1. Vibrations : Introductionhomepages.ulb.ac.be/~aderaema/dynamics/1_DOS-Vibrations_intro.pdf ·...

Dynamics of Structures 2017-2018 1.Vibrations : Introduction

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Dynamics of structures

1. Vibrations : Introduction

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Arnaud Deraemaeker ([email protected])

*Definition of vibrations and vibration mechanismDefinitionSources of vibrationsVibrations in civil engineering structuresMechanism of vibrationsFree/forced vibrationsEffects of vibrations

*A first feeling about vibrations (movies and experiments)

*Tools to describe and deal with dynamic signals :Harmonic signals representationThe discrete Fourier transform (periodic signals)The continuous Fourier transformConvolution integral and convolution theoremTheorem of Parseval

Outline of the chapter

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Definition and mechanism of vibrations

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Vibrations : definition

Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road. (from wikipedia)

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pendulum Mass on a spring


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Vibrations are all around us

Vibrator in cell phoneTools Rotating machines

Sound Shaver5

Tram

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Mechanism of vibrations

Transfer between kinetic energy and potential energy

spring

mass


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Mechanism of vibrations

Transfer between kinetic energy and potential energy

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string

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Sources of excitation

Free vibrations Forced Vibrations

Short initial excitation Continuous excitation


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Free vibrations

Initial displacement

Shock

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Harmonic force signal

Random force signal

Periodic force signal

Forced vibrations : types of input forces


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Harmonic excitation

Mobile phone vibrator 11

The signal is in the form of a sine or/and cosine function

Periodic excitation

Power generator

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The signal repeats itself


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Random excitation

•Wind•Traffic•Waves•Earthquakes

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No structure in the signal

Vibrations sources in civil engineering

EXTERNAL SOURCESSeismic activitySubway, road and rail systems, airplanesConstruction equipmentWind, Waves

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INTERNAL SOURCESVentilation systemsElevator and conveyance systemsFluid pumping equipmentMachines and generatorsAerobics and exercise rooms – human activity


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Vibrations in civil engineering structures

Traditionnally, vibrations has not been a big concern in civil engineering, except for high level vibrations due to earthquakes

But …..

-Vibration sources are increasing-Comfort demands are increasing-Health issues are appearing-In some cases, high precision technologies require very low vibration levels-New designs make some structures more susceptible to vibrations

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Civil engineering structures have evolved towards slender structures with low level of damping, where vibrations become an issue

The Millau viaductAn old arch bridge

This trend is also visible in other areas (automotive, aerospace) : reduction of weigth for optimal use of material results in higher levels of vibrations

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A first example : the Millenium bridge in London

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Tate modern musuem of Art


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Opening june 10, 2000


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Cables vibrations (wind)

A second example : Dongting cable-stayed bridge (China)

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cables


A third example : high rise buildings

Oscillatory motion due to strong winds-> Problems of safety and comfort

Tuned mass damperto reduce motion

Taipei 101 (509 m), Tapei, Taiwan20


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A fourth (catastrophic) example: Takoma Narrows bridge, USA, 1940

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Undesirable effects of vibrations

FatigueNoiseComfortHealthPerformances…(collapse)

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Positive effects of vibrations

- High frequency vibrations to decrease friction in engines (formula 1)- Electric tooth brush, sander- Musical instrument- Loudspeaker

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A first feeling about vibrations through experiments and movies

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Simple Harmonic Motion

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Mass-spring system


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Tools to describe and deal with dynamic signals

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Harmonic signals

A periodic vibration of which the amplitude can be described by a sinusoidal function:

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, is called an harmonic vibration with:

•amplitude a•angular frequency = 2 f •frequency f •period T = 1/f or f = 1/T•phase angle at t=0•total phase angle t +


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Representation in the complex plane:

Projection of the rotating vector on the real axis is a cosine

Projection of the rotating vector on the imaginary axis is a sine

Harmonic signals

Independent of time

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the phase angle of u(t) is 90° behind v(t)

the phase angle of v(t) is 90° behind a(t)

Displacement

Velocity

Acceleration

Harmonic signals


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Harmonic signals

Harmonic analysis : the Fourier transform

Let u(t) be a periodic function of period T

u(t) can be decomposed into a discrete Fourier series of the form

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Period T u(t)

t


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Period T u(t)


…

…

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t


Alternative formulation

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Amplitudes and phases

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*

Complex formulation

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Complex formulation

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Complex formulation

is complex and carries the phase and amplitude information of the nth

component of the Fourier transform

and are complex conjugates so that u(t) is real

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Complex formulation : alternative formulation

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Examples of Fourier transform of periodic signals

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Interest of frequency representation : many signals are concentrated in givenfrequency bands. The response needs to be computed only in this frequencyband

Examples of Fourier transform of periodic signals

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Non periodic signals

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Continuous Fourier transform for non-periodic signals

Continuous Fourier Transform of u(t)

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Continuous Fourier Transform

Continuous Inverse Fourier Transform

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Alternative formulation

Pulsation (rad/s) Frequency (Hz)

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Examples of continuous Fourier transform


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(definition)

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Properties of the Fourier transform

Even function = f(-t)=f(t) Odd function = f(t)=-f(-t)

k>1 compression of time axis

t0>0 time shifting to the right

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The convolution integral of two time functions x(t) and h(t) yields a newtime function y(t) defined as:

Convolution integral

Take the two functions x(t) and h(t) and replace t by the dummy variable

Mirror the function h() against the ordinate, this yields h(-)

Shift the function h(-) with a quantity t

Determine for each value of t the product of x() with h(t-)

Compute the integral of the product y(t)

let t vary from (or a value small enough to make the product zero) to (or a value of t that is big enough)


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Shift h():

1st function Mirrored 2nd function


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Property:


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Convolution in the time domain corresponds with a multiplication in the frequency domain:

The convolution theorem

Proof:

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The convolution theorem

In the same way, one can prove that :

A multiplication in the time domain is a convolution in the frequency domain

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The energy of a signal computed in the time domain equals to the energycomputed in the frequency domain:

The theorem of Parseval

Proof:

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h(t) is real

Fourier transform of

with


And f’ is the frequency variable

PropertiesF.T.


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Time domain


Frequency domain Angular frequency domain

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1. Vibrations : Introductionhomepages.ulb.ac.be/~aderaema/dynamics/1_DOS-Vibrations_intro.pdf ·...

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