Generalized Classical Ritz Method and Its Applications for Design Optimizations of Smart Structures

Abstract

In the field of structural dynamics, the classical Ritz method (CRM) is well known for its computational efficiency and has been used for modeling numerous types of structures. However, applications of the method have been limited to structures with simple configurations as the method specializes in modeling single-body structures. To overcome this limitation, in this study a generalized version of the CRM is developed to conduct geometrically non-linear analyses of flexible multibody systems having general topologies. To verify the accuracy and efficiency of the proposed approach, five numerical examples were solved using the proposed approach and the conventional finite element formulation. The results obtained from the proposed method are at least as accurate as those from the conventional finite element method, and the proposed method is more efficient because it requires fewer degrees of freedom (DOFs) to achieve converged solutions. Numerical models for two smart structures—(1) auxetic structures and (2) piezoelectric energy harvesters (PEHs)—were developed as applications of the proposed generalized CRM (GCRM), and design optimizations were conducted to maximize their performance.

The first design optimization problem addresses auxetic structures. Auxetic structures with a negative Poisson’s ratio (NPR) are known for their novel mechanical properties. Recent studies have shown that these properties can be tailored using numerical simulations based on the finite element method where plane or solid elements are applied, combined with optimization techniques. However, an optimization procedure based on beam theory, which is expected to be more computationally efficient than the prevalent plane or solid element formulations, has not been developed thus far. In this study, a GCRM-based numerical model was developed for the design optimization of bowtie-shaped auxetic structures (BASs). For a systematic design process, effective stiffness, NPR, maximum stress, and volume are defined as the primary performance indices. Two design parameters—the initial shape of the centerline and the thickness profile represented by B-spline curves—are used to maximize the performance of the system. Our study confirmed that auxetic structures can be tailored to yield a minimum value of the NPR, stress, and volume without a loss of effective stiffness. 3D-printed prototypes were tested to verify the accuracy of the proposed model and the optimization of the design.

The second design optimization problem addresses vibration-based PEHs. Piezoelectric energy harvesting has the advantage of having a wider range of voltages than other energy harvesting mechanisms. Furthermore, well-established manufacturing technology facilitates the manufacture of macro- and micro-scale PEHs. However, piezoceramics (one of the most popular materials used for piezoelectric energy harvesting) have the disadvantage that the material is so brittle that the durability of the harvester cannot be guaranteed under harsh excitation conditions. Also, regardless of how much piezoelectric material is used, power generation will not occur unless the piezoelectric material is deformed. Therefore, when designing PEHs, the piezoelectric material should be placed at the point where deformation is expected to occur actively on the structure. PEHs typically have the form of cantilever beams, which have moment distributions that are inherently unsuitable for piezoelectric energy harvesting. When the free-end load is applied to the cantilevered PEH (CPEH), a relatively large moment occurs near the fixed end, but a zero moment (theoretically) occurs near the free end. The piezoelectric material attached near the free end is hardly deformed and therefore hardly participates in power generation. This phenomenon is not preferable in terms of power generation performance of the energy harvester or economy—since the piezoelectric material (the most expensive material in the PEH) is wasted.

In this paper, we propose a gamma-shaped PEH (GPEH) that realizes a uniform moment distribution by simple geometric modification of the conventional cantilevered PEH. We also developed an electromechanically-coupled GCRM for numerical analysis of the system. For a fair comparison between the proposed GPEH and CPEH, a design optimization that considered various performance indices—power, maximum stress value, piezoelectric material volume, mass of the system, and space occupied by the system—was performed. Our study confirmed that the power of GPEH is much larger than that of CPEH. Finally, prototypes for the optimized GPEH and CPEH were fabricated using 3D printing, the frequency response functions for voltage were experimentally obtained, and the results were compared with those from the numerical models to verify the accuracy of the developed numerical models.