Investigation of body motion in the trough of a vibratory conveyor with asymmetric oscillations of walls

2.1. Theoretical research

We consider a vibratory-conveyor configuration schematized in Fig. 1. It comprises a horizontal platform (1) that performs harmonic oscillations along the $Y$-axis and two plates (2, 3) between which the body 4 is located. To determine the mean conveying speed, which is the main performance indicator for conveyors, we first analyse body motion on platform 1 moving at velocity $V_0$. Plate 2 moves at $U_0$, whereas plate 3 is fixed.

As the platform moves, friction at the platform-body interface presses the body against plate 2; simultaneously, friction at the body-plate interface drives motion along the $X$-axis. In the proposed device, the trough bottom undergoes only transverse oscillations, whereas the walls execute asymmetric longitudinal oscillations. Accordingly, the bottom alternately presses the body against the wall that, at a given instant, moves in the conveying direction. Friction with that wall produces purely longitudinal motion without transverse oscillation, so input energy is spent on translation alone, reducing the process energy demand.

The velocity of the body on the platform surface is given by Fig. 2:

where $U$ is the absolute velocity of the body along the $X$-axis; $V=V_0$ is the body’s velocity relative to the platform (opposite the platform motion).

Fig. 1Schematic of the vibratory-conveyor design: 1 – platform; 2, 3 – plates; 4 – body

Fig. 2Free-body diagram of the forces acting on the body during motion in the trough: V0 – platform velocity; U0 – plate velocity; W – body velocity relative to the platform; U – absolute body velocity

Resolving frictional force into components gives:

where $\cos \alpha =\frac{V}{\sqrt{V^2+U^2}}$, $\sin \alpha =\frac{U}{\sqrt{V^2+U^2}}$, $f_1$ is the coefficient of friction between the platform and the body.

The plate-body friction is:

The equation of motion reads:

i.e.:

With the substitution:

and constant $V$, we obtain:

Combining with Eq. (5) yields:

hence:

Integrating Eq. (9) gives $\alpha \left(t\right)$ and the body velocity. If $f_2>\mathrm{t}\mathrm{g}\alpha$, then $\dot{\alpha }>0$ and $\alpha \to {\alpha }_{max}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}{f}_2$ so:

Hence, in this regime $U$ is independent of $U_0$ and of the platform-body friction. The upper bound $U_{max}$ cannot exceed the plate velocity $U_0$.

For small $\alpha$, using $\mathrm{t}\mathrm{g}\alpha =\alpha$ and $\mathrm{c}\mathrm{o}\mathrm{s}\alpha =1$ leads to:

Eq. (11) was integrated with the condition $t=$0, $\alpha =$0:

In the second scenario, plate 2 moves at $U_0$, plate 3 remains fixed and the platform velocity is:

where $A$ and$\omega$ are the amplitude and frequency of the platform oscillations. In this case, the differential equation of motion of the body becomes:

Since Eq. (14) is highly nonlinear, it can only be integrated numerically to obtain the displacement of the body as a function of time and hence the average transportation velocity. For this purpose, we used the fourth-order Runge-Kutta method. The integration step was 0.001 s and the integration limits were $t_0=$0 s, $\dot{x_0}=$0 m/s.

In the third scenario, both plates execute asymmetric oscillations with $U_0=B\omega \sin \omega t$, where $B$ and $\omega$ are the amplitude and frequency of the plate oscillations.

Eq. (14) is integrated over $t_1$ when the body is in contact with plate 2 $0\ll t_1\ll T/2$, where $T=2\pi /\omega$ is the period of plates oscillation.

Over the interval $T/2\ll t_2\ll T$, when the platform changes its direction of motion, the body is pressed against plate 3, whose velocity is also directed along the $X$-axis and the body continues to move in the desired direction.

2.2. Experimental research

To validate the theoretical results, a laboratory test rig was developed, whose general view is shown in Fig. 3. It consists of a platform (1) capable of translation in the horizontal plane along the $Y$-axis and two walls (2, 3). Wall 2 is fixed, whereas wall 3 is mounted to translate along the $X$-axis. A vibration exciter (4) is attached to platform 1. A test body (load) (5) is placed on the platform between the two walls.

Fig. 3General view of the laboratory setup: 1 – platform; 2, 3 – walls; 4 – vibration exciter; 5 – body

In the first stage of the experiments, platform 1 moved at a constant velocity $V_0$, while wall 3 moved at a constant velocity $U_0$. The movement of the body in the trough was recorded on video and analysed frame-by-frame. The displacement-measurement error was 0.5 mm. Each test was performed in triplicate. Average velocity was obtained from the measured displacement over a known time interval. The coefficients of friction were measured by the inclined-plane method. Because the kinetic coefficient depends on sliding speed and is difficult to determine accurately, the static and kinetic coefficients were taken equal for the purposes of this study.

In the second stage, using the vibration exciter (4), platform 1 performed harmonic oscillations with velocity $V_0=\mathrm{А}\omega \sin \omega t$, while wall 3 moved at a constant velocity $U_0$. The exciter frequency was measured with a laser tachometer.

3.1. Results of the theoretical research

As it is shown in theoretical research, when both the platform and the trough walls move at constant velocities, the body’s displacement and velocity vary with time in a manner that depends on the friction coefficients at the platform and wall surfaces (Fig. 4, Fig. 5).

The graphs demonstrate that the body velocity approaches its maximum asymptotically $U_{max}=f_2{V}_0$. The platform-body friction coefficient affects only the time required to reach this limit: at $f_1=$0.2, 95 % of $U_{max}$ is attained in $t=$0.3 s, whereas at $f_1=$0.8 the same level is reached in $t=$0.19 s. Clearly, the body velocity cannot exceed the wall velocity.

For harmonic platform motion $V_0=A\omega \sin \omega t$ computed velocity-time histories for two amplitudes are shown in Fig. 6.

Fig. 4Body displacement versus time for several values of the friction coefficient between the body and the platform surface (f2= 0.5, V0= 0.3 m/s)

Fig. 5Body velocity versus time for several values of the friction coefficient between the body and the platform surface (f2= 0.5, V0= 0.3 m/s)

Fig. 6Body velocity versus time for different platform oscillation amplitudes

a)$f_1=$0.6, $f_2=$0.6, $\omega$= 10 rad/s

b)$f_1=$0.6, $f_2=$0.6, $\omega$= 20 rad/s

The curves indicate a strong dependence of body speed on both platform-oscillation amplitude and frequency. Fig. 7 compares velocity-time responses for several wall-friction coefficients.

Fig. 7Body velocity versus time for different wall-friction coefficients

a)$A=$0.011 m, $f_1=$0.6, $\omega$= 10 rad/s

b)$A=$ 0.011 m, $f_1=$0.6, $\omega$= 20 rad/s

The peak velocity scales nearly linearly with the wall-friction coefficient, consistent with the constant-velocity regime.

Figs. 8-9 show mean conveying speed versus wall-friction coefficient for multiple platform-friction levels (Fig. 8) and for multiple oscillation amplitudes (Fig. 9).

Fig. 8Average body speed versus wall-friction coefficient for different platform-friction coefficients (A= 0.011 m, ω = 10 rad/s)

Fig. 9Average body speed versus wall-friction coefficient for different oscillation amplitudes (f1= 0.6, ω = 10 rad/s)

The results show that mean conveying speed is governed mainly by the wall-friction coefficient and platform-oscillation amplitude, with negligible sensitivity to platform-body friction.

Speed rises with amplitude and frequency; however, at $\omega$= 30 rad/s and $A=$ 0.02 m. it decreases because within a half-period the body cannot reach its peak velocity. From Fig. 7, doubling platform frequency (from 10 rad/s to 20 rad/s) raises the peak velocity only by 1.6 times, so the growth of the average speed with frequency is sub-linear at higher frequencies.

With asymmetric wall oscillations, the body speed cannot exceed the instantaneous wall speed $U_0=B\omega \sin \omega t.$ Fig. 10 compares the body and wall velocities.

Fig. 10Body and wall velocities versus time

3.2. Results of the experimental research

The experimental studies conducted at the laboratory scale plant resulted in obtaining the dependences of an average conveying speed versus oscillation frequency 10 ≤ $\omega$ ≤ 30 (rad/s), amplitude of the platform.0.001 $\le A\le$ 0.021 (rev/m) and friction coefficient between the body and the platform surface 0.2 ≤ $f_2$ ≤ 1 in a technological line, which are presented in the form of response surfaces as a functional а) $V=f(A,f_2)$ at $\omega$ = 20 rad/s; b) $V=f(\omega ,f_2)$ at $A$ = 0.011 m; с) $V=f(A,\omega )$ at $f_2$ = 0.6 (Fig. 11).

Analysis of the response-surface sections indicates that the mean speed versus wall-friction coefficient approaches its asymptote $U_{max}=f_{2}V$. The amplitude and frequency of oscillations exert the dominant influence on the mean conveying speed; notable gains are observed for $\omega$ ≥ 20 rad/s and $A\ge$ 0.011 m.

Fig. 12 compares theoretical predictions with experimental data for constant-velocity motion $V_0=$0.3 m/s and the moving wall at $U_0=$0.3 m/s.

Under the assumption of equal static and kinetic friction coefficients at all interfaces, theory and experiment agree satisfactorily (error within 14-20 %).

Fig. 11Response surfaces of average conveying speed versus as a functional

а) $V=f(A,f_2)$ at $\omega$ = 20 rad/s

b)$V=f(\omega ,f_2)$ at $A=$ 0.011 m

с) $V=f(A,\omega )$ at $f_2$= 0.6

Fig. 12Body velocity versus the coefficient of friction at the surface of the moving wall