Finite Element Static, Vibration and Impact-Contact Analysis of Micromechanical Systems

Transcription

1 Finite Element Static, Vibration and Impact-Contact Analysis of Micromechanical Systems Alexey I. Borovkov Eugeny V. Pereyaslavets Igor A. Artamonov Computational Mechanics Laboratory, St.Petersburg State Polytechnical University, Russia Eugeny N. Pyatishev Laboratory of Microtechnology and MicroElectroMechanical Systems, St.Petersburg State Polytechnical University, Russia Abstract In the present job multivariant finite element (FE) dynamical and structural analysis of sensitive micromechanical gyroscopes (MMG) and micromechanical accelerometers (MMA) was carried out. Natural frequencies and mode shapes, forced oscillations and analysis of transient processes in MMG were analyzed; amplitude-frequency characteristics and dependence of MMG oscillations amplitude on measured angular velocity (relative to "output" axis) were determined. The effect of stressed state on natural frequencies and influence of technological and temperature factors on the sensors accuracy were analysed. Multivariant analysis of MMG and MMA under shock and vibrational loading was carried out. Redesign of 3D sensitive elements geometry was done to improve sensors characteristics. Computation of dynamical stresses was done with consideration of geometrical non-linearity and contact interaction of silicon plate with glass substrate. Analysis of dynamics and statics of micromechanical gyroscopes Modal analysis: In the current paper structural schemes of MMG are being analyzed that are employed in actual inertial navigation systems and serving as angular velocimeters. Sensitive elements or oscillators of rotary (to measure angular velocity in the plane of device) and linear (to measure angular velocity perpendicular to the plane of device) MMG are shown in Figures 1, 2, where the areas of oscillators and glass substrate fastening are marked with green. Operating principle of the considered as well as other schemes of MMG based on gyroscopic effect is described in [1-3]. Single-crystalline silicon that belongs to crystals with cubic symmetry is used as structural material. Elastic properties of such materials in the coordinate system defined by crystallographic axes are defined by three independent elastic moduli [4]. Sensitive element of MMG is manufactured with use batch fabrication methods based on MEMS technology (silicon technology) [2]. On the basis of solid models (see Figures 1, 2) developed within the framework of ANSYS software [5] 3D FE models of oscillators were created (Figures 3, 4; NE number of finite elements in the model, NDF number of degrees of freedom in the considered mechanical system). Creation of FE model was done with use of 3D 20-node solid brick elements SOLID95.

2 Figure 1. 3D solid model of the rotary MMG Figure 2. 3D solid model of the linear MMG

3 Figure 3. 3D FE model of the rotary MMG Figure 4. 3D FE model of the linear MMG

4 Computation of natural frequencies and mode shapes of sensitive elements was carried out to evaluate operating regimes of MMG. In Figures 5, 6 natural modes are shown that are conforming to operating regimes of vibration sensors. For the linear MMG mode shapes are corresponding to multiple natural frequency. Figure 5. First and second natural modes of the rotary MMG corresponding to operating regimes of the sensor

5 Figure 6. Natural modes of the linear MMG corresponding to operating regimes of the sensor

6 Modification of sensitive elements solid models was carried out then in order to improve sensors characteristics. At evaluation of quasi-optimal structural parameters the following optimum criteria were chosen: for the sensitive rotary MMG element maximum juxtaposition of two lower natural frequencies; for the sensitive linear MMG element maximum drop of the multiple natural frequency that corresponds to sensor operational regimes. The juxtaposition of two lower natural frequencies of rotary MMG comes to hand during multivariant FE computations by thickening of springing elements (torsions) to the value of 1.5 µ m. For the linear MMG scheme multifold drop of the operating frequency was achieved by change in torsions geometry and glass substrate fastening areas. Geometry of oscillators with improved characteristics is presented in Figures 7, 8. Comparison of initial and optimized designs for both MMG schemes is presented in Table 1. It will be observed that in the case of the linear MMG multiple natural frequency shifted from ninth to second position in the spectrum and its value decreased 3.7 times. Figure 7. 3D solid model of the rotary MMG sensitive element with optimized parameters

7 Figure 8. 3D solid model of the linear MMG sensitive element with optimized parameters Table 1 MMG scheme Initial design Optimized design Rotary MMG f 1 f 1 f 2 = f 1 f 2 = f 1 Linear MMG f 9 f 2 = f 9 At the next stage of research the influence of stressed state originating due in various thermal regimes on the natural frequencies spectrum was analyzed. At the solution of these problems FE models of vibration detectors were created that considers technological (out of the perpendicular oscillators walls due to manufacturing faults) and thermal factors. Consideration of thermal factors includes modeling of glass substrate, electro-thermo-compressional welding of silicon with glad at Т = 400 С, as well as operating regime temperature. In Figures 9, 10 total displacement vector modulus Usum fields are presented, originating after welding of silicon with glass substrate and cooling to Т = 20 С. Deformed states are

8 shown for optimized MMG designs in comparison with initial (not deformed) state. Results of modal analysis depending on pre-stressed state are presented in Table 2. Analysis of these results enables to conclude that technological and thermal factors don t influence much on values of rotary MMG natural frequencies. Comparison of natural frequencies spectrums with account of pre-stressed state of considered schemes of rotary and linear MMG sensitive elements as well as other oscillators designs [6, 7] enables one to conclude that application of rotary will lead to most robust and error-free performance of the control system on the whole. Figure 9. Initial stressed-strained state of the rotary MMG sensitive element, distribution of Usum total displacements vector modulus

9 Figure 10. Initial stressed-strained state of the linear MMG sensitive element, distribution of Usum total displacements vector modulus Table 2 MMG scheme Number of natural frequency Without preloading With account of thermal stresses Т = -55 С Т = 20 С Т = 75 С Rotary MMG 1 f f f f 1 2 f f f f 2 Linear MMG 1 f f f f 1 2 f f f f 1 Analysis of dynamic processes: One of the most important stages in the analysis of MMG sensitive elements is the analysis of forcedoscillations regime, transient process and definition of output characteristics of the devices, i.e. dependence of forced oscillations amplitude on measured angular velocity. Forasmuch as operational principles of rotary and linear MMG is the same the analysis of dynamic processes was carried out only for rotary MMG. In the considered scheme of the rotary MMG (see Figure 7) oscillator is set in oscillations with use of collar electrostatic actuators in z-direction by the following law: M z = M 0 sin(2πf 1 t), where M z vibration

10 actuator moment, f 1 first natural frequency. In Figure 11 gain-frequency characteristic in the vicinity of first natural frequency corresponding to torsional modes of sensitive mass ( ϕ 1 resonance amplitude on the first natural frequency) is presented. Sharp resonance peak should be noted that could be explained by high quality factor ( ) of silicon oscillator. Establishment process of forced oscillations in z-direction is presented in Figure 12. Figure 11. Amplitude-frequency characteristic of the rotary MMG in the vicinity of first natural frequency Figure 12. Transition of forced vibrations of the rotary MMG At rotation of vibration detector with constant angular velocity ω y around device sensitive axis x opposite Coriolis forces arise that cause angular oscillations of the sensitive element relative to x-axis. Amplitudes of these oscillations are proportional to measured angular velocity ω y. In Figures 13, 14 output

11 characteristics of rotary MMG with account of technological and thermal factors are presented ( ϕ 2 forced oscillations amplitude of the sensitive element in x-direction for the given angular velocity). It is seen from the adduced graphs that account of technological factors, i.e. not vertical position of the walls (shift of the center of mass towards z-axis) leads to the increase of oscillations amplitude in comparison with ideal output characteristics (~ 10-15%). The result of the construction center of mass shift by value x 0 ~ µ m with account of non-vertical walls is insignificant increase of amplitude (~ 0.2%). Consideration of thermal factors leads to increase of oscillations amplitude by constant value what can be explained by thermal deformations originating at welding of silicon with glass. Figure 13. Output characteristics of the rotary MMG with account of technological factors influence

12 Figure 14. Output characteristics of the rotary MMG with account of thermal factors influence Impact and vibration loadings: During exploitation at high-speed and highly maneuverable objects MMG are gone through intensive impact and vibration shock loadings. Analyses of MMG elements dynamical behavior show that devices of such types can withstand impact action equivalent to several thousands g. The most critical direction of impact force for rotary and plane MMG is z-axis. Maximum stresses are observed in zone of fastening of torsions with sensitive mass. Consider several variants of impact and vibration shock loading in rotary MMG. As an impact loading a single impact of 1000g intensity was considered varying by the following law: F у = F у0 sin(π t/t 0 ), where F 0 = m 1000g amplitude of impact force, m mass of oscillator, T 0 = 2 ms impact duration. Structural analysis of the oscillator was carried out with account of technological factors and for all directions x, y, z. In Figure 15 variation of relative equivalent von Mises stresses σ i / σu during impact is shown for the most critical direction ( σ u ultimate stress of silicon). Distribution of these stresses and relative z-axis displacement modulus field Uz/R (R silicon plate radius, R 10 3 µm) for the time moment corresponding to maximum stresses and displacements are shown in Figure 16. On the basis of the analysis results obtained with use of ANSYS FE software [5] for all directions of the impact with given intensity it is possible to state that stresses, strains and displacements don t exceed limiting values.

13 Figure 15. Variation of relative equivalent von Mises stresses σ i / σ u during impact Figure 16. Distribution of relative equivalent von Mises stresses σ i / σ u (a) and field of relative displacements vector Uz/R (b) at the time moment corresponding to maximum stresses and displacements Structural analysis of the MMG sensitive element under vibrational shock with account of technological factors is carried out for the following parameters of vibration loading: F v = F v0 sin(2π ft), where F v0 = m a amplitude of vibrational loading, a = 24g root-mean-square of vibration acceleration, f [0; 500] Hz frequency range. Results of analysis are listed in Table 3. At computations oscillator model with account of

14 technological factors was utilized. During analysis of the influence of vibrations on sensitive MMG element only vibrations were of interest, so thermal factors were not considered. Table 3 Direction of vibration loading Umax/R σ max / σ u x y z The obtained results give evidence that with the given design of MMG sensitive elements stresses, strains and displacements are insignificant in comparison with, for example, thermal stresses, strains and displacements. In the end of the current part of research it should be noted that analyses of dynamical characteristics, impact and vibration loadings were carried out for the rotary MMG as well. Analysis of dynamics and strength of micromechanical accelerometers Modal and static analyses: The study of natural frequencies of MMA sensitive elements that are used in inertial systems as linear accelerations measurers was carried out for two schemes: in the first scheme sensor sensitive axis lies in the plane of device (MMA-xy, Figure 17); in the second scheme perpendicular to the plane of device (MMAz, Figure 18).

15 Figure 17. 3D solid model of the MMA-xy sensitive element Figure 18. 3D solid model of the MMA-z sensitive element

16 As the constructive material as well as for MMG manufacturing single-crystalline silicon is used. 3D FE models of MMA, created with use of 20-node solid brick structural elements, are presented in Figures 19, 20. Figure 19. 3D FE model of the MMA-xy sensitive element Figure 20. 3D FE model of the MMA-z sensitive element

17 Computation of natural frequencies and mode shapes of MMA sensitive elements was carried out by way of determination of sensors operative ranges. Results of FE analysis first natural modes are presented in Figures 21, 22. Figure 21. First natural mode of the MMA-xy sensitive element Figure 22. First natural mode of the MMA-z sensitive element

18 To lower sensor structure stiffness in the direction of measured accelerations the modification of 3D geometry of sensitive elements was performed. Choice of quasi-optimal structural parameters of MMA was done by means of variation of the geometry of torsions and sensitive masses. At the current stage of research it was required to match MMA elastic elements parameters so that at maximum acceleration from the measuring range displacement of the elastic elements were equal to the maximum possible value equal to 10 µ m. At this all essential technological requirements are to be kept. Computations were carried out at first for initial design of MMA sensitive elements and then for optimized designs. For MMA-z the required result was achieved by variation (increasing) of torsions length. At analysis of MMA-xy together with variation of size and shape of torsions parameters of the sensitive mass were changed as well. As the loading 75g acceleration was applied in the direction of the measured acceleration. Geometrical layouts of MMA sensitive elements with improved operation characteristics and computation results are presented in Figures At this, displacements of MMA sensitive masses at maximum accelerations increased in 6.4 and 7.5 times for ММА-xy and ММА-z consequently. Figure 23. 3D solid model of the ММА-xy sensitive element with optimized parameters of flexible elements

19 Figure 24. 3D solid model of the ММА-z sensitive element with optimized parameters of flexible elements Figure 25. Displacements Uy field at loading of the ММА-xy by acceleration a y = 75g

20 Figure 26. Displacements Usum field at loading of the ММА-z by acceleration a z = 75g For all sensitive elements of MMA the analysis of influence of pre-stressed state originating at various operative regimes on the spectrum of natural frequencies was analysed. Results of computations are listed in Table 4. Table 4 MMA scheme Number of natural frequency Without preloading With account of thermal stresses Т = -55 С Т = 20 С Т = 75 С MMA-xy 1 f f f f 1 2 f f f f 2 MMA-z 1 f f f f 1 2 f f f f 2 Analysis of dynamical behavior of MMA sensitive element under impact loading: In the present part of the paper multi-variant FE analysis of MMA sensitive element dynamics was carried out with se of ANSYS/LS-DYNA [5, 8] FE software for the MMA with axis of sensitivity lying in the plane of device MMA-xy. 3D solid and FE models of MMA sensitive element are presented in Figures 27, 28, where upper part of the construction is silicon plate, lower part glass substrate. Analysis of dynamics of silicon plate was done for impact loading in the direction of z-axis. In Figure 29 impact process acceleration depending on time is shown.

21 Figure 27. 3D solid model of the ММА-xy sensitive element and glass substrate Figure 28. 3D FE model of the ММА-xy sensitive element and part of glass substrate with account of symmetry plane YZ

22 Figure 29. Characteristics of impact loading acting perpendicular to the ММА-xy measuring axis Evaluation of dynamical stresses and strains in MMA elements was carried out with account of contact interaction between silicon plate and glass substrate, internal damping of silicon and geometrical nonlinearity (large stains and deflections). Expectedly maximum values of stresses originate in the sensor torsions. In Figures 30, 31 distribution of equivalent von Mises stress σ i is shown for time moment t = 0.06 ms and dependence of Uz displacements on time in the point А (see Figure ). Time dependence of displacements demonstrates decay of oscillations after removal of impact loading (t = 5 ms). Figure 30. Distribution of equivalent von Mises stresses σ i at time moment t = 0.06 ms

23 Figure 31. Dependence of Uz displacements on time in the point A Conclusion In the present paper the results of multi-variant FE analysis of MMG and MMA micromechanical characteristics are presented. As a result of analysis new designs of micro-sensors with quasi-optimal geometrical characteristics were obtained. These results were used by Laboratory of Microtechnology and MicroElectroMechanical Systems (SPbSPU, Russia) for designing and manufacturing of ММГ and ММА with improved operation characteristics. References 1. Lestiev A.M., Popova I.V., State-of-the-art of theory and practical developments of micromechanical gyroscopes, Gyroscopy and Navigation, No. 3 (22), 81-94, 1998 (in Russian). 2. Lestiev A.M., Popova I.V., Pyatyshev E.N., et al. Design and research of a micromechanical gyroscope, Gyroscopy and Navigation, No. 2 (25), 3-10, 1999 (in Russian). 3. Raspopov V.Ya. Micromechanical devices. Tula State University. Russia p. (in Russian). 4. Shermegor T.D. Theory of elasticity for micro-heterogeneous media. Moscow, Nauka p. (in Russian). 5. ANSYS Theory Reference. Eleventh edition. SAS IP, Inc. 6. Borovkov A.I., Pyatishev E.N., Lurie M.S., Pereyaslavets E.V., et al., Microscale effects of silicon-on-glass micromechanical devices. Experimental results and 3D finite element modeling, in Fourth International workshop on Nondestructive Testing and Computer Simulations in Science and Engineering, Alexander I. Melker, Editor, Proceedings of SPIE, Vol. 4348, 2001, Borovkov A.I., Pereyaslavets E.V. 3D Finite Element Modeling and Analysis of Micromechanical Sensors. ANSYS 2002 Conference papers. Pittsburgh, LS-DYNA, Keyword User s Manual, Version 960, Livermore Software Technology Corporation, CA, USA, 2001.