26
L. Cveticanin and D. Cveticanin
1 Introduction
The problem of mass variable body is known for a long time (see [1]). In the seventeenth century, Galileo discovered the anomaly in the Moon motion which he believed
is the function of the system mass variation. Lately, Laplace theoretically proved the
phenomena of the secular acceleration of the Moon according to the influence of
mass variation [2]. Dufour [3] explained that the mass of the Earth varies due to the
falling shooting stars and also due to combustion or spending in the atmosphere.
He found that the dust of shooting stars which falls on the surface of France in one
year can cover a volume of 0.1 m
3 . Oppalzer [4] was the first to analyze the reason
for secular acceleration of the Moon as the result of Earth and Moon mass increase.
Namely, during a hundred year a 2.8 mm dust layer is formed on the Earth.
Since that time, a significant number of publications appear considering the mass
variable systems. Mass variation may be continual or discontinual.
For the discontinual mass variation two processes are typical: mass addition and
mass separation. The first process is usually considered as the plastic impact of two
bodies. However, there are only few investigations in the discontinual mass separation
[5–12].
For continual mass variation, Meshchersky [13] extended Newton’s equation
with the so-called reactive force. Solving this equation, results were obtained which
were of crucial interest in developing of the rocket theory (see [14–18]). Nowadays,
Meshchersky’s equation is widely applied in systems with time-variable mass [12,
19] and specially in time-variable oscillators [20–25]. However, the equation is not
convenient for explaining the dynamics of the body with simultaneous mass and
geometry (i.e., moment of inertia) variation.
The aim of this paper is to extend the dynamics of the mass variable system to
the dynamics of the mass variable body. The variation of the moment of inertia of
the body is taken into consideration. Equations of general motion of the body with
variable mass are formed.
As the special motion, the vibration of the body with variable mass is considered. The influence of mass variation on the dynamic properties is analyzed. As an
example, vibrations of a sieve during mass separation are considered. The problem is
mathematically described. The equation is solved analytically and numerically. The
solutions are compared.
2 Discontinual Mass Separation
The term ‘discontinual mass separation’ means dividing of an initial rigid body into
two rigid ones: a separating body and a remaining body (Fig. 1). The separation is
assumed to be in a very short time during which the position of bodies is invariable
[26]. However, linear and angular velocities of the initial, separating and remaining
bodies differ. To calculate these values, the dynamics of bodies is divided into three
L. Cveticanin and D. Cveticanin
1 Introduction
The problem of mass variable body is known for a long time (see [1]). In the seventeenth century, Galileo discovered the anomaly in the Moon motion which he believed
is the function of the system mass variation. Lately, Laplace theoretically proved the
phenomena of the secular acceleration of the Moon according to the influence of
mass variation [2]. Dufour [3] explained that the mass of the Earth varies due to the
falling shooting stars and also due to combustion or spending in the atmosphere.
He found that the dust of shooting stars which falls on the surface of France in one
year can cover a volume of 0.1 m
3 . Oppalzer [4] was the first to analyze the reason
for secular acceleration of the Moon as the result of Earth and Moon mass increase.
Namely, during a hundred year a 2.8 mm dust layer is formed on the Earth.
Since that time, a significant number of publications appear considering the mass
variable systems. Mass variation may be continual or discontinual.
For the discontinual mass variation two processes are typical: mass addition and
mass separation. The first process is usually considered as the plastic impact of two
bodies. However, there are only few investigations in the discontinual mass separation
[5–12].
For continual mass variation, Meshchersky [13] extended Newton’s equation
with the so-called reactive force. Solving this equation, results were obtained which
were of crucial interest in developing of the rocket theory (see [14–18]). Nowadays,
Meshchersky’s equation is widely applied in systems with time-variable mass [12,
19] and specially in time-variable oscillators [20–25]. However, the equation is not
convenient for explaining the dynamics of the body with simultaneous mass and
geometry (i.e., moment of inertia) variation.
The aim of this paper is to extend the dynamics of the mass variable system to
the dynamics of the mass variable body. The variation of the moment of inertia of
the body is taken into consideration. Equations of general motion of the body with
variable mass are formed.
As the special motion, the vibration of the body with variable mass is considered. The influence of mass variation on the dynamic properties is analyzed. As an
example, vibrations of a sieve during mass separation are considered. The problem is
mathematically described. The equation is solved analytically and numerically. The
solutions are compared.
2 Discontinual Mass Separation
The term ‘discontinual mass separation’ means dividing of an initial rigid body into
two rigid ones: a separating body and a remaining body (Fig. 1). The separation is
assumed to be in a very short time during which the position of bodies is invariable
[26]. However, linear and angular velocities of the initial, separating and remaining
bodies differ. To calculate these values, the dynamics of bodies is divided into three