PSI - Issue 29

Michela Monaco et al. / Procedia Structural Integrity 29 (2020) 134–141 Michela Monaco et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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The analytical problem examined in this paper is the dynamic behaviour of two stacked rigid bodies. The two rigid bodies can represent respectively the statue and its pedestal. The case study is a marble statue in the Archeological Museumof Paestum. It is shown that an optimal position of theartifact canbe assessed. 2. The general problem Here in the following the configuration considered for the rocking problem of museum artefacts is presented. A human body-like shaped rigid block with rectangular base, restingon a horizontal basement, this last one slidingon the ground plane with uniformly distributedmass (centre of gravityG). The dynamic behaviour of two stacked rigid bodies subjected to sinusoidal ground acceleration is considered: the slender artifact and thesquat rigid base inserted between the floor and the artifact. The analytical model combines two elementary motions: the rocking of a single slender rigid block and the slidingof a single squa t block on a movingbase. The geometrical characteristics and the friction coefficients (Gesualdo et a l., 2018a) of the sta tue-pedestal problem are reported in Figure 1, while more information on the problem can be desumed by Gesualdo et a l. (2018c). The system has two degrees of freedom, namely therotation (clockwise positive) of block2 (statue) and the centroid position 1 of block 1 (pedestal). ( ) P t P ( ) P t P

(1)

(1)

P r

P r

y

y

2 G

2 G

2 ( ) G t

2 ( ) G t

1 R

1 R

2 R

2 R

y 

y 

y y 

y y 

y

y

 

 

1 ( )  t

1 ( )  t

x 

x 

O 

O 

x

x

O O

O O

1 G

1 G

( ) g y t &&

( ) g y t &&

( ) g x t &&

( ) g x t &&

Figure 1: Double blocks at rest position (left) and with upper block in rocking mode (right). Let the base acceleration be ̈ ( ) , 1 and 2 the masses of the two blocks whose center of masses are 1 and 2 , = 1 + 2 is the tota l mass of the system and ( ( ), ( )) are the components of the base motion. The position vectors of 2 rela tive to ′ and O in the initia l configurationare r ' 2 and r 2 . Their components in the two Cartesian reference systems ℛ 1 and ℛ 2 arepicted in Figure 3 and are given by: r ' 2 = [ ] , r 2 = [ − ] (1) The hypothesis of slidingmotion for the statue lead to null vertical component of its rela tive motion with respect to the base. The position of the center of mass 1 : x 1 ( ) = [ ( ) + x 1 ( ) ( ) ] (2) and that of 2 is: x 2 ′ ( ) = x 1 ( ) + ∘ ( )r ' 2 , ( ) < 0 (3) x 2 ( ) = x 1 ( ) + ∘ ( )r 2 , ( ) > 0 so tha t the actual positionof the keypoint is: