- Date:22 Nov 2020
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The externally imposed electrostatic field is,operating here over a gap of 7 micro meters to. excite a specific vibration mode shape in the Y, direction see fig 2 while the rate of turn of these. devices creates by means of a coupling Coriolis,Fex 2mYe e 1. where ex e y ez are unit vectors in the direction of. the coordinate system in figure 2,Figure 4 Top view of the device under a. microscope,Figure 3 shows a photograph of the device where.

the proof mass the cantilever beam and a part of the. electronics can be clearly observed Figure 4,provide a broader view of the enclosure providing. the required vacuum level The sensor can observed,through the sapphire window in the middle. The overall behaviour of the sensor evidently, Figure 2 Principle of operation depends on several physical factors that need to be. compared with the numerical model Indeed in our, experiment it was found that several parameters in. Both modes the excited and sensed ones have particular mass and stiffness much like in full scale. close natural frequencies to allow for a sufficient structure were important Here due to the small. sensitivity of the device namely sufficiently large size the damping of the squeezed air film and some. vibration levels in the X and Y directions Equation 1 unexpected cross coupling effects caused by the. shows that sensed mode develops a deflection electrostatic field were observed. serving as a direct indication for the rate of turn Fully coupled analysis of the 3D electric. potential and the resulting electrostatic force which. is coupled to the vibration of the device require a. heavy computational load In fact such a simulation. may last for many hours or even more for a full, transient analysis For this reason we have used in.

this work an efficient modelling procedure making,use of the small gap between the electrodes. 2 Theoretical background,This section outlines the theory and the. assumptions behind the numerical model that was, Figure 3 Top view of the device under a 2 1 The structural part. microscope,The structural part consists of a material. anisotropic finite element model imported from,Ansys into Matlab by using SDT 3 The.

software utility allowed us to import the mass and. stiffness matrices as well as the complete geometry. of the structure that was used in the custom,electrostatic model. equation 2 can be computed by integration on the,volume which is contained only between the. electrodes This assumption practically means that,any fringing effects of the electric field are. neglected and the computation procedure is thus,greatly simplified. The volume over which the integration of equation, 2 is performed changes with the deformation of the.

upper electrode therefore the surface of the FE grid. is being used for the computation of the electrostatic. Figure 5 FE mesh on the upper electrode,As can be seen in right hand side of figure 5. each element had 3 DOFs for each of the 8 nodes, and the isotropic material properties that provided. nearly identical results to the anisotropic properties. are provided in Table 1,Figure 6 Air gap between the electrodes under a. deformed state,Table 1 Mechanical properties of silicon. From equation 3 we can obtain an expression for,the electric potential energy where a volume.

2 2 The electric part element can be expressed as h dx dy leading to. In this section we derive a closed form expression. for the generalised electrostatic forces acting at the V 2 dx dy. nodes For this purpose the electric potential energy 2 A h x y t. that is contained in the volume between the,electrodes is expressed as. Here h x y t is the air gap between the, U E D EdV 2 electrodes as shown in figure 2 The gap can be. 2v directly expressed using the structural shape,When D E 6 is substituted it is possible to. functions of the kth element, express the total potential energy as a function of. hk x y t h0 k i x y qi t 6,the electric field E i,1 Where h0 the initial gap between electrodes k i.

U E E 2 dv 3,2v structural shape function qi structural DOF. Having defined the mechanical degrees of freedom,we are able to obtain an expression for the. electrostatic force The generalized electrostatic force can be expressed. using equation 4 as,gi e i 1K n 4 V2 x y dxdy,qi gk t k i 2 7. 2 i A h x y t, Here gi is the generalised force that is related to the. Here the summation is around a single node of the,qi mechanical degree of freedom.

contribution of all surrounding elements see fig 7. The electric field is created by a potential,difference V t across the electrodes due to this. voltage an electrostatic force is developed This,force operates over a gap between the top and. bottom electrodes that remains much smaller than,its width and length Under this assumption see. 6 7 it is assumed that the total potential energy in. is achieved by a applying a scaling procedure to the. parameters followed by an order reduction scheme, converting a model with typically several thousands. degrees to 10 DOFs,A numerical model order reduction scheme leads.

to the form,Mq Cq Kq g q V 11,where M C K are the mass damping and stiffness. Figure 7 Electrode s deformation due to a virtual matrices while q is the vector of DOFs typically. motion of the kth DOF several thousands g q V stands for the. electrostatic force and V is the externally applied. An efficient method for a numerical evaluation of voltage. the electrostatic force is required and for this reason The order reduction scheme follows two stages. a semi analytical numerical integration scheme is where in the first stage a standard Craig Bampton. introduced approach 5 is taken while the following step. The structural shape functions on the electrode applies a balancing transformation 8 The. can be computed by as shown in fig 5 the relation balancing transformation has improved the. between unit displacement and coefficients Figure numerical conditioning considerably while detecting. 3 non controllable thus non affecting degrees of,1i freedom. The scaling procedure acts by defining new state,k i x y 1 x y xy 8 variables and a new time scale. Q T T t 12, For each element we have xn yn n 1K 4 and where is a reference frequency leading to. the shape function can be computed after solving,1 x1 y1 x1 y1 1i 1.

M 1C M 1 K,M 1 g Q V T,1 x y x y 0 2 h0 2,1 x3 y3 x3 y3 3i 0 where here Q Q. 1 x4 y4 x4 y4 4i 0 The new system has natural frequencies that are. scaled down by allowing the integration, The generalised force is computed at every time algorithm to take large time steps compared with the. step by means of a Guassian quadrature original time scaling. approximating the expression in equation 10,k i x y dxdy. 2 4 Coupled model computational,2 i ymin xmin h0 k i x y qi t 2. It is worthy of mentioning that the expression for The computational scheme can be visualised by the. the nodal electrostatic force depends on the block diagram in figure 8 In this figure the linkage. generalised displacements qk t and thus is a between the structural and the electrostatic part is. function of the mechanical deformation state highlighted Indeed for every newly obtained. deformation state a new distribution of the, 2 3 Model order reduction and electrostatic force must be recomputed.

The speed of execution and the appropriate scaling. of the parameters are important factors in the,transient analysis Efficiency is essential in. particular in light of the fact that equation 10 has to. be computed at every time step Greater efficiency,Figure 8 Computational scheme. Figure 10 Laboratory set up, The nonlinear nature of the coupling effect and the. electrostatic force distribution computed with In this set up a computer controlled excitation. equation 10 is demonstrated in figure 9 In this voltage is being used to excite vibrations in the. figure 3 formation states belonging to 3 different device while it is contained in a vacuum preserving. modeshapes are shown For each modeshape the enclosure The device was placed under a. matching electrostatic force distribution as microscope equipped with a scanning laser sensor. computed by equation 10 is shown The vacuum level in the enclosure see fig 4 was. controlled by an external vacuum pump and,monitored by a special gauge. A parametric study was carried out and the results. are depicted below, Firstly the effect of DC and AC voltage levels was.

examined It is a well known phenomenon in,electrodynamics that the DC and AC amplitudes. affect the dynamics of such devices but is worthy of. an experimental study since this effect is difficult to. quantify with a great precision when replying on a. model The precision of the device which is used as. a sensor depends to a great extent on a successful. Figure 9 Mode shapes and Electrostatic force calibration and characterisation. distribution,Figure 9 demonstrates the nonlinear dependency of. the electrostatic force magnitude on the local air. 3 Experimental work, The laboratory set up is schematically depicted in. Figure 11 Response 1st harmonic under Vdc 1 3 V,Indeed figure 11 shows that in addition to. obtaining larger response amplitude higher DC,voltage shifts the average resonance frequency by.

about 2 Hz This fact can be explained by, examination of equation 7 When an input voltage For an excitation voltage with a single harmonic. of the form as in equation 14 we have measured,V Vdc Vac sin t 14. is being used the force is proportional to sin t,V 2 Vdc 2 Vac2 2VdcVac sin t Vac2 cos 2 t 15. It can be seen that the amplitude of the first, harmonic of the force is directly proportional to Vdc. The change in the natural frequency can be 2, explained by considering a simplified model of the.

electrostatic actuator in the form of a parallel plates 1. capacitor In this device the force behaves,according to FN or. 500 1000 1500,AV 2 AV 2 Hz, The second term in equation 15 behaves like a Figure 13 Decomposition of the measured response. negative stiffness term thus having a softening into harmonics. The second harmonic in figure 13 seem to have,A similar effect is achieved when the AC voltage. been excited at half the first natural frequency of the. is increased,device which behaves in a linear manner by itself. The effect of vacuum level on the damping,Indeed the harmonic content of the response was.

appears in figure 12 Here the increase in pressure. investigated for the entire frequency range of,adds air that serves as spring and damper. interest and a typical output under an excitation of. simultaneously indeed larger pressure stiffens the. 585Hz is analysed in figure 14, devices and an increases in the resonance frequency. f 585 DC 3 AMP 2, can be observed Damping can be seen to increase 0 4 0 2. under a larger pressure Indeed pressure elevation 0 2 0 15. causes a decrease in response amplitude,0 200 400 600 01234 56789. 0 04 0 02 0 0 02 0 04 0 4 0 2 0 0 2 0 4,Figure 14 Top left The measured the fitted.

Figure 12 Effect of pressure level,response and the residual Top right relative. amplitudes of the various harmonics Bottom left, An anomaly compared with ordinary structures is amplitude distribution of the residual Bottom. that the effective excitation force and the response right Amplitude distribution of the response. are characterised for this device by means of several. harmonics A numerical study showed us that the,Figure 14 shows that the Fourier series model. response can be modelled as a Fourier series of the. provides a good fit to the measured response This,fact is being strengthened by the probably. y H i cos i t i Residual 16 distribution function of the residual that seems to. i 1 posses a Guassian character The top right plot. shows that not only the DC 0th harmonic and the P 0 01 mbar No 1st harmonic was present in this. first two harmonics participate but also the 3rd and case. 4th harmonics are dominate at the specific operating. frequency This is contrary to the capacitor like, lumped model often used in the context of MEMS The measured response showed an additional.

The response amplitude distribution shown on spike that corresponded to a vibration in the x. the lower right of fig 14 is typical for sinusoidal direction see figure 2 This mode should not have. signals with slight asymmetry that can be attributed been excited by the electrostatic force acting solely. to the asymmetric nature of the electrostatic force in the y direction An investigation of this. phenomenon is described in the following section,3 1 Comparison with simulations. Comparisons of the measured response with 4 Internal resonance model. two models were conducted The models consisted, of the above mentioned finite element model A fine frequency scan steps of 0 1 Hz was. having initially 1400 DOFs that were reduced to conducted to allow for a verification of this. 10 and a simplified model for the lower modes spurious peak. treating the devices a as a two DOF structure,Out of plane 1175 Hz. In plane 1273,Frequency Hz, Figure 15 Two DOF model of the device Figure 17 A fine frequency scan showing a sharp. peak alongside with the main resonance,The simulated response for this case showed a.

good fit in terms of frequency and some inaccuracy The fact that this peak seemed repeatable and. On the dynamics and modelling of a micro electro mechanical structure MEMS I Bucher 1 A Elka and E Balmes2 1Dynamics Laboratory 2MSSMat Faculty of Mechanical Engineering Ecole Centrale Paris Technion Haifa 32000 Israel 92295 Chatenay Malabry Cedex France e mail bucher technion ac il e mail balmes mss ecp fr Abstract This paper describes a numerically efficient modelling procedure

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