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
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