Chaotic Oscillations of Diaphragm Supported by Nonlinear Springs with Hysteresis
This paper describes vibration analysis using the finite
element method for a small earphone, especially for the diaphragm
shape with a low-rigidity. The viscoelastic diaphragm is supported by
multiple nonlinear concentrated springs with linear hysteresis
damping. The restoring forces of the nonlinear springs have cubic
nonlinearity. The finite elements for the nonlinear springs with
hysteresis are expressed and are connected to the diaphragm that is
modeled by linear solid finite elements in consideration of a complex
modulus of elasticity. Further, the discretized equations in physical
coordinates are transformed into the nonlinear ordinary coupled
equations using normal coordinates corresponding to the linear natural
modes. We computed the nonlinear stationary and non-stationary
responses due to the internal resonance between modes with large
amplitude in the nonlinear springs and elastic modes in the diaphragm.
The non-stationary motions are confirmed as the chaos due to the
maximum Lyapunov exponents with a positive number. From the time
histories of the deformation distribution in the chaotic vibration, we
identified nonlinear modal couplings.
[1] T. Kondo, T. Sasaki, and T. Ayabe, "Forced vibration analysis of a
straight-line beam structure with nonlinear support elements," Trans. Jpn.
Soc. Mech. Eng., vol. 67, no. 656C, pp. 914-921, 2001.
[2] E. Pesheck, N. Boivin, C. Pierre, and S. W. Shaw, "Non-linear modal
analysis of structural systems using multi-mode invariant manifolds,"
Nonlinear Dyn., no. 25, pp. 183-205, 2001.
[3] T. Yamaguchi, K. Nagai, S. Maruyama, and T. Aburada, "Finite element
analysis for coupled vibrations of an elastic block supported by a
nonlinear spring," Trans. Jpn. Soc. Mech. Eng., vol. 69, no. 688C, pp.
3167-3174, 2003.
[4] T. Yamaguchi, N. Nakahara, K. Nagai, S. Maruyama, and Y. Fujii,
"Frequency response analysis of elastic blocks supported by a nonlinear
spring using finite element method," Trans. Jpn. Soc. Mech. Eng., vol. 70,
no. 696C, pp. 2219-2227, 2004.
[5] T. Yamaguchi, T. Saito, K. Nagai, S. Maruyama, Y. Kurosawa, and S.
Matsumura, "Analysis of damped vibration for a viscoelastic block
supported by a nonlinear concentrated spring using FEM," J. Syst. Des.
Dyn., vol. 4, no. 1, pp. 154-165, 2011.
[6] E. H. Dowell, "Flutter of a buckled plate as an example of chaotic motion
of a deterministic autonomous system," J. Sound Vib., vol. 85-3, pp.
333-344, 1982.
[7] A. H. Nayfeh and R. A. Raouf, "Nonlinear Forced Response of Infinitely
Long Circular Cylindrical Shells," J. Appl. Mech., vol. 54, pp. 571-577,
1987.
[8] X. L. Yang and P. R. Sethna, "Non-linear phenomena in forced vibrations
of a nearly square plate: Antisymmetric case," J. Sound Vib., vol. 155-3,
pp. 413-441, 1992.
[9] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining
Lyapunov exponents from a time series," Physica, vol. 16D, pp. 285-317,
1985.
[10] O. C. Zienkiewicz and Y. K. Cheung, The Finite Element Method in
Structural and Continuum Mechanics. New York: MacGraw-Hill, 1967.
[1] T. Kondo, T. Sasaki, and T. Ayabe, "Forced vibration analysis of a
straight-line beam structure with nonlinear support elements," Trans. Jpn.
Soc. Mech. Eng., vol. 67, no. 656C, pp. 914-921, 2001.
[2] E. Pesheck, N. Boivin, C. Pierre, and S. W. Shaw, "Non-linear modal
analysis of structural systems using multi-mode invariant manifolds,"
Nonlinear Dyn., no. 25, pp. 183-205, 2001.
[3] T. Yamaguchi, K. Nagai, S. Maruyama, and T. Aburada, "Finite element
analysis for coupled vibrations of an elastic block supported by a
nonlinear spring," Trans. Jpn. Soc. Mech. Eng., vol. 69, no. 688C, pp.
3167-3174, 2003.
[4] T. Yamaguchi, N. Nakahara, K. Nagai, S. Maruyama, and Y. Fujii,
"Frequency response analysis of elastic blocks supported by a nonlinear
spring using finite element method," Trans. Jpn. Soc. Mech. Eng., vol. 70,
no. 696C, pp. 2219-2227, 2004.
[5] T. Yamaguchi, T. Saito, K. Nagai, S. Maruyama, Y. Kurosawa, and S.
Matsumura, "Analysis of damped vibration for a viscoelastic block
supported by a nonlinear concentrated spring using FEM," J. Syst. Des.
Dyn., vol. 4, no. 1, pp. 154-165, 2011.
[6] E. H. Dowell, "Flutter of a buckled plate as an example of chaotic motion
of a deterministic autonomous system," J. Sound Vib., vol. 85-3, pp.
333-344, 1982.
[7] A. H. Nayfeh and R. A. Raouf, "Nonlinear Forced Response of Infinitely
Long Circular Cylindrical Shells," J. Appl. Mech., vol. 54, pp. 571-577,
1987.
[8] X. L. Yang and P. R. Sethna, "Non-linear phenomena in forced vibrations
of a nearly square plate: Antisymmetric case," J. Sound Vib., vol. 155-3,
pp. 413-441, 1992.
[9] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining
Lyapunov exponents from a time series," Physica, vol. 16D, pp. 285-317,
1985.
[10] O. C. Zienkiewicz and Y. K. Cheung, The Finite Element Method in
Structural and Continuum Mechanics. New York: MacGraw-Hill, 1967.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:50695", author = "M. Sasajima and T. Yamaguchi and Y. Koike and A. Hara", title = "Chaotic Oscillations of Diaphragm Supported by Nonlinear Springs with Hysteresis", abstract = "This paper describes vibration analysis using the finite
element method for a small earphone, especially for the diaphragm
shape with a low-rigidity. The viscoelastic diaphragm is supported by
multiple nonlinear concentrated springs with linear hysteresis
damping. The restoring forces of the nonlinear springs have cubic
nonlinearity. The finite elements for the nonlinear springs with
hysteresis are expressed and are connected to the diaphragm that is
modeled by linear solid finite elements in consideration of a complex
modulus of elasticity. Further, the discretized equations in physical
coordinates are transformed into the nonlinear ordinary coupled
equations using normal coordinates corresponding to the linear natural
modes. We computed the nonlinear stationary and non-stationary
responses due to the internal resonance between modes with large
amplitude in the nonlinear springs and elastic modes in the diaphragm.
The non-stationary motions are confirmed as the chaos due to the
maximum Lyapunov exponents with a positive number. From the time
histories of the deformation distribution in the chaotic vibration, we
identified nonlinear modal couplings.", keywords = "Nonlinear Vibration, Finite Element Method,Chaos ,Small Earphone.", volume = "6", number = "11", pages = "2313-7", }