Performance of an oscillating conveyor driven through a spring

Mech. Mach. TheoryVol. 32, No. 7, pp. 835-842, 1997

Pergamon PII: S01D4-114X(97)00005~

© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 009,1-114X/97 $17.00 + 0.00

PERFORMANCE OF AN OSCILLATING CONVEYOR DRIVEN THROUGH A SPRING RAAFAT A. ABOU-ELNASR Faculty of Engineering, Prod. & Mech. Design Department, Menoufia University, Shebin El-kom, Egypt

M A H M O U D A. MOUSTAFA Faculty of Engineering, Mechanical Engineering Department, Alexandria University, Alexandria, Egypt

(Received 17 January 1995; in revisedform 23 September 1996; receivedfor publication 1997)

Abstract--Equations of motion for a conveyor driven through a spring carrying a load have been derived taking into consideration the friction between the conveyed load and its vibrating table. A quick-return mechanism is used to drive the table of the conveyor through a spring. Effect of velocity and the time ratio of the driving mechanism, spring stiffness and the damping criterion of the suggested system on the relative motion of the conveyed load with respect to the table are studied. © 1997 Elsevier Science Ltd

1. I N T R O D U C T I O N

Conveyors are conveying machines used in several industries to transport items. One type is the vibrating or oscillating conveyor. Vibrating conveyors are used for scalping and screening or picking (size separation) and to handle most bulk materials. Some of the disadvantages of such conveyors are the relatively short conveyor lengths, the limited capacities and the material degradation [1]. The characteristics of two different kinds of vibrating conveyors for incinerator ash handling systems, in relation to performance, maintenance and abrasive wear, are compared in Ref. [2]. A corrugated vibrating conveyor (grain pan) was tested for the effect of some various design variables on layer thickness and layer velocity of straw-chaff material and particles [3]. Several studies on the relative motion of the conveyed load with the trough of the oscillating conveyors are given in Refs [4-6]. The performance of oscillating conveyors is measured as the conveying advance of the conveyed load is given in Ref. [5]. Study of the effect of different selected through motions on the performance of oscillating conveyors is presented in Ref. [6]. The study showed that the amount of the advance depends upon the difference between the forward and backward motions of the trough. This work aims to study the performance of an oscillating conveyor in which the trough (or table) is driven by a quick-return mechanism through a spring and compare it with when the table is driven directly by the same mechanism.

2. E Q U A T I O N S

OF MOTION

The suggested conveyor consists of a table of mass M, two springs of stiffnesses k~ and k2, a dash-pot with damping coefficient c and a quick-return mechanism as a driving mechanism, see Fig. 1. The table is assumed to move on a frictionless horizontal plane. The quick-return mechanism drives the table via the spring of stiffness k2. The conveyed load has a mass m. The static and kinetic coefficient of friction between the load and the table are #s and #k, respectively. The motion of the load has two phases. The first is pure riding where there is no relative motion between the conveyed load and the table. The second is pure sliding where there is sliding motion between the load and 835

836

R. A. Abou.Elnasr and M. A. Moustafa

Stroke "s" ) i

=

i~

_1 ~ I Y

.

.

.

. _ 2 _.

r0

,,

l

AAAAA~

~/~/

VVVVV I

,'\

~'h-'~t~.'/"

rl

.-~ ! ~ / d ,'

. Load

~

~'ff/¢~-~

i ou ", J

Forward direction

!

-J/ K I 2 { K2 .

X

~,

i i

Backward direction ~

i

K, ,~, '

I

M

Table

'

~

t, AA/(A/~/ I------d'3-~

////////////./////////////////////"

Quick-return mechanism

Fig. 1. The suggested conveyor array.

~

3°f A 20

..~

-30 -40

I 0.5

I 1.0

I 1.5

I 2.0

o = 5 rad/sec Tr = 1.8 I I I 3.0 3.5 4.0

I 2.5

Time (see) Fig. 2. Table acceleration when the conveyor is driven directly by the quick-return mechanism.

21

C a

"'x

//

f

/z i

\\\

i!

ii

\\x ///

E e

-21 ;>

0

0.5

1.0

I 1.5

I 2.0

Table velocity ....... Load velocity I I [ I 2.5 3.0 3.5 4.0

Time (scc) Fig. 3. Table and load velocities when the conveyor is driven directly by the quick-return mechanism.

Oscillating conveyor driven through a spring

837

0.5~ R

E

-0.5

.~

-I.0

,,z

-I.5

a

~.

o

St

R

b

R - Riding S i - S l i d i n g in o n e d i r e c t i o n $ 2 - S l i d i n g in o t h e r d i r e c t i o n

-2,0 0

h

I

I

I

I

I

J

I

I

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (sec) Fig. 4. R e l a t i v e d i s p l a c e m e n t

when the conveyor

0.3-

0.2-

is d r i v e n d i r e c t l y b y t h e q u i c k - r e t u r n

t.o = 4 r a d / s e c kl = 1 0 0 0 N / m k2 = 3 0 0 N / m

o.I R

o

-o, i o

-0.2 -

R - Riding S] - S l i d i n g in o n e d i r e c t i o n $ 2 - S l i d i n g in o t h e r d i r e c t i o n

-0.3

I l

1 2

J 3

4

Time (sec) Fig. 5. R e l a t i v e d i s p l a c e m e n t

800

w h e n the c o n v e y o r

. . . . . . . kl = 300 N/m ..... k i = 400 N/m - - - - kl = 500 N/m kl = 600 N/m ..... k I = 800 N/m .... k i= 1000N/m

600

400

~-. / / / ,I. .f

/

/

@

1 /

200

=~

is d r i v e n w i t h a s p r i n g .

~ 2 . " : ~ ' - - .

o

<

-.

tO = l 0 rad/sec

Tr=2 k2 = 1500 N/m c =0

-200 -400

0

I I0

i 20

I 30

l 40

[ 50

J 60

N u m b e r o f c y c l e s (N) F i g . 6. Effect o f t h e s p r i n g stiffness k~ o n t h e a d v a n c e

of the load.

mechanism.

838

R.A. 800

Abou-Elnasr and M. A. Moustafa

• k2 = ..... k2 = -- - - k 2 = k2 = ..... k2= .... k2 =

600

'~m 400

300 N/m 600 N/m 800 N/m 1200 N/m 1500 N/m 1800 N/m

/ ," t" / t'" s, S

js iI i

200

~ <

_~.: ~- "

o

~

~

......................

o[ -200

~ = = 2l0 rad/sec k I = 500 N/m c =0

-40

I 10

0

I 20

I 30

I 40

h 50

I 60

Number of cycles (N) Fig. 7. E f f e c t o f t h e s p r i n g stiffness k: o n t h e a d v a n c e o f t h e l o a d .

800 r-

600

.c=0 ..... c = 10 -- - - c = 20 c = 30 ..... c = 40

N.sec/m N.sec/m N.sec/m N.sec/m

400 Q

200

>

0 ~o = 10 rad/sec Tr=2 kl = 500 N/m k 2 = 1500 N/m

< -200

-400 0

I 10

I 20

I 30

I 40

I 50

I 60

Number of cycles (N) Fig. 8. Effect o f t h e d a m p i n g coefficient c o n t h e a d v a n c e o f t h e l o a d .

50 4O

i

2o

=~lOo

L

< -10

......

-20 0

b 1

L 0

L 20

Motion through a spring Motion directly by the quick-return mechanism I L L I 2 3 4 5

I 6

Z

I 40

L 60 to

I 80

I 100

I 120

Fig. 9. E f f e c t o f f r e q u e n c y r a t i o z o r v e l o c i t y to o n t h e a d v a n c e a t T, = 1.5.

Oscillating conveyor driven t h r o u g h a spring

839

160 -

,.-.,~,~ 120

-~ so 40

st""

0. "~

t

-40 -S0

.

.

......

0

I l

I 0

I 20

.

.

.

Motion through a spring Motion directly by the quick-return m e c h a n i s m I I I I 2 3 4 5 z I 40

1 60 t.o

I 80

I 100

I 6

I 120

Fig. 10. Effect o f frequency ratio z or velocity co on the advance at Tr = 1.8.

the table. Therefore, the equations of motion of the suggested system will be dependent on these phases as follows:

2.1. Riding phase In this case the table and the load move as one body. This phase occurs under the following condition: ~g/> I~1 and ~cg = 0 where g a n d / / a r e gravitational and table acceleration respectively, ~CRis the relative velocity of the load relative to the table. Let y be the exciting motion applied to one end of the spring by means of the quick-return mechanism. From the geometric configuration of the system, see Fig. 1, one can deduce that Y=

(r0 + rt +

r l ) r : sin tot r2 COS t o t '

(1)

250 2O0 150 ,~

100

N

50

~

<

o-50 -lO0

{ 0

. ..... I 1 I 20

Motion through a spring Motion directly by the quick-return m e c h a n i s m I [ I I 2 3 4 5 Z [ [ [ I 40 60 80 100 to

I 6 I 120

Fig. 1 !. Effect o f frequency ratio z or velocity co on the advance at Tr = 2.1.

840

R . A . A b o u - E l n a s r a n d M. A. M o u s t a f a 250

150

.~ too

~

50

•~

0

. . - ................

,'

/

"

< -5o

Motion through a spring M o t i o n directly by the quick-return mechanism I [ I I 2 3 4 5 z

......

0

I 1

I 0

I 20

-I0O

L 40

I 60 0)

L 80

L. I00

I 6 I 120

Fig. 12. Effect o f frequency r a t i o z or velocity co o n the a d v a n c e at T, -- 2.4.

to, r~, r2 are dimensions of the quick-return mechanism and to is the angular velocity of the crank r2. The stroke of the ram of the mechanism " s " can be expressed as 2r2(ro + rl)

s

(2)

and the time ratio "T," is defined as (forward time/backward time) and is given by: Tr

7~

(3)

COS- t r2 to

The equation of motion of the system for this case is given by: (4)

( M + m)/~ + cx + (kt + k2)x = k2y

where, x and ~c are response and velocity of the table, respectively. 250 -

,-,

200 -

"~

150

< 50

1.2

M o t i o n through a s p r i n g f r e q u e n c y ratio z ~ 0.5-1.0 I 1,4

I 1.6

L I I 1.8 2.0 2.2 T i m e ratio (Tr)

I 2.4

I 2.6

Fig. 13. Effect o f the time ratio o n the a d v a n c e of the load.

Oscillating conveyordriven through a spring

841

2.2. Sliding phase The equation of motion of the system in the sliding phase is given by: MS? + cJc + (k, +

k2)x + I~kmg sgn(.;c -- JCL)= k:y

(5)

where the equation of motion of the load can be written as

m~L = lakmg sgn(Jc -- JCL)

(6)

:CL is the velocity of the load. The acceleration of the load relative to the table £R is given by: £R = 5? --/~kg sgn(Jc -- Jet)

(7)

Integrating equation (7) twice with respect to the time yields the relative velocity ,;caand the relative displacement x~ as follows: JCR= JC--/Agt sgn(Jc - .~CL)"1- CI 1

2

x~ = x -- ~/zkgt sgn(Jc -- -;eL)+ Ct t + 6'2

(8) (9)

Where C, and C2 are the integrating constants obtained by applying the initial conditions which are JR(0) = 0 and xR(0) = XRi, X~ is the initial value of the relative displacement XR and equal to the final value of the previous phase. Since y is complicated, Runge--Kutta method [7] is used to solve equations (4) and (5).

2.3. Conditions for riding and sliding phases The riding phase occurs when I£1~<#sg

and

JR=0,

[£l>/~sg

and

JR=0

while the sliding starts when

and ends when JCR= 0. In order to demonstrate these actions, consider the case when the table is driven directly by the quick-return mechanism. Figure 2 represents the table acceleration and includes two lines representing _+/~g. Figure 3 represents the table and load velocities. Before point " a " , riding occurs since the magnitude of the table acceleration is less than #~g. Sliding starts at point " a " and ends at point "b". At point "b", since the magnitude of the table acceleration is larger than /~g, backward slip occurs and ends at point "c". From " e " to " d " riding occurs and so forth. The displacement of the load relative to the table is shown in Fig. 4. It should be noted that the backward slip can be prevented if the value of the acceleration of the table at point "b" is less than/~sg. 3. RESULTS AND DISCUSSION In order to demonstrate the performance of the suggested system and the effects of its elements, consider the following data; M = 5kg, m = 0.5 kg, k, = 300, 400, 500, 600, 800, 1000 N/m, k2 = 300, 600, 900, 1200, 1500, 1800 N/m, c = 0, I0, 20, 30, 40 Nm/s, Tr = 1.5, 1.8, 2.1, 2.5,/as = 0.3,/zK = 0.15, to = 5-100 rad/s (corresponding to frequency ratio z = 0.25-5). Frequency ratio is equal to to divided by the natural frequency ton of the system, where the natural frequency without load is given by:

X/-•

+ k2

~n

--"

m

'

842

R.A. Abou.Elnasr and M. A. Moustafa

The stroke of the ram of the quick-return mechanism is considered unity in all cases. The displacement of the load relative to the table is shown in Fig. 5. The results are presented in terms of the advance of the load per cycle. Figures 6 and 7 show that the advance increases with increasing stiffnesses kl and k2 up to values 500 and 1500 N/m, respectively, after which the advance decreases. The effect of the damping coefficient c on the advance is shown in Fig. 8. Increasing c decreases the advance. A comparison between the advance of the conveyor in which the table is driven by the quick-return mechanism through a spring and when the table is driven directly by the same mechanism are presented in Figs 9-12 at different frequency ratio z (or different angular velocity co) and at different time ratio Tr. The values of kl, k2 and c were in this comparison 500, 1500 N/m and zero, respectively. The advance was computed after 20 cycles for each value of the frequency ratio. Figure 13 shows the effect of the time ratio on the advance. The advance increases with increasing the time ratio to a value of 2.4 and then the advance decreases. From the results it is clearly seen that the spring driven conveyor is preferred over that without a spring particularly at low frequencies. 4. C O N C L U S I O N

The performance of an oscillating conveyor driven through a spring is presented. The effect of the velocity, the time ratio, spring stiffnesses and damping coefficient of the suggested system on the advance of the load are studied. The following conclusions can be drawn: (1) The spring driven conveyor is preferred over that without a spring at low frequency ratio ranges. (2) The advance increases with the time ratio up to a certain value after which it decreases. (3) The damping has negative effect on the performance of the conveyor. Thus, it is better to dispense with it. (4) The springs affect greatly the performance of the system so they should be carefully selected. REFERENCES 1. Colijn, H., Chemical Engineering Progress, 1991, 87, 54-59. 2. Sullivan, J. F., Mayo, W. M. and Sullivan, J. M., National Waste Processing Conf., 14th Biennial Con., Long Beach, CA, U.S.A., 3-6 Jan. 1990. 3. Persson, S. P. E. and Megnin, M. K., Transactions of the ASAE, 1992, 35 (2), 395-400. 4. Parameswaran, M. A., Mechanism and Machine Theory, 1979, 14 (2). 5. EI-Shakery, S. A., Abou-Elnasr, R. A. and Khidr, K. M., Alex. Int. Engineering J., 1990, 30 (4). 6. Abou-Elnasr, R. A., Alex. Int. Engineering J., 1993, 32 (1). 7. Jams, M. L., Smith, G. M. and Wolford, J. C., Applied Numerical Methods for Digital Computation. WU, C-C, 1987.