Ch1-Fundamental of Vibration

Mechanical Vibrations

Chapter 1: Fundamental of Vibration

Dr. Azma Putra

Faculty of Mechanical Engineering

Department of Structure and Materials

Semester I/2014-2015

Lecture Note, Mechanical Vibrations, Semester I/2014-2015

2

What you will get from this course

Nothing

Something

Lots of thing


3

Introduction

Most engineering structures vibrate:

machines, cars, aircraft, etc

Materials becoming lighter, more flexible, engines becoming faster

need to model, design, analyze, understand, treat

Vibration is usually (relatively) small, oscillatory motion about a static equilibrium position


4

Effect of vibration: Structural failure

Fatigue

*Courtesy of Mobius ILearn interactive

Machine failure


5

Effect of vibration: Structural failure

Fatigue in pipeline

Case study:

PETRONAS Malaysia LNG, Bintulu, Sarawak

Cracked pipe, one module had to be shut down

Loss RM25 million/day


6

Effect of vibration: Noise

Piling Railway


7

Effect of vibration: Instabilities

Flutter


8

Effect of vibration: Health

White finger: Hand-arm vibration syndrome


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What causes vibration?

Imperfection in machine or structure

Design

Manufacturing defect

Installation

Assembly

Operations

Maintenance


10

How to quantify the vibration?

1. Measurement and/or Simulation

2. Analysis: frequency, amplitude and phase


11

Where to apply the knowledge?

1. Vibration isolation


12

2. Aircraft – Ground vibration test


13

3. Structural vibration

Noise, Vibration and Harshness (NVH)


14

Which one requires more treatment to control vibration?

Comfort - Customer demand


15

4. Maintenance/Vibration monitoring

Rotating machinery Plant piping system: oil and gas

Most industrial problems !!


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5. Building design

Millenium Bridge – London (2000)


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Most vibrations are undesirable, but there are many instances where vibrations are useful:

Tooth cleaning

Massage chair

Music instrument

Heartbeat

Vibration energy harvesting

VIBRATION: CAN IT BE FRIEND?


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Terminologies and Definitions

Mass - store of kinetic energy

Stiffness - store of potential (strain) energy

Damping - dissipate energy

Force - provide energy

Ingredients in vibration:

Can you identify mass, stiffness and damping from a piece sheet of a panel?


19

Most contents of this course assume that the force and the resulting motion are time harmonic

i.e. a periodic motion that repeats at regular interval.


20

Three important results from vibration measurement:

1. Amplitude - HOW MUCH

2. Frequency - HOW FAST

3. Phase - HOW IT IS VIBRATING

*Courtesy of Mobius ILearn interactive


21

What is amplitude?

Amplitude is the level of vibration from the ‘equilibrium position’.

It can be displacement, velocity or acceleration.

maxy A

( )oy t t B

maximum amplitude:

instantaneous amplitude:


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Am

pli

tud

e

Time Peak-Peak

Peak RMS Average

Amplitude: Time signal Desciptor

2

0

1( )

T

x t dtT

x(t)

RMS

0

1| ( ) |

T

x t dtT

Average

T


23

What is frequency?

Frequency is the number of ‘cycles’ repeated per second.

The unit is Heartz (Hz)

1f

T

T is the period,

i.e. the time required to complete one cycle


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Low frequency vs. High frequency

Which light has the highest flashing frequency?

A

C

B

Which wave has the highest frequency?


25

What is phase?

Blue curve leads the green curve by π/2 radians


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Example of a sinusoidal signal

General expression of a sinusoidal signal: ( ) sin( )x t A ωt φ

( ) 0.2sin(150 3.14)x t t

Amplitude (peak value) = 0.2 m

Frequency = 150 rad/s or 23.9 Hz

Peak-peak = 0.4 m

Phase = 3.14 rad or 180o

Period = 0.04 s

Average = 0.1 m

RMS = 0.14 m


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Complex Exponential Notation

cos sinjωte ωt j ωt

Write time harmonic quantity as ( ) jωtx t Xe

amplitude (usually complex)

frequency

where

However in ‘real world’ we see Re x(t)

( ) cos( )x t X ωt φ ( ) jωtx t XeFor short, we just write

phase magnitude (absolute, real)


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Suppose X a jb

2 2X a b

Show that:

jφX X e

(1).

(2).

(3). ( ) cos( )x t X ωt φ

| |X = max. magnitude 1tanb

φa

= phase

HOW?


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a. A time harmonic motion at 1 kHz has a peak acceleration of 100g (1 g = 9.8 ms-2). What is the peak displacement? (Ans. 0.0248 mm)

b. A time harmonic force f(t) = (4 + j3)ejωt produces a response x(t) = (2 - j)ejωt. Calculate the magnitude and the phase of the force. Calculate the magnitude and the phase of the displacement. Find the velocity per unit force magnitude, assuming ω = 5. (Ans. 5 N, 0.644 rad, 2.24 m, -0.464, 0.447 N/m, 2.24 m/s/N)

Problem examples


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jeXX = |X|cos (ϕ) + j|X|sin (ϕ)


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Makes life easy but introduce complex numbers

Summary of Complex Exponential Notation

tωjXetx )(

Used widely in frequency response methods

Implicitly carries phase information


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Vibration units: Definitions

Displacement:

The distance travel by the mass, how far up and down it is moving

Velocity:

Rate change of displacement. For example, how far it can cover in one second

Acceleration:

Rate change of velocity. How quickly the mass is speeding up or slowing down


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Converting vibration units

( ) jωtx t Xe

Displacement:

Velocity:

( )( )

dx tv t x

dt

Acceleration: 2

2

( ) ( )( )

dv t d x ta t x

dt dt

*Note phase difference between them

Velocity leads displacement by 90o

Velocity lags acceleration by 90o

jωtjωXe

2 jωtω Xe

Figs%5Cshm_wav3%5Cshm_3waveforms.html


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( ) jωtx t Xe

( ) jωtv t jωXe

2( ) ( ) jωta t j j ω Xe

Lead

Lag

Note on the imaginary sign


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90o

270o

180o90

o

270o

180o90

o

270o

180o ωt

Displacement

Velocity

Acceleration


36

displacement probe

B

C

A

D

Maximum

displacement

If the sensor measures maximum displacement at point A, at which location is the maximum acceleration?

The maximum velocity?


37

Free vibration - no external forces act on the system

Forced vibration - forces act on the system

Damped/undamped system - damping does/doesn’t exist

What is the phase difference between the two systems?


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FREE VIBRATION

( )x t

System vibrates at its natural frequency

Natural frequency

( ) sin( )nx t A ω t


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FORCED VIBRATION

( )x t

System vibrates at its forcing frequency

Forcing frequency

( ) sin( )fx t A ω t

( )F t


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We deal with only linear system.

same frequency as input

magnitude change

output proportional

to input

superposition holds

all components linear linear vibration (idealisation)


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Modeling

Degree of Freedom (DOF): number of independent coordinates to describe the motion.

Coordinates may be:

displacement of some points

rotation

other (modal amplitudes, waves, etc)

Number depends on:

how complex the system is

modeling simplifications + assumptions

(how we choose to model it)


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Example:

Modelling vibration of a motor cycle

suspension design (1 DOF)

Bounce +wheel hop (3 DOF)

Effect of motorcycle vibration to rider comfort (4 DOF) Simplification (2 DOF)


43

Could you try to model this system?

Missile carrier: isolating the vibration from the ground

Mass of missile

Mass of truck

Suspension

Isolator

road input


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Stiffness and Flexibility

Stiffness arises from any structural component that deforms (elastically)

under action of forces.

F kx

x : deformation (m) k : spring constant (N/m)


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Examples

vibration mounting

Engineering structures

?


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Spring in series

Carry same force !!

1 ,F k y 2( )F k x y 21

Fk x F

k

1 2

1 1 1

eq

x

k F k k

Extension = sum of individual extension


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Spring in parallel

Have same extension !!

Total force = sum of forces in each spring

1 1F k x

2 2F k x

1 2F F F

1 2eq

Fk k k

x


48

The equivalent stiffness can be found by considering:

force-deformation relation

potential energy-deformation relation (Chapter 2)

Fk

x


49

Cantilever beam

For cantilever with length L and a tip load F giving displacement x

3

3EIF x

L

EI : bending stiffness

Note that this is the stiffness only at the tip of the cantilever 3

3EIk

L

L F

x


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Series or Parallel?


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Shaft

Torque T produces rotation θ

GJT θ

L

G : Shear modulus

J : Polar moment of area of

shaft cross section

GJk

L


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Helical spring

4

364

Gdk

nR

G : Shear modulus of spring material

d : Diameter of wire

2R : Diameter of turn

n : Number of turn


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STIFFNESS TABLE


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STIFFNESS TABLE

(cont.)


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Linearisation

sinx r θ

If small displacement, i.e. θ is small:

3

sin ...3!

θθ θ θ

x rθThus

xθ

r


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bx 1

ax 2

cx 3

1 ?x

2 ?x

3 ?x


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POTENTIAL ENERGY

2120 0

( )x x

V Fdx kx dx kx

‘Linear’ spring equivalent stiffness ‘Torsional’ spring equivalent stiffness

= work done from unstreched position

212 eqV k x 21

2 eqV k θ

2

21 Rkkkeq 22

1 kR

kkeq

**note: we should relate x and θ

More discussion in Chapter 2


58

References

B. R. Mace, ISVR Lecture Notes, Univ. of Southampton

D. J. Inman, Engineering Vibrations, 3rd Ed., Pearson

Animations courtesy of Dr. Dan Russell, Kettering University, USA

S. Rao, Mechanical Vibrations, 5th Ed., Pearson

Documents

Ch1-Fundamental of Vibration