PSI - Issue 44

Gaspar Auad et al. / Procedia Structural Integrity 44 (2023) 1474–1481 Gaspar Auad et al. / Structural Integrity Procedia 00 (2022) 000–000 nodal motion of the structure and the nodal motion of the device = ? (0) ; (1) A (being (0) = ? " (0) $ (0) ' (0) " (0) $ (0) ' (0) A / and (1) = ? " (1) $ (1) ' (1) " (1) $ (1) ' (1) A / ), are linearly related: = (4) in which the symbol represents the kinematic transformation matrix. If small rotations in nodes are assumed, and infinitesimal terms of order higher than one are neglected, the displacement and rate of displacement of the device can be computed as: = M ( ) (5) ̇ = 2 2 5 5 6 = O ( ) ̇ (6) being: M( ) = P 0 0 1 0 1 0 1 0 0 0 − , 0 , 0 0 0 0 0 −1 0 0 0 −1 0 0 0 −1 0 −( ' + + ) $ ' + + 0 − " − $ " 0 R (7) " ( ) = ⎣ ⎢ ⎢ ⎡ 1 ! (#) % (#) − ! (#) 1 & (#) % (#) & (#) 1 0 − ' 0 ' 0 0 0 0 0 −1 − ! (#) % (#) ! (#) −1 − & (#) − % (#) & (#) −1 0 −( ! + ( ) % ! + ( 0 − & − % & 0 ⎦ ⎥ ⎥ ⎤ (8) in which, + = D + (0) − + (1) E represents the relative displacements between nodes J and I along the i-direction. 2.3. Action-deformation The total force transmitted by a frictional isolator can be determine as the sum of the pendular force (7) , the frictional force (8) and the impact force (+) (i.e., = (7) + (8) + (+) ). The pendular force developed in a variable curvature isolator with an ellipse-based sliding surface depends on the magnitude of the normal force generated between the slider and the sliding surface and the vector that describes its direction V : (7) = V( ) = 9 : " $ #! % # " $ &! % # (" ( * ) % *) ! W ; # % ! , ; & % ! , (; ( ) ()) ! X / (9) The frictional force (8) can be determined using the unitary vector in the tangential direction of sliding Z , the friction coefficient 5 , the state vector of the Park-Wen model (Park et al. 1986), its modulus ̅ = ‖ ‖ , and the angle α between (8) and the local xy-plane: (8) = ̅ 5 Z = ̅ 5 W < # ‖ ‖ ( ) , < & ‖ ‖ ( ), ( )X ; = i @ ‖ , # ‖ " $ #! # ‖ , & ‖ " $ & ! A *) * ! " ( j (10) 1477 4