174
L. Igumnov et al.
12.1 Introduction
The present work is a direct continuation of investigations carried out in (Igumnov
et al. 2017), relating to the study of nonlinear dynamics of new designs of eccentric shock–vibration mechanisms (ESVM) with a crank-sliding bar vibration exciter
(CSVE) (Shilkov et al. 2005; Nagaev 1978; Kobrinskiy and Kobrinskiy 1973; Babitskiy and Krupenin 1985; Biderman 1972; Bogodukhov et al. 2005; Vagapov et al.
2008; Zakrzhevskiy 1980; Babitsky 1998; Cveticanin 2002; Pavloaskaia and Wiercigroch 2003; Vagapov et al. 2007; Luo and Ma 2008; Leine and Heimsch 2012;
Goebel et al. 2008; Zheleztsov 1949).
The mechanism has quite a wide range of applications: It is effective for vi-broimpact compaction of various types of soil, sand and concrete in strained industrial
conditions; for breaking the ice and other harder objects; driving piles and structures,
etc. It also can be used for performing experiments: testing new artificial microstructured metamaterials (Del Vescovo and Giorgio 2014; Barchiesi et al. 2018) and
investigating their mechanical response under heavy periodic loads. An example of
mechanical metamaterials is pantographic structure (Placidi et al. 2016; dell’Isola
et al. 2019). Designing of such metamaterials is based on generalized continuum
approaches (Alibert et al. 2003; dell’Isola et al. 2012; Auffray et al. 2013; Rahali
et al. 2015; dell’Isola et al. 2015, 2016a; b, c) and widely uses numerical simulations
for identification of constitutive parameters and validation of theoretical predictions
(dell’Isola et al. 2016a, b, c; Giorgio 2016; dell’Isola et al. 2017; Placidi et al. 2017).
In (Igumnov et al. 2017), a model of ESVM with a CSVE and an arbitrary number
of piston-strikers is presented. Stability regions of periodic motion and a bifurcation
diagram for specific set of parameters are given for the simplest mechanism with
one piston-striker (PS). In the case of an absolutely inelastic impact interaction of PS
mechanism with a fixed stopper, new results of investigation of the point mapping
of a circle into itself are presented. In particular, analytical equations of inaccessible
boundaries are given, in the neighborhood of which there is a countable set of unstable
periodic motions. The present work gives a detailed study of nonlinear dynamics
of a vibro-impact mechanism with CSVE and two PSs. To study complex modes
of motion with an arbitrary number of impacts, including chaotic ones, an original
numerical–analytical method of the research based on the point mapping method has
been developed. As a result, it was possible to propose simple engineering formulas
for finding in the parameter space the region with different qualitative mechanism
behavior, convenient for tuning the mechanism for the required mode of operation.
The created bifurcation diagrams allow us to distinguish in the parameter space the
regions of arbitrarily complex motion modes, including chaotic ones. It is clearly
shown that dynamic systems with vibro-impact interaction belong to the class of
highly nonlinear systems that are used in various mechanisms (Masri and Caughey
1966; Babitsky 2013). The difficulty in studying the nonlinear dynamics of these
systems is due to their very rich, complex and highly nonlinear behavior under
deterministic loading (Pavlovskaia et al. 2015; Bernardini and Litak 2016; Tusset
et al. 2017; Ing et al. 2010; Liu et al. 2015; Neimark 2010; Feigin 1994). Thus, the
L. Igumnov et al.
12.1 Introduction
The present work is a direct continuation of investigations carried out in (Igumnov
et al. 2017), relating to the study of nonlinear dynamics of new designs of eccentric shock–vibration mechanisms (ESVM) with a crank-sliding bar vibration exciter
(CSVE) (Shilkov et al. 2005; Nagaev 1978; Kobrinskiy and Kobrinskiy 1973; Babitskiy and Krupenin 1985; Biderman 1972; Bogodukhov et al. 2005; Vagapov et al.
2008; Zakrzhevskiy 1980; Babitsky 1998; Cveticanin 2002; Pavloaskaia and Wiercigroch 2003; Vagapov et al. 2007; Luo and Ma 2008; Leine and Heimsch 2012;
Goebel et al. 2008; Zheleztsov 1949).
The mechanism has quite a wide range of applications: It is effective for vi-broimpact compaction of various types of soil, sand and concrete in strained industrial
conditions; for breaking the ice and other harder objects; driving piles and structures,
etc. It also can be used for performing experiments: testing new artificial microstructured metamaterials (Del Vescovo and Giorgio 2014; Barchiesi et al. 2018) and
investigating their mechanical response under heavy periodic loads. An example of
mechanical metamaterials is pantographic structure (Placidi et al. 2016; dell’Isola
et al. 2019). Designing of such metamaterials is based on generalized continuum
approaches (Alibert et al. 2003; dell’Isola et al. 2012; Auffray et al. 2013; Rahali
et al. 2015; dell’Isola et al. 2015, 2016a; b, c) and widely uses numerical simulations
for identification of constitutive parameters and validation of theoretical predictions
(dell’Isola et al. 2016a, b, c; Giorgio 2016; dell’Isola et al. 2017; Placidi et al. 2017).
In (Igumnov et al. 2017), a model of ESVM with a CSVE and an arbitrary number
of piston-strikers is presented. Stability regions of periodic motion and a bifurcation
diagram for specific set of parameters are given for the simplest mechanism with
one piston-striker (PS). In the case of an absolutely inelastic impact interaction of PS
mechanism with a fixed stopper, new results of investigation of the point mapping
of a circle into itself are presented. In particular, analytical equations of inaccessible
boundaries are given, in the neighborhood of which there is a countable set of unstable
periodic motions. The present work gives a detailed study of nonlinear dynamics
of a vibro-impact mechanism with CSVE and two PSs. To study complex modes
of motion with an arbitrary number of impacts, including chaotic ones, an original
numerical–analytical method of the research based on the point mapping method has
been developed. As a result, it was possible to propose simple engineering formulas
for finding in the parameter space the region with different qualitative mechanism
behavior, convenient for tuning the mechanism for the required mode of operation.
The created bifurcation diagrams allow us to distinguish in the parameter space the
regions of arbitrarily complex motion modes, including chaotic ones. It is clearly
shown that dynamic systems with vibro-impact interaction belong to the class of
highly nonlinear systems that are used in various mechanisms (Masri and Caughey
1966; Babitsky 2013). The difficulty in studying the nonlinear dynamics of these
systems is due to their very rich, complex and highly nonlinear behavior under
deterministic loading (Pavlovskaia et al. 2015; Bernardini and Litak 2016; Tusset
et al. 2017; Ing et al. 2010; Liu et al. 2015; Neimark 2010; Feigin 1994). Thus, the