The present paper develops a nonlinear stiffness sensor for measuring cubic nonlinear elasticity. The measurement system consists of a vibrator with a control circuit. We apply linear-plus-nonlinear feedback to actuate the vibrator attached to a measurement object for inducing van der Pol type self-excited oscillation so that the response amplitude of the oscillation can be set arbitrarily by changing the nonlinear feedback gain. We focus on the fact that the nonlinear elasticity of the measurement object causes a natural frequency shift related to the magnitude of vibration amplitude of the vibrator. We can set the response amplitude to various values by changing the nonlinear feedback gain and measuring the shift of the response frequency depending on the magnitude of the response amplitude. As a result, based on the bend of the experimentally obtained backbone curve, the nonlinear elasticity of the measurement object is identified.