A MEMS gravimeter with multi-axis gravitational sensitivity
Richard P. Middlemiss, Paul Campsie, William Cunningham, Rebecca Douglas, Victoria McIvor, James Hough, Sheila Rowan, Douglas J. Paul, Abhinav Prasad, G. D. Hammond
AA MEMS gravimeter with multi-axis gravitationalsensitivity
Richard P. Middlemiss ([email protected]), ∗ †
Paul Campsie ([email protected]), ∗ William Cunningham ([email protected]), ∗ Rebecca Douglas( [email protected]), ∗ Victoria McIvor ([email protected]), ∗ James Hough ([email protected]), ∗ Sheila Rowan ([email protected]), ∗ Douglas J. Paul ([email protected]), † Abhinav Prasad ([email protected]), ∗ and G. D. Hammond ([email protected]) ∗ Corresponding authors: Richard P. Middlemiss (+44(0)141 330 5241), and Giles D. Hammond(+44(0)141 330 2258) ∗ Institute for Gravitational Research, S.U.P.A., School of Physics and Astronomy, University of Glasgow, KelvinBuilding, University Avenue, Glasgow, G12 8QQ, U.K. † James Watt School of Engineering, University of Glasgow, Rankine Building, Oakfield Avenue, Glasgow, G128LT, U.K. a r X i v : . [ phy s i c s . a pp - ph ] F e b bstract A single-axis Microelectromechanical system gravimeter has recently been developed at the Uni-versity of Glasgow. The sensitivity and stability of this device was demonstrated by measuringthe Earth tides. The success of this device was enabled in part by its extremely low resonantfrequency. This low frequency was achieved with a geometric anti-spring design, fabricated usingwell-established photolithography and dry etch techniques. Analytical models can be used to cal-culate the results of these non-linear oscillating systems, but the power of finite element analysishas not been fully utilised to explore the parameter space before now. In this article, the results ofprevious analytical solutions are replicated using finite element models, before applying the sametechniques to optimise the design of the gravimeter. These computer models provide the abilityto investigate the effect of the fabrication material of the device: anisotropic < > crystallinesilicon. This is a parameter that is difficult to investigate analytically, but finite element mod-elling is demonstrated here to provide accurate predictions of real gravimeter behaviour by takinganisotropy into account. The finite element models are then used to demonstrate the design of athree-axis gravimeter enabling the gravity tensor to be measured - a significantly more powerfulsurveying tool than the original single-axis device. Gravimeters have applications in air and land-based oil and gas exploration , sinkhole analysis ,the detection of subterranean tunnels and cavities , CO sequestration , geothermal reservoirmonitoring , archaeology , hydrology , and volcanology . Commercial gravimeters areall expensive ( $100,000 USD) and use a range of different technologies for different applications.The commercially available gravimeters all require levelling, and this is carried out manually or be2ncorporating additional components to automate this process.In previous work , the development of a low frequency microelectromechanical system(MEMS) gravimeter with a sensitivity of 4 × − ms − / √ Hz was discussed. This device has sincebeen miniaturised and undergone field-testing . A series of sensors are currently being builtfor integration within the NEWTON-g volcano gravity imager at Mt Etna, Sicily . The deviceis capable of high acceleration sensitivity in part because of its extremely low resonant frequencyMEMS resonator. This resonant frequency was achieved via the use of a geometric anti-springdesign for the mass-on-spring system. A low resonant frequency means that the ratio is minimisedbetween the spring constant, k , and the mass of the proof mass, m , giving a larger displacementfor a given acceleration, and thus greater potential sensitivity for the gravimeter.An anti-spring can be characterised as having a negative or at least partially negative restoringforce. As an anti-spring is extended, the spring constant decreases. One way to design an anti-spring is by using curved monolithic cantilevers, connected at a central point to constrain themotion vertically . Monolithic geometric anti-springs are used as low frequency seismic isolatorsin the VIRGO gravitational wave detector . With the aim of achieving a high accelerationsensitivity for the MEMS gravimeter a monolithic geometric anti-spring configuration was chosen.A monolithic geometry was important because this allowed for the device to be fabricated from asingle silicon chip.An elegant analytical description of monolithic geometric anti-springs (as used in the VIRGOdetector) was written by Cella et. al. . In this mathematical solution, the authors devise sim-plified models to describe the behaviour of this non-linear spring – taking a Lagrangian approach.They do so in order to avoid ‘the brute force of finite element analysis’. It is our belief, however,that there is also value in conducting finite element (FE) analysis on anti-spring systems. It allowsone to investigate the physical consequences of making very small adjustments to the design. Fur-3hermore, with the advance in computing power since 2005 – when Cella et. al. published – onehas the option to be more carefree with computer models that require significant processing power.Little work has been conducted in the FE analsysis of geometric anti-springs since this time. Oneexception is the recent work of Zhang et. al. , who have explored the use of FE analysis to assessthe thermal behaviour of geometric anti-springs. Zhang et. al., however, did not demonstrate thatthe computational results were self-consistent with earlier analytical works in the literature.Here, the results of Cella et. al. are replicated, first analytically, and then in an FE model.After demonstrating that these two models are in agreement, it is demonstrated that FE modellingoffers great advantages to the design process. The crystal structure of the fabrication material – < > silicon – can be included in the FE model. Since this material has an anisotropic Young’sModulus, modelling such a parameter in an analytical model would be extremely difficult. Finally,it is demonstrated that by tuning the MEMS resonators in the FE model it is possible to con-struct a system of three identical devices in a triaxial configuration that would have accelerationsensitivity in x , y , and z . A system that enables the measurement of the vector components ofgravity would not need the same stringent levelling before measurements are undertaken. Thiswould improve the practical use of the gravimeter for a range of applications. In order to verify the FE models of the MEMS based geometric anti-spring, a simple analyticalmodel – based on the work of Cella et.al. – was created. This model is displayed in figure 1a.The model outlines the parameters of a single geometric anti-spring blade. Such blades are neverused individually; the proof mass is suspended from two or more symmetric blades, constraining it4igure 1 The geometries of the springs used in the ANSYS finite element models. a) A simplified1-dimensional analytical schematic diagram of the geometric anti-spring. b) The single springmodel with boundary conditions applied to the free end to fix the displacement and rotation.A point mass of 8 × − kg is applied to the free end of the spring. This system is modelledusing beam elements. c) The same model as a) , but with two symmetric springs and a pointmass of 1 . × − kg. In this model the boundary conditions on the free end are removed. d) The same design as 1c but with solid elements. In all cases a load step of 1 is equal to applyingan acceleration of 9.8066 m/s . e) The final design of the 4-spring geometric anti-spring MEMSsensor, as modelled in ANSYS. The results of models conducted with this geometry would onlyconverge if a dense mesh density was utilised over the springs.5o move along a vertical axis; thus reducing the analytical problem to one dimension. The springis clamped at the base with a launch angle of θ and is constrained at the proof mass, or springtip, by an angle θ L . This results in a boundary value problem that can be conveniently solvedin MATLAB with the bvp4c algorithm . In order to achieve this solution the system must bebroken down into a pair of coupled first order equations, which can be defined in terms of thecurvilinear coordinate along the blade, l . dpd(cid:15) = G x sin θ − G y cos θ (1) dθd(cid:15) = γ ( (cid:15) ) p (2)where θ is the angle that the spring arc makes with the neutral axis at a given position ( x, y ); (cid:15) = l/L (where L is the total arc length of the spring); γ ( (cid:15) ) = w (0) /w ( L ) is the spring’s profilewhere w () is the spring width at the base (0) and at the end ( L ) respectively. The ‘width’of the springs correspond to the dimension into the page in figure 1a. The springs utilised ingravitational wave seismic isolation systems are triangular in order to maintain a constant stressalong their length i.e. their width varies along the length of the spring. Due to the limitationsof etching silicon, however, the MEMS springs are rectangular and thus γ ( (cid:15) ) = 1 for this work.The parameters G x and G y are dimensionless forces that are applied to the spring tip. These arerelated to the real forces via: G i = 12 L Ed w (0) F i with i = x, y (3)where E is the Young’s modulus, and d is the thickness of the spring. Finally, the ( x, y ) coordinates6f the spring profile are determined from: x = L (cid:90) sin( θ ( (cid:15) )) d(cid:15) (4) y = L (cid:90) cos( θ ( (cid:15) )) d(cid:15) (5)As mentioned above, the tip of the geometric anti-spring is constrained to move along a verticalline. This is equivalent to requiring that the horizontal position of the tip always maintains aconstant value. This constraint is defined via the compression ratio x com = x tip /L . As the verticaldisplacement and/or compression ratio increases, the 2 nd spring provides the force G x , whichintroduces the negative component of the spring constant (the 2 nd spring is not displayed in figure1a, but it is symmetrically mirrored in the vertical plane that intersects the spring tip). This is thereason why the term “geometric anti-spring” is used; the anti-spring nature comes from geometryalone. As demonstrated in Cella , for certain compression ratios the spring exhibits a bifurcationin its force-displacement curve resulting in two stable operating points. This is not a desirablesituation, and so in this paper only solutions that exhibit a single operating point are considered.In a displacement-force graph this will be the position of maximum gradient.The MEMS devices were fabricated with springs with zero initial tension and the loadingis provided via gravity alone. A different approach is therefore taken to solving the equationsdescribed in Cella ; at the initial starting point the forces must be G x = G y = 0. For the simpleanalytical model, the following values were used: E = 169 GPa, d = 7 µ m, w = 240 µ m and L = 5 .
48 mm. This matches the typical MEMS gravimeters previously fabricated . The FEsolutions of such systems will be described later in section 2.2.4. A launch angle of θ = π/ θ L = π/
6. For zero initial starting force and an arc length of7 mm the resulting compression ratio is 0.933, which as demonstrated in Cella produces a stablesystem. For the MEMS system, operating at smaller compression ratios can be achieved simplyby increasing the launch angle such that the arc length increases while the tip remains at the sameposition. The solution for zero initial force ( G x = G y = 0) results in a spring which follows asimple arc. For an arc length of 6 mm, θ = π/ θ L = π/ .
58 mm.The procedure for analytically generating the force-displacement graph is as follows. A verticalload component is selected and the boundary value problem solved. The value of the horizontalforce is then adjusted until the desired spring tip horizontal compression is achieved. For thesemodels a tolerance was chosen of 1 × − . The value of the vertical force and vertical spring tipwere then stored, and the process repeated for different values of vertical force. Figure 2 displaysthe results of the analytical model together with those of the FE model, which will be describedin the following section. In this figure the y-axis is the calculated vertical spring tip displacement,and the x-axis is the equivalent load step. In this case a load step of 1 is equivalent to a force of7 . × − N (since a point mass of 8 × − kg corresponds to the mass of previously fabricatedMEMS devices) .The reduction in the gradient of this graph is equivalent to the system softening around a work-ing point of 0 . f , of the system using equation 6: f = 12 π (cid:114) km (6)where k is the spring constant and m is the mass suspended from the spring. k can be determinedby numerical differentiation of the force-displacement curve in figure 2. The results of this differ-8 −3 Load Step S p r i ng T i p D i s p l a c e m en t ( m ) Analytical ModelFEA Model: single spring using beam elements (see Fig. 4a)FEA Model: double spring using beam elements (see Fig. 4b)FEA Model: double spring using solid elements (see Fig. 4c)
Figure 2 FE Vs. Analytical Force-Displacement Comparison. A displacement-force graph com-paring the MATLAB analytical model (red series) and the ANSYS finite element models. Thecyan series is the FE model for a single spring, modelled using ANSYS beam elements. The blueseries is the FE model for a double spring, again using beam elements. The green series is also adouble spring configuration, but ANSYS solid elements have been used instead. The position ofmaximum gradient corresponds to the stable oscillation point.9 F r equen cy ( H z ) Analytical ModelFEA Model: double spring using beam elements (see Fig. 4b)FEA Model: double spring using solid elements (see Fig. 4c)
Figure 3 FE Vs. Analytical Frequency Comparison. A comparison of the resonant frequencies ofthe MATLAB analytical model (red) and the ANSYS finite element models. The blue series isthe FE model for a double spring, modelled using ANSYS beam elements. The green series is adouble spring configuration, modelled using ANSYS solid elements. It can be observed that as theloading on the spring increases, the resonant frequency decreases, goes through a minimum, andthen increases again.entiation are displayed in the red series of figure 3 (along with the results of the corresponding FEmodels that are discussed in section 2.2.1). A stable operating point is demonstrated at a loadstep of around 0.55, where it can be observed the data goes through a turning point.
In order to test the validity of the analytical model a series of simple spring systems were built usingANSYS workbench v17 software. The computer aided drawing (CAD) package within the software– ANSYS Design Modeller – was used to generate these systems. The geometric parameters (launchangle etc.) of the springs in each model were identical to those used in the analytical system10escribed in section 2.1. Three initial FE simulations were performed to verify that the ANSYSmodel could correctly determine the displacement-force curves of the geometric anti-spring system.The use of both beam elements and solid elements were tested. These models are displayed insubfigures 1b,c and d).In the first instance a single spring was designed (see figure 1B) and a point mass applied to thefree end. The tip of the spring had boundary conditions on the free end that ensured that therewas no rotation or horizontal displacement. These boundary conditions were needed to model theconstrained geometric anti-spring. A fixed clamp was applied to the base and a point mass of8 × − kg attached at the spring tip. A simple vertical acceleration was then applied to thesystem with maximum value equal to 9 . to simulate a gravitational loading. A force-displacement plot was generated using the results of this simple model. This data is displayed inthe light blue series of figure 2.The second model to be analysed was a more realistic test: a full two-spring system. Thesystem was built using both beam elements (figure 1c) and solid elements (figure 1d). The use ofbeam elements allows the model to converge quickly, and with 1 µ m wide beams the model onlytakes 30 s to run. The solid elements are much more computationally intensive, requiring severalminutes to execute. The general approach in modelling these systems is to fix the geometry withthe analytical MATLAB model and only utilise the full FE simulation to extract stress, investigateetching tolerances, and to investigate the behaviour of crystalline silicon. Since this system includedtwo springs, the boundary condition used to fix the horizontal position of the tip and the springrotation was removed. A single point mass of 1 . × − kg was applied to the centre point ofthe springs and the system was allowed to evolve with a ramped acceleration. For all models auniform load step was maintained for the solver and split into 20 separate iterations. The verticalforce was determined from the value of the acceleration and the point mass, while the position of11he vertex of the spring was used to determine the vertical position. As expected, the horizontalposition of the spring remained unchanged due to the symmetry of the system. This data for bothof these models is displayed in figure 2. The dark blue series represents the beam element model,and the green series represents the solid element model.The force-displacement data for the two double spring models were used in turn to calculate theresonant frequency as a function of load step. This was carried out via numerical differentiation.This data is displayed in figure 3. Alongside the analytical data (red series), the beam elementmodel is represented by the dark blue series, and the solid element model is represented by thegreen series.It can be observed from figures 2 and 3 that there is excellent agreement between the analyticaland FE models. This agreement is evidence that the simple analytical model can be utilised totest and optimise new geometries. Conversely, the full FE model is useful for determining theeffect of varying spring geometries caused by non-ideal etching tolerances; in addition to exploringthe stress in the springs and the effect of crystalline silicon (and its non-axisymmetric Young’smodulus). Silicon is a crystalline material and thus exhibits a Young’s modulus that depends on the orientationof the crystal axis to the etched device. It is particularly important to utilise the correct Young’smodulus in FE simulations in order to accurately predict the ultimate displacement/resonantfrequency of the MEMS device since the moduli can vary by up to 45% depending on the axis.Furthermore, FE simulations also have the benefit of fully modelling the shear/strain propertiesof a spring, although the effect of shear in thin springs (such as those of the MEMS device) isnegligible. The MEMS devices are fabricated from the < > plane of a silicon wafer, which12rovides the lowest Young’s modulus of any of the possible orientations. The modulus tensorprovided by Hopcroft et. al. was utilised, which is given by: . . . . . . . . . . . . (7)where each element has units of GPa. It is important to note that ANSYS differs in its standarddefinition of the stifness tensor compared to that of Hopcroft. This can be remedied by switchingelements 44 and 66. For gravimetry applications there is a desire to develop low frequency resonators that providestable behaviour. It would therefore be beneficial to have a simple means of tuning the resonantfrequency for a fixed proof mass. This is generally achieved by reducing the compression ratio ofthe spring system x com = x tip /L . For MEMS systems, however, in which the springs are etched,there is no capability to actually change the horizontal compression ratio as this is just set by theinitial geometry of the MEMS mask. As mentioned earlier, a convenient way to alter the frequencyis to change the launch angle of the spring. This has the effect of increasing the arc length of thespring, L , which in-turn reduces the compression ratio for a given x tip . Figure 4 shows the effecton the resonant frequency for a selection of launch angles modelled via ANSYS. A two-springsystem with a mass of 1.6 × − kg was utilised with a maximum acceleration of 9.8066 m/s . As13 F r equen cy ( H z ) ° Launch45 ° Launch50 ° Launch55 ° Launch
Figure 4 Launch Angle Comparison. The resonant frequency of a double spring system for differentlaunch angles (see figure 1a). Increasing the arc length, via the launch angle, is a convenient methodto provide a lower compression ratio in a MEMS geometry that can only be fabricated with a singlefixed horizontal spring extent. The greater the launch angle, the lower the resonant frequency.the launch angle increases the resonant frequencies drop. Although the 55 ◦ launch angle does notreach a minimum by load step 1, running this model with a larger point mass results in a minimumfrequency of 5 Hz.An additional means of tuning the system is to alter the ratio of k/m . By reducing the springthickness or increasing the mass, the level of loading at which the oscillator reaches its minimumfrequency can be changed. Figure 5 demonstrates that changing the thickness of the spring (whilstkeeping the mass constant) does not change the minimum frequency of the system substantially(compared to the change that can be induced by altering the launch angle). The following protocolcould therefore be followed in order to tune the design of a MEMS geometric anti-spring. First, thelaunch angle could be chosen to determine the minimum frequency. With this launch angle fixed,the thickness of the springs could then be altered in order to set the loading at which this minimum14 F r equen cy ( H z ) Spring thickness 8.50 µ mSpring thickness 8.75 µ mSpring thickness 9.00 µ mSpring thickness 9.25 µ m Figure 5 Spring Thickness Comparison. The resonant frequency of the double spring system fordifferent spring thicknesses (whilst keeping the launch angle constant). The level of loading atwhich the frequency minimum occurs can be tuned by altering the thickness of the spring.frequency is achieved. For situations in which changes in vertical gravity need to be measured, itwould be necessary for the frequency minimum to occur at full loading (i.e. suspended verticallyin the Earth’s gravitational field). It is not always the case that gravimeters would be operated insuch a vertical configuration, as will be discussed in Triaxial MEMS Gravimeter section.
The next stage of the investigation was to model a full MEMS gravimeter. Such a device requiresfour springs to support a central mass. With only two springs the system would only be stablewhen vertical: if rotated sideways torsional stresses would break the springs. Four springs areutilised so that the system is self-supporting in any orientation. Although the simpler devicesanalysed in this paper were drawn in ANSYS Design Modeller, these more complex designs weredrawn in SolidWorks and exported using the STEP2014 format into ANSYS Modeller via the15xternal Geometry Import. This also works well and offers a more convenient interface to modelcomplex geometries and parametrically combine dimensions.The final design, comprised of four springs, is displayed in figure 1e. This design is essentiallya double version of the system presented in figure 1d. The spring launch angle has been increasedto 60 ◦ , which provides a low frequency but stable oscillator. The arc length for this system is 8.33mm which results in a compression ratio of 0.87. While smaller compression ratios are possible, acompromise was chosen between the lowest resonant frequency and the total stress in the spring.This final design has a proof mass of 3 × − kg suspended from the four springs. The boundarycondition applied to the system is a fixed base on the reverse of the wafer. Again, the accelerationwas ramped but rather than requiring a constant load step the ANSYS programme it was allowedto determine the most appropriate steps. Typically a step size of 1 × − was used at first (where1 is equal to the full load step of 9.8066 m/s ). As the model converged, the step was typicallyincreased. Around the area of maximum deflection, however, the model often bisects and the loadstep is reduced. This bisection occurs because the spring system becomes more non-linear at thisstage, due to the effect of the geometric anti-spring.In this model a minimum element size of 4.25 × − m was utilised. This system has a minimumresonant frequency when vertical of around 5 Hz and a maximum equivalent stress in the springof 200 MPa. From our previous studies, and measurements of the breaking stress in in thin siliconsuspension beams this seems an appropriate level of stress to provide a robust device. A verydense mesh is required across the springs of the model, especially where the springs meet theframe and the proof mass. This is where most of the motion and therefore most of the stress isconcentrated. Tests were carried out on the mesh density to confirm that the model converges.16 .3 Experimental Verification of Finite Element Modelling To test the efficacy of the finite element modelling, a comparison was made between the resultsof these computer models and some real devices. Two FE computer models were run, both witha 60 ◦ launch angle, the same arc length of 8.33 mm, and the same corresponding compressionratio of 0.87. One model, however, utilised the anisotropic Young’s Modulus tensor presented inequation 7; and the other model used a more simplistic option: instead of an elasticity that changeswith crystal axis, an isotropic value of 169 GPa was used. This value was selected as the bestapproximation of a global Young’s modulus for a device fabricated from < > silicon. For eachof these models, the variation of frequency as a function of spring thickness was calculated. Theresulting data from the anisotropic model is presented in black in figure 6, and the data from thesimplistic isotropic model is presented in red on the same graph.A series of 7 MEMS devices were fabricated using the design presented in figure 1e. Usingthe same launch angle of 60 ◦ , the same arc length of 8.33 mm, and the same correspondingcompression ratio of 0.87; this design was patterned onto the surface of a photoresist-coated, 240 µ m thick piece of silicon wafer using standard photolithography techniques. The silicon was thenthrough-wafer etched using a highly anisotropic deep reactive ion etch (DRIE). Once complete, theMEMS devices were each mounted vertically in a bracket. In turn, each MEMS device was excitedusing a high impulse tap on the workbench; the resultant oscillation was filmed to ascertain theprimary resonant frequency of each device. The films were recorded using a video camera witha 210 frame per second recording rate (the frame rate of the camera was confirmed by filming astopwatch for a period of 30 s). For each of the 7 videos, 20 oscillations were counted and thenumber of frames recorded. The corresponding frequencies of each of the 7 MEMS devices wascalculated: the samples had a mean frequency of 12.92 Hz, with a standard deviation of 0.36 Hz.17he sample with the closest resonant frequency to the mean of the group (12.88 Hz) wasanalysed with an optical microscope and measurements were made of the spring thickness. Ameasurement was made at both ends and the middle of each of the four springs. Measurementswere taken on both sides of the spring (top and bottom) to account for any anisotropy in the DRIEetch. A further measurement was made of the profile of a broken spring using a scanning electronmicroscope (SEM). From this image it was clear that a slight undercut occurred during the etch,meaning that the measurements of the top of the spring were 1.2 µ m larger than the true thicknessof of the spring. The SEM indicated that the spring thickness remained constant from top tobottom apart from the final few microns, where it was observed that a small amount of notching had occurred. This made the optical measurement of the bottom thickness an unreliable indicatorof the true spring thickness. It was therefore decided that the best indicator of the spring thicknesswas the optical measurement of the top of the spring, with a statistical correction made to accountfor the consistent undercut profile. The mean spring thickness was taken to be 8.86 ± µ m,where the error is given by the standard error of the mean of the population (of 12 measurements).This experimental result is presented as a blue data point in figure 6, with the standard error of thethickness measurement indicated by a horizontal error bar. The fact that the anisotropic siliconFE model lies within the bounds of the experimental result (whereas the isotropic FE model doesnot) provides validation that taking into account a variable Young’s Modulus is important for theprediction of the physical behaviour of a fabricated device. Furthermore, since this anisotropyis not possible to model analytically, this result is also validation that FE modelling provides animprovement in predictive accuracy compared to present analytical models18 .577.588.599.51010.5510152025 Spring Thickness ( µ m) F r equen cy ( H z ) FE Model Using Anisotropic Young’s ModulusFE Model Using Isotropic Young’s ModulusExperimental Result
Figure 6 Experimental Comparison to FE Model. The red series is a FE model in which anisotropic Young’s Modulus of 169 GPa has been used. The black series shows a the results of a FEmodel in which the physical anisotropy of silicon has been accounted for: the Young’s Modulus isinputted to the model as a tensor (see equation 7) since Young’s Modulus varies between crystalaxes. The blue data point represents a real sample, fabricated using the same design as that usedin the FE models. The error bar on this data point is the standard error of the 12 spring thicknessmeasurements. 19
Discussion
The Design of Future Triaxial MEMS Gravimeters
After experimental verification FE model accuracy, more complex designs of gravimeters could beconsidered. All previous works on this topic by the authors have concentrated on gravimeters thatoperate in a vertical configuration. These cannot be used to measure variations in accelerationother than those in the vertical, z , component of gravity. It is possible, however, to design agravimeter that has sensitivity to the x , y , and z components of gravity, not just z . Such agravimeter has the advantage that providing the orientation of the device is known, the levellingrequirements are not as stringent as those for a one dimensional gravimeter measuring only the z -component. Making such a device is only possible if the angle at which the minimum frequency(where the optimum acceleration sensitivity occurs) is tunable. As has been demonstrated infigures 4 and 5, it is possible to tune both the minimum frequency, and the loading at which thisfrequency occurs. Since the angle of the device is just a proxy for loading (along with springthickness, and the mass value), one could easily design a device that would reach its minimumfrequency at a specific angle. In such a device three tuned MEMS devices could be placed in theGalperin configuration at an angle of θ = 54 . ◦ from the horizontal, and separated azimuthallyfrom each other at an angle of 120 ◦ (see Fig. 7). The Galperin configuration was designed to allowthree identical sensors to measure gravity (or seismic activity) in three dimensions. Conversely,if one wanted to mount three sensors parallel to x , y and z , then two different sensor geometrieswould be required. This is because the sensors in x and y would be perpendicular to the 1 g field,and the sensor in z would be parallel to it. The devices in x and y would consequently experiencedifferent forces to the device in z .For a triaxial device such as that presented in figure 7, the acceleration in each of the three20igure 7 A Future Gravimeter Design. A computer generated image of a three-axis MEMS gravime-ter in a Galperin configuration . 21xes would be given by equations 8, 9 and 10: g z = ( g + g + g ) sin θ g x = (( g + g ) cos α ) cos θ − g cos θ (9) g y = (( g − g ) sin α ) cos θ (10)To test the susceptibility of such a triaxial model to tilt, a geometry was tuned so that itsminimum frequency would occur at the Galperin angle. Three identical models of this kind werethen set up to match the configuration detailed in figure 7. The acceleration vectors of thesemodels were then varied to simulate the effect of a tilting base plate. The displacement andresonant frequencies of each of the proof masses were measured, and a value of gravitationalacceleration calculated using g i = ω x (where g i is the output of one of the individual sensors).The total parasitic acceleration in g z (see eq. 8) due to tilt was then plotted to ascertain whethertilt susceptibility was better in this configuration than for a single MEMS chip oriented verticallyin a gravitational field. Figure 8 shows the results of this test. It can be seen that the triaxialconfiguration is less sensitive to tilt than a single vertical sensor. Since the triaxial system offersmulti-axis sensitivity, as well as a reduced tilt susceptibility, it is clear that this design providepractical benefits in the field of gravimetry. 22igure 8 Tilt Susceptibility Comparison. A comparison between the tilt susceptibility of a singlevertical sensor, and that of three devices placed in a triaxial configuration.23 Methods
The fabrication of the MEMS devices used in these experiments were fabricated by Kelvin Nan-otechnology. A similar fabrication process is discussed in detail in Middlemiss et al. . Here we taken the design of geometrical anti-springs from an analytical model, through a sequenceof tests within a FE modelling environment to validate the power of this novel spring design. Al-though it is possible to construct analytical models of these complex non-linear systems, FE modelsoffer superior power when designing systems for which multiple parameters need to be optimised.This is particularly the case when modelling the effect of the anisotropic crystalline structure ofthe fabrication material; something not possible with an equivalent analytical calculation. Theability to tune the frequency of these systems – and in particular the angle at which the minimumin frequency occurs – has enabled the design of a tri-axial MEMS gravimeter with accelerationsensitivity in all three axes. The FE simulations demonstrate that this configuration is less suscep-tible to tilt and from a practical gravimetry perspective it allows gravity measurements withoutthe requirement to level the gravimeter before undertaking measurements which can reduce thetime required to complete gravity surveys. 24 cknowledgements
The authors would like to thank Kelvin Nanotechnology who fabricated the final MEMS sensorsdiscussed in this paper. They would also like to thank the staff and other users of the James WattNanofabrication Centre for help and support in undertaking the MEMS fabrication development.This work was funded by the Royal Society Paul Instrument Fund (STFC grant numberST/M000427/1), and the UK National Quantum Technology Hub in Quantum Enhanced Imaging(EP/M01326X/1).Dr R. P. Middlemiss is supported by the Royal Academy of Engineering under the ResearchFellowship scheme (Project RF/201819/18/83).
Author Declaration
The authors declare no conflict of interest.
Author contributions • Richard Middlemiss led the methodology of the etch process for the MEMS gravimeter. Heconducted finite element modelling of the geometrical anti-spring. He led the writing of themanuscript. • Paul Campsie contributed to the initial design of the geometrical anti-spring flexures, andconducted preliminary finite element analysis. • William Cunningham provided consultancy on various technical aspects of the finite elementmodelling. 25
Rebecca Douglas contributed to the tuning of the geometrical anti-spring design. • Victoria McIvor contributed to the modelling of the effect of crystal orientation. • James Hough developed the methodology of utilizing geometric anti-springs for the MEMSgravimeter system and commented on the manuscript. • Sheila Rowan was responsible for the resources that were necessary to complete the projectand commented on the manuscript. • Douglas Paul coordinated the fabrication of the MEMS gravimeter, and contributed to themanuscript preparation. • Abhinav Prasad conducted the validation of the tri-axial modelling, and contributed to themanuscript preparation. • Giles Hammond had the initial concept of the MEMS gravimeter, and had oversight of designprocess. He led the formulation of the analytical modelling, and conducted finite elementmodelling. He co-wrote parts of the manuscript with Richard Middlemiss.
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