Modelling and Optimization of the Dynamic Behaviour of Sprayer Booms

Copyright © IFAC Intelligent Components for Vehicles, Seville, Spain, 1998

MODELLING AND OPTIMIZATION OF THE DYNAMIC BEHAVIOUR OF SPRA YER BOOMS.

Patrik Kennes (0), Jan Antbonis, Herman Ramon

K. U.Leuven, Faculty ofAgricultural and Applied Biological Sciences Department ofAgro-Engineering and -Economics Kardinaal Mercierlaan 92, B-3001 Heverlee, BELGIUM (*)Tel: +32-16-32.14. 78;fax: +32-16-32.19.94 email: [email protected]

Abstract: Finite element simulations show that existing suspensions approximately halve vertical sprayer boom movements compared to a fixed boom. The reduction of horizontal boom vibrations however is negligible. This shows the need for a performant horizontal suspension. Because of the high mass and inertia of sprayer booms active suspensions would require too much engine power. Therefore the possibilities of a passive suspension are investigated. Optimal suspension settings (spring stiffness, damping constant) are determined by optimization of the singular value plot derived from a parameterized linear multi-body model of the boom and suspension. Validation experiments confirm that the used modelling techniques are reliable. Copyright © 1998 IFAC Keywords: passive suspension, modelling, optimization, fmite elements, dynamics, validation

1. INTRODUCTION

they reduce the unwanted boom motions and thus increase the homogeneity of the spray pattern, just as the suspension of a vehicle influences its driving comfort and road-holding. Finite element models for different existing (mainly vertical) boom suspensions reveal their effectiveness for reducing vertical boom movements, while they are ineffective for the horizontal boom movements. Horizontal movements are much more difficult to reduce by a passive suspension than vertical vibrations because there is no reference level (as the soil is for vertical movements), and there is no gravity force that defmes the static horizontal equilibrium position of the boom. Just as for a vehicle suspension, a sprayer boom suspension should pass the low frequent tractor motions (e.g. turning at the headland of the field), while high-frequent tractor movements, which cause boom deformations, have to be filtered. Active suspensions would require much engine power

Chemical products for crop protection are usually distributed as liquids by a field sprayer. Unwanted movements of the sprayer boom, effected by soil unevenness, create local under- and overdoses of spray liquid due to a varying velocity and position of the nozzles with respect to the ground. Vertical vibrations give rise to variations in spray deposit between 0 % and 1000 % and horizontal vibrations between 20 % and 600 %, while 100 % is ideal (Langenakens et aI, 1995). To obtain the desired response on the total field, the advised dose is in excess of that required to make sure that there is enough product in each area. However, environmental and economical reasons demand that farmers use less chemicals which must be better distributed to maintain their effectiveness. This requires more performant boom suspensions because

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because of the large weight of the spray er boom. Therefore the possibilities of a passive boom suspension are investigated.

2. MODELLING BOOM VIBRATIONS

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A non-linear Finite Element Modelling software package (Samcef®) is used to simulate boom movements of a sprayer during field operation. This requires knowledge of the soil profile under the wheels and data of the geometric and physical properties of the tractor and the sprayer (stiffness of the tires, friction in hinges, .. . ). In the first stage, the sprayer and tractor body are represented as a rigid bodies with corresponding mass and inertia characteristics. Numerical values of tractor mass and inertia, the tire stiffness and damping are based on data given in the article of Crolla et al (\990) . Contact between the wheels and the ground is modelled by flexible wheel elements. The tires can have radial deformation, while the ground is assumed to be rigid. The front axle is connected to the tractor body by a hinge that allows it to pendulate. Movements at the hitchpoint of the boom suspension are calculated for a tractor driving across a standardized bumpy road (according to the ISO norm 5008) at a speed of 5 kmlh . In a next stage, these disturbances are applied as input excitations for the sprayer models to simulate boom movements for several types of existing vertical suspensions (pendulum, single and double trapezium). The modelled sprayer boom has a working width of 24 meters and consists of five parts, which allow the boom to be fold for transportation (figure I).

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Figures 2 and 3 show the vertical movements of the boom tip for a 24 meters width spray boom respectively fixed to the sprayer frame and with a pendulum suspension. It is clear that the suspension filters out high frequency boom movements, so boom deformation and stress in the beams are reduced. The low-frequent sinusoidal signal that can be found in the boom tip motion is due to the rigid body rolling motion of the boom with respect to the pivot point of the pendulum. The amplitude of the vertical motion of the boom tip with suspension is almost halve of the motion of the boom tip without suspension. The reduction of horizontal boom vibrations by existing suspensions however is negligible (figures 4 and 5), while very small horizontal vibrations with an amplitude of 30 cm can cause large fluctuations in spray distribution (Ramon, 1993). From this it can be concluded that the development of a performant horizontal suspension is necessary. Because of the high mass and inertia of sprayer booms (up to 36 meters wide) active suspensions would require too much engine power. Therefore the possibilities of a passive boom suspension are investigated.

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3. OPTIMIZATION OF THE BOOM SUSPENSION

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3.1 . Description of the experimental suspension

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An experimental improvement of the performance of the boom suspension would involve costly and timeconsuming prototyping and measurement efforts. Therefore a parametric linear multi-body model of a boom with a passive horizontal suspension is built. This implies that the important characteristics of the boom and its suspension, such as the mass and moments of inertia of the boom and the properties of applied springs and dampers are represented by parameters in the equations of motion, so that their value can be easily adapted. The suspension has two degrees of freedom, one translational and one rotational, to isolate the boom from respectively jolting and yawing tractor vibrations. Practically the translation of the boom is realized by mounting it on a sledge that can slide on two linear bearings. On this sledge there is a vertical axle around which the boom can rotate. The modelled suspension is worked out on a test rig as showed schematically in figure 6. The springs are necessary to defme the static equilibrium position of the boom because there is no returning force as gravity for a pendulum suspension. Just as for a vehicle suspension, a sprayer boom suspension should pass the low frequent tractor motions (e.g. turning at the headland of the field), while high frequent tractor movements, which cause boom deformations, have to be filtered. An optimization routine points out which are the optimal numerical values of the suspension parameters to reach this aim as good as possible.

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Fig. 6. Scheme of the spray boom with passive horizontal suspension. The equations of motion with force input (eq. 1) can be transformed to position input according to the method described by Anthonis (1998). This results in MU

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The advantage to have position-input is clear because the hydraulic actuators of the test rig to reproduce horizontal tractor movements are position controlled. The delivered force contrarely is unknown, unless a force cell is mounted.

3.3. Optimization procedure

The advantage to have linear equations of motion (eq. 2) is that they can be written in state space form as:

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The vector up represents the input translation and rotation of the tractor, while the output vector y consists of the boom translation and rotation (absolute with respect to the ground). From this, a singular value (SV) plot of the system is calculated. Besides this, it is also possible to defme a desired SV plot. For low frequencies, the boom should follow the tractor motions, which requires SV's of 1. This happens for example when the tractor turns at the headland or accelerates slowly. For high frequencies contrarily the boom should be isolated from the tractor to reduce boom deformation and fatigue. This corresponds with low SV's. Based upon field measurements (CRAFT-project, 1997) a breaking frequency can be defmed beyond which tractor vibrations must be filtered. Another important thing to be considered is the resonance peaks which must be avoided because they induce large boom movements. The difference between the desired and modelled SV's is the goal function that has to be minimized by adapting the parameters k), kz, c) and Cz (figure 7). The used optimization algorithm is a quasi-Newton method with the simplex search method of Nelder and Mead. The initial SV's and the SV's after optimization are plotted in figure 7, from

3.2. Linear multi-body models

Based on the modelling technique of Ramon (1993), the linearized equations of motion of a rigid spray boom with the passive horizontal suspension as described earlier are determined: M.q +C.q+K.q = W.w (I) with: M = mass matrix C = damping matrix K = stiffness matrix W = input distribution matrix q = vector of generalized co-ordinates w = input forces The mass matrix M is function of the mass and moment of inertia of the boom for a rotation about the pivot point and of the mass of the sledge. The matrices C and K depend on the values (k), k 2) and (Cl> cz) respectively for translational and rotational spring stiffness and damping.

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speed of 5 km/h. The suspension filters high frequency translation and yawing, while the global tractor movement is followed.

which it can be seen that the proposed dynamic characteristics of the suspension are reached. MeroplirTisation: "-9.3 m4z27.S 1<4:330 1<1...,0 k2a17.2 01_ c:2-1035 Before opirrisation: "-9.3 m4=27.S 1<4:330 kl =2000 1<2=1155 01=33 c2=35 Cl)

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3.4 Validation of the linear multi-body models

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The multi-body models are validated by experiments on a hydraulic test rig that can reproduce jolting and yawing tractor motions. Hydraulic actuators impose a swept sine translation excitation from 0.1 to 5 Hz. Meanwhile, movements of the centre of mass of the boom with the passive (not yet optimized) suspension are registered. From this measurements the frequency response function (FRF) is calculated. The simulated SV plot and the measured FRF can be compared in this way (figure 8). The slight difference just before resonance is due to stick-slip effects of the linear bearings, which is not included in the linear model.

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4. PRACTICAL IMPLEMENTATION OF THE OPTIMIZATION RESULTS The optimization procedure described above determines the optimal stiffness and damping constant for the used springs and dampers. In reality a normal spring behaves like in the model (F=k.q). A practical problem occurs when we have to choose the right damper, because most (oil)dampers have a nonnegligible friction and don't correspond with the Friction and modelled behaviour F = c.q corresponding stick-slip effects cause deformations of the boom at its eigenfrequencies and should be avoided as much as possible. Therefore a low friction adjustable air damper is developed. The working principle is like an air cylinder. Linear bearings at both ends supply a low friction motion of the damper. Changing the orifice area for the incoming/outgoing air (figure I 1) alters the damping constant. The same test procedure as for the model validation is executed, but now with the air damper mounted on the suspension. FRF's for translation of the boom with mounted air damper are given in figure 12 for two different orifice areas. The resonance peak disappeared when the correct orifice opening is applied (critically damped system).

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3.5 Results

After validation, the models are used to simulate the effect of adaptations to the suspension set-up. Figures 9 and 10 show respectively the simulated translations and rotations of the boom centre for a 12-m spray boom with an optimized passive horizontal suspension. The input signal is calculated from the non-linear fmite element model of a tractor with mounted sprayer that is driven across a field at a

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ACKNOWLEDGEMENT

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This paper was elaborated within the IWONL Project D 1I4-6115/5702A.

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REFERENCES Fig. 11. Scheme of the air damper

Anthonis, 1., Ramon, H. (1998) Generalized procedure to derive the linearized equations of motion of position controlled mechanical sy stems. Paper in preparation CRAFT project F A-S2-9009 (1997), Progression Report. Crolla, D . A. , Horton, D. N. , Stayner, R . M. (1990) . Effect of tire modelling on tractor ride vibration predictions. Journal of Agricultural Engineering Research, 47,55-77 . Langenakens, J., Ramon, H., De Baerdemaeker 1. (1995) . A model for measuring the effect of tire pressure and driving speed on the horizontal sprayer boom movements and spray patterns.

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Fig. 12. FRF's with air damper

5. CONCLUSIONS

Transactions of the ASAE, 65-72. Nelder, 1. A., Mead, R. A simplex method for function minimization. Computer Journal, 7, 308-313. Norme Internationale, ISO 5008 (1979) (F). Organisation Internationale de Normalisation. Ramon, H. (1993). A design procedure for modern

Finite element models are a useful method to simulate the dynamic behaviour of sprayers. Simulations show that a pendulum or trapezoidal suspension of the boom decrease the rolling motion significantly. The sprayer models can also be used to simulate the effect of structural adaptations on the sprayer, without the need for time-consuming experiments . A lot of work and material costs can be saved in this way.

control algorithms on agricultural machinery applied to active vibration control of a spray boom, Ph.D. thesis NR. 231 , Faculteit der Landbouwwetenschappen, K .U .Leuven.

The singular value decomposition of the state space representation of the linearized equations of motion for a boom with suspension is a valuable tool for design optimization. Simulations show that an optimized passive horizontal suspension of the boom . acts as a low pass filter, decreasing the yawing and jolting motions significantly for the high frequencies, while the low frequencies are passed. This enables the boom to follow field undulations and turning of the tractor at the headland, while boom vibrations and deformations are strongly reduced . First practical tests with a low friction damper prototype are promising for building a prototype suspension.

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