)1) Prof. Emrit., Ag. Eng., Ain-Shams U.

(2) Lec., Structural Eng. Dept., Fac. Eng., Cario U.

(3) Lec., Ag. Eng., Ain-Shams U.

Finite Element Analysis of Tree Shaking during

Harvesting El Awady, M.N.(1), Tagel-Din, H. (2), El Attar, M.Z.(3)

ABSTRACT

Simulation of tree shaking harvesting process is carried out.

Steps were taken to build an electronic model, which leads to the

development of principles governing the behavior of vibrating tree

harvesting systems. Finite Element Analysis software tool was used

to represent, analyze, and simulate the olive-tree model structure.

Model data entry represent an approximation of olive-tree physical

system and its responses to harvesting actions by means of tree

shaker. Simulation results agreed with results of field studies..

Analysis has indicated that it is impossible to determine a unique

optimum applied frequency or even to choose a very narrow

frequency range that is particularly effective in tree harvesting.

Simulation results also emphasize that tree clamping point at 40 cm

above ground. In addition, tree must be trained and pruned in vase-

shape to increase the harvesting efficiency and to avoid developing

fruit near the ground. Tree harvesting numerical model is capable of

simulating different tree harvesting actions for different tree types

and analysis of the results establish roles of designing and

constructing tree harvester mechanisms.

1. INTRODUCTION

Mechanical harvesting is employed when the overall cost of

doing so is less than for manual harvesting of a product. Fridley et

2

al.(1991) summarized factors which tend to increase the difficulty of

achieving good fruit removal in: (1)Small fruit,(2)Long limber fruit

stems, (3)Long limber fruit-bearing branches, (4)Poor fruit

abscission at harvest time, (5)Brittle trees that cannot withstand the

required vibration without excessive damage.

Mayo (1997) found that the biggest problem for mechanical

harvesting is the structure of the harvested trees. The majority of

trees are: (1)multitrunked, having three or four trunks per tree which

make tree shaking almost impossible, and because of the shape and

size of the limbs and the terrain of the orchards. Mayo added that

commercial use of harvest machinery also slowed because of

several technological hurdles. He concluded that Mechanical

harvesting technology may be adopted by one production area, but

not by another because of the unique differences in factors such as

(2) climate, (3)soil, (4)terrain, (5)labor, (6)crop mixture, (7)market,

(8)utilization, (9)variety, and (10)tree or plant type.

Lack of uniform maturity and differences in criteria for readiness

of harvest between different horticultural crops, and even between

species and varieties, made it very complicated to substitute

machines for human judgment and dexterity. Selecting only mature

product for harvested fruits and vegetables is still very difficult to

achieve with machines.

Tsatsarlis (1987) mentioned that the high cost (50 to 60% of the

total production cost) using manual harvesting of olive-tree makes

mechanical harvesting desirable. He also discussed several factors

that contribute to the difficulty of removing olive, and stated that:

3

The olive is small fruit attached strongly to the limb. As a result, the

ratio of removal force to weight (f/w) is high.

Fruit is generally born on long willowy branches which hang

downward isolating fruit from applied vibration forces.

The tree structure is also willowy. Many trees are old and have

brittle branches weakened by disease or grafts.

A further difficulty is that many trees have been cultivated on hilly

or mountainous areas making the mechanization difficult.

2. MATERIALS AND METHODS

In investigation on wood mechanical properties, by the

resonance vibration method, Vobolis et al. (2002) described wood

as polymeric materials and put them in the category of viscoelastic

materials. So wood mechanical properties are defined in terms of

elastic solids and of viscous liquids. Dynamic vibrating system

(fig.1), may be expressed as mentioned by Sharkawy and Awady

(1970): tFzckzzm sin.

,

Fig. 1: Scheme of the investigated specimen.

where "m" is equivalent concentrated mass, "F" the applied

shaking force, "c", is the coefficients of damping, and k, mean spring

resistance of the specimen.

Tree limb or branch can be represented by a mechanical

element with an infinite number of degrees of freedom. In this case,

K C

m

K C

m

4

the mass of the bar is also distributed along its length, and its

transverse oscillations equation can be written as reviewed by

Vobolis et al.,(2003): 02

2

4

4

tEI

zS

x

z (This equation is not

general equation) tfuKuCuM where is the

density of wood, S is the cross-sectional area of the bar, E is the

modulus of elasticity, I is the moment of inertia of the cross section,

and x, z are oscillation coordinates.

By solving this equation, an infinite sequence of frequencies is

obtained. The basic frequency of free vibration is calculated as:

352.3

mL

EIf where E is the modulus of elasticity, I is the moment

of inertia of the cross-section, f is frequency of the mass free

oscillations, L is length of the element, m is mass of the element.

The natural frequency is determined according to the

relationship: m

kfn reviewed by El Attar et al., (2004). where

"k" is the appropriate elasticity value (spring rate) "m" is the

appropriate mass value (fig.2).

Fig. 2: The model of olive-fruit vibratory system.

SIMULATING TREE SHAKER HARVESTER:

SAP2000, an "Integrated Finite Element Analysis and Design of

Structures" computer software tool, was used to represent, analyze,

and simulate the olive-tree model structure as illustrated in fig(3).

Olive fruit

Spur

5

Fig. 3: Finite-element computer program block diagram.

Data entry:

Data were entered according to simulation program procedure

as: wood material, physical properties, geometric properties and

section stiffness. Basic geometric properties are used, together with

the material properties, to generate the stiffness of the section.

PRE-PROCESSOR

INPUT DATA

Control data, Materials, Node and Element Definition, Boundary Conditions, Loads

FORM ELEMENT [K]

Read Element Data, Calculate Element

Stiffness Matrix, [K]

FORM SYSTEM [K]

Assemble Element [K]s to Form the Overall

Stiffness Matrix, [K]

APPLY DISPLACEMENT BOUNDARY

CONDITIONS (READ MY NOTE)

COMPUTE DISPLACEMENTS

Solve the System Equations [K]{D}={F} for the displacements {D}=[K]-1{F}

COMPUTE STRESSES

Calculate Stresses and Output Files for Postprocessor Plotting

POST-PROCESSOR

Element File

Load File

Element File

Load File

Displacement

Stress File

Solid circular section shear

2r 9.0 areashearEffective

6

Elements of the cross-sectional area, and the moment of inertia are

shown in fig.(4).

Fig. 4: Corresponding bending stiffness of the Section.

Formulae for calculating the shear areas of typical sections are

given in: (1)Tree frame-elements, representing trunk, branches,

leaves, and fruit, (2)Joints that represent connections of tree

elements, (3)Restraints and springs that support the joints, and

(4)Loads, including self-weight (wood, stem, leave, and fruit mass),

vibration force, and others. After analyzing tree skeleton, the model

also includes displacements, stresses, and reactions due to the

loads.

Model description: (Fundamental Assumptions)

Data were recorded over a short period of time "1sec", so that the

properties of the limb were assumed constant throughout the test

period.

Tree structure was evaluated for steady-state forced vibration.

Tree limb elements were considered to be truncated conic

segments, with length and radius of curvature very large compared

to their diameter.

The tree was simulated by a number of elements that formulate a

trunk with three branches for simplicity, each supporting secondary

branches which support fruits in return.

Un-damped mode shapes and natural frequencies were found

using the "Eigenvector" analysis procedure, considering

proportional damping

Rotary inertia and shear effects are neglected.

7

External damping was neglected (This part will bring a lot of

criticism to your work. Are you sure we do not have any damping?

I guess we neglected the external damping for eigen value analysis

only not for external excitation model).

Deflection and loads occurred in a three dimensional space.

Olive-tree model considered Z direction as the vertical axis, with

+Z being upward. Local coordinate systems for joints, elements, and

ground-acceleration loading are defined with respect to this upward

direction. Self-weight loading (wood, leaves, and fruits) always acts

downward.

Definitions: Definitions are set to create named entities that are not

part of the geometry of the model. These entities include:

Material properties: Material properties of olive-tree wood which

are mainly: (a) mass per unit volume, (b) mass per unit volume, and

(c) modulus of elasticity.

Mass Density, Mass density, w, is used for calculating the self-

weight "Ws" of the element. The total weight of the element is

VWWs , Where "V" is the volume of the element.

This weight is apportioned to each joint of the element. Self-

weight is activated using Self-weight Load and Gravity Load.

Tree frame properties: Data were added by defining the

dimensions and or properties manually. Elements could be arranged

at any convenient angle to create curved branches; fig.(5).

8

Fig. 5: Frame section "truncated cone" definition.

(SAP2000 Software)

Model analysis:

Completing structural model, and operations above were used

to determine the resulting displacements, stresses, and reactions.

However, before analyzing, options must be set up for the tree

model including the following:

Degrees of freedom:

Model was assigned to 6 degrees of freedom. Translations are

denoted U1, U2, and U3. The joint may also rotate about its two local

axes (R1, R2 and R3); fig.(6) The rotation around Z was restrained

to preserve overall system equations stability.

Olive T

run

k Frame Element

Element length

Ele

men

t ba

se

Element tip

Joint load components Global origin

Z

X Y

uz

ux uy

9

Fig. 6: The six degrees of freedom in the joint local

coordinate system.

Model Damping:

Proportional modal damping is assumed with respect to the total

stiffness matrix, K, which includes the effective stiffness from the

nonlinear elements

One-dimensional spring system:

Fig.(7) illustrates the fundamentals of finite element method

analysis in one-dimensional spring system by Sap2000 manual

(Better refer to a FEM book not SAP. This is general idea about

FEM). When applying F force at node (3). To find the node

displacement and spring forces, a general element must be

formulated. Element p with nodes i and j, assumes positive

displacement components of ui, at node i and uj at node j. the

element spring constant k, and forces at node result, when these

displacements occur.

When force fip acts on node i due to the node displacements of

element p it could be illustrated at equilibrium form

10

Fig. 7: One-dimensional spring (a)structure, (b) element.

jpippj

jpipip

ukukf

ukukf

And it could be solved according to matrix form as

pj

ip

j

i

pp

pp

f

f

u

u

kk

kk , and fdk

where [k] is the element stiffness matrix, {d} is the element node

displacement vector, and {f} is the element node internal force vector

332

22221

111

0 3 nodeat

0 2 nodeat

0 1 nodeat

Ffforces

Fffforces

Ffforces

Eigenvector analysis used by simulation software, determines

the undamped free-vibration mode shapes and frequencies of the

system. These natural modes provide an excellent insight into the

behavior of structures. They can also be used as the basis for

response-spectrum analyses; eigenvector analysis involves the

solution of the generalized eigenvalue problem: 0 MK

where K is the stiffness matrix, M is the diagonal mass matrix; is

the diagonal matrix of eigenvalues, is the matrix of eigenvectors s

mode shapes.

The modes are identified by numbers from 1 to 40 in the order

in which the modes are found by the program. The Eigenvalue is the

square of the circular frequency, for that Mode. The cyclic

(a) (b)

11

020406080

0-100

100-200

200-300

Heig

hts

ab

ov

e g

rou

nd

"cm

"

Leaves mass distribution % Fruit mass %

frequency, f, and period, T, of the Mode are related to by:

2

1 fand

fT

4-RESULTS AND DISCUSSION Data of olive-tree geometry, wood physical properties, masses

of different tree parts, and load distribution are represented in fig.(8).

SAP2000 analysis results (fig.9), agreed with results of field studies

(fig.10), reviewed by El Attar et al. (2004). Analysis has indicated

that: (1)it is impossible to determine a unique optimum applied

frequency or even to choose a very narrow frequency range that is

particularly effective in tree harvesting, (2)simulation results, also

emphasize the tree clamping point to be at 40 cm above ground as

indicated from fig.(11), (3)tree must be trained and pruned in vase-

shape to increase the harvesting efficiency and to avoid developed

fruit near the ground. Tree harvesting electronic model and

simulation is capable of similar different tree harvesting actions for

different tree types and analysis of the results establish roles of

designing and constructing tree harvester mechanisms.

Elevation

12

Fig.(8):Olive-tree geometry output by SAP2000 finite element

analysis tool.

Fig. (9): Actual branch displacement compared with

simulation output at shaking frequency of 6.9Hz.

-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1

Bra

nc

h d

isp

lac

em

en

t "c

m"

Shaking time "sec"

Shaking frequency " 6.9 Hz"

0-100 100-200

200-300 Shaking frequency

Heights above ground "cm"

Top view 3-dim. view

13

Fig. (10): Actual branch displacement compared with

simulation output at shaking frequency of 6.9Hz

Fig.(11): Olive-tree branch displacement response to shaking

point at 40 cm above ground.

4.85.7

11

5.74.8 4.43

11.6

5.97

0

2

4

6

8

10

12

14

Inputdisplacement

0-100 100-200 200-300

Bra

nch

dis

pla

cem

ent "c

m"

Heights above ground "cm"

Shaking frequency " 6.9 Hz"

Actual displacement Simulation output

0

5

10

15

20

25

30

35

5 6.9 8.125 8.83 10.2 7.36 10.44 16.15 22.02 7.23 14.7 20.17

Dis

pla

ce

me

nt

"c

m".

Shaking frequency "Hz".

0-100 100-200 200-300

Shaking displacement5cm

Shaking displacement10cm

Shaking displacement 15cm

Heights above ground "cm"

14

5- CONCLUSIONS The developed simulation model of tree harvesting based on

finite element analysis agreed with the field tests. Simulation model

analysis showed that tree response varied with different shaking

actions according to fruit location on the olive-tree due to differences

in physical and mechanical properties. Concluded

recommendations of simulating harvesting cases are:

1. Tree must be trained and pruned in vase-shape to limit the

branches in the bottom portion of the olive-tree giving inefficient

properties for fruit removal by shaking.

2. Attaching the clamping device of the tree shaker harvester at

height 40cm above the ground gives good results of shaking

olive-tree branches giving high fruit removal.

3. Using simulation model for other conditions of tree harvesting is

foreseen by applying changes to values of physical properties,

tree skeleton, stiffness, and mass distribution.

This study iterates demand for further studies or simulating

the changes in the physical properties for different tree

developing stages by environmental, climatic, and biological

factors, etc...

REFERENCES

El Attar, M.Z.; El Awady, M.N.; Rashwan, M.; M.A.I. Genaidy., 2004, Physical properties effects on shaker-model harvesting of olive-trees, 10th MSAE conf.:

Fridley, R.B.; Adrian, P.A.; Claypool, L.L.; Rizzi, A.D.; and Leonard, S.J., 1991, Mechanical harvesting of cling peaches. Calif. Ag. Exp. Sta., Bull: 851.

15

Mayo, D.,1997, Mechanical harvesting, Adapted from Australian Olive Grower, Olive Agencies, 16 McGarva Rd, Grantham Q4347, Australia. (http://www.oliveaustralia.aust.com)

SAP2000 manual,2000 , SAP2000 an Integrated Finite Element Analysis and Design of Structures,

Sharkawy, M.K. ; and Awady, M.N. , 1970, Theory of machine, Ain-Shams Press: 321-324

Tsatsarelis; C.A., 1987, Vibratory olive harvesting: The response of the fruit-stem system to fruit removing action, J. Agric. Eng., 83:77-90.

Vobolis, J.; and Aleksiejunas, M., 2003, Investigation of wood mechanical proparties by the resonance vibration method, ISSN 1392-1320 Materials Scie., 9(1):139-142.

16

محاكاة هز األشجار بواسطة نموذج الكترونى (3)، العطار، م.ز.(2)، تاج الدين ، ح.(1)العوضى، م.ن.

البحث صلخستم

جرى محاكاة حصاد األشجار بالهز. لذا تم تكوين نموذج الكترونى يعتمد على

( لتحليل العناصر المحدودة فى تقييم المنشآت. وتمثل SAP2000نهج )برنامج

المعطيات لهذا النهج تقريبا للخصائص الطبيعية لشجرة الزيتون، واستجابتها

نتائج المحاكاة والدراسات الحقلية. وأظهرت هذه النتائج ما اتفقتلالهتزاز. وقد

ارتفاع( أمثل 2) ( عدم إمكان تحديد تردد أمثل بعينة للحصاد االهتزازى ،1يلى: )

لنقطة الربط على األشجار وتقليمها على شكل "كأس مفرغ" لزيادة كفاءة الحصاد

أن المحاكى وأتضحالمنخفض على الشجرة قرب األرض. راألثمابتفادى

االهتزاز ألنواع األشجار، وتحليل النتائج أوجهااللكترونى قادر على تمثيل مختلف

الهتزازية.لتصميم وتطوير آليات الحصاد ا

( أ. متفرغ، هندسة زراعية، ك.الزراعة، ج. عين شمس، 1) ( مدرس، هندسة مدنية، ك. الهندسة، ج. القاهرة،2) ( مدرس، هندسة زراعية، ك. الزراعة، ج. عين شمس.3)

17

General comments: 1- I think the fonts are very big and also margins. Try to specify

the Journal name and reshape the paper in their format 2- Make all sentences in present form 3- Do not say a name of a commercial program like SAP2000.

Please replace it with “Structural Analysis software”. The program SAP2000 is the least among tens of advanced structural analysis software. If you like to say it, just once is sufficient inside the context with reference to it as you did

4- Note that you draw a picture of mass-spring model. I am afraid the reader think this is what we did. We are using multi-degree of freedom lumped mass system. It is better also to write the general dynamic equation we use for analysis

5- Figure 3 is very important. Can we illustrate it in more details inside the text.

6- This equation tfuKuCuM is the one

we used for our simulation. So, it is better to state it clearly and define its terms. Most of equations you are stating are for Eigen Value analysis to obtain the tree frequencies which is correct. But for solving the tree for a specific input displacement history, we used this equation

7- We should state more words about the way you applied the load by exciting the base of the tree. It is a great part of your work

8- Please state more details about the FEM analysis we did for example:

a. Number of frame elements b. Number of joints c. Number of degrees of freedom (6 * number of joints) d. Number of cross sections

9- When you talk about One Dimensional spring-mass model, it will confuse the reader about what we did. We did 3-D FEM model using frame elements not 1-D models using series of mass-springs

10- In brief, re arrange the paper to be 1- review about what people did, review about FEM model and eigen value calculations, talk about our model in SAP with more details about joints, elements, etc. We have two models with SAP, one for Eigen value analysis and one for external excitation, right? Make that very clear

18

11- Finally, reduce the fonts to be 10 Arial for text and 12 for titles and reduce the margins to be 2.5 cm from all directions