VTAF 01 – Sound in Buildings and Environment

Lab 1 – Measuring sound and vibrationNIKOLAS  VARDAXISDIVISION  OF  ENGINEERING  ACOUSTICS,  LTH,  LUND  UNIVERSITY

2021.04.27

Outline

Part 1 – Guitar string

Part 2 – Beam and plate

Part 1 – Eigenfrequencies in a guitar string (I)

First  step:  variation  of  the  frequencies  of  a  guitar  string

Theory:  -­ Wave  equation  in  a  string  

-­

Eigenfrequencies in a guitar string

- Wave equation in a string - Frequency = f(length, density, tension)- Fourier transform (FFT): time à freq. domain

In general:

λ=2L/nfn=n·v/2L

v=(tension/mass-length) 1/2

Part 1 – Eigenfrequencies in a guitar string (II)

• Notes of provided code readsound.mat

y t = 3 sin 2π4t

FFT

Part 1 – Eigenfrequencies in a guitar string (III)

• Example of postprocessed data with readsound.mat

Outline

Part 1 – Guitar string

Part 2 – Beam

Modal analysis – Intro

Modal analysis is the field of measuring and analysing the dynamic response of structures and/or fluids during excitation.

• Eigenfrequencies

• Mode shapes

Photo  from  an  actual  study:Grzegorz Litak et.al.,  Vertical  beam  modal  response  in  a  broadband  energy  harvester  (2017)

Notes on modal superposition (I)

Source: http://signalysis.com

Notes on modal superposition (II)

VibrationResponse

1st

Mode2nd

Mode - - - - - - - - -3rd

Mode

SDOF – Complex representation (Freq. domain)

• Complex representation of a damped SDOF

Mü(t) + Rů(t) + Ku(t) = F(t)

F(t) = Fdriv·cos(ωt)

u(t)

u.(ω) =F2345

K −Mω9 +RiωDisplacement  response

SDOF – Frequency response functions (FRF)

• In general, FRF = transfer function, i.e.: ‒ Contains system information

‒ Independent of outer conditions

• Different FRFs can be obtained depending on the measured quantity‒ Excitation measured with a force transducer in the hammer

‒ Response measured with accelerometers

𝐻=> 𝜔 =𝑠=A (𝜔)𝑠>A (𝜔)

=outputinput

SDOF – Frequency response functions (FRF) – (I)

• In general, FRF = transfer function, i.e.: ‒ Contains system information

‒ Independent of outer conditions

• Different FRFs can be obtained depending on the measured quantity

Measured quantity FRFAcceleration (a) Accelerance = Ndyn(𝜔) = a/F Dynamic Mass = Mdyn(𝜔) = F/a

Velocity (v) Mobility/admitance = Y(𝜔) = v/F Impedance = Z(𝜔) = F/v

Displacement (u) Receptance/compliance = Cdyn(𝜔)= u/F Dynamic stiffness = Kdyn(𝜔) = F/u

𝐻=> 𝜔 =𝑠=A (𝜔)𝑠>A (𝜔)

=outputinput

Part 2 – Modal analysis of a beam [Example]

First  step:  variation  of  the  frequencies  of  a  guitar  string

Theory:  -­ Wave  equation  in  a  string  -­ Frequency  =  fct (length,  density,  tension)-­ Fourier  transform  (FFT):  time  scale  -­>  freq.  

scale

Modal analysis of a steel beam

- Bending waves in a beam- Eigenfrequency = f(E, I, density, length)- Fourier transform (FFT): time à freq. domain

Part 2 – Modal analysis of a steel plate [Example]

Modal analysis of a steel plate

- Bending waves in a steel plate

- Eigenfrequency = f(E, I, density, length, width)

- Fourier transform (FFT): time à freq. domain

Part 2 – Experimental modal analysis: Lab setupFrontend

PC with B&K Pulse (software)New version: BK Connect

Note on frequency content of excitation

Source: AgilentTechnologies

Experimental Modal Analysis (EMA)

Source: AgilentTechnologies

Experimental Modal Analysis (EMA) – Examples

Part 2 – Modal analysis of a steel plate [Lab setup]

1

2

3

• Excitation with impacthammer at point 1

• Force input

– (driving point)

• Accelerometers at points 1, 2, 3

Part 2 – Modal analysis of a steel plate [Lab setup]

• Study the input force and accelerometer positions signals for the steel plate.

• Perform the Fast Fourier Transformation (FFT) using Matlab.

• Plot together in the same graph all acceleration frequency response functions (FRF). Comment on:

– the first Eigenfrequencies

– resonances and anti-resonances (phase differences ?)

Part 2 – Modal analysis of a steel plate [Lab setup]

• Study the input force and accelerometer positions signals for the steel plate.

• Do the same with a decibel scale on the y-axis this time. You can use acceleration normalized to 1, expressed as 10*log10(a).

• Plot together the mechanical impedance and mobility FRFs in decibel again,

– Comment again on the Eigenfrequencies of the system.

Reminder: The mechanical impedance Z is the ratio of input force over velocity,while mobility Y is the inverse of Z.

𝑍 = EF , Y= F

E

Thank you for your attention!nikolas.vardaxis@construction.lth.se