A Note on Stability of Longitudinal Vibrations of an Inhomogeneous Beam

ABSTRACT

In this paper, we have considered an inhomogeneous beam with a damping distributed along the length of the beam. The beam is clamped at both ends and is assumed to vibrate longitudinally. We have estimated the total energy of the system at any time t. By constructing suitable Lyapunov functional, it is established directly that the energy of this system decays exponentially.

In this paper, we have considered an inhomogeneous beam with a damping distributed along the length of the beam. The beam is clamped at both ends and is assumed to vibrate longitudinally. We have estimated the total energy of the system at any time t. By constructing suitable Lyapunov functional, it is established directly that the energy of this system decays exponentially.

Cite this paper

P. Nandi, G. Gorain and S. Kar, "A Note on Stability of Longitudinal Vibrations of an Inhomogeneous Beam,"*Applied Mathematics*, Vol. 3 No. 1, 2012, pp. 19-23. doi: 10.4236/am.2012.31003.

P. Nandi, G. Gorain and S. Kar, "A Note on Stability of Longitudinal Vibrations of an Inhomogeneous Beam,"

References

[1] G. Chen, “Energy Decay Estimates and Exact Boundary-Value Controllability for the Wave Equation in a Bounded Domain,” Journal de Mathématiques Pures et Appliquées, Vol. 58, 1979, pp. 249-273.

[2] G. Chen, “A Note on the Boundary Stabilization of the Wave Equation,” SIAM Journal of Control and Optimization, Vol. 19, No. 1, 1981, pp. 106-113. doi:10.1137/0319008

[3] J. Lagnese, “Note on Boundary Stabilization of Wave Equations,” SIAM Journal of Control and Optimization, Vol. 26, No. 5, 1988, pp. 1250-1256. doi:10.1137/0326068

[4] J. Lagnese, “Decay of Solutions of Wave Equations in a Bounded Region with Boundary Dissipation,” Journal of Differential Equations, Vol. 50, No. 2, 1983, pp. 163-182. doi:10.1016/0022-0396(83)90073-6

[5] J. L. Lions, “Exact Controllability, Stabilization and Perturbations for Distributed Systems,” SIAM Review, Vol. 30, No. 1, 1988, pp. 1-68. doi:10.1137/1030001

[6] V. Komornik, “Rapid Boundary Stabilization of Wave Equations,” SIAM Journal of Control and Optimization, Vol. 29, No. 1, 1991, pp. 197-208. doi:10.1137/0329011

[7] P. K. Nandi, G. C. Gorain and S. Kar, “Uniform Exponential Stabilization for FLexural Vibrations of a Solar Panel,” Applied Mathematics, Vol. 2, No. 6, 2011, pp. 661-665. doi:10.4236/am.2011.26087

[8] Y. J. Ye, “On the Exponential Decay of Solutions for Some Kirchoff-Type Modelling Equations with Strong Dissipation,” Applied Mathematics, Vol. 1, No. 6, 2010, pp. 529-533. doi:10.4236/am.2010.16070

[9] V. Komornik and E. Zuazua, “A Direct Method for Boundary Stabilization of the Wave Equation,” Journal de Mathématiques Pures et Appliquées, Vol. 69, No. 1, 1990, pp. 33-54.

[10] K. Ammari and M. Tuesnak, “Stabilization of Bernoulli-Euler Beams by Means of a Point FEedback Force,” SIAM Journal of Control and Optimization, Vol. 39, No. 4, 2000, pp. 1160-1181. doi:10.1137/S0363012998349315

[11] K. Liu and Z. Liu, “Exponential Decay of Energy of the Euler-Bernoulli Beam with Locally Distributed Kelvin-Voigt Damping,” SIAM Journal of Control and Optimization, Vol. 36, No. 4, 1998, pp. 1086-1098. doi:10.1137/S0363012996310703

[12] K. Nagaya, “Method of Control of Flexible Beams Subject to Forced Vibrations by Use of Inertia Force Cancellations,” Journal of Sound and Vibration, Vol. 184, No. 2, 1995, pp. 184-194. doi:10.1006/jsvi.1995.0311

[13] R. Rebarbery, “Exponential Stability of Coupled Beams with Dissipative Joints: A Frequency Domain Approach,” SIAM Journal of Control and Optimization, Vol. 33, No. 1, 1995, pp. 1-28. doi:10.1137/S0363012992240321

[14] D. S. Mitrinovi?, J. E. Pe?ari? and A. M. Fink, “Inequalities Involving Functions and Their Integrals and Derivatives,” Kluwer, Dordrecht, 1991.

[15] G. C. Gorain, “Exponential Energy Decay Estimate for the Solutions of n-Dimensional Kirchhoff Type Wave Equation,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 235-242. doi:10.1016/j.amc.2005.11.003

[16] G. C. Gorain and S. K. Bose, “Exact Controllability and Boundary Stabilization of Torsional Vibrations of an Internally Damped Flexible Space Structures,” Journal of Optimization Theory and Applications, Vol. 99, No. 2, 1998, pp. 423-442. doi:10.1023/A:1021778428222

[1] G. Chen, “Energy Decay Estimates and Exact Boundary-Value Controllability for the Wave Equation in a Bounded Domain,” Journal de Mathématiques Pures et Appliquées, Vol. 58, 1979, pp. 249-273.

[2] G. Chen, “A Note on the Boundary Stabilization of the Wave Equation,” SIAM Journal of Control and Optimization, Vol. 19, No. 1, 1981, pp. 106-113. doi:10.1137/0319008

[3] J. Lagnese, “Note on Boundary Stabilization of Wave Equations,” SIAM Journal of Control and Optimization, Vol. 26, No. 5, 1988, pp. 1250-1256. doi:10.1137/0326068

[4] J. Lagnese, “Decay of Solutions of Wave Equations in a Bounded Region with Boundary Dissipation,” Journal of Differential Equations, Vol. 50, No. 2, 1983, pp. 163-182. doi:10.1016/0022-0396(83)90073-6

[5] J. L. Lions, “Exact Controllability, Stabilization and Perturbations for Distributed Systems,” SIAM Review, Vol. 30, No. 1, 1988, pp. 1-68. doi:10.1137/1030001

[6] V. Komornik, “Rapid Boundary Stabilization of Wave Equations,” SIAM Journal of Control and Optimization, Vol. 29, No. 1, 1991, pp. 197-208. doi:10.1137/0329011

[7] P. K. Nandi, G. C. Gorain and S. Kar, “Uniform Exponential Stabilization for FLexural Vibrations of a Solar Panel,” Applied Mathematics, Vol. 2, No. 6, 2011, pp. 661-665. doi:10.4236/am.2011.26087

[8] Y. J. Ye, “On the Exponential Decay of Solutions for Some Kirchoff-Type Modelling Equations with Strong Dissipation,” Applied Mathematics, Vol. 1, No. 6, 2010, pp. 529-533. doi:10.4236/am.2010.16070

[9] V. Komornik and E. Zuazua, “A Direct Method for Boundary Stabilization of the Wave Equation,” Journal de Mathématiques Pures et Appliquées, Vol. 69, No. 1, 1990, pp. 33-54.

[10] K. Ammari and M. Tuesnak, “Stabilization of Bernoulli-Euler Beams by Means of a Point FEedback Force,” SIAM Journal of Control and Optimization, Vol. 39, No. 4, 2000, pp. 1160-1181. doi:10.1137/S0363012998349315

[11] K. Liu and Z. Liu, “Exponential Decay of Energy of the Euler-Bernoulli Beam with Locally Distributed Kelvin-Voigt Damping,” SIAM Journal of Control and Optimization, Vol. 36, No. 4, 1998, pp. 1086-1098. doi:10.1137/S0363012996310703

[12] K. Nagaya, “Method of Control of Flexible Beams Subject to Forced Vibrations by Use of Inertia Force Cancellations,” Journal of Sound and Vibration, Vol. 184, No. 2, 1995, pp. 184-194. doi:10.1006/jsvi.1995.0311

[13] R. Rebarbery, “Exponential Stability of Coupled Beams with Dissipative Joints: A Frequency Domain Approach,” SIAM Journal of Control and Optimization, Vol. 33, No. 1, 1995, pp. 1-28. doi:10.1137/S0363012992240321

[14] D. S. Mitrinovi?, J. E. Pe?ari? and A. M. Fink, “Inequalities Involving Functions and Their Integrals and Derivatives,” Kluwer, Dordrecht, 1991.

[15] G. C. Gorain, “Exponential Energy Decay Estimate for the Solutions of n-Dimensional Kirchhoff Type Wave Equation,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 235-242. doi:10.1016/j.amc.2005.11.003

[16] G. C. Gorain and S. K. Bose, “Exact Controllability and Boundary Stabilization of Torsional Vibrations of an Internally Damped Flexible Space Structures,” Journal of Optimization Theory and Applications, Vol. 99, No. 2, 1998, pp. 423-442. doi:10.1023/A:1021778428222