Investigation of body motion in the trough of a vibratory conveyor under harmonic oscillations in two mutually perpendicular directions

2.1. Theoretical research

To investigate the transport process, we consider the motion of a body in the trough of a vibratory conveyor, consisting of a base and two side walls (Fig. 1).

Fig. 1Schematic diagram of the body motion in a vibratory conveyor trough: 1 – material being conveyed; 2, 3 – side walls of the trough; 4 – trough base

It is assumed that the trough undergoes harmonic oscillations in the $XOY$ plane simultaneously in two mutually perpendicular directions:

where $A$ and $B$ are the amplitudes of the oscillations along the $X$ and $Y$ axes respectively, $\omega$ is the angular frequency of the oscillations, and $t$ is time.

Fig. 2Dependence of the velocity and acceleration of the vibratory conveyor trough on dimensionless time

This combination of oscillations produces a periodic change in the magnitude and direction of the inertial forces, which determine the nature of the interaction between the transported body and the surfaces of the trough.

Differentiation of Eqs. (1) and (2) gives the expressions for the velocity and acceleration of the trough’s oscillations as functions of time:

Fig. 2 shows the velocity and acceleration profiles of the trough as a function of dimensionless time $\tau$ ($\tau =\omega t$).

The following forces act on the body: $P$ is gravity, $N_1$ is the reaction from the bottom of the trough, $F_1$ is the frictional force between the body and the bottom of the trough, $N_2$ is the reaction from the wall of the trough, $F_2$is the frictional force between the body and the wall of the trough, ${Ф}_x$, ${Ф}_y$ are inertial forces.

If the body moves in a horizontal plane, then:

where $f_1$ is the coefficient of friction between the body and the bottom of the trough:

where $f_2$ is the friction coefficient between the body and the wall of the trough. If:

then, due to the friction $F_2$ the body will move along with the trough until the inertial force ${Ф}_x$ exceeds the friction force $F_2$ ($f_1^0$, $f_2^0$ – static coefficients of friction). When:

The instant ${\tau }_1^{*}$, at which the inertial force equals the frictional force and the body begins to slide along the surface of wall 2 is determined under the condition:

On section (${\tau }_1^{*}\le \tau \le {\tau }_2^{*}$) the body will move under the influence of inertial force, and the differential equation of relative motion will take the form:

Integrating the differential Eq. (14) with the initial conditions $\tau ={\tau }_1^{*}$, ${\dot{x}}_0=0$, $x_0=0$ we obtain the dependences of the relative velocity and the movement of the body along the surface of the chute on time. At the time $\tau ={\tau }_2^{*}$ the relative velocity will again be equal to 0, the body is pressed against wall 3 and again moves together with the chute.

Considering the time of relative motion along the surface of the gutter $t_{ }^{*}=t_2^{*}-t_1^{*}$ and the displacement of the body $x$, we determine the average velocity of the body.

Thus, the mathematical model developed allows us to describe all the main stages of the body's motion in the trough of a vibratory conveyor, including joint motion with the trough, the onset of sliding, movement along the bottom, contact with the side walls, and the body’s re-engagement with the trough’s motion. The analytical relationships obtained can be used to determine the parameters of the body’s motion under various conveyor operating modes and form the basis for further experimental verification of the mathematical model's adequacy.

2.2. Experimental research

To verify the results of the theoretical studies, a laboratory setup was developed; its schematic diagram is shown in Fig. 3. It consists of a stationary base 1 and a carriage 2, which is mounted so as to be able to move in the horizontal plane along the $X$-axis by means of a connecting rod 3 hinged to a crank 4.

Fig. 3Laboratory vibrating conveyor installation: 1 – stationary base; 2 – carriage; 3 – connecting rod; 4 – crank; 5 – movable carriage; 6 – rollers; 7 – trough; 8 – body; 9 – motor

The base 1 has grooves on its sides, into which the movable carriage 5 is guided by rollers 6. A trough 7 is rigidly attached to the movable carriage, along which the body 8 moves. The grooves in the base are designed such that, as the rollers 6 move along them, the carriage performs harmonic oscillations along the $Y$-axis. As the crank 4 rotates, the trough 7 performs mutually perpendicular movements, the amplitude of which depends on the radius of the crank and the dimensions of the grooves in the base 1. In our case, the amplitude of the oscillations along the $X$-axis was 2.5 mm, and along the $Y$-axis 22.5 mm. The rotational movement of the crank 4 is driven by motor 9, the rotational speed of which can be varied. A plastic block measuring 50×35×15 mm is used as the transported object. To vary the coefficients of friction between the surface of the transported object and the trough wall, strips of different materials were affixed.

The coefficients of static friction of the object against the surfaces of the trough floor and walls were determined using the inclined plane method. The time taken for the body to travel along the trough surface was determined using video recording.

3.1. Results of the theoretical research

The results of theoretical studies of the average velocity of the body moving along the trough surface are shown in Fig. 4.

Fig. 4Average body velocity as a function of the oscillation amplitude A and oscillation frequency ω of the vibratory conveyor trough: B= 0.04 m, f1= 0.2, f2= 0.7

The graphs indicate that the average velocity of the body, and consequently the productivity of the vibratory conveyor, is significantly influenced by the amplitude and frequency of the longitudinal oscillations. Thus, at $\omega =$85 rad/s, $A=$0.005 m, $V_c=$0.22 m/s, whilst at $\omega =$85 rad/s, $A=$0.008 m, $V_c=$0.55 m/s. The results of theoretical studies determining the dependence of the relative velocity of a body moving along the surface of a vibratory conveyor trough on the dimensionless time (Fig. 5).

Fig. 5Dependence of the body’s velocity on the dimensionless time: B= 0.03 m, f1= 0.2, f2= 0.7

a) ω=50 rad/s

b) 100 rad/s

Analysis of the obtained relationships leads to the conclusion that the relative motion of the body along the surface of the trough occurs over the time interval $\frac{\pi }{3}<\tau <\pi$.

3.2. Results of the experimental research

Fig. 6 shows the graphical dependences of the average speed of movement of the body in the trough of the vibrating conveyor on the frequency of oscillations obtained experimentally. The maximum relative error between the theoretical and experimental values of the body’s average speed is 23 %. The obtained experimental values of the average velocity of the body are lower than the theoretical ones, since in the mathematical modeling of the body motion in the conveyor chute we assumed that the coefficients of static and sliding friction between the body and the bottom and walls of the conveyor chute are the same.

Fig. 6Experimental values of the average body velocity as a function of the oscillation frequency ω of the vibratory conveyor trough: A= 0.0025 m, B= 0.0225 m, f1= 0.26, f2= 0.57