Stability of Negative Stiffness Viscoelastic Systems

We analytically investigate the stability of a discrete viscoelastic system with negative stiffness elements both in the time and frequency domains. Parametric analysis was performed by tuning both the amount of negative stiffness in a standard linear solid and driving frequency. Stability conditions were derived from the analytical solutions of the differential governing equations and the Lyapunov stability theorem. High frequency response of the system is studied. Stability of singularities in the dissipation tan δ is discussed. It was found that stable singular tan δ is achievable. The system with extreme high stiffness analyzed here was metastable. We established an explicit link for the divergent rates of the metastable system between the solutions of differential governing equations in the time domain and the Lyapunov theorem. 1. Nomenclature. M , m1, m2: mass. K, k1, k2: stiffness for positive stiffness elements. c, c1, c2: damping coefficient. α, α1, α2: ratio between the two spring elements in a standard linear solid. α = α1 when α2 = 0. γ1, γ2: ratio of damping coefficient of a standard linear solid to that of a damper parallelconnected to the standard linear solid, i.e., γ = η/c. δ: phase angle. Define tan δ = (k∗)/ (k∗), where k* denotes the dynamic complex spring constant. κ1, κ2: ratio of stiffness of the series spring in a standard linear solid to that of a spring parallel-connected to the standard linear solid, i.e., κ = y/k. Received December 15, 2003. 2000 Mathematics Subject Classification. Primary 74B10; Secondary 74C10, 74D05. E-mail address: [email protected] c ©2004 Brown University