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BrazilProblem 1 – Invent Yourself

Team of BrazilBRAZILIYPT 2018

Problem 1

Invent YourselfReporter: Victor Cortez

Construct a simple seismograph that amplifies a local disturbance by mechanical, optical or electrical methods. Determine the typical response curve of your device and investigate the parameters of the damping constant. What is the maximum amplification that you can achieve?

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BrazilProblem 1 – Invent Yourself

CONTENTS

BrazilProblem 1 – Invent Yourself

1. Theoretical Introduction

Basic Concepts

The Seismograph

Theoretical Model

2. Experiments

Experimental Materials

Experimental Set-up

Experiments

3. Conclusion

Summary

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

Vibrations Amplification

Mechanical

Optical

Electrical

3

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

The Seismograph

D

d

Amplification Optical

ScreenLaser

Figure 1: Laser scheme.

4

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

D

d

Mirror

Laser

Screen

Figure 2: Scheme of an optical system with multiple mirrors.

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The Seismograph

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

D

d

MirrorLaser

AProjection

D

Screen

Mirror

Virtual Laser Image

Figure 3: Scheme of the optics of the problem.

6

The Seismograph

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

F

Young’s Modulus

Figure 4: Damped mass-spring system.

7

Theoretical Model

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model 8

Theoretical Model:

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

Overdamped

Critically damped

UnderdampedTime

Po

siti

on

Underdamped

Critically Damped

Overdamped

9

Theoretical Model

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

Figure 5: Function x(t) using the indicated parameters.

10

Theoretical ModelA

mp

litu

de

(u.a

.)

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

General case:

11

Theoretical Model

BrazilProblem 1 – Invent Yourself

Introduction The Seismograph Theoretical Model

Damping Sensibility Response

12

Theoretical Model

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

1 - Well polished mirrors held by stable supports;

2 - Powerful green laser beam;

3 - Stable supports;

4 - High frame rate camera (240 fps);

5 - White screen;

6 - Computer for data analysis.

Figure 6: Mirrors held by stable supports.

Figure 7: Seismograph.

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Experimental Materials

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Analyze dataCause

perturbationsFilm the experiment

Place the cameraAlign the beamPlace the Mirrors

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Experimental Procedure

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Processing

Figure 9: Laser projection over the screen.

Figure 8: Scheme for experimental set-up.

Mirrors

Screen

CameraLaser

15Figure 10: Typical signal.

Experimental Procedure

Time (s)

Rel

ativ

e p

osi

tio

n

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Luminosity scale

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Experimental Procedure: Data Analysis

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Figure 11: Seismograph response to random perturbations.

Time (s)

Rel

ativ

e p

osi

tio

n

Random vibrations captured through the Y axis.

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Experiment 1: Random vibrations

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Controlled Vibration

C

2 Kg weight

Concrete table

h = 0.1m

Laser

D = 0.8 m

Measuring parameters

Stable support

Figure 12: Experimental scheme.

2 Kg

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Experiment 2: Controlled vibrations

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Figure 13: Response curve to a controlled vibration.

Controlled vibrations captured through the X axis

Time (s)

Rel

ativ

e p

osi

tio

n

19

Experiment 2: Controlled vibrations

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Experimental

Theoretical

Rel

ativ

e p

osi

tio

n o

n t

he

X a

xis

Time (s)

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P wave:

S wave:

Experiment 2: Controlled vibrations

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Controlled distance

C5 Kg Sandbag

Concrete table

h = 0.3m

Laser

D

Measurement of the decay

Stable support

Figure 14: Experiment scheme.

h = 0.5m

h = 0.7m

D = 1.00 mD = 2.00 mD = 3.00 mD = 4.00 m

Laser trajectory: 39.03 m

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Experiment 3: Controlled distances

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Specific vibrations

C

ExperimentalTheoretical

Rel

ativ

e am

pli

tud

e o

n t

he

X a

xis

Distance (m)

Figure 15: Computational fitting for the experimental plot from 50cm launches.

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Experiment 3: Controlled distances

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Laser

DC Motor

Frequency Generator Oscilloscope

Camera

Screen

C DFT

Response curve

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Experiment 4: Response curve

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Frequency (Hz)

No

rmal

ized

res

po

nse

am

pli

tud

e

24

Experiment 4: Response curve

BrazilProblem 1 – Invent Yourself

Experimental Materials Experimental Procedure Experiments

Relative amplitude as a function of the laser’s physical path.

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Experiment 4: Controlled vibrations

[5]

BrazilProblem 1 – Invent Yourself

Dd

MirrorLaser

Screen

Underdamped

Critically Damped

Overdamped

P wave

S wave

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Summary: Theory

Time

Po

siti

on

BrazilProblem 1 – Invent Yourself

ExperimentalTheoretical

Rel

ativ

e p

osi

tio

n o

n X

axi

s

Time (s)

Experimental

Theoretical

Rel

ativ

e am

pli

tud

e o

n t

he

X a

xis

Distance (m)

Mirrors

Screen

CameraLaser

Frequency (Hz)

Response Curve

No

rmal

ized

res

po

nse

am

pli

tud

e

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Summary: Experiment

BrazilProblem 1 – Invent Yourself

Bibliography

[1] Kim, Dong-Soo, and Jin-Sun Lee. “Propagation and Attenuation Characteristics of Various Ground Vibrations.” Soil Dynamics and Earthquake Engineering, vol. 19, no. 2, 2000, pp. 115–126, doi:10.1016/s0267-7261(00)00002-6.

[2] G. E. General Seismology. Lecture. Retrieved March 26, 2018, from https://www.youtube.com/watch?v=2wiV0Ow5oT0&t=1684s.

[3] H. Moysés Nussenzveig, Curso de Física Básica, vol. 2, Editora Edgard Blücher, LTDA (1999).

[4] Gutowski, T. G., & Dym, C. L. (1976). Propagation of ground vibration: A review. Journal of Sound and Vibration, 49(2), 179–193. https://doi.org/10.1016/0022-460x(76)90495-8

[5] Novotny, Oldrich(1999). Seismic Surface Waves, Lecture notes for post-graduate students [Pdf File]. Retrieved from http://geo.mff.cuni.cz/vyuka/Novotny-SeismicSurfaceWaves-ocr.pdf

[6] Romney, C. (1959). Amplitudes of seismic body waves from underground nuclear explosions. Journal of Geophysical Research, 64(10), 1489–1498. https://doi.org/10.1029/jz064i010p01489

Figure: Oleg Alexandrov, 2007

BrazilProblem 1 – Invent Yourself

Thank you!

BrazilProblem 1 – Invent Yourself

Appendix: Experiment 3 - Controlled distances

C C

C

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