PSI - Issue 29

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Michela Monaco et al. / Structural Integrity Procedia 00 (2019) 000 – 000

Michela Monaco et al. / Procedia Structural Integrity 29 (2020) 134–141

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′ ( ) = ∘ ( )r (1) , ( ) < 0

(4) ( ) = ∘ ( )r (2) , ( ) > 0 where ∘ ( ) is the rotationmatrix (13) that takes into account the rockingmotionof block2 and the position vectors r (1) and r (2) are represented in Figure 5. The total kinetic energy of the system is: ( ) = 1 ( )+ 2 ( ) (5) with 1 ( ) and 2 ( ) the kinetic energies of pedestal and statue: 1 ( ) = 1 2 1 ẋ 1 ⋅ ẋ 1 , 2 ( ) = 1 2 [ 2 ̇ 2 ( ) + 2 ẋ 2 ′ ( ) ⋅ ẋ 2 ′ ( )], ( ) < 0 (6) 1 ( ) = 1 2 1 ẋ 1 ⋅ ẋ 1 , 2 ( ) = 1 2 [ 2 ̇ 2 ( ) + 2 ẋ 2 ( ) ⋅ ẋ 2 ( )], ( ) > 0 being 2 the centroidmoment of inertia of the statue. The potential energy of the system is given by: ( ) = 1 ( ) + 2 ( ) with 1 ( ) and 2 ( ) the potential energies of pedestal and statue: 1 ( ) = 1 x 1 ⋅ j, 2 ( ) = 2 x 2 ′ ⋅ j, ( ) < 0 (7) 1 ( ) = 1 x 1 ⋅ j, 2 ( ) = 2 x 2 ⋅ j, ( ) > 0 being j the unit vector of y axis. The friction forcea t the base of pedestal during the slidingmovement is given by: , = − ( + ̈ ( )) ̈ ( ) ( ̇ ( )) (8) so tha t the Lagrangian formulation of the problemstates: ( ) = ( ) − ( ) (9) The motion is governedby two differential equations derivedby the Euler-Lagrange relation: ∂ 2 ∂ ∂ ( ̇ ) − ∂ ∂ ( ) = ( ), = 1,2 (10) Where 1 ( ) = 1 ( ), 2 ( ) = ( ) are the two Lagrangian parameters and ( ) is the genera lized non conservativeforce dual to ( ) . The systemassumes twodifferent expressionsaccording to thesign of ( ) . In view of equations (6)-(10), the DAEs canbe expressedas: { ̈( ) − 2 [ − | |]( ̈ ( )+ ̈ 1 ( )) + 2 ( ( )) [ − | |] = 0, ( ) ≠ 0 ( ̈ ( ) + ̈ 1 ( )) + ( ( )){− 2 [ ( − | |) ̇ 2 ( )− ( − | |) ̈( )] + } = 0, ( ) ≠ 0 ̇ + ( ) = r ̇ − ( ), ( ) = 0 The motion problem (11) is composed by two ordinary nonlinear differential equations and a single a lgebraic one, tha t involves thepre-and-post-impact angular velocityof the statueduring therockingmotion. Uncouplingof the two differential equations is not possible. The DAEs (11) govern the motions of the two blocks, while the rockingof the top block when the flat block is a t rest with respect to the ground is governed by the following rela tion: ̈( )− 2 ( − | |) ̈ ( ) + 2 ( ( )) ( − | |) = 0 . (12) derived from (11 1 ) with the condition 1 ( ) = 1 ( ) = 0 . Equation (11 2 ) describes the sliding motion of the two blocks:

(11)