Dynamics of a torsional type inertia shaker

J. ugric . E~gng Res. ( 1974) 19, 213-225

Dynamics

of a Torsional

Type

Inertia

Shaker

C. R. TUCK*; F. R. BROWNS An equation of motion for a torsional type inertia shaker is formed which describes the separate angular motions of the eccentric masses and the driven unit in terms of the geometry of their component parts. In the absence of a satisfactory theoretical relationship between the respective angular motions of the two components performance data from an existing Techanism were analysed to establish an empirical relationship. An expression of the form Q .I,.;, in which ,f’is dependent upon both the moment of inertia of the total assembly and that of the eccentric weights, was established and tested. Use of the mathematical model in the design of inertia shakers is discussed. Attention is drawn to the need to confirm the validity of the empirical relationship in applications where the critical dimensions of the inertia shaker and in particular their ratios differ significantly from those on which the relationship was based. Finally a simplified expression defining peak displacement is formulated and tested. 1.

Introduction

The design of the inertia shakers used in the N.I.A.E. experimental blackcurrant harvester was evolved by empirical methods and provided a ready means of adjusting the displacement of the vibrating tines. Once the desired range of displacement had been established further design changes, for example to increase the reliability of the mechanism, were checked by direct measurement before field trials commenced to ensure that they did not affect the amplitude. When preparing the design specification for a commercial machine considerable advantage was foreseen in deriving a mathematical model of the inertia shaker assembly. Such a model would permit rapid assessment of the existing design in relation to the optimum arrangement and could be used to compute the necessary change in eccentric mass to compensate for a proposed increase in the inertia of the driven unit.

KEY TO SYMBOLS Symbols

in main text

displacement, velocity and acceleration of inertia shaker assembly at time t displacement, velocity and acceleration of central drive pulley at time t displacement, velocity and acceleration of eccentric weight centre line at time I one eccentric mass displacement of centre of gravity of eccentric mass displacement of centre of gravity of very small angle sector distance between rotational axies of inertia shaker and the eccentric weight inertia of inertia shaker assembly torque produced by one eccentric weight

2. Inertia shaker The basis of the inertia shaker is shown in Fig. I. The two eccentrically mounted weights rotate in the same direction but 180” out of phase. The resulting centrifugal force can be seen to produce a clockwise and counter-clockwise couple during each revolution of the eccentric weights. This alternating couple produces the torsional vibrations of the inertia shaker, the frequency of which will be equal to the frequency of rotation of the eccentric weights. * Forage Conservation Department, N.I.A.E., Wrest Park, Silsoe, Bedford t Vegetable and Soft Fruit Department, N.I.A.E.. Wrest Park, Silsoe, Bedford

713

214

DYNAMICS

OF

A TORSIONAL

TYPE

INERTIA

SHAKER

(b)

\ .__-____-. Senseof

coup42 3

Zerocouple Cd)

Sense of couple3

(balanced)

_________

Zero couple

(balanced)

Drive /

(e)

Fig. I. Principle of’the inertia shaker

3.

Mathematical

analysis

Fig. 2 is a simplified representation of the inertia shaker with the sectionally shaped eccentric weights in their most inward position. This instant was considered to be time r = 0 and the resultant motion investigated. If the eccentric weights rotate in a clockwise direction the inertia shaker as a whole can also be assumed to have clockwise rotation at this instant, since the torque produced during the preceding half revolution of the weights would be mainly in a clockwise direction. Fig. 3 represents half the inertia shaker at time t when the inertia shaker as a whole is considered to have angular displacement 8, the central drive pulley D and the eccentric weight centre line a. A mathematical analysis (see Appendix I) gives the expression for the torque T produced by one eccentric mass at time 1. T = M[8(2D, d, cos a-d:--dE)+&(D,

d, cos a-dT)--D,

d,k2 sin a -2&gD, d, sin n]

. . (1)

Since the two eccentric weights are identical, the total torque developed in the inertia shaker will be twice this expression. x

Angle of sector =2y OA=d,=OA’

x

;”

r\

A

(-_-,(> (%) Y

AY

Y

Y

0

F-

”

”

Fig. 2. Inertia shaker at time t = 0

A’

Fig. 3. Inertia shaker crt time t

If the inertia shaker assembly is assumed to be perfectly rigid, it can be represented by inertia I made up from the inertia shaker inertia, Is, and the inertia of the driven unit Iti. For the inertia shaker as a whole 2T= IQ From Eqn (I) [/-~:-2M(d’4fd$--20,

d, cos a)]8-2M(Dl

d, cos a-d;)ii +2MD, d,(t2 sin a--4MD,

d,&d

sin a = 0

. . . (2)

To solve this equation it is necessary to establish the relationship between the angular mations of the eccentric weights (a) and inertia shaker (0). If it is assumed that the drive between the central pulley and the outer pulley is non-elastic and the pulleys are both of the same diameter then Q == Q-0 &z:&_e ;ic= n--J 80 -

5 40E 8 %

30-

$

-

0’ 0 25

Predicted Measured

?? VoAO

g 20,0 ;: IO$ I

performonce pOlntS

I

I

I

0.3 Total

assembly

I

I

05

04 Inertia

(kg 6)

Fig. 4. Predicted performance for f = 0.7 compared with crctual perfi,rmonc.e fbr diflerent eccerrtric weigh/s.

216

DYNAMICS

OF

A TORSIONAL

TYPE

INERTIA

SHAKER

TABLE I F-values

Total M = assembly DI = inertia (kg ml) Jcvahe

2082g

51.0 mm

-

I

--

Range for ho.5 mm error

M = 2944g DI = 50.5 mm

-

J

1‘-value

0.735 0.286

0.755

0.647

0.650

0751 0,803 0.748 0.753

0.762

-

0.611

0.697

0.637 0.664

0.592 0.621

0.708

0.645

0.688

0.668

0.596

0,655 0,685

0.630

0.723

0.656

0.709

0.648

0.600

0.676 0.705

0.633

0.714

-

0.635

0.708

0.655

0.760

-

0,583

0.668 0.690

0.685

0.820

-

0.642

0.682

0.644

0,706 0.735

0.599 0.623

0.699

0.774

0.637

0.644

0.698

0.807

0.795

0.613

0.664

0.73 1

0.782

0,603 0,621

0.662

0.752

0.630

0.634

0.683

0.729 -

error 0,596

0.687

0.675

0.700 0.729

f-value

_I

0.615

0.656

0.747

Range for to.5 mm

0.657

0.673

0.800

0.780

error

0.682

0.728

Range for $0.5 mm

0.630

0.705

0.780

I ~vulue

0.678

0.680

M = 4182g DI =- 53.3 mm

-

0.672

0,625

0.781

I

M = 3361 g DI = 53.6 mm

0.642

0,673

0.757

0.507

JCvalue

-

_Range for f 0.5 mm error

0.653

0.798

0.476

-

0.615

0.772

0.760

0.436

I

0.696

0.788

0.397

M = 3212g DI = 526 mm

0.660

0.754

0.357

Range for * 0.5 mm error __0.653

0.674 0.770

0.318

points

Eccentric weight description

-

-

for measured

0,661

-

These equations suggest that the relationship between a and D will depend upon the relative values of the inertias of the shaker and driven unit. For example as the inertia of the driven unit increases so its motion will decrease and G-+&j). If it is assumed that d = f6

hence [1+2M(dg+df(l-CC,)-(2-C&D,&

cos a)]8+2MD,

d,k2 sin a-4MD,

d,did

sin a = 0

. . . (4)

C.

K.

‘,‘UCK;

I:.

R.

BROWN

I

0 0 25

I

I 0.4

I

0.3

I

I

0.5

Total assembly inertia (kg m2)

Fig. 5. Best fit curves of form 0 = aIb for practically

obtained points

Provided f is known this equation can be solved by a numerical step by step solution on a digital computer with the aid of the additional relationships k2 = &,+&!dt a2 = al+&,dt 4.

The derivation off

No satisfactory theoretical derivation was obtained and therefore an empirical approach was adopted in which the relationship between a and Q was examined as a function of time and variation in the assembly inertias. By inserting specific standard functions (e.g. .f = constant,

0.6 02

03

0.4

05

I 0.6

I

I

I

I

I

0.7

00

09

IO

I.1

Total assembly inertia (kg mZ)

Fig. 6. Trend lines for f-values

to the regression

curve

I2

218

DYNAMICS

TABLE

F-values

Total assembly inertia (kg maI

OF

A TORSIONAL

TYPE

INIlRTlA

SHAKER

II

for regressed

Eccentric

points

weight description

M = DI = _____-

0.286 0.318 0.357 0.397 0.436 0.476 0.507

2082g 51.0 mm

M DI

0.769 0.761 0.768 0.773 0.78 1 0.790 0.798

= =

2944g 50.5 mm

M DI

= =

0.664 0.668 0.682 0.700 0.721 0.745 0.756

3212g 52.6 mm

M DI

0.667 0,658 0.662 0.669 0.677 0.692 0.703

=== 3361 g --= 53.6 mm 0.672 0.666 0.663 0,668 0.678 0.691 0.702

M DI

= 4182g == 53.3 mm 0.629 0,616 0,610 0.609 0,620 0,633 0,650

f = a sin 52 t) into the computer programme it was found possible in the case off’= constant to compute the cyclic variation in displacement as recorded on the experimental rig. An investigation was then carried out using the test rig to try to define a general relationship fort: The test rig permitted variation of the eccentric weight in the inertia shaker and of the inertia of the driven mechanism. This allowed seven points at different assembly inertias to be obtained for each of five different weights used in the test rig. Performance was measured as the peak to peak displacement at 254 mm radius. With the aid of the computer programme, thef-values necessary to give the performance as measured on the rig were calculated. These,f-values are tabulated in Table 1. The points do not present a simple relationship although there is an upward trend off with decreasing mass and to a lesser extent with increasing inertia. Since the possible error in measuring the displacement at 254 mm radius could be +0*5 mm, Table I also gives the corresto this error. ponding permissible range for each value off corresponding The approximate mean value offfor all the points obtained was 0.7 and the computed peak to peak displacements obtained using this value are compared with the measured data in Fig. 4. Although the largest error of prediction was 13.0 mm or about 4% it was considered necessary to search for possible relationships betweenfand eccentric mass or total assembly inertia before using the mathematical model to predict performance. To reduce the influence of experimental error in determining specific value of J’a regression analysis was carried out on the measured performance points. Curves of the form y = UX’were found to be a good fit as can be seen in Fig. 5 and the correspondingfvalues are given in Table II. Although these f values follow a pattern not enough of the curve is known to define any particular relationship. If however, the regression curves shown in Fig. 5 are extrapolated to permit calculation off values for total assembly inertias beyond the range of the measured performance data then a particular trend becomes noticeable as seen in Fig. 6. This trend lends TABLE

Coefficient

Eccentric

weight

III

values for each eccentric

“a” coejf:

“b” coeff.

0.9563 0.9348 0.9347 0.9296 0.9127

0.8325 2.5011 2.0267 2.0363 3.4991

M=2082 g, M=2944 g, M=3212 g, M=3361 g, M===4182g,

01~5 1 .O mm DI=50.5 mm Dl=52.6 mm D1=53.6 mm D1=53.3 mm

weight

“k,” coeff. _____ OGIO7308 I OTJOI2588 0mO91835 OX)0091 902 OW105273

493 539 689 743 980

402.0 077.0 150.0 676.0 142.0

<

R.

I-Uc’h:

I-.

R.

RROWN

A-

-

Mothematlcal

z;;t

?? O -

I 2

I

I

04

06

fit

from regrewon

+Q 5 mm boundary to regression curve points

---

I

I

IZ

I.0

0.8

Totalassembly Inertia (kg m2) Kg. 7.

Comptrrison of mnthemutical

b=O

fit to points, and actual points.fi,r

&values for two particular eccentric weights

,o - k,=25068

2120e~~~o9~&

“?

s?

0e6’95’Jo

9-

x

2:

a-

5

7-

$%j -2”

.

i

6-

051 010

’

012

’

014

I

016

I

01’3

I

020

I

022

I

024

Eccentric

Fig, 8. Individual f-equation

coefficients

41 0

I

0 12

0.14

016

018

0.20

weight inertia

ugainst total eccentric

weight inerfia

022

220

DYNAMICS

OF

A TORSIONAL

TYPE

INERTIA

SHAKER

80 %

a 2 -i-o: 5

60-

z z E

50-

z 40E 8 0 0 300 :: “a 200 s d

IOI o-3

0 0 25

I 0.4

I

Total assembly

Fig. 9. Computer

predicted

I

I 0.5

inertia

curves using the f-equation

with modified coqficient

justification to the use of extrapolation in that the curves appear to be asymptotic to unity or a value below unity and from Eqn (3) this can be seen to be an impossible value forf. The data in Fig. 6 can be fitted with the mathematical relationship a

f=

_(k,I

. . . (5)

+ k,lI?

1+be to a high degree of accuracy as can be seen in Fig. 7, which shows two sets of points and the boundary conditions for errors in displacement of ItO- mm. The values of the coefficients for

0.9 -

f.f=l

1619

fcp-

Simple

Fig. 10.

Comparison

of f-values

obtained from

formula

f-values

computer programme formula

and corresponding

f-values

used in the simpfe

(‘.

R.

‘I UCK;

F.

R.

BKOWN

01 0 25

I

Prediction

I

using modified f-equation

assembly

J

I

05

04 Total

Fig. 11.

I

I

0.3

Inertia

in conjunction

(kg rn?

with the simple formula

compared

to actual points

each eccentric weight are given in Table III. Errors in these curves at low inertias are of little concern since the total assembly inertia must exceed that of the shaker mechanism. From the previous curves and tables it would appear that the data at 2944 g eccentric weight is inconsistent with the other sets. The inconsistency is shown clearly in Fig. 4 where there appears to be a discontinuity part way through the data set. This is most likely due to some experimental error and repeat measurements were not possible because of changes in the experimental equipment. For this reason that set of points was ignored in the following analysis. Regression analyses on each of the four coefficients in thef equation indicated that they were simply related with the inertia of the eccentric weights about the common axis of rotation as shown in Fig. 8. In Fig. 9 comparisons are drawn between the predicted values of performance using these relationships to determinefand the actual measured performance. These predictions are highly satisfactory; differences between individual measured values and the corresponding prediction were within the accuracy of the measurements throughout the whole range. Relationships between the four coefficients of the f equation and eccentric mass instead of eccentric inertia were considered likely to give almost equally good predictions of performance. However, mass does not fully describe the system since no account of the eccentricities is made and, with insufficient variety in some of the variables to help ascertain which of these two parameters was the best, eccentric inertia was chosen as the more logical. In fact, it may be, that with TABLE IV

Simple formula predictions of displacement Displacement d, (mm)

f-value Total inertiu

Computer

Simple formula

Simple formula prediction

Actual

Error in d

0.525 0.674 0.824

0.679 0.143 0.799

0.636 0.711 0.176

51.0 42.7 37.1

50.5 41.5 35.0

0.5 I.2 2.1

f,

f2

222

DYNAMICS

OF

A TORSIONAL

TYPE

INERTIA

more variation of D, and d, the relationship could be shown to be a more complex m, D, and d, than that described by the inertia. 5.

SJiAKtR

function

of

Application of the model

The mathematical model was applied in the development of the blackcurrant harvester in two ways. Field experience with the early machines suggested the need to lengthen the vibrating cylinders and increase the number of tines in each ring. This proposal was implemented without loss of performance by calculation of the increase in eccentric mass required to compensate for the change in total inertia. The model was also used to established the influence of detail design features on performance. For example the relationship between the included angle of the sector of the circle describing the shape of the eccentric weights, their mass and performance was studied to establish the optimum shape of the weights. Providing the expression for f is valid beyond the range considered then it would appear possible to predict the performance of any inertia shaker assembly. In its present form solution of the equation of motion depends upon access to a computer and for universal application a simpler process would be desirable. For low values off, 0.5 or below, the displacement time function is almost sinusoidal and asf’increases so the function deviates from the pure sinusoid. If it were assumed that this function was always sinusoidal and could be represented by 0 =-- B0 sin at where 8, is the peak amplitude of the vibration then a computer is no longer necessary to calculate the peak displacement since the equation of motion reduces to 2MDr d, ” = 1+2M[d;t-d&LC,)] -1 where C, = l-f

.

(6)

as before.

Compensation for the error in the estimate of peak displacement resulting from assuming the motion is sinusoidal can be made by adjustment of the value off. Fig. IO shows the relationship between thef-values derived from consideration of the eccentric inertia of the shaker unit (Eqn (5)) and those required for substitution in the simplified equation. The relationship between these two sets off-values is clearly linear except for those values off corresponding to arrangements of the inertia shaker assembly in which the driven inertia is less than half the inertia shaker inertia. The exception is of little consequence since it is unlikely to occur in any practical application. Two comparisons of actual and predicted performance, using the simplified formula, were made. The first shown in Fig. I1 gives calculated and actual values for the five sets of data used in the derivation of,fi The second comparison in Table IV shows the error in the prediction when the driven inertia was progressively increased to more than twice that used in the derivation of,f: 6.

Conclusions

To describe the angular motion of a torsional type inertia shaker in terms of the geometry of its component parts it is necessary to supplement theory with an analysis of the performance of such a unit to establish an empirical relationship between the motion of the eccentric masses and that of the driven unit. The resultant mathematical model can then be used to develop the inertia shaker. Considerable confidence was gained in using this approach to effect major changes in the total inertia of the shaker use on the blackcurrant harvester. This experience however, cannot be regarded as a complete verification of the empirical relationship used in the model. The possible limitation in using this model to design an inertia shaker of widely different configuration may be nullified by the requirement to use the first experimental unit to establish the desired amplitudes of vibration for the particular application. Under these circumstances a

(

K.

: ll(

ii:

f:.

K.

717 __.

HKOWN

series of eccentric weights are likely to be required. If this is so then those weights specified for the middle of the range could be used to confirm or modify the empirical relationship before proceeding to manufacture the full range. Use of the simplified equation, for which it is assumed that the motion is sinusoidal. provides a rapid means of examining the effect of component changes on performance without any signiticant loss in accuracy over the full equation.

Appendix I Derivation of the torque equatiorz Consider the inertia forces, SF, and 6F,,, developed by the small sector position /_I from the central line of the eccentric weight and of included angle S/I (Fig. 3). This small sector can be replaced by a point mass 6m of equal magnitude acting through its centre of gravity B where AB = 4. For this small sector

(Al) where r /,

= outside

radius of weight

= mass per unit sector. d

and From Newton’s

1;

?!

=

--iima,

b42!

3

second law of motion 6F,

. . . (A31 . . (A41

6F,, = 6ma, The acceleration rotating axes:

of the mass can be obtained tiB := a,+&X+OX(W


where d, = [t?d, cos e-&d,sin tire, = [Gdl sin (u+O+p)-%dl

from the general

equation

for motion

fi,,, +a,,,

B]i+[&‘d,sin

O+@d,cos

(A3 e]j

cos (u+O+/l)]i

I (A61 (A71

--[&” dl cos (a+O-t-[j)+Gdl c WI =

relative to

sin ((I- 0 . /O]j

;Id, cos (a t-f--tp)i--iYdl sin (a+O+B)j y-

-d, sin (u-+-e+b)i+d,

cos (a-. 0 :-p)j

w = tik d = ok &~;r == -d,t?cos wx(c;,xP)

(~_l-O+@i-d~#sin

(a to-L-p)j

= d,~2sin(u+~+~)i-d,~2cos(u-tB+~)j

26 x tire, = 2&d, sin (u+e+/3)i-2dl&~

cos (a+O+B)j

. . . (A8) (A9) . . . (AlO)

224

DYNAMICS

OF A TORSIONAL

TYPE

INERTIA

SHAKER

From Eqns (A5-AIO) a, = 0[d, cos 19-d, cos (f_~+Q+P)]--8~d, sin 8-Gd, cos (a+Q+/?) +d, sin (a+8+P) uY = B’[d,sin 6d,

sin (~+8+/?)]+8~d~

cos

sin

e-a,

(~%+ti)~ . . . A(1 1)

(a+e+p) -dl

cos (a+e+p)(k+-0)2

...

(A12)

From Eqns (A3) and (All) 610, = 6m{B[d, cos

(a+e+j3)--d2

cos

B]+@d2 sin B+;;d,

cos (~+e+p)

-4

sin (u+~+~)(&+~)2}

. . . (A13)

From Eqns (A4) and (A12) 6F,, = &z{B’[d, sin O-d, sin (a+B+P)J+82d2

cos B-&d, sin (u+fl+p) --dl cos (u-~O+/?)(&+8)2}

. . . (A14)

If the torque generated by the small sector about 0 in the anti-clockwise direction is 6T then 6T = 6F,[d, cos e-d,

cos (u+B+/3)]-6F,,[d2

sin O-d, sin (u+O+fi)]

Substitution of Eqns (A13) and (A14) gives 6T = 6m{8[2d,d2 cos (u+/3)-df-d;]+B[d,d,

cos (u+P)-df] -dld2k2 sin (a+/?)-2&6d,d,

sin (u+b)

. . (A15)

If the torque generated by the complete eccentric weight is T, T=

y @ -Y s

T=

’ ;p{ -?Js

= gp[28d,d,

8[2d,d, cos (a+/?-d,“-d;]+?i[d,d,cos

(a+j3-d:] -djd2di2 sin (u+P)--2&6d,d,

sin (u+P)-8/3(dF+di)+&d,d,

sin (u+B) -&df/?+&“dld2

= sp{2Bd,d,[sin

(a- Y)]-2&d:

cos (at-,!l)]-vY

y

+&2d,d2[cos (a+ y)-cos =pr2(8[2dld2

cos (u+/?)+2&~d,d,

(a-Y)]-2(dt+dz)8Y

(u+y)-sin

+&d,d,[sin (a+ y)-sin

sin (u+P)}dP

sin y cos a-(d2

+di)Y]+&[dld2

(a- Y)]+2&8d,d,[cos

(a+ Y)-cos

(a- Y)]

cos a sin Y-d,2Y] -dld2di2 sin a sin y--2&d,d,

sin a sin y}

sin y COSU-df Y

-d,d2k2

sin uy

-2&d,d,

sin a y)

. .

(Al61

(

R.

I-UC K:

F.

R.

‘25

BROWN

If the total mass of the eccentric weight is M and the eccentricity is D, M := pr”y

.1.

(A17)

D = 2r sin Y L

3

‘i’

From Eqn (A2) D,

=

dl

sini’

. . (A18)

2’

:. from Eqns (A16), (A17) and (A18) 7’= 114[&2D,d, cos a-dt-dz)+bi(D,d,

cos ~-d;)-D~d,&~

sin a-2&D,d,

sin a]

. . (A19)