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Non-stationary analytic solution of the stochastic Bagley-Torvik equation
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Fan KONG1, Yijian XU1, Wenjie GUO2, Xu HONG1, Hongyou CAO3
Journal of Vibration Engineering | 2025, 38(7) : 1432 - 1440
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Journal of Vibration Engineering | 2025, 38(7): 1432-1440
Non-stationary analytic solution of the stochastic Bagley-Torvik equation
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Fan KONG1, Yijian XU1, Wenjie GUO2, Xu HONG1, Hongyou CAO3
Affiliations
  • 1.College of Civil Engineering，Hefei University of Technology，Hefei 230009，China
  • 2.School of Transportation Engineering，East China Jiaotong University，Nanchang 330013，China
  • 3.School of Civil Engineering and Architecture，Wuhan University of Technology，Wuhan 430070，China
Published: 2025-07-10 doi: 10.16385/j.cnki.issn.1004-4523.202309028
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The Bagley-Torvik（B-T） equation is a differential equation of motion with fractional （3/2）-order derivative terms that is applied to describe the motion of a rigid plate in Newtonian，viscous fluid. In this paper，we develop non-stationary analytic solutions of the B-T equation whose inhomogeneous term is a stochastic process. The B-T equation is transformed into a half-order state-space equation in matrix form and eigen-analysis is performed to obtain complex eigenvalues and eigenvectors. Subsequently，the generalized coordinate transformation is introduced to decouple the equation into a system of independent 1/2-order differential equations which are solved by Laplace transform to obtain the solution in generalized coordinates； The generalized coordinate solution is converted into a natural coordinate solution to obtain the impulse or step response function. When the inhomogeneous term of the equation is a stochastic process，the Laplace transform can be used to derive the time-varying frequency response function from which the analytical solution of the non-stationary stochastic response can be obtained by relying on the relationship between the excitation and the response power spectral density. The correctness of the method is verified by numerical cases using the Spanos-Solomos fully non-statoionary stochastic excitation as an example.

random vibration  /  fractional derivative  /  Bagley-Torvik equation  /  fully non-stationary  /  Mittag-Leffler function
Fan KONG, Yijian XU, Wenjie GUO, Xu HONG, Hongyou CAO. Non-stationary analytic solution of the stochastic Bagley-Torvik equation[J]. Journal of Vibration Engineering, 2025 , 38 (7) : 1432 -1440 . DOI: 10.16385/j.cnki.issn.1004-4523.202309028
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Year 2025 volume 38 Issue 7
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Article Info
doi: 10.16385/j.cnki.issn.1004-4523.202309028
  • Receive Date：2023-09-11
  • Online Date：2026-02-09
  • Published：2025-07-10
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History
  • Received：2023-09-11
  • Revised：2023-12-07
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Affiliations
    1.College of Civil Engineering，Hefei University of Technology，Hefei 230009，China
    2.School of Transportation Engineering，East China Jiaotong University，Nanchang 330013，China
    3.School of Civil Engineering and Architecture，Wuhan University of Technology，Wuhan 430070，China
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表12种不同金属材料的力学参数
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Family		 属数
Number of
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Number of
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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