Issues in the Theory of Vibratory Conveying

G. V. SURKOV

(Presented by the scientific seminar of the Departments of Gyroscopic Instruments and Devices and Mechanics)

The Process of Vibratory Conveying

Vibratory conveying is carried out on special installations (vibrating conveyors), whose working member (a trough) performs periodic oscillations of high frequency with a relatively small amplitude.

Let us consider the interaction of trough 1 (Fig. 1) with a unit object, which is represented in the form of an inelastic material point. The trough moves in time \(t\) according to the law \(A(1-\cos\omega t)\) at the vibration angle \(\beta\). The working surface of the trough is set at an angle \(\alpha\) to the horizon. The coefficient of friction of the object on the trough is \(f=\operatorname{tg}\rho\). The trough moves with acceleration \(a\), which is imparted to the object in the process of their joint motion. The force of gravity of the object \(G\) and the inertial force \(P\) give the resultant \(R\). The direction of the resultant relative to the normal is determined by the angle \(\psi\); from Fig. 1 we have:

Fig. 1. Diagram of the interaction of the object with the trough.

\[ \operatorname{tg}\psi=\frac{-A\omega^2\cos\omega t-g\sin\alpha}{A\omega^2\sin\omega t+g\cos\alpha}. \]

Dividing by \(\operatorname{tg}\rho\), after transformation we obtain:

\[ \varepsilon=\frac{\operatorname{tg}\psi}{\operatorname{tg}\rho} =-\,\frac{\gamma+\operatorname{tg}\alpha\,\operatorname{tg}\beta}{(\gamma+1)\operatorname{tg}\rho\,\operatorname{tg}\beta}. \tag{1} \]

Here

\[ \gamma=\Gamma\cos\omega t,\qquad \Gamma=\frac{A\omega^2\sin\beta}{g\cos\alpha}\,[1]. \tag{2} \]

Equation (1) characterizes the behavior of the object on the trough, that is, determines the regime of vibratory conveying. Thus, when \(|\varepsilon|\geq 1\)

there will be sliding of the object relative to the tray. The condition for forward sliding is

\[ \varepsilon \geqslant 1. \tag{3} \]

Backward sliding: \(|-\varepsilon| \geqslant 1\). The condition for separation of the object from the tray is

\[ \varepsilon = \infty. \tag{4} \]

From equation (1), condition (4) is satisfied, if \(\operatorname{tg}\beta \operatorname{tg}\rho \ne 0\), only when \(\gamma=-1\). The minus sign indicates that separation of the object is possible only for a negative direction of acceleration, i.e., for retarded motion of the tray forward. On the basis of the fact that in equation (1) condition (3) is fulfilled before condition (4), one may conclude: a regime of vibratory conveying with tossing is impossible without preliminary forward sliding.

The occurrence of forward sliding from (1), (3), for the value

\[ -\gamma_{(-)} \geqslant \frac{\operatorname{tg}\rho+\operatorname{tg}\alpha}{\operatorname{ctg}\beta+\operatorname{tg}\rho} \]

and, analogously, backward sliding will occur when

\[ \gamma_{(+)} \geqslant \frac{\operatorname{tg}\rho-\operatorname{tg}\alpha}{\operatorname{ctg}\beta-\operatorname{tg}\alpha}. \tag{5} \]

The acceleration at which sliding of the object along the tray forward arises, from (2) and (5), is equal to

\[ a_{(-)}=A\omega^{2}\cos \omega t_c'= g\frac{\sin(\alpha+\rho)}{\cos(\beta-\rho)}. \]

The acceleration at which backward sliding arises is

\[ a_{(+)}=A\omega^{2}\cos \omega t_c''= g\frac{\sin(\rho-\alpha)}{\cos(\beta+\rho)}. \tag{6} \]

Here \(t_c'\) and \(t_c''\) are the values of the time at which forward and backward sliding begins. The relation \(a_{(-)}=a_{(+)}\) gives the limiting value of the vibration parameters for which directed displacement is impossible; from (6) we have \(\operatorname{tg}\alpha=\operatorname{tg}^{2}\rho\,\operatorname{tg}\beta\). Thus, for the existence of directed forward motion, it is necessary, along with the known condition \(\operatorname{tg}\alpha<\operatorname{tg}\rho\), to satisfy the requirement

\[ \operatorname{tg}\alpha<\operatorname{tg}^{2}\rho\,\operatorname{tg}\beta. \]

For the vibration angle \(\beta=0^\circ\), the ratio is

\[ \frac{a_{(-)}}{a_{(+)}}=\frac{\sin(\alpha+\rho)}{\sin(\rho-\alpha)}, \]

whence it follows that \(a_{(-)}>a_{(+)}\), that is, with a symmetric law of oscillations and \(\beta=0^\circ\), motion is possible only down an incline. If, in addition, \(\alpha=0^\circ\), then \(a_{(-)}=a_{(+)}\), which means that there is no possibility of directed motion. To create displacement of the object along the tray in this case, the latter must be given an asymmetric law of oscillations; we then obtain the so-called rocking conveyor.

Figure 2 gives a diagram of the regimes of the vibratory-conveying process, obtained as a result of solving equation (1) with \(\alpha=0^\circ\). The character of the relative motion of the object along the tray in Fig. 2 is indicated by regions: I—the region of joint forward motion; II—joint backward motion; III—forward sliding; IV—backward sliding; V—the region of motion with separation. It is clear from the diagram that

With a symmetric law of oscillations, conveying occurs in the presence of friction only when the vibration angle is not equal to zero. Forward sliding (region III), for any nonzero value of the vibration angle, begins earlier than backward sliding. It is impossible to obtain a tossing mode without forward sliding.

Fig. 2. Diagram of vibratory-conveying modes.

Vibratory conveying in the tossing mode is characterized by the fact that, in the course of motion, the object remains in a suspended state for a considerable time, owing to which high feed velocities can be obtained, wear of the trough is reduced, and the influence of the loaded mass of material on the oscillatory process is decreased.

Vibratory conveying with tossing

1. General equations. Various combinations of vibration parameters and the characteristics of the conveyed material yield a great variety of schemes for the motion of the object [1, 2, 3]. From the standpoint of productivity and reduction of trough wear, the most effective mode is one in which there are two phases of motion: joint motion of the object and trough, during which their velocities equalize, and free-flight motion (a throw). Moreover, the smaller the share of joint motion in the entire cycle, the more effective the mode. The projections of the displacement, velocity, and acceleration of the trough and the object onto a fixed coordinate system for a two-phase mode are shown in Fig. 3.

The time at which sliding of the object along the trough begins forward, from (6), is equal to

\[ t_c'=\frac{1}{\omega}\arccos \frac{g}{A\omega^2}\frac{\sin(\alpha+\rho)}{\cos(\beta-\rho)}. \]

In Fig. 3 the instant of sliding is marked by the point \(t_c'\). The velocity of the object and of the trough at the instant when sliding begins is

\[ \dot{x}(t_c')=A\omega \sin \omega t_c' \cos\beta . \]

The sliding velocity of the object relative to the trough up to the instant of separation is

\[ \dot{x}(t_c'\div t_0)=\dot{x}(t_c')-\int_{t_c'}^{t_0}\ddot{x}(t_c'\div t_0)\,dt, \tag{7} \]

where \(\ddot{x}(t_c \div t_0)=\dfrac{gF}{G}\) is the deceleration of the object on the trough due to the friction force \(F\).

The time and phase angle of the instant of separation from (2) are

\[ t_0=\frac{1}{\omega}\arccos\left(-\frac{1}{\Gamma}\right);\qquad \varphi_0=\arccos\left(-\frac{1}{\Gamma}\right). \tag{8} \]

Fig. 3. Displacement, velocities, and accelerations of the object and the trough.

At the instant of separation the velocity of the trough is

\[ \dot{x}_{\text{tr}}(t_0)=A\omega\sin\omega t_0. \tag{9} \]

From (7) and (9) it is evident that the difference between the sliding velocity of the object and the velocity of the trough at the instant of separation decreases with increasing friction and with the vibration-regime parameter. To estimate the difference between the sliding and separation velocities, take the ratio

\[ \frac{\dot{x}(t_c')}{\dot{x}_{\text{tr}}(t_0)} = \frac{\sin\omega t_c'}{\sin\omega t_0}. \]

Substituting the values of \(t_c'\) and \(t_0\), after transformation we obtain

\[ \frac{\dot{x}(t_c')}{\dot{x}_{\text{tr}}(t_0)} = \sqrt{ \frac{ \Gamma^2- \left[ \dfrac{\sin\beta}{\cos\alpha} \dfrac{\sin(\alpha+\rho)}{\cos(\beta-\rho)} \right]^2 }{ \Gamma^2-1 } }. \]

Let, for example, \(A=0.75\ \text{mm}\), \(\beta=20^\circ\), \(\alpha=0^\circ\), \(\omega=314\ \dfrac{1}{\text{s}}\), \(\rho=23^\circ\); then

\[ \frac{\dot{x}(t_c')}{\dot{x}_{\text{tr}}(t_0)}=1.07, \]

that is, the difference in velocities is \(7\%\).

In reality, for the given case the relative difference is even smaller, since the example does not take into account the decrease in sliding velocity by the instant of separation due to friction forces (7).

From the instant of separation the object moves in the medium according to the law described by the equation of free fall. The flight time is equal to the sum of the ascent time \(t_1\) and the descent time \(t_2\).

\[ t_{\text{f}}=t_1+t_2. \tag{10} \]

The time \(t_{\mathrm{p}}\) is determined under the assumption that the separation velocity of the object is equal to the trough velocity at the moment of separation, and that there is no air resistance:

\[ t_1=\frac{A\omega \sin t_0 \sin(\alpha+\beta)}{g}. \]

The time of fall of the object is \(t_2=\sqrt{\frac{2h}{g}}\), where the height of fall is (Fig. 3)

\[ h=h_1+h_2-x_{\mathrm{p}}\sin\alpha. \]

Substituting the values of \(t_1\), \(t_2\) into (10) and replacing \(\omega t_0=\varphi_0\), \(\omega t_{\mathrm{p}}=\varphi_{\mathrm{p}}\), the angle of flight, after transformation we obtain

\[ 0.5\,\varphi_{\mathrm{p}}=\varphi_{\mathrm{v}}\sin\varphi_0+[-\cos\varphi_0+\cos(\varphi_0+\varphi_{\mathrm{p}})]. \tag{11} \]

If we take \(\varphi_{\mathrm{p}}=2\pi n\), where \(n\) is the fraction of the flight time in the total duration of \(2\pi\) oscillations, then (11), after substituting the separation angle and a number of transformations, takes the form

\[ \Gamma=\sqrt{\left(\frac{2\pi^2 n^2+\cos 2\pi n-1}{2\pi n-\sin 2\pi n}\right)^2+1}. \tag{12} \]

The dependence \(\Gamma=f(n)\) is already known in [4].

1. Velocity of vibratory conveying. For a steady two-phase regime with tossing, the conveying velocity of the object in the direction of the trough is equal to (Fig. 3)

\[ v=(x_{\mathrm{p}}+x_{\mathrm{l}})\frac{\nu}{k}, \tag{13} \]

where \(x_{\mathrm{p}}\) is the displacement of the object along the \(x\)-axis during the flight time,
\(x_{\mathrm{l}}\) is the same during the time of joint motion with the trough,
\(\nu\) is the oscillation frequency of the trough in Hz,
\(k\) is the number of complete periods of trough oscillation during which one cycle of motion occurs.

Under the assumptions made earlier,

\[ x_{\mathrm{p}}=\frac{A\omega\sin\varphi_0\cos(\alpha+\beta)}{\cos\alpha}\,t_{\mathrm{p}}, \]

\[ x_{\mathrm{l}}=x_{\mathrm{l}}(t_0)-x(t_{\mathrm{p}})=A[-\cos\varphi_0+\cos(\varphi_0+\varphi_{\mathrm{p}})]\frac{\cos(\alpha+\beta)}{\cos\alpha}. \]

Equation (13), after substituting \(x_{\mathrm{p}}\) and \(x_{\mathrm{l}}\), \(\varphi_{\mathrm{p}}\) and \(\Gamma\), and after transformations taking (11) into account, assumes the form

\[ v=\frac{g n^2 \cos(\alpha+\beta)}{2k\nu\sin\beta}. \tag{14} \]

For \(\alpha=0^\circ\), expression (14) becomes the formula previously obtained by other authors without taking into account the trough inclination angle [1].

Since formula (14) was derived with a number of assumptions, the actual vibratory conveying velocity is determined as

\[ v=\eta\,\frac{g n^2 \cos(\alpha+\beta)}{2k\nu\sin\beta}, \tag{14'} \]

where \(\eta\) is an experimental correction coefficient.

When analyzing the motion according to Fig. 3, it is evident that there is a range of values of the flight angle in which the object may fall onto the trough in the zo-

not \(\ddot y_l(t_n) \leqslant g\cos\alpha\), that is, when the object separates from the trough before it begins to move with the velocity of the trough. From (2) and (4) we have

\[ \sqrt{\gamma-\Gamma\cos(\alpha_0+\alpha_n)} \leqslant -1 \]

Transformation of this expression, taking (8) into account, gives the condition for abnormal course of the vibratory-conveying process

\[ \gamma=-\sqrt{\Gamma^2-1}\sin\varphi_n+\cos\varphi_n \leqslant -1 . \]

In Fig. 4 the solution of this equation is given. It has been established that when the parameter \(\Gamma\) changes from 3.3 to 4.6; from 6.3 to 7.79, and in other limits of \(\Gamma\), equation (14) is unsuitable for determining the velocity.

Fig. 4. Function \(\gamma=f(\Gamma)\).

Formula (14) is derived from the assumption that the impact is inelastic or, if rebound of the object occurs in the process of collision, that its magnitude is insignificant and the velocity of the object is compared with the velocity of the trough up to the moment of separation. In Fig. 3 this process is shown conventionally by significant oscillations of acceleration and velocity. In reality, especially when piece goods are being conveyed, elasticity plays a noticeable role and is the cause of disruption of the process at high collision velocities [3].

The collision velocity of the object with the trough is

\[ v_y=\dot y(t_n)-\dot y_l(t_n), \]

where \(\dot y(t_n)\) is the component of the velocity of the object at the moment of impact,

\(\dot y_l(t_n)\) is the component of the velocity of the trough at the same moment.

The equation for determining the collision velocity, after substitution of the components and transformations, will take the form:

\[ v_y=\frac{g\cos\alpha}{\omega}\left\{\left[\sqrt{\Gamma^2-1}(1-\cos\varphi_n)+\sin\varphi_n\right]-\varphi_n\right\}. \]

The limiting collision velocities at which the motion is still stable are determined experimentally.

To estimate the influence of the coefficient of friction on vibratory conveying, let us determine the angle of collision of the object with the trough through

\[ \operatorname{tg}\delta=\frac{v_x^0}{v_y}, \]

where

\[ v_x^0=\dot x(\varphi_0)-\dot x_l(\varphi_0+\varphi_n)=\operatorname{ctg}\beta\cos\frac{g}{\omega}\left[\sqrt{\Gamma^2-1}(1-\cos\varphi_n)+\sin\varphi_n\right]. \]

The magnitude of the collision angle

\[ \tg \delta = \ctg \beta - \frac{\left[\sqrt{\Gamma^2-1}\,(1-\cos \varphi_{\mathrm{п}})+\sin \varphi_{\mathrm{п}}\right]} {\left[\sqrt{\Gamma^2-1}\,(1-\cos \varphi_{\mathrm{п}})+\sin \varphi_{\mathrm{п}}\right]-\varphi_{\mathrm{п}}}. \tag{15} \]

From the solution of (15) it has been established that the collision angle in regimes with a parameter close to \(\Gamma=3.3;\ 6.3;\ 9.4;\ 12.6\) has a small value. Regimes with \(\Gamma=3.3;\ 6.3\), etc., have a collision angle equal to zero.

It is known that, when the collision angle is smaller than the angle of friction, there will be no relative motion between the colliding inelastic bodies; therefore it may be concluded that there are vibratory-transport regimes with \(\Gamma=3.3;\ 6.3\), etc., in which the velocity theoretically does not depend on the coefficient of friction.

1. Optimal vibration parameters. From (14) it is clear that a specified velocity can be obtained with various combinations of vibration parameters: amplitude, frequency, and the angles \(\alpha\) and \(\beta\). However, not all combinations are rational from the standpoint of the requirements imposed on the vibratory conveyor and on the process.

Fig. 5. Optimal value of the vibration angle.

Fig. 6. Effect of vibration amplitude and frequency on velocity.

In vibratory conveying, conditions may be imposed for minimal disturbance of the surface of the trough or of the objects themselves, and for the possibility of transport when the load has special properties. In operation, requirements are imposed on the spectrum of vibration frequencies, operating noise, dust formation, and energy consumption. A special condition for the design may be the requirement of increased reliability and durability of operation, which, other conditions being equal, can be achieved only by reducing the dynamic coefficient of the vibratory conveyor,

\[ K=\frac{A\omega^2}{g} \]

[1]. Combined requirements are often imposed on the conveyor.

In evaluating the effectiveness of vibratory conveying, one may proceed from the condition of the minimum dynamic coefficient, as the most general indicator determining the preservation of the load and the stress state of the conveyor’s elastic system.

From (14) it may be concluded that an increase in the angle of inclination of the trough, all other conditions being equal, causes a decrease in velocity; therefore, insofar as possible, one should strive to reduce the angle of inclination of the trough to a minimum.

The dependence of the optimal vibration angle on the dynamic coefficient is shown in Fig. 5; the graph was obtained by processing formula (14) for $\alpha = 0^\circ$.

Figure 6 gives the solution of equation (14) in amplitude–frequency coordinates. It is evident from the diagram that, at a smaller value of the dynamic coefficient, low-frequency vibratory conveyors can provide the highest feed rates.

REFERENCES

1. I. F. Goncharevich, P. A. Sergeev. Vibratory Machines in Construction. Mashgiz, Moscow, 1963.

2. I. I. Blekhman. Study of the process of vibroseparation and vibratory conveying. Engineering Collection of the USSR Academy of Sciences, vol. 2, 1952.

3. V. A. Povidailo. Calculation and Design of Vibratory Feeders. Mashgiz, Moscow, 1962.

4. K. H. Wehmeier. Untersuchungen zum Fördervorgang auf Schwingrinnen. Fördern und Heben, No. 5, pp. 317–327, 1961.