DESIGN OF VISCO-ELASTIC SUPPORTS FOR TIMOSHENKO CANTILEVER BEAMS

Abstract

The appropriate design of supports, upon which beams are usually placed as structural components in many engineering scenarios, has substantial significance in terms of both structural efficacy and cost factors. When beams experience various dynamic vibration effects, it is crucial to contemplate appropriate support systems that will effectively adapt to these vibrations. The present work investigates the most suitable support configuration for a cantilever beam, including viscoelastic supports across different vibration modes. Within this particular framework, a cantilever beam is simulated using beam finite elements. The beam is positioned on viscoelastic supports, which are represented by simple springs and damping elements. These supports are then included in the overall structural model. The equation of motion for the beam is first formulated in the temporal domain and then converted to the frequency domain via the use of the Fourier Transform. The basic equations used in the frequency domain are utilized to establish the dynamic characteristics of the beam by means of transfer functions. The determination of the ideal stiffness and damping coefficients of the viscoelastic components is achieved by minimizing the absolute acceleration at the free end of the beam. In order to minimize the objective function associated with acceleration, the nonlinear equations derived from Lagrange multipliers are solved using a gradient-based technique. The governing equations of the approach need partial derivatives with respect to design variables. Consequently, analytical derivative equations are formulated for both the stiffness and damping parameters. The present work introduces a concurrent optimization approach for both stiffness and damping. Passive constraints are established inside the optimization problem to impose restrictions on the lower and higher boundaries of the stiffness and damping coefficients. On the other hand, active constraints are used to ascertain the specific values of the overall stiffness and damping coefficients. The efficacy of the established approach in estimating the ideal spring and damping coefficients of viscoelastic supports and its ability to provide optimal support solutions for various vibration modes have been shown via comparative experiments with prior research.

Keywords

• Timoshenko Beam
• Transfer Function
• Visco-Elastic Support
• Beam Vibration
• Damping
• Spring

References

1. S. I. Timoshenko, D. H. Young and W. Weaver, “Vibration Problems in Engineering”, in John Wiley & Sons, 1974.
2. H. Wei, and Z. Yida, “The dynamic response of a viscoelastic Winkler foundation-supported elastic beam impacted by a low velocity projectile,” Computers & Structures, vol. 52, no.3, Aug., pp. 431-436, 1994.
3. J. H. Chung, W. H. Joo, and K. C. Kim, “Vibration and Dynamic Sensitivity Analysis of a Timoshenko Beam-Column with Elastically Restrained Ends and Intermediate Constraints,” Journal of Sound and Vibration, vol. 167, no. 2, Oct., pp. 209-215, 1993.
4. Y. H. Chen, and J. T. Sheu, “Beam on viscoelastic foundation and layered beam,” Journal of Engineering Mechanics, vol. 121, no. 2, Feb., pp. 340-344, 1995.
5. A. V. Metrikine, and H. A. Dieterman,” Instability of vibrations of a mass moving uniformly along an axially compressed beam on a viscoelastic foundation,” Journal of Sound and Vibration, vol. 201, no. 5, Apr., pp. 567-576, 1997.
6. B. K. Lee, J. S. Jeong, G. F. Li, and T. K. Jin, “Free Vibrations of Tapered Piles Embedded Partially in An Elastic Foundation,” Chinese Journal of Geotechnical Engineering, vol. 21, no. 5, Sep., pp. 609- 613, 1999.
7. F. Zhen-yu, W. Zhong-min, and F. Li-jian, “Dynamic Stability Analysis of Visco-elastic Pile with Point Visco-elastic Supports,” China Journal of Highway and Transport, vol. 19, no. 1, Jan., pp. 67-70, 2006.
8. Y. H. Chen, and Y. H. Huang, “Dynamic stiffness of infinite Timoshenko beam on viscoelastic foundation in moving co‐ordinate,” International Journal for Numerical Methods in Engineering, vol. 48, no.1,Mar., pp. 1-18, 2000.

9. Y. H. Chen, Y. H. Huang, and C. T. Shih, “Response of an infinite Timoshenko beam on a viscoelastic foundation to a harmonic moving load,” Journal of Sound and Vibration, vol. 241 no. 5, Apr., pp. 809-824, 2001.
10. M. Ansari, E. Esmailzadeh, and D. Younesian, “Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load,” Nonlinear Dynamics, vol. 61, no. 1, Jan., pp. 163-182, 2010.
11. B. Zhen, J. Xu, and J. Sun, “Analytical solutions for steady state responses of an infinite Euler-Bernoulli beam on a nonlinear viscoelastic foundation subjected to a harmonic moving load,” Journal of Sound and Vibration, vol. 476, Jun., pp. 1-21, 2020.
12. D. Y. Zheng, F. T. K. Au, and Y. K. Cheung, “Vibration of vehicle on compressed rail on viscoelastic foundation,” Journal of Engineering Mechanics, vol. 126, no. 11, Nov., pp. 1141-1147, 2000.
13. A. V. Vostroukhov, and A. V. Metrikine, “Periodically supported beam on a visco-elastic layer as a model for dynamic analysis of a high-speed railway track,” International Journal of Solids and Structures, vol. 40, no. 21, Oct., pp. 5723-5752, 2003.
14. A. V. Metrikine, “Steady state response of an infinite string on a non-linear visco-elastic foundation to moving point loads,” Journal of Sound and Vibration, vol. 272, no. 3-5, May, pp. 1033-1046, 2004.
15. C. E. Majorana, and B. Pomaro, “Dynamic stability of an elastic beam with visco‐elastic translational and rotational supports,” Engineering Computations, vol. 28, no. 2, Mar., pp. 114-129, 2011.
16. D. Basu, and N. S. V. Kameswara Rao, “Analytical solutions for Euler–Bernoulli beam on visco‐elastic foundation subjected to moving load,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 37, no. 8, Jan., pp. 945-960, 2013.
17. D. Froio, E. Rizzi, F. M. Simões, and A. Pinto da Costa, “DLSFEM–PML formulation for the steady-state response of a taut string on visco-elastic support under moving load,” Meccanica, vol. 55, no. 4, Oct., pp. 765-790, 2020.
18. Z. Dimitrovová, “Dynamic interaction and instability of two moving proximate masses on a beam on a Pasternak viscoelastic foundation,” Applied Mathematical Modelling, vol. 100, Dec., pp. 192-217, 2021.
19. G. Rozvany, “Optimization of Unspecified Generalized Forces in Structural Design,” Journal of Applied Mechanics, vol. 41, no. 4, Dec., pp.1143–1145, 1974.
20. Z. Mróz, and G. I. N. Rozvany, “Optimal Design of Structures with Variable Support Positions,” Journal of Optimization Theory and Applications, vol. 15, Jan., pp. 85–101, 1975.
21. W. Prager, and G. Rozvany, “Plastic Design of Beams: Optimal Locations of Supports and Steps in Yield Moment,” International Journal of Mechanical Sciences, vol. 17, no. 10, Oct., pp. 627–631, 1975.
22. B. Akesson, and N. Olhoff, “Minimum Stiffness of Optimally Located Supports for Maximum Value of Beam Eigenfrequencies,” Journal of Sound and Vibration, vol. 120, no. 3, Feb., pp. 457–463, 1988.
23. J. W. Hou, and C. H. Chuang, “Design Sensitivity Analysis and Optimization of Vibration Beams with Variable Support Locations,” In Proc. Design Automation Conference ‘16, 1990, Chicago, pp. 281–290.
24. B. P. Wang, “Eigenvalue Sensitivity with Respect to Location of Internal Stiffness and Mass Attachment,” American Institute of Aeronautics and Astronautics Journal, vol. 31, no. 4, May, pp. 791–794, 1993.
25. B. P. Wang, and J. L. Chen, “Application of Genetic Algorithm for the Support Location Optimization of Beams,” Computers and Structures, vol. 58, no. 4, Feb., pp. 797–800, 1996.
26. K. M. Won, and Y. S. Park, “Optimal Support Position for a Structure to Maximize Its Fundamental Natural Frequency,” Journal of Sound and Vibration, vol. 213, no. 5, Jun., pp. 801–812, 1998.
27. J. D. Aristizabal-Ochoa, “Static, Stability and Vibration of Non-Prismatic Beams and Columns,” Journal of Sound and Vibration, vol. 62, no. 3, Apr., pp. 441 455, 1993.
28. J. K. Sinha, and M. I. Friswell, “The Location of Spring Supports from Measured Vibration Data,” Journal of Sound and Vibration vol. 244, no. 1, Jun., pp. 137–153, 2001.
29. E. Aydin, “Minimum dynamic response of cantilever beams supported by optimal elastic springs,” Structural Engineering and Mechanics, vol. 51, no. 3, Aug., pp. 377-402, 2014.
30. E. Aydin, M. Dutkiewicz, B. Öztürk, and M. Sonmez, “Optimization of elastic spring supports for cantilever beams,” Structural and Multidisciplinary Optimization, vol. 62, no. 1, Jan., pp. 55-81, 2020.
31. E. Aydın, B. Öztürk, and M. Dutkiewicz, “Determination of optimal elastic springs for cantilever beams supported by elastic foundation,” In Proc. International Conference on Engineering Mechanics ‘24, 2018, pp. 33-36.
32. D. Wang, and M. Wen, “Vibration Attenuation of Beam Structure with Intermediate Support under Harmonic Excitation,” Journal of Sound and Vibration, vol. 532, Aug., pp. 1-19, 2022.
33. G. Rozvany, and Z. Mróz, “Column Design: Optimization of Support Conditions and Segmentation,” Journal of Structural Mechanics, vol. 5, no. 3, Dec., pp. 279–290, 1977.
34. N. Olhoff, and J. E. Taylor, “Designing Continuous Columns for Minimum Total Cost of Material and Interior Supports,” Journal of Structural Mechanics, vol. 6, no. 4, Feb., pp. 367–382, 1978.
35. N. Olhoff, and B. Akesson, “Minimum Stiffness of Optimally Located Supports for Maximum Value of Column Buckling Loads,” Structural optimization, vol. 3, Sep., pp.163–175, 1991
36. B. K. Lee, J. K. Lee, T. E. Lee, and S. G Kim, “Free Vibrations of Tapered Beams with General Boundary Conditions,” KSCE Journal of Civil Engineering, vol. 6, no. 3, Sep., pp. 283-288, 2002.
37. T. C. Huang, and C. C. Huang, “Free vibrations of viscoelastic Timoshenko beams,” Journal of Applied Mechanics, vol. 38, no. 2, Jun., pp. 515-521, 1971.
38. Z. S. Liu, H. C. Hu, and D. J. Wang, “New Method for Deriving Eigenvalue Rate with Respect to Support Location,” American Institute of Aeronautics and Astronautics Journal, vol. 34, no. 4, May, pp. 864–866, 1996.
39. I. Takewaki, “Optimal Damper Positioning in Beams for Minimum Dynamic Compliance,” Computer Methods in Applied Mechanics and Engineering, vol. 156, no. 1-4, Apr., pp. 363-73, 1998.
40. L. Sun, “A closed-form solution of beam on viscoelastic subgrade subjected to moving loads,” Computers & Structures, vol. 80, no. 1, Jan., pp. 1-8, 2002.
41. M. H. Kargarnovin, D. Younesian, D. J. Thompson, and C. J. C. Jones, “Response of beams on nonlinear viscoelastic foundations to harmonic moving loads,” Computers & Structures, vol. 83, no. 23-24, Sep., pp. 1865-1877, 2005.
42. F. F. Çalım, “Dynamic analysis of beams on viscoelastic foundation,” European Journal of Mechanics-A/Solids, vol. 28, no. 3, May-Jun., pp. 469-476, 2009.
43. T. Mazilu, “Using the green's function method to analyse the response of an infinite wire on visco-elastic support under moving load,” Acta Technica Corviniensis-Bulletin of Engineering, vol. 6, no. 2, Apr.-Jun.,pp. 35-38, 2013.
44. S. M. Abdelghany, K. M. Ewis, A. A. Mahmoud, and M. M. Nassar, “Dynamic response of non-uniform beam subjected to moving load and resting on non-linear viscoelastic foundation,” Beni-Suef University Journal of Basic and Applied Sciences, vol. 4, no. 3, Sep., pp. 192-199, 2015.
45. Z. Dimitrovová, “Complete semi-analytical solution for a uniformly moving mass on a beam on a two-parameter visco-elastic foundation with non-homogeneous initial conditions,” International Journal of Mechanical Sciences, vol. 144,Aug., pp. 283-311, 2018.
46. S. Roy, G. Chakraborty, and A. DasGupta, “Coupled dynamics of a viscoelastically supported infinite string and a number of discrete mechanical systems moving with uniform speed,” Journal of Sound and Vibration, vol. 415, Feb., pp. 184-209, 2018.
47. Z. Dimitrovová, “New semi-analytical solution for a uniformly moving mass on a beam on a two-parameter visco-elastic foundation,” International Journal of Mechanical Sciences, vol. 127, Jul., pp. 142-162, 2017.
48. Z. Dimitrovová, “Semi-analytical solution for a problem of a uniformly moving oscillator on an infinite beam on a two-parameter visco-elastic foundation,” Journal of Sound and Vibration, vol. 438, Jan., pp. 257-290, 2019.
49. W. Huang, and Y. D. Zou, “The dynamic response of an elastic circular plate on a viscoelastic Winkler foundation impacted by a moving rigid body,” JSME international journal, Ser. 3, Vibration, control engineering, engineering for industry, vol. 35, no. 2, Jul., pp. 274-278, 1992.
50. E. Aydın, B. Adıyaman, and Y. E. Kebeli, “Visko-elastik mesnetler üzerine oturan Timoshenko konsol kirişlerinin minimum titreşimi,” In Proc. Uluslararası Türk Dünyası Fen Bilimleri ve Mühendislik Kongresi ‘04, 2022, pp. 1221-1235.
51. G. P. Cimellaro, “Simultaneous stiffness–damping optimization of structures with respect to acceleration, displacement and base shear, Engineering Structures, vol. 29, no. 11, Nov., pp. 2853-2870, 2007.