 23.5 4.2 2.5

Length(m) 0.255 0.006 0.006 0.006 0.005 0.007 0.235 0.042 0.025

Table 3 above shows the dimensions of the Beam. This was originally recorded during the stress and strain lab. The average of the thickness was taken from T1-T5 and used for calculations.

Beam 2 1.5

Voltage(V)

1 0.5 0 0

0.5

1

1.5

2

2.5

3

-0.5 -1 -1.5 -2

Time(s) Data

C0-Zero 0.10119213

C1-Amp 1.55359118

C2-Time 1.21169383

C3-Frq 465.891807

Regressison

C4-Phase -1.282903

Figure 1 shows the data and the regression line for the Beam with no weights along with the calculated C values.

3.5

Beam and 1 mass 0.9 0.8 0.7

Voltage(V)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

Time(s) Data

Regression

C0-Zero

C1-Amp

C2-Time

C3-Frq

C4-Phase

0.72278503

-0.1075856

0.88398062

168.868147

1.52425542

Figure 2 shows the data and the regression line for the Beam with 1 weight along with the calculated C values.

3.5

Beam and 2 masses 2 1.8 1.6

Voltage(V)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5 Data

2

2.5

3

Regression

C0-Zero

C1-Amp

C2-Time

C3-Frq

C4-Phase

1.21558418

0.65783128

1.77250058

132.944754

4.38318044

Figure 3 shows the data and the regression line for the Beam with 2 weights along with the calculated C values.

3.5

Beam and 3 masses 1 0 0

0.5

1

1.5

2

2.5

3

Voltage(V)

-1 -2 -3 -4 -5

Time(s) Data

Regression

C0-Zero

C1-Amp

C2-Time

C3-Frq

C4-Phase

-2.3166148

2.39727443

2.10996178

113.683047

4.00976549

Figure 4 shows the data and the regression line for the Beam with 3 weights along with the calculated C values.

3.5

Beam and 4 masses 1 0 0

0.5

1

1.5

2

2.5

3

Voltage(V)

-1 -2 -3 -4 -5

Time(s) Data

Regression

C0-Zero

C1-Amp

C2-Time

C3-Frq

C4-Phase

-1.6896231

2.34690054

0.94879869

102.914901

2.04766877

Figure 5 shows the data and the regression line for the Beam with 4 weights along with the calculated C values.

3.5

Beam and 5 masses 3 2 1

Voltage(V)

0 0

1

2

3

4

5

6

-1 -2 -3 -4 -5

Time(s) Data

Regression

C0-Zero

C1-Amp

C2-Time

C3-Frq

C4-Phase

-1.0385295

2.91028646

0.69975073

92.4835797

-0.5371882

Figure 6 shows the data and the regression line for the Beam with 5 weights along with the calculated C values.

Weight

End Mass

Wn

Damping ratio

1/Wn^2

Stiffness

Youngs Modulus

Beam Beam and hanger W1 W2 W3 W4 W5

0

465.8918073

0.002600805

4.60712E-06

0

0.00E+00

0.0498 0.1761 0.2944 0.3942 0.5109 0.6379

168.8681474 132.9447539 113.6830466 102.9149014 92.48357975

0.005234739 0.01333261 0.018560039 0.009219255 0.007566216

3.50675E-05 5.65793E-05 7.73764E-05 9.44155E-05 0.000116915

5021.74706 5203.316155 5094.575793 5411.185558 5456.094269

6.17E+10 6.39E+10 6.26E+10 6.65E+10 6.70E+10

Table 4 above shows the end mass, Wn, Damping ratio, 1/Wn^2, Stiffness, Young’s Modulus. Starting with the end mass is the total mass that is hanging off the end of the beam. So each run a weight was added which is why it increases. Wn is the natural frequency. This was taken from

the graphs above using the C3 constant. The damping ratio was also taken from the graphs above by dividing C2 by C3. The 1/Wn^2 was calculated using the natural frequency. Stiffness was calculated by dividing the end mass by the 1/Wn^2. Young’s modulus was calculated using the following formula 𝐸 =

H67 18

Where k is the stiffness. The calculations for the beam and hanger

were not computed as the data was not able to be obtained.

1/ω^2 vs M 0.00014 y = 0.0002x + 5E-06 R² = 0.9988

0.00012

1/ω^2 (Ω)

0.0001 0.00008 0.00006 0.00004 0.00002 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M(kg)

Figure 7 above was plotted using the natural frequency and the mass. This gave us a linear regression line with an R squared value of 0.9988 which Is very close to 1. The equation for the line of best fit tells us a few things. The slope 0.0002 is equal to 1/K so the stiffness is 5000. This can be used to calculate the Youngs Modulus. The y intercept is used to calculate the effective mass. Meff 0.028088744

Stiffness(k) 5000

Moment of Inertia 4.50E-10

Length 0.255

Youngs Modulus 6.14E+10

Volume 0.00003825

Mass of beam 0.115734422

Table 5 above shows the calculations using the graph above. The effective mass was calculated by multiplying the stiffness by the y intercept. The stiffness was calculated using the slopes inverse. The moment of inertia is from the previous lab as well as the length. The volume was

Density 2693.02

calculated using L*W*H and the mass is equal to the Mass effective divided by 0.2427. the Youngs modulus was calculated using the formula 𝐸 = 1*8 C being the slope. 67

Weight

End Mass

Mb+Mass

mass effective

Density

Average Density

Beam Beam and hanger W1 W2 W3 W4 W5

0 0.0498 0.1761 0.2944 0.3942 0.5109 0.6379

0.115734422 0.165534422 0.291834422 0.410134422 0.509934422 0.626634422 0.753634422

0.028123464 0.040224864 0.070915764 0.099662664 0.123914064 0.152272164 0.183133164

3025.736515 4327.6973 7629.658084 10722.46854 13331.61887 16382.59926 19702.8607

10731.80561

Table 6 above contains the end mass from a previous table. The total mass was calculated by summing the end mass with the mass of the beam from table 5. The effective mass is the end mass multiplied by 0.2427. The density was calculated by the total mass by volume for each weight. The table also contains the average of the densities. Conclusion/Discussion Following all the calculations it is determined that the Young’s Modulus is 61.4 meaning it is below the expected value by an 11% error. This is small though compared to all the assumptions made during this lab. During the lab we had major issues with the computer and the probe not reading or responding to the vibrations. Therefore, it was ignored and not calculated as we were told to ignore it. If we were able to calculate using that data, if it were correct, our results may have been more accurate. The calculated error will be shown below for a 95% confidence interval. The natural frequency is proportionate to the amount of weight the more weight the smaller the natural frequency got meaning the larger the inverse squared became. This lab showed us a third way to calculate Young’s Modulus though it was the least accurate way it still was close to 69 GPa. 2693.02 kg/m^3 was the dencity calculated the actual density is 2700 kg/m^3.

Questions:

1. What are the 95% confidence intervals for Young’s modulus and beam density for your results?

2. Is there any measurable trend in the relationship between end mass and damping ratio?

Damping Ratio vs End Mass

Damping Ratio

0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004

y = -0.0868x2 + 0.0657x + 0.001 R² = 0.6505

0.002 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

End Mass(kg)

As you can see in the graph above it does look like there is a polynomial regression line to the 2nd degree. If you increase the degree of the polynomial the R squared value moves closer to 1 but I feel as if the damping ratio increases then decreases as the mass increases as seen above with the 2nd degree polynomial. References [1]2019, "MODULUS OF ELASTICITY FOR METALS", AmesWeb [Online]. Available: https://www.amesweb.info/Materials/Modulus-of-Elasticity-Metals.aspx. [Accessed: 18- May- 2019]. [2]2019, "What Is the Density of Aluminum?", WorldAtlas [Online]. Available: https://www.worldatlas.com/articles/what-is-the-density-of-aluminum.html. [Accessed: 29- May- 2019].

Appendices (i)

All calculations are shown above....