vibration measurements2

Periodic StructureMaged Mostafa

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Vibration of Structures



Vibration Measurements



Course ContentsSDOF M-DOFCables/String Bars Shafts Vibration Attenuation Beams Membranes & Thin Plateso Vibration Testing and Measurements



Objectives

• Vibration Measurements Hardware• Introduction to signal Processing• Vibration Measurements Terminology



Measurements Hardware



Measurements Hardware

Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network, Smart Mater. Struct. 16 (2007) 1208–1217



Measurements Hardware• Transducers• Exciters• Signal Conditioning• Analyzer



Measurements HW: Transducers• Load cell: a piezoceramic device

configured to produce a voltage proportional to force

• Accelerometer: (piezoceramic) produce a signal proportional to acceleration at the point of attachment

• Proximity probes: magnetic device gives a signal proportional to local displacement



• Laser vibrometers: give a signal proportional to local velocity (or scanned to give velocity field)

• Strain gauges: (local strain or strain rate)

Measurements HW: Transducers



• Electromagnetic shakers: which may apply a force through a range of frequencies (harmonic or random inputs)

• Instrumented hammers: which simulate an impact

Measurements HW: Exciters



• The direct out put of a transducer not usually well suited for input into an analyzer: Impedance miss matched, voltage or current levels too low

• SC is a charge or voltage amp designed to take an accelerometer signal and match it to the input requirements of the analyzer

Measurements HW: SC



• Electronic boxes (dedicated computers) gathers signals and manipulate them mathematically

• Modern analyzers have evolved almost in to chip sized devices

• Main source of manipulation is Digital Fourier Transforms for manipulating the vibration data in the frequency domain

Measurements HW: Analyzer



Objectives

• Vibration Measurements Hardware• Introduction to Signal Processing• Vibration Measurements Terminology



Fourier Series: Physical Meaning



Fourier Series of F(t) of period T

,....3,2,1

)sin()(2 ,)cos()(2

)(2

2

)sin()cos(2

)(

00

00

1

0

n

dttntFT

bdttntFT

a

dttFT

a

T

tnbtnaatF

T

Tn

T

Tn

T

T

nTnTn



Fourier Series: Physical Meaning



Digital Signal Processing

The analyzer takes signals form the transducer and puts the signal into a form that can be mathematically `manipulated



Analog-to-Digital Converter

Analog signal x(t) is sampled at manyequally spaced time intervals to producethe digital record:

{x(t1), x(t2), ….x(tN)}where x(tk) is the discrete value of x(t) at

time tk and N is the number of samples

taken



Analog-to-Digital ConverterWhat happens if Sampling rate is too

large?

sec1

)4

7sin(

)4

sin(

2

1

T

tx

tx




What happens if Sampling rate is too large?




• Aliasing: When sampling rate is too large, the digital record does not catch the details of the analog signal

• What is the optimum Sampling rate?• Min. 2 Samples / Cycle• Experience: 2.5 Samples / Cycle

Shannon’s sampling theorem




• Antialiasing Filter: low-pass Filter(only allows low frequencies through).

• The filter cuts off frequencies higher than about half the maximum frequency of interest: (ωmax , Nyquist frequency).

• Most modern digital analyzers provide built-in antialiasing filters.



Digital Fourier Transform

,....3,2,1

2sin1 ,2cos1

1

2sin2cos2

)(

11

10

2/

1

0

n

tTnx

Nbt

Tnx

Na

xN

a

tTnbt

Tnaatxx

N

kkkn

N

kkkn

N

kk

N

nknknkk

N is fixed by hardware (a power of 2)



Digital Fourier TransformAbove becomes the matrix equation

x=Ca Where:C: contains the sin and cos termsx: is the vector of samples anda: is the vector of Fourier Coefficients• Analyzer computes the coefficientsin the DFT formula:

a=C-1x



Digital Fourier TransformWhat happens if Samples do not match the periods?



Digital Fourier TransformLeakage: If the signal is not periodic in N samples (signal cut off mid period) the DFT willproduce extra frequencies



Digital Fourier TransformWindowing: multiply signal by a function which is zero at the end points



Radom Signals• Autocorrelation: Tells how fast x(t) is

changing

T

Tkxx dtxtxT

tR0

)()(1lim)(

dexRS jxxxx )(

21)(

• Power Spectral Density (PSD): Frequency domain of Autocorrelation [ FT(Rxx) ]



Radom Signals• Crosscorrelation: Tells how fast x(t) is

changing w.r.t. f(x)

T

Tkxf dtftxT

tR0

)()(1lim)(

dexRS jxfxf )(

21)(

• Cross Spectral Density (PSD): Frequency domain of Autocorrelation FT(Rxf)

vibration measurements2

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