Stability of chains of oscillators with negative stiffness normal, shear and rotational springs

We consider chains of particles with fixed ends in planar movement such that each particle has three degrees of freedom: two translational and one rotational. The particles are connected by one normal (longitudinal), one shear (transverse) and one rotational spring with a possibility that stiffness of some springs can be negative. We showed that the necessary condition of stability is that such a chain is allowed no more than three negative stiffness springs. The allowable negative stiffness springs can be arranged in two ways: (i) the chain contains one normal, one shear and one rotational negative stiffness springs or; (ii) the chain contains one normal and two rotational negative stiffness springs. The absolute values of the negative stiffnesses should not exceed certain threshold values that depend upon the stiffnesses and the number of the other (positive) springs. The positions of normal and shear negative stiffness springs do not affect the critical values, however the positions of the negative rotational stiffness springs are found to be important. The modal frequencies reduce when one of the negative stiffnesses tends to the critical value; the smallest frequency tends to zero. Damped chains also exhibit similar decrease of damping frequencies, but the lowest frequency tends to zero while the chain is still stable. At this point the damping bifurcates and produces two brunches: one increases with the increase in the value of negative stiffness, the other decreases. No giant damping is observed.