Measurement of softening cubic nonlinear and negative linear stiffness using van der Pol type self-excited oscillation

In this paper, a stiffness sensor for measuring the softening cubic nonlinear and negative linear stiffness is developed. The measurement system consists of a measurement object, an oscillator and a feedback circuit, and achieves the dynamics of van der Pol oscillator by the positive linear velocity feedback and the nonlinear feedback proportional to the squared position and the velocity. Then, the tuning of the nonlinear feedback gain can realize any setting of the magnitude of the response amplitude. Therefore, from the relationship between response amplitudes set by various nonlinear feedback gains and the corresponding response frequencies, we can experimentally describe the backbone curve which exhibits the dependency of the natural frequency on the amplitude in nonlinear mechanical systems. It has been experimentally verified by our research group that the hardening cubic nonlinear and positive linear stiffness can be measured by using the backbone curve. In this research, we assess the potential for measuring the softening cubic nonlinear and negative linear stiffness by employing a magnetic attraction model. The softening cubic nonlinear and negative linear stiffness are also measured by setting the sign of the coefficient of the nonlinear feedback gain appropriately to produce the stable limit cycle.