Vibration of Elastically Connected Multiple Timoshenko Beams under a Moving Six Degrees of Freedom System
Authors
Abstract
In this article, the vibration analysis of a set of parallel Timoshenko beams connected by intermediate flexible connections, with arbitrary numbers, is studied. The moving load is a vehicle, which is modeled by a two-axle six degrees of freedom system, as a mass-spring-damper system, in a plane motion. For the solution, a new method is proposed which uses a change of variables strategy to decouple the system of differential equations. For this purpose, the stiffness matrix obtained from each column of intermediate connections should have the same normalized eigenvectors. The displacements and the bending moments of the beams and the vehicle due to changes in the stiffness of connections and changes in speeds will be examined. Finally, the validity of the results are measured.
Keywords
1. Willis, R., “Report of the Commissioners Appointed
to Inquire into the Application of Iron to Railway
Structures”, London, Stationary Office, 1949.
2. Mamandi, A., Kargarnovin, M. H., and Farsi, S., “An
Investigation on Effects of Traveling Mass with
Variable Velocity on Nonlinear Dynamic Response
of an Inclined Timoshenko Beam with Different
Boundary Conditions”, International Journal of
Mechanical Sciences, Vol. 52, pp. 1694-1708,
2010.
3. Sharbati, E., and Szyszkowski, W., “A New FEM
Approach for Analysis of Beams with Relative
Movements of Masses”, Finite Elements in Analysis
and Design, Vol. 47, pp. 1047-1057, 2011.
4. Esmailzadeh, E., and Ghorashi, M., “Vibration
Analysis of Beams Traversed by Uniform Partially
Distributed Moving Masses”, Journal of Sound
Vibration, Vol. 184, pp. 9-17, 1995.
5. Esmailzadeh, E., and Ghorashi, M., “Vibration
Analysis of Timoshenko Beam Subjected to a
Traveling Mass”, Journal of Sound and Vibration,
Vol. 199, pp. 615-628, 1997.
6. Eftekhar Azam, S., Mofid, M., and Afghani-
Khoraskani, R., “Dynamic Response of Timoshenko
Beam under Moving Mass”, Scientia Iranica
Transactions A: Civil Engineering, Vol. 20, pp. 50-
56, 2013.
7. Lin, Y. H., and Tretheway, M. W., “Finite Element
Analysis of Elastic Beams Subjected to Moving
Dynamic Loads”, Journal of Sound and Vibration,
Vol. 136, pp. 323-342, 1990.
8. Stancioiu, D., Ouyang, H., and Mottershead, J. E.,
“Vibration of a Continuous Beam with Multiple
Elastic Supports Excited by a Moving Two-axle
System with Separation”, Meccanica, Vol. 44, pp.
293-303, 2009.
9. Wu, J. J., “Free Vibration Analysis of Beams
Carrying a Number of Two-Degree-of-Freedom
Spring-Damper-Mass Systems”, Finite Elements in
Analysis and Design, Vol. 40, pp. 363-381, 2004.
10. Esmailzadeh, E., and Jalili, N., “Vehicle-Passenger-
Structure Interaction of Uniform Bridges Traversed
by Moving Vehicles”, Journal of Sound and
Vibration, Vol. 260, pp. 611-635, 2003.
11. Lou, P., and Au, F. T. K., “Finite Element Formulae
for Internal Forces of Bernoulli–Euler Beams under
Moving Vehicles”, Journal of Sound and Vibration,
Vol. 332, pp. 1533-1552, 2013.
12. Yang, Y. B., and Wu, Y. S., “A Versatile Element
for Analyzing Vehicle-Bridge Interaction Response”,
Engineering Structures, Vol. 23, pp. 452-469, 2001.
13.Vu, H. V., Ordonez, A. M., and Karnopp, B. H.,
“Vibration of a Double-beam System”, Journal of
Sound and Vibration, Vol. 229, pp. 807-822, 2000.
14.Abu-Hilal, M., “Dynamic Response of a Double
Euler-Bernoulli Beam due to a Moving Constant
Load”, Journal of Sound and Vibration, Vol. 297,
pp. 477-491, 2006.
15.Ariaei, A., Ziaei-Rad, S., and Ghayour, M.,
“Transverse Vibration of a Multiple-Timoshenko
Beam System with Intermediate Elastic Connections
Due to a Moving Load”, Archive of Applied
Mechanics, Vol. 81, pp. 263-281, 2010.
16. Oniszczuk, Z., “Forced Transverse Vibrations of an
Elastically Connected Complex Simply Supported
Double-beam System”, Journal of Sound and
Vibration, Vol. 264, pp. 273-286, 2003.
17. Stojanovic, V., Kozic, P., Pavlovic, R., and Janevski,
G., “Effect of Rotary Inertia and Shear on Vibration
and Buckling of a Double Beam System under
Compressive Axial Loading”, Archive of Applied
Mechanics, Vol. 81, pp. 1993-2005, 2011.
18.Dadfarnia, M., Jalili, N., and Esmailzadeh, E., “A
Comparative Study of the Galerkin Approximation
Utilized in the Timoshenko Beam Theory”, Journal
of Sound and Vibration, Vol. 280, pp. 1132-1142,
2005.
19. Fryba, L., “A Rough Assessment of Railway Bridges
for High Speed Trains”, Journal of Engineering
Structures, Vol. 23, pp. 548-556, 2001.
to Inquire into the Application of Iron to Railway
Structures”, London, Stationary Office, 1949.
2. Mamandi, A., Kargarnovin, M. H., and Farsi, S., “An
Investigation on Effects of Traveling Mass with
Variable Velocity on Nonlinear Dynamic Response
of an Inclined Timoshenko Beam with Different
Boundary Conditions”, International Journal of
Mechanical Sciences, Vol. 52, pp. 1694-1708,
2010.
3. Sharbati, E., and Szyszkowski, W., “A New FEM
Approach for Analysis of Beams with Relative
Movements of Masses”, Finite Elements in Analysis
and Design, Vol. 47, pp. 1047-1057, 2011.
4. Esmailzadeh, E., and Ghorashi, M., “Vibration
Analysis of Beams Traversed by Uniform Partially
Distributed Moving Masses”, Journal of Sound
Vibration, Vol. 184, pp. 9-17, 1995.
5. Esmailzadeh, E., and Ghorashi, M., “Vibration
Analysis of Timoshenko Beam Subjected to a
Traveling Mass”, Journal of Sound and Vibration,
Vol. 199, pp. 615-628, 1997.
6. Eftekhar Azam, S., Mofid, M., and Afghani-
Khoraskani, R., “Dynamic Response of Timoshenko
Beam under Moving Mass”, Scientia Iranica
Transactions A: Civil Engineering, Vol. 20, pp. 50-
56, 2013.
7. Lin, Y. H., and Tretheway, M. W., “Finite Element
Analysis of Elastic Beams Subjected to Moving
Dynamic Loads”, Journal of Sound and Vibration,
Vol. 136, pp. 323-342, 1990.
8. Stancioiu, D., Ouyang, H., and Mottershead, J. E.,
“Vibration of a Continuous Beam with Multiple
Elastic Supports Excited by a Moving Two-axle
System with Separation”, Meccanica, Vol. 44, pp.
293-303, 2009.
9. Wu, J. J., “Free Vibration Analysis of Beams
Carrying a Number of Two-Degree-of-Freedom
Spring-Damper-Mass Systems”, Finite Elements in
Analysis and Design, Vol. 40, pp. 363-381, 2004.
10. Esmailzadeh, E., and Jalili, N., “Vehicle-Passenger-
Structure Interaction of Uniform Bridges Traversed
by Moving Vehicles”, Journal of Sound and
Vibration, Vol. 260, pp. 611-635, 2003.
11. Lou, P., and Au, F. T. K., “Finite Element Formulae
for Internal Forces of Bernoulli–Euler Beams under
Moving Vehicles”, Journal of Sound and Vibration,
Vol. 332, pp. 1533-1552, 2013.
12. Yang, Y. B., and Wu, Y. S., “A Versatile Element
for Analyzing Vehicle-Bridge Interaction Response”,
Engineering Structures, Vol. 23, pp. 452-469, 2001.
13.Vu, H. V., Ordonez, A. M., and Karnopp, B. H.,
“Vibration of a Double-beam System”, Journal of
Sound and Vibration, Vol. 229, pp. 807-822, 2000.
14.Abu-Hilal, M., “Dynamic Response of a Double
Euler-Bernoulli Beam due to a Moving Constant
Load”, Journal of Sound and Vibration, Vol. 297,
pp. 477-491, 2006.
15.Ariaei, A., Ziaei-Rad, S., and Ghayour, M.,
“Transverse Vibration of a Multiple-Timoshenko
Beam System with Intermediate Elastic Connections
Due to a Moving Load”, Archive of Applied
Mechanics, Vol. 81, pp. 263-281, 2010.
16. Oniszczuk, Z., “Forced Transverse Vibrations of an
Elastically Connected Complex Simply Supported
Double-beam System”, Journal of Sound and
Vibration, Vol. 264, pp. 273-286, 2003.
17. Stojanovic, V., Kozic, P., Pavlovic, R., and Janevski,
G., “Effect of Rotary Inertia and Shear on Vibration
and Buckling of a Double Beam System under
Compressive Axial Loading”, Archive of Applied
Mechanics, Vol. 81, pp. 1993-2005, 2011.
18.Dadfarnia, M., Jalili, N., and Esmailzadeh, E., “A
Comparative Study of the Galerkin Approximation
Utilized in the Timoshenko Beam Theory”, Journal
of Sound and Vibration, Vol. 280, pp. 1132-1142,
2005.
19. Fryba, L., “A Rough Assessment of Railway Bridges
for High Speed Trains”, Journal of Engineering
Structures, Vol. 23, pp. 548-556, 2001.
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