PSI - Issue 37

Rogério Lopes et al. / Procedia Structural Integrity 37 (2022) 115–122 R. F. Lopes et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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transducers and load cells (Melo et al., 2006). The algorithm comprises a set of equations that allow the calculation of the velocity and acceleration vectors of the structure in test, so defining its updated configuration (De Melo, Carneiro, Lopes, Dias Rodrigues, & Gomes, 2001). The test being presented next, is a dynamic model approach having only one degree of freedom, showing the condensation of the structural dynamic behavior to a point of contact between the structure and the equivalent mass that represents the impactor, see in Fig. 1. The stiffness of the structure K is calculated from the relationship between the applied force F (t) and the consequent displacement at the contact point.

Fig. 1. Equivalent dynamic structure system subject to a single load.

This pseudo-dynamic problem approaches the event of a rigid solid body striking another body, where this last one is deformable. During the time increments, the initial speed of the rigid solid decreases and the deformation of the deformable solid increases at the point of contact. The numerical formulation to characterize this problem is based on the Theorem of Quantity of Motion or Momentum Theorem, due to Newton. The developed procedure can be synthetized as follows: At an instant , Momentum Theorem sets that the Force Impulse is equal to the variation of the quantity of motion of the impacted structure × ∆ = × ∆ , (1) Obtain the speed decrement (in fact, the striker speed must be reduced each time step): ∆ = ( × ∆ )/ , (2) Now update the speed for the new time step: +∆ = − ∆ , (3) Now calculate the displacement, from an average speed between current and updated values: +∆ = + 0.5 ∙ ∆ ( +∆ + ) , (4) (note: the displacement starts with U = 0 and t = 0, when the rigid mass starts contacting the deformable structure at an initial speed ). This displacement to the structure is prescribed and + ∆ is obtained (it does not need to be necessarily linear and elastic, it may result of the structure “ response ” to the prescribed displacement effort. The answer will be read in the load cell. Now + is identified as being : = +∆ , (5) The new speed to start the cycle in the equation (1) is identified: = +∆ (6) Proceed to step (1) to obtain ∆ and continue until the final speed +∆ = 0 . The total displacement obtained in step (4) is that which the structure presents by the deformation due to the impact. At any time in the program it is possible to register, depending on the time, the speed, the internal reaction and even register the internal force as a function of the displacement.