We study the problem of optimizing the performance of a nonlinear spring-mass-damper attached to a class of multiple-degree-of-freedom systems. We aim to maximize the rate of one-way energy transfer from primary system to the attachment, and focus on impulsive excitation of a two-degreeof-freedom primary system with an essentially nonlinear attachment. The nonlinear attachment is shown to be able to perform as a 'nonlinear energy sink' (NES) by taking away energy from the primary system irreversibly for some types of impulsive excitations. Using perturbation analysis and exploiting separation of time scales, we perform dimensionality reduction of this strongly nonlinear system. Our analysis shows that efficient energy transfer to nonlinear attachment in this system occurs for initial conditions close to homoclinic orbit of the slow time-scale undamped system, a phenomenon that has been previously observed for the case of single-degree-of-freedom primary systems. Analytical formulae for optimal parameters for given impulsive excitation input are derived. Generalization of this framework to systems with arbitrary number of degrees-offreedom of the primary system is also discussed. The performance of both linear and nonlinear optimally tuned attachments is compared. While NES performance is sensitive to magnitude of the initial impulse, our results show that NES performance is more robust than linear tuned-massdamper to several parametric perturbations. Hence, our work provides evidence that homoclinic orbits of the underlying Hamiltonian system play a crucial role in efficient nonlinear energy transfers, even in high dimensional systems, and gives new insight into robustness of systems with essential nonlinearity.
With advances in technology, hyperelastic materials are seeing use in varied applications ranging from microfluidic pumps, artificial muscles to deformable robots. Development of such complex devices is leading to increased use of hyperelastic materials 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 made of hyperelastic polymers. Since the strain energy potentials of various hyperelastic material models have nonlinearities present in them, 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 Mooney–Rivlin material's strain energy potential has quadratic nonlinearities, 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 1:2 ratio. Once a topology of the structure is achieved, the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete nonlinear Lagrangian formulation of the hyperelastic structure can be used to develop a nonlinear two-mode dynamic 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 nonlinear strain energy potential for the Mooney–Rivlin material makes it possible for internal resonance to occur in such structures. The effect of nonlinear material parameters on the dynamic response is investigated.
Nonlinear phenomena such as internal resonances have significant potential applications in micro electro mechanical systems (MEMS) for increasing the sensitivity of biological and chemical sensors and signal processing elements in circuits. While several theoreti-cal systems are known which exhibit 1:2 or 1:3 internal resonances, designing systems that have the desired properties required for internal resonance and that are physically realizable as MEMS devices is a significant challenge. Traditionally, the design process for obtaining resonant structures exhibiting an internal resonance has relied heavily on the designer’s prior knowledge and experience. However, with advances in computing power and topology optimization techniques, it should be possible to synthesize structures with the required nonlinear properties (such as having modal interactions) computation-ally. In this work, a preliminary work on computer based synthesis of structures consist-ing of beams for desired internal resonance is presented. The linear structural design is accomplished by a Finite Element Method (FEM) formulation implemented in MATLAB to start with a base structure and iteratively modify it to obtain a structure with the desired properties. Possible design criteria are having the first two natural frequencies of the structure in some required ratio (such as 1:2 or 1:3). Once a topology of the structure is achieved that meets the desired criterion, the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the nonlinear elastic structure can be used to develop a nonlinear two-mode model of the structure. The reduced-order model is expected to capture the appropriate resonant dynamics associated with modal interactions between the two modes. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. Many candidate structures are synthesized that meet the desired modal frequency criterion and their nonlinear responses are compared. [DOI: 10.1115/1.4024845
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