Technical Brief

Partial Stochastic Linearization of a Spherical Pendulum With Coriolis Damping Produced by Radial Spring and Damper

[+] Author and Article Information
L. D. Viet

Institute of Mechanics,
Vietnam Academy of Science and Technology,
264 Doi Can,
Hanoi 10000, Vietnam
e-mails: laviet80@yahoo.com; ldviet@imech.ac.vn

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 12, 2014; final manuscript received May 16, 2015; published online June 15, 2015. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 137(5), 054504 (Oct 01, 2015) (6 pages) Paper No: VIB-14-1388; doi: 10.1115/1.4030663 History: Received October 12, 2014; Revised May 16, 2015; Online June 15, 2015

This study considers the stochastic analysis of a spherical pendulum, whose bidirectional vibration is reduced by spring and damper installed in the radial direction between the point mass and the cable. Under sway motion, the centrifugal force results in the radial motion, which in its turn produces the Coriolis force to reduce sway motion. In stochastic analysis and design, the problem is that the Monte Carlo simulation is time-consuming, while the full stochastic linearization totally fails to describe the effectiveness of the spring and damper. We propose the partial linearization applied to the Coriolis damping to overcome the disadvantages of two mentioned methods. Moreover, the proposed technique can give the analytical solution of partial linearized system. A numerical simulation is performed to verify the proposed approach.

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Shen, J., Sanyal, A. K., Chaturvedi, N. A., Bernstein, D., and Mc-Clamroch, N. H., 2004, “Dynamics and Control of a 3D Pendulum,” 43rd IEEE Conference on Decision and Control (CDC), Nassau, Bahamas, Dec. 14–17, pp. 323–328. [CrossRef]
Cho, S., Shen, J., and McClamroch, N. H., 2003, “Mathematical Models for the Triaxial Attitude Control Test Bed,” Math. Comput. Model. Dyn. Syst., 9(2), pp. 165–192. [CrossRef]
Cho, S., and McClamroch, N. H., 2002, “Feedback Control of Triaxial Attitude Control Testbed Actuated by Two Proof Mass Devices,” 41st IEEE Conference on Decision and Control (CDC), Las Vegas, NV, Dec. 10–13, pp. 498–503. [CrossRef]
Viet, L. D., and Park, Y. J., 2011, “Vibration Control of the Axisymmetric Spherical Pendulum by Dynamic Vibration Absorber Moving in Radial Direction,” J. Mech. Sci. Technol., 25(7), pp. 1703–1709. [CrossRef]
Anh, N. D., Matsuhisa, H., Viet, L. D., and Yasuda, M., 2007, “Vibration Control of an Inverted Pendulum Type Structure by Passive Mass-Spring-Pendulum Dynamic Vibration Absorber,” J. Sound Vib., 307(1–2), pp. 187–201. [CrossRef]
Viet, L. D., 2012, “Semi-Active On-Off Damping Control of a Dynamic Vibration Absorber Using Coriolis Force,” J. Sound Vib., 331(15), pp. 3429–3436. [CrossRef]
Proppe, C., Pradlwarter, H., and Schueller, G., 2003, “Equivalent Linearization and Monte Carlo Simulation in Stochastic Dynamics,” Probab. Eng. Mech., 18(1), pp. 1–15. [CrossRef]
Socha, L., 2008, Linearization Methods for Stochastic Dynamic Systems (Lecture Notes in Physics, Vol. 730), Springer, Berlin.
Lutes, L. D., and Sarkani, S., 2004, Random Vibration: Analysis of Structural and Mechanical Systems, Elsevier, Amsterdam.
Ricciardi, G., 2007, “A Non-Gaussian Stochastic Linearization Method,” Probab. Eng. Mech., 22(1), pp. 1–11. [CrossRef]
Pradlwarter, H. J., 2001, “Non-Linear Stochastic Response Distributions by Local Statistical Linearization,” Int. J. Non-Linear Mech., 36(7), pp. 1135–1151. [CrossRef]
Guo, S. S., 2014, “Probabilistic Solutions of Stochastic Oscillators Excited by Correlated External and Parametric White Noises,” ASME J. Vib. Acoust., 136(3), p. 031003. [CrossRef]
Elbeyli, O., and Sun, J. Q., 2002, “A Stochastic Averaging Approach for Feedback Control Design of Nonlinear Systems Under Random Excitations,” ASME J. Vib. Acoust., 124(4), pp. 561–565. [CrossRef]
Zhu, W. Q., Huang, Z. L., and Suzuki, Y., 2001, “Equivalent Non-Linear System Method for Stochastically Excited and Dissipated Partially Integrable Hamiltonian Systems,” Int. J. Non-Linear Mech., 36(5), pp. 773–786. [CrossRef]
Er, G. K., Guo, S. S., and Iu, V. P., 2012, “Probabilistic Solutions of the Stochastic Oscillators With Even Nonlinearity in Displacement,” ASME J. Vib. Acoust., 134(5), p. 054501. [CrossRef]
Chang, R. J., and Lin, S. J., 2004, “Statistical Linearization Model for the Response Prediction of Nonlinear Stochastic Systems Through Information Closure Method,” ASME J. Vib. Acoust., 126(3), pp. 438–448. [CrossRef]
Spanos, P., Failla, G., and Di Paola, M., 2003, “Spectral Approach to Equivalent Statistical Quadratization and Cubicization Methods for Nonlinear Oscillators,” J. Eng. Mech., 129(1), pp. 31–42. [CrossRef]
Sun, J. Q., 2006, Stochastic Dynamics and Control, Vol. 4, Elsevier, Amsterdam.


Grahic Jump Location
Fig. 2

Geometrical description of the axisymmetric spherical pendulum attached with the spring–damper and two swing angles

Grahic Jump Location
Fig. 1

Spherical pendulum attached with radial spring–damper

Grahic Jump Location
Fig. 3

Performance index versus frequency ratio, in the case ϕ0 = θ0 = 10 deg

Grahic Jump Location
Fig. 4

Performance index versus frequency ratio, in the case ϕ0 = 2θ0 = 20 deg

Grahic Jump Location
Fig. 5

Damping error versus effective damping ratio in the case ϕ0 = 2θ0 = 20 deg, α = 2. Markers denote the effective damping obtained analytically.



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