Mechanical and Fluidic Signals

  • Arokia Nathan
  • Henry Baltes
Part of the Computational Microelectronics book series (COMPUTATIONAL)


Simulation of the static and dynamic behavior of solid structural and fluid mechanical variables, e.g., stress, strain, strain-rate, displacement, force, and velocity, is critical to the design and analysis of microsensors and microactuators in the mechanical domain. For example, in Chapt. 6, we saw how electrical transport is modified by piezoresistance. This, in addition to deflection-induced capacitance change, can be effectively utilized for conversion of signals from the mechanical to the electrical domain. Alternatively, as we will see in Chapt. 8, a micromechanical structure subject to an electrical, thermal, magnetic, or mechanical excitation signal, gives rise to micro-actuation in the mechanical domain. In this chapter, we deal with model equations and constitutive relations relevant to: simulation of mechanical (e.g., pressure) microsensors; computation of velocity profiles relevant to flow microsensors (needed in Chapt. 5) and selected microfluidic systems; computation of mechanical stresses induced by packaging or encapsulation of microtransducers or integrated circuits (needed in Chapt. 6); and simulation of mechanical microactuators, including fluidic damping effects (needed in Chapt. 8).


Residual Stress Flow Channel Inlet Velocity Fluidic Signal Squeeze Film 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    McDonald, P., Continuum Mechanics, Boston: PWS Publishing Co., 1996.Google Scholar
  2. [2]
    Timoshenko, S. P., Goodier, J. N., Theory of Elasticity, 3rd Ed., New York: McGraw-Hill, 1970.MATHGoogle Scholar
  3. [3]
    Schlichting, H., Boundary Layer Theory, New York: McGraw-Hill, 1968.Google Scholar
  4. [4]
    Nye, J. F., Physical Properties of Crystals, Oxford: Oxford University Press, 1957.MATHGoogle Scholar
  5. [5]
    Fung, Y. C., A First Course in Continuum Mechanics, 3rd Ed., New Jersey: Prentice Hall, 1994.Google Scholar
  6. [6]
    Sokolnikoff, I. S., Tensor Analysis, Theory and Applications to Geometry and Mechanics of Continua, 2nd Ed., New York: Wiley, 1964.MATHGoogle Scholar
  7. [7]
    Landau, L. D., Lifshitz, E. M., Fluid Mechanics, 2nd Ed., New York: Pergamon, 1989.Google Scholar
  8. [8]
    Middelhoek, S., Audet, S. A., Silicon Sensors, New York: Academic Press, 1989.Google Scholar
  9. [9]
    Sze, S. M., (Ed.), Semiconductor Sensors, Wiley, New York, 1994.Google Scholar
  10. [10]
    Bin, T. Y., Huang, R. S., CAPSS: A Thin Diaphragm Capacitive Pressure Sensor Simulator, Sensors and Actuators, 11 (1987), 1–22.CrossRefGoogle Scholar
  11. [11]
    Bouwstra, S., Geijselaers, B., On the Resonance Frequencies of Microbridges, Digest of Technical Papers, Transducers’ 91, San Francisco, 1991, pp. 538-542.Google Scholar
  12. [12]
    Elgamel, H. E., Closed-Form Expressions for the Relationships Between Stress, Diaphragm Deflection, and Resisitance with Pressure in Silicon Piezoresistive Pressure Sensors, Sensors and Actuators A, 50 (1995) 17–22.CrossRefGoogle Scholar
  13. [13]
    Steinmann, R., Friemann, H., Prescher, C., Schellin, R., Mechanical Behaviour of Micromachined Sensor Membranes Under Uniform External Pressure Affected by In-Plane Stresses Using a Ritz Method and Hermite Polynomials, Sensors and Actuators A, 48 (1995), 37–46.CrossRefGoogle Scholar
  14. [14]
    Meng, Q., Mehregany, M., Theoretical Modeling of Microfabricated Beams with Elastically Restrained Supports, J. of Microelectromechanical Systems, 2 (1993), 128–137.CrossRefGoogle Scholar
  15. [15]
    Gerlach, G., Schroth, A., Pertsch, P., Influence of Clamping Conditions on Micro-structure Compliance, Sensors and Materials, 8 (1996), 79–98.Google Scholar
  16. [16]
    Lee, K. W., Modeling and Simulation of Solid-State Pressure Sensors, Ph.D. Dissertation, University of Michigan, Ann Arbor, USA, 1982.Google Scholar
  17. [17]
    Korvink, J., An Implementation of the Adaptive Finite Element Method for Semiconductor Sensor Simulation, Ph.D. Dissertation, ETH Zurich, No. 10143, Switzerland, 1993.Google Scholar
  18. [18]
    Zhang, Y., Wise, K. D., Performance of Non-Planar Silicon Diaphragms Under Large Deflections, J. of Microelectromechanical Systems, 3 (1994), 59–68.CrossRefGoogle Scholar
  19. [19]
    Mallon Jr., J. R., Pourahmadi, F., Petersen, K., Barth, P., Vermeulen, T., Bryzek, J., Low-Pressure Sensors Employing Bossed Diaphragms and Precision Etch-Stopping, Sensors and Actuators, A21-A23 (1990), 89–95.Google Scholar
  20. [20]
    Bergqvist, J., Finite-Element Modelling and Characterization of a Silicon Condenser Microphone with a Highly Perforated Backplate, Sensors and Actuators A, 39 (1993), 191–200.CrossRefGoogle Scholar
  21. [21]
    Pourahmadi, F., Barth, P., Petersen, K., Modeling of Thermal and Mechanical Stresses in Silicon Microstructures, Sensors and Actuators, A21-A23 (1990), 850–855.Google Scholar
  22. [22]
    Bessho, M., Tsuru, Y., Horiike, H., Jinmon, M., Yamagami, K., Wataya, S., High Reliability Absolute Semiconductor Pressure Sensor, SAE Special Publication, 536 (1983), 55–59.Google Scholar
  23. [23]
    Suzuki, S., Yamada, K., Nishihara, M., Hachino, H., Minorikawa, S., Structural Analysis of a Semiconductor Pressure Sensor, Proc, The 1st Sensor Symp., Japan, 1981, pp. 131-133.Google Scholar
  24. [24]
    Suzuki, S., Yagi, Y., Optimum Design of Silicon Pressure Sensor by Nonlinear Finite Element Method, Proc, The 2nd Sensor Symp., Japan, 1982, pp. 163-165.Google Scholar
  25. [25]
    Barth, P. W., Pourahmadi, F., Mayer, R., Poydock, J., Petersen, K., A Monolithic Silicon Accelerometer with Integral Air Damping and Overrange Protection, Technical Digest, IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Is., 1988, pp. 35–38.Google Scholar
  26. [26]
    Puers, B., Peeters, E., Sansen, W., CAD Tools in Mechanical Sensor Design, Sensors and Actuators, 17 (1989), 423–429.CrossRefGoogle Scholar
  27. [27]
    Tschan, T., de Rooij, N., Characterization and Modelling of Silicon Piezoresistive Accelerometers Fabricated by a Bipolar-Compatible Process, Sensors and Actuators A, 25-27 (1991), 605–609.CrossRefGoogle Scholar
  28. [28]
    Tschan, T., de Rooij, N., Bezinge, A., Analytical and FEM Modeling of Piezoresistive Silicon Accelerometers: Predictions and Limitations Compared to Experiments, Sensors and Materials, 3 (1992), 189–203.Google Scholar
  29. [29]
    Kovaács, A., Stoffel, A., Mechanical Analysis of Polycrystalline and Single-Crystalline Silicon Microstructures, Sensors and Actuators A, Vol. 41-42 (1994), 672–679.CrossRefGoogle Scholar
  30. [30]
    Yamada, K., Kuriyama, T., A Novel Degree of Freedom Separation Technique in a Multi-Axis Accelerometer, Sensors and Actuators A, 43 (1994), 120–127.CrossRefGoogle Scholar
  31. [31]
    Lee, Y-T., Seo, H.-D., Takano, R., Matsumoto, Y., Ishida, M., Nakamura, T., Design Consideration for Silicon Rectangular Diaphragm Pressure Sensor with Single-Element Four-Terminal Strain Gauge, Sensors and Materials, 7 (1995), 53–63.Google Scholar
  32. [32]
    Marco, S., Samitier, J., Morante, J. R., Gotz, A., Esteve, J., Novel Structures for Miniature Pressure Transducers Obtained by Electrochemical Etch-Stop on Diffused Membranes, Sensors and Materials, 7 (1995), 331–345.Google Scholar
  33. [33]
    Hein, S., Schlichting, V., Obermeier, S. E., Piezoresistive Silicon for Very Low Pressures Based on the Concept of Stress Concentration, Digest of Technical Papers, Transducers’ 93, Yokohama, 1993, pp. 628-631.Google Scholar
  34. [34]
    Fotheringham, G., Simulation Methods for Multi-Chip Modules, Sensors and Actuators A, 30 (1992), 157–165.CrossRefGoogle Scholar
  35. [35]
    Lin, Y.-C., Hesketh, P. J., Schuster, J. P., Finite-Element Analysis of Thermal Stresses in a Silicon Pressure Sensor for Various Die-Mount Materials, Sensors and Actuators A, 44 (1994), 145–149.CrossRefGoogle Scholar
  36. [36]
    Pourahmadi, F., Petersen, K., Package Design of Silicon Micromachined Sensors Using Finite Element Modeling, Digest of Technical Papers, Transducers’ 93, Yokohama, 1993, pp. 774-778.Google Scholar
  37. [37]
    Koen, E., Pourahmadi, F., Terry, S., A Multilayer Ceramic Package for Silicon Micromachined Accelerometers, Digest of Technical Papers, Vol. 1, Transducers’ 95, Stockholm, 1995, pp. 273–276.Google Scholar
  38. [38]
    Washizu, K., Note on the Principle of Stationary Complimentary Energy Applied to Free Vibration of an Elastic Body, Int. J. Solids and Structures, 2 (1969), 27–35.Google Scholar
  39. [39]
    Baltes, H., Korvink, J. G., Paul, O., Numerical Modelling and Materials Characterization for Integrated Micro Electro Mechanical Systems, Simulation of Semiconductor Devices and Processes, Vol. 6, Ryssel, H., Pichler, P. (Eds.), Wien-New York: Springer-Verlag, 1995, pp. 1–9.CrossRefGoogle Scholar
  40. [40]
    Hodge, Jr., P. G., Plastic Analysis of Structures, New York: McGraw-Hill, 1959.MATHGoogle Scholar
  41. [41]
    Korvink, J. G., Baltes, H., Microsystem Modelling, Chapt. 6, Sensors Update, Baltes, H., Göpel, W., Hesse, J., (Eds.), Weinheim: VCH, 1996, pp. 181–209.Google Scholar
  42. [42]
    Washizu, K., Variational Methods in Elasticity and Plasticity, 3rd Ed., Oxford: Pergamon Press, 1982.MATHGoogle Scholar
  43. [43]
    Chau, K., Allegretto, W., Ristic, L., Simulation of Silicon Microstructures, Sensors and Materials, 2 (1991), 253–264.Google Scholar
  44. [44]
    Timoshenko, S., Woinowsky-Krieger, S., Theory of Plates and Shells, New York: McGraw-Hill, 1959.Google Scholar
  45. [45]
    Clark, S. K., Wise, K. D., Pressure Sensitivity in Anisotropically Etched Thin-Diaphragm Pressure Sensors, IEEE Trans. Electron Devices, ED-26 (1979), 1887–1896.CrossRefGoogle Scholar
  46. [46]
    Benaissa, K., Integrated Silicon Opto-Mechanical Sensors, Ph.D. Dissertation, Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, 1996.Google Scholar
  47. [47]
    Benaissa, K., Nathan, A., IC Compatible Optomechanical Pressure Sensors Using Mach-Zender Interferometry, IEEE Trans. Electron Devices, 43 (1996), 1571–1582.CrossRefGoogle Scholar
  48. [48]
    Gorman, D. J., Free Vibration Analysis of Rectangular Plates, New York: Elsevier, 1982.MATHGoogle Scholar
  49. [49]
    Lee, K. W., Wise, K. D., SENSIM: A Simulation Program for Solid-State Pressure Sensors, IEEE Trans. Electron Devices, ED-29 (1982), 34–41.Google Scholar
  50. [50]
    Stavsky, Y., Hoff, N. J., Mechanics of Composite Structures, Composite Engineering Laminates, Dietz, A. G. H, (Ed.), Cambridge: MIT Press, 1969.Google Scholar
  51. [51]
    Senturia, S. D., Microfabricated Structures for the Measurement of Mechanical Properties and Adhesion of Thin Films, Digest of Technical Papers, Transducers’ 87, Tokyo, 1987, pp. 11-16.Google Scholar
  52. [52]
    Allen, M. G., Mehregany, M., Howe, R. T., Senturia, S. D., Microfabricated Structures for the in situ Measurement of Residual Stress, Young’s Modulus, and Ultimate Strain of Thin Films, Appl. Phys. Letts., 51 (1987), 241–243.CrossRefGoogle Scholar
  53. [53]
    Maseeh, F., Schmidt, M. A., Allen, M. G., Senturia, S. D., Calibrated Measurements of Elastic Limit, Modulus, and the Residual Stress of Thin Films Using Micro-machined Suspended Structures, Technical Digest, IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Is., 1988, pp. 84–87.Google Scholar
  54. [54]
    Tabata, O., Kawahata, K., Sugiyama, S., Igarashi, I., Mechanical Property Measurement of Composite Rectangular Membrane, Sensors and Actuators A, 20 (1989), 135–141.CrossRefGoogle Scholar
  55. [55]
    Puers, B., Vergote, S., A Subminiature Capacitive Movement Detector Using a Composite Membrane Suspension, Sensors and Actuators A, 31 (1992), 90–96.CrossRefGoogle Scholar
  56. [56]
    Guckel, H., Randazzo, T., Burns, D. W., A Simple Technique for the Determination of Residual Stress in Thin Films with Application to Polysilicon, J. Appl. Phys., 57 (1985), 1671–1675.CrossRefGoogle Scholar
  57. [57]
    Campbell, D. S., Handbook of Thin Films Technology, New York: McGraw Hill, 1970.Google Scholar
  58. [58]
    Peterson, K. E., Guarnieri, C. R., Young’s Modulus Measurements of Thin Films Using Micromechanics, J. Appl. Phys., 50 (1979), 6761–6766.CrossRefGoogle Scholar
  59. [59]
    Zhang, L. M., Uttamchandani, D., Culshaw, W., Measurement of the Mechanical Properties of Silicon Microresonators, Sensors and Actuators A, 29 (1991), 79–84.CrossRefGoogle Scholar
  60. [60]
    Pratt, R. I., Johnson, G. C., Howe, R. T., Chang, J. C., Micromechanical Structures for Thin Film Characterization, Digest of Technical Papers, Transducers’ 91, San Francisco, 1991, pp. 205-208.Google Scholar
  61. [61]
    Osterberg, P. M., Gupta, R. K., Gilbert, J. R., Senturia, S. D., Quantitative Models for the Measurement of Residual Stress, Poisson Ratio and Young’s Modulus Using Electrostatic Pull-In of Beams and Diaphragms, Technical Digest, IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Is., 1994, pp. 184–188.Google Scholar
  62. [62]
    Pan, J. Y., Lin, P., Maseeh, F., Senturia, S. D., Verification of FEM Analysis of Load-Deflection Methods for Measuring Mechanical Properties of Thin Films, Technical Digest, IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Is., 1990, pp. 70–73.Google Scholar
  63. [63]
    Seino, Y., Nagai, S., Temperature Dependence of the Young’s Modulus of Diamond Thin Film Prepared by Microwave Plasma Chemical Vapor Deposition, J. Mat. Sc. Lett., 12 (1993), 324–325.CrossRefGoogle Scholar
  64. [64]
    Bourgeois, C., Hermann, J., Blanc, N., de Rooij, N. F., Rudolf, F., Determination of the Elastic Temperature Coefficients of Monocrystalline Silicon, Digest of Technical Papers, Vol. 2, Transducers’ 95, Stockholm, 1995, pp. 92–95.Google Scholar
  65. [65]
    Han, M. Y., Jou, J. H., Determination of the Mechanical Properties of RF-Magnetron-Sputtered Zinc Oxide Thin Films on Substrates, Thin Solid Films, 260 (1995), 58–64.CrossRefGoogle Scholar
  66. [66]
    Klein, C. A., Anisotropy of Young’s Modulus and Poisson’s Ratio in Diamond, Mat. Res. Bull, 27 (1992), 1407–1414.CrossRefGoogle Scholar
  67. [67]
    Maier-Schneider, D., Köprülülü, A., Obermeier, E., Elastic Properties and Micro-structure of LPCVD Polysilicon Films, J. of Micromechanics and Microengineering, 5 (1995), 121.CrossRefGoogle Scholar
  68. [68]
    Kahn, H., Stemmer, S., Nandakumar, K., Hever, A. H., Mullen, R. L., Ballarini, R., Huff, M. A., Mechanical Properties of Thick, Surface Micromachined Polysilicon Films, Proc. IEEE MEMS, San Diego, 1996, pp. 343-348.Google Scholar
  69. [69]
    Biebl, M., Brandi, G., Howe, R. T., Young’s Modulus of in situ Phosphorus-Doped Polysilicon, Digest of Technical Papers, Vol. 2, Transducers’ 95, Stockholm, 1995, pp. 80–83.Google Scholar
  70. [70]
    Obermeier, E., High Temperature Microsensors Based on Polycrystalline Diamond Thin Films, Digest of Technical Papers, Vol. 2, Transducers’ 95, Stockholm, 1995, pp. 178–181.Google Scholar
  71. [71]
    Kuschnereit, R., Fath, H., Kolomenskii, A. A., Szabadi, M., Hess, P., Mechanical and Elastic Properties of Amorphous Hydrogenated Silicon Films Studied by Broad Band Surface Acoustic Wave Spectroscopy, Appl. Phys. A, 61 (1995), 269–276.CrossRefGoogle Scholar
  72. [72]
    Jean, A., El Khakani, M. A., Chaker, M., Boily, S., Gat, E., Kieffer, J. C., Pepin, H., Biaxial Young’s Modulus of Silicon Carbide Thin Films, Appl. Phys. Lett., 62 (1993), 2200–2202.Google Scholar
  73. [73]
    Windischmann, H., Intrinsic Stress and Mechanical Properties of Hydrogenated Silicon Carbide Produced by Plasma-Enhanced Chemical Vapor Deposition, J. Vac. Sci. Tech., A9 (1991), 2459–2463.Google Scholar
  74. [74]
    Watkins, T. R., Green, D. J., Ryba, E. R., Determination of Young’s Modulus in Chemically Vapor-Deposited SiC Coatings, J. Am. Ceram. Soc., 76 (1993) 1965–1968.CrossRefGoogle Scholar
  75. [75]
    Walsh, D., Culshaw, B., Optically Activated Silicon Microresonator Transducers: An Assessment of Material Properties, Sensors and Actuators A, 25-27 (1991), 711–716.CrossRefGoogle Scholar
  76. [76]
    Stewart, R. A., Kim, J., Kim, E. S., White, R. M., Muller, R. S., Young Modulus and Residual Stress of LPCVD Silicon-Rich Silicon Nitride Determined from Membrane Deflection, Sensors and Materials, 2 (1991), 285–298.Google Scholar
  77. [77]
    Tsukahara, Y., Ohira, K., Yanaka, M., Inaba, M., Satoh, A., Elastic Properties Measurement of Glass Layers Fabricated on Silicon Wafers for Microelectronics and Micromachines, IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, 42 (1995), 387–391.CrossRefGoogle Scholar
  78. [78]
    Maier-Schneider, D., Ersoy, A., Maibach, J., Schneider, D., Obermeier, E., Influence of Annealing on Elastic Properties of LPCVD Silicon Nitride and LPCVD Poly-silicon, Sensors and Materials, 7 (1995), 121–129.Google Scholar
  79. [79]
    Wells, G. M., Chen, H. T. H., Wallace, J. P., Engelstad, R. L., Cerrina, F., Radiation Damage-Induced Changes in Silicon Nitride Membrane Mechanical Properties, J. Vac. Sci. Technol. B, 13 (1995), 3075–3077.CrossRefGoogle Scholar
  80. [80]
    Jou, J., Chen, L., Relaxation and Thermal Expansion Coefficient of Polyimide Films Coated on Substrates, Appl. Phys. Lett., 59 (1991), 46–47.CrossRefGoogle Scholar
  81. [81]
    Fan, L.-S., Tai, Y.-C., Muller, R. S., Integrated Movable Micromechanical Structures for Sensors and Actuators, IEEE Trans. Electron Devices, ED-35 (1988), 724–730.CrossRefGoogle Scholar
  82. [82]
    Lin, Y.-C., Hesketh, P. J., Schuster, J. P., Finite-Element Analysis of Thermal Stresses in a Silicon Pressure Sensor for Various Die-Mount Materials, Sensors and Actuators A, 44 (1994), 145–149.CrossRefGoogle Scholar
  83. [83]
    Reichl, H., Packaging and Interconnection of Sensors, Sensors and Actuators A, 25-27 (1991), 63–71.Google Scholar
  84. [84]
    Peterson, K. E., Silicon as a Mechanical Material, Proc. IEEE, 70 (1982), 420–457.CrossRefGoogle Scholar
  85. [85]
    Hälg, B., On a Nonvolatile Memory Cell Based on Micro-Electro-Mechanics, Proc. IEEE MEMS, Napa Valley, 1990, pp. 172-176.Google Scholar
  86. [86]
    Lide, D. R., Handbook of Chemistry and Physics, 72nd Ed., Boston: Chemical Rubber Publishing Co., 1992.Google Scholar
  87. [87]
    Wur, D. R., Davidson, J. L., Kang, W. P., Kuiser, D. L., Polycrystalline Diamond Pressure Sensor, IEEE J. of Microelectromechanical Systems, 4 (1995), 34–41.CrossRefGoogle Scholar
  88. [88]
    Mehregany, M., Tong, L., Matus, L. G., Larkin, D. J., Internal Stress and Elastic Modulus Measurements on Micromachined 3C-SiC Thin Films, IEEE Trans. Electron Devices, 44 (1997), 74–79.CrossRefGoogle Scholar
  89. [89]
    Thangaraj, D., Nathan, A., Two Dimensional Analysis of Incompressible Viscous Flow in Ducts Using a Rotated Difference Scheme, Sensors and Materials, 8 (1996), 13–22.Google Scholar
  90. [90]
    Nagata, M., Swart, N., Stevens, M., Nathan, A., Thermal Based Micro Flow Sensor Optimization Using Coupled Electrothermal Numerical Simulations, Digest of Technical Papers, Vol. 2, Transducers’ 95, Stockholm, 1995, pp. 447–450.Google Scholar
  91. [91]
    Mastrangelo, C. H., Muller, R. S., A Constant-Temperature Gas Flowmeter with a Silicon Micromachines Package, Technical Digest, IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Is., 1988, pp. 43–46.Google Scholar
  92. [92]
    Engel, P. A., Chen, W. T., (Eds.), Advances in Electronic Packaging, Proc. ASME Int. Electro Packaging Conf., Vols. 1 and 2, 1993.Google Scholar
  93. [93]
    Middleman, S., Hochberg, A. K., Process Engineering Analysis in Semiconductor Device Fabrication, New York: McGraw-Hill, 1993.Google Scholar
  94. [94]
    American Institute of Physics Handbook, 3rd Ed., New York: McGraw-Hill, 1972.Google Scholar
  95. [95]
    Starr, J.B., Squeeze-Film Damping in Solid-State Accelerometers, Technical Digest, IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Is., 1990, pp. 44–47.Google Scholar
  96. [96]
    van Kampen, R. P., Vellekoop, M. J., Sarro, P. M., Wolffenbuttel, R. F., Application of Electrostatic Feedback to Critical Damping of an Integrated Silicon Capacitive Accelerometer, Digest of Technical Papers, Transducers’ 93, Yokohama, 1993, pp. 818-821.Google Scholar
  97. [97]
    Cho, Y.-H., Pisano, A. P., Howe, R. T., Viscous Damping Model for Laterally Oscillating Microstructures, J. of Microelectromechanical Systems, 3 (1994), 81–87.CrossRefGoogle Scholar
  98. [98]
    Zhang. X., Tang, W. C., Viscous Air Damping in Laterally Driven Microresonators, Sensors and Materials, 27 (1995), 415–430.Google Scholar
  99. [99]
    Hosaka, H., Itao, K., Kuroda, S., Evaluation of Energy Dissipation Mechanisms in Vibrational Microactuators, Proc. IEEE MEMS, 1994, pp. 193-198.Google Scholar
  100. [100]
    Reuther, H. M., Weinmann, M., Fischer, M., von Münch, W., Aßmus, F., Modeling Electrostatically Deflectable Microstructures and Air Damping Effects, Sensors and Materials, 8 (1996), 251–269.Google Scholar
  101. [101]
    Tang, W. C., Lim, M. G., Howe, R. T., Electrostatic Comb Drive Levitation and Control Method, J. Micro electromechanical Systems, 1 (1992), 170–178.Google Scholar
  102. [102]
    Langlois, W. E., Isothermal Squeeze Films, Quart. Appl. Math., XX (1962), 131–150.Google Scholar
  103. [103]
    Langlois, W. E., Slow Viscous Flow, New York: Macmillan, 1964.Google Scholar
  104. [104]
    Yang, Y.-J., Senturia, S. D., Numerical Simulation of Compressible Squeezed-Film Damping, Technical Digest, IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Is., 1996, pp. 76–79.Google Scholar
  105. [105]
    Sadd, M. H., Stiffler, A. K., Squeeze Film Dampers: Amplitude Effects at Low Squeeze Numbers, J. Eng. Indust., Trans. of the ASME, B97, (1975), 1366–1370.CrossRefGoogle Scholar
  106. [106]
    Morgan, K., Periaux, J., Thomasset, F. (Eds.), A GAMM Workshop, Notes on Numerical Fluid Dynamics, Braunschweig: Vieweg, 1984.Google Scholar
  107. [107]
    Denis, S. C. R., Chang, G.-Z., Numerical Solutions for Steady Flow Past a Circular Cylinder at Reynolds Numbers up to 100, J. Fluid Mech., 42 (1970), 471–489.CrossRefGoogle Scholar
  108. [108]
    Hamielec, A. E., Raal, J. D., Numerical Studies of Viscous Flow Around Circular Cylinders, Phys. of Fluids, 12 (1969), 11–17.MATHCrossRefGoogle Scholar
  109. [109]
    Acrivos, A., Leal, L. G., Snowden, D. D., Pan, F., Further Experiments on Steady Separated Flows Past Bluff Objects, J. Fluid Mech., 34 (1970), 25–48.CrossRefGoogle Scholar
  110. [110]
    Roache, P. J., Computational Fluid Dynamics, Albuquerque: Hermosa, 1976.Google Scholar
  111. [111]
    Peyret, R. T., Taylor, T. D., Computational Methods for Fluid Flow, New York: Springer-Verlag, 1983.MATHGoogle Scholar
  112. [112]
    Patankar, S. V., Numerical Heat Transfer and Fluid Flow, New York: Hemisphere Publishing Co., 1980.MATHGoogle Scholar
  113. [113]
    Thangaraj, D., Wu, H., Jayaram, S., Stream Function Distribution of Petroleum Liquids in Relaxation Tanks, Proc. IEEE-IAS 27th Meeting, Denver, 1994, pp. 1676-1681.Google Scholar
  114. [114]
    Raithby, G. D., Skew Upstream Differencing Schemes for Problems Involving Fluid Flow, Computer Methods in Applied Mechanics and Engineering, 9 (1976), 153–164.MATHMathSciNetCrossRefGoogle Scholar
  115. [115]
    Rice, J. G., Schnipke, R. J., A Monotone Streamline Upwind Finite Element Method for Convection-Dominated Flows, Computer Methods in Applied Mechanics and Engineering, 48 (1985), 313–327.MATHCrossRefGoogle Scholar
  116. [116]
    Roache, P. J., A Comment on the Paper “Finite Difference Methods for the Stokes and Navier-Stokes Equations” by J. C. Strickwerda, Int. J. Num. Meth. in Fluids, 8 (1988), 1459–1463.MathSciNetCrossRefGoogle Scholar
  117. [117]
    Gresho, P. M., Sani, R. L., Introducing Four Benchmark Solutions, Int. J. Num. Meth. in Fluids, 11 (1990), 951–952.CrossRefGoogle Scholar
  118. [118]
    Sani, R. L., Gresho, P. M., Résumé and Remarks on the Open Boundary Condition Minisymposium, Int. J. Num. Meth. in Fluids, 18 (1994), 983–1008.MATHMathSciNetCrossRefGoogle Scholar
  119. [119]
    Small, M. K., Vlassak, J. J., Powell, S. F., Daniels, B. J., Nix, W. D., Accuracy and Reliability of Bulge Test Experiments, Proc. MRS, 308 (1993), 159–164.CrossRefGoogle Scholar
  120. [120]
    Ziebart, V., Paul, O., Münch, U., Baltes, H., A Novel Method to Measure Poisson’s Ratio of Thin Films, Proc. MRS, 505 (1998), 27–32.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag/Wien 1999

Authors and Affiliations

  • Arokia Nathan
    • 1
  • Henry Baltes
    • 2
  1. 1.Dept. of Electrical and Computer EngineeringUniversity of WaterlooWaterlooCanada
  2. 2.Physical Electronics LaboratoryETH HoenggerbergZürichSwitzerland

Personalised recommendations