University of New HavenDepartment of Mechanical Engineering

ME 315March 24, 2005

Shock Test Analysis Report

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Author: Franco PezzaPartners: Mudar Al-Bayat, Tareq Al-Saflan, Derek Shoemaker

Table of Content

Abstract 3

Introduction 3

Theory and Analysis 4

Equipment 5

Procedure 7

Results and Discussions 7

Conclusions 11

Recommendations 11

Reference 11

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Abstract

Correlating dynamic testing with simple beam deflection and strain measurements was accomplished by using a rotating hammer that produced a measurable deformation in a cantilevered rectangular aluminum beam. The deformation was then used to find the potential and kinetic energies of the falling hammer.

The maximum impact strain was 1876 -strain that occurred at 20o from vertical. At that angle, the maximum calculated impact strain was 2291-strain. The time of impact was determined to be 0.023 s, while the natural frequency of oscillation was found to be 76.92 Hz. The energy loss of the system had a constant value of 7.89%. The maximum stress due to the shock test that acted on the beam was 18760.69 Psi, which is 40.43% of the yield stress for Aluminum.

Introduction

Shock tests are used to make sure that a design can manage transient vibrations that may occur during its operational use. The shock test determines how much stress, strain and deflection actually develops in the test specimen which may then be used to modify the design. An example of a design that would undergo a shock test would be an automobile’s bumper. The manufacturer would adjust the design of the bumper so that it may withstand impacts up to a certain speed.

(Fig.1: Schematic of the shock test system)

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In figure 1 is represented the apparatus used for the experiment in which a rotating hammer hits a cantilevered rectangular beam of aluminum, creating deformation. A string gage measures the deformation created and the data obtained is used for calculations.

Theory & Analysis

Shock tests are designed to study direct impact loading at a short period of time. The load is characterized as transient and should produce decaying motion due to the damping characteristics of the specimen involved.

Shock testing produces damages which levels can be divided into low-energy and high-energy, depending on the acceleration produced and the length of the time interval. Figure 2 shows the categories of the mechanical shock. The low-energy shock test may result in low velocities whereas the high-energy shock test results in high velocities.

(Figure 2: Characteristics of Mechanical Shock)

Using the conservation of energy and momentum theories, the velocity of the

specimen and the loading device can be calculated before and after contact.

Conservation of Energy:

Conservation of Momentum:

where V1 is the initial velocity, V2 is the velocity after impact, m1 is the mass of the hammer and m2 is the mass of the rubber stop.

Also, using the Principle of Work and Energy, the energy absorbed by the beam can be found.

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where U2 is the energy of the system after impact. The value of U2 can then be used to find the maximum force applied to the

specimen.

where Fmax is the maximum force applied, E is the modules of elasticity, I is the moment of inertia of the beam, and L is the length of the beam.

Having found the maximum force, the strain can be found using the Flexure formula because the specimen is a cantilever beam. The maximum strain will then be used to find the maximum stress using Hooke’s Law. Flexure Formula:

Hooke’s Law:

where M is the bending moment due to Fmax and c is the distance from the neutral axis to the surface of the beam.

Using the data obtained from the testing, the period of free vibration is found by simply reading the time between two consecutive peaks. The reciprocal of the period of free vibration determines the natural frequency of the beam, fn

Equipment

The equipment used for the experiment was the following:

1. Custom built shock test setup (Figure 3).

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(Figure 3: Shock Test System)

2. Computer with DAQ and LabVIEW 7 (Figure 4).

(Figure 4: LabVIEW Scheme)

3. Vishay strain gage conditioner (Figure 5).

(Figure 5: Vishay Strain Gage Conditioner)

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4. NI ELVIS Interface (Figure 6).

(Figure 6: NI ELVIS Interface)

5. Micrometer caliper and digital weight scale

Procedure

The conditioner was first calibrated following steps 4.9 thru 4.13 of the lab manual. Next, the NI ELVIS interface was wired by connecting the BNC plug from the strain gage conditioner to one of the connectors on the NI ELVIS.

The breadboard was wired to connect the BNC to one of the six available analog in/out channels. Excessive noise was eliminated by using the differential voltage measurements. The dimensions of the beam as well as the masses of the plastic disk and hammer were taken.

The procedure for the shock test itself was as follows:

1. The shock test machine was setup and the leveling screws were adjusted2. A LabVIEW program was created to collect the data and store it to an

Excel file3. Three samples were taken for angles of 5, 8, 10, 12, 15, 18, and 20

degrees.

Results

Using the Flexure equation, the maximum value for stress was found to be 18760.69 +/- 40.9 psi at an angle of 20o +/- 0.5o. This value is 40.43% of the yield stress for Al 6061. The values for strain ranged from 506 +/- 38.38 -strain at 5o +/- 0.5o to 1876 +/- 4.09 -strain at 20o +/- 0.5o.

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The energy loss in the system was determined about 28.5%. The natural frequency of the beam was found to be 76.92 Hz and the period of the free beam was read as 0.013 s. The time of contact between the hammer and the beam was determined to be 0.023 s.

AngleTrial #

1Trial #

2Trial #

3 AverageStrain

Measureddegrees Volts Volts Volts Volts µ-strain

5 1.00 1.04 1.11 1.05 5068 1.55 1.60 1.65 1.60 774

10 2.01 2.00 2.05 2.02 97512 2.48 2.39 2.41 2.43 117315 2.98 3.00 2.99 2.99 144518 3.49 3.42 3.51 3.47 167820 3.89 3.88 3.89 3.88 1876

(Table 1:Recorded Data)

Max Stress = 18760.69 psi

y = 46.4 Kpsi

% Yield = 40.43 %

T = 0.013 s

f = 76.92 Hz

k = 149.9 lbf/in

tc = 0.0231 s

Angle H V1 V2 U2 Fmax M2

Strain Calculated Strain Based

degrees inches in/s in/s in.lbf lbf in.lbf µ-strain Angle (deg)5 0.05 6.00 5.53 0.08 5.02 16.95 575 4.408 0.12 9.59 8.84 0.22 8.03 27.11 920 6.73

10 0.19 11.99 11.04 0.34 10.04 33.87 1150 8.4812 0.27 14.38 13.24 0.48 12.04 40.62 1379 10.2115 0.42 17.95 16.53 0.75 15.03 50.73 1722 12.5918 0.60 21.51 19.82 1.08 18.01 60.80 2064 14.6320 0.74 23.88 22.00 1.33 20.00 67.49 2291 16.38

(Table 2: Calculated Results)

AngleStrain

MeasuredStrain

CalculatedStrain

Difference U U

Degrees µ-strain µ-strain % -strain Degrees5 506 575 12.04 +/- 38.38 +/- 0.58 774 920 15.93 +/- 35.38 +/- 0.5

10 975 1150 15.21 +/- 18.16 +/- 0.512 1173 1379 14.93 +/- 33.44 +/- 0.5

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15 1445 1722 16.08 +/- 8.90 +/- 0.518 1678 2064 18.70 +/- 32.11 +/- 0.520 1876 2291 18.11 +/- 4.09 +/- 0.5

Average Difference = 15.86%

(Table 3: Uncertainty)

Angle Fmax U1 U2 Energy Lossdegrees lbf in.lbf in.lbf in.lbf

5 1.50 0.09 0.07 0.0178 2.29 0.23 0.17 0.05910 2.88 0.36 0.28 0.08812 3.46 0.52 0.40 0.12415 4.27 0.82 0.61 0.21018 4.96 1.18 0.82 0.35620 5.54 1.45 1.02 0.424

(Table 4: Energy Loss)

Discussion

From a typical graph obtained from the data recorded for each of the trials, we were able to make few considerations and observations:

The maximum impact strain was observed to be proportional to the impact energy (see graph 1).

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Strain Vs. Angle

0

500

1000

1500

2000

2500

0 5 10 15 20 25

Angle (Degrees)

Str

ain

( -

stra

in)

Strain Measured µ-strainStrain Calculated µ-strain

(Graph 1: Strain vs. Initial Angle)

The impact energy of the hammer was also proportional to the deformation of the beam itself (see table 2).

The contact time of the hammer was proportional to the initial angle and was measured being 0.013 seconds.

The energy loss between the potential initial energy to the energy after impact was proportional to the initial angle (see table 4), and the percent energy loss respect the initial potential energy U1= mgh was almost 28.5% (see Graph 2).

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Final Energy Vs. Initial Energy

y = 0.7149x

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

Initial Energy (in.lbf)

Fin

al E

ner

gy

(in

.lb

f)

(Graph 2. Energy Loss Graph)

The natural frequency of oscillation of the beam was observed to be 76.92 Hz. The damping effect of the beam vibration was observed by the smaller amplitude

over time (excluding the first impact peak) with a damping constant Lambda value of about 2.49 (see graph 3).

Damping Curve

y = 1.1588e-2.4947x

-2

-1

0

1

2

3

4

5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time (seconds)

Am

pli

tud

e (

sc

ala

r)

(Graph 3: Typical response curve obtained from the experiment)

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The value of the strain obtained from the experimental data is out of the acceptable range calculated with uncertainties. Few considerations may be made on the fact that we did not take into consideration several factors such as:

The energy loss due to the impact caused by not perfectly elastic materials. In fact, some energy is absorbed by the bodies as plastic and thermal energy.

The friction due to the bearing of the hammer’s attachment, air resistance and the parallax error in leveling the unit and reading the initial angle.

The error due to materials specification data utilized for the calculations and the approximation in the significant figures utilized.

Conclusion

In this experiment, we were able to verify the relationship of a dynamic impact of a body and the relative deformation in the beam which absorb the kinetic energy. We were able to measure the strain of the beam, and the time of contact. The relationship between the initial angle and the energy absorbed by the beam was also verified.

The maximum value for stress was found to be 18760.69 +/- 40.9 psi at an angle of 20o +/- 0.5o. This value is 40.43% of the yield stress for Al 6061. The values for strain ranged from 506 +/- 38.38 -strain at 5o +/- 0.5o to 1876 +/- 4.09 -strain at 20o +/- 0.5o.

The energy loss in the system was determined about 28.5%. The natural frequency of the beam was found to be 76.92 Hz and the period of the

free beam was read as 0.013 s. The time of contact between the hammer and the beam was determined to be

0.023 s.

Recommendations

In this experiment we were able to obtain and record data which was analyzed and discussed above. In addition, it will be interesting also to analyze the behavior of different beam materials and the relationship of them with the impact hammer. Another recommendation is the opportunity to observe and analyze the damping effect of the beam.

References

Mechanical measurements 5th edition, Bechwith-Marangoni- Lienard. –Addison-Wesley Publishing Company, 1995

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