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The design and performance of a gimbal-type mounting for sprayer booms 2. Optimization model and validation
J. agric. Engng Res. (1987) 36, 247-260
The Design and Performance of a Gimbal-type Mounting for Sprayer Booms 2. Optimization
Model and Validation H. J. NATION*
An empirical approach had already indicated that the gimbal-type mounting is effective in reducing movement of a sprayer boom. This mounting involves springs and viscous dampers and it was concluded that the effects of these would more readily be optimized by calculation than by further experiments. An analysis was made of the dynamics of the gimbal mounting and a computer model developed by means of which the value of viscous damping could be optimized for a spring stiffness by searching for the minimum value of an output function integrated over a range of components of an input power spectrum. Power spectra for roll and yaw were calculated from data recorded at three speeds over the standard bumpy test track. The polar moment of inertia of the boom was determined by bifilar suspension. Optimum values of damping were computed for ranges of natural frequencies. The experimental gimbal mounting was rebuilt to facilitate tests at ranges of polar spring stiffness and polar damping coefficient. The conditions giving minima in the response obtained were found to agree well with the values predicted by the computer model.
1. Introduction Previous work examining the dynamic behaviour of field sprayer booms indicated the need for a flexible form of mounting between the boom and sprayer frame to reduce errant boom movements.’ An examination of the effectiveness of different types of boom mounting
led to the development of the gimbal-type of mounting with mutually-perpendicular pivots through the centre of gravity of the boom.’ This development had evolved from the pendulum type of suspension and theory showed that for this to be effective the polar radius of gyration should be large in comparison with the length of the pendulum. Experiment showed a similar benefit of a large polar moment of inertia with the gimbal mounting. With the horizontal pivot through the centre of gravity, no restoring force is provided by the weight of the boom, and springs were adopted to restrain the boom in a rest position parallel to the machine axle. Natural oscillation of such a system when suitably excited by sprayer rolling motion could be controlled by viscous dampers. However, determination of optimum values of viscous damping by empirical means would be a laborious way to obtain optimum performance. A mathematical model was indicated as the most appropriate approach. 2. Method 2.1. Experimental procedures The spectral densities to be examined by the computer program were obtained using the same sprayer as in the earlier experiments. The boom was removed and replaced by a weight * National Institute of Agricultural Engineering, Wrest Park, Silsoe, Bedford MK45 4HS, UK Received
8 March
1985; accepted
in revised form 6 May 1986
247 OO? I-Xh34’87
040247 + 14 %03.00,‘0
‘.r: 1987 The British Souety
for Research in Agricultural
Engineering
248
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2
Notation a
length of bifilar suspension wires polar damping coefficient L polar spring stiffness radius to attachment point r’ moment of inertia K radius of gyration M mass of boom T periodic time Y angle of displacement of boom in motion
l-
Fourier representation of time history of boom movement 4 angle of displacement of sprayer frame in motion CD Fourier representation of time history of sprayer movement W angular frequency resonant frequency WR
of equivalent mass but with relatively low polar inertia clamped to the frame at the attachment point. At the same place an instrumented gyroscope was fastened to measure roll and yaw.’ Several runs were made over the standard bumpy test track4 at speeds of approximately 9, 12 and 16 km/h. The recorded data were digitized and the spectral densities calculated.’ These were taken in turn as input to the optimization program for a range of natural frequencies. To obtain the appropriate values of damping from the results it was necessary to determine the polar moment of inertia of the boom. This was done experimentally using bifilar suspension. The moment of inertia, I = MK’; and K can be obtained from p-
T2r2g 4n2a ’
These observations were first made for the boom as it operated in yaw, and then repeated with the additional parts which move with the boom in roll. The gimbal mounting included slides for the fixtures to which the springs and dampers were attached to facilitate choice of settings for each (Fig. 1). Load versus extension relationships for the springs were obtained experimentally in a tension testing machine and the individual linear constants converted to polar stiffness for each pair. The characteristics of the viscous dampers were examined in a rig in which they could be compressed or extended under constant load, their displacement being measured by linear transducers connected to a chart recorder, from the output of which velocity was calculated. In the experiments with the boom different effective damping values were obtained by use of different radius settings.
2.2. Theoretical analysis In normal use on agricultural surfaces a sprayer frame is subjected to a complex combination of translational and rotational disturbances. From the aspect of dynamic boom behaviour the most important of these are the rolling and yawing rotational disturbances. These belong to a certain class of non-recurring disturbance, the “stationary random” type,
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spray
Rolling beam’
Fig. I. Revised arrangement of gimbal mounting
for which conventional frequency analyses are not appropriate. For these disturbances, where average characteristics do not change with time, a frequency analysis is possible by means of the so-called “power spectral density function”. From a time history of, for example, the rolling motion of the sprayer, consisting of individual measurements of angular displacement at equal intervals of time, it is possible to construct a Fourier transform, consisting of terms representing the individual harmonic components at increasing frequencies. From the Fourier transform it is possible to calculate the power spectral density which may be looked upon as providing a measure of the disturbance energy per unit time and this has come to be regarded as a measure of power. This is particularly useful when dealing with linear systems where a simple relationship exists between the spectrum of an exciting disturbance and the spectrum of the system response. The generalized form of the relationship is:
where Q,(o) = the output spectrum, Qi(o) = the input spectrum, and T(w) = the frequency response function of the system. An analysis of the response to input angular disturbances of a boom subject to linear spring rate restoring torques and viscous damping in rotation about its centre of gravity is given in the Appendix. An expression is derived representative of the power spectrum of the output motion in response to an input power spectrum of rotation of the frame to which the
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restoring springs and dampers are attached. By means of an iterative computer program values of the ratio of damping coefficient to polar moment of inertia, c/I, are searched to minimize the integral
where S(o) is the power spectrum of the input motion. In this derivation there is no reference to the effect of gravity as the boom is in a state of neutral equilibrium and the analysis is therefore equally applicable to the behaviour of the boom in horizontal yaw. So, for both planes of motion of a gimbal-mounted boom, this is a
MAX E CI DC K
= = = = =
Read and wte boom details Read mitral data for model Maximum number of !+erot!o”s Requwed standard of accuracy Tr10 value of C/Z Trial increment 1” CI Number of resonant frequency
loops
I
Read spectrum 1025 terms
doto
Read resonant frequency. WR Read number of terms to use
I i
I
Calculate /OLD”
functlo”
value
N=l
1
I
I-------l Cl=CItDC Calculate N=Ntl
NEW function
DIFF =“OLD”-
value
Calculate Increment, (t or-_) Re-owgn”OLD”
DC
NEW
I
Print
out results
I
tl END
Fig. 2. Flow diagram for the damping value program
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technique by means of which, for a value of natural frequency-itself dependent upon spring rate and moment of inertia-an optimum value of damping may be determined. A computer program was written to carry out a search for a value of damping which will produce a minimum response. In running this program a trial value of the ratio, c/Z, is supplied in the input data, together with a trial increment in its value. The value of the above integral is evaluated in a subroutine, for each of two values of c/I and the results are compared. In another subroutine an iterative process makes a search along the axis of c/Z for that value which produces the minimum value of the above integral. Trapezoidal integration is used in the function evaluation, conforming to the discrete spectrum and quadratic interpolation in the search for the minimum. The program also receives values of natural frequency, the frequency increment between the spectral density values, the number of spectral density terms to be included, an accuracy limit in the value of c/Z derived and a maximum permitted number of iterations to be tried in the search. The derived value of c/Z is reported as well as the residual value of the function. The process can be repeated for any number of natural frequencies. A flow diagram for the computer program is shown in Fig. 2.
2.3. Validation of derived values Experiments were carried out over the standard bumpy test track at speeds of approximately 9, 12 and 16 km/h with the sprayer tank empty and using different combinations of effective spring rate and damper settings. The profile of the bumpy test track, modelled on measurements made of an agricultural surface, had been previously shown to provide boom excitation similar to that experienced in field spraying, though approximately 30% more severe than typical cereal spraying conditions.’ Similar instrumentation to that used in the preliminary experiments provided records of horizontal and vertical acceleration at the boom tip from which displacement values were derived.
3. Results 3.1. Moments of inertia of the boom The characteristics of the boom are given in Table 1. The radius of gyration values given are the means of several determinations using the bifilar suspension system. The calculated value of radius of gyration in roll is less than that for yaw because the additional mass of the assembly which rolls is concentrated near the centre.
3.2. Spring stij%ess The mean linear stiffness values obtained from the tension testing machine and the calculated polar stiffness values for the pairs of springs when at one effective radius are given in Table 2. Table 1 Radius of gyration and moment of inertia of boom Yaw
Boom mass, kg Mean radius of gyration, m Moment of inertia, kgm’
Roll 215.3 2,201 1042
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Table 2 Stiffness of spring pairs
I Linear stiffness, N/m Polar stiffness, set at 200 mm radius, Nm/rad
Yaw
I
Roll
18330
33 750
1460
2700
3.3. Damping coeficien t
The damping coefficients calculated from the chart recorder traces were found to vary with velocity, the widest variation being 30% over a 4 : 1 range of speed. Mean values were taken, the dampers paired and the polar coefficients for each pair calculated (Table 3). 3.4. Spectral density data Traces of the power spectral densities for roll and yaw at the three speeds, calculated from the experimental data for rotational movement without boom, are shown in Fig. 3. The full lines show the variation in components up to 23 Hz and the broken line expands along the same x-axis the components up to 4.6 Hz, since the most significant portion for boom response is that at very low frequencies. The analysis of these data showed that in roll the greatest input movement differs little across the three speeds but in yaw the maximum was clearly at the highest speed. 3.5. Validation experiments The standard deviation of measured boom tip movement was the characteristic used in assessing agreement with the computer-predicted conditions for optimum performance. For presenting the results, the best impression was given by a three-dimensional graphical plot. The two horizontal and mutually-perpendicular axes comprising the sides of the base plane represent damping and stiffness. The vertical ordinates represent the standard deviation of the measured vertical movements of the boom tip. A typical example for measured vertical behaviour at a speed of 12 km/h, is shown in Fig. 4. In this illustration the general concavity of the upper surface may be seen, implying the existence of combinations of spring rate and damping coefficient for which movement tends to minimum values. The computer program for optimization was run for a range of natural frequencies from 0.85 to 2.25 rad/s with the power spectral densities for roll and for natural frequencies from 0.61 to 1.8 rad/s for yaw. For each chosen spring stiffness value, and its corresponding natural frequency, the calculated value of damping coefficient for which the computer model gave a minimum value
Table 3 Polar damping coefficient, Nms/rad Damper radius. mm
Roll
Yaw
Damper radius, mm
Roll
Yaw’
190 250 320
358 620 1017
427 742 1216
400 550 750
1589 3005 5587
1900 3592 6678
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Frequency,
Hz
Fig. 3. Power spectral densities of motion of sprayer frame (left) in roll and (right) in yaw, at (top) 9 km/h, (centre) 12 km/h and (bottom) 16 km/h on the standard bumpy test track, up to 23 Hz (__) and expanded up to 4.6 Hz ( - - - )
of the output function, was plotted on the base of the three-dimensional illustration and the relationship is shown by the chain-dot line. Correlation between the optimum damping values obtained from the computer model and those indicated by the experimental results was investigated by fitting a quadratic curve to the measured displacement values along each line of spring stiffness to find the minimum displacement value and the value of damping at which it occurred. The value of displacement on this curve corresponding to the computer-derived value of damping for minimum response was also calculated. The results of these for the vertical data are given in Table 4 and the positions of the calculated minima of the experimental results are plotted on the base of Fig. 4 for comparison with the chain-dot curve of the theoretical minima. There is a trend for improvement in vertical performance of the gimbal-mounted boom with increase in forward speed indicated by the experimental results in Table 4. The improvement with respect to the performance of the fixed boom is apparent in Fig. 4 in which a dashed line around the perimeter is drawn at a height indicating the standard deviation of boom tip movement measured with the boom fixed firmly to the sprayer chassis.
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RelationshIp between stiffness and damping for theoretlcol mmlmum response
Fig. 4. Standard
deviation of vertical displacement of boom tip with d@erent spring and damper at 12 km/h. a. positions of minima of experimental results
Table 4 Comparison between minimum values of standard deviation of measured vertical boom tip movement and those at modelderived values of damping At model-derived value of damping Polar stiffness, Nmlrad
r
At minimum of experimental results
Damping, Nmslrad
Movement, m
Damping, Nmslrad
Movement, m
At 9 km/h 1730 2700 3880 8270 13670
2160 2560 3010 4360 5700
0.074 0.090 0.097 0.134 0.160
1450 1640 3040 4880 4810
0.068 0.080 0,097 0.132 0.157
At 12 km/h 1730 2700 3880 8270 13670
1940 2470 2870 4OOa 5140
0.039 0.056 0.072 0.116 0.195
1910 2170 2340 3940 4870
oa41 0.048 0.065 0.116 0.190
At 16 km/h 1730 2700 3880 8270 13670
1900 2470 2950 4160 5030
OaO 0.048 0.055 0.085 0.160
1830 1950 3000 3490 5070
oa41 0.047 0.058 0.075 0.153
settings,
2
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255
.I. NATION Table 5 Calculated values of spring stiffness for minimum vertical displacement
Polar stiffness, Nmlrad
Calculated minimum displacement, m
1020 1590
780 930
0.062 0.064
At 12 km/h 1020 1590
1290 550
0.053 0.038
At 16 km/h 1020 1590
1670 1200
O@l4 0.040
Polar damping, Nmsjrad ~--_____ At 9 km/h
Whilst the optimum values of damping are within the range of the results given, this may not be true of the values of spring stiffness. At the lowest values it is evident that further decrease in boom movement may be obtainable at lower spring stiffnesses. This was investigated by fitting a quadratic curve to the values along lines of constant damping, where this situation appeared to exist. These results are given in Table 5, indicating some possible further improvement by use of a spring stiffness between 900 and 1200 Nm/rad. However, such values were not investigated experimentally because of practical difficulties in positioning the springs for such low settings. If these projected minimum displacement values could be obtained, they would represent reductions in movement compared with a fixed boom of 49x, 57% and 73% respectively, at 9, 12 and 16 km/h. The standard deviation values of the measured horizontal boom tip displacements relative to the sprayer were plotted in a similar three-dimensional manner, and those for 12 km/h are Relotlonship between stiffness and damplng for theoretlcal mmlmum response
Fig. 5. Standard
deviution of horizontal displacement of boom tip with d$ferent spring and damper settings, at 12 km/h. ?? , positions of minima of experimental results
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Table 6 Comparison between minimum values of standard deviation of measured horizontal boom tip movement and those at model-derived vales of damping
Polar stljhess, Nmlrad
T
At model-derived value of damping
1
At minimum of experimental results
Damping, Nmslrad
Movement, m
Damping, Nmslrad
Movement, m
At 9 km/h 940 1470 2110 4490 7420
1100 1360 1940 3250 4400
0,049 0,052 0,052 0,052 0.078
1150 1360 2010 3870 4290
0.045 0.052 0.051 0.052 0.077
At 12 km/h 1470 2110 4490 7420
1060 1400 2680 3920
0.039 0.041 0.041 0.047
970 1560 2750 4090
0.037 0.038 0041 0.045
At 16 km/h 1470 2110 4490 7420
980 1330 2850 4340
0.029 0.029 0.030 0.031
N.B. Missing values indicate range of the observed results
that those derived
3260 3860 by quadratic
0.031 0.027 fitting were outside
the
presented in Fig. 5. This presentation of horizontal performance also shows a concavity in the upper surface, and the course of the minimum values was determined by quadratic fitting, for each spring stiffness value. These results are given in Table 6, from which some values are missing because the calculated conditions for minimum values using quadratic fitting lay outside the range of the observed results. The minimum horizontal displacements observed or calculated represented reductions in movement of 58x, 71% and 66% respectively, at 9, 12 and 16 km/h, in comparison with the horizontal boom tip movement of the boom when fixed firmly to the sprayer. 4. Discussion 4.1. Vertical behaviour With a spring stiffness of 1200 Nm/rad, by interpolation between the observed values, minimum values of standard deviation of vertical movement of 0.063, O-048 and 0.043 m would be obtained at the three speeds. Reference to more comprehensive results shows that these values can be associated with ranges of vertical movement of the boom tip of 0.333, 0.270 and 0.240 m respectively. The practical significance of this is that for a boom set at a nominal height above the target of 0.54 m, the minimum boom tip height will be 0.37 m. This can be interpreted in terms of its theoretical effect on spray distribution and, if fan spray nozzles are assumed, the worst instantaneous coefficient of variation of lateral distribution, due to error in spray overlap will be less than 20%. Likewise, the deviation from the mean of the maximum local deposit will be less than 40% and the deviation of the lowest deposit will be less than 20%. Such errors in distribution are dramatically less than those measured in typical farm spraying.* On average, however, half the standard deviaticn
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of the movement gives 0.032 m as the average downwards movement of the boom, leading to an average coefficient of variation of transverse distribution of only about 2%. 4.2. Horizontal behaviour Differing from the vertical behaviour, the measured horizontal behaviour of the gimbal mounting was superior to that of the fixed boom over the whole range of stiffness and damping examined as indicated by the dashed line in Fig. 5. This also was true for the two other speeds for which the diagrams are not presented. It was also evident that no further improvement in performance would be obtained by use of lower values of spring stiffness. Again, the minimum values of standard deviation of movement decreased with increase in speed, 0.045 m, 0.031 m and 0.031 m at speeds of 9, 12 and 16 km/h respectively. Taking 0.045 m, fuller details show that this typically corresponds to a range of movement of about 0.25 m, a coefficient of variation of deposit along a line in the direction of travel under fan spray nozzles of 5.9% and deviations of maximum and minimum deposits of about 19% and 15%. Thus, the horizontal performance of the gimbal mounting with the boom and sprayer examined was slightly superior to the vertical performance. 4.3. The computer model The agreement indicated in Tables 4 and 6 confirms the appropriateness of the procedures incorporated in the computer model for calculating the value of damping to minimize the response for a given value of spring stiffness. It provides a powerful tool for the designer of such passive suspensions for sprayer booms, provided he also has the facilities to obtain the power spectral density of the sprayer input to the suspension and the polar moment of inertia of the boom. The time histories of the roll and yaw motions of the sprayer frame, from which the input power spectral densities were obtained, consisted of 2048 observations. The full spectral densities therefore consisted of 1025 terms, and all of these were included in the integration in the searches made. A special series of computer runs was made, in view of the rapid fall in the power spectrum with increasing frequency, to investigate the significance of the terms. The calculations in the model are carried out in double precision and it was found that with low values of natural frequency as few as 30 terms provided the same accuracy in the first three significant figures of the value of c/I obtained. Use of this model to optimize damping should be possible on a microcomputer, provided means are available to obtain a power spectrum of the input disturbance up to a maximum frequency of approximately 1.5 Hz. 4.4. Hillside working A useful practical feature of the gimbal mounting, in contrast to that of the simple pendulum suspension, is the ability of the boom to remain parallel to the tractor (or trailer) axle in rear elevation when operating across a slope. However, the requirement is that the boom should remain parallel to the ground, and it has been shown, particularly in work at the Scottish Institute of Agricultural Engineering,’ that because of weight transfer towards the downhill wheel this tyre will suffer additional deflection and sinkage into the ground, whilst the uphill wheel will experience less. Results are available to show how the angle between the tractor axle and the ground varies as the gradient increases. For three tractors it was found that the slope of the axle was approximately 20% greater than that of the ground. Thus, the tip of a 12 m spray boom of a sprayer on a 10” slope would be O-2 m closer to the ground at the downhill end than at the centre-an error greater than that due to the range of dynamic movement noted above.
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A practical solution to this operational difficulty is to raise the horizontal pivot above the centre of gravity until the pendulum effect restores the boom parallel to the ground in opposition to, and in equilibrium with, the polar spring stiffness. Taking as an example the boom used in the present experiments, and assuming that it is restrained with a polar stiffness of 1200 Nm/rad, the required effect is obtained if the pivot is raised about 0.1 m above the centre of gravity. This is only 5% of the radius of gyration, and consequently it affects the dynamic behaviour very little.’ This feature is already incorporated successfully in some of the commercial embodiments of the centre-pivot and gimbal mountings. 5. Conclusions From the above, the following can be concluded: (1) The gimbal mounting is an effective method of reducing transmission to the spray boom of the rapid rolling and yawing movements of the sprayer. (2) The effectiveness of the mounting improves with increase in forward speed; for example, increase of speed from 9 to 16 km/h improves the reduction in vertical boom tip movement from 49% to 73% (calculated values) and the reduction in horizontal movement from 58% to 66% (observed values). (3) The model developed to determine damping values which minimize the output gives results in good agreement with observed behaviour and constitutes a powerful design tool. (4) The model theory illustrates the importance of a high value of polar moment of inertia. (5) The gimbal is suitable for use on a side slope, and the practical disadvantage of additional inclination experienced by the sprayer can be compensated for by introducing a small pendulum effect. Acknowledgements D. C. Lawrence, A. Lockwood, D. Stutchfield (Sandwich Student) and M. R. Holden assisted with the practical work. Dr P. F. Davis gave valuable mathematical assistance particularly in the development of the model. J. Fairey transcribed the recorded data.
References ’ Nation, H. J. The dynamic behaviour of field sprayer booms. Journal of Agricultural Engineering
Research 1982, 27: 61-70 * Nation, H. J. The design and performance of a gimbal-type mounting for sprayer booms. I. Development procedure. Journal of Agricultural Engineering Research 1986, 36: 233-246. 3 Nation, H. J. The dynamic behaviour of field sprayer booms: A technique for field measurement. Departmental Note DN/S/814/09005, National Institute of Agricultural Engineering, Silsoe, 1978 (unpubl.) 4 Anon. Method of test for seats on agricultural wheeled tractors. British Standard 4220: Part I: Tests on an artificial track 5 Nation, H. J. The dynamic behaviour of field sprayer booms: The suite of computer programs. Departmental Note DN/S/801/09005, National Institute of Agricultural Engineering, Silsoe 1976 (unpubl.) 6 Nation, H. J.; Holden, M. R. The dynamic behaviour of field sprayer booms: A comparison between boom tip behaviour in the field and on a standard bumpy test track. Departmental Note DN/S/538/1905, National Institute of Agricultural Engineering, Silsoe 1975 (unpubl.) 7 Nation, H. J. Spray nozzle performance and effect of boom height on distribution. Departmental Note DN/S/777/1925, National Institute of Agricultural Engineering, Silsoe 1976 (unpubl.)
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Spray application for the continuous cereal grower. Proceedings 9th British Weed Control Conference 1968, pp. 214-220 g Owen, G. M. Personal communication 1982
8 Nation, H. J.
Appendix. Analysis of behaviour of a centre-point pivot supported hoom A simpler analysis for the case of a pivot through the centre of gravity is possible, than for the general case of the pendulum. Use of the differential equations of motion lead directly to a solution. Consider the motion of a spray boom rolling in response to variation in slope of the sprayer axle as the wheels rise and fall independently and randomly in following the field surface (Eig. Al). The boom is supported on the pivot at 0 and restrained by springs having a polar constant k, and viscous dampers of polar coefficient c. The angle of slope of the sprayer axle is 4 and that of the boom is y. The basic equation of rotational motion then is k(y-@)+c(j-$)+Zj;
or
Zjj+cj+ky
= 0
= ctj+k&
(1) (2)
Expressing the time histories of the angles as Fourier series y=
and after differentiating,
s
reiwt and
4 =
@eiwt s
the above equation of motion can be written (-Zo2+ci~+k)I-
= (ciw+k)@.
(3)
Squaring with the conjugate gives (k + cio)(k - cio)
r2
II 5
= (k + ciw - Zw2)(k - ciw - lo’) (k2 + w2c2) = ((k-Zco2)‘+co2c2)
’
(4)
The spring stiffness in this expression can be replaced, with the knowledge that the resonant frequency so
r2
so
Iz I=
o;z2 + co2c2 [(o;-021)2+o%2]’
(5) From this, if o is small compared with wR, that is, the rate of change of slope of the sprayer is very slow, e.g. in slowly undulating fields or at very slow forward speeds, then
that is, the boom tends to follow the sprayer movements.
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260 Whee
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Is
I
Mean ground surface ntaneous transverse ground surface proflle
Fig. Al. Diagrammatic rear view of boom on centre-pivot mounting
If w = wR, that is, the input disturbance is mainly at the resonant frequency of the boom mounting system, then III2 > IW or, the motion will be amplified. However, if the input consists mostly of frequencies considerably in excess of the resonant frequency, i.e. W>>WR, then (II2 <<[@I2 if c is small compared with I. Thus, this again points to the need for a high polar moment of inertia. As before, since )@I2is proportional to the power spectrum of 4, let this be S(w) which may be substituted in (5) and then the average least square value of y can be obtained by appropriate choice of the ratio, c/Z, to minimize the integral
,-
[w’+w26~l 2
(w;-w2)2+w2
S(w)dw,
; 01
In the above analysis, the effect of gravity has played no part, although motion in a vertical plane was considered. Thus, the result will be equally applicable to any non-vertical plane and can be used for the horizontal movement of a boom on the gimbal mounting.
The Design and Performance of a Gimbal-type Mounting for Sprayer Booms 2. Optimization
Model and Validation H. J. NATION*
An empirical approach had already indicated that the gimbal-type mounting is effective in reducing movement of a sprayer boom. This mounting involves springs and viscous dampers and it was concluded that the effects of these would more readily be optimized by calculation than by further experiments. An analysis was made of the dynamics of the gimbal mounting and a computer model developed by means of which the value of viscous damping could be optimized for a spring stiffness by searching for the minimum value of an output function integrated over a range of components of an input power spectrum. Power spectra for roll and yaw were calculated from data recorded at three speeds over the standard bumpy test track. The polar moment of inertia of the boom was determined by bifilar suspension. Optimum values of damping were computed for ranges of natural frequencies. The experimental gimbal mounting was rebuilt to facilitate tests at ranges of polar spring stiffness and polar damping coefficient. The conditions giving minima in the response obtained were found to agree well with the values predicted by the computer model.
1. Introduction Previous work examining the dynamic behaviour of field sprayer booms indicated the need for a flexible form of mounting between the boom and sprayer frame to reduce errant boom movements.’ An examination of the effectiveness of different types of boom mounting
led to the development of the gimbal-type of mounting with mutually-perpendicular pivots through the centre of gravity of the boom.’ This development had evolved from the pendulum type of suspension and theory showed that for this to be effective the polar radius of gyration should be large in comparison with the length of the pendulum. Experiment showed a similar benefit of a large polar moment of inertia with the gimbal mounting. With the horizontal pivot through the centre of gravity, no restoring force is provided by the weight of the boom, and springs were adopted to restrain the boom in a rest position parallel to the machine axle. Natural oscillation of such a system when suitably excited by sprayer rolling motion could be controlled by viscous dampers. However, determination of optimum values of viscous damping by empirical means would be a laborious way to obtain optimum performance. A mathematical model was indicated as the most appropriate approach. 2. Method 2.1. Experimental procedures The spectral densities to be examined by the computer program were obtained using the same sprayer as in the earlier experiments. The boom was removed and replaced by a weight * National Institute of Agricultural Engineering, Wrest Park, Silsoe, Bedford MK45 4HS, UK Received
8 March
1985; accepted
in revised form 6 May 1986
247 OO? I-Xh34’87
040247 + 14 %03.00,‘0
‘.r: 1987 The British Souety
for Research in Agricultural
Engineering
248
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Notation a
length of bifilar suspension wires polar damping coefficient L polar spring stiffness radius to attachment point r’ moment of inertia K radius of gyration M mass of boom T periodic time Y angle of displacement of boom in motion
l-
Fourier representation of time history of boom movement 4 angle of displacement of sprayer frame in motion CD Fourier representation of time history of sprayer movement W angular frequency resonant frequency WR
of equivalent mass but with relatively low polar inertia clamped to the frame at the attachment point. At the same place an instrumented gyroscope was fastened to measure roll and yaw.’ Several runs were made over the standard bumpy test track4 at speeds of approximately 9, 12 and 16 km/h. The recorded data were digitized and the spectral densities calculated.’ These were taken in turn as input to the optimization program for a range of natural frequencies. To obtain the appropriate values of damping from the results it was necessary to determine the polar moment of inertia of the boom. This was done experimentally using bifilar suspension. The moment of inertia, I = MK’; and K can be obtained from p-
T2r2g 4n2a ’
These observations were first made for the boom as it operated in yaw, and then repeated with the additional parts which move with the boom in roll. The gimbal mounting included slides for the fixtures to which the springs and dampers were attached to facilitate choice of settings for each (Fig. 1). Load versus extension relationships for the springs were obtained experimentally in a tension testing machine and the individual linear constants converted to polar stiffness for each pair. The characteristics of the viscous dampers were examined in a rig in which they could be compressed or extended under constant load, their displacement being measured by linear transducers connected to a chart recorder, from the output of which velocity was calculated. In the experiments with the boom different effective damping values were obtained by use of different radius settings.
2.2. Theoretical analysis In normal use on agricultural surfaces a sprayer frame is subjected to a complex combination of translational and rotational disturbances. From the aspect of dynamic boom behaviour the most important of these are the rolling and yawing rotational disturbances. These belong to a certain class of non-recurring disturbance, the “stationary random” type,
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spray
Rolling beam’
Fig. I. Revised arrangement of gimbal mounting
for which conventional frequency analyses are not appropriate. For these disturbances, where average characteristics do not change with time, a frequency analysis is possible by means of the so-called “power spectral density function”. From a time history of, for example, the rolling motion of the sprayer, consisting of individual measurements of angular displacement at equal intervals of time, it is possible to construct a Fourier transform, consisting of terms representing the individual harmonic components at increasing frequencies. From the Fourier transform it is possible to calculate the power spectral density which may be looked upon as providing a measure of the disturbance energy per unit time and this has come to be regarded as a measure of power. This is particularly useful when dealing with linear systems where a simple relationship exists between the spectrum of an exciting disturbance and the spectrum of the system response. The generalized form of the relationship is:
where Q,(o) = the output spectrum, Qi(o) = the input spectrum, and T(w) = the frequency response function of the system. An analysis of the response to input angular disturbances of a boom subject to linear spring rate restoring torques and viscous damping in rotation about its centre of gravity is given in the Appendix. An expression is derived representative of the power spectrum of the output motion in response to an input power spectrum of rotation of the frame to which the
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restoring springs and dampers are attached. By means of an iterative computer program values of the ratio of damping coefficient to polar moment of inertia, c/I, are searched to minimize the integral
where S(o) is the power spectrum of the input motion. In this derivation there is no reference to the effect of gravity as the boom is in a state of neutral equilibrium and the analysis is therefore equally applicable to the behaviour of the boom in horizontal yaw. So, for both planes of motion of a gimbal-mounted boom, this is a
MAX E CI DC K
= = = = =
Read and wte boom details Read mitral data for model Maximum number of !+erot!o”s Requwed standard of accuracy Tr10 value of C/Z Trial increment 1” CI Number of resonant frequency
loops
I
Read spectrum 1025 terms
doto
Read resonant frequency. WR Read number of terms to use
I i
I
Calculate /OLD”
functlo”
value
N=l
1
I
I-------l Cl=CItDC Calculate N=Ntl
NEW function
DIFF =“OLD”-
value
Calculate Increment, (t or-_) Re-owgn”OLD”
DC
NEW
I
out results
I
tl END
Fig. 2. Flow diagram for the damping value program
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technique by means of which, for a value of natural frequency-itself dependent upon spring rate and moment of inertia-an optimum value of damping may be determined. A computer program was written to carry out a search for a value of damping which will produce a minimum response. In running this program a trial value of the ratio, c/Z, is supplied in the input data, together with a trial increment in its value. The value of the above integral is evaluated in a subroutine, for each of two values of c/I and the results are compared. In another subroutine an iterative process makes a search along the axis of c/Z for that value which produces the minimum value of the above integral. Trapezoidal integration is used in the function evaluation, conforming to the discrete spectrum and quadratic interpolation in the search for the minimum. The program also receives values of natural frequency, the frequency increment between the spectral density values, the number of spectral density terms to be included, an accuracy limit in the value of c/Z derived and a maximum permitted number of iterations to be tried in the search. The derived value of c/Z is reported as well as the residual value of the function. The process can be repeated for any number of natural frequencies. A flow diagram for the computer program is shown in Fig. 2.
2.3. Validation of derived values Experiments were carried out over the standard bumpy test track at speeds of approximately 9, 12 and 16 km/h with the sprayer tank empty and using different combinations of effective spring rate and damper settings. The profile of the bumpy test track, modelled on measurements made of an agricultural surface, had been previously shown to provide boom excitation similar to that experienced in field spraying, though approximately 30% more severe than typical cereal spraying conditions.’ Similar instrumentation to that used in the preliminary experiments provided records of horizontal and vertical acceleration at the boom tip from which displacement values were derived.
3. Results 3.1. Moments of inertia of the boom The characteristics of the boom are given in Table 1. The radius of gyration values given are the means of several determinations using the bifilar suspension system. The calculated value of radius of gyration in roll is less than that for yaw because the additional mass of the assembly which rolls is concentrated near the centre.
3.2. Spring stij%ess The mean linear stiffness values obtained from the tension testing machine and the calculated polar stiffness values for the pairs of springs when at one effective radius are given in Table 2. Table 1 Radius of gyration and moment of inertia of boom Yaw
Boom mass, kg Mean radius of gyration, m Moment of inertia, kgm’
Roll 215.3 2,201 1042
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Table 2 Stiffness of spring pairs
I Linear stiffness, N/m Polar stiffness, set at 200 mm radius, Nm/rad
Yaw
I
Roll
18330
33 750
1460
2700
3.3. Damping coeficien t
The damping coefficients calculated from the chart recorder traces were found to vary with velocity, the widest variation being 30% over a 4 : 1 range of speed. Mean values were taken, the dampers paired and the polar coefficients for each pair calculated (Table 3). 3.4. Spectral density data Traces of the power spectral densities for roll and yaw at the three speeds, calculated from the experimental data for rotational movement without boom, are shown in Fig. 3. The full lines show the variation in components up to 23 Hz and the broken line expands along the same x-axis the components up to 4.6 Hz, since the most significant portion for boom response is that at very low frequencies. The analysis of these data showed that in roll the greatest input movement differs little across the three speeds but in yaw the maximum was clearly at the highest speed. 3.5. Validation experiments The standard deviation of measured boom tip movement was the characteristic used in assessing agreement with the computer-predicted conditions for optimum performance. For presenting the results, the best impression was given by a three-dimensional graphical plot. The two horizontal and mutually-perpendicular axes comprising the sides of the base plane represent damping and stiffness. The vertical ordinates represent the standard deviation of the measured vertical movements of the boom tip. A typical example for measured vertical behaviour at a speed of 12 km/h, is shown in Fig. 4. In this illustration the general concavity of the upper surface may be seen, implying the existence of combinations of spring rate and damping coefficient for which movement tends to minimum values. The computer program for optimization was run for a range of natural frequencies from 0.85 to 2.25 rad/s with the power spectral densities for roll and for natural frequencies from 0.61 to 1.8 rad/s for yaw. For each chosen spring stiffness value, and its corresponding natural frequency, the calculated value of damping coefficient for which the computer model gave a minimum value
Table 3 Polar damping coefficient, Nms/rad Damper radius. mm
Roll
Yaw
Damper radius, mm
Roll
Yaw’
190 250 320
358 620 1017
427 742 1216
400 550 750
1589 3005 5587
1900 3592 6678
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Frequency,
Hz
Fig. 3. Power spectral densities of motion of sprayer frame (left) in roll and (right) in yaw, at (top) 9 km/h, (centre) 12 km/h and (bottom) 16 km/h on the standard bumpy test track, up to 23 Hz (__) and expanded up to 4.6 Hz ( - - - )
of the output function, was plotted on the base of the three-dimensional illustration and the relationship is shown by the chain-dot line. Correlation between the optimum damping values obtained from the computer model and those indicated by the experimental results was investigated by fitting a quadratic curve to the measured displacement values along each line of spring stiffness to find the minimum displacement value and the value of damping at which it occurred. The value of displacement on this curve corresponding to the computer-derived value of damping for minimum response was also calculated. The results of these for the vertical data are given in Table 4 and the positions of the calculated minima of the experimental results are plotted on the base of Fig. 4 for comparison with the chain-dot curve of the theoretical minima. There is a trend for improvement in vertical performance of the gimbal-mounted boom with increase in forward speed indicated by the experimental results in Table 4. The improvement with respect to the performance of the fixed boom is apparent in Fig. 4 in which a dashed line around the perimeter is drawn at a height indicating the standard deviation of boom tip movement measured with the boom fixed firmly to the sprayer chassis.
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RelationshIp between stiffness and damping for theoretlcol mmlmum response
Fig. 4. Standard
deviation of vertical displacement of boom tip with d@erent spring and damper at 12 km/h. a. positions of minima of experimental results
Table 4 Comparison between minimum values of standard deviation of measured vertical boom tip movement and those at modelderived values of damping At model-derived value of damping Polar stiffness, Nmlrad
r
At minimum of experimental results
Damping, Nmslrad
Movement, m
Damping, Nmslrad
Movement, m
At 9 km/h 1730 2700 3880 8270 13670
2160 2560 3010 4360 5700
0.074 0.090 0.097 0.134 0.160
1450 1640 3040 4880 4810
0.068 0.080 0,097 0.132 0.157
At 12 km/h 1730 2700 3880 8270 13670
1940 2470 2870 4OOa 5140
0.039 0.056 0.072 0.116 0.195
1910 2170 2340 3940 4870
oa41 0.048 0.065 0.116 0.190
At 16 km/h 1730 2700 3880 8270 13670
1900 2470 2950 4160 5030
OaO 0.048 0.055 0.085 0.160
1830 1950 3000 3490 5070
oa41 0.047 0.058 0.075 0.153
settings,
2
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.I. NATION Table 5 Calculated values of spring stiffness for minimum vertical displacement
Polar stiffness, Nmlrad
Calculated minimum displacement, m
1020 1590
780 930
0.062 0.064
At 12 km/h 1020 1590
1290 550
0.053 0.038
At 16 km/h 1020 1590
1670 1200
O@l4 0.040
Polar damping, Nmsjrad ~--_____ At 9 km/h
Whilst the optimum values of damping are within the range of the results given, this may not be true of the values of spring stiffness. At the lowest values it is evident that further decrease in boom movement may be obtainable at lower spring stiffnesses. This was investigated by fitting a quadratic curve to the values along lines of constant damping, where this situation appeared to exist. These results are given in Table 5, indicating some possible further improvement by use of a spring stiffness between 900 and 1200 Nm/rad. However, such values were not investigated experimentally because of practical difficulties in positioning the springs for such low settings. If these projected minimum displacement values could be obtained, they would represent reductions in movement compared with a fixed boom of 49x, 57% and 73% respectively, at 9, 12 and 16 km/h. The standard deviation values of the measured horizontal boom tip displacements relative to the sprayer were plotted in a similar three-dimensional manner, and those for 12 km/h are Relotlonship between stiffness and damplng for theoretlcal mmlmum response
Fig. 5. Standard
deviution of horizontal displacement of boom tip with d$ferent spring and damper settings, at 12 km/h. ?? , positions of minima of experimental results
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Table 6 Comparison between minimum values of standard deviation of measured horizontal boom tip movement and those at model-derived vales of damping
Polar stljhess, Nmlrad
T
At model-derived value of damping
1
At minimum of experimental results
Damping, Nmslrad
Movement, m
Damping, Nmslrad
Movement, m
At 9 km/h 940 1470 2110 4490 7420
1100 1360 1940 3250 4400
0,049 0,052 0,052 0,052 0.078
1150 1360 2010 3870 4290
0.045 0.052 0.051 0.052 0.077
At 12 km/h 1470 2110 4490 7420
1060 1400 2680 3920
0.039 0.041 0.041 0.047
970 1560 2750 4090
0.037 0.038 0041 0.045
At 16 km/h 1470 2110 4490 7420
980 1330 2850 4340
0.029 0.029 0.030 0.031
N.B. Missing values indicate range of the observed results
that those derived
3260 3860 by quadratic
0.031 0.027 fitting were outside
the
presented in Fig. 5. This presentation of horizontal performance also shows a concavity in the upper surface, and the course of the minimum values was determined by quadratic fitting, for each spring stiffness value. These results are given in Table 6, from which some values are missing because the calculated conditions for minimum values using quadratic fitting lay outside the range of the observed results. The minimum horizontal displacements observed or calculated represented reductions in movement of 58x, 71% and 66% respectively, at 9, 12 and 16 km/h, in comparison with the horizontal boom tip movement of the boom when fixed firmly to the sprayer. 4. Discussion 4.1. Vertical behaviour With a spring stiffness of 1200 Nm/rad, by interpolation between the observed values, minimum values of standard deviation of vertical movement of 0.063, O-048 and 0.043 m would be obtained at the three speeds. Reference to more comprehensive results shows that these values can be associated with ranges of vertical movement of the boom tip of 0.333, 0.270 and 0.240 m respectively. The practical significance of this is that for a boom set at a nominal height above the target of 0.54 m, the minimum boom tip height will be 0.37 m. This can be interpreted in terms of its theoretical effect on spray distribution and, if fan spray nozzles are assumed, the worst instantaneous coefficient of variation of lateral distribution, due to error in spray overlap will be less than 20%. Likewise, the deviation from the mean of the maximum local deposit will be less than 40% and the deviation of the lowest deposit will be less than 20%. Such errors in distribution are dramatically less than those measured in typical farm spraying.* On average, however, half the standard deviaticn
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of the movement gives 0.032 m as the average downwards movement of the boom, leading to an average coefficient of variation of transverse distribution of only about 2%. 4.2. Horizontal behaviour Differing from the vertical behaviour, the measured horizontal behaviour of the gimbal mounting was superior to that of the fixed boom over the whole range of stiffness and damping examined as indicated by the dashed line in Fig. 5. This also was true for the two other speeds for which the diagrams are not presented. It was also evident that no further improvement in performance would be obtained by use of lower values of spring stiffness. Again, the minimum values of standard deviation of movement decreased with increase in speed, 0.045 m, 0.031 m and 0.031 m at speeds of 9, 12 and 16 km/h respectively. Taking 0.045 m, fuller details show that this typically corresponds to a range of movement of about 0.25 m, a coefficient of variation of deposit along a line in the direction of travel under fan spray nozzles of 5.9% and deviations of maximum and minimum deposits of about 19% and 15%. Thus, the horizontal performance of the gimbal mounting with the boom and sprayer examined was slightly superior to the vertical performance. 4.3. The computer model The agreement indicated in Tables 4 and 6 confirms the appropriateness of the procedures incorporated in the computer model for calculating the value of damping to minimize the response for a given value of spring stiffness. It provides a powerful tool for the designer of such passive suspensions for sprayer booms, provided he also has the facilities to obtain the power spectral density of the sprayer input to the suspension and the polar moment of inertia of the boom. The time histories of the roll and yaw motions of the sprayer frame, from which the input power spectral densities were obtained, consisted of 2048 observations. The full spectral densities therefore consisted of 1025 terms, and all of these were included in the integration in the searches made. A special series of computer runs was made, in view of the rapid fall in the power spectrum with increasing frequency, to investigate the significance of the terms. The calculations in the model are carried out in double precision and it was found that with low values of natural frequency as few as 30 terms provided the same accuracy in the first three significant figures of the value of c/I obtained. Use of this model to optimize damping should be possible on a microcomputer, provided means are available to obtain a power spectrum of the input disturbance up to a maximum frequency of approximately 1.5 Hz. 4.4. Hillside working A useful practical feature of the gimbal mounting, in contrast to that of the simple pendulum suspension, is the ability of the boom to remain parallel to the tractor (or trailer) axle in rear elevation when operating across a slope. However, the requirement is that the boom should remain parallel to the ground, and it has been shown, particularly in work at the Scottish Institute of Agricultural Engineering,’ that because of weight transfer towards the downhill wheel this tyre will suffer additional deflection and sinkage into the ground, whilst the uphill wheel will experience less. Results are available to show how the angle between the tractor axle and the ground varies as the gradient increases. For three tractors it was found that the slope of the axle was approximately 20% greater than that of the ground. Thus, the tip of a 12 m spray boom of a sprayer on a 10” slope would be O-2 m closer to the ground at the downhill end than at the centre-an error greater than that due to the range of dynamic movement noted above.
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A practical solution to this operational difficulty is to raise the horizontal pivot above the centre of gravity until the pendulum effect restores the boom parallel to the ground in opposition to, and in equilibrium with, the polar spring stiffness. Taking as an example the boom used in the present experiments, and assuming that it is restrained with a polar stiffness of 1200 Nm/rad, the required effect is obtained if the pivot is raised about 0.1 m above the centre of gravity. This is only 5% of the radius of gyration, and consequently it affects the dynamic behaviour very little.’ This feature is already incorporated successfully in some of the commercial embodiments of the centre-pivot and gimbal mountings. 5. Conclusions From the above, the following can be concluded: (1) The gimbal mounting is an effective method of reducing transmission to the spray boom of the rapid rolling and yawing movements of the sprayer. (2) The effectiveness of the mounting improves with increase in forward speed; for example, increase of speed from 9 to 16 km/h improves the reduction in vertical boom tip movement from 49% to 73% (calculated values) and the reduction in horizontal movement from 58% to 66% (observed values). (3) The model developed to determine damping values which minimize the output gives results in good agreement with observed behaviour and constitutes a powerful design tool. (4) The model theory illustrates the importance of a high value of polar moment of inertia. (5) The gimbal is suitable for use on a side slope, and the practical disadvantage of additional inclination experienced by the sprayer can be compensated for by introducing a small pendulum effect. Acknowledgements D. C. Lawrence, A. Lockwood, D. Stutchfield (Sandwich Student) and M. R. Holden assisted with the practical work. Dr P. F. Davis gave valuable mathematical assistance particularly in the development of the model. J. Fairey transcribed the recorded data.
References ’ Nation, H. J. The dynamic behaviour of field sprayer booms. Journal of Agricultural Engineering
Research 1982, 27: 61-70 * Nation, H. J. The design and performance of a gimbal-type mounting for sprayer booms. I. Development procedure. Journal of Agricultural Engineering Research 1986, 36: 233-246. 3 Nation, H. J. The dynamic behaviour of field sprayer booms: A technique for field measurement. Departmental Note DN/S/814/09005, National Institute of Agricultural Engineering, Silsoe, 1978 (unpubl.) 4 Anon. Method of test for seats on agricultural wheeled tractors. British Standard 4220: Part I: Tests on an artificial track 5 Nation, H. J. The dynamic behaviour of field sprayer booms: The suite of computer programs. Departmental Note DN/S/801/09005, National Institute of Agricultural Engineering, Silsoe 1976 (unpubl.) 6 Nation, H. J.; Holden, M. R. The dynamic behaviour of field sprayer booms: A comparison between boom tip behaviour in the field and on a standard bumpy test track. Departmental Note DN/S/538/1905, National Institute of Agricultural Engineering, Silsoe 1975 (unpubl.) 7 Nation, H. J. Spray nozzle performance and effect of boom height on distribution. Departmental Note DN/S/777/1925, National Institute of Agricultural Engineering, Silsoe 1976 (unpubl.)
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Spray application for the continuous cereal grower. Proceedings 9th British Weed Control Conference 1968, pp. 214-220 g Owen, G. M. Personal communication 1982
8 Nation, H. J.
Appendix. Analysis of behaviour of a centre-point pivot supported hoom A simpler analysis for the case of a pivot through the centre of gravity is possible, than for the general case of the pendulum. Use of the differential equations of motion lead directly to a solution. Consider the motion of a spray boom rolling in response to variation in slope of the sprayer axle as the wheels rise and fall independently and randomly in following the field surface (Eig. Al). The boom is supported on the pivot at 0 and restrained by springs having a polar constant k, and viscous dampers of polar coefficient c. The angle of slope of the sprayer axle is 4 and that of the boom is y. The basic equation of rotational motion then is k(y-@)+c(j-$)+Zj;
or
Zjj+cj+ky
= 0
= ctj+k&
(1) (2)
Expressing the time histories of the angles as Fourier series y=
and after differentiating,
s
reiwt and
4 =
@eiwt s
the above equation of motion can be written (-Zo2+ci~+k)I-
= (ciw+k)@.
(3)
Squaring with the conjugate gives (k + cio)(k - cio)
r2
II 5
= (k + ciw - Zw2)(k - ciw - lo’) (k2 + w2c2) = ((k-Zco2)‘+co2c2)
’
(4)
The spring stiffness in this expression can be replaced, with the knowledge that the resonant frequency so
r2
so
Iz I=
o;z2 + co2c2 [(o;-021)2+o%2]’
(5) From this, if o is small compared with wR, that is, the rate of change of slope of the sprayer is very slow, e.g. in slowly undulating fields or at very slow forward speeds, then
that is, the boom tends to follow the sprayer movements.
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260 Whee
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Is
I
Mean ground surface ntaneous transverse ground surface proflle
Fig. Al. Diagrammatic rear view of boom on centre-pivot mounting
If w = wR, that is, the input disturbance is mainly at the resonant frequency of the boom mounting system, then III2 > IW or, the motion will be amplified. However, if the input consists mostly of frequencies considerably in excess of the resonant frequency, i.e. W>>WR, then (II2 <<[@I2 if c is small compared with I. Thus, this again points to the need for a high polar moment of inertia. As before, since )@I2is proportional to the power spectrum of 4, let this be S(w) which may be substituted in (5) and then the average least square value of y can be obtained by appropriate choice of the ratio, c/Z, to minimize the integral
,-
[w’+w26~l 2
(w;-w2)2+w2
S(w)dw,
; 01
In the above analysis, the effect of gravity has played no part, although motion in a vertical plane was considered. Thus, the result will be equally applicable to any non-vertical plane and can be used for the horizontal movement of a boom on the gimbal mounting.