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This paper examines the effect of viscoelasticity on the periodic response of a lumped parameter viscoelastic von Mises truss. The viscoelastic system is described by a second-order equation that governs the mechanical motion coupled to a first-order equation that governs the time evolution of the viscoelastic forces. The viscoelastic force evolves at a much slower rate than the elastic oscillations in the system. This adds additional time scales and degrees of freedom to the system compared to its viscous counterparts. The focus of this study is on the system’s behavior under harmonic loading, which is expected to show both regular and chaotic dynamics for certain combinations of forcing frequency and amplitude. While the presence of chaos in this system has already been demonstrated, we shall concentrate only on the periodic solutions. The presence of the intrawell and interwell periodic oscillations is revealed using the Harmonic Balance method. The study also looks at the influence of parameter changes on the system’s behavior through bifurcation diagrams, which enable us to identify optimal system parameters for maximum energy dissipation. Lastly, we formulate an equivalent viscous system using an energy-based approach. We observe that a naive viscous model fails to capture the behavior accurately depending on the system and excitation parameters, as well as the type of excitation. This underscores the necessity to study the full-scale viscoelastic system. 

Abstract High frequency excitation (HFE) is known to induce various nontrivial effects, such as system stiffening, biasing, and the smoothing of discontinuities in dynamical systems. These effects become increasingly pertinent in multi-stable systems, where the system’s bias towards a certain equilibrium state can depend heavily on the combination of forcing parameters, leading to stability in some scenarios and instability in others. In this initial investigation, our objective is to pinpoint the specific parameter ranges in which the bistable system demonstrates typical HFE effects, both through numerical simulations and experimental observations. To accomplish this, we utilize the method of multiple scales to analyze the interplay among different time scales. The equation of slow dynamics reveals how the excitation parameters lead to a change in stability of equilibrium points. Additionally, we delineate the parameter ranges where stabilizing previously unstable equilibrium configurations is achievable. We demonstrate the typical positional biasing effect of high-frequency excitation that leads to a shift in the equilibrium points as the excitation parameter is varied. This kind of excitation can enable the active shaping of potential wells. Finally, we qualitatively validate our numerical findings through experimental testing using a simplistic model made with LEGOs. 

Abstract This paper examines the effect of viscoelasticity on the dynamic behavior of a bistable dome-shaped structure represented as a lumped parameter viscoelastic von Mises truss. The viscoelastic system is governed by a third order jerk equation. The presence of viscoelasticity also introduces additional time scales and degrees of freedom into the problem when compared to their viscous counterparts, thus making the study of these systems in the presence of harmonic loading even more interesting. It is highly likely that the system would exhibit non-regular behavior for some combination of forcing frequency and forcing amplitude. With this motivation, we start by studying the dynamics of a harmonically forced von Mises truss in the presence of viscous damping only. This leads to a Duffing type equation with an additional quadratic non-linearity. We demonstrate some of the rich dynamic behavior that this system exhibits in some parameter ranges. This provides useful insight into the possible behavior of the viscoelastic system. The viscous damper is then replaced by a viscoelastic unit. We show that the system can exhibit both regular as well as chaotic behavior. The threshold limit for the chaotic motion has been determined using Melnikov’s criteria and verified through numerical simulations using the largest Lyapunov exponent.