Design for 1:2 Internal Resonances in In-Plane Vibrations of Plates With Hyperelastic Materials Available to Purchase

With advances in technology, hyperelastic materials are seeing increased use in varied applications ranging from microfluidic pumps, artificial muscles to deformable robots. They have also been proposed as materials of choice in the construction of components undergoing dynamic excitation such as the wings of a micro-unmanned aerial vehicle or the body of a serpentine robot. Since the strain energy potentials of various hyperelastic materials are more complex than quadratic, exploration of their nonlinear dynamic response lends itself to some interesting consequences. In this work, a structure made of a Mooney-Rivlin hyperelastic material and undergoing planar vibrations is considered. Since the stresses developed in a Mooney-Rivlin material are at least a quadratic function of strain, a possibility of 1:2 internal resonance is explored. A Finite Element Method (FEM) formulation implemented in Matlab is used to iteratively modify a base structure to get its first two natural frequencies close to the ratio 1:2. Once a topology of the structure is achieved, the linear modes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the hyperelastic structure can be used to develop a nonlinear two-mode model of the structure. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. It is shown that the strain energy potential for the Mooney-Rivlin material makes it possible for internal resonance to occur in such structures.