MS12.5: Nonlinear Dynamics for Engineering Design

Linear and geometrically nonlinear vibrations of functionally graded sandwich porous shallow shells and plates are studied using the R-functions theory and methods by Ritz, Bubnov-Galerkin and Runge-Kutta. The first order shear deformation theory (FSDT) is applied to construct mathematical model. The power- and sigmoid law models are employed to determine the effective material properties of the functionally graded materials made of mixture of metal and ceramics. Effect of the boundary conditions, shape of the plan, type of constituent materials, the layup of layers and porosity on behaviour of linear and nonlinear frequencies is analysed. Analytical expressions for elements of the matrices, which determine force resultant in-plane, bending and twisting moments resultant and transverse shear force resultants are obtained. New results for plates and shallow shells with a complex shape including cut-out are reported.

Artificial muscles are dielectric elastomeric system that mimic the response of natural muscle and are helpful in many bio-mimicking applications. The existing literature are based on the assumption that electrodes are compliant and are focused on materials with greater extensibility than biological matter. The present work deals with the modeling and nonlinear dynamic analysis of dielectric elastomers considering the elastomer-electrode inertia and damping for bio-mimicking. The distinguishing point is the appearance of terms related to the elastomer-electrode inertia in the governing equation. Equilibrium stretch variation exhibits cusp catastrophe and effect of electrode inertia on equilibria is examined. Single-well and double-well potential energy characteristics are obtained and equilibria's sensitiveness to initial condition and damping is illustrated by basin of attraction. Multi-frequency response exists and resonant frequency variation with geometry and inertia is presented. Time-varying voltage may lead to chaotic behavior illustrated through the bifurcation diagram and chaos is justified by the presence of strange attractor and positive Lyapunov exponent. Additionally, chaos is obtained in the case of single-well and double-well potential. The existence of similar attractors with same fractal dimension is shown. The inertia significantly changes the dynamic characteristic from chaotic to non-chaotic or vice-versa. The mechanical and electrical load may lead to chaotic or non-chaotic behavior depending on the geometry of the elastomer. Our results show that electrode inertia significantly impacts the system under quasi-static and dynamic conditions. The proposed work gives a foundation for precisely designing artificial muscle for practical applications like soft robotics, artificial arms, and prosthetic implants.

In contrast to the past, when nonlinear phenomena arising in structures of practical interest were seen as almost pathological, today very often slender and highly flexible structures are used in engineering design to take advantage of the beneficial effects of nonlinear dynamics. Needless to say, a key point in correctly describing the dynamics of such structures is the appropriate writing of the relevant equations of motion. Although such equations can be written following different approaches, it is desirable that whatever derivation strategy is chosen, the equations are the same. Restricting attention to the nonlinear dynamics of deformable beams satisfying the Euler-Bernoulli assumptions, and with a generic initial configuration, the question that arises is: What conditions must the constitutive assumptions about generalized forces satisfy in order for the equations of motion obtained by the Newton approach to be identical to the Euler-Lagrange equations derived from an appropriate Lagrangian, whether natural or virtual? The goal of this work is to try to answer this basic question.

Metamaterials are recognized as artificially microstructured media, exhibit distinct effective material properties, and can be tailored to achieve negative properties that inhibit the propagation of acoustic or elastic waves. Traditional linear metamaterials (LMs), relying on resonance mechanisms, are constrained by a narrow frequency band. In response to this, nonlinear metamaterials (NLMs) emerge as a solution, offering expanded bandwidth and advanced wave phenomena, surpassing the limitation of linear systems. This paper introduces an innovative approach for achieving an ultra-broad bandgap in NLMs, using a chain of resonator triplets, which exploits a novel combination of hardening and softening nonlinearities.