PSI - Issue 71

Faisal Hussain et al. / Procedia Structural Integrity 71 (2025) 248–255

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With the exception of the irregularities in the stiffening factors, the modeling implies that both the B and C subsystems behave linearly in their FRFs. Test-derived FRF information with corresponding linked parameters are used to depict the interaction between modules. Although the Harmonic Balance Method (HBM) is used on the presumption of continuous overall reaction, minor deviations have been taken into account by considering substance features as uniform and isotropic, focusing on stiffness-dominated behaviour. Higher-order chaotic is neglected and connectivity factors were considered for the sake of analytical simplicity.

(a)

(b) Fig.1. (a) Cantilever beam bolted at one end; (b) Sub-system B and C

The B Subsystem Matrix is shown as a diagonal matrix [ ] [ ] = [ β 11 β 12 β 21 β 22 ]= [ 1 1 0 0 1 2 ∗ ]

(1)

Where, β 11 , β 22 are direct receptance for subsystem B and β 12 , β 21 are cross receptance for subsystem B. Subsystem C [ γ ] FRF Matrix is represented as γ 11 = ( − 3 )( 1 λ 3 )( 5 3 )

γ 22 = ( )( 1 γ 12 = ( 2 )( 1

λ )( 6 3 ) λ 2 )( 1 3 )=

(2) (3)

γ 21

(4)

The frequency [non-dimensional – λ] is given as, λ = [ ω 2 ρAL 4 EI ] 1⁄4 The nonlinear K 2 * is defined as, 2 ∗ = 2 + 4

(5) (6)

Using HBM, we get, K 2 From Eq. (6) and Eq. (7), we get, 4 ≅ 3 4 4 θ θ 2 θ( t)+ K 4 θ 3 (t)=F(t)

(7)

(8) The equation of frequency for the system is derived by equating receptance variables of the denominator ( ∆) as zero. ∆=( β 11 + γ 11 )( β 22 + γ 22 )−( β 12 + γ 12 ) 2 =0 (9) Here, β 11 = 1 1 ∗ , β 22 = 1 2 ∗ and β 12 =0 ( 1 1 ∗ + γ 11 )( 1 2 ∗ + γ 22 )−( γ 12 ) 2 =0 ( 1+ γ 11 1 ∗ 1 ∗ )( 1+ γ 22 2 ∗ 2 ∗ )−( γ 12 ) 2 =0 (1+ γ 11 1 ∗ )(1+ γ 22 2 ∗ )−( γ 12 ) 2 1 ∗ 2 ∗ =0 (1+ γ 22 2 ∗ )+ γ 11 1 ∗ + γ 11 γ 22 1 ∗ 2 ∗ −( γ 12 ) 2 1 ∗ 2 ∗ =0 [ γ 11 γ 22 −( γ 12 ) 2 ] 1 ∗ 2 ∗ + γ 11 1 ∗ + γ 22 2 ∗ +1=0 [ γ 11 γ 22 −( γ 12 ) 2 ] ( 1 + 3 )( 2 + 4 )+ γ 11 ( 1 + 3 )+ γ 22 ( 2 + 4 )+1=0 (10)