12 The Dynamics of Eccentric Vibration Mechanism (Part 2)
175
dynamics of such systems should be studied with the help of numerical–analytical
methods.
12.2 Problem Setting
The new constructive solution is based on the “inverted vibrator” principle (Igumnov
et al. 2017; Shilkov et al. 2005). The power pulse, transmitted to the processed
medium (fixed obstacle, soil, piles, structures, etc.), arises both due to the thrust
against the shoulder of the eccentric shaft and to the drop in the kinetic energy of
the operating element. It is evident (Nagaev 1978) that a dense and, at the same
time, strong structure of the processed medium can be achieved only when specific
pressure on the contact surface of an operating element with the processed medium
increases gradually. Its lower limit is ditial state (before competermined by the physical properties of the medium in its inaction process), while the upper one by the
ultimate strength of the medium or technological conditions. Thus, the parameters
of such machines and mechanisms must be determined from the conditions close
to quasi-plastic interaction (Nagaev 1978). Such multi-pulse loading mode can be
realized using multi-impact ESVM with CSVE, designed to easily regulate working
modes by changing the geometry of kinematic connections.
This paper presents a detailed study of nonlinear dynamics of the mechanism
with two PSs. It is shown that the used mathematical apparatus of the point mapping
method for studying the dynamics of the mechanism allows us to obtain rather simple
analytical relations for geometrical parameters of the mechanism, which indicate in
the parameter space, the regions of existence of impact motions of the mechanism
after the stroke either by just one PS or alternately by two PSs. Multidimensional
parameter space of the model is divided into regions of stable periodic motion modes
with both two impacts of PS and a large number of PS impacts. The developed
software written in high-level language makes it possible to calculate bifurcation
diagrams for the main parameters.
Figure 12.1 presents the scheme of the mechanism under consideration, where 1 is
the frame of the mechanism, 2 is a flywheel, 3 are cranks with adjustable eccentricities
r i , 4 are connecting rods (of the length l i ) with a stationary phase shift φ, 5 are pistonstrikers, 6 is a fixed stopper, 7 is eccentric shaft, 8 is a guide sleeve, 9 is a frame
guide rod, 10 is an anvil.
Neglecting the masses of PSs, rods and cranks, the equation of free (without
impact) motion of the system for y pi > 0 can be written in the following form:
M
d
2 y
dt 2 = −Mg,
where y is coordinate of the mass center of the body, counted from fixed anvil
block, g is free-fall acceleration. Let y pi be the distance of the base ith PS from an anvil
(i = 1, 2). When one of the PSs contacts the anvil, a momentary impact interaction
175
dynamics of such systems should be studied with the help of numerical–analytical
methods.
12.2 Problem Setting
The new constructive solution is based on the “inverted vibrator” principle (Igumnov
et al. 2017; Shilkov et al. 2005). The power pulse, transmitted to the processed
medium (fixed obstacle, soil, piles, structures, etc.), arises both due to the thrust
against the shoulder of the eccentric shaft and to the drop in the kinetic energy of
the operating element. It is evident (Nagaev 1978) that a dense and, at the same
time, strong structure of the processed medium can be achieved only when specific
pressure on the contact surface of an operating element with the processed medium
increases gradually. Its lower limit is ditial state (before competermined by the physical properties of the medium in its inaction process), while the upper one by the
ultimate strength of the medium or technological conditions. Thus, the parameters
of such machines and mechanisms must be determined from the conditions close
to quasi-plastic interaction (Nagaev 1978). Such multi-pulse loading mode can be
realized using multi-impact ESVM with CSVE, designed to easily regulate working
modes by changing the geometry of kinematic connections.
This paper presents a detailed study of nonlinear dynamics of the mechanism
with two PSs. It is shown that the used mathematical apparatus of the point mapping
method for studying the dynamics of the mechanism allows us to obtain rather simple
analytical relations for geometrical parameters of the mechanism, which indicate in
the parameter space, the regions of existence of impact motions of the mechanism
after the stroke either by just one PS or alternately by two PSs. Multidimensional
parameter space of the model is divided into regions of stable periodic motion modes
with both two impacts of PS and a large number of PS impacts. The developed
software written in high-level language makes it possible to calculate bifurcation
diagrams for the main parameters.
Figure 12.1 presents the scheme of the mechanism under consideration, where 1 is
the frame of the mechanism, 2 is a flywheel, 3 are cranks with adjustable eccentricities
r i , 4 are connecting rods (of the length l i ) with a stationary phase shift φ, 5 are pistonstrikers, 6 is a fixed stopper, 7 is eccentric shaft, 8 is a guide sleeve, 9 is a frame
guide rod, 10 is an anvil.
Neglecting the masses of PSs, rods and cranks, the equation of free (without
impact) motion of the system for y pi > 0 can be written in the following form:
M
d
2 y
dt 2 = −Mg,
where y is coordinate of the mass center of the body, counted from fixed anvil
block, g is free-fall acceleration. Let y pi be the distance of the base ith PS from an anvil
(i = 1, 2). When one of the PSs contacts the anvil, a momentary impact interaction