Dynamic Analysis of a Microgripper and Its Components

This paper is focused on the dynamic analysis and simulation of a novel microgripper and its components: microcantilever and V-shape thermal actuator. These devices are designed on silicon and implemented on Professional Autodesk Inventor 2014. Simulations were realized using Ansys Workbench Software. This analysis is common for cantilever, and RF MEM devices, but it is has not been widely realized for other structures, such V-actuator and microgrippers. The analytical response was acquired with Steady-State Thermal, Static Structural, Modal and Harmonic Response modules. The dynamic behavior, resonance frequencies of each modal shape and the harmonic behavior with different damping factors of these devices are presented. Parameters as actuation forces, displacements, natural frequencies and specific displacement corresponding to each modal shape in all devices, are also considered in the analysis. The simulation results show the modal shapes of all analyzed devices, determining their respective modal frequencies and harmonic response. Damping factors of 1% to 10% were employed. The phase angle (±90º to ±105º) and attenuation levels due to damping were also obtained.


INTRODUCTION
Nowadays, design engineers and many enterprises are interested in evaluate the mode shapes in which a device or structural element behaves in accordance to dynamics input caused by vibrations [1]. This development is an important step in the manufacture of dynamic or static devices. In Micro-Electro-Mechanical Systems (MEMS) technology, there is an important interest in the characterization of these devices using dynamic analysis, which is a technique to determine the dynamic behavior of a structure or component, involving the time, the inertia and possibly damping of the structure [2]. Their dynamic behavior can be calculated with the dynamic analysis, deformations or displacements, actuation forces, modal shapes, natural frequencies, harmonic responses, spectrum responses, random vibrations and transient analyses [3][4][5][6][7][8]. A powerful tool that facilitates the analysis and calculation, using finite element method (FEM), or finite element analysis (FEA), is based on the idea of building a complicated object with simple blocks, or, dividing a complicated object into small and manageable pieces. The application of this simple idea can be found in daily life as well as in engineering [9][10][11][12][13][14].
Mechanical sensor produce a mechanical response, a bending or deflection, when temperature changes are given [15,16].
A cantilever is fixed at one end, while the other one is free to move when it experiences some stress (Fig. 1a). A Microcantilever is a device that can be used as physical, chemical or biological sensor by detecting the changes in cantilever bending or vibrational frequency. These microcantilevers produce a deflection at the free end when force is applied. As their dimensions are in micrometers and the amount of applied stress is slight, the deflection will be also in micrometers. This device is frequently used in complexes systems [7][8][9][10]17,18].
Chevron-type electrothermal V-actuators use an array of silicon beams facing each other in pairs to generate one directional shuttle. Heating of beam-pairs causes them to expand and ultimately buckle. The beams are designed with a pre-bend angle, so buckling has a tendency to move in-plane (parallel to the substrate) as depicted in Fig. 1b. These actuators typically exhibit forces ranging from just tenths to hundreds of µN. Buckle-beam arrays can develop forces in the order of mN. These devices incorporate the benefits of a chevron actuator in terms of high output force, low operating voltages, sub-micrometer resolution in shuttle positioning, linear movement without deformation of the shuttle, etc. [2,5,10,19].
Manipulation of microparticles can be done using several physical principles and methods. For manipulation of a microstructure, specific ambient conditions are considered, such as in liquid; suction, cryogenic, electrostatic, and friction are the most often considered methods. In recent years, microgrippers have been widely studied due to their application areas, such as advanced microassembly, micromanipulation, microrobotics, minimally invasive and living cell surgery. Microgrippers are fabricated using integrated circuits (IC) or IC compatible technology, electrostatic, piezoelectric or electrothermal actuation [3,11,12]. This paper is organized as follows: Section 2 describes the MEM structures under analysis. In the same section, the used modules of Ansys Workbench are described. Physical and mechanical properties of silicon are also given. In section 3, the equations used for simulation of modal and harmonic response analyses are presented. Section four gives the simulation results of each considered device. Finally, in section 5, some concluded remarks are given.

METODOLOGY
In the implementation of structures, Professional Autodesk Inventor 2014 is used. The finite element analysis is considered by specific Ansys-Workbench modules [5] such as: Geometry, Steady-State Thermal, Static Structural, Modal and Harmonic Response. In Fig. 2, the schematic sequence of analysis process is described.

Fig. 2. Schematic project of analysis implemented in Ansys-Workbench
Temperature boundary conditions of MEM devices shown in Fig. 1 are considered:  For the cantilever (Fig. 1a), the fixed end is fed using a thermal source at 100°C. The free end of beam has room temperature (22°C), as it was realized in [20].  The V-actuator is formed on the base of an array of symmetric cantilevers, [5,7,[19][20][21], Fig. 1b. In order to increase actuation force and displacements. 8 symmetric beams with 0.9º inclination angle were used. Its anchor is fixed at 100ºC and the shuttle has fixed at room temperature (22ºC).  On the base of V-actuator a microgripper is proposed for specific applications, (Fig. 1c). Its boundary conditions are: The anchor and inferior part are fixed at 100°C, the free gripper at room temperature (22ºC). Only upper gripper is analyzed.
These devices were simulated under the mentioned physical and mechanical properties give in Table 1, in order to compare their behavior. The analysis process in the Schematic Project of Ansys-Workbench, sequence was carried out as follows:  At first, geometry is imported from Inventor to Project Schematic.  Later, in the Steady-State Thermal module, materials and temperatures (power supply and room temperature) are determined and applied.  In the Static Structural module, the temperature distribution and effects of the structural deformations (total displacement) are evaluated.  With this deformation value, the actuation force analysis is also realized using the Static Structural module. These parameters allow obtaining the modal shapes and frequency responses of each mode, by means of Modal module and Harmonic response, respectively. Six modals forms were calculated, which will be presented in section 4 (as it is described in [22][23][24][25]).  Finally, with constant values of damping factor in the range 1% to 10% of the total response of the analyzed devices (as it is realized in [26]) and with Harmonic Response module the central frequency and phase angle of all devices is obtained.
In Table 2 the dimensions of cantilever are shown.

Description of Modal Analysis
A modal analysis is a technique used to determine the vibration characteristics of structures [26][27][28][29][30][31][32][33]. The vibration of bodies are divided in three main categories: free, forced, and self-excited vibrations.

Free vibration
Free vibration occurs in the absence of a long term, external excitation force. It is the result of some initial conditions imposed on the system, such as a displacement from the system's equilibrium position, for example. Free vibration produces motion in one or more of the system´s natural frequencies and, because all physical structures exhibit some form of damping (or energy dissipation), it is seen as a decaying oscillation with a relatively short duration [1,13,24].

Forced vibration
Forced vibration takes place when a continuous, external periodic excitation produces a response with the same frequency as the forcing function (after the decay of initial transients). While free vibration is often represented in the time-domain, forced vibration is typically analyzed in the frequency and enables the convenient identification of natural frequencies. A typical source of forced vibration in mechanical systems is rotating imbalance. Large vibrations occur when the forcing frequency, ω, is near to the system natural frequency, ω n [1,2,16,24,34].

Self-excited vibration
In self-excited vibration, a steady input force is present, as in the case of forced vibration. However, this input is modulated into vibration at one of the system's natural frequencies, as with free vibration. The physical mechanism, that provide this modulation, are varied. Common examples of self-excited vibration include playing a violin, flutter in airplane wings, and chatter in machining [16,24].
With modal analysis in Ansys-Workbench Software, the characteristic vibrations can be calculated: Natural frequencies and Modal shapes.

Mathematical modeling of modal analysis
This analysis considered the motion equation for an undamped linear (with constant stiffness and mass) system, which is used for natural frequency and modal shape determination.
the global stiffnes, mass and force matrices, respectively [35]. Note that the global mass matrix is assembled in the same manner as the global stiffness matrix. Equation 1 represents a set matrix equations discretized with respect to space. Discretization in time is also necessary to solve it [36]. Which can be solved on the base of eq. 1-4, using one of two solvers available in workbench mechanical software.
In most cases, the program-controlled option selects the optimal solver automatically.

Harmonic Response Analysis
This technique is used to determine the steady state response of a structure to harmonic loads of known frequency. This analysis solves the time dependent equations of motion for linear structures subject to steady state vibration [26], with:

Input
-Harmonic loads (forces, pressures, and imposed displacements) of known magnitude and frequency.
-Multiple loads may be at the same frequency. Forces and displacements can be in phase or out of phase. Body loads can only be specified with a phase angle of zero.

Output
-Harmonic displacements at each DoF, usually out of phase with the applied loads. -Other derived quantities, such as stresses and strains.

Restrictions of harmonic response analysis [26]
Restrictions -The entire structure has constant or frequency-dependent stiffness, damping and mass effects. -All loads and displacements vary sinusoidal, at the same known frequency (although not necessarily in phase) -Acceleration, modals and moment loads are assumed to be real (in phase) only.

Mathematical modeling of harmonic response analysi
General equation of motion for a structural system is considered [26]: Eq. 7 Where: [ Eq. 8 Note: The symbols Ω and ω differentiate the input from the output: Ω: input circular frequency ω: output circular frequency Taking two time derivatives of u: Eq. 9 Substituting equation 8 and 9, in equation 6, and simplifing: Eq. 10 Equation 10 can then be solved using one of the following two methods: The full method solves the system of simultaneous equations directly using a static solver designed for complete arithmetic solution: Where: -C denotes a complex matrix or vector Or the mode-superposition method, which expresses the displacements as a linear combination of modal shapes.

SIMULATION RESULTS
All results were obtained using Ansys Workbech, on the base of two mathematical models presented in section 3. The boundary mechanical and thermal conditions, geometrical and physical data and restrictions for each device were provided, for each corresponding simulation.
The displacement, actuation force, modal frequencies, and harmonic response for cantilever, V-shape actuator and microgripper are summarized in Tables 3-5.
Modal forms and Harmonic Response are given for each device in Tables 6-8.    From Tables 3-5, the following observations were  obtained: -About displacements, the aperture of microgripper is bigger than the case of cantilever, due to the contribution of Vshape actuator. The aperture level is determinant in its possible application, such us microcellular manipulations. -The actuation force of microgripper is also bigger than the other cases (309 µN). This data can be also of interest for certain applications. -The conventional use of cantilever corresponds to the first modal form, in this case, given at 90.589 KHz. -In the case of V-shape actuator, its conventional use corresponds to 2 nd modal shape, in this case given at 140 KHz.
-For the microgripper, the knowledge of modal shapes allow us to determine the frequency to operate it. In this case, the appropriate modal shape corresponds to 6 th mode, given at 108.9 kHz. The geometry is determinant in the dynamic response of devices.
-Constant damping factors are considered on the total response of each device. It was observed a decrement on the displacements produced by these factors. In real environments, damping factors can be produced as effect of the medium where device is actuating. Each response can be observed in the graphs of harmonic response, frequency versus Amplitude, given in Tables 6-8. In all cases, the bigger amplitude corresponds to 1% of damping factor. -In addition, this analysis allow us to determine the maximum amplitude or displacement versus frequency. -About phase angle, in the case of cantilever, V-shape and microgripper, their angles are in the range 100º, -90º, and -105º. -As a result of simulation, the movement of each structure in each modal shape give us information about the behavior of the total structure.
In the case of the microgripper, it is possible to choose the more adequate frequency avoiding unappropriated aperture. About design, it is possible to add other structural elements in order to improve its performance.
In Tables 6-8, the critical nodal points are clearly observed. In all cases, displacement is inversely proportional to the damping factor.

CONCLUSION
In each analyzed device, fundamental natural frequencies and their corresponding modal shapes, using FEA, were determined. Steady-State Thermal, Static Structural, Modal and Harmonic Response modules were used, but for further development it could be necessary include Response Spectrum Analysis, Random Vibration Analysis and Transient Analysis.
Parameters of force and displacement of each device were calculated in order obtain the complete response of the microgripper. In addition, it was demonstrate, by means of FEA that the behavior of these structures with constant damping factor produce a phase angle shift of 100º, -90º, and -105º, for the case of cantilever, V-shape actuator and microgripper, respectively. A decrement on the displacement is also observed, accordingly to the damping factor increasing. The resonance frequency for each device is the same for all considered values of the damping factor.
Information obtained with dynamic analysis, for the case of the microgripper, makes possible to choose the more adequate frequency avoiding unappropriated aperture. About design, it is possible to add other structural elements in order to improve its performance.
The modal shapes analysis allows to understand the performance of the analyzed devices under each characteristic frequency, which determine their performance, and gives us the opportunity to choose the more adequate modal shape to complex systems operation and its use for specific applications of complex systems.
The large actuation force (309 µN) of upper microgripper allows it to be used in applications that needs this range of force and low frequency response (resonance frequency 100 kHz). The rear microgripper performance and its possible applications will be analyzed in future work.