waves in structures - · pdf filewaves in structures wolfgang kropp 10.1 chalmers university...

Waves in Structures Wolfgang Kropp 10.1

CHALMERS UNIVERSITY OF TECHNOLOGY

Division of Applied Acoustics

E-mail: [email protected]

Vibration of structures Noise for instance from traffic is transmitted through façades, windows, doors into houses. Vibration for instance form trains or heavy vehicles is

propagating trough the ground into the fundament of houses. From there it is propagating trough the building structure and radiated into room where people are living. In all these processes vibration of structures are involved. There are two main types of waves important:

longitudinal waves and bending waves. These are in the following briefly described:

Longitudinal waves

were described as the deformation of a particles volume, i.e. as a compression of a particle. They are identical with waves in air. Just the speed of sound is

different. For longitudinal waves in beams:

�

cL,beam =E

!

E is the Young's modulus, ρ the density of the material. When we have longitudinal waves on a beam the cross section changes during propagation (although very little, the picture is very idealised and the cross





contraction extremely exaggerated). You can demonstrate this behaviour of change in cross section demonstrate with a soft piece of rubber. The situation becomes different when considering a plate of an infinite elastic space (e.g. inside the earth). In this case each element has neighbours, which also wants to move. Only at the borders of the material, a cross contraction of the particles is possible. As a consequence the material is softer when cross contraction is possible than when it is not and the speed of

sound will be higher in solids than in beams

�

cL,solid

=1

!

E 1 " µ( )1 + µ( ) 1 "2µ( ) .

In plates the speed of sound is between both values

�

cL,plate =1

!

E

1 " µ 2( ). µ is

the Poisson's number. All what we have learned about waves in air is also valid here. It becomes different when looking at the second type, the bending waves. (the most important group of waves discussed in this lecture).

They are important due to two reasons. First of all they are mainly responsible for the radiation of sound from vibrating structures since they

have a displacement component in the normal direction to the surface of the structure. Secondly, it is the most common wave type when dealing with structure borne sound. We





will not derive the wave equation in detail. The description of bending waves presented in this chapter is called Euler - Bernoulli theory for beams or Kirchhoff theory for plates.

Both theories contain simplifications, which are mainly the assumption of an npn-deformed cross section (i.e. an infinite high shear stiffness) and the

neglecting of rotational inertia.

These 'shortcomings' lead to erroneous results when the bending wavelength becomes comparable with the thickness of the plate or the beam as it will be shown later.

To compensate for this fact, extended theories (i.e. Timoshenko or Mindlin theory) are developed which hold further up in frequency but are also limited. However, in most cases (beside thick concrete walls and who is working with this ) the simple Euler - Bernoulli theory is sufficient and therefore preferable. Euler - Bernoulli theory In the figure a beam is shown where bending waves (often also called flexural waves) are propagating. The displacement in the normal direction (i.e. y -

direction) is !. When analysing the figure one observes that a cross section is also rotating around the neutral line.





With simple geometrical considerations one can conclude that the bending

angle

�

! is

�

! ="#

"x (i.e. the change of the normal displacement over the length

of the beam). The bending angle will cause a certain bending moment

�

M which depends on the bending stiffness

�

Bof the material and the change of the bending angle

over x

�

M = !B"#

"x= !B

" 2$

"x2

where the bending stiffness is

�

B = EI (do you

remember from courses on mechanics, statics?). Imagine a body where two different moments are applied on both sides. The difference in the moment will lead to a net force. Consequently, the force at a cross section in the y

direction is

�

F = !"M

"x= B

" 3#

"x 3. The force is also changing

over x. A particle is accelerated due to the net force applied to it. This means that Newton's law will lead to

�

!"F

"x= m

" 2#

"t2

. m is the mass per unit length, with the cross

section S and the density ρ ,

�

m = !S . Finally we get the equation for Bending waves:

�

B! 4"

!x 4+ m

! 2"

!t2

= 0

This wave equation differs substantially from all the other wave equations discussed before due to a fourth derivative in space. You might wonder what is special about this: Suddenly we will see that waves at different frequencies are not propagating with the same speed anymore!





TASK : Make clear for you what that’s mean. Think about listening to music. How will this music arrive to the listener when the different frequencies travel with different speed? Anyway, lets try with the standard solution:

�

! x,t( ) = !Ae" jkxe j#t : which will give

�

B! 4"

!x 4+ m

! 2"

!t 2= B jk( )

4

"Ae# jkxe j$t+ m j$( )

2

"Ae# jkxe j$t= 0

Such equations are called dispersion equation. They give us the relation

between frequency and wavelength (wavenumber):

�

B jk( )4

+ m j!( )2

= 0

�

k4

=!

2m

B This has for solutions. Lets call

the wavenumber

�

kB as

�

kB =m!

2

B

4

the solutions are:

�

jkB

!jkB

kB

!kB

or

�

! x,t( ) = !A +e" jkB xe j#t

! x,t( ) = !A"e+ jkB xe j#t

! x,t( ) = !A + je"kxe j#t

! x,t( ) = !A" jekxe j#t

Propagating waves

Near field





or a sum of all these four solutions

�

! x,t( ) = !A +e" jkB xe j#t

+!A"e+ jkB xe j#t

+!A + je"kxe j#t

+!A" jekxe j#t .

TASK: Test the same for sound in air! While the first two terms are propagating waves, as we are used to, the last

two terms are deformations of the beam, which decay exponentially with distance from their excitation point. (Obs! these are not static deformations, but they are oscillating with the

excitation frequency). These near fields can be caused by an excitation but also by the presence of boundary conditions as will be shown later. The figure shows the influence of the near fields when exciting an infinite beam in the middle by a point force. Please observe how fast the contribution of the near fields disappear with increasing distance from the excitation point. How fast are these bending waves. It is just using the dispersion relation and

calculate the speed:

�

k =!

c or

�

c =!

k. In our case:

�

cB = !B

m!2

4 =!

2B

m

4

The speed of the bending waves increases with the





root of the frequency There is no upper limit for the speed, which violates physics. The reason for this failure is due to the simplifications when deriving the theory. At higher frequencies the bending waves change to transversal waves which give the upper limit for the speed of sound. The error for the sound speed is about 10 % when the wavelength is on the order of six times the thickness of the beam. In opposite to the wave types presented before, bending waves have a phase velocity (speed of sound), which depends on the frequency. Such a wave type is called dispersive. For a non-dispersive medium all frequencies travel with the same speed. Therefore the shape of the signal will not change. For bending waves on a beam, lower frequencies travel slower than the higher frequencies and the form of the signal will

therefore be distorted. Imagine a source, which sends out a pulse. After a certain distance the pulse will be smeared out. First the high

frequencies will arrive and then the low frequencies. Kirchhoff theory for plates The Kirchhoff theory for bending waves in plates is based on the same simplifications as the Euler - Bernoulli theory. The wave equation is





�

Bx

! 4"

!x 4+ 2Bxz

! 2"

!x 2

! 2"

!z 2+ Bz

! 4"

!z4

+ m! 2"

!t 2= 0

�

Bx and

�

Bz are the bending stiffness in the x- and z-directions.

�

Bxz is a mixed term. For isotropic plates,

�

Bx and

�

Bz are identical while for orthotropic plates they can be different. A typical example of an orthotropic plate is a corrugated steel plate which is substantially stiffer in one direction (z- direction). An approximation

often used for the mixed term is

�

Bxz = BzBx .

The solution for the free waves on plates is similar to the solutions for beams. The physically properties are identical.

Torsional waves

In addition to the waves described in the previous text there are other wave types, e.g. membrane waves, torsional waves, Rayleigh waves, Lamb waves, etc..

Since the torsional waves are important in machinery especially due to the fact the torque (load) on rotating parts (e.g. crank shafts) is not constant over time. The wave speed for these torsional waves depend beside on material properties strongly on the geometry ( the rotational inertia depends strongly on the shape of the cross section) The wave equation for torsional waves is identical with the equation for transversal waves, but the dynamic quantities have different names

!2"

!x2=

1

ctor

2

!2"

!t 2 . (5)

The transversal displacement is replaced by a twisting angle ! , instead of a force a moment is acting on the structure. The speed of torsional waves can be calculated as





ctor

=T

! (6)

where ! is the polar moment of inertia and T the torsional stiffness. Both depend on the geometry. The wave speed can be written as

c

tor= G ! I

TI

p where

I

T is the radius of

inertia for the stiffness and I

p the radius of inertia for the polar moment.

To obtain these values can be tricky sometimes, but in the case of a circular cross section

I

T and

I

p are identical and the speed becomes

c

tor= G !





Bending waves - addition

As seen there is a limit for the theory of thin plates (

�

!B > 6h , h: thickness of the plate). There is no problem for thin steel plates, gypsum plates, etc. More difficult for concrete. Example 16 cm concrete, E=32-40 GPa, ρ=2300 kg/m3,

µ=0.15-0.2. The sound speed is

�

cB =Eh

3

12h!"2

4 or

�

cB = hf2!

12

cL " hf1.8cL

The speed of longitudinal waves on plates is

�

E

! 1 " µ 2( ). For concrete this is

about 3800 m/s that the speed of the bending waves is about

which means

�

cB = 0.16 2000 1.8 3800 ! 1480 m/s The wavelength is in this case

�

! = c f = 1480 2000 = 0.74m which is not bigger than 6 times the thickness.

What to do? Mr Mindlin made a more exact theory that allows for a correction of the speed for bending waves:

�

1

! c B3

=1

cB

3+

1

" 3cS

3 where

�

cS =G

! is the speed of shear waves and G the shear

modulus

�

G =E

2 1 + µ( )

The factor

�

! depends on the Poisson number

µ 0.2 0.3 0.4 0.5

! 0.689 0.841 0.919 0.955





Finite plates

Exactly as rooms or other limited spaces, plates will have resonances, i.e. there will be certain frequencies where the propagating waves fit perfectly to the boundary conditions (remember the tube, where we talked about modes).

What are typical boundary conditions? A wall could be considered as simply supported, i.e. there is no displacement at the edges but the cross section can rotate. Another possibility would be clamped. In this case the plate cannot rotate and there will be no displacement on the edges.

In reality often it can be assumed that the boundary conditions of walls are similar to simply supported.

In room acoustics we had as resonance (or eigenfrequency)

�

fn,m,l =cluft

2

n

Lx

!

"

# #

$

%

& &

2

+m

Ly

!

"

#

#

$

%

&

&

2

+l

Lz

!

"

# #

$

%

& &

2

For bending waves on a plate with the dimension Lx and Lz we get a little more difficult result (due to the more complicated speed of sound and the dispersion of the bending waves)

�

fnx ,nz=!

2

B

m

n x

Lx

"

#

$ $

%

&

' '

2

+nz

Lz

"

#

$ $

%

&

' '

2(

)

*

*

*

+

,

-

-

-





In the case of simply supported plates the vibration pattern at the resonance frequencies (often called mode-shapes) has the form

�

v nx ,nzx,z,t( ) = Anx ,nz

sinn x!

Lx

x"

#

$ $

%

&

' ' sin

nz !

Lz

z"

#

$ $

%

&

' ' e j(t

examples are

And the total field is a sum of all modes

�

v x,z,t( ) = Anx ,nzsin

n x!

Lx

x"

#

$ $

%

&

' ' sin

nz !

Lz

z"

#

$ $

%

&

' ' nz

Nz

(nx

N x

( e j)t

Even more complicated will it be for orthotropic plates and this we often have in practise (e.g. corrugated plates). For the two directions we can have different Young’s modulus (Ex an Ez), different Poisson number (µ� �� µ� ), different bending stiffness Bx and Bz.

1,1 mode

2,1 mode

2,2 mode





�

Bx =E xh

3

12 1 ! µ x µ z( ) and

�

By =E y h3

12 1 ! µ x µ z( )

The eigenfrequencies are

�

fnx ,nz=!

2

1

m

n x

Lx

"

#

$ $

%

&

' '

4

Bx +nz

Lz

"

#

$ $

%

&

' '

4

By + 2n x n y

LxLy

"

#

$

$

%

&

'

'

2

Bxy

(

)

*

*

*

+

,

-

-

-

1

2

A good approximation is

�

Bxy = BxBy The first resonances are important when looking on the radiation of sound insulation of structures. Often these resonances are very “audible”.

Response of structures

When dealing with sound the impedance is important saying what velocity response one expect for a given pressure

In structural acoustics the mobility is used instead.

�

Y x,z,!( ) =

u x,z,!( )F x,z,!( )

As long as only One Degree of Freedom (1DOF) is used mobilities and

impedances are related as

�

Z !( ) =1

Y !( )

The next two figures show results for mobilities measured on a simply supported plate. The first picture displays a mobility for a plate with little





damping. One sees clearly the resonances as peaks. As smaller the damping as higher these peaks. There also dips visible. These are the so-called antiresonances. At these frequencies it will be difficult to get energy into the plate. At low frequencies (below the first resonance, the response is like you work against a spring. Increasing the damping will lead to lower peaks and less visible anti-resonances. It is also visible that the mobility varies around some value. In the case of a plate this value is constant and represents the

mobility of an infinite plate

�

Yinfinite =1

8 Bm

Have in mind that the values for B and m have to have the right units. For a plate we need a mass per unit area for instance.

The mobility is a measure how sensitive a point of the structure is for the excitation by an external force. When mounting for instance a fan on a floor it is of advantage to find such points with a low mobility.

The mobility could locally increased by adding additional mass or by stiffening the structure. However, the last is often dangerous, as we will see in the next chapter on sound radiation.





Problems to section 10 2.1 Emil excites an infinite plate with a hit from a hammer and

measures the response 20m from the excitation position. How much later arrives a bending wave of the frequency 50 Hz

compared to a bending wave of 500 Hz?

The material data for the plate is, E=210 GPa, ν=0.3, ρ=1200 Kg/m3 and thickness=3mm.

2.2 Determine the phase speed (wave speed) for:

a) Longitudinal waves in an infinite body.

b) Quasi-longitudinal waves in a plate.

c) Transversal waves in a plate.

d) Bending waves in a plate.

2.1 Sketch the wave pattern for the different wave types

2.3 Bending waves propagates in a 15 cm thick concrete plate with the following material data: E = 2,5 !1010 N/m2, ρ=2300 kg/m3, ! " 0,3 .

a) Calculate the bending wavelength for the octave bands with centre frequencies between 63 and 1 kHz.

b) Above what frequency is not the simple bending wave equation valid any more and why?

waves in structures - · pdf filewaves in structures wolfgang kropp 10.1 chalmers university...

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