PSI - Issue 75

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

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Fig. 3. Coupled damped oscillators - graphical representation of the mathematical model

The number of bearing spokes n 1 and is the number of mounting pads n 2 are design parameters. The current work considers a case in which: n 1 = n 2 = 4. The stiffness k 1 is formulated as a parametric function of the spoke second moment of area about the y -axis: I y . Approximating the spoke deflection x 1 – x 2 witha simply supported cantilever beam of the length L one obtains [33], [34]: 1 ( ) = ( 1 − 2 ) = ( 2 3 + 2(1 + ) ) −1 (4) The quantity E in (4)is the Young’s modulus , ν is the Poisson’s ratio. The parameter A is the cross-section area of the H-beam: = + ( − )ℎ (5) where a, b and H are constants indicating the web thickness, the width, and the depth of the H-profile accordingly (see Fig. 1). The parameter h in (5)is a distance between the fillets. It is constructed in as a function of I y : ℎ( )=( 12 − 3 − ) 1/3 (6) The quantity κ s in (4) is the Timoshenko corrective shear coefficient of a H-profile: = 10(1 + )(1 + 3 ) 2 (12 + 72 + 150 2 +90 3 ) + (11 + 66 + 135 2 +90 3 )+30 2 ( + 2 ) + 5 2 (8 + 9 2 ) (7) where: = 2 −+ ℎℎ (8) = 2 +ℎ (9) A lumped mass approach of the considered mechanical components leads to a description of the system behavior, here expressed using a set of differential equations: 1 ̈ 1 + 1 ( ̇ 1 − ̇ 2 )+ 1 ( 1 − 2 )= ( ) 2 ̈ 2 + 2 ̇ 2 + 2 2 − 1 ̇ 1 − 1 1 + 1 ̇ 2 + 1 2 =0 (10) Equations (10) can be shorten after the Laplace transform to: 1 1 = + 1 2 ( 2 + 1 ) 2 = 1 1 (11) where: x 1 (t), x 2 (t) ⊶ X 1 (s), X 2 (s) and P 1 , P 2 , Q 1 are the functions of a complex number s according to: