Stress Wave Propagation Lab Report

David Browne S0900461 NT090461M Applied Mechanics ME4211 Lab Report

Objectives The objectives of these experiments are to investigate wave propagation. The first experiment will investigate wave velocity, the relationship between wave velocity and impact force and the behavior of wave reflection as it reaches a free end boundary. The second experiment will investigate the stress wave that occurs due to an impact between bars. The third experiment will investigate the behavior of a wave passing through an impedance change boundary.

Equipment & Method The first experiment will be conducted by creating a compressive impulse wave on a thin rod with two free ends by hitting one end with a hammer. The second experiment will be conducted by projecting one rod into a second rod. The initial velocity of the first rod will be recorded by recording the time it takes to pass 2 laser beams across its path. The third experiment will use a ‘striker bar’ to impact the end of a second bar made up of 2 parts with different impedances. For the experiment is desirable to observe each stage of the stress wave’s propagation individually. For this reason the strain gage is placed in the middle of each section of the bar. The striker bar is then required to be less than half the length of each section of the bar so that the length of each stress wave is shorter than double the distance between the strain gage and a boundary. This allows the strain gage to record the whole incident wave before it records the reflected wave. In each experiment the stress/strain with respect to time will be recorded at a set of specific locations along the rods. The strain will be calculated by placing a Strain Gage at each location which will send a voltage versus time signal to an Oscilloscope as it experiences a strain.

Results Experiment 1 L=57.5cm Free end 1

Free end 2

Impact Gage A

Gage B

T2= 120

2.00E-04 1.50E-04 1.00E-04

T1= 110

Strain

5.00E-05 0.00E+00 -5.00E-05

0

200

Diagram 1 – Apparatus for experiment 1

Strain signal from Gage A Strain signal from Gage B

Graph 1 – Strain vs Time result for experiment 1

T3= 110 400

600

800

1000

-1.00E-04 -1.50E-04 -2.00E-04

Time (Micro second)

Analysis of Experiment 1 The Strain vs Time graph shows a compressive sinusoidal wave caused by the initial impulse passing through Gage A at around 200μs and then passing through Gage B at around 310μs. Once the compressive wave has passed Gage B it reaches ‘free end 2’ of the rod. The free end of the rod can be considered to have zero impedance and therefore the wave reflects back as a tensile sinusoidal wave. This tensile wave is then seen to pass Gage B at 550μs and then Gage A at around 670μs. Finally, after the wave passes Gage A it reaches ‘free end 1’ of the rod again, where as previously before, it is reflected back with an opposite sign as a compressive sinusoidal wave. The compressive wave is then seen to pass Gage A at around 750μs and then Gage B at around 860μs. The experimental wave velocity (C) can be calculated using the time (T) taken for the wave to pass through a distance (L) between Gage A and Gage B; 

=

… (1)

T is calculated from the average of the time taken for the wave to pass between Gage A and Gage B on the 3 separate occasions in the experiment as described previously =

   

=

110+120+110 3

= 113.3

.∗

 = .∗ = 5075 

… (2) … (3)

The theoretical wave velocity in a rod can be calculated through its relationship to the rod material elastic modulus (E) and density (ρ) (Appendix 1); 

= =

∗! "

= 5063.7 −1

… (4)

Experiment 2

V=5.084−1

Rod 1

Diagram 2 – Apparatus for experiment 2

Rod 2 Gage A

L1=40cm

Calculated stress at Gage A

80.0 60.0

Graph 2 – Stress vs Time result for experiment 2

Stress (Mpa)

40.0

T1 =145

20.0 0.0

-20.0 0 -40.0 -60.0

100

T2=305

200

300

400

500

600

-GH = 60

-80.0 Time (Micro second)

Analysis of experiment 2 The Stress vs Time graph shows a compressive square stress (% ) wave of magnitude -60MPa, caused by the impact of Rod 1 into Rod 2, beginning to passing Gage A at around 150μs until around 300μs. The impedance of each Rod can be calculated from its cross sectional area (A) and its materials density (ρ) and elastic modulus (E); & = '( = ()'* 

& =

+∗, .∗- .

& =

+∗,.3∗- .

/



/

… (5) ∗ √2700 ∗ 69 ∗ 103 = 1729.03

… (6)

∗ √7800 ∗ 200 ∗ 103 = 4392.85

… (7)

The theoretical stress in Rod 2 (% ) can be calculated by considering the impedance of Rod 1 and Rod 2 together with the difference in initial velocity of Rod 1 (6 ) and Rod 2 (6 ) (Appendix 2); 8 8

(;;9 ) < 

% = 8 9 8 9

1729.03∗4392.85

= 1729.03+4392.85 ∗

(0−5.084) =11.9∗10−3 >

+∗

2

= −56.72@AB

… (8)

4

The experimental duration of the stress wave can be read off the stress-time graph;  = 2 − 1 = 305 − 145 = 160

… (9)

The theoretical duration of the stress wave can be calculated by considering how long the two bars are in contact during their impact; =

9 C9

=

9 D

E9 9

=

2∗40∗10−2 69∗10

9

2700

= 150

… (10)

Experiment 3

Rod 2a/2b boundary

V

Diagram 3 – Apparatus for experiment 3 Rod 1

Rod 2a

Rod 2b Gage A

Gage B

Calculated Stress at Gage A Calculated Stress at Gage B

30.0 20.0 10.0

Graph 3 – Stress vs Time result for experiment 3

Stress (Mpa)

0.0 -10.0 0 -20.0

200

% T = -20

400

600

800

1000

% R=-27.5

-30.0 -40.0

% I = -55

-50.0 -60.0 -70.0

Time (Micro second)

Analysis of Experiment 3 The Stress vs Time graph shows the incident compressive square stress wave (%8 ) of magnitude -55MPa passing through Gage A at around 200μs. The wave then continues to the Rod 2a/2b boundary. For help in reading the rest of the Stress vs Time graph it is first useful to compute the expected sign of the reflected stress wave (%I ) that occurs at the rod 2a/2b boundary; 8 8

%I =  9 %8 ∴ if & > & the reflected stress wave will be same sign as the incident wave 8 8 

9

… (11)

From (5); & = 4392.8 & & = 16886.9 (Appendix 3) & > & ∴ Reflected stress wave will be compressive. It can then be seen that a compressive stress wave of magnitude -27.5MPa is reflected at the Rod 2a/2b boundary back towards Gage A. At the same time a compressive stress wave of magnitude -20MPa is transmitted through the Rod 2a/2b boundary towards Gage B. %8 = −55@AB %I = −27.5@AB = 0.5 %8 % = −20@AB = 0.36 %8

The theoretical incident, reflected and transmitted stress can be calculated using the equations; % =

<9  M %8 8 89

%I =

8 89 % 8 89 8

= 0.34%8

= 0.59%8

… (12) … (13)

Evaluation In all 3 experiments the experimental results had some differences with the expected theoretical result. One reason for this is the experimental ‘noise’ on the stress/strain vs time graphs. This noise means that the exact instant when a stress/strain wave passes a gage is distorted, making it difficult to precisely read the duration of a wave from the graph. Experiment 1 yielded an experimental value for wave speed very similar to the theoretical value;

∗(N.) ∗ 100  N.

= 0.22% difference

… (14)

Experiment 2 also yielded an experimental value close to the theoretical value for the magnitude of stress;

∗(NN. ) ∗ N N.

100 = 5.62% difference

… (15)

Experiment 3 gave the largest difference between experimental and theoretical values. This could be due to the greater potential for human error in reading the graphs as the reflected stress wave had an appearance closer to a quarter sinusoidal cycle as opposed to the square incident stress wave. This made it more difficult to choose a definite magnitude to the wave. 0.59−0.5

2 ∗ 0.59+0.5 ∗ 100 = 16.5% difference

… (16)

Appendix 1 Steel bar:

Modulus Density= Diameter Distance between gage A and gauge B:

200 7800 11.90 57.5

GPa Kg/m3 mm cm

BAR 2: Steel

Modulus Density Diameter

200 Gpa 7800 Kg/m3 11.9 mm

BAR 1: Aluminum

Modulus Density Diameter Length Velocity

Appendix 2

69 2700 12.7 40 5.084746

Gpa Kg/m3 mm cm m/s

Appendix 3 BAR 1 : Steel

BAR 2 : Copper

Modulus Density Diameter Distance between gage A and steel/copper bars interface Length of striker bar

200 Gpa 7800 Kg/m3 11.9 mm

Modulus Density Diameter Distance between gage B and steel/copper bar interface

120 Gpa 8900 Kg/m3 25.65 mm

67.5 Cm 60.5 Cm

64 Cm