A Fruit Tree Stability Model for Static and Dynamic Loading

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Biosystems Engineering (2003) 85 (4), 461–466 doi:10.1016/S1537-5110(03)00092-8 PH}Postharvest Technology

A Fruit Tree Stability Model for Static and Dynamic Loading Zolt!an L!ang Szent Istv!an University, H-1118 Budapest, Vill!anyi u! t 31, Hungary; e-mail of corresponding author: [email protected] (Received 31 October 2002; accepted in revised form 7 May 2003; published online 16 June 2003)

Static and dynamic tests were carried out on the trunk and main roots of cherry trees to obtain their important elastic and viscous properties. The horizontal and vertical displacement of the stem was measured at horizontal forces 80 cm above ground level. Force relaxation tests were carried out in the same way for 15 min. Main root samples were taken from a felled tree for bending, stressing and force relaxation tests. Accelerations of pre-stressed and released trunks were recorded for whole trees and for trunks when the limb was totally removed above 85 cm. From the static and dynamic test results a common model was composed of two Maxwell elements, a damping element and a spring, all coupled in parallel. During static loading, i.e. at force relaxation, the damping of the two Maxwell elements was active. During dynamic loading, when the trunk is swinging at its own frequency, only the single damping element acts together with the three parallel springs. The damping of the Maxwell elements is too high to influence the oscillations at the natural frequency. # 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd

1. Introduction The purpose of the investigation was to obtain more information on the fruit tree–soil connection, to determine root and trunk behaviour during static loading, to examine input energy dissipation during free oscillations caused by an external force, and to establish a static and dynamic model to describe the tree–soil connection for horizontal loading of the tree. Few investigations have been carried out to describe the tree–soil connection, especially during external loading (applied by strong winds or during shaker harvesting of fruit trees). Whitney et al. (1990) modelled the tree trunk as a vertical cantilever. They found that it absorbs very little energy and it acts nearly as a pure spring. Horv!ath and Sitkei (2000) calculated the mass of soil which absorbs the energy loss during shaking. In their model, a large mass of soil vibrates with the trunk during shaker harvesting. This, however, cannot handle the static loading and the visco-elastic connection between soil and trunk. Other investigations were carried out to find the variation of strength along the root length (Stokes & Mattheck, 1996). For each of the three determined root systems, longitudinal compression tests and lateral bending tests of the root material were carried out. It 1537-5110/03/$30.00

is thought that bending and compression at the bottom surface of the roots are the main loading. Along the roots, local maximum values were found for both compression and bending strength. 2. Materials and methods 2.1. Bending and force relaxation tests of the trunk Bending tests of cherry tree trunks were carried out by stressing them horizontally at the usual shaking height (80 cm). The experiments took place in ten years old cherry orchards on sandy soil (plantation spacing of 5 m by 6 m). A light pointer (1) was fixed vertically to the bottom of the trunk (2) in the plane of bending (Fig. 1). An easily recognisable marker was fixed to the pointer (1) and to the trunk (3) at 80 cm. A steady-state digital camera captured the image of the trunk deformation relative to the pointer and the turning of the pointer at different bending forces. The images were taken immediately after the stressing forces were applied. Force relaxation tests of the tree trunks were carried out with the above arrangement. The process was captured over a period of 15 min. Force data were recorded automatically every 0
02 s.

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# 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd

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Notation c F F0 Fbend, Fpull k

spring stiffness, mm N1 remaining force, N initial force at force relaxation, N bending and pulling forces, respectively, N difference between initial and remaining forces at force relaxation, N

2.2. Root tensile and force relaxation tests A whole cherry tree was pulled out, and then 100 mm long samples were taken next to the broken root crosssections. Their diameter without bark varied between 5 and 14 mm. An Instron testing machine was used to carry out tensile tests on the samples at a crosshead speed of 50 mm min1. Special jaws had to be made to grip the

l 0, l 1 and l2 r x, y x0

dimensions, m damping coefficient, N s m1 displacement in the X–Y plane, mm initial deformation, m

roots firmly. Load, displacement, strain and stress at break were recorded, and Young’s modulus was calculated. Force relaxation versus time curves were recorded with similar root samples in the Instron machine. Force data were automatically recorded in every 0.05 s.

2.3. Root bending tests Bending tests of main roots of the pulled out tree were carried out on the spot with a dynamometer acting at 40 cm distance from the centreline of the stem. The diameter of samples where the force was applied varied between 45 and 65 mm. Force and displacement, perpendicular to the axis of the root were recorded.

2.4. Dynamic tests Horizontal acceleration data of free swinging trees at 80 cm above ground level were recorded at a frequency of 4
8 kHz until they diminished to zero. The oscillations were induced by quickly pre-stressing the main branches at a height of 1
8 m with a force of approximately 150 N and then releasing them. Horizontal acceleration of only a free swinging trunk at 80 cm was also recorded (at 85 cm height, all the limbs were removed to leave only the lower part of the trunk). The oscillations were induced by quickly pre-stressing the trunk with a 3 kN horizontal force, and then releasing it. Data were captured with the above frequency.

3. Results 3.1. Bending and force relaxation tests of the trunk Fig. 1. The arrangement of the test: (1) pointer; (2) attachment of pointer at base of trunk; and (3) marker on trunk at height of 80 cm

Typical x and y translation co-ordinates of the stem at 80 cm above the soil surface, at the top and bottom of the pointer during bending are shown in Tables 1 and 2.

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FRUIT TREE STABILITY MODEL

Table 1 Horizontal and vertical displacements x and y for a cherry tree trunk with an average diameter of 13 cm Force,

Displacement, mm

N

Pointer at soil level

2000 3000

Pointer at 80 cm

Trunk at 80 cm

x1

y1

x2

y2

x3

y3

3
1 5
7

1
6 1
6

11
4 19
2

1
5 1
5

15
6 25
0

1
6 1
6

Table 2 Horizontal and vertical displacements of a cherry tree trunk with an average diameter of 18 cm Force,

Displacement, mm

N

Pointer at soil level

3550 6950 11650

Pointer at 80 cm

Trunk at 80 cm

x1

y1

x2

y2

x3

y3

0 4
1 4
1

–4
2 0
5 1
0

2
0 7
1 12
2

–4
1 0
5 1
0

3
1 9
2 15
3

–4
1 0
6 1
0

Table 3 Main data of the root tensile tests Sample

Diameter, mm

Load at break, N

Displacement at break, mm

Strain at break, %

Stress at break, N mm2

Young’s modulus, N mm2

5
0 14 13
5

679 1180 3400

4
0 0
9 2
6

16
00 3
60 18
4

34
6 7
7 23
8

216
3 213
0 228
8

1 2 3

3.4. Force relaxation of roots

Table 4 Main data of the root bending tests

Test 1 2 3

Force, N

Displacement, mm

Spring stiffness, mm N1

Young’s modulus, N mm2

390 715 810

47 83 96

0
120 0
116 0
119

395 408 398

3.2. Root tensile tests Table 3 shows the load, displacement, strain and stress at rupture for tree root samples. It also contains their calculated Young’s modulus.

Figure 2 shows a typical measured force relaxation curve of a root sample. The initial tensile force was 515 N, which decreased after 5 min to 95 N.

3.5. Force relaxation of the tree trunk Figure 3 shows the force relaxation of a 15 cm diameter cherry tree trunk at 80 cm. The initial horizontal tensile force was 3
03 kN, which decreased after 10 min to 2
45 kN. The apparent spring stiffness at the pre-stressing process was 0
02 mm N1.

3.6. Horizontal acceleration of the trunk 3.3. Root bending tests Table 4 shows the acting forces, displacements and the apparent spring stiffness at the root bending tests.

Figure 4 shows the horizontal displacement, twice integrated from the acceleration measured on the trunk of a freely swinging cherry tree at 80 cm.

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Acceleration, m s2 Velocity, × 100 m s-1 Deformation, × 10 mm

600

Force, N

500 400 300 200 100 0

400 200 0 - 200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

- 400 - 600 - 800

0

50

100

200 Time, s

150

250

300

350

Fig. 2. Measured (–––––) and calculated (-------) force relaxation curves of a root sample

-1000 Time, s

Fig. 5. Free swinging of a trunk: ––––– , acceleration; – – – , velocity; ----- deformation

3500 3000

Force, N

2500 2000 1500 1000 500 0 100

0

200

300

400 Time, s

500

600

700

Fig. 6. Translation and rotation of the trunk and two main roots with bending in the XY plane: left, unloaded position; right, bending of the trunk and roots when a force F is acting on the trunk at 80 cm horizontally

Fig. 3. Measured (–––––) and calculated (-------) force relaxation curves for horizontal trunk loading

4. Discussion

Amplitude, mm

5

0 0.0 -5

0.2

0.4

0.6

Time, s

Fig. 4. Trunk amplitude at 80 cm

The main characteristics of the oscillations are: frequency of 8
3 Hz; logarithmic decrement of 1
02; and damping coefficient of 2360 N s m1. Note that the low amplitude is due to the mode of induction. In the case of this test, the whole limb was swinging in the same phase as the trunk. At higher amplitudes it is very difficult to achieve this situation. Horizontal acceleration, speed and displacement of only a freely swinging trunk (with all limbs removed) at 80 cm height is shown in Fig. 5. The main characteristics of this swinging are: frequency of 25 Hz; logarithmic decrement of 0
75; and damping coefficient of 480 N s m1.

The translation of the upper and the bottom end of the pointer fixed to the bottom of trunk defines the rotation angle of the stump. The difference in translation of the marker on the trunk and on the upper end of the pointer gives the relative bending of the trunk. The bending process can be regarded as a superposition of turning of the stump with the main roots around a horizontal axis, the longitudinal strain of those and a pure bending of the trunk relative to the stump with the main roots (Fig. 6). This leads to the conclusion that the deformation of the trunk is composed of at least three parts: the bending of the trunk, the stress, and the bending of the main roots (Fig. 7). One may draw a simple dynamic model and establish the equation for equilibrium: F0 l0 ¼ Fbend l1 þ Fpull l2

ð1Þ

where: F0, Fbend and Fpull are the forces acting on the trunk, the generated bending and pulling forces respectively in N; and l0 , l1 and l2 are dimensions in m.

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Trunk bending

Force acting on the trunk

Table 5 Main parameters for the composite root model for force relaxation

Root bending Root pulling

Axis of rotation

Fig. 7. Forces generated by horizontal loading of the trunk

Element Maxwell 1 Maxwell 2 Maxwell 3 Spring

Difference between the initial and remaining forces (kn), N

Product of spring stiffness and damping (c1nrn), s

Remaining force (x0/c2), N

35 180 200 }

499
95 49
95 4
95 }

} } } 90

Fig. 8. The model of the root for force relaxation, built of spring and damping elements: c11, c12, c13, and c2, spring stiffnesses; r1, r2, and r3 , damping coefficients

Fig. 9. The model of the trunk for static and dynamic loading, built of spring and damping elements: c11, c12 and c2, spring stiffnesses; r1, r2, and r3, damping coefficients

For the evaluation of the force relaxation curve of root samples, visco-elastic behaviour was assumed. First, a simple three-element model (Poynting–Thomson standard) was established, as characterised by x0 ð2Þ F ¼ ket=c1 r þ c2

rooting system in both loading situations. Accordingly, one of the Maxwell elements represents the stressing, the other the bending of the main roots, a single damping element r3 is included for the dynamic loading, which, together with the spring system of the model takes part in the swinging of the reduced system mass (Fig. 9). To get the model parameters, the following considerations were made.

where F is the remaining force in N; x0 is the initial deformation in m; c1 and c2 are spring stiffnesses in m N1; r is the damping coefficient in N s m1; t is time in s; and x0 k ¼ F0  ð3Þ c2 denotes the difference between the initial force and the force remaining after a considerable time period in N. As it did not follow the measured force relaxation curve, the number of model elements was increased to the level as shown in Fig. 8. The parameters were freely chosen, as no data were available for the inner structure of the root, with Table 5 containing a well-matched combination. The calculated and measured curves are shown in Fig. 2. Similar to the example of the root tests, the force relaxation diagram of the tree trunk in Fig. 3 was also regarded a composite model of four parallel elements. Here, however the model has to replace the trunk

(1) It is assumed that the same number and size of main roots are deflected and stressed. (2) In the force relaxation test, r3 does not play any role. Its value is 480 N s m1, taken from the trunk oscillation test. (3) The sum of the three spring stiffnesses in the model is the apparent spring stiffness for the pre-stressing process of the force relaxation test (0.02 mm N1). The sum of the spring stiffnesses c11 and c12 may be divided according the Young’s Modulus for root bending and root tension. (4) The force remaining after an extended time period x0/c2 is equal to 2.45 kN. (5) The sum of the values of k for the two Maxwell elements is given by the force relaxation diagram (580 N) and may be divided between them also according to their Young’s moduli. Equation (3)

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Table 6 Data for the composite trunk – rooting system model

Element

Difference between the Initial and remaining forces (kn), N

Spring stiffness of the Maxwell elements (c1n), mm N1

380 200

8
3  103 1
6  102

Maxwell 1 Maxwell 2 Damping element Spring

Damping of the Maxwell elements (rn), N s m1 2
4  107 1
2  105 480

0
02

shows how the force decreases in the time: F ¼ k1 e

Stiffness of the Single spring (c2), mm N1

t=c11 r1

þ k2 e

t=c12 r2

þ x0 =c2

ð4Þ

Using the values of the parameters in Table 6, the calculated curves are presented in Fig. 3 and compared with the measured values. The model in Fig. 9 explains the different damping for static and dynamic loading. In the static case for the force relaxation test, the damping elements r1, r2 are influential. In the dynamic case, when the trunk oscillates at its own frequency, only the damping element r3 acts with the three parallel springs c11, c12 and c2. The damping of the elements r1 and r2 are so high that they do not influence the oscillations at the frequency measured.

5. Conclusions The result of static bending tests of the tree trunk indicates that: (a) the momentary axis of bending is outside the trunk axis; (b) the trunk both translates and rotates; (c) the tensile and bending strength of the roots are mainly responsible for the stability of a tree; (d) the roots themselves and the rooting with the trunk can be modelled as composite visco-elastic materials; and (e) a momentum equilibrium can be established for the acting and reacting forces and their ratio can be roughly defined. As a result of dynamic tests carried out on whole trees suggests that: (a) at low amplitude, the trunk without limbs above 80 cm in height behaves nearly as a simple single

spring, single mass and single damper model, with damping being speed related; (b) as expected, a 85 cm high trunk itself gave lower a value for damping than a whole tree (both measured at the same height); (c) knowing the damping of a post as a cantilever, the approximate damping of the tree parts are 50 N s m1 for a trunk of 14 cm diameter; 480 N s m1 for the trunk-rooting system; and 2360 N s m1 for the whole tree; and (d) using static and dynamic data, the force relaxation diagram of the trunk can be nearly replaced by two Maxwell models, a damping element and a spring. A four-element model explains the changing damping behaviour of the trees at different loadings.

Acknowledgements Special thanks to the colleagues at the Hungarian Institute of Agricultural Machinery, at the Szent Istv!an University Research Orchard and to Kevefrukt Ltd. for their help in carrying out the above experiments.

References Horv!ath E; Sitkei G (2000): Energy consumption of selected tree shakers under different operational conditions. EurAgEng Paper No. 00-PM-048, AgEng2000 Warwick, UK Stokes A; Mattheck C (1996): Variation of wood strength in tree roots. Journal of Experimental Botany, 47(298), 693–699 Whitney J D; Smerage G H; Block W A (1990): Dynamic analysis of a trunk shaker-post system. Transactions of the ASAE, 33(4),1066–1068