Exact and nonlocal solutions for vibration of multiply connected bar-chain system with direct and indirect neighbouring interactions

This paper is concerned with the free vibration problem of multiply connected bar-chain (MCB) system with direct and indirect neighbouring interactions. MCB is a line of rigid bars connected by multiple elastic rotational springs. The so-called direct or indirect neighbouring interactions are realized by connections between two adjacent bars or two distant bars by the rotational springs. The free vibration frequencies of two-neighbouring interacted and p-neighbouring interacted MCB are analytically obtained for simply supported end restraints. A nonlocal continuum model is built based on this lattice model by continualizing the discrete equations. This continualized nonlocal model (CNM) is found to be equivalent to the Eringen's stress gradient nonlocal model. The expression of the length scale of CNM is figured out and it is found to be dependent on the rotational spring stiffnesses of MCB. Exact nonlocal vibration frequencies are predicted with the calibrated length scale. The comparison between CNM and MCB shows that the scale effect of a generalized lattice system including multiple interactions could be captured by the simple nonlocal continuum model.