Embedding negative structures to model holes and cut-outs

It has now been established that geometric boundary conditions and continuity conditions can be modelled by using either positive or negative stiffness or inertia type penalty term [1-5]. The experience of working with negative stiffness and inertial parameters has led to the question: what if both stiffness and mass were to be taken as negative? Changing the sign of all stiffness and inertial terms of a structure is simply equivalent to multiplying both sides of an eigenvalue equation by minus one, which does not change its frequencies or modes. Basically, a negative structure in such a sense has the same vibratory properties as that of its positive counterpart, although the structure itself may not have a physical meaning. However, an interesting question emerges about the behaviour of a structure formed by attaching such a negative structure to a larger positive structure: would it be possible to effect a hole or a cut-out in a plate by embedding a “negative plate ” to a plate without the hole? The idea is that the negative plate to be attached must have the same shape as that of the hole and the same magnitude of stiffness and mass distribution but with opposite sign (see Figure 1). The negative plate unit would be bonded to the larger (uncut) plate by using distributed penalty stiffness over the area of bonding to prevent any relative motion between the negative part and the positive part. To see if this concept works, a cantilever beam of length L, flexural rigidity EI, and mass per unit length m was made into a free-free beam of length L/2, by attaching another cantilever beam of length L/2, flexural rigidity –EI and mass per unit length –m. The two beams were connected by means of a uniform distribution of elastic springs of high stiffness k per uni