Dynamic behaviours of spinning pre-twisted Rayleigh micro-beams
DOI:
https://doi.org/10.1080/17797179.2017.1354576Keywords:
propagation, spectrum curve, beam theory, elastic wave, spinning dynamics, Rayleigh beam theory, Rayleigh–Ritz method, modified couple stress theory, Pre-twisted micro-scaleAbstract
Developments in micro-systems’ operations rely on identifying and controlling the dynamic responses of micrometre-scale elements. Pre-twisted micro-elements feature prominently in these systems, yet very little is known about the interacting effect of motley factors on their behaviours. Presented here is a model of pre-twisted micro-beams that accounts for the coupling of scale-dependent, rotary inertia and spinning effects in vibratory and wave propagation analyses. In tackling the problem, the system’s physical domain is transformed into the mathematical realm via a scale-dependent microcontinuum theory, while its time evolution is captured by Hamiltonian mechanics. Analytic expressions are obtained for the long-wavelength limit and dispersion relations. The spectrum relations of the waves are established from an octic characteristics equation (which, being in violation of the Abel– Ruffini theorem is treated numerically). Splitting of the waves within the system is observed, and the frequencies of the split shift for altered values of the pre-twist angle and rotary inertia effect. Thinner, highly pre-twisted micro-scale beams experience a widening (narrowing) of the difference between the two phase speeds (group velocities) of the waves. Further, pre-twisting lowers the natural frequencies associated with odd-numbered modes of vibration of the element, while the small-scale effect strongly affects the higher vibration modes.
Downloads
References
Abramowitz, M., & Stegun, I. A. (1965). Handbook of mathematical functions. New York, NY:
Dover.
Aifantis, E. C. (1984). On the microstructural origin of certain inelastic models. Journal of
Engineering Materials and Technology, 106, 326–330.
Aifantis, E. C. (1992). On the role of gradients in the localization of deformation and fracture.
International Journal of Engineering Science, 30, 1279–1299.
Ataei, H., Beni, Y. T., & Shojaeian, M. (2016). The effect of small scale and intermolecular forces
on the pull-in instability and free vibration of functionally graded nano-switches. Journal
of Mechanical Science and Technology, 30, 1799–1816.
Balhaddad, A. S., & Onipede, D., Jr (1998). Three-dimensional free vibration of pretwisted
beams. AIAA Journal, 36, 1524–1528.
Banerjee, J. R. (2001). Free vibration analysis of a twisted beam using the dynamic stiffness
method. International Journal of Solids and Structures, 38, 6703–6722.
Banerjee, J. (2001). Free vibration analysis of a twisted beam using the dynamic stiffness
method. International Journal of Solids and Structures, 38, 6703–6722.
Banerjee, J. R. (2001). Free vibration analysis of a twisted beam using the dynamic stiffness
method. International Journal of Solids and Structures, 38, 6703–6722.
Banerjee, J. R. (2004). Development of an exact dynamic stiffness matrix for free vibration
analysis of a twisted Timoshenko beam. Journal of Sound and Vibration, 270, 379–401.
Carnegie, W. (1957). Static bending of pre-twisted cantilever blading. Proceedings of the
Institution of Mechanical Engineers, 171, 873–894.
Carnegie, W. (1959). Vibrations of pre-twisted cantilever blading. Proceedings of the Institution
of Mechanical Engineers, 173, 343–374.
Carnegie, W. (1964). Vibrations of pre-twisted cantilever blading allowing for rotary inertia
and shear deflection. Journal of Mechanical Engineering Science, 6, 105–109.
Challamel, N. (2013). Variational formulation of gradient or/and nonlocal higher-order shear
elasticity beams. Composite Structures, 105, 351–368.
Chen, W.-R., & Chen, C.-S. (2015). Parametric instability of twisted Timoshenko beams with
localized damage. International Journal of Mechanical Sciences, 100, 298–311.
Chen, W.-R., & Keer, L. (1993). Transverse vibrations of a rotating twisted timoshenko beam
under axial loading. Journal of Vibration and Acoustics, 115, 285–294.
Chin, W. C. (2014). Overview and fundamental ideas. In Wave propagation in drilling, well
logging and reservoir applications (pp. 1–49). Hoboken, NJ: Wiley.
Cosserat, E., & Cosserat, C. F. (1909). Théorie des Corps Déformables. Paris: A. Hermann et Fils.
Dawson, B. (1968). Coupled bending‐bending vibrations of pre‐twisted cantilever blading
treated by the rayleigh ritz energy method. Journal of Mechanical Engineering Science, 10,
–388.
Dehrouyeh-Semnani, A. M., Dehrouyeh, M., Torabi-Kafshgari, M., & Nikkhah-Bahrami,
M. (2015). An investigation into size-dependent vibration damping characteristics of
functionally graded viscoelastically damped sandwich microbeams. International Journal
of Engineering Science, 96, 68–85.
Dehrouyeh-Semnani, A. M., Mostafaei, H., & Nikkhah-Bahrami, M. (2016). Free flexural
vibration of geometrically imperfect functionally graded microbeams. International Journal
of Engineering Science, 105, 56–79.
Deng, Y., Peng, L., Lai, X., Fu, M., & Lin, Z. (2017). Constitutive modeling of size effect on
deformation behaviors of amorphous polymers in micro-scaled deformation. International
Journal of Plasticity, 89, 197–222.
Doyle, J. F. (1997). Wave propagation in structures : Spectral analysis using fast discrete Fourier
transforms (2nd ed.). New York, NY: Springer.
Eringen, A. C., & Edelen, D. G. B. (1972). On nonlocal elasticity. International Journal of
Engineering Science, 10, 233–248.
Filiz, S., & Ozdoganlar, O. B. (2011). A three-dimensional model for the dynamics of microendmills
including bending, torsional and axial vibrations. Precision Engineering, 35, 24–37.
Fleck, N. A., & Hutchinson, J. W. (1993). A phenomenological theory for strain gradient effects
in plasticity. Journal of the Mechanics and Physics of Solids, 41, 1825–1857.
Fleck, N. A., Muller, G. M., Ashby, M. F., & Hutchinson, J. W. (1994). Strain gradient plasticity:
Theory and experiment. Acta Metallurgica et Materialia, 42, 475–487.
Fu, C. (1974). Computer analysis of a rotating axial-turbomachine blade in coupled bendingbending-
torsion vibrations. International Journal for Numerical Methods in Engineering, 8,
–588.
Ghayesh, M. H., Farokhi, H., & Gholipour, A. (2017). Oscillations of functionally graded
microbeams. International Journal of Engineering Science, 110, 35–53.
Ghorbani Shenas, A., Malekzadeh, P., & Ziaee, S. (2017). Thermal buckling of rotating pretwisted
functionally graded microbeams with temperature-dependent material properties.
Acta Mechanica, 228, 1115–1133.
Ghorbani Shenas, A., Ziaee, S., & Malekzadeh, P. (2016). Vibrational behavior of rotating pretwisted
functionally graded microbeams in thermal environment. Composite Structures,
, 222–235.
Gong, Y., Ehmann, K. F., & Lin, C. (2003). Analysis of dynamic characteristics of micro-drills.
Journal of Materials Processing Technology, 141, 16–28.
Goodier, J. N., & Griffin, D. (1969). Elastic bending of pretwisted bars. International Journal
of Solids and Structures, 5, 1231–1245.
Guglielmino, E., & Saccomandi, G. (1996). On the bending of pretwisted bars by a terminal
transverse load. International Journal of Engineering Science, 34, 1285–1299.
Houbolt, J. C., & Brooks, G. W. (1957). Differential equations of motion for combined flapwise
bending (TN 3905). NASA. Retrieved from Name website: https://ntrs.nasa.gov/search.
jsp?R=19930092334
Hu, X. X., & Tsuiji, T. (1999). Free vibration analysis of curved and twisted cylindrical thin
panels. Journal of Sound and Vibration, 219, 63–88.
Huang, B.-W., & Kuang, J.-H. (2007). The parametric resonance instability in a drilling process.
Journal of Applied Mechanics, 74, 958–964.
Ilanko, S., Monterrubio, L., Mochida, Y. (2015). The Rayleigh-Ritz method for structural analysis.
Hoboken, NJ: Wiley.
Karp, B., & Durban, D. (2005). Evanescent and propagating waves in prestretched hyperelastic
plates. International Journal of Solids and Structures, 42, 1613–1647.
Lee, J., & Lee, J. (2016). Development of a transfer matrix method to obtain exact solutions for
the dynamic characteristics of a twisted uniform beam. International Journal of Mechanical
Sciences, 105, 215–226.
Leissa, A., MacBain, J., & Kielb, R. (1984). Vibrations of twisted cantilevered plates – Summary
of previous and current studies. Journal of Sound and Vibration, 96, 159–173.
Leung, A. Y. T., & Fan, J. (2010). Natural vibration of pre-twisted shear deformable beam systems
subject to multiple kinds of initial stresses. Journal of Sound and Vibration, 329, 1901–1923.
Leung, A., & Fan, J. (2010). Natural vibration of pre-twisted shear deformable beam systems
subject to multiple kinds of initial stresses. Journal of Sound and Vibration, 329, 1901–1923.
Liang, L.-N., Ke, L.-L., Wang, Y.-S., Yang, J., & Kitipornchai, S. (2015). Flexural Vibration of an
Atomic Force Microscope Cantilever Based on Modified Couple Stress Theory. International
Journal of Structural Stability and Dynamics, 15, 1540025.
Liao, C. L., & Dang, Y. H. (1992). Structural characteristics of spinning pretwisted orthotropic
beams. Computers and Structures, 45, 715–731.
Liew, K., Lim, C., & Ong, L. (1994). Vibration of pretwisted cantilever shallow conical shells.
International Journal of Solids and Structures, 31, 2463–2476.
Lim, M. C., & Liew, K. (1995). Vibration of pretwisted cantilever trapezoidal symmetric
laminates. Acta Mechanica, 111, 193–208.
Lin, S.-M., Wang, W.-R., & Lee, S.-Y. (2001). The dynamic analysis of nonuniformly pretwisted
Timoshenko beams with elastic boundary conditions. International Journal of Mechanical
Sciences, 43, 2385–2405.
Liu, K.-C., Friend, J., & Yeo, L. (2009). The axial–torsional vibration of pretwisted beams.
Journal of Sound and Vibration, 321, 115–136.
Love, A. E. H. (1927). A treatise on the mathematical theory of elasticity. Cambridge: Cambridge
Academic Press.
MacBain, J. C., Kielb, R. E., & Leissa, A. W. (1985). Vibrations of twisted cantilevered plates –
Experimental investigation. Journal of Engineering for Gas Turbines and Power, 107, 187–196.
Mindlin, R. D., & Tiersten, H. F. (1962). Effects of couple-stresses in linear elasticity. Archive
for Rational Mechanics and Analysis, 11, 415–448.
Mohammadimehr, M., Farahi, M. J., & Alimirzaei, S. (2016). Vibration and wave propagation
analysis of twisted micro-beam using strain gradient theory. Applied Mathematics and
Mechanics, 37, 1375–1392.
Mustapha, K. (2015). Coupled extensional-flexural vibration behaviour of a system of elastically
connected functionally graded micro-scale panels. European Journal of Computational
Mechanics, 24, 34–63.
Mustapha, K., & Hawwa, M. A. (2015). Eigenanalyses of functionally graded micro-scale beams
entrapped in an axially-directed magnetic field with elastic restraints. International Journal
of Structural Stability and Dynamics, 16, 1550022.
Mustapha, K. B., & Wong, B. T. (2016). Torsional frequency analyses of microtubules with
end attachments. ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschrift für
Angewandte Mathematik und Mechanik, 96, 824–842.
Mustapha, K., & Zhong, Z. (2012). A new modeling approach for the dynamics of a micro
end mill in high-speed micro-cutting. Journal of Vibration and Control, 19, 901–923.
doi:10.1177/1077546312439912
Mustapha, K., & Zhong, Z. (2012). Wave propagation characteristics of a twisted micro scale
beam. International Journal of Engineering Science, 53, 46–57.
Mustapha, K., & Zhong, Z. (2013). A hybrid analytical model for the transverse vibration
response of a micro-end mill. Mechanical Systems and Signal Processing, 34, 321–339.
Pai, P. F., Qian, X., & Du, X. (2013). Modeling and dynamic characteristics of spinning Rayleigh
beams. International Journal of Mechanical Sciences, 68, 291–303.
Prasolov, V. V. (2009). Polynomials (vol. 11). Berlin: Springer Science & Business Media.
Qatu, M. S., & Leissa, A. W. (1991). Vibration studies for laminated composite twisted cantilever
plates. International Journal of Mechanical Sciences, 33, 927–940.
Rao, D. K. (1976). Transverse vibrations of pre-twisted sandwich beams. Journal of Sound and
Vibration, 44, 159–168.
Razavilar, R., Alashti, R., & Fathi, A. (2014). Investigation of thermoelastic damping in
rectangular microplate resonator using modified couple stress theory. International Journal
of Mechanics and Materials in Design, 12, 1–13.
Reddy, J. N. (2002). Energy principles and variational methods in applied mechanics (2nd ed.).
Hoboken, NJ: Wiley.
Reddy, J. N. (2011). Microstructure-dependent couple stress theories of functionally graded
beams. Journal of the Mechanics and Physics of Solids, 59, 2382–2399.
Reissner, E., & Wan, F. Y. M. (1971). On stretching, twisting, pure bending and flexure of
pretwisted elastic plates. International Journal of Solids and Structures, 7, 625–637.
Rosen, A. (1991). Structural and dynamic behavior of pretwisted rods and beams. Applied
Mechanics Reviews, 44, 483–515.
Sahu, S., Asha, A., & Mishra, R. (2005). Stability of laminated composite pretwisted cantilever
panels. Journal of Reinforced Plastics and Composites, 24, 1327–1334.
Shojaeian, M., Beni, Y. T., & Ataei, H. (2016). Electromechanical buckling of functionally
graded electrostatic nanobridges using strain gradient theory. Acta Astronautica, 118, 62–71
Şimşek, M., & Reddy, J. N. (2013). A unified higher order beam theory for buckling of a
functionally graded microbeam embedded in elastic medium using modified couple stress
theory. Composite Structures, 101, 47–58.
Sinha, S. K., & Turner, K. E. (2011). Natural frequencies of a pre-twisted blade in a centrifugal
force field. Journal of Sound and Vibration, 330, 2655–2681.
Song, O., Jeong, N.-H., & Librescu, L. (2000). Vibration and stability of pretwisted spinning
thin-walled composite beams featuring bending–bending elastic coupling. Journal of Sound
and Vibration, 237, 513–533.
Subrahmanyam, K., Kulkarni, S., & Rao, J. (1981). Coupled bending-bending vibrations of pretwisted
cantilever blading allowing for shear deflection and rotary inertia by the Reissner
method. International Journal of Mechanical Sciences, 23, 517–530.
Tekinalp, O., & Ulsoy, A. (1989). Modeling and finite element analysis of drill bit vibrations.
Journal of Vibration Acoustics Stress and Reliability in Design, 111, 148–155.
Toupin, R. A. (1964). Theories of elasticity with couple-stress. Archive for Rational Mechanics
and Analysis, 17, 85–112.
Troesch, A., Anliker, M., & Ziegler, H. (1954). Lateral vibrations of twisted rods. Quarterly of
Applied Mathematics, 12, 163–173.
Wajchman, D., Liu, K. C., Friend, J., & Yeo, L. (2008). An ultrasonic piezoelectric motor utilizing
axial-torsional coupling in a pretwisted non-circular cross-sectioned prismatic beam. IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 55, 832–840.
Watson, B., Friend, J., & Yeo, L. (2009). Micromotor of less than 1 mm3 volume for in vivo
medical procedures. In Quantum, Nano and Micro Technologies, 2009 ICQNM ‘09. Third
International Conference on, 2009 (pp. 81–85).
Yang, J. F. C., & Lakes, R. S. (1982). Experimental study of micropolar and couple stress elasticity
in compact bone in bending. Journal of Biomechanics, 15, 91–98.
Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. (2002). Couple stress based strain gradient
theory for elasticity. International Journal of Solids and Structures, 39, 2731–2743.
Yardimoglu, B., & Inman, D. J. (2004). Coupled bending-bending-torsion vibration of a rotating
pre-twisted beam with aerofoil cross-section and flexible root by finite element method.
Shock and Vibration, 11, 637–646.
Young, T., & Gau, C. (2003). Dynamic stability of spinning pretwisted beams subjected to axial
random forces. Journal of Sound and Vibration, 268, 149–165.
Zeighampour, H., & Beni, Y. T. (2015). A shear deformable cylindrical shell model based on
couple stress theory. Archive of Applied Mechanics, 85, 539–553.
Zeighampour, H., Beni, Y. T., & Mehralian, F. (2015). A shear deformable conical shell
formulation in the framework of couple stress theory. Acta Mechanica, 226, 2607–2629.
Zhang, W.-M., & Meng, G. (2006). Stability, bifurcation and chaos of a high-speed rub-impact
rotor system in MEMS. Sensors and Actuators A: Physical, 127, 163–178.
Zickel, J. (1952). Bending of pretwisted beams. DTIC Document.
