PSI - Issue 75

Robert Goraj et al. / Procedia Structural Integrity 75 (2025) 691–708 Goraj / StructuralIntegrity Procedia (2025)

704 14

is a periodic function. The period equals 2T, where T is restricted according to: = 2 , ∈ +

(39)

Using Laplace transform one obtains: Noticing, that:

( ) = ∫ − ( ) ∞ 0

= 1 − 1 −2 ∫ − ( ) 2 0

(40)

⋁ ( ) ∈< ,2 > =0 0 ⏟ 1 + 1 = 1− −

(41)

It follows:

( ) = 1 − 1 −2 ( ℎ ∫ −

∫ − ( ) 0 ⏟ 2 ) −∫ (− −1 ⏟ ( )) 0

(42)

For the integral I 1 in (42) one obtains:

(43)

For the integral I 2 in (42) an integration by parts in applied: 2 =∫ − ⏟ ⏟ ( ) 0 =[ − ⏟ (− −1 ⏟ ( )) ] 0 Evaluating (44) results in:

(− )

− ⏟

(44)

2 = 1− − − ∫ − ( ) 0 ⏟ 2

(45)

For the integral I 2a in (45) an integration by parts in applied: 2 =∫ − ⏟ ⏟ ( ) 0 = [ − ⏟ −1 ⏟ ( ) ] 0 The integral (46) simplifies to: Inserting (47) into (45) results in: 2 = 2 + 2 (1− − ) Inserting (48) and (43) into (42) results in Noticing that: 2 = 0 + ∫ ( ) − 0

−∫ −1 ⏟ ( ) 0

(− ) − ⏟

(46)

(47)

(48)

( ) = ( ℎ 1 + 2 + 2 ) 1− − 1− −2 1− − 1− −2 = 1− − (1+ − )(1− − ) = 1+ 1 −

(49)

(50)

Inserting (50) into (49) results in the transfer function the propeller excitation (cf. (3)): ( ) = ( ℎ 1 + 2 + 2 ) 1+ 1 −

(51)