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LAB 1 Determination of the Elastic Modulus by Wave Propagation

Summary

The purpose of this lab is to determine the elastic modulus of steel, utilizing an accelerometer to obtain

the period of time for a stress wave to propagate along a sample of known length and density.

Background on theory

o Nondestructive testing technique used to determine the elastic modulus of a material through an

impact. This impact is accompanied by a stress wave that propagates through the impacting

bodies away from the point of impact. For metals where the impact is within the elastic region,

Hooks law, Poissons ratio, Newtons second law of motion, and other basic laws of mechanics

and physics are applicable.

Glossary of terms

o Accelerometer: an electromechanical device that measures acceleration forces

o Elastic Modulus: ratio of the axis stress to strain within the limits where Hooke's law holds

o Flexural wave: a lateral bending motion perpendicular to the axis of propagation

o Longitudinal wave. A wave that propagates along an axis and causes the particles along its path

to compress and expand with the passage of the wave.

Experimental setup consisted of a steel rod 0.912m in length with and accelerometer affixed at one end.

The steel rod was placed horizontally upon two supports with minimal contact (as seen in photo below).

A hammer was used to strike the flat end of the rod opposite the accelerometer, causing a stress wave to

propagate along the steel rod and then reflect. Software converted the electrical impulses generated by

the accelerometer to produce the accompanying graphs and period intervals.

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Procedure consisted of clicking the OK Rearm Trigger button on the software display and then striking

the end of the steel rod opposite the accelerometer with a small hammer. This process was conducted a total

of six times, each occurrences corresponds to one of the below graphs rendered by the software program.

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Calculations and Graphs

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Stress wave propagation evaluated over a small element of the rod resulting in the longitudinal motion, letting

(alpha) be the displacement of the element.

Applying Newtons Second Law of Motion, Force (net) = Mass * Acceleration

= 0 ( +

) = () (

2

2)

=

2

2

Strain-Displacement equation

=

= =

=

2

2

One dimensional wave equation (combining the preceding two equations)

2

2=

2

2 (, ) = ( ) + ( + )

L= length of rod. Thus, 2L is the distance travelled by one wave pulse between reflecting on the rod end with

the accelerometer and then reflecting on the opposite end and again registering on the accelerometer.

T = period of time elapsed between wave pulses.

= =2

=

= (

2

)2

Density of steel = 7.85x103 kg/m3 (provided in the lab packet) Theoretical elastic modulus of steel E = 200 GPa (provided in the lab packet)

Length of steel rod specimen L = 0.912 m (provided in the lab packet)

Results and Discussion

+

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The period (T) between pulses was recorded for each of the six trials as:

1) 0.00038s 2) 0.000364s 3) 0.000384s

4) 0.00038s 5) 0.000388s 6) 0.00038s

The experimentally derived mean elastic modulus of the steel rod was found to be 181.717 GPa, by comparison

the theoretical elastic modulus of steel is 200 GPa

Evaluating each of the periods in the derived equation for elastic modulus and then comparing that value to the

theoretical elastic modulus obtained the following % error values. Test:

1) 9.568% 2) 1.443207% 3) 11.4421875%

4) 9.568% 5) 13.25870975% 6) 9.568%

The mean percentage error for the six tests is 9.14135%

Possible sources of error include Density of the steel sample being presumed to be 7.85x103 kg/m3, the density of steel depends on the

alloying constituents. Steel density is typically in the range of 7.750x103 kg/m

3 to 8.050x10

3 kg/m

3.

The length of the steel rod was given as 0.912 m, this was not confirmed during the experiment

The degree to which the hammer blow was in-line with the linear axis of the shaft may have resulted in non-longitudinal waves such as flexural bending.

Notes regarding the shapes of the graphs obtained Graphs 1, 2, 3, and 6 all have the same general sharp peak and U shaped troughs. Save for graph 2, the

other graphs all depict amplitudes across a broader range.

Graphs 2, 4, and 5 appear to share the trait of having their successive peaks and troughs being notably lower in amplitude in each successive wave cycle. The range of amplitudes of one wave cycle is so narrow that the

true scale is skewed. This may indicates that the accelerometer was over or under stimulated for the

sensitive level selected for this lab. It was noted during the testing that the steel rod in both graph 2 and 5

had some noticeable non-linear displacement (sideways motion was observed).

Graphs 2, 4, and 5 have unsmooth curves. This is possibly due to secondary impacts between the rod and the platform/rests it was placed on. Or as mentioned above, an issue du to the sensitivity level the equipment

was set at.

Learned from this lab Learned to verify the information provided in lab handouts, both the length and density of the steel

rod could have been readily verified during the lab. Re-try the experiment with mild-variations to better understand what and how the data is interpreted

and depicted Adjust the sensitivity level Strike harder and/or softer Adjust sampling interval Change orientation from horizontal to vertical

Recommendation Repeating the tests with a pendulum hammer could minimize the off axis impacts Rubber dampeners between the rod and supports could minimize the graph noise from secondary impacts Comparing the results on the 0.01s - 0.012s time interval with a later interval such as 0.06s 0.062s, may

reveal additional insights