Waves in Structures
Wolfgang Kropp
10.1
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: c L,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
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10.2
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
c L,solid =
( )
! (1 + µ )(1 " 2µ ) .
1
E 1"µ
In plates the speed of sound is between both values c L,plate =
1
(
E
! 1 " µ2
)
. 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
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10.3
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.
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10.4
With simple geometrical considerations one can conclude that the bending
"#
angle ! is ! =
(i.e. the change of the normal displacement over the length
"x
of the beam).
The bending angle will cause a certain bending moment M which depends on
the bending stiffness B of the material and the change of the bending angle
"#
" 2$
over x M = !B
where the bending stiffness is B = EI (do you
= !B
"x
"x 2
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
"M
" 3#
direction is F = !
. The force is also changing
=B
3
"x
"x
over x. A particle is accelerated due to the net force
applied to it. This means that Newton's law will lead to
"F
" 2#
. m is the mass per unit length, with the cross
!
=m
"x
"t 2
section S and the density ρ , m = !S .
Finally we get the equation for Bending waves:
! 4"
! 2"
B
+m
=0
4
2
!x
!t
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!
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10.5
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 = !A e " jkx e j#t : which will give
B
! 4"
!x 4
+m
! 2"
!t 2
( )
= B jk
4
( )
2
"A e # jkx e j$t + m j$ "A e # jkx e j$t = 0
Such equations are called dispersion equation. They give us the relation
( )
between frequency and wavelength (wavenumber): B jk
k4 =
! 2m
B
4
( )
+ m j!
2
=0
This has for solutions. Lets call
the wavenumber k B as k B =
jk B
the solutions are:
! jk B
kB
!k B
4
m! 2
( )
!(x, t ) = ! e
or
!(x, t ) = ! e
!(x, t ) = ! e
! x, t = !A +e " jkB x e j#t
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B
A"
+ jkB x
A+ j
A" j
"kx
kx
Propagating waves
e j#t
e j#t
e j#t
Near field
Waves in Structures
Wolfgang Kropp
10.6
or a sum of all these four solutions
( )
! x, t = !A +e " jkB x e j#t + !A "e + jkB x e j#t + !A + j e "kx e j#t + !A " j e kx e 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 = or c = . In our case:
c
k
2
B
! B
cB = ! 4
=4
m
m! 2
The speed of the
bending waves
increases with the
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10.7
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
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10.8
Bx
! 4"
! 2" ! 2"
! 4"
! 2"
+ 2B xz
+ Bz
+m
=0
!x 4
!x 2 !z 2
!z 4
!t 2
B x and B z are the bending stiffness in the x- and z-directions. B xz is a mixed
term. For isotropic plates, B x and B z 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 B xz = B z B x .
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"
!x 2
=
1 ! 2"
2
c tor
!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
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10.9
c tor =
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 T I 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 !
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10.10
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 c B = 4
Eh 3
12h!
" 2 or c B = hf
The speed of longitudinal waves on plates is
(
E
! 1"µ
2
)
2!
12
c L " hf 1.8c L
. For concrete this is
about 3800 m/s that the speed of the bending waves is about
which means c B = 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
3
B
c!
=
1
c
3
B
+
1
3 3
S
" c
where c S =
G
!
is the speed of shear waves and G the shear
modulus G =
E
( )
21+µ
The factor ! depends on the Poisson
number
µ 0.2
!
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0.3
0.4
0.5
0.689 0.841 0.919 0.955
Waves in Structures
Wolfgang Kropp
10.11
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)
f n ,m,l
2
c luft ! n $
# &
=
# &
2 " Lx %
!
m
+#
#L
" y
$2 !
& +# l
& #" L
z
%
$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)
f n x ,nz
(
2
"
%
"n
! B * nx
z
$
'
=
*$ ' + $$
2 m *# Lx & # Lz
)
+
%2 ' '
& ,
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Wolfgang Kropp
10.12
In the case of simply supported plates the vibration pattern at the resonance
frequencies (often called mode-shapes) has the form
(
)
v n x ,nz x, z , t = A n x ,nz
"n ! % "n ! %
sin$$ x x '' sin$$ z z ''e j(t
# Lx & # Lz &
examples are
2,1 mode
1,1 mode
2,2 mode
And the total field is a sum of all modes
(
)
N x Nz
v x, z , t = ( ( A n x ,nz
n x nz
"n ! % "n ! %
sin$$ x x '' sin$$ z z ''e j)t
# Lx & # Lz &
Even more complicated will it be for orthotropic plates and this we often have
in practise (e.g. corrugated plates). For the two directions we can have
different Young’s modulus (Ex an Ez), different Poisson number (µ
µ ),
different bending stiffness Bx and Bz.
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10.13
Bx =
(
E xh3
12 1 ! µ x µ z
)
and B y =
(
E y h3
12 1 ! µ x µ z
The eigenfrequencies are f n x ,nz
)
(
4
"n n
" n %4
! 1 *" n x %
x
y
z
=
*$$ '' B x + $$ '' B y + 2$$
2 m *# Lx &
# Lz &
# Lx Ly
)
1
%
'
'
&
2
+2
B xy ,
A good approximation is B xy = B x B y
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
u x, z, !
In structural acoustics the mobility is used instead. Y 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
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10.14
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 Y infinite =
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.
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10.15
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.
Above what frequency is not the simple bending wave equation valid
any more and why?
b)
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