A Commentary on the Paper Titled “Nonlinear Dynamics of a Micromechanical Torsional Resonator: Analytical Model and Experiments“ by Dario Antonio and Hernan Pastoriza

UniversitA˜ degli Studi di Genova

MEMS (EMARO 09/10)

A commentary on the paper titled “Nonlinear dynamics of a micromechanical torsional resonator: analytical model and experiments“ by Dario Antonio and Hernan Pastoriza

Professor: Prof. M. Zoppi

Authors: Alex Jahya Ali Paikan Komal Rauf

January 17, 2010 Abstract This report has been written to fulfill the requirements for the course MEMs. The aim of this report is to study in detail the chosen reference and to simulate the mathematical model presented in [1] for the purpose of understanding and verification. This report compares the analytical, simulated and experimental results for the torsional oscillator discussed in the chosen reference and verifies the design and analysis carried out by the authors for the dynamics of a micromechanical torsional resonator.

1 Introduction

The paper chosen for study [1] is a fairly recent paper which presents a complete mathematical description of an electrostatically actuated torsional oscillator along with the comparison of the model to the experimentally measured oscillator response. Several softwares have been explored and utilized to achieve this task as the secondary purpose of the report was to acquaint the authors with the state of the art tools present for mathematical modeling and analysis of Microelectromechanical Structures. Microelectromechanical structures have gained a lot of importance in the recent years due to their vast applications. The need for reliable modeling and simulation is more in micro structures because their fabrication is difficult and costly and optimization in their design is required to obtain the desired characteristics. Several analysis tools are available and it is the need of the hour to acquaint ourselves with them. As described in [1] the studied mechanical resonator is inherently linear, and the nonlinearities are introduced by the electrostatic driving and detection however, for the purpose of this report we only analyze the oscillator in the linear mode. We first study the mathematical derivation of equations by the author and then the oscillator is implemented in two CAD softwares. The motion of the oscillator is then simulated and compared to the analytical and experimentally measured oscillator response presented in [1] using modal analysis from the two softwares. The remainder of the report is organized as follows. Section 2 describes the analytical analysis and mathematical model verification carried out of the resonator in Maple Software. Section 3 describes the geometrical and modal analysis of the oscillator as carried out in the Salome Software. In Section 4 we present the geometrical and modal analysis of the oscillator carried out in CATIA Software and the comparison of the results obtained through both the SALOME and the CATIA simulations. It also discusses the results obtained through simulation and their comparison to the results presented by the authors in [1]. Finally, Section 5 contains some concluding remarks.

2 Analytical Analysis

In this section the objective was to analytically verify the contents of the paper. The Mathematical Modeling and the Mechanical Description/Properties of the oscillator discussed in [1] were studied in detail and were re-calculated/veriﬁed using the Maple Software.

1 Figure 1: Image of the Mechanical Torsional Oscillator

2.1 Mechanical Properties As described in [1] the polysilicon mechanical microresonator consists of 106× 44m2 suspended central plate that is anchored to the substrate by means of two serpentine springs as shown in ﬁgure 1. The microresonator was designed for applications requiring torsional oscilla- tions then the elastic constant k of the serpentine springs corresponding to this mode of vibration was calculated analytically in [1] according to the following formula;

Eωt3 K = (1) 24(N + 1)l0

In this case E = 158GP a, w = 1.7m, l0 = 28m, t = 1.5m and N = 2 then according to the equation (1) k = 4.496 · 10−10N · m. Alternatively, for the torsional mode of vibration the elastic constant of the spring may also be calculated as T K = (2) θ where θ is the rotation angle in radians and T is the applied torque. To verify if both the formulas yield the same result we analyzed the deformation of the single spring in simulation, one end of the spring was constrained while a pure moment corresponding to a small rotation angle, approximately 5 degrees, was applied to its free end. From the rotation angle obtained from simulation and the equation (2) we have the value of the spring constant, K = 4.49·10−10N ·m verifying the elastic constant. The moment of inertia I for the full structure is approximately the same as the moment of inertia of the plate and as such may be employed to verify the value presented in the paper. The moment of inertia of a rectangular plate for this torsional mode is given by

m(w2 + t2) I = (3) 12 The density of polysilicon = 2330kg/m3, and the volume of the plate is calculated as V = l × b × h = 6.96 · 10−15, then the mass of the polysilicon plate may be calculated as m = density · V = 1.63 · 10−11. Using these values and the

2 dimensions of the plate as given by the authors, we obtain I = density · V = 2.63 · 10−21

2.2 Mathematical Model Veriﬁcation The authors detail the analytical model for the dynamics of an electrostatically actuated torsional oscillator in [1] and our aim was to study the complete mathematical description and solve/verify them using Maple.

2.2.1 Electrical Drive and Detection The mechanical resonator is actuated and its motion is detected using under- lying electrodes. It can be easily veriﬁed that when there is no motion of the plate, zero voltage condition, it is parallel to the electrodes, forming two parallel plate capacitors of the form ε ld C = 0 (4) g

where ε0 is the permittivity of the free space, l is the length of the moving plate, d is half the width of the moving plate and g is the initial separation between the plates of the capacitor. Applying a voltage to one of the electrodes causes the moving plate to rotate by an angle θ. The plates and the capacitor are no longer parallel however, we can approximate this capacitor by a combination of N parallel-plate capacitors all connected in parallel with the total capacitance given by

d Z cos θ C = ε0l( dx) (5) gx 0

where gx = g0 + xsin(θ) for the excitation capacitor and gx = g0 − xsin(θ) for the detection capacitor. Developing the equation (5) in Maple yields the following result

ε lcos(θ) g + dsin(θ) C = 0 log( ) (6) e sin(θ) g however according to [1] the result of the integration of the equation (5) for the excitation capacitor is given as

ε l g + dsin(θ) C = 0 log( ) (7) e sin(θ) g clearly there is a conflict between the results presented in [1] and the results as derived from maple. We tried to find a way to prove that both equations (6) and (7) are equivalent but we were not able to find an explanation for that. Similarly for the detection capacitors we have the following results given by maple and the paper respectively

ε lcos(θ) g C = 0 log( ) (8) d sin(θ) g − dsin(θ)

3 ε l g C = 0 log( ) (9) d sin(θ) g − dsin(θ) These equations were further utilized for developing the mathematical model for the motion of the oscillator. We tried to use both the equations (6) and (7) for the excitation capacitor and equations (8) and (9) for the detection capacitor in the development of the mathematical model for motion of the oscillator for the purpose of comparison and clariﬁcation.

2.2.2 Mathematical Modeling The torsional oscillator obeys the following equation of motion:

d2θ dθ I + Γ + kθ = τ(θ, t) (10) dt2 dt where I is the moment of inertia of the oscillator, Γ is a damping constant, k is the linear elastic constant and τ(θ, t) is the applied torque which may be calculated for both the actuation and the detection electrodes as:

V (t)2 dC V 2 dC τ(θ, t) = e e + b d (11) 2 dθ 2 dθ where Ve is the ac voltage applied to the actuation electrode and Vb is the dc bias for the detection electrode. Replacing equations (7) and (9) in the equation (11), calculating the deriva- tives and expanding the result out to the cubic order in θ we obtain the following expression for the total torque:

−1 lε d2(V 2 − V 2) 1 ld3(V 2 + V 2) τ(θ, t) = 0 e b + e b θ 4 g2 3 g3 1 lε d2(g2 − 3d2)(V 2 − V 2) 2 lε d3(5g2 − 9d2)(V 2 + V 2) + 0 e b θ2 − 0 e b θ3 (12) 8 g4 45 g5 The result (12) is similar to that obtained in [1]. Replacing equations (6) and (8) in the equation [11], calculating the deriva- tives and expanding the result out to the cubic order in θ we obtain the following expression for the total torque:

−1 lε d2(V 2 − V 2) −1 ld(3g2 − 2d2)(V 2 + V 2) τ(θ, t) = 0 e b + e b θ 4 g2 6 g3 1 lε d2(4g2 − 3d2)(V 2 − V 2) 180 lε d(15g4 − 100g2d2 + 72d4)(V 2 + V 2) + 0 e b θ2+ 0 e b θ3 8 g4 1 g5 (13) This result (13) is equivalent to the result obtained in [1] in the linear case, however, for the nonlinear case it has some additional terms which are not addressed in [1]. For the linear case, where |θ|d << g, we can drop the terms with the factor θ in (13) and [12] yielding the same result τ(θ, t) = τ0cos(wt) where τ0 = 2 2 2 −ε0ld Vac /8g . The solution to the diﬀerential equation given in (10) is then obtained the same as detailed in the paper by the authors.

4 3 Analyzing the resonator in SALOME

In this step we analyze some parts of the structure such the elastic constant of a single spring, two springs, eﬀect of the inertia as well as the natural frequency of the whole structure in an open-source software called SALOME. In section 2, the results from the simulation has been compared to the equation given in the paper. Therefore here we just detail more on F.E simulation and modal analysis aproaches in SALOME.

3.1 What is SALOME? SALOME is an open-source software that provides a generic platform for Pre- and Post-Processing for numerical simulation. It is based on an open and ﬂexible architecture made of reusable components. SALOME is a cross- platform solution which is distributed as open-source software under the terms of the GNULGPL license. SALOME can be used as standalone application for generation of CAD models, their preparation for numerical calculations and post-processing of the calculation results. It can also be used as a platform for integration of the external third-party numerical codes to produce a new application for the full life-cycle management of CAD models [2]. Salome-Meca-2009 (the software we used in this study) contains the newly inte- grated software suite combining Salome v.4.1.6 Pre-Post processor & Code-Aster v9.4 FE solver [4] and includes the easy to use Code-Aster module for Salome and its Analysis Wizards [3]. Further information about Salome-Meca and its features can be found on the oﬃcial website addressed in [3].

3.2 Calculating the elastic constant of spring The torsional natural frequency of the structure is highly depended on elastic constant of serpentine springs. To conﬁrm that the value of elastic constant of the spring given in the paper by Equation (1) is correct, we recalculate the constant by analyzing it in the software. Considering the constant of a serpentine spring can be calculated by dividing the torque over the angular rotation, we constrain one side of the spring while applying torque to the other side as shown in Figure (2). To check the accuracy of the constant of a serpentine spring given by Equation (1), (K = 4.496x10−10), we apply a desired torque to the free side of the spring and measure the rotational angle from simulation. To generate the desired torque around desired axis (here x axis), we add two small sections to the free side of the spring to apply reverse pressure on the surfaces pointed by arrows in Figure (2.a). the area of each surface is A = −12 −6 2.25x10 and the distance between two surfaces is ds = 4.5x10 . Considering Equation (2) and knowing K = 4.496x10−10, to rotate the spring around x axis by θ = 0.0873rad (or 5deg) we need to apply τ = 3.92x10−11. Having A and ds, the amount of the pressure should be applied to the surfaces is P = 3.85x106. As shown in Figure (2.b), to measure the rotational angle of deformed spring, we need to measure the translation of the side surfaces of the spring along the vertical axis (z axis). Measuring this translation and having the width of the dz spring, we can simply measure the angle of rotation from θ = arcsin( l ) where l is the half length of the spring.

5 Figure 2: a) applying two inverse forces to one side of a spring generating torque around desired axis. b) front view of deformation of the spring due to applied torque.

To precisely measure the translation of the deformed surface on the side of the spring, we use the Cutlines feature of the software which gives the translation of the nodes on the desired surface after F.E analysis. the Cutline surface also shown in Figure (2.a) as dotted surface. As shown in Figure (3), the most deformed area is the last loop of the spring close to the free side. The lower x position of the nodes belongs to the area close to the constrained side which is clear to have less deformation than other parts. From the simulation results, −6 averaging the most translated nodes along the z axis, we obtain dz = 1.21·10 and θ = 0.0865rad which is almost same to the desired rotational angle. This simulation has been repeated for diﬀerent value of θ to conﬁrm the value of elastic constant of the spring given in the paper is equal to what obtained from simulation.

3.3 Cheking the phenomenon of summing the elastic constants and the effect of inertia Equation (1) given in the paper to calculate the elastic constant of the serpentine spring, is only for one spring. Then the author simply multiplies it by two for two serpentine springs. Having the doubt that whether the constant of two springs connected by a plate can be summed, we remove the plate from the structure discussed in the paper and connected two springs directly to each other as shown in figure (4.a). The natural frequency of this structure resulted by simulation is 514Khz. Going one step forward, as shown in figure (4.b), we connect the springs by a beam which has the same length as the plate in the structure. Simulation result shows that the natural frequency is 532Khz. In- creasing the frequency in this manner cannot have any logical reason and it can be said that it is due to the difference in meshing and the accuracy of the modal analysis software in two run of simulation. Morever, the inertia of connection beam in this case is negligible and it cannot have so much effect on the natural frequency since the beam is in the center of torsional resonating.

To study the eﬀect on the inertia on natural frequency of the resonators,

6 Figure 3: Cutlines on deformed surface. Hotizental axis shows the position of the node on the x axis and the vertical axis shows the translation of the node on deformed surface along z axis.

we increase the width of the beam to double which increases the moment of inertia consequently and the result obtained shows the frequency decreases to 250Khz, and (shown in Figure (4.c)). In the last experiment the width of the connection beam increased by ﬁve times while the natural frequency decreased by two times. Considering that the change in the natural frequency is not q K linearly proportional to change of the inertia and follows the formula ω = I , it make sense to obtain lower frequency (proportional to square root) while increasing the inertia.

3.4 The effect of different meshing During the first runs of modal analysis software, it did not result the natural frequency exactly same as value given in the paper. The effect of different meshing algorithm could be a reason to this conflict. Therefore, Clarifying this doubt, we tried to study the effect of different meshing and increasing the number of nodes. Mesh creator of SALOME can apply different algorithm for meshing but depend on the size of the structure and the parameters chosen for the algorithm, it might failed or end up with some error rate in meshing. Table (3.4) lists the different meshing algorithms supported by SALOME. First we tried almost all the possible meshing algorithm to mesh the structure more precisely varying from different number of elements and nodes. The results from applying different meshing algorithm were not so different from each others. Increasing the number of elements and nodes could have some small effects on the accuracy of the results but another problem arises due to lack of memory and computational capability. Therefor, using submeshing and compound meshed, we mesh the edges more precisly than other parts such as plate in which we cannot find big different results of meshing. Figure (5) shows the two preferred

7 Figure 4: a) two springs, b) two srings connected by a beam, c) two springs connected by a wider beam, d) two springs connected by a plate.

meshing method for the structure. After trying diﬀerent meshing algorithm and the number of elements, we concluded that NetGen meshing is more preferable than others due to accuracy and simplicity during the modal analysis.

1D 2D 3D Composite side discretization Triangle Hexahedron Projection 1D Netgen 1D-2D Netgen 1D-2D-3D Wire discretization Netgen 2D Tetrahedron Use existing edges Projection 2D 3D extrusion - Quadrangle Projection 3D - Use existing faces Radial prism 3D

Table 1: Meshing algorithms supported by SALOME

3.5 Modal analysis of the whole structure Clarifying the eﬀect of the inertia and the phenomenon of summing the elastic constants as well as understanding diﬀerent meshing, we can analyze the resonator in the SALOME. NetGen algorithm chosen for meshing the resonator which produce 6320 nodes connected in volume network. As shown in Figure

8 Figure 5: a) Applying very ﬁne NetGen meshing to the resonator. b) Using separated meshing for plate and springs

(4.d), free sides of the both springs are constrained. Young module and the density of the material are taken from the [1] (E = 158GP , RHO = 2330kg/m3) and the poisson ratio is estimated NU = 0.3. The natural frequency of the resonator results for modal analysis is 88Khz which is almost equal to 80Khz given by the paper.

4 Analyzing the resonator in CATIA

In the second part of the exercise, we are going to use CATIA and its built-in FEA features to compare the result obtained by the program “SALOME” in Linux. However, in CATIA environment V5, its FEA is only applicable for element in the minimum size of 10−4 m. Thus, because of this limitation, we are going to amplify the structure 1000x. Therefore, we are going to work in the scale of mm instead of µm. The first part of using the FEA analysis in CATIA is to use its sketcher and Part Design Workbench to create the desired structure. Since the two end support structures are not going to contribute significant change to the end result, the structure of the mechanical torsional oscillator can be simplified into the following structure in figure (6) with boundary conditions shown at the two ends of the springs. Having created the desired structures in mm scale, we can then proceed forward to use the Analysis for Free Frequency Analysis. In this analysis, similar to ANSYS, we have to create mesh of the structures for computation. In CA- TIA, there are two types of solid elements available in CATIA V5: linear and parabolic. Both are referred to as tetrahedron elements and shown below in figure (7). The linear tetrahedron elements are faster computationally but less accurate. On the other hand, the parabolic elements require more computational resources but lead to more accurate results. Another important feature of parabolic elements is that they can fit curved surfaces better. In general, the analysis of bulky objects requires the use of solid elements. In this analysis, we are using the parabolic tetrahedron meshing of element size 5 mm with absolute sag value of 1 mm. The concept of element size is

9 Figure 6: Microstructure shown with the constraints

Figure 7: Tetahedron elements. Left is linear and right is parabolic

self-explanatory. A smaller element size leads to more accurate results at the expense of a larger computation time. Thus, we should create a balance between the accuracy and computation time. The “sag” terminology is unique to CATIA. In FEA, the geometry of a part is approximated with the elements. The surface of the part and the FEA approximation of a part do not coincide. The “sag” parameter controls the deviation between the two. Therefore, a smaller “sag” value could lead to better results. The resulting meshed structure is displayed in the figure (8). Prior to the computation of the free frequency of the structure, we have defined as well according to the paper the structure material properties. In this structure, we are using E = 158 GPa and ρ = 2330 kg/m. The resulting stress analysis, maximum displacement and displacement with nodes ( figure (9), (10), and (11) respectively) can then be obtained and studied. For this analysis, we have specified for calculation for 200 iterations and 10 modal frequency analysis. The modal frequency result from CATIA is 94kHz. From figure (11), we know that the maximum displacement of the structure is displacement = scaling factor · max displacement = 0.0086 · 12.2 m = 9.76 · 10−3 m.

10 Figure 8: Structure with its mesh

Figure 9: Structure with Von Misses Stress

4.1 Justiﬁcation of the results obtained The result of the analysis can be justiﬁed as per following. From the previous exercise that has been done to check the validity of the spring constant formula, we know that the formula given in the paper is valid. Thus, we know that scaling of the structure of the spring will be governed by

k ∼ t3 (14) In this case the value of k will be increased by the factor of 10−9. Corre- sponding k value of the two serpentine springs is 8.94 · 10−1N · m. The next step is to ﬁnd out the inertia value, I of the serpentine springs. Using the I value of a rectangular cuboid, 1 I = · ρ · v · (h2 + w2) (15) 12 Thus, the value of

I ∼ v · (h2 + w2) (16)

11 Figure 10: Maximum displacement of the structure

Figure 11: Structure with displacement nodes

The value of I will be increased by the value of 1015. Therefore using the formula of r k ω = (17) o I

the value of ωo will be scaled down by 1000Hz compared to the value produced by the SALOME programme since r 109 ω = = 10−3 (18) o 1015 The actual value produced by CATIA is 94kHz and in comparison the value produced by the SALOME is 88kHz.

5 Conclusion

In conclusion, we have veriﬁed the experiment done in the paper with two diﬀerent FEA analyzers as well as with Maple for the analytical modeling. There

12 is some difference between the analytical model obtained by the authors and the one given in the reference paper for the nonlinear range, however, for the linear range both the analysis yield the same results. In SALOME, we have re-created the structure in micro scale and verified the result of the experiment virtually. The result obtained is then compared with CATIA in the mm scale since we encounter the limitation of the CATIA in the MEMS scale. However, despite different softwares and scales used, the result of the analyzer comes to a very close range. The result differs only by 6% with result from SALOME coming to 88 kHz and from CATIA coming to 94 kHz.

References

[1] D. Antonio and H. Pastoriza: “Nonlinear dynamics of a micromechanical torsional resonator: analytical model and experiments”, Journal of microelectromechanical systems, VOL.18, NO.6, December 2009

[2] The open source integeration platform for numerical simulation: SALOME key features, http://www.salome-platform.org/

[3] Open-source powered engineering: SALOME-MECA 2009, http://www.caelinux.com/

[4] Code-Aster oﬃcial website: About Code-Aster, http://www.code-aster.org/