New Consistent Numerical Modelling of a Travelling Accelerating Concentrated Mass

Affiliation(s)

Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland.

Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland.

ABSTRACT

This paper deals with vibrations of structures subjected to moving inertial loads. In literature structures are usually subjected to massless forces. In numerical applications, however, the direct influence of the inertia of a moving object is usually neglected since the characteristic matrices, although simple, can not be easily derived. The paper presents a direct, non-iterative treatment of the motion of a mass along the finite element edge. The general characteristic matrices of finite elements that carry an inertial particle are given and can be applied directly to almost all types of structures. Numerical tests and a comparison with examples from the literature and especially with analytical results, prove the efficiency and accuracy of our analysis.

This paper deals with vibrations of structures subjected to moving inertial loads. In literature structures are usually subjected to massless forces. In numerical applications, however, the direct influence of the inertia of a moving object is usually neglected since the characteristic matrices, although simple, can not be easily derived. The paper presents a direct, non-iterative treatment of the motion of a mass along the finite element edge. The general characteristic matrices of finite elements that carry an inertial particle are given and can be applied directly to almost all types of structures. Numerical tests and a comparison with examples from the literature and especially with analytical results, prove the efficiency and accuracy of our analysis.

Cite this paper

B. Dyniewicz and C. Bajer, "New Consistent Numerical Modelling of a Travelling Accelerating Concentrated Mass,"*World Journal of Mechanics*, Vol. 2 No. 6, 2012, pp. 281-287. doi: 10.4236/wjm.2012.26034.

B. Dyniewicz and C. Bajer, "New Consistent Numerical Modelling of a Travelling Accelerating Concentrated Mass,"

References

[1] M. M. Stanisic, J. A. Euler and S. T. Montgomery, “On a Theory Concerning the Dynamical Behavior of Structures Carrying Moving Masses,” Archive of Applied Mechanics, Vol. 43, No. 5, 1974, pp. 295-305. doi:10.1007/BF00537218

[2] S. Sadiku and H. H. E. Leipholz, “On the Dynamics of Elastic Systems with Moving Concentrated Masses,” Archive of Applied Mechanics, Vol. 57, No. 3, 1987, pp. 223-242.

[3] H. P. Lee, “On the Dynamic Behaviour of a Beam with an Accelerating Mass,” Archive of Applied Mechanics, Vol. 65, No. 8, 1995, pp. 564-571. doi:10.1007/BF00789097

[4] H. P. Lee, “Transverse Vibration of a Timoshenko Beam Acted on by an Accelerating Mass,” Applied Acoustics, Vol. 47, No. 4, 1996, pp. 319-330. doi:10.1016/0003-682X(95)00067-J

[5] V. A. Krysko, J. Awrejcewicz, A. N. Kutsemako and K. Broughan, “Interaction between Flexible Shells (Plates) and a Moving Lumped Body,” Communications in Nonlinear Science and Numerical Simulation, Vol. 11, No. 1, 2006, pp. 13-43. doi:10.1016/j.cnsns.2004.06.006

[6] J. J. Wu, A. R. Whittaker and M. P. Cartmell, “The Use of Finite Element Techniques for Calculating the Dynamic Response of Structures to Moving Loads,” Computers and Structures, Vol. 78, No. 6, 2000, pp. 789-799. doi:10.1016/S0045-7949(00)00055-9

[7] P. Lou, G. Dai and Q. Zeng, “Dynamic Analysis of a Timoshenko Beam Subjected to Moving Concentrated Forces Using the Finite Element Method,” Shock and Vibration, Vol. 14, No. 6, 2007, pp. 459-468.

[8] J. Hino, T. Yoshimura, K. Konishi and N. Ananthanarayana, “A Finite Element Prediction of the Vibration of a Bridge Subjected to a Moving Vehicle Load,” Journal of Sound and Vibration, Vol. 96, No. 1, 1984, pp. 45-53. doi:10.1016/0022-460X(84)90593-5

[9] Y. H. Lin and M. W. Tretheway, “Finite Element Analysis of Elastic Beams Subjected to Moving Dynamic Loads,” Journal of Sound and Vibration, Vol. 136, No. 2, 1990, pp. 323-342. doi:10.1016/0022-460X(90)90860-3

[10] Y. S. Cheng, F. T. K. Au and Y. K. Cheung, “Vibration of Railway Bridges under a Moving Train by Using Bridge-Track-Vehicle Element,” Engineering Structures, Vol. 23, No. 12, 2001, pp. 1597-1606. doi:10.1016/S0141-0296(01)00058-X

[11] A. V. Pesterev, L. A. Bergman, C. A. Tan, T.-C. Tsao and B. Yang, “On Asymptotics of the Solution of the Moving Oscillator Problem,” Journal of Sound and Vibration, Vol. 260, No. 3, 2003, pp. 519-536. doi:10.1016/S0022-460X(02)00953-7

[12] D. M. Yoshida and W. Weaver, “Finite-Element Analysis of Beams and Plates with Moving Loads,” Journal of the International Association Bridge and Structural Engineering, Vol. 31, No. 1, 1971, pp. 179-195.

[13] F. V. Filho, “Finite Element Analysis of Structures under Moving Loads,” The Shock and Vibration Digest, Vol. 10, No. 8, 1978, pp. 27-35. doi:10.1177/058310247801000803

[14] A. O. Cifuentes, “Dynamic Response of a Beam Excited by a Moving Mass,” Finite Elements in Analysis and Design, Vol. 5, No. 3, 1989, pp. 237-246. doi:10.1016/0168-874X(89)90046-2

[15] J. R. Rieker, Y. H. Lin and M. W. Trethewey, “Discretization Considerations in Moving Load Finite Element Beam Models,” Finite Elements in Analysis and Design, Vol. 21, No. 3, 1996, pp. 129-144. doi:10.1016/0168-874X(95)00029-S

[16] C. I. Bajer and B. Dyniewicz, “Space-Time Approach to Numerical Analysis of a String with a Moving Mass,” International Journal for Numerical Methods in Engineering, Vol. 76, No. 10, 2008, pp. 1528-1543.

[17] C. I. Bajer and B. Dyniewicz, “Virtual Functions of the Space-Time Finite Element Method in Moving Mass Problems,” Computers and Structures, Vol. 87, No. 7-8, 2009, pp. 444-455. doi:10.1016/j.compstruc.2009.01.007

[18] H. P. Lee, “The Dynamic Response of a Timoshenko Beam Subjected to a Moving Mass,” Journal of Sound Vibration, Vol. 198, No. 2, 1996, pp. 249-256. doi:10.1006/jsvi.1996.0567

[19] B. Dyniewicz and C. I. Bajer, “New Feature of the Solution of a Timoshenko Beam Carrying the Moving Mass Particle,” Archive of Mechanics, Vol. 62, No. 5, 2010, pp. 327-341.

[20] B. Dyniewicz and C. I. Bajer, “Paradox of the Particle’s Trajectory Moving on a String,” Archive Applied Mechanics, Vol. 79, No. 3, 2009, pp. 213-223. doi:10.1007/s00419-008-0222-9

[21] B. Dyniewicz and C. I. Bajer, “Discontinuous Trajectory of the Mass Particle Moving on a String or a Beam,” Machine Dynamics Problems, Vol. 31, No. 2, 2007, pp. 66-79.

[1] M. M. Stanisic, J. A. Euler and S. T. Montgomery, “On a Theory Concerning the Dynamical Behavior of Structures Carrying Moving Masses,” Archive of Applied Mechanics, Vol. 43, No. 5, 1974, pp. 295-305. doi:10.1007/BF00537218

[2] S. Sadiku and H. H. E. Leipholz, “On the Dynamics of Elastic Systems with Moving Concentrated Masses,” Archive of Applied Mechanics, Vol. 57, No. 3, 1987, pp. 223-242.

[3] H. P. Lee, “On the Dynamic Behaviour of a Beam with an Accelerating Mass,” Archive of Applied Mechanics, Vol. 65, No. 8, 1995, pp. 564-571. doi:10.1007/BF00789097

[4] H. P. Lee, “Transverse Vibration of a Timoshenko Beam Acted on by an Accelerating Mass,” Applied Acoustics, Vol. 47, No. 4, 1996, pp. 319-330. doi:10.1016/0003-682X(95)00067-J

[5] V. A. Krysko, J. Awrejcewicz, A. N. Kutsemako and K. Broughan, “Interaction between Flexible Shells (Plates) and a Moving Lumped Body,” Communications in Nonlinear Science and Numerical Simulation, Vol. 11, No. 1, 2006, pp. 13-43. doi:10.1016/j.cnsns.2004.06.006

[6] J. J. Wu, A. R. Whittaker and M. P. Cartmell, “The Use of Finite Element Techniques for Calculating the Dynamic Response of Structures to Moving Loads,” Computers and Structures, Vol. 78, No. 6, 2000, pp. 789-799. doi:10.1016/S0045-7949(00)00055-9

[7] P. Lou, G. Dai and Q. Zeng, “Dynamic Analysis of a Timoshenko Beam Subjected to Moving Concentrated Forces Using the Finite Element Method,” Shock and Vibration, Vol. 14, No. 6, 2007, pp. 459-468.

[8] J. Hino, T. Yoshimura, K. Konishi and N. Ananthanarayana, “A Finite Element Prediction of the Vibration of a Bridge Subjected to a Moving Vehicle Load,” Journal of Sound and Vibration, Vol. 96, No. 1, 1984, pp. 45-53. doi:10.1016/0022-460X(84)90593-5

[9] Y. H. Lin and M. W. Tretheway, “Finite Element Analysis of Elastic Beams Subjected to Moving Dynamic Loads,” Journal of Sound and Vibration, Vol. 136, No. 2, 1990, pp. 323-342. doi:10.1016/0022-460X(90)90860-3

[10] Y. S. Cheng, F. T. K. Au and Y. K. Cheung, “Vibration of Railway Bridges under a Moving Train by Using Bridge-Track-Vehicle Element,” Engineering Structures, Vol. 23, No. 12, 2001, pp. 1597-1606. doi:10.1016/S0141-0296(01)00058-X

[11] A. V. Pesterev, L. A. Bergman, C. A. Tan, T.-C. Tsao and B. Yang, “On Asymptotics of the Solution of the Moving Oscillator Problem,” Journal of Sound and Vibration, Vol. 260, No. 3, 2003, pp. 519-536. doi:10.1016/S0022-460X(02)00953-7

[12] D. M. Yoshida and W. Weaver, “Finite-Element Analysis of Beams and Plates with Moving Loads,” Journal of the International Association Bridge and Structural Engineering, Vol. 31, No. 1, 1971, pp. 179-195.

[13] F. V. Filho, “Finite Element Analysis of Structures under Moving Loads,” The Shock and Vibration Digest, Vol. 10, No. 8, 1978, pp. 27-35. doi:10.1177/058310247801000803

[14] A. O. Cifuentes, “Dynamic Response of a Beam Excited by a Moving Mass,” Finite Elements in Analysis and Design, Vol. 5, No. 3, 1989, pp. 237-246. doi:10.1016/0168-874X(89)90046-2

[15] J. R. Rieker, Y. H. Lin and M. W. Trethewey, “Discretization Considerations in Moving Load Finite Element Beam Models,” Finite Elements in Analysis and Design, Vol. 21, No. 3, 1996, pp. 129-144. doi:10.1016/0168-874X(95)00029-S

[16] C. I. Bajer and B. Dyniewicz, “Space-Time Approach to Numerical Analysis of a String with a Moving Mass,” International Journal for Numerical Methods in Engineering, Vol. 76, No. 10, 2008, pp. 1528-1543.

[17] C. I. Bajer and B. Dyniewicz, “Virtual Functions of the Space-Time Finite Element Method in Moving Mass Problems,” Computers and Structures, Vol. 87, No. 7-8, 2009, pp. 444-455. doi:10.1016/j.compstruc.2009.01.007

[18] H. P. Lee, “The Dynamic Response of a Timoshenko Beam Subjected to a Moving Mass,” Journal of Sound Vibration, Vol. 198, No. 2, 1996, pp. 249-256. doi:10.1006/jsvi.1996.0567

[19] B. Dyniewicz and C. I. Bajer, “New Feature of the Solution of a Timoshenko Beam Carrying the Moving Mass Particle,” Archive of Mechanics, Vol. 62, No. 5, 2010, pp. 327-341.

[20] B. Dyniewicz and C. I. Bajer, “Paradox of the Particle’s Trajectory Moving on a String,” Archive Applied Mechanics, Vol. 79, No. 3, 2009, pp. 213-223. doi:10.1007/s00419-008-0222-9

[21] B. Dyniewicz and C. I. Bajer, “Discontinuous Trajectory of the Mass Particle Moving on a String or a Beam,” Machine Dynamics Problems, Vol. 31, No. 2, 2007, pp. 66-79.