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L. Igumnov et al.
There are various works on vibration absorption and control (Maurini et al. 2004;
Andreaus et al. 2004; dell’Isola et al. 2004; Vidoli and dell’Isola 2001). Some new
microstructures metamaterials have vibration isolation properties (Barchiesi et al.
2019; Di Cosmo et al. 2018).
More information on metamaterials and one of their mechanical examples—
pantographic structures—can be found in (Del Vescovo et al. 2014; Placidi et al.
2016; dell’Isola et al. 2019). And it is also worth to mention that designing metamaterials is based on generalized continuum approaches (Alibert et al. 2003; Auffray
et al. 2013; Abali et al. 2017, 2019; dell’Isola et al. 2012, 2015, 2016).
One of the variants of building foundation protection from vibrations appearing
in the soil is placing vibration-absorbing plates between source of vibration and the
object. Quite a lot of works are dedicated to research of problem of dynamic interaction between object and soil (Umek 1973; Kostrov 1964; Ryl’ko 1977). Homogeneous one-layer sound-absorbing plates in nowadays are well researched. At the
same time, the research of multi-layer plates dynamics, including plates filled with
filler, requiring proper accounting of transverse shear as well as transverse normal
stress in filling layer, in other case the state of dynamic stress can be calculated with
severe mistakes. On the other hand, in common, there is a wave stationary influence
with different configuration influence on barriers being researched.
In this work, the influence of flat and cylindrical wave influence on barrier with
complex structure, accounting all mentioned effects. In practice, this situation can be
caused by vibration, caused by single disturbance, appearing in the soil when train in
underground passing by in a distance, which appears to be flat, or nearby the barrier
which inducting cylindrical wave.
8.2 Problem Statement of Interaction of Harmonic
Spherical Wave Propagating in Continuum
and a Three-Layered Plate
Three-layered plate placed in the soil is acting like a barrier model. It is separating soil
in two parts, «1» and «2». The part «1» is where the wave source is placed. Isotropic
propulsive half space described by propulsion theory equations (Rakhmatulin et al.
1983; Berezhnoi et al. 2004) is acting as a soil. Usage of such half space is justified
for disturbances with little amplitude, caused for example by train passing through
the tunnel. A planar statement of the problem is considered. It is considered that
border conditions are equal to the conditions on the simply supported contour of the
plate. To describe the plate movement, Paimushin V.N. equation system (Ivanov and
Paimushin 1995a, b) is being used. The beginning of coordinates axes in the upper
point of the plate. Axis Ox is directed along middle line of the plate toward ground,
and axis Oz is directed inside of the half space «2». Undisturbed state of the soil is
undeformed. From the negative side of Oz axis («1» environment), there is harmonic
wave with amplitude of normal movement p on front is moving toward the barrier.
L. Igumnov et al.
There are various works on vibration absorption and control (Maurini et al. 2004;
Andreaus et al. 2004; dell’Isola et al. 2004; Vidoli and dell’Isola 2001). Some new
microstructures metamaterials have vibration isolation properties (Barchiesi et al.
2019; Di Cosmo et al. 2018).
More information on metamaterials and one of their mechanical examples—
pantographic structures—can be found in (Del Vescovo et al. 2014; Placidi et al.
2016; dell’Isola et al. 2019). And it is also worth to mention that designing metamaterials is based on generalized continuum approaches (Alibert et al. 2003; Auffray
et al. 2013; Abali et al. 2017, 2019; dell’Isola et al. 2012, 2015, 2016).
One of the variants of building foundation protection from vibrations appearing
in the soil is placing vibration-absorbing plates between source of vibration and the
object. Quite a lot of works are dedicated to research of problem of dynamic interaction between object and soil (Umek 1973; Kostrov 1964; Ryl’ko 1977). Homogeneous one-layer sound-absorbing plates in nowadays are well researched. At the
same time, the research of multi-layer plates dynamics, including plates filled with
filler, requiring proper accounting of transverse shear as well as transverse normal
stress in filling layer, in other case the state of dynamic stress can be calculated with
severe mistakes. On the other hand, in common, there is a wave stationary influence
with different configuration influence on barriers being researched.
In this work, the influence of flat and cylindrical wave influence on barrier with
complex structure, accounting all mentioned effects. In practice, this situation can be
caused by vibration, caused by single disturbance, appearing in the soil when train in
underground passing by in a distance, which appears to be flat, or nearby the barrier
which inducting cylindrical wave.
8.2 Problem Statement of Interaction of Harmonic
Spherical Wave Propagating in Continuum
and a Three-Layered Plate
Three-layered plate placed in the soil is acting like a barrier model. It is separating soil
in two parts, «1» and «2». The part «1» is where the wave source is placed. Isotropic
propulsive half space described by propulsion theory equations (Rakhmatulin et al.
1983; Berezhnoi et al. 2004) is acting as a soil. Usage of such half space is justified
for disturbances with little amplitude, caused for example by train passing through
the tunnel. A planar statement of the problem is considered. It is considered that
border conditions are equal to the conditions on the simply supported contour of the
plate. To describe the plate movement, Paimushin V.N. equation system (Ivanov and
Paimushin 1995a, b) is being used. The beginning of coordinates axes in the upper
point of the plate. Axis Ox is directed along middle line of the plate toward ground,
and axis Oz is directed inside of the half space «2». Undisturbed state of the soil is
undeformed. From the negative side of Oz axis («1» environment), there is harmonic
wave with amplitude of normal movement p on front is moving toward the barrier.