Torsional and flexural wave propagation constants in a periodic three-layered sandwich
grillage: A numerical study
Lt. R. Murali Krishna, Venkata Mangaraju and Venkata R. Sonti
Facility for Research in Technical Acoustics
Department of Mechanical Engineering
Indian Institute of Science, Bangalore
Abstract
Dominantly propagating branches for flexural and torsional wave propagation in a
periodic three-layered grillage are presented. There exist several other branches, which
are unimportant due to their highly attenuated nature.
1. Introduction
One of the typical 2-D periodic systems which serves as a simple model for foundation
structures such as in ships and submarines, hulls of ships, satellite launching booms and
even curved structures such as an aircraft fuselage is a grillage. As shown in figure 1, a
grillage is a structure made of connected cross beams. The cross beams at a junction are
at right angles to each other. This grillage can carry two flexural waves, an in-plane
longitudinal wave and a torsional wave.
In this paper, we present the approximate wave propagation behavior of a threelayered grillage. This grillage carries one out-of-plane flexural wave and a torsional wave
(of the four waves mentioned above). As with the infinite periodic beam, so also in
grillages propagation zones (pass bands) and attenuation zones (stop bands) do occur [1].
However, since two distinct waves are being propagated and within three layers, one
should expect several branches of these propagation constants. In order to include
damping, the central layer can be assumed to be visco-elastic [1].
Figure 1. A periodic grillage.
2. Methodology
1. The differential equations and boundary conditions for a three-layered beam
carrying flexure are obtained using the Hamilton's principle with Timoshenko
kinematic assumptions.
1
2. The differential equations and boundary conditions for a three-layered torsional
beam carrying torsion in each of its layers are obtained using the Hamilton's
principle using some simplified kinematic assumptions.
3. Using the two models above, a plus beam (two beams forming a cross) is
modelled. In turn, a flexural wave and a torsional wave are made to be incident
onto the single plus junction. The resulting reflections, transmissions and
conversions of wave types are computed.
4. The transmission and reflection coefficients from the above calculation are used
in formulating the periodic repetition of the plus beam to form a grillage. The
steady state vibrations in neighboring spans are now related through the
propagation constants.
3. Three-layered flexure beam
Consider a three-layered beam with densities 1, 2, 3, Young's moduli E1, E2, E3, and
shear moduli G1, G2, G3, respectively. The three thicknesses are d1, d2, d3, respectively
and the width of the beam is b. The beam is allowed to have shear strain (Timoshenko
kinematic assumptions) and longitudinal motion in the horizontal plane and transverse
motion in y direction. Each layer has it's own shear deformation and rotation but the
transverse motion at any x location is the same for all layers. As can be seen from Figure
2, the in-plane displacements (also known as longitudinal displacements) and the
corresponding strains in the three layers are given by
and
1   L  y1
 2   L  h1 1  ( y  h1 )  2
d
 3   L  h1  1  d 2  2  ( y  h1  d 2 )  3 and  1  1   ' L  y '1
dx
d

d 2
2 
  ' L  h1  '1  ( y  h1 )  ' 2 ,  3  3   ' L  h1  '1  d 2  ' 2  ( y  h1  d 2 )  ' 3 ,
dx
dx
and the shear strains are given by
d
d
d
d
d
d
1  1 
 1   ' ,
2  2 
 2 ' ,
3  3 
 3  ' .
dy dx
dy
dx
dy
dx
The potential and kinetic energies are given by
l U
l U
3
3
b
b
2
2
Epot =    ( Ei  i  Gi  i )dydx
and
Ekin=    i (i2   2 )dydx .
2 l L i 1
2 l L i 1
Application of the Hamilton's principle, results in a set of partial differential equations
for L, , 1, 2, 3 and the boundary conditions. In the partial differential equations, for
each variable we substitute a wave solution of the form A e j (t  kx ) . The determinant of the
5x5 matrix equations gives the dispersion relation. There are five different free
wavenumbers at a given frequency, but in the frequency range of interest only two of
them are real and so correspond to the propagating waves. Of the two propagating
wavenumbers, one is related to flexure and the other to the longitudinal wave.
Throughout this paper we neglect the longitudinal wave. From the remaining set of 4x4
matrix equations we can find not only the free wavenumbers but these homogenous
algebraic equations give the relation amongst 1, 2, 3 and  at a given frequency and
wavenumber (since different motions are coupled).
2
Figure 2. Schematic for three-layered beam flexure kinematics.
4. Three-layered torsional beam theory
In this section, the kinematics of torsion in a three-layered torsional beam are derived
along with the differential equations and the dispersion relation. All the three rotations
(1, 2, 3) are assumed small. Also the cross section warping due to torsion is neglected.
This assumption has important consequences. The rotation is continuous across the
layers. However, only the horizontal displacement is continuous across the layers and not
the vertical displacement.
Let 1, 2, 3 be the rotations in each of the three layers in the beam of width b
and d1, d2,d3 be the layer thicknesses, respectively (figure 3). Let h1 be the distance of the
top fibre of the first layer from the neutral axis. The cross section along with the torsion
kinematics and coordinate system is shown in figure 3.
u1   z1 , and v1  y1 ,
u2   z 2  h1 ( 2  1 ), and v2  y 2 ,
u3  h11  d 2 2  ( z  h1  d 2 ) 3 , and v3  y 3 ,
The kinetic and potential energies are again formulated and the procedure in step 2 is
followed. For every  three real torsional wavenumbers are obtained, one for each .
The homogenous equations relate the amplitudes of the three s at each wavenumber.
5. Three-layered infinite cross beams
In this section, two three-layered infinite intersecting cross beams are considered (as
shown in figure 4. First a flexural wave at a single frequency (with only the flexural
propagating wavenumber k1) and amplitude A1 is made incident onto the junction along
the beam denoted as section 1 in figure 4. Since we have neglected the longitudinal wave
in this study, there exist one flexure and three rotations in section 1. There will be three
rotations related to this incident flexure at the k1 wavenumber whose relative amplitudes
can be found from the homogenous algebraic equations. This flexural wave gets
transmitted into section 3 of the cross beam as flexure and as rotations with all four
wavenumbers. This flexure also gets transmitted as torsion and flexure onto the
perpendicular section 2. In section 1, the flexure gets reflected as flexure and rotations.
3
Figure 3. Schematic of three-layered torsion kinematics.
The flexure and three rotations in section 1 are given by
w1  A1 (i)e  jk1x  A1 (r )e jk1x  A1 (r11 (r ))e jk2 x  A1 (r 21 (r ))e jk3 x  A1 (r 31 (r ))e jk4 x
11  11 (iA1 (i))e  jk x  11 (rA1 (r ))e jk x  11 (r )e jk x  11 (r 21 (r ))e jk x  11 (r 31 (r ))e jk x
1
1
3
2
4
 21   21 (iA1 (i))e  jk x   21 (rA1 (r ))e jk x   21 (r11 (r ))e jk x   21 (r )e jk x   21 (r 31 (r ))e jk x
1
1
2
3
4
 31   31 (iA1 (i))e  jk x   31 (rA1 (r ))e jk x   31 (r 11(r ))e jk x   31 (r 21 (r ))e jk x   31 (r )e jk x
1
1
2
3
4
The `r' in the brackets implies a reflected wave, ‘I’ represents the incident wave. The
doubled subscripted ij on the left hand side of the equations denote the total rotations in
the three layers. The first subscript `i' denotes the layer and the second subscript `j'
denotes the side (see figure 4). The underlined variables are the unknowns to be
computed. There are four unknowns in section 1. All others variables can be computed
from the homogenous algebraic equations mentioned above once these are known. There
are in all 15 unknowns and 15 continuity equations are needed at the junction in terms of
kinematics and force/moment relations. When a torsional wave is incident there will be
accordingly be 10 unknowns. The solutions give the transmission and reflection
coefficient values needed for the periodic grillage.
6. Wave propagation in a three-layered periodic grillage
Considering a single junction in the grillage, if a flexural or a torsional wave with
amplitude v0 is propagating along the horizontal beam onto the section (n,m) in figure 5,
then the reflected and transmitted amplitudes of various waves can be computed using the
methods given above.
The velocity in a given section (say (n,m)) is the sum total of infinite reflections
of waves propagating within the section (n,m) after initially being transmitted into this
section due to flexure or torsion in the neighbouring sections. These summed velocities
and rotations for a particular section (n,m) are denoted by (see figure 5) vn,m+, vn,m-,
n,m+, and n,m-. The subscripts denote the sections, and v and  denote flexure and
torsion, respectively. The '+' sign indicates a right travelling wave and '-' denotes a left
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traveling wave. We consider a single dominant torsional wavenumber (instead of three
for the original system) and reduce the system order.
Figure 4. Two three-layered cross beams
Figure 5. Schematic of a periodic grillage showing spans.
Let us consider the section (n,m) in figure 5. The waves from neighbouring sections
contributing to vn,m+ which is a wave travelling to the right are (1) transmitted component
of vn-1,m+, (2) right travelling component of vn+1,m-: after one reflection at the left end of
the section (n,m), (3) component of the bending wave (vm-1,n+), (4) component of bending
wave (vm-1,n+1+), (5) component of the bending wave (vm,n+), (6) component of the
bending wave (vm,n+1+), (7) component of the torsional wave (m-1,n+) inducing flexure in
the section (n,m), (8) component of the torsional wave m-1,n+1+, (9) component of the
torsional wave m,n+, and (10) component of the torsional wave m,n+1+. These 10
contributions will go through infinite reflections within the section (n,m). The
reflections form an infinite geometric series and hence are summed up accordingly.
Similarly, all the other 7 waves can be found. Then, periodicity conditions are imposed
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on neighboring spans in terms of propagation constants, resulting in a set of homogenous
matrix equations. The solution of the determinant gives the propagation constants. A case
is presented for a 1cm thick three-layered beam. All three layers have similar material
constants (close to steel). This can be verified by a homogenous grillage.
Figure 6. Real and imaginary part of propagation constants for three-layered grillage.
7. Results and discussion
In a plus beam, like in figure 2, when a flexural wave is incident from section 1 onto the
junction it gets transmitted into section 3 quite easily even at low frequencies [1].
Transmission to flexure in the cross beam is constant with frequency and transmission to
torsion is efficient only when the flexural and torsional impedances are equal. In
constrast, torsional waves do not propagate easily at low frequencies. Most of it remains
reflected and confined to the individual section [1]. In figure 6, there are two real
constants one of which is quite high implying attenuation. We believe that this is related
to the torsional wave because of the reasons mentioned. The other real value follows the
regular flexural curve. Since nearfield waves have been neglected, the very low
frequency values are not accurate. We mention here that only a single dominant torsional
wave has been considered. Actually, each layer propagates a torsional wave.
References
1. M.A.Heckl, “Investigations on the vibrations of grillages and other simple beam
structures.” The Journal of the Acoustical Society of America 36(7) 1335-1343,
(1964).
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