2.2. Theoretical Analysis of the Miniature Model’s Static Falling down Sidelong Angle
The lateral overturning of the tractor is analyzed theoretically by literature review, and the relationship between the original and miniature models is derived based on the analysis result. The main assumptions made to simplify the theoretical analysis are as follows:
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The subject of the analysis is a four-wheel-drive tractor; hence, the weight of the front axle and front wheels cannot be ignored.
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The slope angle increases quasi-statically (ignoring the dynamic effect), and the tire makes point contact with the ground.
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The track widths of the front wheels and rear wheels are the same.
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Slip does not occur between the tire and the ground, and the tire stiffness is ignored.
Figure 3a illustrates a simplified version of the tractor. In the static falling down sidelong angle experiment, the tractor is in a stationary state. Accordingly, the force exerted in the tractor’s moving direction (
) is zero. The free-body diagram of the tractor is illustrated in
Figure 3b.
When the tractor is placed in the direction of the contour of the slope, the ground contact forces of the rear and front wheels located in the upper part of the slope are expressed by Equations (1) and (2), respectively [
10].
Here,
= Rear upslope tire contact force
= Mass of a component body of the tractor at rear (kg)
= Gravitational acceleration (m/s2)
= Track width (m)
= Length of wheel base (m)
= Dimension from rear axle to rear center of gravity (m)
= Height of front-axle pivot (m)
= Slope angle
= Downslope displacement of rear body center of gravity with respect to midplane (m)
= Height of rear body center of gravity above height of pivot (m)
Here,
= Front upslope tire contact force
= Mass of a component body of the tractor at front (kg)
= Downslope displacement of front body center of gravity with respect to mid plane (m)
= Height of front body center of gravity above the height of pivot (m)
When the inclination angle of the slope increases, either the front or rear wheel located in the upper part of the slope is lifted off the ground first, causing a Phase I rollover [
8]. Afterward, the other wheel is lifted off the ground if the inclination angle continues to increase, and the tractor quickly overturns sideways. The moment the wheel begins to lift off the ground, the ground contact force of the wheel becomes zero.
If
is the inclination angle at which the rear wheel starts to lift off the ground, it is the inclination angle when
is zero; hence,
can be derived from Equation (1) as expressed below.
Here,
= Instability angle when the upslope rear wheel is lifted
If the length, width, and height of the tractor are reduced by a similar ratio of
while maintaining the shape of the tractor, the variables included in Equation (3) are also reduced by a similar ratio of
. In the miniature model reduced to 1/
n scale, the angle at which the rear wheel starts to lift off the ground is expressed by Equation (4). Here,
,
, and
are variables related to the size of the tractor body, and
,
, and
are variables related to the position of the center of gravity. When the size of the tractor is reduced, the position of the center of gravity is also reduced by a similar ratio (1/
n).
Here,
= Instability angle of the upslope rear wheel for the 1/n scale-down model
The following result can be obtained by eliminating the constant terms of 1/
n2 in the denominator and numerator from Equation (4). Therefore, if the length, width, and height of the tractor are reduced by the same ratio while maintaining the overall shape of the tractor, the angle at which the rear wheel in the upper part of the slope starts to lift off the ground for the miniature model is equal to that of the original tractor.
In a similar manner, if
is the inclination angle at which the front wheel starts to lift off the ground, it is the inclination angle when
is zero; hence,
can be derived from Equation (2), as expressed below.
Here,
= Instability angle when the upslope front wheel is lifted
In Equation (6),
and
are variables related to the weight of the tractor, and
and
are variables related to the position of the center of gravity. If the length, width, and height of the tractor are reduced by a similar ratio of 1/
n while maintaining the materials and shape of the tractor, the weight is reduced by the ratio of 1/
n3, whereas the position of the center of gravity is reduced by the ratio of 1/
n. Therefore, the angle at which the front wheel starts to lift off the ground is expressed by Equation (7) for the miniature model reduced to 1/n scale.
Here,
= Instability angle of upslope front wheel for 1/n scale-down model
The following result can be obtained by eliminating the constant terms of 1/
n5 in the denominator and numerator from Equation (7). Therefore, if the length, width, and height of the tractor are reduced by the same ratio while maintaining the materials and overall shape of the tractor, the angle at which the front wheel in the upper part of the slope starts to lift off the ground for the miniature model is equal to that of the original tractor.
Therefore, for the miniature model in which the length, width, and height are reduced by the same ratio while maintaining the materials and overall shape of the original tractor, the angle at which the front and rear wheels in the upper part of the slope starts to lift off the ground is equal to that of the original tractor. If both the front and rear wheels lift off the ground, lateral overturning occurs quickly; hence, the static falling down sidelong angles of the original tractor and the miniature model can be assumed to be the same.