Abstract
The generalized Bagley–Torvik system is a linear oscillator whose damping force is described by fractional-order derivative with order between 0 and 2. This paper shows that as a sequential fractional-order differential equation with constant coefficients whose general solution depends on more than two free (independent) constants, the generalized Bagley–Torvik equation actually admits a general solution that involves two free constants only and can be determined fully by the initial displacement and initial velocity.
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References
Podlubny I (1999) Fractional differential equations. Academic, San Diego, CA
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam
Das S (2008) Functional fractional calculus for system identification and controls. Springer, Berlin
Bagley RL, Torvik PJ (1984) On the appearance of the fractional derivative in the behavior of real materials. ASME J Appl Mech 51:294–298
Bagley RL, Torvik PJ (1983) Fractional calculus – a different approach to the analysis of viscoe1astically damped structures. AIAA J 21(5):741–748
Wang ZH, Hu HY (2010) Stability of a linear oscillator with damping force of fractional-order derivative. Sci China: Phys, Mech Astron 53(2):1–8, doi:10.1007/s11433-009-0291-y
Suarez LE, Shokooh A (1997) An eigenvector expansion method for the solution of motion containing fractional derivatives. ASME J Appl Mech 64:629–635
Lorenzo CF, Hartley TT (2008) Initialization of fractional-order operators and fractional differential equations. ASME J Comput Nonlinear Dyn 3:021101
Bonilla B, Rivero M, Trujillo JJ (2007) Linear differential equations of fractional orders. In: Sabatier J, Agrawal OP, Tenreiro Machado JA (eds) Advances in fractional calculus. Springer, Dordrecht, pp 77–91
Acknowledgement
This work was supported by NSF of China under Grant 10825207 and in part by FANEDD of China under Grant 200430.
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Wang, Z.H., Wang, X. (2010). General Solution of a Vibration System with Damping Force of Fractional-Order Derivative. In: Luo, A. (eds) Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5754-2_1
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DOI: https://doi.org/10.1007/978-1-4419-5754-2_1
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