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This paper reports on numerical modeling and simulation of a generalized contact-type MEMS device having large potential in various micro-sensor/actuator applications, which are currently limited because of detrimental effects of the contact bounce phenomenon that is still not fully explained and requires comprehensive treatment. The proposed 2-D finite element model encompasses cantilever microstructures operating in a vacuum and impacting on a viscoelastic support. The presented numerical analysis focuses on the first three flexural vibration modes and their influence on dynamic characteristics. Simulation results demonstrate the possibility to use higher modes and their particular points for enhancing MEMS performance and reliability through reduction of vibro-impact process duration.

Many traditional devices of microelectromechanical systems (MEMS) do not include contacting surfaces. However in recent years there is an increasing interest in various microsensors and microactuators that employ contact interaction in their normal mode of operation. This trend is determined by the new developments in MEMS technology and new market demands. Among such devices, the fast development of microswitches is very promising. However, insufficient mechanical reliability is one of the main obstacles for wider successful application of these microdevices [

A review of the literature on contact bounce in microswitches suggests that extensive research efforts are still needed in this field and that scientific results on underlying dynamical aspects of this detrimental phenomenon are relatively scarce. Modification of electrostatic control mechanism is a predominant approach used for reduction of bouncing however we believe that there is still enough undisclosed potential in the mechanical domain alone, which could be beneficial in tackling the considered problem. Therefore in this paper a contact-type microdevice is analyzed purely from mechanical point of view, thereby concentrating on intrinsic dynamic properties of elastic structures such as natural vibration modes and their advantageous utilization.

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After proper selection of generalized displacements in the inertial system of coordinates, model dynamics is described by the following equation of motion given in a general matrix form:

The initial conditions are defined as:
_{i}_{dM}, β_{dK}_{1} and _{2} that correspond to two unequal natural frequencies of vibration _{1} and _{2} [

The presented FE model of the vibro-impact microsystem was implemented in FORTRAN.

Free impact vibrations of elastic microstructures constitute one of the operation modes of contact-type MEMS devices. Complete vibro-impact process consists of free vibrations of the microstructure in the intervals between the impacts and its vibration during the impacts. Therefore, thorough analysis of free and impact vibrations of elastic microstructures is essential. For this purpose special FORTRAN numerical codes were written and used for running detailed dynamic simulations with the developed FE model of the cantilever microstructure that undergoes impacts against the support.

The modes of natural transverse vibrations of the microstructure (_{0}_{0} from the anchor of the cantilever microstructure and its whole length _{ij}_{ij}

The process of free impact vibrations of the microstructure for the case when the support is located at the free end of the cantilever is presented in

The simulated _{l}_{exp} − _{exp}) × 100. Simulated vibro-impact process in _{exp}

Temporal characteristics that are most typical for the free impact vibrations are: _{p}_{1}—duration of vibrations between two impacts, _{2}—impact duration. The accuracy of simulation results is significantly influenced by the density of the finite elements mesh. _{max} = _{max}/

_{0}/_{0}/_{0}/

_{0}/_{p}_{1}_{p}_{1} - first circular natural frequency of the cantilever) may be reduced. The remaining characteristics are less sensitive to variations of support position.

The points _{0}/^{nd} and the 3^{rd} flexural vibration modes, while _{0}/^{rd} mode. These points will be referred to as particular points of natural vibration modes. The subsequent numerical analysis will be confined to the consideration of the first three modes since they significantly influence the dynamic characteristics of the vibro-impact process. In order to clarify the nature of these characteristics it was necessary to determine vibration modes of the microstructure during the impact on the support.

_{ij}_{ij}_{i}_{i}_{0}_{i}

Simulated curves presented in _{0}/^{nd} nodal point of the 3^{rd} vibration mode of the supported microstructure coincides with the same point for the case of unsupported one (_{0}/^{rd} mode increases resulting in more intensive energy dissipation in the material of the microstructure since it is considered [_{3}/_{1} ≅ 17 times less than in the case of vibrations in the 3^{rd} mode. It is evident that intensification of the amplitude of the 3^{rd} mode by locating the support at its nodal point does not cancel the first two modes. The fact that nodal points _{31} and _{21} coincide in the case when the support is located at point _{0}/^{nd} mode as well. However, the advantages achieved in the considered case are first of all related to the intensification of the amplitude of the 3^{rd} vibration mode (during vibro-impact process cantilever vibrations in a wide frequency range are excited). The advantages achieved when the support is positioned in point _{0}/^{nd} vibration mode amplitude because this is the point in which the nodal points of the 2^{nd} vibration mode of the supported and unsupported microstructure are located (_{0}/_{20} and _{31} when the support is located in point _{0}/

The presented explanation is also confirmed by the dependences of the maximal amplitude points of separate vibration modes on the position of the support (_{11} with respect to support locations (_{0}/^{rd} displacement mode Y_{33} is maximal whereas other amplitudes do not reach their maximal values in this point. Positioning of the support in the point of the maximum amplitude of the 3^{rd} vibration mode (_{0}/_{32} that coincides with the said point of maximum amplitude.

Amplitudes Y_{30} and Y_{31} are increased as well, whereas amplitude Y_{33} is reduced. When the support is positioned in the nodal point of the 2^{nd} displacement mode, the displacement amplitude Y_{22} increases whereas other amplitudes of the 2nd mode decrease. Similarly, the amplitudes of rotational vibration modes Φ_{11} are intensified as well (

After the performed analysis of the behavior of the nodal points and the points of maximum amplitude with respect to the support location, it is important to investigate the dependence of the frequencies of separate vibration modes on the position of the support. _{i}_{iin}^{st} natural frequency of the supported microstructure reaches the maximum value when the support is located in point _{0}/^{nd} and the 3^{rd} natural frequencies reach their maximum values when the support is located in other positions.

Therefore, in order to ensure maximum vibrational stability of a contact-type MEMS device containing a supported cantilever microstructure, the support must be positioned in point _{0}/^{nd} mode of natural vibrations and to dissipate a significant portion of kinematically-transferred energy in the material of the microstructure. Furthermore, when the support is located in point _{0}/^{st} and the 2^{nd} natural frequencies of the supported microstructure is maximum, and by selecting the stiffness of the support to be located in the given point, the 1^{st} natural frequency may be brought closer to its 2^{nd} natural frequency thereby increasing its vibrational stability under external kinematical excitation, which may be very important when microdevice is located on the moving object.

Common contact-type MEMS devices incorporate gaps between compliant and fixed microstructures. However, feasible MEMS designs may be also based on usage of prestress of contacting links. Therefore it is crucial to select the prestress in such a way that minimal rebound amplitudes are achieved resulting in reduced energy consumption during device control. _{max} = _{max}/

As the simulation results in

In addition to the amplitude-frequency characteristics of free impact vibrations, it is essential to determine the velocities and the forces induced during the impact. ^{rd} flexural mode of the cantilever microstructure, a decrease in the velocity and original contact pressure force is observed, which is related to the increase in the dissipated energy in the material.

Simulations results (_{0}/_{0}/

In this paper we have presented a 2-D finite element model of cantilever microstructure impacting against viscoelastic support thereby representing a general case of contact-type MEMS devices. The model was developed within FORTRAN environment. Impact is modeled by means of contact-element approach that uses Kelvin-Voigt rheological element taking into account both contact stiffness and damping. Values of these parameters were selected empirically to match experimentally-obtained vibro-impact trajectories. Results of numerical analysis of characteristic vibro-impact process–free impact vibrations–were reported by considering three stages of the studied process: pre-impact, impact and post-impact. Obtained numerical results are provided in a dimensionless form and therefore are applicable across all scales ranging from macro to nano.

Numerical analysis is centered around the consideration of the first three flexural modes of the cantilever microstructure since they have a major effect on dynamic characteristics of the vibro-impact process. Investigation of influence of support position (along horizontal axis of the microstructure) on maximum post-impact rebound amplitudes indicates that the smallest values are obtained when the support is located in specific points coinciding with the nodal points of the 2^{nd} and the 3^{rd} flexural vibration modes (_{0}/^{rd} mode (_{0}/^{nd} and 3^{rd} modes are amplified when the force of impact is applied to these points. The effect is particularly pronounced in the case of the 2^{nd} nodal point of the 3^{rd} flexural mode (_{0}/^{rd} mode increases resulting in more intensive energy dissipation in the material of the microstructure (energy dissipated is _{3} /_{1} ≅ 17 times larger than in the case of microstructure vibrating in its fundamental mode). Increase of dissipated energy in the material at this particular point is also confirmed by observed reduction of the induced velocity and contact pressure force during impact.

Numerical study of influence of support position on the natural frequencies of separate vibration modes indicates that maximization of vibrational stability of contact-type MEMS devise containing supported microstructure is achieved by placing support at _{0}/^{st} natural frequency of the supported microstructure. By selecting the stiffness of the support to be located in the given point, the 1^{st} natural frequency may be brought closer to its 2^{nd} natural frequency thereby increasing the vibrational stability.

Obtained results of numerical analysis reveal huge potential of advantageous usage of higher vibration modes with their particular points for suppressing harmful bouncing process in contact-type microdevices resulting in improved reliability and performance. Therefore further research efforts are necessary in this field in order to identify different approaches for control of impact-related processes thereby enabling designers to develop innovative MEMS sensors and actuators that operate in contact mode.

Schematic of: (a) generalized model of common electrostatic contact-type MEMS device operating in ambient air, (b) developed 2-D finite element model of impacting cantilever microstructure.

Natural vibration modes of the cantilever microstructure: (a) flexural, (b) rotational. _{0}_{0} from the anchor of the cantilever and its whole length _{ij}_{ij}

Simulated typical process of free impact vibrations of the cantilever with characteristic parameters: _{p}_{1}—duration of vibrations between two impacts, _{2}—impact duration, _{max}

Dependence of dimensionless rebound amplitude of the microstructure _{max} = _{max}/_{0}

Temporal characteristics of free impact vibrations of the microstructure as a function of support position: _{p}_{1}_{p}_{0}_{1}_{p}_{1}_{1} = _{1}_{1}, _{2} = _{1}_{2}. _{1}—first circular natural frequency of the cantilever.

Dependence of nodal points of the displacement (_{ij}_{ij}

Dependence of maximum amplitudes of the flexural (Y_{ij}_{ij}

Dependences of the ratio between the circular natural frequencies of the supported microstructure _{i}_{iin}

Dependence of maximum rebound amplitudes of the cantilever microstructure _{max} = _{max}/

Dependence of impact velocity

Dependence of contact pressure force _{2} = _{1}_{2}, _{2}—impact duration, _{1}—first circular natural frequency of the cantilever.