Numerical Analysis of Unsteady Vibrations of a Plate Resting on an
Elastic Isotropic Half-Space
MARINA V. SHITIKOVA1,2, ANNA S. BESPALOVA1
1Research Center on Dynamics of Structures,
Voronezh State Technical University,
20-Letija Oktjabrja Str. 84, Voronezh 394006,
RUSSIA
2Department of High Mathematics,
National Research Moscow State University of Civil Engineering,
Yaroslavskoye Shosse 26, Moscow 127238,
RUSSIA
Abstract: - The paper is devoted to the numerical solution of the problem of vibrations of an infinite elastic
plate resting on an elastic isotropic half-space using the analytical method based on the ray method with its
numerical realization via the Maplesoft package. Unsteady oscillations are caused by the action of
instantaneous loads on the plate, resulting in the appearance of two plane wave surfaces of strong discontinuity
in the elastic half-space, behind the fronts of which, up to the contact boundary, the solution is constructed
using ray series. The unknown functions entering the coefficients of the ray series and the equation of plate
motion are determined from the boundary conditions of the contact interaction between the plate and the half-
space. Previously, the approximate solution of this problem was obtained analytically without using
mathematical packages, and the dynamic deflection of the plate involving only the first three terms of the ray
series was written down. In this work, a two-layer medium with different properties was investigated using an
algorithm developed to solve contact dynamic problems related to the occurrence and propagation of strong and
weak discontinuity surfaces.
Key-Words: - Dynamic contact, infinitely long elastic plate, ray method, truncated ray series, nonstationary
vibrations, conditions of compatibility, theory of discontinuities.
Received: April 5, 2023. Revised: December 17, 2023. Accepted: February 11, 2024. Published: April 3, 2024.
1 Introduction
Problems devoted to the analysis of the impact
interaction of solids remain relevant to date, as they
are widely used in various fields of science and
technology, [1], [2], [3], [4], [5]. The physical
phenomena involved in the process of impact
interaction include dynamic reactions of contacting
bodies, effects of contact conditions, and wave
propagation. Since such problems belong to the
class of problems of dynamic contact interaction,
their solution is associated with significant
mathematical and computational difficulties, which
include not only the complex equations describing
the dynamic behavior of a continuous medium but
also the variety of boundary conditions on the
contacting surfaces of solids, [6], [7], [8], [9], [10],
[11].
All dynamic contact problems can be divided
into two types. The first type includes the problems
related to the propagation of harmonic vibrations
and waves (bodies are either in constant contact
with each other or in prolonged contact). The
second type involves the problems related to the
propagation of surfaces of strong or weak
discontinuity, as well as the problems leading to
unsteady oscillatory motions (short-term contact of
bodies, or shock interaction), [1].
To solve the problems of the first and second
types, different mathematical methods are used.
Among the main methods for solving problems of
the second type should be mentioned the following:
the method of invariant-functional solutions [3], the
Wiener-Hopf method [3], the method of
characteristics [12], various numerical methods, and
the ray method, [1], [6], [9], based on the theory of
geometrical optics and its generalizations.
The ray method is most effective in solving
problems of propagation and attenuation of transient
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.2
Marina V. Shitikova, Anna S. Bespalova
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Volume 19, 2024
waves carrying jumps of field parameters at the
wavefront, [1].
The zero term of the ray series exactly describes
the changes in the jump of field parameters along
the ray, but the rest of terms within the radius of the
series convergence describe the changes in the field
behind the wavefront, [1], [6].
The question of application of ray series to the
problems of transient wave problems has been
considered by many researchers, and yet, until
recently, the area of its practical applicability, which
is largely determined by the possibility of
calculating a sufficient number of coefficients of the
ray series. sufficient number of ray series
coefficients and the radius of convergence of the ray
series, remained poorly studied, [1], [13].
Since in practice one has to limit oneself to finite
truncated ray series to construct a solution, instead
of the question of convergence, the problem of
uniform convergence of the solved truncated ray
series in the region of existence of wave motion is
most often solved.
All ray series used could be divided into two
main groups [1]. The ray series of the first group are
mainly used in wave dynamics problems for
approximation of physical fields of regular
functions. The second type of ray series is used for
the approximation of physical fields of singular
functions.
The majority of the studies devoted to ray
methods based on ray series of both types deal with
the volume wave investigation. However, ray
methods are also successfully applied to the study of
waves propagating along the free surface, waves
propagating along the interface between two media,
and nonstationary Love waves, [1], [5], [13], [14].
This paper presents a description of an algorithm
developed based on the Maplesoft package for
solving contact dynamic problems related to the
generation and propagation of surfaces of strong and
weak discontinuity by an analytical method based
on the ray method. Numerical investigations of a
two-layer medium with different properties have
been performed based on this algorithm.
2 Problem Formulation
The solution of the problem of unsteady vibrations
of a plate of constant thickness on an elastic
isotropic half-space was constructed in [15], for the
case of a sliding contact between the plate and the
half-space. The ray method, [1], was used as a
method of solution, which allowed one to obtain the
time dependence of the plate displacements in an
analytical form. In this case, the author of [15],
managed to determine only the first three terms of
the ray series "manually".
The purpose of this paper is to develop an
algorithm based on the Maple software package for
solving various contact dynamic problems related to
the generation and propagation of strong and weak
discontinuity surfaces using the ray method. This
approach will make it possible to determine a
sufficiently large number of members of the ray
series, which, in turn, allows one to obtain a solution
with a sufficiently large accuracy.
For this purpose, following [15], let us consider a
layer on a foundation, which is modeled by a half-
space with x, y, z coordinate system (Figure 1).
Fig. 1: Calculation scheme
The layer and the supporting half-space are
homogeneous, isotropic, and linearly elastic. The
plane deformed state is realized, in which the
displacement component in the y-axis direction is
zero, and the displacements along the x and z-axes
are the functions of the x and z coordinates,
respectively.
The dynamic behavior of the layer is modeled by
a classical plate, the equation of motion of which
has the form:
342
1
1
42
1
4μ 2ρ0
3(1 ν)
hWW
hp
xt
, (1)
where W(x, t) is the displacement of the middle
plane of the plate (z=0), 2h is the plate thickness, μ1,
ν1 and ρ1 are shear modulus, Poisson's ratio and
density of the plate material, respectively, and p(x,t)
is the transverse surface load.
The set of equations describing the dynamic
behavior of an isotropic half-space has the form:
2
2
σρ
ij i
j
u
xt
,
σ λ δ μ j
ki
i j i j
k j i
u
uu
x x x
(2)
where are the components of the stress
tensor and displacement vector, respectively, is
the density of the half-space material, λ and μ are
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Lame parameters, δij is the Kronecker symbol, Latin
indices take on values 1 and 3, i.e.
.
The plate and the half-space are in contact at
. For the case of the sliding contact, the boundary
conditions could be written as follows:
33
σ ( , , ) ( , )x h t p x t
,
13
σ ( , , ) 0x z t
,
3( , , ) ( , )u x z t W x t
. (3)
Unsteady plate vibrations are excited by sending
velocities to the points of the plate at the initial
moment of time
()
0
Wcx
tt
, (4)
where c(x) is a given function.
3 Method of Solution
As a result of instantaneous action of velocities (4)
on the plate, two surfaces of strong discontinuity
(volume waves of compression and shear) appear in
the half-space, behind the fronts of which the
solution for desired functions is constructed in the
form of the ray series, [1], [9]
()
0
1
( , , ) ,
!
k
k
k
z h z h
Z x z t Z t H t
k G G
(5)
where are jumps in the k-th
order time derivatives of the functions Z on the
shock wave fronts ∑, i.e. at t=(z-h)/G, H(t) is the
unit Heaviside function, and G is the shock wave
velocity.
To determine the coefficients of the ray series
(5), it is necessary to differentiate the first equation
in (2) k times and the second equation k+1 times in
time, write down their difference on different sides
of the wave surface and apply the conditions of
compatibility, [1], for jumps of k+1 derivatives of
the functions Z (x, z, t):
,( )
,( )
,( 1)
,( )
1
1
,
νν
δ
k
k
k i i
i
k
i
dZ
Z
GZ
x dt
Z
Gx
(6)
where are the components of the unit normal
vector to the wave surface, is the Kronecker
symbol, and d/dt is the Thomas delta-derivative,
[16].
As a result, the following set of recurrence
equations is obtained:
()
2
( 1)
()
( 1) ,
ω
ρ (λ+2μ) ω 2(λ+2μ)
(λ+μ)
k
k
k
k
d
Gdt
w
GF
x
(7)
()
2
( 1) ( 1) ,
ω
ρ μ 2μ (λ μ)
kk
kk
dw
G w G Ф
dt x
(8)
where
( ) ,( )
ων
k i k i
v
,
( ) ,( ) 1
δ
k i k i
wv
,
,(1)ii
vu
,
2
22
( 1) ( 1)
2
( 1) 2
( 1) ,
ωω
(λ 2μ) μ
(λ μ)
kk
k
k
dd
FG
dt dx
w
d
Gdt x
22
( 1) ( 1)
2
( 1) 22
( 1) .
μ (λ 2μ)
ω
(λ μ)
kk
k
k
d w d w
ФG
dt d x
d
Gdt x
Solution of the recurrence equations (7) - (8) at k
= -1,0,1,2... results in the expressions for
determining the velocity of the first wave
2
(1)
ρ (λ+2μ)G
and the second wave
2
(2)
ρμG
, as
well as the values of and on the both
waves up to arbitrary functions, where the upper
index in parentheses denotes the ordinal number of
the wave, and the lower index in parentheses
denotes the ordinal number of the jump.
Since in this paper, unlike [13], mathematical
calculations for finding these quantities are carried
out using the Maple computer algebra system, no
restrictions are imposed on the number of defined
members of the ray series.
The solution is reduced to integration of the
differential equations of motion of the medium (7)-
(8) subjected to the initial conditions at each wave.
Thus, on the first wave
(1)
( 1) 0w
,
(1)
( 1)
ω0
,
(1)
(0) 0w
,
(1)
(0) 0
ω ( )fx
(9)
Next, a loop is written to determine the N values
of and on the first wave:
for i from 1 to N do
(1) (1)
( 1) ( 1)
(1) (1)
() 2
(1)
2 (1) 2 (1)
2
( 2) ( 2)
(1)
22
(1)
( 2)
(1) ,
ω
12μ (λ+μ)
ρμ
μ (λ+2μ)
ω
(λ+μ)
ii
i
ii
i
dw
wG
dt x
G
d w d w
G
dt d x
d
Gdt x
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(1) 2 (1)
( ) ( 1)
(1) (1)
() 2
2 (1) (1)
2( 1) ( 1)
(1) (1)
2
ω
1
ω (λ+μ) (λ+2μ)
2(λ+2μ)
ω
+μ (λ+μ) ( );
ii
i
ii
i
wd
Gxdt
dw
d
G G dt f x
dt x
dx
end do
Similar actions are performed on the second
wave:
(2)
( 1) 0w
,
(2)
( 1)
ω0
,
(2)
(0) 0w
,
(2)
(0) 0
ω ( )gx
(10)
for i from 1 to N do
(2)
( 1)
(2)
() 2
(2)
(2) 2 (2) 2 (2)
2
( 1) ( 2) ( 2)
(2) (2)
22
(2)
( 2)
(2) ;
ω
1
ω 2(λ+2μ)
ρ (λ+2μ)
ωω
(λ+μ) (λ+2μ) μ
(λ+μ)
i
i
i i i
i
d
dt
G
w d d
GG
xdt d x
w
d
Gdt x
(2) 2 (2)
( ) ( 1)
(2) ( 2)
() 2
2 (2) (2)
2( 1) ( 1)
(2) ( 2)
2
ω
1(λ+μ) μ
2μ
ω
(λ+2μ) (λ+μ)
( );
ii
i
ii
i
dw
wG
xdt
dw d
G G dt
dt x
dx
gx
end do
As it is known, [6], [12], behind the front of the
strong discontinuity surface, the solution for the
desired function is constructed in the form of a ray
series (5), wherein the jumps of the k-th order time-
derivatives of the function Z on the wave surface are
calculated at t=(z-h)/G. Therefore, the next action
for each of the defined quantities and is to
replace the parameter t by t=(z-h)/G.
The obtained jumps allow one to define the
desired functions and for the half-space in the
form of truncated ray series
2(α)
1 ( 1)
(α) (α)
α=1 1 (α)
( )/
1
!
k
N
k
kt z h G
z h z h
u t w H t
kGG
(11)
2(α)
3 ( 1)
(α) (α)
α=1 1 (α)
( )/
1ω
!
k
N
k
kt z h G
z h z h
u t H t
kGG
(12)
where N is the number of considered terms in the
ray expansion.
To construct the solution for the plate, the ray
series (11) and (12) and the ray expansions for the
stresses
13
σ
and
33
σ
should be written at the contact
boundary:
33
zh
uu
,
11
zh
uu
,
13 13
σσ
zh
,
33 33
σσ
zh
(13)
At the next stage, the obtained solutions are
spliced together on the contact boundary. For this
purpose, the contact conditions (3), the initial
condition (4) and the equation of motion of the plate
(1) should be considered. The values involved in
equation (1) are written as follows:
()
1
1
!
Nk
k
k
W W t
k
,
()
1
1
!
Nk
k
k
p p t
k
(14)
Substituting relationships (11) and (12) into the
above equations and equating the coefficients at the
same orders of t, arbitrary functions
()
()
a
fx
,
()
()
a
gx
,
( 0,1,2,3...)a
are determined at each step, as well
as the required values . Then, taking
the obtained values into account and considering
(14), the ray series could be constructed to
determine the displacement of the plate and the
reaction force of the half-space.
4 Numerical Results
To carry out numerical investigations, let us assume
for certainty that
0,( ) cos lx
c x W h
(15)
where are some constants.
Then, the four-term truncated ray series for W
will take the form:
3
23
24
(1) 2 (1) 3
1
2
11
11
34 (1)
(1) 2
(2)
1
2 3 3
11 1 1
4
(1) (2) (1)
0
2 μ
ρρ
2ρ 2 2ρ 6
3ρ (1 ν )
2μρ
ρρ
4
2ρ 3ρ (1 ν ) 2ρ
1
4 cos .
26
l
G t G t
Wt hh h
lG
Gl
G
hhh
tl
G G G W x
h
(16)
The main attention in studies of unsteady
vibrations of a plate is paid to its dynamic
displacement, because this value is important for
practice. For this purpose, it is more convenient to
rewrite the resulting expression for the displacement
in the dimensionless form W* using the following
relations:
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(1)
*
0
GW
WhW
,
(1)
*G
tt
h
,
*1
ρ
ρρ
,
*1
μ
μμ
,
*x
xh
,
(2)
*
(1)
G
GG
.
(17)
We will study a two-layer medium with the
following constant parameter values: cos(lx*)=1,
l=1, . The results for four
examples will be evaluated in terms of amplitude
and period depending on different values of
dimensionless quantities of
*
μ
.
First example. Let us consider the first variant of
the two-layered medium: the plate is light and rigid
with
*
μ
(the choosing
ratios of parameters correspond to light concrete and
loam soil with porosity coefficient e=0.45).
The results of calculations obtained with the
developed algorithm based on the Maplesoft
considering different number of terms in the ray
expansion are shown in Figure 2 for N = 4 – 22, 25,
30.
To estimate the amplitude and period of
oscillations during dynamic contact interaction,
characteristic time points were determined, namely:
values of t* at which the dimensionless deflection
W* attains its maximal and minimal magnitudes, i.e.
extrema of the function W*, and magnitudes of t*
corresponding to odd and even half-periods, i.e.
when W*=0 (data are given in Table 1, Table 2,
Table 3 and Table 4 for the first and third
examples).
It is seen from Figure 2 that the curves plotted
for a large number of members of the series almost
coincide and repeat the trajectory, indicating the
convergence of the solution with the increase in the
number of terms of the ray series.
Having analyzed the data from Table 1, we could
draw a conclusion that the difference in the values
of t* when dynamic deflection attains its extremum
magnitude W*=W*max calculated at N=4 (the
number of terms used in the "manual" calculation)
and at N=7 is around 10%, while for the first half-
period, i.e. when W*=0, the difference is 19.5%. It
could be also seen from Figure 2 that the period of
oscillation is approximately around the value of
t*=0.6. So, if we are interested in the period of
oscillation, it is necessary to determine 14 or 16
terms of the ray series. If we need to advance in
time, it is worth using the expansion in terms of a
30-term truncated ray series for a more reliable
picture. And if the task is to determine the
maximum displacement, to find the maximum
stresses to check the local strength, then in principle
it is sufficient to restrict oneself by 4 terms of the
ray expansion or to determine 5 terms for a more
accurate value.
Fig. 2: The dimensionless time t* dependence of the
dimensionless deflection W* for the first example at
different numbers of the ray expansion terms:
-N=4
-N=15
-N=5
-N=16
-N=6
-N=17
-N=7
-N=18
-N=8
-N=19
-N=9
-N=20
-N=10
-N=21
-N=11
-N=22
-N=12
-N=25
-N=13
-N=30
-N=14
Table 1. Magnitudes of time t* at which the
oscillation amplitude attains its extreme values for
the first example according to Figure 2
Number of
terms of
the series
The t* value for W*=W*extr
Ordinal number of extremum
1 2 3 4
N=4
0.129
-
-
-
N=5
0.146
0.267
-
-
N=6
0.144
0.287
-
-
N=7
0.143
-
-
-
N=8
0.143
-
-
-
N=9
0.143
0.378
-
-
N=10
0.143
0.393
-
-
N=11
0.143
-
-
-
N=12
0.143
-
-
-
N=13
0.143
0.429
-
-
N=14
0.143
0.431
-
-
N=15
0.143
0.434
0.601
-
N=16
0.143
0.434
0.634
-
N=17
0.143
0.433
-
-
N=18
0.143
0.433
-
-
N=19
0.143
0.433
0.694
-
N=20
0.143
0.433
0.711
-
N=21
0.143
0.433
0.741
0.785
N=22
0.143
0.433
0.727
0.856
N=25
0.143
0.433
0.724
0.929
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N=30
0.143
0.433
0.724
1.013
The second example: the plate (layer) and the
base have the same density, i.e. , and the
plate is rigid, i.e.
* 665.24
. The results of
calculations considering different number of terms
in the ray expansion are shown in Figure 3 for N = 4
– 22, 25, 30.
Table 2. Summary table of half-periods for the first
example
Number of
terms of
the series
The t* value at W*=0
Ordinal number of the half-period
1 2 3 4
N=4
0.227
-
-
-
N=5
-
-
-
-
N=6
-
-
-
-
N=7
0.282
-
-
-
N=8
0.285
-
-
-
N=9
0.292
0.436
-
-
N=10
0.291
0.461
-
-
N=11
0.291
-
-
-
N=12
0.291
-
-
-
N=13
0.291
0.538
-
-
N=14
0.291
0.555
-
-
N=15
0.291
-
-
-
N=16
0.291
0.591
0.667
-
N=17
0.291
0.579
-
-
N=18
0.291
0.581
-
-
N=19
0.291
0.582
0.767
-
N=20
0.291
0.581
0.803
-
N=21
0.291
0.581
-
-
N=22
0.291
0.581
-
-
N=25
0.291
0.581
0.879
0.966
N=30
0.291
0.581
0.872
1.135
Fig. 3: The dimensionless time t* dependence of the
dimensionless deflection W* at different numbers of
the ray expansion terms (designations are the same
as in Figure 2)
The third example: heavy rigid plate with
(these ratios of parameters
correspond to heavy concrete and loam soil with
porosity coefficient e=0.45). The results of
calculations considering different number of terms
in the ray expansion are shown in Figure 4 for N = 4
– 22, 25, 30. A summary table of t* values at which
the oscillation amplitude attains its extremes and
half-period values are compiled in Table 3 and
Table 4, respectively.
Fig. 4: The dimensionless time t* dependence of
dimensionless deflection W* for the third example
at different numbers of the ray expansion terms
(designations are the same as in Figure 2)
Table 3. Time t* values at which the oscillation
amplitude attains its extremes for the third example
Number of
terms of
the series
The t* value at W*=W* extr
Ordinal number of extremum
1 2 3 4
N=4
0.118
-
-
-
N=5
0.134
0.251
-
-
N=6
0.133
0.260
-
-
N=7
0.131
-
-
-
N=8
0.131
-
-
-
N=9
0.131
0.350
-
-
N=10
0.131
0.358
-
-
N=11
0.131
-
-
-
N=12
0.131
-
-
-
N=13
0.131
0.393
-
-
N=14
0.131
0.394
-
-
N=15
0.131
0.397
0.558
-
N=16
0.131
0.397
0.575
-
N=17
0.131
0.396
-
-
N=18
0.131
0.396
-
-
N=19
0.131
0.396
0.639
-
N=20
0.131
0.396
0.647
-
N=21
0.131
0.396
0.670
0.737
N=22
0.131
0.396
0.665
0.770
N=25
0.131
0.396
0.661
0.858
N=30
0.131
0.396
0.661
0.923
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.2
Marina V. Shitikova, Anna S. Bespalova
E-ISSN: 2224-3429
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Volume 19, 2024
Table 4. Summary table of half-period values for
the third example
Number of
terms of
the series
The t* value at W*=0
Ordinal number of the half-period
1 2 3 4
N=4
0.207
-
-
-
N=5
-
-
-
-
N=6
-
-
-
-
N=7
0.258
-
-
-
N=8
0.260
-
-
-
N=9
0.266
0.407
-
-
N=10
0.265
0.419
-
-
N=11
0.265
-
-
-
N=12
0.265
-
-
-
N=13
0.265
0.497
-
-
N=14
0.265
0.505
-
-
N=15
0.265
-
-
-
N=16
0.265
0.541
0.602
-
N=17
0.265
0.528
-
-
N=18
0.265
0.529
-
-
N=19
0.265
0.530
0.709
-
N=20
0.265
0.530
0.728
-
N=21
0.265
0.530
-
-
N=22
0.265
0.530
-
-
N=25
0.265
0.530
0.799
0.898
N=30
0.265
0.530
0.795
1.026
Analyzing the data, as before, we obtain
differences in amplitude of 10% and in period of
20%.
The fourth example: very heavy and rigid plate
with (these ratios of
parameters correspond to extra heavy concrete and
loam soil with porosity coefficient e=0.45). The
results of calculations considering different number
of terms in the ray expansion are shown in Figure 5
for N = 4 – 22, 25, 30.
Analyzing the results obtained above, it could be
concluded that the smaller the number of members
of the series, the greater the difference in the results
when estimating the period. However, reliable
results when estimating the amplitude are obtained
starting from N=4. Reference to Figure 6 shows that
for a rigid plate the dynamic deflection is practically
insensitive to its weight if calculated with the same
number of ray series terms.
Analyzing the obtained maximum deflection
values for the four examples, it could be concluded
that the difference between magnitudes calculated at
N=4 (manual counting) and N=5 (members of the
series determined with the help of mathematical
software) is 6%. Therefore, this is sufficient to
determine the strength of the structure.
Fig. 5: The dimensionless time t* dependence of
dimensionless deflection W* at different numbers of
the ray expansion terms for the fourth example
(designations are the same as in Figure 2)
Maximum magnitudes of the deflection for all
four examples within the first half-wave are
summarized in Table 5, reference to which shows
that it is sufficient to restrict calculations by four-
term expansions, i.e. via ‘manual’ calculations.
Table 5. Values of maximum deflection within
the first half-wave.
Number
of terms
of the
series
ρ*=0.82
μ*=
557.36
ρ*=1
μ*=
665.24
ρ*=1.43
μ*=
1168.66
ρ*=2.35
μ*=
1546.23
N=4
0.084
0.085
0.078
0.087
N=5
0.089
0.091
0.083
0.093
N=6
0.089
0.091
0.083
0.093
N=7
0.089
0.091
0.082
0.093
N=8
0.089
0.091
0.082
0.093
N=9
0.089
0.091
0.082
0.093
N=10
0.089
0.091
0.082
0.093
N=15
0.089
0.091
0.082
0.093
N=16
0.089
0.091
0.082
0.093
N=20
0.089
0.091
0.082
0.093
N=25
0.089
0.091
0.082
0.093
N=30
0.089
0.091
0.082
0.093
5 Conclusion
This paper describes an algorithm developed based
on the Maple software package for solving contact
dynamic problems related to the generation and
propagation of strong and weak discontinuity
surfaces using an analytical approach based on the
ray method. The efficiency of the constructed
algorithm is illustrated on the example of solving
the problem of unsteady vibrations of an elastic
plate lying on an elastic isotropic half-space, caused
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.2
Marina V. Shitikova, Anna S. Bespalova
E-ISSN: 2224-3429
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Volume 19, 2024
by the action of instantaneous loads on the plate,
resulting in the generation and propagation of two
plane wave surfaces of strong discontinuity in the
elastic half-space. Behind the wave fronts up to the
contact boundary, the solution is constructed by
means of the ray series. The algorithm for solving
such a problem is presented by a program code
written in the programming language embedded in
the system.
Fig. 6: The dimensionless time t* dependence of
dimensionless deflection W* at N=16 and different
values of :
-
-
-
-
Numerical studies have shown that the Maple
software package allows one to solve quite complex
mathematical and engineering problems. In the
considered examples, it is possible to obtain a
solution for a significant number of terms of the ray
series, which was previously not possible with
"manual" calculation, and to demonstrate the
convergence of the solution with the increase in the
number of terms of the series, as well as to analyze
the values of amplitude and period for a two-layer
medium with different combinations of properties. It
has been established that to determine the period of
nonstationary vibrations, it is needed to consider
rather large number of terms of the series. To
determine the maximum amplitude, in principle,
four terms are sufficient, but if the task is to analyze
the behavior of the system considering the time
progression, a series with 30 terms is optimal.
The developed algorithm could be applied to
other types of boundary and initial conditions, as
well as it could be generalized for the analysis of
dynamic contact interaction for more complex
dynamical systems.
Acknowledgement:
The authors thank Mr K.A. Modestov for his
consultations concerning the application of the
Maple software package.
References:
[1] Yu.A. Rossikhin & M.V. Shitikova, Ray
method for solving dynamic problems
connected with propagation of wave surfaces
of strong and weak discontinuities, Applied
Mechanics Reviews, Vol. 48, No 1, 1995, pp.
1-39.
[2] I. Colombaro, A. Giusti, & F. Mainardi, On
transient waves in linear viscoelasticity,
Wave Motion, Vol. 74, 2017, pp. 191–212,
DOI: 10.1016/j.wavemoti.2017.07.008
[3] V.B. Porutchikov, Methods of Dynamic
Theory of Elasticity [in Russian], Nauka,
Moscow, 1986.
[4] W. Goldsmith, Impact. The Theory and
Physical Behaviour of Colliding Bodies,
Dover Publications, New York, 2001.
[5] J.D. Achenbach, Wave Propagation in
Elastic Solids, North-Holland Series in
Applied Mathematics and Mechanics,
Elsevier, 1973.
[6] A.S. Alekseev & B.Ya. Gelchinskiy, Ray
method for calculation of head wave
intensities, Questions of Dynamic Theory of
Seismic Wave Propagation, Vol. 5, 1961, pp.
54-72.
[7] J. D. Achenbach & S. P. Keshava, Free
waves in a plate supported by a semi-infinite
continuum, Transactions of the American
Society of Mechanical Engineers, Journal of
Applied Mechanics, Vol. 34, 1967, pp. 397-
404.
[8] J.D. Achenbach, S.P. Keshava, & G.
Herrmann, Waves in a smoothly joined plate
and half-space, Journal of the Engineering
Mechanics Division, Proceedings of the
American Society of Civil Engineers, Vol.
92, 1966, pp. 113-129.
[9] J.D. Achenbach & D.P. Reddy, Note on
wave propagation in linearly viscoelastic
media, Zeitchtift fr Angewandte Mathematik
und Physik, Vol. 18, 1967, pp. 141-144.
[10] A.G. Gorshkov & D.V. Tarlakovskii,
Resulting reactions in a transient contact
problem for an elastic anisotropic half-space,
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.2
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Volume 19, 2024
Journal of Applied Mathematics and
Mechanics, Vol. 60, issue 5, 1996, pp. 799-
809.
[11] P.V. Kzauklis & L.A. Molotkov, About low-
frequency vibration of a plate on an elastic
half-space, Soviet Applied Mechanics, Vol.
27, 1963, pp. 947-951.
[12] M. Mochihara & Y. Tanaka, Behavior of
plates in the elastic range under transverse
impact, Research Reports of Kagoshima
Technological College, Vol. 23, 1989, pp.
35-44.
[13] J.D. Achenbach, A.K. Gautesen &H.
McMaken, Ray Methods for Waves in Elastic
Solids, Pitman, Boston, 1982.
[14] N.D. Verveiko, Wave propagation in thin
viscoelastic layers, International Applied
Mechanics, Vol. 21, issue 12, pp. 63-67.
[15] M.V. Shitikova, Unsteady vibrations of a
plate lying on an elastic isotropic half-space,
Modern Methods of Static and Dynamic
Calculation of Structures and Constructions,
Interuniversity Collection of Scientific
Works, Vol. I, issue 2, 1993, pp. 92-98 [in
Russian].
[16] T.Y. Thomas, Plastic Flow and Fracture in
Solids, Academic Press, New York, 1961.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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(Attribution 4.0 International, CC BY 4.0)
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
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