Free Vibration Analysis of Functionally Graded Beams with General Elastically End Constraints by DTM

Affiliation(s)

Department of Mechanical Engineering, Mahanakorn University of Technology, Bangkok, Thailand.

Department of Mechanical Engineering, Chulalongkorn University, Bangkok, Thailand.

Department of Mechanical Engineering, Mahanakorn University of Technology, Bangkok, Thailand.

Department of Mechanical Engineering, Chulalongkorn University, Bangkok, Thailand.

ABSTRACT

The differential transformation method (DTM) is applied to investigate free vibration of functionally graded beams supported by arbitrary boundary conditions, including various types of elastically end constraints. The material properties of functionally graded beams are assumed to obey the power law distribution. The main advantages of this method are known for its excellence in high accuracy with small computational expensiveness. The DTM also provides all natural frequencies and mode shapes without any frequency missing. Fundamental frequencies as well as their higher frequencies and mode shapes are presented. The significant aspects such as boundary conditions, values of translational and rotational spring constants and the material volume fraction index on the natural frequencies and mode shapes are discussed. For elastically end constraints, some available results of special cases for isotropic beams are used to validate the present results. The new frequency results and mode shapes of functionally graded beams resting on elastically end constraints are presented.

The differential transformation method (DTM) is applied to investigate free vibration of functionally graded beams supported by arbitrary boundary conditions, including various types of elastically end constraints. The material properties of functionally graded beams are assumed to obey the power law distribution. The main advantages of this method are known for its excellence in high accuracy with small computational expensiveness. The DTM also provides all natural frequencies and mode shapes without any frequency missing. Fundamental frequencies as well as their higher frequencies and mode shapes are presented. The significant aspects such as boundary conditions, values of translational and rotational spring constants and the material volume fraction index on the natural frequencies and mode shapes are discussed. For elastically end constraints, some available results of special cases for isotropic beams are used to validate the present results. The new frequency results and mode shapes of functionally graded beams resting on elastically end constraints are presented.

Cite this paper

N. Wattanasakulpong and V. Ungbhakorn, "Free Vibration Analysis of Functionally Graded Beams with General Elastically End Constraints by DTM,"*World Journal of Mechanics*, Vol. 2 No. 6, 2012, pp. 297-310. doi: 10.4236/wjm.2012.26036.

N. Wattanasakulpong and V. Ungbhakorn, "Free Vibration Analysis of Functionally Graded Beams with General Elastically End Constraints by DTM,"

References

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[2] Z. Zhong and T. Yu, “Analytical Solution of a Cantilever Functionally Graded Beam,” Composites Science and Technology, Vol. 67, No. 3-4, 2007, pp. 481-488. doi:10.1016/j.compscitech.2006.08.023

[3] S. Kapuria, M. Bhattacharyya and A. N. Kumar, “Bending and Free Vibration Response of Layered Functionally Graded Beams: A Theoretical Model and Its Experimental Validation,” Composite Structures, Vol. 82, No. 3, 2008, pp. 390-402. doi:10.1016/j.compstruct.2007.01.019

[4] S. A. Sina, H. M. Navazi and H. Haddadpour, “An Analytical Method for Free Vibration Analysis of Functionally Graded Beams,” Materials & Design, Vol. 30, No. 3, 2009, pp. 741-747. doi:10.1016/j.matdes.2008.05.015

[5] M. Simsek, “Fundamental Frequency Analysis of Functionally Graded Beams by Using Different Higher-Order Beam Theories,” Nuclear Engineering and Design, Vol. 240, No. 4, 2010, pp. 697-705. doi:10.1016/j.nucengdes.2009.12.013

[6] M. Aydogdu and V. Taskin, “Free Vibration Analysis of Functionally Graded Beams with Simply Supported Edges,” Materials & Design, Vol. 28, No. 5, 2007, pp. 16511656. doi:10.1016/j.matdes.2006.02.007

[7] N. Wattanasakulpong, B. G. Prusty, D. W. Kelly and M. Hoffman, “A Theoretical Investigation on the Free Vibration of Functionally Graded Beams,” Proceedings of the 10th International Conference on Computational Structures Technology, Valencia, 14-17 September 2010, p. 285.

[8] E. Alshorbagy Amal, M. A. Eltaher and F. F. Mahmoud, “Free Vibration Characteristics of a Functionally Graded Beam by Finite Element Method,” Applied Mathematical Modelling, Vol. 35, No. 1, 2011, pp. 412-425. doi:10.1016/j.apm.2010.07.006

[9] J. Yang and Y. Chen, “Free Vibration and Buckling Analysis of Functionally Graded Beams with Edge Cracks,” Composite Structures, Vol. 83, No. 1, 2008, pp. 4860. doi:10.1016/j.compstruct.2007.03.006

[10] S. Kitipornchai, L. L. Ke, J. Yang and Y. Xiang, “Nonlinear Vibration of Edge Cracked Functionally Graded Timoshenko Beams,” Journal of Sound and Vibration, Vol. 324, No. 3-5, 2009, pp. 962-982. doi:10.1016/j.jsv.2009.02.023

[11] S. Sahraee and A. R. Saidi, “Free Vibration and Buckling Analysis of Functionally Graded Deep Beam-Columns on Two-Parameter Elastic Foundations Using the Differential Quadrature Method,” Journal of Mechanical Engineering Science, Vol. 223, No. 6, 2009, pp. 1273-1284. doi:10.1243/09544062JMES1349

[12] S. C. Pradhan and T. Murmu, “Thermo-Mechanical Vibration of FGM Sandwich Beam under Variable Elastic Foundations Using Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 321, No. 1-2, 2009, pp. 342-362. doi:10.1016/j.jsv.2008.09.018

[13] J. C. Hsu, H. Y. Lai and C. K. Chen, “Free Vibration of Non-Uniform Euler-Bernoulli Beams with General Elastically End Constraints Using Adomain Modified Decomposition Method,” Journal of Sound and Vibration, Vol. 318, No. 4-5, 2008, pp. 965-981.

[14] H. Y. Lai, J. C. Hsu and C. K. Chen, “An Innovative Eigenvalue Problem Solver for Free Vibration of EulerBernoulli Beam by Using the Adomain Decomposition Method,” Computers & Mathematics with Applications, Vol. 56, No. 12, 2008, pp. 3204-3220. doi:10.1016/j.camwa.2008.07.029

[15] Y. Liu and C. S. Gurram, “The Use of He’s Variational Iteration Method for Obtaining the Free Vibration of an Euler-Bernoulli Beam,” Mathematical and Computer Modelling, Vol. 50, No. 11-12, 2009, pp. 1545-1552. doi:10.1016/j.mcm.2009.09.005

[16] Q. Mao and S. Pietrzko, “Free Vibration Analysis of Stepped Beams by Using Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3429-3441. doi:10.1016/j.amc.2010.09.010

[17] J. K. Zhou, “Differential Transformation and Its Application for Electrical Circuits,” Huazhong University Press, Wuhan, 1986.

[18] M. Malik and H. H. Dang, “Vibration Analysis of Continuous Systems by Differential Transformation,” Applied Mathematics and Computation, Vol. 96, No. 1, 1998, pp. 17-26. doi:10.1016/S0096-3003(97)10076-5

[19] M. O. Kaya and O. O. Ozgumus, “Flexural-Torsional-Coupled Vibration Analysis of Axially Loaded Closed-Section Composite Timoshenko Beam by Using DTM,” Journal of Sound and Vibration, Vol. 306, No. 3-5, 2007, pp. 495-506.

[20] O. O. Ozgumus and M. O. Kaya, “Flapwise Bending Vibration Analysis of Double Tapered Rotating Euler-Burnoulli Beam by Using the Differential Transform Method,” Meccanica, Vol. 41, No. 6, 2006, pp. 661-670. doi:10.1007/s11012-006-9012-z

[21] O. O. Ozgumus and M. O. Kaya, “Vibration Analysis of a Rotating Tapered Timoshenko Beam Using DTM,” Meccanica, Vol. 45, No. 1, 2010, pp. 33-42. doi:10.1007/s11012-009-9221-3

[22] S. C. Pradhan and G. K. Reddy, “Buckling Analysis of Single Walled Carbon Nanotube on Winkler Foundation Using Nonlocal Elasticity Theory and DTM,” Computational Materials Science, Vol. 50, No. 3, 2011, pp. 10521056. doi:10.1016/j.commatsci.2010.11.001

[23] F. Delale and F. Erdogan, “The Crack Problem for a NonHomogeneous Plane,” Journal of Applied Mechanics, Vol. 50, No. 3, 1983, pp. 609-614. doi:10.1115/1.3167098

[1] B. V. Sankar, “An Elasticity Solution for Functionally Graded Beams,” Composites Science and Technology, Vol. 61, No. 5, 2001, pp. 689-696. doi:10.1016/S0266-3538(01)00007-0

[2] Z. Zhong and T. Yu, “Analytical Solution of a Cantilever Functionally Graded Beam,” Composites Science and Technology, Vol. 67, No. 3-4, 2007, pp. 481-488. doi:10.1016/j.compscitech.2006.08.023

[3] S. Kapuria, M. Bhattacharyya and A. N. Kumar, “Bending and Free Vibration Response of Layered Functionally Graded Beams: A Theoretical Model and Its Experimental Validation,” Composite Structures, Vol. 82, No. 3, 2008, pp. 390-402. doi:10.1016/j.compstruct.2007.01.019

[4] S. A. Sina, H. M. Navazi and H. Haddadpour, “An Analytical Method for Free Vibration Analysis of Functionally Graded Beams,” Materials & Design, Vol. 30, No. 3, 2009, pp. 741-747. doi:10.1016/j.matdes.2008.05.015

[5] M. Simsek, “Fundamental Frequency Analysis of Functionally Graded Beams by Using Different Higher-Order Beam Theories,” Nuclear Engineering and Design, Vol. 240, No. 4, 2010, pp. 697-705. doi:10.1016/j.nucengdes.2009.12.013

[6] M. Aydogdu and V. Taskin, “Free Vibration Analysis of Functionally Graded Beams with Simply Supported Edges,” Materials & Design, Vol. 28, No. 5, 2007, pp. 16511656. doi:10.1016/j.matdes.2006.02.007

[7] N. Wattanasakulpong, B. G. Prusty, D. W. Kelly and M. Hoffman, “A Theoretical Investigation on the Free Vibration of Functionally Graded Beams,” Proceedings of the 10th International Conference on Computational Structures Technology, Valencia, 14-17 September 2010, p. 285.

[8] E. Alshorbagy Amal, M. A. Eltaher and F. F. Mahmoud, “Free Vibration Characteristics of a Functionally Graded Beam by Finite Element Method,” Applied Mathematical Modelling, Vol. 35, No. 1, 2011, pp. 412-425. doi:10.1016/j.apm.2010.07.006

[9] J. Yang and Y. Chen, “Free Vibration and Buckling Analysis of Functionally Graded Beams with Edge Cracks,” Composite Structures, Vol. 83, No. 1, 2008, pp. 4860. doi:10.1016/j.compstruct.2007.03.006

[10] S. Kitipornchai, L. L. Ke, J. Yang and Y. Xiang, “Nonlinear Vibration of Edge Cracked Functionally Graded Timoshenko Beams,” Journal of Sound and Vibration, Vol. 324, No. 3-5, 2009, pp. 962-982. doi:10.1016/j.jsv.2009.02.023

[11] S. Sahraee and A. R. Saidi, “Free Vibration and Buckling Analysis of Functionally Graded Deep Beam-Columns on Two-Parameter Elastic Foundations Using the Differential Quadrature Method,” Journal of Mechanical Engineering Science, Vol. 223, No. 6, 2009, pp. 1273-1284. doi:10.1243/09544062JMES1349

[12] S. C. Pradhan and T. Murmu, “Thermo-Mechanical Vibration of FGM Sandwich Beam under Variable Elastic Foundations Using Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 321, No. 1-2, 2009, pp. 342-362. doi:10.1016/j.jsv.2008.09.018

[13] J. C. Hsu, H. Y. Lai and C. K. Chen, “Free Vibration of Non-Uniform Euler-Bernoulli Beams with General Elastically End Constraints Using Adomain Modified Decomposition Method,” Journal of Sound and Vibration, Vol. 318, No. 4-5, 2008, pp. 965-981.

[14] H. Y. Lai, J. C. Hsu and C. K. Chen, “An Innovative Eigenvalue Problem Solver for Free Vibration of EulerBernoulli Beam by Using the Adomain Decomposition Method,” Computers & Mathematics with Applications, Vol. 56, No. 12, 2008, pp. 3204-3220. doi:10.1016/j.camwa.2008.07.029

[15] Y. Liu and C. S. Gurram, “The Use of He’s Variational Iteration Method for Obtaining the Free Vibration of an Euler-Bernoulli Beam,” Mathematical and Computer Modelling, Vol. 50, No. 11-12, 2009, pp. 1545-1552. doi:10.1016/j.mcm.2009.09.005

[16] Q. Mao and S. Pietrzko, “Free Vibration Analysis of Stepped Beams by Using Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3429-3441. doi:10.1016/j.amc.2010.09.010

[17] J. K. Zhou, “Differential Transformation and Its Application for Electrical Circuits,” Huazhong University Press, Wuhan, 1986.

[18] M. Malik and H. H. Dang, “Vibration Analysis of Continuous Systems by Differential Transformation,” Applied Mathematics and Computation, Vol. 96, No. 1, 1998, pp. 17-26. doi:10.1016/S0096-3003(97)10076-5

[19] M. O. Kaya and O. O. Ozgumus, “Flexural-Torsional-Coupled Vibration Analysis of Axially Loaded Closed-Section Composite Timoshenko Beam by Using DTM,” Journal of Sound and Vibration, Vol. 306, No. 3-5, 2007, pp. 495-506.

[20] O. O. Ozgumus and M. O. Kaya, “Flapwise Bending Vibration Analysis of Double Tapered Rotating Euler-Burnoulli Beam by Using the Differential Transform Method,” Meccanica, Vol. 41, No. 6, 2006, pp. 661-670. doi:10.1007/s11012-006-9012-z

[21] O. O. Ozgumus and M. O. Kaya, “Vibration Analysis of a Rotating Tapered Timoshenko Beam Using DTM,” Meccanica, Vol. 45, No. 1, 2010, pp. 33-42. doi:10.1007/s11012-009-9221-3

[22] S. C. Pradhan and G. K. Reddy, “Buckling Analysis of Single Walled Carbon Nanotube on Winkler Foundation Using Nonlocal Elasticity Theory and DTM,” Computational Materials Science, Vol. 50, No. 3, 2011, pp. 10521056. doi:10.1016/j.commatsci.2010.11.001

[23] F. Delale and F. Erdogan, “The Crack Problem for a NonHomogeneous Plane,” Journal of Applied Mechanics, Vol. 50, No. 3, 1983, pp. 609-614. doi:10.1115/1.3167098