Abstract
This paper investigates the nonlinear dynamic responses of the rotating blade with varying rotating speed under high-temperature supersonic gas flow. The varying rotating speed and centrifugal force are considered during the establishment of the analytical model of the rotating blade. The aerodynamic load is determined using first-order piston theory. The rotating blade is treated as a pretwist, presetting, thin-walled rotating cantilever beam. Using the isotropic constitutive law and Hamilton’s principle, the nonlinear partial differential governing equation of motion is derived for the pretwist, presetting, thin-walled rotating beam. Based on the obtained governing equation of motion, Galerkin’s approach is applied to obtain a two-degree-of-freedom nonlinear system. From the resulting ordinary equation, the method of multiple scales is exploited to derive the four-dimensional averaged equation for the case of 1:1 internal resonance and primary resonance. Numerical simulations are performed to study the nonlinear dynamic response of the rotating blade. In summary, numerical studies suggest that periodic motions and chaotic motions exist in the nonlinear vibrations of the rotating blade with varying speed.
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References
Carnegie, W.: Vibrations of rotating cantilever blading: theoretical approaches to the frequency problem based on energy methods. J. Mech. Eng. Sci. 1, 235–240 (1959)
Rao, J.S., Carnegie, W.: Solution of the equations of motion of coupled bending-bending-torsion vibrations of turbine blades by the method of Ritz–Galerkin. Int. J. Mech. Sci. 12, 875–882 (1970)
Anderson, G.L.: On the extensional and flexural vibrations of rotating bars. Int. J. Non-Linear Mech. 10, 223–236 (1975)
Rao, J.S., Banerjee, S.: Coupled bending-torsional vibrations of rotating cantilever blades-method of polynomial frequency equation. Mech. Mach. Theory 12, 271–280 (1977)
Swamlnathan, M., Rao, J.S.: Vibration of rotating, pre-twisted and tapered blades. Mech. Mach. Theory 12, 331–337 (1977)
Subrahmanyam, K.B., Kulkarni, S.V., Rao, J.S.: Coupled bending-bending vibrations of pre-twisted cantilever blading allowing for shear deflection and rotary inertia by the Reissner method. Int. J. Mech. Sci. 23, 517–530 (1981)
Subrahmanyam, K.B., Kulkarni, S.V., Rao, J.S.: Coupled bending-torsion vibrations of rotating blades of asymmetric aerofoil cross section with allowance for shear deflection and rotary inertia by use of the Reissner method. J. Sound Vib. 75, 17–36 (1981)
Chen, L.W., Chen, C.L.: Vibration and stability of cracked thick rotating blades. Comput. Struct. 28, 67–74 (1988)
Wang, J.H., Shieh, W.L.: The influence of a variable friction coefficient on the dynamic behavior of a blade with a friction damper. J. Sound Vib. 149, 137–145 (1991)
Chen, L.W., Jeng, C.H.: Vibrational analyses of cracked pre-twisted blades. Comput. Struct. 46, 133–140 (1993)
Chen, L.W., Peng, W.K.: Dynamic stability of rotating blades with geometric non-linearity. J. Sound Vib. 187, 421–433 (1995)
Yoo, H.H., Kwak, J.Y., Chung, J.: Vibration analysis of rotating pre-twisted blades with a concentrated mass. J. Sound Vib. 240, 891–908 (2001)
Lin, C.Y., Chen, L.W.: Dynamic stability of rotating pre-twisted blades with a constrained damping layer. Compos. Struct. 61, 235–245 (2003)
Sinha, S.K.: Dynamic characteristics of a flexible bladed-rotor with Coulomb damping due to tip-rub. J. Sound Vib. 273, 875–919 (2004)
Fazelzadeh, S.A., Malekzadeh, P., Zahedinejad, P., Hosseini, M.: Vibration analysis of functionally graded thin-walled rotating blades under high temperature supersonic flow using the differential quadrature method. J. Sound Vib. 306, 333–348 (2007)
Liu, T.R., Ren, Y.S.: Vibration of wind turbine blade modeled as composite thin-walled closed-section structure. Adv. Mater. Res. 129–131, 23–27 (2010)
Young, T.H.: Dynamic response of a pretwisted, tapered beam with non-constant rotating speed. J. Sound Vib. 150, 435–446 (1991)
Oh, S.Y., Song, O., Librescu, L.: Effects of pretwist and presetting on coupled bending vibrations of rotating thin-walled composite beams. Int. J. Solids Struct. 40, 1203–1224 (2003)
Turkalj, G., Brnic, J., Jasna, P.O.: Large rotation analysis of elastic thin-walled beam-type structures using ESA approach. Comput. Struct. 81, 1851–1864 (2003)
Mitra, M., Gopalakrishnan, S., Seetharama Bhat, M.: A new super convergent thin walled composite beam element for analysis of box beam structures. Int. J. Solids Struct. 41, 1491–1518 (2004)
Yang, J.B., Jiang, L.J., Chen, D.C.: Dynamic modelling and control of a rotating Euler–Bernoulli beam. J. Sound Vib. 274, 863–875 (2004)
Librescu, L., Oh, S.Y., Song, O.: Spinning thin-walled beams made of functionally graded materials: modeling, vibration and instability. Eur. J. Mech. A, Solids 23, 499–515 (2004)
Yoo, H.H., Lee, S.H., Shin, S.H.: Flapwise bending vibration analysis of rotating multi-layered composite beams. J. Sound Vib. 286, 745–761 (2005)
Hamdan, M.N., El Sinawib, A.H.: On the non-linear vibrations of an inextensible rotating arm with setting angle and flexible hub. J. Sound Vib. 281, 375–398 (2005)
Jarrar, F.S.M., Hamdan, M.N.: Nonlinear vibrations and buckling of a flexible rotating beam: a prescribed torque approach. Mech. Mach. Theory 42, 919–939 (2007)
Choi, S.C., Park, J.S., Kim, J.H.: Vibration control of pre-twisted rotating composite thin-walled beams with piezoelectric fiber composites. J. Sound Vib. 300, 176–196 (2007)
Lin, S.M.: PD control of a rotating smart beam with an elastic root. J. Sound Vib. 312, 109–124 (2008)
Piovan, M.T., Sampaio, R.: A study on the dynamics of rotating beams with functionally graded properties. J. Sound Vib. 327, 134–143 (2009)
Vadiraja, D.N., Sahasrabudhe, A.D.: Vibration analysis and optimal control of rotating pre-twisted thin-walled beams using MFC actuators and sensors. Thin-Walled Struct. 47, 555–567 (2009)
Chandiramani, N.K.: Active control of a piezo-composite rotating beam using coupled plant dynamics. J. Sound Vib. 329, 2716–2737 (2010)
Younesian, D., Esmailzadeh, E.: Vibration suppression of rotating beams using time-varying internal tensile force. J. Sound Vib. 330, 308–320 (2011)
Yao, M.H., Zhang, W.: Multi-pulse Shilnikov orbits and chaotic dynamics for nonlinear nonplanar motion of a cantilever beam. Int. J. Bifurc. Chaos 15, 3923–3952 (2005)
Zhang, W.: Chaotic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Chaos Solitons Fractals 26, 731–745 (2005)
Zhang, W., Yao, M.H.: Theories of multi-pulse global bifurcations for high-dimensional systems and applications to cantilever beam. Int. J. Mod. Phys. B 22, 4089–4141 (2008)
Zhang, W., Yao, M.H., Zhang, J.H.: Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam. J. Sound Vib. 319, 541–569 (2009)
Zhang, W., Yang, X.L.: Transverse nonlinear vibrations of a circular spinning disk with varying rotating speed. Sci. China Ser. G, Phys. Mech. Astron. 53, 1536–1553 (2010)
Lai, S.K., Lim, C.W., Lin, Z., Zhang, W.: Analytical analysis for large-amplitude oscillation of a rotational pendulum system. Appl. Math. Comput. 217, 6115–6124 (2011)
Liew, K.M., Lim, C.W.: A global continuum Ritz formulation for flexural vibration of pretwisted trapezoidal plates with one edge built in. Comput. Methods Appl. Mech. Eng. 114, 233–247 (1994)
Lim, C.W., Liew, K.M.: Vibration of pretwisted cantilever trapezoidal symmetric laminates. Acta Mech. 111, 193–208 (1995)
Lim, C.W.: A spiral model for bending of nonlinearly pretwisted helicoidal structures with lateral loading. Int. J. Solids Struct. 40, 4257–4279 (2003)
Librescu, L., Song, O.: Thin-Walled Composite Beams. Springer, Berlin (2006)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-Interscience, New York (1979)
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Yao, M.H., Chen, Y.P. & Zhang, W. Nonlinear vibrations of blade with varying rotating speed. Nonlinear Dyn 68, 487–504 (2012). https://doi.org/10.1007/s11071-011-0231-z
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DOI: https://doi.org/10.1007/s11071-011-0231-z