BasicsofMechanicalVibration

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Page 1January 5, 2006

Chris K Mechefske

Machine Condition Monitoring

and

Fault Diagnostics


• Introduction to Machine Condition Monitoringand Condition Based Maintenance

• Basics of Mechanical Vibrations

• Vibration Transducers

• Vibration Signal Measurement and Display

• Machine Vibration Standards and AcceptanceLimits (Condition Monitoring)

• Vibration Signal Frequency Analysis (FFT)

Course Overview


• Machinery Vibration Trouble Shooting

• Fault Diagnostics Based on Forcing Functions

• Fault Diagnostics Based on Specific MachineComponents

• Fault Diagnostics Based on Specific Machine Type

• Automatic Diagnostic Techniques

• Non-Vibration Based Machine Condition Monitoring and Fault Diagnosis Methods

Course Overview


Current Topic








Definition

The variation with time of the magnitude of a quantity, which is descriptive of the motion or position of a mechanical system, when the magnitude is alternately greater and smaller than the average value or reference.

Basically – an object oscillates back and forth about an equilibrium point.

Basics of Mechanical Vibration


By Motion:

Simple Harmonic Motion

The simplest form of vibration.

Exact position is predictable from the equation of motion.

Mathematical description:

Classification of Vibration

)sin()( θ+ω= tAtx

2


Terms:

- instantaneous displacement (m)

- maximum amplitude (m)

- angular velocity (Radians/Second)

- phase angle (Radians)

f = frequency,

T = cycle/period,


)sin()( θ+ω= tAtx

)(tx

Aωθ

fπ=ω 2fT /1=



t

T

A

Graphical description of simple harmonic motion


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)

Ampl

itude


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


Graphical description of Random Motion.

Random 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)

Ampl

itude


Combination of random and periodic motion.

Random and repeatable characteristics combined in a non-linear fashion.

Vibration signal contains all frequencies in a given band but not in equal proportions.

Chaotic Motion

3


Any motion other than the above.

Impulsive in nature, but not regularly repeated.

Transient Motion


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 of energy within system.Non-oscillatory energy to oscillatory excitation.

Classification of Vibration by Excitation


Single Degree-of-Freedom System Model

Basic Theory of Vibration

K

Mass M

C

F(t)x(t)


The equation of motion comes from the force balance equation,


)()()()( tFtKxtxCtxM =++ &&&

The total solution to the equation of motion has two parts. The transient solution (x1) and the steady state part (x2). We are usually more interested in the steady state solution, but will consider both here for completeness.


Solution to equation of motion.

Transient state solution ( ).


0)( =tF

tsts BeAetx 21)(1 +=

are initial conditions

is the natural frequency,

is the damping ratio,

BA,

0ω MK=ω0

ζ ( )02 ω=ζ MC

[ ] 02

2,1 1ωζζ −±−=S



There are three special cases of transient vibration.

1. Underdamped

2. Critically Damped

3. Overdamped

1<ζ1=ζ

1>ζ

4


Underdamped


1<ζ

[ ] 02

2,1 1 ωζ−±ζ−= jS

)1sin()( 02

10 ϕ+ωζ−′= ζω− teAtx t

Oscillation with frequency

Amplitude decays exponentially0ω


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, ζ=0.15


Critically Damped


Quick restoration to equilibrium state

teBtAtx 0)()(1ω−+=

1=ζ



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, ζ=1.0


Overdamped


Exponential decaying without oscillation

1>ζ

tsts BeAetx 21)(1 +=

Rss ∈21,



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, ζ =3.0

5



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


Solution to equation of motion.

Steady state solution ( ).


)sin()( 0 tFtF ω=

( ) )sin()( 20

2 φ−ωω−+ω

= tMKC

Ftx

Total solution to equation of motion.

)()()( 21 txtxtx +=


We are primarily interested in the Steady State response of a system due to some continuous forcing function input.

Recall that the equation that describes simple harmonic motion is:

Relationship between Displacement, Velocity and Acceleration

)sin()( θ+ω= tAtx


Velocity (m/s)

- The rate of change of displacement with time

Displacement Velocity

)2

sin()()( π+θ+ωω== tAtxtv &


Acceleration (m/s2)

- The rate of change of velocity with time

Velocity Acceleration

)sin()()( 2 π+θ+ωω== tAtxta &&


Relationship between Displacement, Velocity and Acceleration

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)

Ampl

itude

displacementvelocityacceleration

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Descriptors of Vibration Signals

Mean

Indicates the DC level in the signal and should be subtracted first.

Definition:

∫= dttxT

x )(1



Average

Indicates average vibration level of the signal.

Definition:

∫= dttxT

xav )(1


Vibration Descriptors

Amplitude

time

avxMean



Peak Value (zero-to-peak)

Indicates peak vibration level of the signal.

Definition:

[ ]xtxx p −= )(max



Amplitude

time

px

Mean



Peak-to-Peak

Indicates total fluctuation in the vibration signal.

Definition:

[ ] [ ])(min)(max txtxx pp −=−

7



Amplitude

time

ppx −

Mean



RMS (root mean square)

Value proportional to the energy in the vibration signal.

Definition:

[ ]∫= dttxT

xRMS2)(1



Amplitude

time

RMSxMean



Amplitude

time

px

avx RMSx ppx −

Mean


With respect to A in the original equation of simple harmonic motion:


)sin()( θ+ω= tAtx

Ax p =

Ax pp 2=−

Axav 637.0=

AxRMS 707.0=


Note that equations on the last slide are true for simple harmonic motion only. If the vibration signal has a different character the simplification below does not hold.


AxRMS 707.0≠

But rather, the RMS value must be calculated from...

[ ]∫= dttxT

xRMS2)(1

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Time and Frequency Domains

+ =

F1

F2



Amplitude Frequency

Time

F2

F1



Amplitude

Time



Amplitude

FrequencyF1 F2


A measure of vibration amplitude

• Logarithmic scale

• With respect to a reference value

• Effective in displaying small values together with very large values.

Decibel (dB) units


Definition (Mechanical and Acoustics)

Decibel (dB) units

⎥⎦

⎤⎢⎣

⎡=

ref

rms10log20dB

AA

- RMS value of a parameter- Reference value of the parameter

rmsA

refA

Double amplitude corresponds to an increase of 6 dB

9


Decibel (dB) units

Linear Scale

1 100010010

1 10 100 1000

Logarithmic Scale


Standardised reference values (ISO standard)

Decibel (dB) units

m 101 12−× sm9101 −×

g6101 −×

Parameter Displacement Velocity Acceleration

Reference or 2

6101 sm−×


dB increase Linear Multiplication

6 x 210 x 320 x 1030 x 3040 x 10050 x 30060 x 100070 x 3000

Decibel (dB) units


Decibel (dB) units


Using the Single Degree-of-Freedom System Model

What is Mechanical Vibration?

K

Mass M

C

F(t)

y(t) - output

x(t) - input


How mechanical systems respond to forcing function inputs?

Mechanical Vibration is:

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

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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 function input.

Mechanical Vibration

Consider only one source of potential forcing function input – the road surface.

Also consider the vehicles suspension system as a linear single degree-of-freedom system.


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)


Assume unsprung mass (wheel) is small (but not negligible) compared to that of the vehicle.

K is the spring stiffness (linear). Spring stores energy when 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 and opposes motion proportional to velocity. Zero velocity – zero force.



Response to System Inputs

Road Input Vehicle Output

Ampl.

Time

Ampl.

Time

Evaluation of these plots reveals two important quantities – gain and phase shift.


Gain is the change in amplitude (often an increase) from input to output (often expressed in decibels).

Gain = Output AmplitudeInput 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

Road Input Vehicle Output

Ampl.

Time

Ampl.

Time

Gain

Phase Shift

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

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


Gain(dB)

Freq.Gain Plot

Phase(degrees)

Freq.Phase Plot


When considered together the gain and the phase shift 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 are found from these plots.



Gain(dB)

Freq.

Phase(degrees)

Freq.

The gain at low frequencies is one or close to one.

The frequency shift is zero.



Gain(dB)

Freq.

Phase(degrees)

Freq.

There is little change in the gain as the frequency increases, until the system Natural Frequency is approached where the gain quickly increases with increasing frequency.

The phase shifts towards 90° as the frequency gets close to the Natural Frequency.



Gain(dB)

Freq.

Phase(degrees)

Freq.

Above the Natural Frequency, the gain decreases at a constant rate (usually rapid).

The frequency shift approaches 180°.


As the frequency increases the gain initially increases (until natural frequency) and then decreases (after natural frequency). Note – there may be more than one natural Frequency.


While low frequency inputs are passed through the system (gain equals one), high frequency inputs are attenuated.

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.



An increase in gain and dramatic phase shift occur at the frequency of mechanical resonance.

Mechanical Resonance

Many system responses or forcing 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.


Note:

Mechanical Resonance

The gain remains relatively constant at low frequencies.

Systems natural frequency (resonance) occurs when the phase shift is exactly -90o

The peak gain occurs slightly below the system resonance due to damping.

For frequencies above resonance the gain decreases as the phase shift approaches -180o.

Gain(dB)

Freq.

Phase(degrees)

Freq.


As noted earlier – system damping affects the response of the system.

System Damping

Increased damping results in lower peak gain.

Increased damping results in reduced phase shift slope.

Changes in damping result in only minimal changes in gain and phase shift at low and high frequencies.

Gain(dB)

Freq.

Phase(degrees)

Freq.


Typically only outputs can be measured, not inputs.

To complicate this – a different transfer function exists from each vibration forcing function input to the 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.

Analysis of Mechanical Vibrations


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.

Non-Linearities

Damping Force

Velocity

In reality, the damping force levels off as velocity approaches zero.

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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 of vibration inputs.

Summary

In a simple linear system, the response to a sinusoidal input is a sinusoidal output with the same frequency, but different phase and amplitude.


A system response to vibration input depends on the frequency of the input.

The change in amplitude and phase shift of the output relative to the input is slight at low frequencies, but is dramatic close to the system natural frequency (resonance) and above.

Summary

In vibration analysis it is essential to consider both the specifics of the input and the system characteristics (transfer function) such as resonances and non-linearities.


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