Basic idea - aq.upm.es Vibration.pdfBasic idea • Best conditions for vibration test are: ......

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Course in Non Destructive Testing of Wood 05 Vibration – Pág. 1ETSI Montes, ETS Arquitectura – Universidad Politécnica de Madrid Madrid, Junio 2005

Vibration Methods

Basic idea

• Best conditions for vibration test are:- support sample at the nodal points- hit sample at amplitude maximum, - hitting direction is important- microphone locate at max amplitude location

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Fourier Transformation

( ) ( )∫∞

∞−

−= dtethfH fti π2 1−=i

Discrete Fourier Transformation

( )kTh 1,.......,3,2,1,0 −= Nk

1,.....3,2,1,0 −= Nn( ) NkniN

kekTh

NTnH

π21

0

−−

=∑=⎟

⎠⎞

⎜⎝⎛

T: time between two samples, T=1/fsampling

Fast Fourier Transformation:h function is identical, H is almost the same, but computation time is shorter.

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Dynamic MOE determination by longitudinal vibration

2, VMOE longdyn ρ=

LfV 2=

Dynamic MOE determination by bending vibration

ImL

CfMOE

n

nbendingdyn

32

, ⎟⎟⎠

⎞⎜⎜⎝

⎛=

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Timoshenko beam theory• The effect of shear is included.

01 4

42

22

4

2

2

4

4

=∂∂

+∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛+−

∂∂

+∂∂

tr

GI

txr

GEI

trA

xrEI

βρ

βρρ

where: b - shear factor (1/1.2 for prismatic beams),r - displacement,x - longitudinal coordinate,t - time,A - cross section,ρ - density,I - moment of inertia,E - bending modulus of elasticity, MOEG - shear modulus.

Solution is available by iteration technique only.

Solution of the Timoshenko equation

• Evaluation software presented by Dr. Chui, Canada. Input data are:- length- thickness- density- bending vibration frequency, mode 1- bending vibration frequency, mode 2

Demonstration……… E.exe

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Euler beam theoryNeglecting the shear Timoshenko theory turns to

Euler beam theory

where:fn - frequency measured in the nth mode,Cn - mode factor (see Figure ),m - specimen mass. Mode 1. 2. 3.E - MOEI - Inertia

ImL

CfE

n

n32

⎟⎟⎠

⎞⎜⎜⎝

⎛=

,

Influencing factors are:

- Geometry: Nodal distance/thickness- Damping- Static-dynamic correction

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Nodal distance

Nodal distance is the distance between two neighbouringnodal points.

Nodal point

Nodal distance / beam height

•

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Effect of damping• The measured frequency (f) and the not damped

vibration frequency (fo) is different

Λ is the logarithmic decrement: Λ =β/fβ is damping coefficient

Logarithmic decremet for wood is 0,1-0,01, thedamping correction is minor, less than 0,2%

2

2

0 41

πΛ

+= ff

Damping coefficient: β

x1 x3

x2 x4

x

A

T

t

Ae-βt

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Static - dynamic correction• One order of magnitude difference in the

characteristics time causes a 1.7% change in the measured MOE value.

9,25

9,5

9,75

10

10,25

10,5

-4 -3 -2 -1 0 1 2 3

log(time[s])

MO

E [G

Pa]

stress wavebending vibration, 3rd mode

bending vibration, 1st mode

static, 100 mm/minstatic, 10 mm/min

static, 1 mm/min

static, 0.1 mm/min

Determination of shear modulus by torsion vibration

t

pntorsiondyn K

InLf

Gρ2

,2

⎟⎠⎞

⎜⎝⎛=

B1

T1

B2 T2 B3 T3

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Evaluation chart

MOE and G evaluation example.xls

Portable Lumber Grader setup

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Longitudinal vibration, Portable Lumber Grader (PLG)

PLG screen

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PLG algorithm

( ) ( )50192.02 2 ulfMOEmea += ρρ: density

l: length

f: longitudinal vibration frequency

u: moisture different = actual – service condition

CKDRMOEMOE mea 2.6−=

Concentrated Knot Diameter Ratio

The knot diameter is a distance between the two tangential lines parallel to arises (longitudinal direction) of a lumber surface in which the knot exists. If a knot diameter not less than 2.5 times as much as its smallest diameter, it shall be considered to have one half of its actual measured diameter. The knot diameter ratio (KDR) is a percentage of the diameter of a knot to the width of a lumber surface in which it exists. The concentrated KDR (CKDR) is the sum of KDR concerning the knots existing in any 15 cm length of a piece of the lumber. The highest - considering 4 faces - CKDR represents the piece of lumber. The CKDR value is between 0 and 1.

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PLG decision table

Example, 2 by 4 material

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Demonstration: PLG and rapture test

Basic idea - aq.upm.es Vibration.pdfBasic idea • Best conditions for vibration test are: ......

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