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V = f (D, d b , d i , d o , d m , B, L , ρ, E S , E R , W s , W r , I s , I r , c, δ, T, ,, f s )
(1)
The vibration signals are acquired from the system to study effect of speed, clearance, and unbalance mass. All the above variables considered for formulation and
dimensionless products are given in Eq. (2)
f (π 1 , π 2 , π 3 , . . . , π m ) = 0
( 2 )
In the above mathematical formulation, the total number variables are equal to
26 out of which three variables are not forming dimensionless product; therefore, to
formulate the complete set, 23 − 3 = 20 parameters required.
3.1 System Modeling
The relationship shown in Eq. (1) is modeled with the dimensional analysis; using
dimension analysis techniques, dynamic response modeled can be written as
π 1 = V [R c ]
a [N r ]
b [W u ]
c
(3)
where a, b, and c represent constants which are obtained from fundamental units, as
given in Eq. (4)
= LT
−2 [L]
a
T
−1
b
F L
−1 T
2
c
(4)
The balancing of the fundamental units to find constants a, b, and c, may be done
as,
F
0 L
0 T
0
= F
c L
−1 + a − c T
2 + b + 2c
(5)
The equations may be written as follows
c = 0, 1 + a − c = 0, 2 + b + 2c = 0
Solving equations, values of a, b and c are
a = −1, b = −2, c = 0
Hence, the first dimensionless (π 1 ) group is
π 1 = V [R c ]
1 [N r ]
−2 [W u ]
0