Analytical solution of a mass-spring system containing shape memory alloys: Effects of nonlinearity and hysteresis

Nonlinear dynamics of vibration systems containing NiTi shape memory alloy (SMA) bars has long been obscured by the lack of an analytical solution, like the analytical solution for duffing equation. The problem results from the nonlinear and hysteretic restoring force of the SMA bar. Here we use a piecewise linear hysteretic model to describe the force-displacement relation of the SMA bar and use the averaging method to solve the equation of motion. We thus obtain an approximate analytical solution of the steady-state response of an SMA mass-spring system. The analytical solution can describe both stable and unstable behaviors of the vibration system and therefore offer a comprehensive understanding of the nonlinear responses. It is shown that the phase transition induced softening nonlinearity bends the frequency response curve (FRC) to the left, while the subsequent rehardening of martensite further bends the FRC to the right, leading to multi-valued regions and jump phenomena. The hysteresis is found to have little influence on the bending but it can significantly suppress the response amplitude. Comparison of the analytical results with experimental data validates the piecewise linear hysteretic model and the analytical solutions. This work provides a theoretical tool for design and vibration control of SMA mass-spring systems.