A robust method for nonlinear modal analysis based on NNMs as a family of periodic solutions is presented. The continuation of NNMs is improved by the addition of an opti- mized phase condition in the frequency domain and a damping matrix multiplied by quasi- null coefficient. The concept of NNM is extended to take into account non-conservative equations of motions. It is shown that the non-conservative NNMs can be computed as periodical solutions at iso-energy level by introducing an appropriate fictive force in the system. Similarly to linear modal analysis, two definitions of this fictive force are pro- vided to characterize the phase and the energy resonances of non-conservative NNMs. As a result, non-conservative NNMs can be computed with the same numerical methods as for the conservative case. The stability analysis is performed by considering a shifted ver- sion of the quadratic eigenvalue problem derived from Hill’s method. It results in a robust method for the computation of all conservative or non-conservative NNM branches, their stability and bifurcation points.
Then, a 2-ddl system is analyzed with respect to several designs. With the first de- sign, the complex dynamical behaviors of the system are analyzed under the light of an underlying symmetry. The topology of both tongues of modal interaction and BPs are explained. The robustness of the method is validated by the calculation of all the NNM branches in a single computation. With the second design, techniques of LP continuation are applied to the system in order to visualize the evolution of INNM with respect to a parameter that breaks the symmetry of the system. It is that LP continuation techniques are efficient tools for the analysis of INNM. Finally, the theory describing the energy reso- nance of the displacement vector is applied to the two last designs. First, viscous damping
is considered in the model simulating structural damping. For viscous damping of a few percentages, phase and energy resonances are almost equal. Thus, phase resonance can be taken over energy resonance for simulating system with structural damping. It is also shown that a sufficiently accurate fictive force can target specific points onto NNMs such as modal interaction points or orbits of modal interaction. This opens the possibility of experimental NNM tracking with a controlled fictive force targeting a specific resonance. Finally, a system with a cubic nonlinear damping is considered to validate the energy resonance NNM for more complex nonlinear damping.
Chapter 4
Dynamical analysis of MEMS array
This chapter investigates the mass-sensing potential of an array of identical resonant electrostatically actuated micro beams, as a first step toward the implementation of arrays of
thousands of sensors for mass sensing. The effect of electrostatic coupling is studied on a 2-beam array. Predicted
frequency responses exhibit complex branches of solutions with additional loops and synchronization in frequency of bifurcations due to the coupling between the beams. Then, the
dynamic of a 2-beam array is analyzed with bifurcation tracking and Nonlinear Normal Modes (NNM) after a symmetry-breaking event induced by the addition of an added
mass onto one beam of the MEMS array under symmetric configuration. Phenomena leading to the apparition or the merging of isolated solutions (IS) and isolated NNM (INNM)
are analyzed. Finally, mechanism of detection using 2-beam and 3-beam array are proposed. Depending on the applied voltages, the solutions with and without added mass exhibit a large change in amplitude that can be used for detection. For
symmetric configurations, exploiting the bifurcations of symmetry-breaking type permits improving mass sensing.
4.1.1 Case without added mass . . . 128 4.1.2 Case with added mass . . . 133 4.1.3 Designs of beam array . . . 133 4.1.4 Convergence in terms of Taylor expansion order . . . 133 4.2 Effect of the electrostatic coupling on a 2-beam array responses . . . . 135 4.2.1 Averaging method . . . 135 4.2.2 Response curve analysis . . . 137 4.2.3 Comparison of the responses obtained with HBM+ANM and with
time-integration methods. . . 138 4.3 Dynamics of MEMS array after symmetry-breaking event induced by
an added mass . . . 138 4.3.1 Parametric analysis of NNM . . . 142 4.3.2 Analysis of frequency responses . . . 144 4.4 Mechanisms of detection . . . 160 4.4.1 Based on frequency shift and hysteresis cycle (Design #1) . . . 161 4.4.2 Based on symmetry-breaking event . . . 167 4.4.3 Perspectives for mass detection . . . 180 4.5 Conclusion . . . 180
n-Beam array model
In this chapter builds on the previous chapters and addresses electrostatically actuated resonant MEMS array. An array of n identical clamped-clamped micro beams is consid- ered in order to study the response change due to a very small added mass. The response change is analyzed numerically for various numbers of beams and several configurations of the beam array such as asymmetric and symmetric. The beams of the array are coupled only by electrostatic forces and exhibit complex dynamical behaviors which are used to provide additional methods for mass sensing.
In Section 4.1, a reduced-order model for an array of n clamped-clamped beams is considered. It is obtained through Galerkin expansion onto a finite number of linear eigenmodes and solved numerically by means of the Harmonic Balance Method (HBM) combined with the Asymptotic Numerical Method (ANM). In Section 4.2, the case of a two-beam array is investigated. The response change due to the electrostatic coupling between the beam is analyzed. Approximated solutions are used to distinguish and with- draw the coupling during the computation of the frequency responses. In Section 4.3, a symmetric two-beam array is analyzed before and after symmetry-breaking event in- duced by the addition of a small mass onto the first beam. Nonlinear modal analysis and parametric analysis are performed using the methods presented in the previous Chap- ters. First, the NNMs of the array are analyzed with LP tracking. INNM (INNM) are detected after the symmetry-breaking event. Secondly, the frequency responses are com- puted and analyzed with the previous NNM and LP tracking. Merging and birth of IS with symmetry-breaking event are explained. A localization of motion is also observed when the in-phase mode is excited after the symmetry-breaking event. In Section 4.4, mass de- tection mechanisms are introduced according to the specific dynamical behaviors of the beams. First, mechanism of detection based on hysteresis cycle with asymmetric voltages are proposed. Two- and three-beam arrays are examined. Secondly, mass detection based on symmetry-breaking event are analyzed on a three-beam array. An analysis related to location, possible detection as well as quantification of the added mass is conducted.
Conclusions are drawn in Section 4.5. The originality of this chapter lies in the analy- sis of complex phenomenon specific to MEMS array and their exploitation to provide alternative mechanisms for mass sensing.