Studies on the Experimental Validation of the Theoretical …
177
The origin of the tractor reference system is at the centre of the parallelepiped that
modelling the motor block, with the dimensions of 1.0 × 0.45 × 0.87 m.
2.2 Creating the Mathematical Model
To create the noise level simulation tool in the vicinity of the tractor, we starting from
a simple formula suggested and used by [11] and also applied in [1]. The start formula
contains three parameters, the power of the source, L w and the model parameters,
a, b:
L eq (ξ 1 , ξ 2 , ξ 3 , x 1 , x 2 , x 3 , L w , a, b)
= L w − a ln
(ξ 1 − x 1 )
2
+ (ξ 2 − x 2 )
2
+ (ξ 3 − x 3 )
2
S
− b
(1)
where (ξ 1 , ξ 2 , ξ 3 ) is a point where a source (point source) is located, (x 1 , x 2 , x 3 ) is
the current point in the three-dimensional space, L eq is the noise intensity level, and
S is the external surface of the source (thus, the argument of the logarithmic function
becomes dimensionless and there are no dimensional problems). The introducing
of the constant S was made to satisfy the requirements expressed in [12] and [13].
Therefore, with respect to reference models, based on a point source, this model
uses a spatially distributed source on the surface of a three-dimensional body. By
hypothesis, we assume that the considered noise source is evenly distributed on a
parallelepiped that includes, at the limit, the tractor block. We consider the origin
of the Cartesian axis system at the centre of this parallelepiped and the dimensions
of the parallelepiped given after the measurement, −X min = X max =
L
2
, −Y min =
Y max =
l
2
, −Z min = Z max =
h
2
, where L = 1.0 m, l = 0.45 m, h = 0.87 m.
Taking into account the above considerations, the total noise level results on each
plane portion that is a face of the parallelepiped by summing, for example, the face
x 3 = Z max :
L eqZ max (x 1 , x 2 , x 3 , L w , a, b)
= 10 · log
⎛
⎝
Y max
Y min
X max
X min
10
Leq (ξ1,ξ2,Zmax,x1,x2,x3,Lw,a,b)
10
dξ 1 dξ 2
⎞
⎠
(2)
Similarly, the level of noise generated by distributed sources is uniformly
distributed over the other five faces of the parallelepiped. The noise level of the
source distributed on the six faces of the parallelepiped has the expression:
177
The origin of the tractor reference system is at the centre of the parallelepiped that
modelling the motor block, with the dimensions of 1.0 × 0.45 × 0.87 m.
2.2 Creating the Mathematical Model
To create the noise level simulation tool in the vicinity of the tractor, we starting from
a simple formula suggested and used by [11] and also applied in [1]. The start formula
contains three parameters, the power of the source, L w and the model parameters,
a, b:
L eq (ξ 1 , ξ 2 , ξ 3 , x 1 , x 2 , x 3 , L w , a, b)
= L w − a ln
(ξ 1 − x 1 )
2
+ (ξ 2 − x 2 )
2
+ (ξ 3 − x 3 )
2
S
− b
(1)
where (ξ 1 , ξ 2 , ξ 3 ) is a point where a source (point source) is located, (x 1 , x 2 , x 3 ) is
the current point in the three-dimensional space, L eq is the noise intensity level, and
S is the external surface of the source (thus, the argument of the logarithmic function
becomes dimensionless and there are no dimensional problems). The introducing
of the constant S was made to satisfy the requirements expressed in [12] and [13].
Therefore, with respect to reference models, based on a point source, this model
uses a spatially distributed source on the surface of a three-dimensional body. By
hypothesis, we assume that the considered noise source is evenly distributed on a
parallelepiped that includes, at the limit, the tractor block. We consider the origin
of the Cartesian axis system at the centre of this parallelepiped and the dimensions
of the parallelepiped given after the measurement, −X min = X max =
L
2
, −Y min =
Y max =
l
2
, −Z min = Z max =
h
2
, where L = 1.0 m, l = 0.45 m, h = 0.87 m.
Taking into account the above considerations, the total noise level results on each
plane portion that is a face of the parallelepiped by summing, for example, the face
x 3 = Z max :
L eqZ max (x 1 , x 2 , x 3 , L w , a, b)
= 10 · log
⎛
⎝
Y max
Y min
X max
X min
10
Leq (ξ1,ξ2,Zmax,x1,x2,x3,Lw,a,b)
10
dξ 1 dξ 2
⎞
⎠
(2)
Similarly, the level of noise generated by distributed sources is uniformly
distributed over the other five faces of the parallelepiped. The noise level of the
source distributed on the six faces of the parallelepiped has the expression: