Experimental characteristics and coupled nonlinear forced vibrations of axially travelling hyperelastic beams

Abstract

Comprehensively investigated in this paper is the coupled nonlinear dynamics of hyperelastic beams. Experimental testing is performed to accurately model the hyperelasticity using different hyperelastic energy density models. Samples were fabricated using 3D printing technique for flexible thermoplastic polyurethane filaments following the ASTM D638 guidance and tested using a tensile testing machine. For a proper hyperelastic energy density model, coupled nonlinear equations of motion are obtained via the von Kármán geometrical nonlinearity and Hamilton’s principle. Free natural vibration response of the gyroscopic system is analysed using a generalised modal decomposition method and the influence of different parameters such as the hyperelastic stiffness term and the axial velocity on the natural frequencies is investigated. Complex mode shapes are obtained and the influence of axial velocity on altering the real and imaginary parts is discussed. Furthermore, a comprehensive analysis on the coupled nonlinear dynamics of the system is presented using a combination of Galerkin’s scheme and a dynamic equilibrium technique. The frequency response to different design parameters such as axial speed, axial tension, harmonic external load, hyperelastic parameters and geometrical imperfection is discussed in detail. It is shown that the system shows a wide range of rich nonlinear dynamics including hardening depending on the characteristics of the system.