Nonlinear and Chaotic Motions for a Shock Absorbing Structure Supported by Nonlinear Springs with Hysteresis Using Fast FEA
This paper describes dynamic analysis using proposed
fast finite element method for a shock absorbing structure including a
sponge. The structure is supported by nonlinear concentrated springs.
The restoring force of the spring has cubic nonlinearity and linear
hysteresis damping. To calculate damping properties for the structures
including elastic body and porous body, displacement vectors as
common unknown variable are solved under coupled condition. Under
small amplitude, we apply asymptotic method to complex eigenvalue
problem of this system to obtain modal parameters. And then
expressions of modal loss factor are derived approximately. This
approach was proposed by one of the authors previously. We call this
method as Modal Strain and Kinetic Energy Method (MSKE method).
Further, using the modal loss factors, the discretized equations in
physical coordinate are transformed into the nonlinear ordinary
coupled equations using normal coordinate corresponding to linear
natural modes. This transformation yields computation efficiency. As
a numerical example of a shock absorbing structures, we adopt double
skins with a sponge. The double skins are supported by nonlinear
concentrated springs. We clarify influences of amplitude of the input
force on nonlinear and chaotic responses.
<p>[1] T. Yamaguchi,Y. Kurosawa and S. Matsumura, “FEA for damping of
structures having elastic bodies, viscoelastic bodies, porous media and
gas”, Mechanical Systems and Signal Processing, Vol.21, pp.535-552,
2007.
[2] T. Yamaguchi, Y. Kurosawa and H. Enomoto, “Damped vibration analysis
using finite elementmethod with approximated modal damping for
automotive double walls with a porous material”, Journal of Sound and
Vibration, Vol.325, pp.436-450, 2009.
[3] T. Yamaguchi, Y. Fujii, K. Nagai andS. Maruyama, “FEA forvibrated
structures with non-linear concentrated spring havinghysteresis,”
Mechanical Systems and Signal Processing, vol.20, pp.1905–1922, Nov.
2006.
[4] T.Yamaguchi, Y.Fujii, T.Fukushima, T.Kanai, K. NagaiandS. Maruyama,
“Dynamic responses for viscoelastic shock absorbers to protect a finger
under impact force,”Applied Mechanics and Materials, vol.36,
pp.287-292,Oct.2010.
[5] T. Kondo,T. SasakiandT. Ayabe, “Forced vibrationanalysis of a
straight-line beam structure with nonlinear support elements”,
Transactions of the Japan Society of Mechanical Engineers, Vol.67,
No.656C, pp.914-921, 2001.
[6] E. Pesheck,N.Boivin,C. Pierre and S. W. Shaw, “Non-linear modal
analysis of structural systems using multi-mode invariant manifolds”,
Nonlinear Dynamics, Vol.25, pp.183-205, 2001.
[7] A. Craggs, “A finite element model for rigid porous absorbing materials”,
Journal of Sound andVibration, Vol. 61-1, pp.101-111, 1978.
[8] Y. Kagawa,T.Yamabuchi and A. Mori, “Finite element simulation of an
axisymmetric acoustictransmissionsystem with a sound absorbing wall”,
Journal of Sound and Vibration, Vol.53-3, pp.357-374, 1977.
[9] H. Utsuno,T. W. Wu, A. F. Seybert and T. Tanaka, “Prediction of sound
fields in cavities with sound absorbing materials”, AIAA
Journal,Vol.28-11, pp.1870-1875, 1990.
[10] Y. J. Kang and S. Bolton, “Finite element modeling of isotropic elastic
porous materials coupled with acoustical finite elements”, Journal of the
Acoustical Society of America, Vol.98- 1, pp.635-643, 1995.
[11] N. Attala, R. Panneton and P. A. Debergue, “A mixed
pressure-displacement formulation for poroelastic materials”, Journal of
the Acoustical Society of America, Vol.104-3, pp.1444-1452, 1998.
[12] M. A. Biot, “Theory of propagation of elastic waves in a fluid-saturated
porous solid”, Journal of the Acoustical Society of America, Vol.28-2,
pp.168-178, 1955.
[13] C. D. Johnson and D. A. Kienholz, “Finite element prediction of damping
structures with constrained viscoelastic layers”, AIAA Journal, Vol.20-9,
pp.1284-1290, 1982.
[14] B. A. Ma and J. F. He,“A finite element analysis of viscoelastically
damped sandwich plates”, Journal of Sound and Vibration,Vol.152-1,
pp.107-123, 1992.
[15] O. C. Zienkiewicz and Y. K. Cheung,The finite element method in
structural and continuum mechanics,MacGraw-Hill, 1967.
[16] E. L. Wilson,R. L. Taylor,W. P. Doherty and J. Ghaboussi, Incompatible
displacement methods, in numerical and computer methods in structural
mechanics,Academic Press, 1973.
[17] T. YamaguchiandK. Nagai, “Chaotic vibration of a cylindrical shell-panel
with an in-plane elastic support at boundary”,Nonlinear Dynamics,
Vol.13, pp.259-277, 1997.
[18] A. Wolf,J. B. Swift, H. L. Swinny and J. A. Vastano, “Determining
Lyapunov exponents from a timeseries”, Physica16D, pp.285–317, 1985.</p>
<p>[1] T. Yamaguchi,Y. Kurosawa and S. Matsumura, “FEA for damping of
structures having elastic bodies, viscoelastic bodies, porous media and
gas”, Mechanical Systems and Signal Processing, Vol.21, pp.535-552,
2007.
[2] T. Yamaguchi, Y. Kurosawa and H. Enomoto, “Damped vibration analysis
using finite elementmethod with approximated modal damping for
automotive double walls with a porous material”, Journal of Sound and
Vibration, Vol.325, pp.436-450, 2009.
[3] T. Yamaguchi, Y. Fujii, K. Nagai andS. Maruyama, “FEA forvibrated
structures with non-linear concentrated spring havinghysteresis,”
Mechanical Systems and Signal Processing, vol.20, pp.1905–1922, Nov.
2006.
[4] T.Yamaguchi, Y.Fujii, T.Fukushima, T.Kanai, K. NagaiandS. Maruyama,
“Dynamic responses for viscoelastic shock absorbers to protect a finger
under impact force,”Applied Mechanics and Materials, vol.36,
pp.287-292,Oct.2010.
[5] T. Kondo,T. SasakiandT. Ayabe, “Forced vibrationanalysis of a
straight-line beam structure with nonlinear support elements”,
Transactions of the Japan Society of Mechanical Engineers, Vol.67,
No.656C, pp.914-921, 2001.
[6] E. Pesheck,N.Boivin,C. Pierre and S. W. Shaw, “Non-linear modal
analysis of structural systems using multi-mode invariant manifolds”,
Nonlinear Dynamics, Vol.25, pp.183-205, 2001.
[7] A. Craggs, “A finite element model for rigid porous absorbing materials”,
Journal of Sound andVibration, Vol. 61-1, pp.101-111, 1978.
[8] Y. Kagawa,T.Yamabuchi and A. Mori, “Finite element simulation of an
axisymmetric acoustictransmissionsystem with a sound absorbing wall”,
Journal of Sound and Vibration, Vol.53-3, pp.357-374, 1977.
[9] H. Utsuno,T. W. Wu, A. F. Seybert and T. Tanaka, “Prediction of sound
fields in cavities with sound absorbing materials”, AIAA
Journal,Vol.28-11, pp.1870-1875, 1990.
[10] Y. J. Kang and S. Bolton, “Finite element modeling of isotropic elastic
porous materials coupled with acoustical finite elements”, Journal of the
Acoustical Society of America, Vol.98- 1, pp.635-643, 1995.
[11] N. Attala, R. Panneton and P. A. Debergue, “A mixed
pressure-displacement formulation for poroelastic materials”, Journal of
the Acoustical Society of America, Vol.104-3, pp.1444-1452, 1998.
[12] M. A. Biot, “Theory of propagation of elastic waves in a fluid-saturated
porous solid”, Journal of the Acoustical Society of America, Vol.28-2,
pp.168-178, 1955.
[13] C. D. Johnson and D. A. Kienholz, “Finite element prediction of damping
structures with constrained viscoelastic layers”, AIAA Journal, Vol.20-9,
pp.1284-1290, 1982.
[14] B. A. Ma and J. F. He,“A finite element analysis of viscoelastically
damped sandwich plates”, Journal of Sound and Vibration,Vol.152-1,
pp.107-123, 1992.
[15] O. C. Zienkiewicz and Y. K. Cheung,The finite element method in
structural and continuum mechanics,MacGraw-Hill, 1967.
[16] E. L. Wilson,R. L. Taylor,W. P. Doherty and J. Ghaboussi, Incompatible
displacement methods, in numerical and computer methods in structural
mechanics,Academic Press, 1973.
[17] T. YamaguchiandK. Nagai, “Chaotic vibration of a cylindrical shell-panel
with an in-plane elastic support at boundary”,Nonlinear Dynamics,
Vol.13, pp.259-277, 1997.
[18] A. Wolf,J. B. Swift, H. L. Swinny and J. A. Vastano, “Determining
Lyapunov exponents from a timeseries”, Physica16D, pp.285–317, 1985.</p>
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:50902", author = "T. Yamaguchi and Y. Kurosawa and S. Maruyama and K. Tobita and Y. Hirano and K. Yokouchi and K. Kihara and T. Sunaga", title = "Nonlinear and Chaotic Motions for a Shock Absorbing Structure Supported by Nonlinear Springs with Hysteresis Using Fast FEA", abstract = "This paper describes dynamic analysis using proposed
fast finite element method for a shock absorbing structure including a
sponge. The structure is supported by nonlinear concentrated springs.
The restoring force of the spring has cubic nonlinearity and linear
hysteresis damping. To calculate damping properties for the structures
including elastic body and porous body, displacement vectors as
common unknown variable are solved under coupled condition. Under
small amplitude, we apply asymptotic method to complex eigenvalue
problem of this system to obtain modal parameters. And then
expressions of modal loss factor are derived approximately. This
approach was proposed by one of the authors previously. We call this
method as Modal Strain and Kinetic Energy Method (MSKE method).
Further, using the modal loss factors, the discretized equations in
physical coordinate are transformed into the nonlinear ordinary
coupled equations using normal coordinate corresponding to linear
natural modes. This transformation yields computation efficiency. As
a numerical example of a shock absorbing structures, we adopt double
skins with a sponge. The double skins are supported by nonlinear
concentrated springs. We clarify influences of amplitude of the input
force on nonlinear and chaotic responses.
", keywords = "Dynamic response, Nonlinear and chaotic motions,
Finite Element analysis, Numerical analysis.", volume = "7", number = "7", pages = "1355-12", }
