In this study we deal with a microelectromechanical system (MEMS) and develop a dynamical integrity analysis to interpret and predict the experimental response. The device consists of a clamped-clamped polysilicon microbeam, which is electrostatically and electrodynamically actuated. It has non-negligible imperfections, which are a typical consequence of the microfabrication process. A single-mode reduced-order model is derived and extensive numerical simulations are performed in a neighborhood of the first symmetric natural frequency, via frequency response diagrams and behavior chart. The typical softening behavior is observed and the overall scenario is explored, when both the frequency and the electrodynamic voltage are varied. We show that simulations based on direct numerical integration of the equation of motion in time yield satisfactory agreement with the experimental data. Nevertheless, these theoretical predictions are not completely fulfilled in some aspects. In particular, the range of existence of each attractor is smaller in practice than in the simulations. This is because these theoretical curves represent the ideal limit case where disturbances are absent, which never occurs under realistic conditions. A reliable prediction of the actual (and not only theoretical) range of existence of each attractor is essential in applications. To overcome this discrepancy and extend the results to the practical case where disturbances exist, a dynamical integrity analysis is developed. After introducing dynamical integrity concepts, integrity profiles and integrity charts are drawn. They are able to describe if each attractor is robust enough to tolerate the disturbances. Moreover, they detect the parameter range where each branch can be reliably observed in practice and where, instead, becomes vulnerable, i.e. they provide valuable information to operate the device in safe conditions according to the desired outcome and depending on the expected disturbances.

An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response

https://re.public.polimi.it/bitstream/11311/1022251/1/JSV_Caps%20et%20Al_R4_low%20res.pdf

Stability of dynamic response of suspension bridges

Harvesting electricity from low frequency vibration sources such as human motions using piezoelectric energy harvesters (PEH) is attracting the attention of many researchers in recent years. The energy harvested can potentially power portable electronic devices as well as some medical devices without the need of an external power source. For this purpose, the piezoelectric patch is often mechanically attached to a cantilever beam, such that the resonance frequency is predominantly governed by the cantilever beam. To increase the power generated from vibration sources with varying frequency, a multiresonant PEH (MRPEH) is often used. In this study, an attempt is made to enhance the performance of MRPEH with the use of a cantilever beam of optimised shape, i.e., a cantilever beam with two triangular branches. The performance is further enhanced through optimising the design of the proposed MRPEH to suit the frequency range of the targeted vibration source. A series of parametric studies were first carried out using finite-element analysis to provide in-depth understanding of the effect of each design parameters on the power output at a low frequency vibration. Selected outcomes were then experimentally verified. An optimised design was finally proposed. The results demonstrate that, with the use of a properly designed MRPEH, broadband energy harvesting is achievable and the efficiency of the PEH system can be significantly increased.

https://www.mdpi.com/1996-1073/12/14/2770/pdf

Performance enhancement of a multiresonant piezoelectric energy harvester for low frequency vibrations

A rotor system with asymmetry in its bending flexibility as well as in its support bearings has many stable and unstable speed ranges. The dynamics of such a system is governed by differential equations with time-varying parametric coefficients which lead to parametric instability in certain rotor speed ranges. In this study, an asymmetric rotor shaft having a rotor disc placed firmly on its centre is considered. The rotor shaft is mounted on two bearings at its two ends having stiffness asymmetry. A permanent magnet-type DC motor is used to drive the rotor. When rotor approaches the lower limit of any unstable speed range, the rotor spin speed is captured at that speed with increasing whirl amplitudes and the rotor speed does not respond to an increase in the motor power, unless there is sufficient surplus power to accelerate the rotor through the corresponding unstable speed range. With this surplus power, the escape from lower instability limit speed to a much higher stable speed takes place as a nonlinear jump phenomenon. In certain situations, escape from one unstable speed range may lead to capture at another unstable speed range or simultaneous escape from the next unstable range. This specifically occurs due to the presence of the non-ideal drive and is termed here as the Sommerfeld effect of second kind. Unlike regular Sommerfeld effect (of first kind) where the power scarcity at the resonance is the cause of speed capture; there is, ideally, no need for a residual rotor unbalance in the Sommerfeld effect of second kind. In both the cases of speed capture, the excess motor power is spent to increase the whirl amplitudes. The Sommerfeld effect of the first kind relates to the resonance at the synchronous rotor whirl (critical speeds), whereas that of the second kind relates to instability of the rotor whirl. Due to such nonlinear jumps, some of the speed ranges where the rotor is theoretically stable under ideal or mathematical conditions may not be reached in practice, i.e. with a real drive which is naturally non-ideal. The dynamics of this rotor system is analytically and numerically studied in this article. The numerical simulations are performed with a multi-energy domain bond graph model which guarantees energetic consistency of the model.

The Sommerfeld effect of second kind: passage through parametric instability in a rotor with non-circular shaft and anisotropic flexible supports

The present research study deals with an electrically actuated MEMS device. An experimental investigation is performed, via frequency sweeps in a neighbourhood of the first natural frequency. Resonant behavior is explored, with special attention devoted to jump and pull-in dynamics. A theoretical single degree-of-freedom spring-mass model is derived. Classical numerical simulations are observed to properly predict the main nonlinear features. Nevertheless, some discrepancies arise, which are particularly visible in the resonant branch. They mainly concern the practical range of existence of each attractor and the final outcome after its disappearance. These differences are likely due to disturbances, which are unavoidable in practice, but have not been included in the model. To take disturbances into account, in addition to the classical local investigations, we consider the global dynamics and explore the robustness of the obtained results by performing a dynamical integrity analysis. Our aim is that of developing an applicable confident estimate of the system response. Integrity profiles and integrity charts are built to detect the parameter range where reliability is practically strong and where it becomes weak. Integrity curves exactly follow the experimental data. They inform about the practical range of actuality. We discuss the combined use of integrity charts in the engineering design. Although we refer to a particular case-study, the approach is very general.

https://orb.binghamton.edu/cgi/viewcontent.cgi?article=1001&context=mechanical_fac

Theoretical prediction of experimental jump and pull-in dynamics in a MEMS sensor.

This contribution investigates the influence of parametric excitation on the dynamic stability of a microelectrome- chanical system. In systems with just a single degree of freedom, parametric excitation causes the oscillator to exhibit unstable behavior within certain intervals of the parametric excitation frequency. In multi-degree of freedom systems on the other hand, unstable behavior is caused within a wider range of intervals of the parametric excitation frequency. Moreover, such systems show frequency intervals of enhanced stability, an effect known as anti-resonance phenomenon. Both types of phenomena, the parametric resonance and anti-resonance, are modeled and studied for a microelectromechanical system with two degrees of freedom and some novel results are discussed.

/pdf/parametric-excitation-in-a-two-degree-of-freedom-mems-system-2jgce59nf7.pdf

Parametric Excitation in a Two Degree of Freedom MEMS System

Microelectromechanical systems (MEMS) are becoming important for all kinds of industrial applications. Among them are filters in communication devices, due to the growing demand for efficient and accurate filtering of signals. In recent developments single degree of freedom (1-dof) oscillators, that are operated at a parametric resonances, are employed for such tasks. Typically vibration damping is low in such MEM systems. While parametric excitation (PE) is used so far to take advantage of a parametric resonance, this contribution suggests to also exploit parametric anti-resonances in order to improve the damping behavior of such systems. Modeling aspects of a 2-dof MEM system and first results of the analysis of the non-linear and the linearized system are the focus of this paper. In principle the investigated system is an oscillating mechanical system with two degrees of freedom x = [x1x2]T that can be described by Mx+Cx+K1x+K3(x2)x+Fes(x,V(t)) = 0. The system is inherently non-linear because of the cubic mechanical stiffness K3 of the structure, but also because of electrostatic forces (1+cos(ωt))Fes(x) that act on the system. Electrostatic forces are generated by comb drives and are proportional to the applied time-periodic voltage V(t). These drives also provide the means to introduce time-periodic coefficients, i.e. parametric excitation (1+cos(ωt)) with frequency ω. For a realistic MEM system the coefficients of the non-linear set of differential equations need to be scaled for efficient numerical treatment. The final mathematical model is a set of four non-linear time-periodic homogeneous differential equations of first order. Numerical results are obtained from two different methods. The linearized time-periodic (LTP) system is studied by calculating the Monodromy matrix of the system. The eigenvalues of this matrix decide on the stability of the LTP-system. To study the unabridged non-linear system, the bifurcation software ManLab is employed. Continuation analysis including stability evaluations are executed and show the frequency ranges for which the 2-dof system becomes unstable due to parametric resonances. Moreover, the existence of frequency intervals are shown where enhanced damping for the system is observed for this MEMS. The results from the stability studies are confirmed by simulation results.

Stability analysis of a time-periodic 2-dof MEMS structure

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem.As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods.Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly.Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.Copyright © 2013 by ASME

Observations Regarding Numerical Results Obtained by the Floquet-Method

This paper presents a nonlinear parametrically excited cantilever beam with electromagnets. A parametrically excited two-degree-of-freedom (2-DOF) system with linear time-varying stiffness, nonlinear cubic stiffness, nonlinear cubic parametric stiffness and nonlinear damping is considered. In previous studies the stability and bifurcation of the nonlinear parametrically excited 2-DOF were investigated through analytical, semi-analytical and numerical methods. Unlike previous studies, in this contribution the system's response amplitude and phase at parametric resonance and parametric combination resonance are demonstrated experimentally and some novel results are discussed. Experimental and analytical amplitude-frequency plots are presented to show the stable solutions. Solutions for the system response are presented for specific values of parametric excitation frequency and the energy transfer between modes of vibrations is observed. The results presented in this paper prove that the bifurcation point and hence the bandwidth of the parametric resonance can be predicted correctly with the proposed analytical method. The proposed nonlinear parametrically excited 2-DOF can be used to design Micro ElectroMechanical Systems (MEMS) actuators and sensors. Validating the experimental results with the theory can improve the efficiency of these electrical systems.

Parametrically excited nonlinear two-degree-of-freedom electromechanical systems

An increasing number of MEMS devices use parametric excitation (PE) to outperform conventional designs. These systems are operated at parametric resonance. Different to standard cases of resonance, at parametric resonances the vibration's amplitudes increase faster and within a smaller interval. The amplification is much larger and only limited by non-linearities. So far the focus has mostly been on one degree of freedom (1DOF) systems. This is partly caused by a lack of methods to investigate and design multi degree of freedom (MDOF) systems time-efficiently. Restricting the systems to 1DOF ignores the opportunity to make use of PE effects only available in MDOF systems: parametric combination resonances and parametric anti-resonances, where an enhanced damping behaviour can be observed. The paper demonstrates how to approximate non-linear MDOF PE systems with 1DOF models. This leads to a generalised, dimensionless model applicable to many systems. Approaches are presented for investigating such a reduced non-linear 1DOF PE model analytically and semianalytically at parametric resonances using averaging methods. For a 2DOF system the results are validated numerically by continuation methods and time simulations. Limits of both the analytical and the semi-analytical approaches are discussed.

https://iopscience.iop.org/article/10.1088/1742-6596/744/1/012126/pdf

Till Jochen Kniffka

Papers

Parametric Excitation in a Two Degree of Freedom MEMS System

Stability analysis of a time-periodic 2-dof MEMS structure

Observations Regarding Numerical Results Obtained by the Floquet-Method

Parametrically excited nonlinear two-degree-of-freedom electromechanical systems

Studies of parametrically excited non-linear MDOF systems at parametric resonances