Free vibrations of a mass grounded by linear and nonlinear springs in series

In many technical applications spring-like flexible elements or real springs connected in series are used. The operation range of these components determines whether the system behaviour has a linear or nonlinear characteristic. In the relevant literature, there exists the knowledge how the equivalent spring is obtained for linear springs connected serially. However, some cases occur in which one linear and one nonlinear spring arranged in series are used. In such cases, it is not possible to define an equivalent spring rate. This study is concerned with such a system that consists of a mass grounded two springs, one of them linear and the other nonlinear. Two methods are developed to analyze the dynamic behaviour of system. One method makes use of a set of differential-algebraic equations (DAE in short). The other is based on getting a single equation of motion using relative displacement variables. For the second method, analytical solutions are also obtained by means of the Lindstedt and the harmonic balance techniques. It is observed, that numerical and analytical solutions found for both methods are in very good agreement when v0⩽1v0⩽1 and 0.1⩽ξ⩽100.1⩽ξ⩽10 where v0v0 is the initial deflection of nonlinear spring and ξξ is the ratio of linear portion coefficient of the nonlinear spring to that of the linear spring.