mech826-week02-basicsofmechanicalvibrations

8/9/2019 MECH826-Week02-BasicsofMechanicalVibrations

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Page 2September 22, 2010

Current Topic

Introduction to Machine Condition Monitoring

and Condition Based Maintenance

Basics of Mechanical Vibrations

Vibration Transducers

Vibration Signal Measurement and Display

Machine Vibration Standards and Acceptance

Limits (Condition Monitoring)

Vibration Signal Frequency Analysis (FFT)


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Definition

The variation with time of the magnitude of aquantity, which is descriptive of the motion or position

of a mechanical system, when the magnitude is

alternately greater and smaller than the averagevalue or reference.

Basically an object oscillates back and forth about

an equilibrium point.

Basics of Mechanical Vibration


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Vibration in machines is a result of dynamic forces in

machines which have moving parts and in structures which areconnected to the machine. Different parts of the machine will

vibrate with various frequencies and amplitudes. Vibration

causes wear and fatigue. It is often responsible for the ultimate

breakdown of the machine.

Basics of Mechanical Vibration


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It is a benign, non-intrusive method, which can beperformed without having to take the machine off-line.

It is applicable to a broad range of machinery andmechanical components, including bearings, motors, andgears.

It is applicable to a broad range of fault types, including:

wear, fracture, unbalance, misalignment, flow problems. The associated hardware used to measure and recordmechanical vibrations is relatively economical andreliable, is commonplace in industry, and is relatively

straight-forward to use. Vibration signals are well suited for continuous on-line

monitoring.

Why Measure Mechanical Vibration


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Dominant condition monitoringtechnology

Good for rotating and cyclical machines. Various levels of analysis

broadband trending (maximum overall

level) narrow band trending (max. level in a

specific frequency range)

spectrum (signature) analysis (details

of frequency spectrum)

Vibration Monitoring


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Vibration is usually identified as a maintenanceproblem if

machine reliability/maintainability is affected (short termor long term)

machine operation is unacceptable

Vibration can be categorized according to thesource of vibration rotary mechanical

other mechanical hydraulic

electrical

Typical Vibration Problems


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Mass Unbalance (Imbalance) non-uniform mass distribution on rotating part (typically

a shaft) Caused by loss or build-up of material, poor design, etc.

Coupling Misalignment centre-lines of coupled machines

maintenance often requires uncoupling of driver anddriven machines

Bent Shaft/Rotor heavy rotors bend if stationary for long periods

shafts are prone to damage during operation Bearings

Gears, Belts, Pulleys, ...

Rotary Mechanical Vibration


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Looseness loose parts hit against limits of movement

off-harmonic frequency components: 1.5, 2.5, etc.

Soft Foot hold-down bolts can result in distortion of machine

casing - hence change in stiffness this stiffness change shifts the resonant frequency

and/or causes misalignment of machines

remedy by loosening hold-down bolts and shim base of

machine, then tighten bolts. Structural Resonance

Other Mechanical Vibration


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Spring-Mass-Damper model of vibration

Simple Harmonic Motion

Wave Fundamentals amplitude, frequency,

phase

Displacement, Velocity and Acceleration and how

they relate

Vibration Measurement (sensors and techniques)

Basic Machinery Vibrations


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By Motion:


The simplest form of vibration.

Seen as principal component in most rotating equipment

vibration signals.

Exact position is predictable from the equation of motion.

Mathematical description:

Classification of Vibration

)sin()( += tAtx


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Terms:

- instantaneous displacement (m)

- maximum amplitude (m)

- angular velocity (Radians/Second)

- phase angle (Radians)

f = frequency,

T= cycle/period,


)sin()( += tAtx

)(tx

A

f= 2fT /1=


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Phase measurement related to a trigger

360

reference signal

phase

lag

Vibration

Signal

phase

lead



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Motion repeats itself in equal time periods.

Includes harmonic motion, pulses, etc.

Periodic Motion

0 10 20 30 40 50 60 70 80 90-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (ms)

Amplitude


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Periodic Motion


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Periodic Motion


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Motion is not deterministic (That is, not

repeatable).

Statistics of motion history may be well defined, but

exact location as a function of time is not obtainable.

Vibration signal contains all frequencies in a given

band.

Often generated by machine looseness.

Random Motion


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Any motion other than the above.Impulsive in nature, but not regularly repeated.

Transient Motion


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Free vibration:

Oscillation occurs at natural frequency after an

initial force input has disappeared.

Forced vibration:

Oscillation occurs at the frequency of a driving

force input.

Self-induced vibration:

Vibration of a system resulting from conversion ofenergy within system.

Non-oscillatory energy to oscillatory excitation.

Classification of Vibration by Excitation


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Single Degree-of-Freedom System Model

Basic Theory of Vibration

K

Mass M

C

F(t)x(t)


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Solution to equation of motion.

Transient state solution ( ) general form.


0)( =tF

tstsBeAetx 21)(1 +=

are initial conditions

is the natural frequency,

is the damping ratio,

BA,

0 MK=0 ( )02

=M

C

2

1,2 01S =


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There are three special cases of transient vibration.

1. Underdamped

2. Critically Damped

3. Overdamped

1


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Underdamped


1


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Responses of free vibration versus damping

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

x(t)

Time (s)

Underdamped,=0.15


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Critically Damped


Quick restoration to equilibrium state

teBtAtx 0)()(1

+=

1=


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0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

x(t)

Time (s)

Critically damped, =1.0

i f i i


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Overdamped


Exponential decaying without oscillation

1>

1 2

1( ) s t s t

x t Ae Be = +

Rss 21,

R f f ib i d i


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0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

x(t)

Time (s)

Overdamped, =3.0

R f f ib ti d i


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0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

x(t)

Time (s)

Underdamped, =0.15Critically damped, =1.0Overdamped, =3.0

B i Th f Vib ti


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Solution to equation of motion.

Steady state solution ( ).


)sin()( 0 tFtF =

( )

)sin()(2

0

2 += t

MKC

Ftx

Total solution to equation of motion.

)()()( 21 txtxtx +=

Relationship between Displacement


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We are primarily interested in the Steady State

response of a system due to some continuous

forcing function input.

For now, recall that the equation that describessimple harmonic motion is:

Relationship between Displacement,

Velocity and Acceleration

)sin()( += tAtx



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Measures of Simple Harmonic Motion

Displacement

Velocity

Acceleration



Displacement Velocity


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Velocity (m/s)

- The rate of change of displacement with time

Displacement Velocity

)2

sin()()(

++== tAtxtv &

Velocity Acceleration


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Acceleration (m/s2)

- The rate of change of velocity with time

Velocity Acceleration

)sin()()( 2 ++== tAtxta &&



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0 5 10 15 20 25 30 35 40-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (ms)

Amp

litude

displacementvelocityacceleration

Descriptors of Vibration Signals


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Average

Indicates average vibration level of the signal.Definition:

= dttxTxav )(1

Vibration Descriptors


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Amplitude

time

avx


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Amplitude

time

px



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Peak-to-Peak

Indicates total fluctuation in the vibration signal.Definition:

[ ] [ ])(min)(max txtxx pp =



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Amplitude

time

ppx



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p g

RMS (root mean square)

Value proportional to the energy in the vibrationsignal.

Definition:

[ ]= dttxTxRMS2

)(1



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Amplitude

time

RMSx



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p

Amplitude

time

px

avx RMSx

ppx



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With respect to A in the original equation of

simple harmonic motion:

p g

)sin()( += tAtx

Axp=

Ax pp 2=

Axav 637.0=

AxRMS

707.0=



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Note that equations on the last slide are true for

simple harmonic motion only. If the vibration

signal has a different character the simplificationabove does not hold.

AxRMS 707.0

But rather, the RMS value must be calculated

from...

[ ]= dttxTxRMS2

)(1


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Time and Frequency Domains


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Amplitude Frequency

Time

F2

F1


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Amplitude

FrequencyF1 F2


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Component Frequencies of a Square Waveform



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Component Frequencies of a Square Waveform



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( )

=

++=1

0 sincos)(

n

nn tnBtnAAty

Fourier series.

+++= tttty10

10sin5

4

10

6sin3

4

10

2sin4)(

Example Fourier series: T = 10 (-5 to +5), A0= 0.

+++= tttty 5sin

5

13sin

3

1sin

20)(

What is Mechanical Vibration?


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Using the Single Degree-of-Freedom System Model

K

Mass M

C

F(t)

y(t) - output

x(t) - input

Mechanical Vibration is:


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How mechanical systems respond to forcing

function inputs?

Consider an everyday example the motor vehicle.

A wide range of different inputs can cause

vibrations in motor vehicles.Wind Engine Combustion

Road surface Mechanical Imbalance

Engine Fan Misalignment

Mechanical Vibration


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All vibrations experienced by the driver and other

occupants are the result of mechanical dissipation

of energy in response to some forcing functioninput.

Consider only one source of potential forcingfunction input the road surface.

Also consider the vehicles suspension system as

a linear single degree-of-freedom system.



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Using the single Degree-of-Freedom System

model for the suspension system

K spring stiffness

Mass of

vehicle, M

C shock damping

F(t)

y(t) output(vehicle vibration)

x(t) input

(road surface)



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Assume unsprung mass (wheel) is small (but not

negligible) compared to that of the vehicle.

K is the spring stiffness (linear). Spring stores energywhen stretched or compressed and acts to oppose

motion proportional to position. Unstretched or

uncompressed spring no force.C is the damping coefficient of the shock absorber,

which is modeled as a viscous damper. The shock

absorber dissipates energy rather than storing it andopposes motion proportional to velocity. Zero velocity

zero force.

Response to System Inputs


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Road Input Vehicle Output

Ampl.

Time

Ampl.

Time

Evaluation of these plots reveals two importantquantities gain and phase shift.



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Gain is the change in amplitude (usually a

decrease, but often an increase) from input to

output (often expressed in decibels).

Gain = Output Amplitude

Input Amplitude

The phase shift is the change in the position of the

output vibration signal relative to the input

vibration signal.

The frequency of the output does not change

relative to the input.

Response to System Inputs


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Road Input Vehicle Output

Ampl.

Time

Ampl.

Time

Gain

Phase Shift



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Consider now the gain and phase shift of a systemover a range of frequencies.

In order to do this we need to introduce what isknown as the Transfer Function (TF).

Gain(dB)

Freq.

Gain Plot

Phase(degrees)

Freq.

Phase Plot



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When considered together the gain and the phaseshift plots represent the Transfer Function of a

particular mechanical system.

Gain

(dB)

Freq.

Phase

(degrees)

Freq.

The Gain and Phase Shift at

any particular frequency arefound from these plots.



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Gain

(dB)

Freq.

Phase

(degrees)

Freq.

The gain at low

frequencies is oneor close to one.

The frequency shift

is zero.



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Gain

(dB)

Freq.

Phase

(degrees)

Freq.

There is little change inthe gain as the frequency

increases, until the

system Natural Frequencyis approached where the

gain quickly increases

with increasing frequency.

The phase shifts

towards 90 as the

frequency gets close tothe Natural Frequency.



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Gain

(dB)

Freq.

Phase

(degrees)

Freq.

Above the Natural

Frequency, the

gain decreases at

a constant rate

(usually rapid).

The frequency shiftapproaches 180.



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As the frequency increases the gain initially increases (untilnatural frequency) and then decreases (after natural

frequency). Note there may be more than one natural

Frequency.

While low frequency inputs arepassed through the system

(gain equals one), high

frequency inputs areattenuated.

Such a system is called a low

pass filter.

Gain

(dB)

Freq.

Phase

(degrees)

Freq.



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All mechanical systems act as low pass filters for

two reasons.

High frequencies require higher speeds to reach

the same amplitudes as lower frequencies

All machines have a maximum velocity (due to

inertia). Once the maximum velocity is reached,

higher frequencies can only be reached by

reducing the amplitude.

Mechanical Resonance


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An increase in gain and dramatic phase shift

occur at the frequency of mechanical resonance.

Many system responses orforcing function frequencies exist

at or close to resonance.

It is essential to consider the

existence of these resonances

when designing new machines

and when maintaining existing

machines.

Gain

(dB)

Freq.

Phase

(degrees)

Freq.


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Typically only outputs can be measured not inputs

Analysis of Mechanical Vibrations


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Typically only outputs can be measured, not inputs.

To complicate this a different transfer function

exists from each vibration forcing function input tothe point where the output is measured.

Not all (if any) transfer functions are known due to

their complex nature.

As a result separately analyzing transfer functions

and inputs is extremely challenging.

D i i ll d l d li

Non-Linearities


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Damping is usually modeled as linear.

Using this model - as velocity slows the

damping force goes to zero.

This is, of course, not true in real systems.

DampingForce

Velocity

In reality, the damping force levels off as velocity

approaches zero.

Summary


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Vibration is the mechanical dissipation of energy in

response to a mechanical input.

All mechanical systems act as low pass filters ofvibration inputs.

In a simple linear system, the response to asinusoidal input is a sinusoidal output with the

same frequency, but different phase and

amplitude.

A system response to vibration input depends on

Summary


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A system response to vibration input depends on

the frequency of the input.

The change in amplitude and phase shift of theoutput relative to the input is slight at low

frequencies, but is dramatic close to the system

natural frequency (resonance) and above.

In vibration analysis it is essential to consider both

the specifics of the input and the system

characteristics (transfer function) such asresonances and non-linearities.




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We are primarily interested in the Steady State

response of a system due to some continuous

forcing function input.


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Decibel (dB) units


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A measure of vibration amplitude

Logarithmic scale With respect to a reference value

Effective in displaying small values togetherwith very large values.

Definition (Mechanical and Acoustics)

Decibel (dB) units


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Definition (Mechanical and Acoustics)

= ref

rms

10log20dB A

A

- RMS value of a parameter

- Reference value of the parameter

rmsA

refA

Double amplitude corresponds to an increase of 6 dB

Decibel (dB) units

Linear Scale


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Linear Scale

1 100010010

1 10 100 1000

Logarithmic Scale

Decibel (dB) units


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5 000 Hz


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Dynamic Range


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gear mesh

4 gs @ 4500 Hz

5 000 Hz

gear mesh - 4 gs


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Dynamic Range


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gear mesh

4 gs @ 4500 Hz

5 000 Hz

gear mesh - 4 gs


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dB increase Linear Multiplication

Decibel (dB) units


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6 x 2

10 x 3

20 x 10

30 x 30

40 x 100

50 x 300

60 x 1000

70 x 3000


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Next Time

Introduction to Machine Condition Monitoring

d C diti B d M i t


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and Condition Based Maintenance

Basics of Mechanical Vibrations Vibration Transducers

Vibration Signal Measurement and Display

Machine Vibration Standards and Acceptance

Limits (Condition Monitoring)

Vibration Signal Frequency Analysis (FFT)

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