Vibrations of Lumped-Parameter Systems

Journal of Micromechanics and Microengineering, 2014

resonators, labeled A and B in figure 1, consisting of a square mass and two folded torsional springs attached to the proof mass, allow for the differential sensing of the acceleration. Driving and sensing of the resonators is obtained through two parallel electrodes attached to the substrate, as shown by dashed lines in figure 2 and also represented in figure 3. When an external out-of-plane acceleration is applied, the proof mass tilts and the frequency of the resonators change due to the variation of the electric stiffness induced by the gap variation. To obtain a high sensitivity the mechanical torsional stiffness of the resonators should be low but this can result in nonlinear dynamic behavior which is studied in this paper. The paper is organized as follows. In section 2, Hamilton's principle is formulated for the mechanical description of nonlinear oscillations of an electrostatically actuated torsional resonator. In section 3, Hamilton's principle is used as a basis for a one-degree-of-freedom formulation and analytical solutions are obtained. Section 4 discusses the influence of the geometry of the resonator and of the fabrication imperfections on the nonlinear behavior. The experimental results obtained both in the linear and nonlinear regimes on the resonators of the resonant out-of-plane accelerometer [3] are presented and discussed in section 5. The experimental data concerning the

The attainable resolution of inertial sensors is ultimately limited by the cumulated noise level generated in both the mechanical domain (mechano-thermal noise) and the frontend of the electrical readout circuit, provided that deterministic errors, such as quadrature errors in the case of gyroscopes, are kept under control. Improving the resolution performance of MEMS structures mounts to being able to either increase the minimum detectable signal through an increased sensitivity, or to improve the signal-tonoise ratio (SNR). This paper reports on parametric amplification and damping employed in a MEMS gyroscope. Experiments confirm that parametric modulation through electro-mechanical coupling leads to both an increase in spectral selectivity and a reduction of the equivalent input noise angular rate (from 0.0046 • /(s √ Hz) to 0.0026 • /(s √ Hz) for a parametric gain of 5). In a more general analysis of a MEMS resonant structure, electro-mechanical parametric amplification decreases the mechano-thermal noise associated with the resonant mode motion -the equivalent input noise acceleration was diminished from 0.033 m s −2 to 0.022 m s −2 for a parametric gain of 5. Either signal amplification or an attenuation of undesired signal components can be achieved by tuning the phase difference between the driving force and the parametric coupling. Therefore, the technique can be applied as well to reduce the quadrature error signal, which strongly constrains the maximum gain of the sensing circuit. Our experiments show a 2.2 improvement factor in SNR using a parametric amplification with a gain of 25. . His research interest is focused on MEMS-based microsystems design for automotive and biomedical applications, and on rapid microfabrication technologies. He has received in 2005 the "Tudor Tanasescu" (Science and Information Technology) award from Romanian Academy, for a group of papers in the field of microsystems, co-authored with L. Rocha and R.F. Wolffenbuttel.

Journal of Sound and Vibration, 2006

This paper investigates the dynamic response of a class of electrostatically driven microelectromechanical (MEM) oscillators. The particular systems of interest are those which feature parametric excitation that arises from forces produced by fluctuating voltages applied across comb drives. These systems are known to exhibit a wide range of behaviors, some of which have escaped explanation or prediction. In this paper we examine a general governing equation of motion for these systems and use it to provide a complete description of the dynamic response and its dependence on the system parameters. The defining feature of this equation is that both the linear and cubic terms feature parametric excitation which, in comparison to the case of purely linear parametric excitation (e.g. the Mathieu equation), significantly complicates the system's dynamics. One consequence is that an effective nonlinearity for the overall system cannot be defined. Instead, the system features separate effective nonlinearities for each branch of its nontrivial response. As such, it can exhibit not only hardening and softening nonlinearities, but also mixed nonlinearities, wherein the response branches in the system's frequency response bend toward or away from one another near resonance. This paper includes some brief background information on the equation of motion under consideration, an outline of the analytical techniques used to reach the aforementioned results, stability results for the responses in question, a numerical example, explored using simulation, of a MEM oscillator which features this nonlinear behavior, and preliminary experimental results, taken from an actual MEM device, which show evidence of the analytically predicted behavior. Practical issues pertaining to the design of parametrically excited MEM devices are also considered. r

IEEE Transactions on Circuits and Systems I-regular Papers, 2010

The aim of this paper is to show that it is possible to excite selectively different mechanical resonant modes of a MEMS structure using pulsed digital oscillators (PDOs). This can be done by simply changing the working parameters of the oscillator, namely its sampling frequency or its feedback filter. A set of iterative maps is formulated to describe the evolution of the spatial modes between two sampling events in PDOs. With this lumped model, it is established that under some circumstances PDO bitstreams related to only one of the resonances can be obtained, and that in the anti-oscillation regions of the PDO the mechanical energy is absorbed into the electrical domain on average. The possibility of selecting for a given resonant frequency the oscillation and anti-oscillation behavior allows one to obtain oscillations at any given resonant mode of the MEMS structure.

Active and Passive Smart Structures and Integrated Systems 2016, 2016

Proc Spie, 2007

Micro Electro Mechanical Systems (MEMS) are among the new and emerging technologies of the future and have many applications in different disciplines. This study presents the dynamic characterization methods that we use to identify the modal parameters of a MEMS device and also the techniques that can be implemented to change the modal parameters. A micro scanner mirror was chosen as the case study to demonstrate the developed methodologies. Initially, the micro mirror was dynamically characterized using experimental modal analysis techniques to identify the modal parameters such as resonance frequencies and mode shapes. Then, it was introduced in a velocity feedback control loop to alter the effective damping of the structure. This method proves to be a very efficient method to alter the modal damping of a micro structure, especially when high quality factors are required for MEMS applications.

Journal of Vibroengineering, 2007

The paper deals with finite element analysis of damped modal vibrations Q-factor values determined by thermal-elastic damping in micro-electrical-mechanical systems (MEMS). Mathematically the problem is formulated as a complex eigenvalue problem. Verification problems have been solved by using several computational environments and different presentations of model equations in order to comprehend and reduce the influence of rounding errors. The analysis of damped modal properties of selected real MEMS resonator revealed the main features of thermal-elastic damping by taking into account 3D geometry of the resonator and anchoring (clamping) structure.

2011 IEEE SENSORS Proceedings, 2011

In this paper it is presented a study on two key parameters of MEMS resonant accelerometers, resolution and start-up dynamics. A uniaxial differential accelerometer, built in the surface micromachining ThELMA process of ST Microelectronics, is used in the tests together with a discrete components readout circuit. The implemented circuit is an oscillator constituted by a Transimpedance stage (TIA) followed by a gain stage and an amplitude-limiting circuit (ALC). In the present work, it is shown how the sensor resolution can be related to the phase noise of the oscillator output signal and how the start-up dynamics depends on the biasing voltage of the resonator. A sensing resolution of 98μg/√Hz is demonstrated through experimental measurements and a well-defined relationship between the resonators biasing voltage and the start-up time is observed. I.

With advances in fabrication technologies lowering cost, MEMS gyroscopes are being used in an ever wider variety of applications, including those requiring operation at/or beyond the manufacturer's recommended temperature range. In these high temperature applications, such as deep water energy exploration and down-hole drilling, extensive lab testing is required to assess the effects of temperature on the response of a MEMS gyroscope. The objective of this paper is to develop a method to simulate the behavior of a MEMS vibratory gyroscope at various temperature conditions. The MEMS vibratory gyroscope is a two degree-of-freedom spring-mass-damper system. With known values of mass, spring stiffness and damping coefficient in the drive and sense direction, the characteristic equations of motion of the MEMS vibratory gyroscope can be solved using the first order approach developed in this paper. It is also shown, by comparing simulation results with experimental results, that this approach can accurately simulate the temperature-dependent characteristics of a MEMS vibratory gyroscope.

2020

In this work, the vibration analysis, dynamic characteristics, and fundamentals of optimized and traditional Micro electro mechanical systems (MEMS) gyroscope is reviewed based on different types of modeling and theories including modified couple stress theory (MCST), nonlocal elasticity theory, modified coupled displacement theory, nonclassical theories with mutable and changing nonclassical parameter. Several conditions affecting the performance and operation of MEMS gyroscopes including temperature effects, effects of the rotations and angular velocity, constrained relations coupling the translational and rotational motions, and further considerations are reported and gathered. It is reported by several research articles that consideration and scrutiny of the mentioned parameters in the design procedure is an important and influential key step in order to design the most optimal MEMS devices such as MEMS gyroscopes. Optimization of MEMS gyroscopes is highly important to the indus...

In this paper, a parametrically resonated MEMS gyroscope is considered, and the effect of its parameters on the system stability is studied. Unlike the general case of MEMS gyroscopes with harmonic excitation, in this new class of gyroscopes with parametric excitation, the origin is one stationary point of the system. The study starts with the stability analysis of the origin, and then it goes on to analyze the effect of each parameter on the stability of periodic orbits. Stabilities are studied by means of Floquet theory. As the results indicate, presence of a non-trivial response for the system is closely interconnected to the stabilities (and instabilities) of the system. It is demonstrated that the stability of the origin always contributes to a zero response for the sensor, and hence the instability of origin is required for the occurrence of parametric resonance. In contrast, stability of a periodic orbit does not necessarily guarantee a resonant response for the gyroscope, and again it is the instability of the origin which is required for this purpose. Because in the case of linear stiffness—lin-ear parametric excitation the instability of the origin results in instability of the system, it is concluded that nonlinearities are required for a parametrically actuated gyroscope.

2005

Due to the position-dependent nature of electrostatic forces, many microelectromechanical (MEM) oscillators inherently feature parametric excitation. This work considers the nonlinear response of one such oscillator, which is electrostatically actuated via non-interdigitated comb drives. Unlike other parametricallyexcited systems, which feature only linear parametric excitation in their equation of motion, the oscillator in question here exhibits parametric excitation in both its linear and nonlinear terms. This complication proves to significantly enrich the system's dynamics. Amongst the interesting consequences is the fact that the system's nonlinear response proves to be qualitatively dependent on the system's excitation amplitude. This paper includes an introduction to the equation of motion of interest, a brief, yet systematic, analysis of the equation's nonlinear response, and experimental evidence of the predicted behavior as measured from an actual MEM oscillator.

Sensors, 2014

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 Sensors 2014, 14 17090 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.

International Journal of Applied Mathematical Research, 2013

This paper presents a simplified mathematical model for the purpose of studying the resonant responses of a nonlinear dynamical system, (micro -electro -mechanical systems (MEMS)), which represented by a Van-der Pol equation subjected to a weakly non-linear parametric and forcing excitations. Using Multiple scales method, the Van-der Pol equation is transformed to a system of second order differential equation up to first order of small parameter ε. Three types of resonances are studied (harmonic resonance and subharmonic resonances of even order (onehalf and one -fourth )). The modulation equations for each resonances, steady state solutions, frequency-response equations, stability analysis are determined. Numerical analysis for frequency-response equations and stability conditions are carried out. Results are presented graphically by group of figures. Finally discussion for these figures are given.

IEEE Transactions on Circuits and Systems I-regular Papers, 2010

The aim of this paper is to show that it is possible to excite selectively different mechanical resonant modes of a MEMS structure using pulsed digital oscillators (PDOs). This can be done by simply changing the working parameters of the oscillator, namely its sampling frequency or its feedback filter. A set of iterative maps is formulated to describe the evolution of the spatial modes between two sampling events in PDOs. With this lumped model, it is established that under some circumstances PDO bitstreams related to only one of the resonances can be obtained, and that in the anti-oscillation regions of the PDO the mechanical energy is absorbed into the electrical domain on average. The possibility of selecting for a given resonant frequency the oscillation and anti-oscillation behavior allows one to obtain oscillations at any given resonant mode of the MEMS structure.