Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation
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ORIGINAL PAPER
Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation Shinpachiro Urasaki
· Hiroshi Yabuno
Received: 28 February 2020 / Accepted: 5 September 2020 © Springer Nature B.V. 2020
Abstract This study presents an experimental method for identification of the backbone curves of cantilevers using the nonlinear dynamics of a van der Pol oscillator. The backbone curve characterizes the nonlinear stiffness and nonlinear inertia of the resonator, so it is important to identify this curve experimentally to realize high-sensitivity and high-accuracy sensing resonators. Unlike the conventional method based on the frequency response under external excitation, the proposed method based on self-excited oscillation enables direct backbone curve identification, because the effect of the viscous environment is eliminated under the linear velocity feedback condition. In this research, the method proposed for discrete systems is extended to give an identification method for continuum systems such as cantilever beams. The actuation is given with respect to both the linear and nonlinear feedbacks so that the system behaves as a van der Pol oscillator with a stable steady-state amplitude. By varying the nonlinear feedback gain, we can produce the selfexcited oscillation experimentally with various steadystate amplitudes. Then, using the relationship between these steady-state amplitudes and the corresponding experimentally measured response frequencies, we can detect the backbone curve while varying the nonlinear feedback gain. The efficiency of the proposed method S. Urasaki (B) · H. Yabuno Tsukuba City, Japan e-mail: [email protected] H. Yabuno e-mail: [email protected]
is determined by identifying the backbone curves of a macrocantilever with a tip mass and a macrocantilever subjected to atomic forces, which are representative sources of hardening and softening cubic nonlinearities, respectively. Keywords Backbone curve · Self-excited oscillation · Nonlinear feedback control · Van der Pol-type oscillator · Cantilever beam
1 Introduction The resonances of micro- and nanocantilevers are used widely in measurement devices, e.g., in mass sensing, viscometers, and atomic force microscopy (AFM) [1–5]. Because the measurement principle is mainly based on linear vibration theory, the accuracy of these measurements is not ensured when the nonlinearity of the resonator appears (for example, see reference [6]) because of the growth of the resonance amplitude. However, the nonlinear normal modes (NNMs) [7,8] provide numerous insights to enable prediction of the nonlinearity that is essentially included in mechanical systems and the responses to this nonlinearity [9,10]. Therefore, realization of high-sensitivity and high-accuracy sensing is expected through application of these NNMs. Kumar et al. proposed a highsensitivity mass sensing principle using the jump phenomenon of the nonlinear frequency responses [11], while Tadokoro et al. proposed a nanoelec
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