Lang_Dynamic Modelling Structure of a Fruit Tree for Inertial Shaker System Design

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Biosystems Engineering (2006) 93 (1), 35–44doi:10.1016/j.biosystemseng.2005.09.003PM—Power and Machinery

Dynamic Modelling Structure of a Fruit Tree for Inertial Shaker System Design

Zolta ´ n La ´ ng

Technical Department, Corvinus University, Budapest Villa ´ nyi u. 31, 1118 Budapest, Hungary; e-mail: [email protected]

(Received 15 December 2004; accepted in revised form 5 September 2005; published online 28 November 2005)

Idling and working power equations for a shaker with rotating eccentric weights and for a slider crank type

were established and compared. The tree structure model comprised both trunk and main roots. Using static

equations, the location of virtual turning centre versus shaking height function was defined. Based on this

function, a simple three-element model composed of reduced mass, spring constant and kinetic dampingcoefficient was transferred from a trunk cross-section to all others. Replacing these reduced parameters into

the power equation of the two types of shaker and into the equation of trunk amplitude, the power

consumption and amplitude versus shaking height curves were drawn. These functions, with the free

combination of eccentricity, shaking frequency and mass parameters of the shakers help to optimise machine

design. Calculating the relationship between power consumption and trunk amplitude for all trunk cross-

sections, the most efficient clamping heights of the shakers were found.r 2005 Silsoe Research Institute. All rights reserved

Published by Elsevier Ltd

1. Introduction

One of the aims of fruit tree modelling is to supply

reliable data for the shaker harvester design.

In the differential equation of the fruit tree–shaker

system of Fridley and Adrian (1966) the tree was

replaced by a three-element model, which was vibrated

by a sinusoidal changing force, generated by unbalanced

masses (Fig. 1):

M t €xM þ k _xM þ1

cxM ¼ mro2 sinot (1)

where: M t is the total mass of the limb–shaker system in

kg; €xM is the limb acceleration in m sÀ2; k is the viscous

damping coefficient of the limb in N s mÀ1; _xM is the

limb velocity in m sÀ1; c is the apparent spring constantof the limb in mNÀ1; xM is the limb displacements in

horizontal direction in m; m is the total unbalanced mass

of the shaker in kg; r is the eccentricity of the

unbalanced masses in m; o is the shaking frequency in

rad sÀ1; and t is the time in s.

The momentary power input for a rotating type

shaker as shown in Fig. 1 (Ludvig, 1973) is

P r ¼mro3X

2sin 2ot À jð Þ þ sinj½ (2)

where: X is the amplitude of displacement in horizontal

direction in m; and j is the phase angle in rad.

For the calculation of the trunk displacement

amplitude X , the following well-known equation can

be used:

X ¼mro2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

cÀ M to

2

2

þ k oð Þ2

s (3)

For the phase angle:

tanj ¼k oc

1 À M tco2(4)

With the assumption that the shaking frequency ismuch higher than the fundamental mode frequency of

the limb (o ) on), the calculation of the limb peak-to-

peak stroke S in m was simplified: it depended on the

masses of the system and of the eccentricity of

unbalanced masses:

S ffi2mr

M t(5)

This widely used Eqn (5) of Fridley and Adrian (1966)

however does not take into account the elasticity and the

ARTICLE IN PRESS

1537-5110/$32.00 35 r 2005 Silsoe Research Institute. All rights reserved

Published by Elsevier Ltd



damping property of the tree, neither of the shaking

frequency.

An attempt to include the above parameters in the

model was made by Whitney et al . (1990). The reduced

mass, elasticity and the viscous damping coefficient were

measured individually on a wooden post fixed to theground as a vertical cantilever. The data achieved were

controlled in shaking experiences: an inertia-type trunk

shaker was clamped to the post, and displacement and

acting force were measured and calculated.

Comparing measured and calculated data, Whitney

et al. (1990) found that the post acted nearly as a

pure spring at the frequencies employed. They found

that the trunk itself is rather elastic; a great part of

the input energy during shaking harvest is absorbed

elsewhere.

Horva ´ th and Sitkei (2001) presumed that during

shaking the input energy is mostly absorbed in the soil

through the rooting system and, therefore, the trunk

cannot be regarded as a vertical cantilever. It translates

and turns during shaking and vibrates a certain amount

of soil around the base. They measured the translationsof the tree by shaking the trunk at different heights, then

calculated the virtual centre of turning. It was found

that the location of this centre changes with the height of

shaking, and so does the reduced mass measured at the

clamping points. Their conclusion was that the increase

in the reduced mass means increasing mass in soil

vibrating with the trunk. Evaluating run-out accelera-

tion curves of a trunk shaker, the logarithmic decre-

ments for different trunk cross-sections were defined and

compared with data obtained by a presumption, based

ARTICLE IN PRESS

Notation

A, B nodes of the main root in the soil

b horizontal distance of nodes A and B

from the centre line of the trunk, m

C the virtual turning centre of the tree

model,

c apparent spring constant of the limb at

the spot of shaking, m NÀ1

c1, c2, c0 apparent horizontal and vertical spring

constants of the main roots, m NÀ1

F force in scalar form, N

F Ax1, F Ax2,

F Bx1, F Bx2

virtual forces acting at the end of main

roots horizontally, N

F Ay, F By virtual forces acting at the end of main

roots vertically, N

f force in vector form, N

h the vertical depth of the nodes A and B

underneath the soil surface, mk viscous damping coefficient of the limb at

the spot of shaking, N s mÀ1

M mass of the shaker body, kg

M M reduced mass of the tree limb at the

clamping point of the shaker, kg

M r reduced mass of the mass of M M to the

nodes A and B, kg

M r total mass of the limb–shaker system,

including M M , M and m, kg

m total unbalanced masses, kg

O trunk position on the soil surface,

P power consumption of the shaker, kWr eccentricity of unbalanced masses, m

S peak-to-peak stroke of the shaken limb, m

T time of a cycle, s

t time, s

U elastic energy, N m

X amplitude of displacement in horizontal

direction, m

x displacements in horizontal direction, m

_x velocities in the horizontal direction,

m sÀ1

€x accelerations in the horizontal direction,

m sÀ2

y vertical distance from ground level, m

v velocity, m sÀ1

W energy input, N m

a turning angle of the tree trunk, rad

_a angular velocity of the tree trunk, rad sÀ1

j Phase angle, rad

r vertical distance of the virtual turning

centre C from O, m

o shaking frequency, rad sÀ1

on fundamental mode frequency, rad sÀ1

Subscripts

av average

c spring related

def defined value

i idle

k viscous damping related

M limb

m unbalanced masses

r rotating eccentric mass-type shaker

red reduced values slider crank-type shaker

th Theoretical

Z. LA ´ NG36



on the relation of reduced masses of soil and canopy

(Horva ´ th & Sitkei, 2002).

Lang (2003, 2005) presented a tree model, in which

the main roots were included. Their external ends were

fixed to the soil through joints. The translation, rotation

and bending of real trunks were measured applying

different static loads. As a result, virtual centres of turning were defined, which changed with the force

applied. Apparent static spring constant and viscous

damping coefficient of the whole trunk and also of root

samples was measured and defined. Dynamic damping

coefficient of the acceleration curve of the free swinging

limb was calculated. Finally a static and dynamic model

was suggested, composed of reduced mass, damping and

elastic elements.

The objective of the investigations described below is

to describe mathematically the power consumption,

generated amplitude and specific power for each trunk

cross-section to be able to design shaker harvesters moreprecisely.

2. Materials and methods

2.1. Power requirement of inertia shaker

Equation (5) of Fridley and Adrian (1966) for the

peak-to-peak stroke of the limb does not differ from

the stroke of a mass (M +M M ), the sum of the mass of

the shaker body M and the reduced mass of the tree limb

at the clamping point M M , both in kg, to which

an unbalanced rotating or periodically translating

mass m is clamped (La ´ ng, 2004). In these cases, the

total mass in kg

M t ¼ M þ m þ M M (6)

includes the mass of the shaker body, the exciting mass

and the reduced tree mass (Fig. 1).

The theoretical power consumption of a rotating

eccentric–weight mechanism P r,th in kW and of a slider

crank shaker mechanism P s,th in kW, coupled to a one-

degree-of-freedom mass M M , was compared by La ´ ng

(2004). Because of the somewhat different working

principle of the two units, the power equations are

different.

For the shaker with rotating eccentric–mass [Fig. 2(a)]:

P r;th ¼ m

2

r

2

o

3

sin2ot2ðM þ M M þ mÞ

(7)

For the slider crank-type one [Fig. 2(b)],

P s;th ¼mðM þ M M Þr2

o3 sin2ot

2ðM þ M M þ mÞ(8)

Note that in these equations, the limb is regarded as a

pure reduced mass; elasticity and damping are neglected.

According to Fridley and Adrian (1966), Eqn (5) and,

consequently, Eqn (7) gave no exact results all around the

limb.

Equations (7) and (8), however, can be used to define

the idling power consumption of the two types of shakers. In these cases, M M ¼ 0 (Fig. 2).

It is interesting to compare the trends for the two

cases:

if M -N, then P r,i -0, but

P s;i !mr2

o3 sin2ot

2(9)

By increasing the mass of the shaker body the idle

power consumption of the rotating eccentric–weight-

type shaker decreases and that of the slider crank-type

shaker tends to the value in Eqn (9).

Substituting M M , c and k data measured on real treesinto the equation of the three-element model coupled

with inertia-type shakers [Eqn (1), Fig. 2], more realistic

trunk displacement and power requirement data can be

achieved for a given trunk cross-section.

The momentary displacement of the limb in this case is

xM ¼ X sin ðot À jÞ (10)

The momentary velocity of it is

_xM ¼ X o cos ðot À jÞ (11)

ARTICLE IN PRESS

ω

ω

M

m

c

k

2

m2

M M

Fig. 1. The model of the tree-shaker system (La ng, 2004): M,mass of the shaker; m, total unbalanced masses; M M , reduced mass of the tree limb; k, viscous damping coefficient of the limb;

c, apparent spring constant of the limb

DYNAMIC MODELLING STRUCTURE OF A FRUIT TREE 37



And the acceleration is

€xM ¼ ÀX o2 sin ðot À jÞ (12)

For the limb displacement amplitude X , Eqn (3) gives

the values for both the rotating- and the alternating-type

shaker.

The average effective power needed to drive a rotating-

type shaker can be calculated as follows. The energy input

W r in N m for a cycle (Fridley & Adrian, 1966) is

W r ¼Z

P rdt ¼Z T ¼2p=o

0

mro3X

2 sin ð2ot À jÞdt

þ

Z T ¼2p=o

0

mro3X

2sinjdt ¼

mro3XT

2sinj

ð13Þ

where T is the time of a cycle in s.

The average effective driving power is

P r;av ¼W r

T ¼

mro3X

2sinj (14)

In the case of the alternating or slider crank-type

shaker (Fig. 3) the calculation of the power input can bedone by summarising the powers P M and P m needed to

drive the masses M +M M and m, respectively.

The equilibrium of the forces acting on the mass

M +M M as shown in Fig. 4(b) is

f M ¼ ðM þ M M Þ €x1 À f c À f k (15)

where f M , f c and f k are the vectors of limb inertia, limb

elasticity and limb damping forces, respectively, all in N.

For the translation, velocity, acceleration and phase

angle of M +M M Eqns (10), (11), (12) and (4) apply.

Equation (15) can be written in scalar form as follows:

F M ¼ À ðM þ M M ÞX o2 sin ðot À jÞ

þ kX o cos ðot À jÞ þ1

cX sin ðot À jÞ ð16Þ

or

F M ¼ X 1

cÀ ðM þ M M Þo

2

!sin ðot À jÞ

þ k o cos ðot À jÞ

!ð17Þ

ARTICLE IN PRESS

ω

ω

M

M

(a) (b)

m

2

m

2

mr

Fig. 2. Models built to define the idle power consumption of the two types of inertia shakers: (a) rotating eccentric–weight; and (b)a slider crank shaker mechanism; M, mass of the shaker; m, total unbalanced masses

c

k

M M M

m

Fig. 3. The slider crank-type shaker coupled to the three-element tree model: M, mass of the shaker; m, unbalanced mass;M M , reduced mass of the tree limb; k, viscous dampingcoefficient of the limb; c, apparent spring constant of the limb

Z. LA ´ NG38



The power needed to drive the mass M+M M is then

P M ¼ F M _xM

¼ X 2o1

2

1

cÀ ðM þ M M Þo

2

!sin 2ðot À jÞ½

þ k o cos2ðot À jÞ! ð18Þ

The equilibrium of forces acting on mass m can be

written in the following way [Fig. 4(a)]:

f m ¼ Àm €xm (19)

whereby its displacement, velocity and acceleration are

xm ¼ xM þ r sinot (20)

_xm ¼ _xM þ ro cosot (21)

€xm ¼ €xM À ro2 sinot ¼ ÀX o2 sin ðot À jÞ À ro2 sinot

(22)

The driving force is

F m ¼ m X o2 sin ðot À jÞ þ ro2 sinotÄ Å

(23)

The power requirement is then

P m ¼ F mvm ¼ m X o2 sin ðot À jÞ þ ro2 sinotÄ Åð _xM þ ro cosotÞ ð24Þ

where vm is the velocity of m in msÀ1.

Substituting _xM from Eqn (11) instead of vm, P m takes

the following form:

P m ¼1

2mo3½X 2 sin2ðot À jÞ þ r2 sin2ot

À 2Xr sin ð2ot À jÞ ð25Þ

The power consumption of the slider crank-type

shaker is then

P s À P M þ P m (26)

The effective average power needed to drive the slider

crank-type shaker can be calculated as follows. The

energy input for a cycle is

W s ¼

Z ðP M þ P mÞ dt (27)

As the integral of P m is zero, the energy input is

W s ¼

Z T ¼2p=o

0

X 2o

2

1

cÀ ðM þ M M Þo

2

!sin2ðot À jÞ dt

þ Z T ¼2p=o

0

X 2k o2 cos2ðot À jÞ dt

¼ 0 þ X 2k o2 p

oð28Þ

The average effective power consumption of the slider

crank-type shaker T ¼ 2p=oÀ Á

is

P s;av ¼W s

T ¼

1

2X 2k o2 ¼

1

2

ðmro2Þ2k o2

1

cÀ ðM tÞo2

!2

þ ðk oÞ2

(29)

2.2. A new tree structure model

For the simple tree model of La ´ ng (2003), built of

trunk and main roots (Fig. 5), the following equations

can be written:

F ¼ F Ax2 þ F Bx2 À F Ax1 À F Bx1 ¼ 2x

c(30)

Fy ¼ bðF Ay þ F ByÞ þ hðF Ax1 þ F Bx1Þ À hðF Ax2 þ F Bx2Þ

(31)

where: F Ay and F By are the virtual forces acting at the

end of main roots vertically in N; F Ax1, F Ax2, F Bx1, F Bx2are the virtual forces acting at the end of main roots

horizontally in N; y is the vertical distance from groundlevel in m; b is the horizontal distance of nodes A and B

from the centre line of the trunk in m; h is the vertical

depth of the nodes A and B underneath the soil surface

in m.

Presuming that F Ay ¼ F By, F Ax1 ¼ F Bx1 and

F Ax2 ¼ F Bx2,

2hðF Ax1 À F Ax2Þ ¼ Fy À 2bF Ay (32)

F Ax1 À F Ax2 ¼Fy À 2bF Ay

2h(33)

2bF Ay ¼ F ð y þ hÞ (34)

and, from Eqns (33) and (34),

F Ax1 À F Ax2 ¼F

2(35)

Let x be the translation of O. Presuming that for

the apparent horizontal and vertical spring constants of

the main roots c1, c2, c0 (all in mNÀ1), c1 ¼ c2 ¼ c0,

then:

x ¼c0F

2(36)

ARTICLE IN PRESS

m

x 2 x 1

f m

(a) (b)

M+M M

f m

f c

f k

Fig. 4. (a) The force balance of the exciting mass m and (b) of the excited masses (M+M M ): M, mass of the shaker; m,unbalanced mass; M M , reduced mass of the tree limb; x1, the

displacements of m; x 2, the displacements of (M+M M )




‘The turning angle a in rad of the trunk around O is

given by

a ffiF Ayc0

b¼

F ð y þ hÞc0

2b2(37)

Finally, the vertical distance r in m of the virtual turning

centre C from O is

r ffix

a¼

c0F

2 c0F ð yþhÞ

2b2

¼b2

y þ h(38)

It can be seen that besides the geometrical sizes h andb of the main roots, r depends on the vertical position y

of the force F .

The change of r in function of y (0o yo1100 mm) is

shown in Fig. 6 , together with the measured r values

(’) of Horva ´ th and Sitkei (2001). The best-fitting

values of the parameters for the calculation of r were as

follows: h of 100 mm and b of 680mm.

With the help of Eqn (37), the reduced trunk mass,

spring constant and the viscous damping of a cross-

section can be reduced to any other.

First let a defined reduced mass M red of the tree trunk

cross-section yred be transferred to A and B (Fig. 7 ). Let

rdef be the calculated value of r belonging to ydef ,

keeping the kinetic energy unchanged:

1

2M def ð ydef þ rdef Þ

2_a

2 ¼1

2M red rdef À h

2

þ b2

!_a

2

(39)

where _a is the angular velocity of the tree trunk, and

M red ¼ð ydef þ rÞ2

ðrdef À hÞ2 þ b2M def (40)

Now for any cross-section of the trunk the reduced

mass M(y) can be calculated the following way:

1

2M ð yÞð y þ rÞ2

_a2 ¼

1

2M red ðr À hÞ2

Âþ b2

Ã_a

2 (41)

M ð yÞ ¼ð ydef þ rdef Þ

2

ð y þ rÞ2

ðr À hÞ2 þ b2

ðrdef À hÞ2 þ b2M def (42)

Figure 8 shows the change of calculated reduced

masses along the trunk. The values of the parameters forthe calculation were as follows: ydef of 800 mm, rdef of

463 mm, h of 100 mm and b of 680 mm.

La ´ ng (2005) measured the reduced mass of cherry

trees at 80 cm trunk height using Rayligh’s method. For

a tree of 13 cm trunk diameter the reduced mass M def was 130 kg. These data were replaced into Eqn (42) and

the M(y) values were calculated and plotted for

different trunk heights (Fig. 8). To be able to check

the ability of the new model, the M def value measured by

Horva ´ th and Sitkei (2001) was also replaced into

ARTICLE IN PRESS

c1

c2

c1

bb

h

y

F

AB

c2

F Ay

F Bx 2F Bx 1F Ax 2F Ax 1

F Ay

b

x

b

h

y

F

α

α

α

AB

C

0 0

Fig. 5. The static equilibrium of the simple model built of trunk and main roots: A and B, nodes of the main root in the soil; F, forceacting on the trunk; F Ax1, F Ax2, F Bx1, F Bx2, virtual forces acting at the end of main roots horizontally; F Ay, F By, virtual forces actingat the end of main roots vertically; b, h, y, coordinates of nodes A and B; x, the displacement of O; a, turning angle of the tree trunk;C, the virtual turning centre of the tree model; r, the vertical distance of the virtual turning centre C from O; c1, c 2, the apparent

horizontal and vertical spring constants of the main roots

0

500

1000

1500

2000

2500

3000

0 400 800 1200

Position of the shaking force F above ground, mm

P o s i t i o n o f C b e l o w g r o u n d

l e v e l , m m

Fig. 6. The vertical position of the virtual turning centre C as a function of y, the position of F above ground

Z. LA ´ NG40



Eqn (42) (M def ¼ 190 kg, Fig. 8). Both curves and the

discrete reduced mass values of Horva ´ th and Sitkei

(2001) are shown in Fig. 8.

The energy stored in the springs can be calculated as

the sum of the energy stored in the horizontal and

vertical springs in Fig. 5:

U ¼ 12

ððr À hÞaÞ2

cdef

þ 12

ðbaÞ2

cdef

!2 ¼ ðr À hÞ

2

þ b2

cdef

a2 (43)

The elastic energy U in N m of a cross-section at height y

above ground level is

U ¼1

2

ð y þ rÞ2

cð yÞa

2 (44)

Solving Eqns (43) and (44), the spring constant c0 can

be transferred to any cross-section the following way:

cð yÞ ¼cdef

2

ð y þ rÞ2

ðr À hÞ2 þ b2(45)

Substituting the value for cdef of 7 Â 10À6 m NÀ1 in

Eqn (45) [the measured value in La ´ ng (2003) for a 13 cm

trunk diameter cherry tree at 80 cm trunk height], the

change of spring constant along the trunk can be drawn,

as shown in Fig. 9.

The k(y) values can be calculated similarly to the

spring constant reduction:

k ð yÞ ¼ðr À hÞ2 þ b2

ð y þ rÞ2k def (46)

La ´ ng (2003) calculated the value for k def of

2360NsmÀ1 for the same 13 cm trunk diameter cherry

tree at 80 cm trunk height. Figure 10 shows the change of

Lang’s viscous damping coefficient along the trunk(below), together with a curve (above), calculated using

k def ¼ 10400N smÀ1 as an approach to the measured

discrete values of Horva ´ th and Sitkei (2002).

3. Results

Replacing the three-element model parameters from

Figs 8–10 into Eqns (14) and (29), the average power

ARTICLE IN PRESS

0

200

400

600

0 200 400 600 800 1000 1200

Height above ground, mm

R e d u c e d m a s s

M ( y ) , k g

Fig. 8. Measured reduced masses by Horva th and Sitkei (2001)’, calculated values, based on the measured data of Horva thand Sitkei, 2001 (above), and of La ng, (2003) (below), all

along the tree trunk

0.0000.0020.0040.0060.0080.011

0 200 400 600 800 1000 1200


S p r i n g c o n s t a n t ,

m m N − 1

Fig. 9. Calculated spring constant values along the tree trunk

M def

y d e f

b

b

hA B

C

M red

2

M red

2

0

Fig. 7. The transfer of reduced mass M def to nodes A and B asM red /2; M def , a defined (measured) trunk mass value; M red ,reduced mass from the trunk to nodes A and B; b,h; coordinatesof nodes A and B; ydef , the vertical coordinate of M def ; r, the

vertical distance of the virtual turning centre C from O

0

10

20

30

40

50

0 200 400 600 800 1000 1200

Height above ground, mm V i s

c o u s d a m p i n g c o e f f i c i e n t ,

k N s m − 1

Fig. 10. Measured viscous damping coefficients of Horva th and Sitkei (2002), marked by’, calculated values using the data of Horva th and Sitkei, (2002) (above) and of La ng (2003)

(below), all along the tree trunk




versus trunk height curves can be drawn. Figure 11

shows those for a slider crank-type shaker and for a

rotating eccentric–weight-type one.

The values for the parameters in Eqn (14) were taken

from the data sheet of a Kilby tree shaker, the exiting

masses of which are rotating and are m of 64 kg, r of 105

mm and o of 100Á5radsÀ1. The mass of the shaker body

is 680 kg.The parameters for Eqn (29) were taken from the data

sheet of a Schaumann shaker machine, the exiting mass

are of which are alternating and are m of 135 kg, r of

22 mm and o of 100Á5radsÀ1. The mass of the shaker

body is 34 kg.

From the point of view of fruit removal, the

amplitude of the shaken cross-section may be of interest.

Replacing the parameters of the three-element model

and shaker machine into Eqn (3) the amplitude versus

clamping height functions can be drawn (Fig. 12).

To judge the efficiency of the shaker type the specific

power (power needed to 1 mm trunk displacement)

can be introduced. Figure 13 shows the specific power

values for the alternating-type and rotating-type tree

shaker.

4. Discussion

The setup of the idle power equations for rotating

eccentric–weight-type and slider crank-type shakers

enables us to study the effect of shaker body mass on

the power consumption and optimise it.

Equation (29) set up for the average effective power

consumption of the slider crank-type shaker is the same

as the equation of the rotating-type one.

The simple tree structure model in Fig. 5 proves to be

appropriate to explain the phenomenon of changing

virtual turning centre location of the tree when shakingit at different clamping heights. For the calculation of

the turning centre’s location Eqn (38) can be used, which

is composed of geometrical parameters.

The diagram in Fig. 6 shows the vertical position

of the turning centre as a function of the shakers

clamping height. It also shows the measured discrete

values of Horva ´ th and Sitkei (2001) which fit well on the

curve.

Based on Eqn (38) a measured reduced mass M 0 of a

trunk cross-section can be transferred to any other trunk

cross-sections. The calculation can be made in two steps:

in the first one the kinetic energy of M 0 is transferred to

the end point of the roots, in the second phase it can be

reduced from there to any trunk cross-section. Using

one measured reduced mass value of La ´ ng (2003) all

others were calculated. Figure 8 shows this diagram

together with another one, which was calculated using

the measured discrete values of Horva ´ th and Sitkei

(2001).

Similar to the mass reduction, the spring constant of a

trunk cross-section, based on the principle of elastic

energy conservation, can be reduced to any other. Figure

9 shows how it changes along the trunk.

With the help of Eqn (38) the viscous damping

coefficient of a trunk cross-section was also reduced toall other trunk cross-sections. In the diagram of Fig. 10,

two curves, calculated on different k 0 values, are shown

together with the calculated discrete values of Horva ´ th

and Sitkei (2002). As their calculations were based on

one measured value only, the difference between the

curve and the discrete values is not crucial.

Hence the reduction of the parameters was carried out

at unchanged energy values, the total shaken mass of

limb, rooting and soil can be regarded as unchanged

independent of the height of shaking. In other words,

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0.000

0.005

0.010

0.015

0 200 400 600 800 1000

Height above ground level, mm

A m p l i t u d e , m

Rotating type

Slider crank type

Fig. 12. Shaken cross-section amplitude of a rotating-type and aslider crank-type tree shaker as a function of trunk height

0

1

2

3

4

5

0 200 400 600 800 1000

Height above ground level, mm

A v e r a g e p o w e r

c o n s u p t i o n ,

k W Slider crank type

Rotating type

Fig. 11. Average power consumption of a slider crank a type treeshaker and a rotating-type one as a function of trunk height

00.10.20.30.40.50.60.70.8

0 100 200 300 400 500 600 700 800 900 1000


S p e c i f i c p o w e r ,

k W / m m Slider crank type

Rotating type

Fig. 13. Specific power needed when shaking with alternating-type and rotating-type tree shaker

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the mass of soil vibrated when shaking at different

heights seems to be the same, only its amplitude

changes.

Replacing the equations of the tree trunk parameters

into the power equations of the two shaker types the

average effective power consumption versus shaking

height curves can be drawn. The diagrams generated bya simple personal computer program are shown in Fig.

11, with unchanged frequency of shaking. It seems clear

that the tendency of the curves is influenced by both the

machine and tree trunk parameters.

Also, the amplitude of the shaken trunk cross-section

can be calculated by replacing the trunk parameters into

Eqn (3). The difference between the working principle

and parameters of the two shaker types results in

different shapes of their diagrams.

The average effective power consumption and trunk

amplitude versus shaking height curves for the two types

of shakers are useful for the machine design. With thefree combination of m, r, o and M , the amplitude X for

the best fruit removal can be set up and the theoretical

power needed to drive the harvester machine can be

calculated along the trunk y. Note that due to the

mechanical and hydraulic losses the real power will be

higher.

To judge the efficiency of the shaker types at different

shaking heights, the specific power (the power needed to

generate 1 mm trunk amplitude) can be calculated for

each trunk cross-section. As can be seen in Fig. 13, when

shaking the tree near soil level the slider crank-type

shaker is more efficient. Clamping the machine higher

on the trunk the situation changes: the rotating-type

shaker seems to get more and more efficient.

Note that the diagrams above were calculated using

real shaker machine parameters. As Eqns (14) and (29)

are equivalent, by changing the values of the parameters

m, M and r the diagram shapes will be influenced,

independent of machine type.

5. Conclusions

Different equations were established for the calcula-

tion of idle power consumption of the rotatingeccentric–weight-type and the slider crank-type fruit

tree shakers, which may be explained by the somewhat

different working principle of the two shaker machine

types.

The simple three-element model for the replacement

of the shaken trunk cross-section of the tree seemed to

be a good model approach.

The equations for the calculation of average effective

power consumption in work were found to be equivalent

for the two shaker types. The different tendencies in

power consumption and amplitudes along the trunk

when shaking similar trees with the two types of shakers

is the result of differences in machine data.

The tree structure model introduced above explains to

the change of location of the virtual turning center in as

a function of shaking height.

Although the function describing the vertical positionof the turning centre is composed only of geometrical

elements, it also explains why the changing shaking

force results in changing the value of the vertical

position of the virtual turning centre. That is because

the change in the force acting on the trunk horizontally

influences the geometrical location of the bending of the

main roots: those bending nearer to the trunk if a larger

force is applied. With the change of the geometrical

location of bending, the virtual turning centre of the tree

changes as well.

The function describing the location of the virtual

turning centre can be used to transfer the measuredmass, spring constant and viscous damping coefficient

values of one trunk cross-section to any other.

Replacing the changing reduced mass, spring constant

and viscous damping values into the power equation of

the shakers, power consumption versus trunk cross-

section curves can be drawn.

Replacing the changing reduced mass, spring constant

and viscous damping values into the equation of the

trunk amplitude, the change of amplitude versus trunk

cross-section curves can be drawn.

Calculating and drawing the specific power values for

each trunk cross-section, the most efficient clamping

height of a given shaker machine can be found.

Acknowledgements

The author wishes to thank La ´ szlo ´ Csorba, associate

professor of Szent Istvan University in Go ¨ do ¨ llo+ for the

valuable advice and discussions. Thanks are also due to

the staff of the Hungarian Institute of Agricultural

Engineering, Go ¨ do ¨ llo+ , and of Kevefrukt Co. for their

cooperation in field experiments.

References

Fridley R B; Adrian P A (1966). Mechanical harvestingequipment for deciduous tree fruits. California AgriculturalExperiment Station. Bulletin, 825, 56

Horva ´ th E; Sitkei G (2001). Energy consumption of selec-ted tree shakers under different operational conditions.Journal of Agricultural Engineering Research, 80(2),191–199

Horva ´ th E; Sitkei G (2002). Damping properties of fruit treesshaking at their trunks. Proceedings of the Symposium on

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Fruit and Nut production Engineering. Bornim, Germanypp. 83–88

La ´ ng Z (2003). A fruit tree stability model for staticand dynamic loading. Biosystem Engineering, 85(4),461–465

La ´ ng Z (2004). Comparison of inertia and positive displace-ment type fruit tree shakers. Proceedings of MTA-AMB

Kutata ´ si e ´ s fejleszte ´ si tana ´ cskoza ´ s. Go ¨ do ¨ llo+, Hungary

La ´ ng Z (2005). Optimal shaking frequencies of inertia typefruit tree shakers. Proceedings of MTA-AMB Kutatasi esfejleszte ´ si tana ´ cskoza ´ s. Go ¨ do ¨ llo+, Hungary

Ludvig G (1973). Ge ´ pek dinamika ´ ja. [Dynamics of machines.].Mu+szaki Kiado ´ , Budapest p. 556

Whitney J D; Smerage G H; Block W A (1990). Dynamicanalysis of a trunk shaker-post system. Transactions of the

ASAE, 33(4), 1066–1068

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Documents

Lang_Dynamic Modelling Structure of a Fruit Tree for Inertial Shaker System Design