328
8 Thermoelastic Vibrations of Timoshenko Microbeams
Fig. 8.13 Comparison of the eigenfrequencies for the (s − s) beam: a F = 0; b F = 0.260417
References
1. Awrejcewicz, J., Krysko, V.A., Pavlov, S., Zhigalov, M.V., Kalutsky, L.A., Krysko, A.V.:
Thermoelastic vibrations of a Timoshenko microbeam based on the modified couple stress
theory. Nonlinear Dyn. 99, 919–943 (2020)
2. Zener, C.: Internal friction in solids II. General theory of thermoelastic internal friction. Phys.
Rev. 53(1), 90–99 (1938)
3. Lifshitz, R., Roukes, M.L.: Thermoelastic damping in micro- and nanomechanical systems.
Phys. Rev. B 61(8), 5600–5609 (2000)
4. Yasumura, K.Y., Stowe, T.D., Chow, E.M., Pfafman, T., Kenny, T.W., Stipe, B.C., Rugar, D.:
Quality factors in micro- and submicron-thick cantilevers. J. Microelectromech. Syst. 9(1),
117–125 (2000)
5. Khisaeva, Z.F., Ostoja-Starzewski, M.: Thermoelastic damping in nanomechanical resonators
with finite wave speeds. J. Therm. Stres. 29, 201–216 (2006)
6. Houston, B., Photiadis, D.M., Marcus, M.H., Bucaro, J.A., Liu, X., Vignola, J.F.: Thermoelastic loss in microscale oscillators. Appl. Phys. Lett. 80, 1300–1302 (2002)
7. Yang, J., Ono, T., Esashi, M.: Energy dissipation in submicrometer thick single-crystal silicon
cantilevers. J. Microelectromech. Syst. 11(6), 775–783 (2002)
8. Duwel, A., Gorman, J., Weinstein, M., Borenstein, J., Ward, P.: Experimental study of thermoelastic damping in MEMS gyros. Sens. Actuator A 103, 70–75 (2003)
9. Tilmans, H.A.C., Elwespoek, M., Fluitman, J.H.J.: Micro resonant force gauges. Sens. Actuator A 30, 35–53 (1992)
10. Lothe, J.: Aspects of the theories of dislocation mobility and internal friction. Phys. Rev. 117,
704–708 (1960)
11. Zhang, C.L., Xu, G.S., Jiang, Q.: Analysis of the air-damping effect on a micromachined
beam resonator. Math. Mech. Sol. 8, 315–325 (2003)
12. Tortonese, M., Barrett, R.C., Quate, C.F.: Atomic resolution with an atomic force microscope
using piezoresistive detection. Appl. Phys. Lett. 62(8), 834–836 (1993)
13. Barnes, J.R., Stephenson, R.J., Welland, M.E., Gerber, Ch., Gimzewski, J.K.: Photothermal
spectroscopy with femtojoule sensitivity using a micromechanical device. Nature 372, 79–81
(1994)
14. Yurke, B., Greywall, D.S., Parqellis, A.N., Busch, P.A.: Theory of amplifier-noise evasion in
an oscillator employing a nonlinear resonator. Phys. Rev. A 51, 4211–4229 (1995)
15. Cleland, A.N., Roukes, M.L.: Fabrication of high frequency nanometer scale mechanical
resonators from bulk Si crystals. Appl. Phys. Lett. 69, 2653–2655 (1996)
16. Lun, F.Y., Zhang, P., Gao, F.B., Jia, H.G.: Design and fabrication of micro-optomechanical
vibration sensor. Microfab. Technol. 120(1), 61–64 (2006)
8 Thermoelastic Vibrations of Timoshenko Microbeams
Fig. 8.13 Comparison of the eigenfrequencies for the (s − s) beam: a F = 0; b F = 0.260417
References
1. Awrejcewicz, J., Krysko, V.A., Pavlov, S., Zhigalov, M.V., Kalutsky, L.A., Krysko, A.V.:
Thermoelastic vibrations of a Timoshenko microbeam based on the modified couple stress
theory. Nonlinear Dyn. 99, 919–943 (2020)
2. Zener, C.: Internal friction in solids II. General theory of thermoelastic internal friction. Phys.
Rev. 53(1), 90–99 (1938)
3. Lifshitz, R., Roukes, M.L.: Thermoelastic damping in micro- and nanomechanical systems.
Phys. Rev. B 61(8), 5600–5609 (2000)
4. Yasumura, K.Y., Stowe, T.D., Chow, E.M., Pfafman, T., Kenny, T.W., Stipe, B.C., Rugar, D.:
Quality factors in micro- and submicron-thick cantilevers. J. Microelectromech. Syst. 9(1),
117–125 (2000)
5. Khisaeva, Z.F., Ostoja-Starzewski, M.: Thermoelastic damping in nanomechanical resonators
with finite wave speeds. J. Therm. Stres. 29, 201–216 (2006)
6. Houston, B., Photiadis, D.M., Marcus, M.H., Bucaro, J.A., Liu, X., Vignola, J.F.: Thermoelastic loss in microscale oscillators. Appl. Phys. Lett. 80, 1300–1302 (2002)
7. Yang, J., Ono, T., Esashi, M.: Energy dissipation in submicrometer thick single-crystal silicon
cantilevers. J. Microelectromech. Syst. 11(6), 775–783 (2002)
8. Duwel, A., Gorman, J., Weinstein, M., Borenstein, J., Ward, P.: Experimental study of thermoelastic damping in MEMS gyros. Sens. Actuator A 103, 70–75 (2003)
9. Tilmans, H.A.C., Elwespoek, M., Fluitman, J.H.J.: Micro resonant force gauges. Sens. Actuator A 30, 35–53 (1992)
10. Lothe, J.: Aspects of the theories of dislocation mobility and internal friction. Phys. Rev. 117,
704–708 (1960)
11. Zhang, C.L., Xu, G.S., Jiang, Q.: Analysis of the air-damping effect on a micromachined
beam resonator. Math. Mech. Sol. 8, 315–325 (2003)
12. Tortonese, M., Barrett, R.C., Quate, C.F.: Atomic resolution with an atomic force microscope
using piezoresistive detection. Appl. Phys. Lett. 62(8), 834–836 (1993)
13. Barnes, J.R., Stephenson, R.J., Welland, M.E., Gerber, Ch., Gimzewski, J.K.: Photothermal
spectroscopy with femtojoule sensitivity using a micromechanical device. Nature 372, 79–81
(1994)
14. Yurke, B., Greywall, D.S., Parqellis, A.N., Busch, P.A.: Theory of amplifier-noise evasion in
an oscillator employing a nonlinear resonator. Phys. Rev. A 51, 4211–4229 (1995)
15. Cleland, A.N., Roukes, M.L.: Fabrication of high frequency nanometer scale mechanical
resonators from bulk Si crystals. Appl. Phys. Lett. 69, 2653–2655 (1996)
16. Lun, F.Y., Zhang, P., Gao, F.B., Jia, H.G.: Design and fabrication of micro-optomechanical
vibration sensor. Microfab. Technol. 120(1), 61–64 (2006)