396
9 Topologic Optimization of Vibrations of Size-Dependent Beams
Table 9.13 Dynamical characteristics of the beam at temperature θ=-100: (a) time series w(0.5; t);
(b) Fourier spectra S(ω); (c) 2D wavelet spectra; (d) phase portraits ˙
w [w(t)]; (e) Poincaré maps
w t+T [w t ]; (f) values of LLEs (q = 642) [reprinted with permission from International Journal of
Non-Linear Mechanics publishers]
References
1. Gorman, D.J.: Free Vibration Analysis of Beams and Shafts. Wiley, New York (1975)
2. Aydogdu, M., Taskin, V.: Free vibration analysis of functionally graded beams with simply
supported edges. Mater. Des. 28, 1651–1656 (2007)
3. Kapuria, S., Bhattacharyya, M., Kumar, A.N.: Bending and free vibration response of layered
functionally graded beams: a theoretical model and its experimental validation. Compos.
Struct. 82(3), 390–402 (2008)
4. Pradhan, K.-K., Chakraverty, S.: Free vibration of Euler and Timoshenko functionally graded
beams by Rayleigh-Ritz method. Compos. Part B: Eng. 51(4), 175–184 (2013)
5. Simsek, M., Kocaturk, T., Akbas, S.D.: Static bending of a functionally graded microscale
Timoshenko beam based on the modified couple stress theory. Compos. Struct. 95, 740–747
(2013)
6. Simsek, M.: Fundamental frequency analysis of functionally graded beams by using different
higher-order beam theories. Nucl. Eng. Design. 240(4), 697–705 (2010)
7. Sina, S.A., Navazi, H.M., Haddadpour, H.: An analytical method for free vibration analysis
of functionally graded beams. Mater. Design. 30(3), 741–747 (2009)
8. Thai, H.T., Vo, T.P.: Bending and free vibration of functionally graded beams using various
higher order shear deformation beam theories. Int. J. Mech. Sci. 62, 57–66 (2012)
9. Rajasekaran, S.: Differential transformation and differential quadrature methods for centrifugally stiffened axially functionally graded tapered beams. Int. J. Mech. Sci. 74(3), 15–31
(2013)
10. Shahba, A., Rajasekaran, S.: Free vibration and stability of tapered Euler-Bernoulli beams
made of axially functionally graded materials. Appl. Math. Model. 36(7), 3094–3111 (2012)
11. Calio, I., Elishakoff, I.: Closed-form solutions for axially graded beam-columns. J. Sound
Vib. 280(3–5), 1083–1094 (2009)
9 Topologic Optimization of Vibrations of Size-Dependent Beams
Table 9.13 Dynamical characteristics of the beam at temperature θ=-100: (a) time series w(0.5; t);
(b) Fourier spectra S(ω); (c) 2D wavelet spectra; (d) phase portraits ˙
w [w(t)]; (e) Poincaré maps
w t+T [w t ]; (f) values of LLEs (q = 642) [reprinted with permission from International Journal of
Non-Linear Mechanics publishers]
References
1. Gorman, D.J.: Free Vibration Analysis of Beams and Shafts. Wiley, New York (1975)
2. Aydogdu, M., Taskin, V.: Free vibration analysis of functionally graded beams with simply
supported edges. Mater. Des. 28, 1651–1656 (2007)
3. Kapuria, S., Bhattacharyya, M., Kumar, A.N.: Bending and free vibration response of layered
functionally graded beams: a theoretical model and its experimental validation. Compos.
Struct. 82(3), 390–402 (2008)
4. Pradhan, K.-K., Chakraverty, S.: Free vibration of Euler and Timoshenko functionally graded
beams by Rayleigh-Ritz method. Compos. Part B: Eng. 51(4), 175–184 (2013)
5. Simsek, M., Kocaturk, T., Akbas, S.D.: Static bending of a functionally graded microscale
Timoshenko beam based on the modified couple stress theory. Compos. Struct. 95, 740–747
(2013)
6. Simsek, M.: Fundamental frequency analysis of functionally graded beams by using different
higher-order beam theories. Nucl. Eng. Design. 240(4), 697–705 (2010)
7. Sina, S.A., Navazi, H.M., Haddadpour, H.: An analytical method for free vibration analysis
of functionally graded beams. Mater. Design. 30(3), 741–747 (2009)
8. Thai, H.T., Vo, T.P.: Bending and free vibration of functionally graded beams using various
higher order shear deformation beam theories. Int. J. Mech. Sci. 62, 57–66 (2012)
9. Rajasekaran, S.: Differential transformation and differential quadrature methods for centrifugally stiffened axially functionally graded tapered beams. Int. J. Mech. Sci. 74(3), 15–31
(2013)
10. Shahba, A., Rajasekaran, S.: Free vibration and stability of tapered Euler-Bernoulli beams
made of axially functionally graded materials. Appl. Math. Model. 36(7), 3094–3111 (2012)
11. Calio, I., Elishakoff, I.: Closed-form solutions for axially graded beam-columns. J. Sound
Vib. 280(3–5), 1083–1094 (2009)