Closed-form solutions and uncertainty quantification for gravity-loaded beams
Tóm tắt
Typically, the cantilever non-uniform gravity-loaded Euler–Bernoulli beams are numerically modeled as the governing equation for free vibration analysis does not yield an exact solution. We show that, for certain polynomial variations of the mass and stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation for gravity-loaded beams. An inverse problem approach is used to find an infinite number of such beams, with various mass and stiffness distributions, which share the same fundamental frequency. The derived distributions are demonstrated as test functions for a p-version finite element method. The functions can also be used to design gravity-loaded cantilever beams having a pre-specified fundamental natural frequency. Examples of such beams with rectangular cross section are presented. The bounds for the pre-specified fundamental frequency and its variation for beams of different lengths are also studied. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is found analytically. This analytical solution can serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the stiffness distribution for known probability distributions of the frequencies.
Tài liệu tham khảo
Schäfer B (1985) Free vibrations of a gravity-loaded clamped-free beam. Ingenieur-archiv 55(1):66–80
Ganesh R, Ganguli R (2013) Stiff string approximations in Rayleigh–Ritz method for rotating beams. Appl Math Comput 219(17):9282–9295
Alley VL, Leadbetter SA (1963) Prediction and measurement of natural vibrations of multistage launch vehicles. AIAA J 1(2):374–379
Krauthammer T (1987) A numerical study of wind-induced tower vibrations. Comput Struct 26(1):233–241
Bracci JM, Reinhorn AM, Mander JB (1995) Seismic resistance of reinforced concrete frame structures designed for gravity loads: performance of structural system. ACI Struct J 92(5):597–610
Güler K (1998) Free vibrations and modes of chimneys on an elastic foundation. J Sound Vib 218(3):541–547
Chmielewski T, Górski P, Beirow B, Kretzschmar J (2005) Theoretical and experimental free vibrations of tall industrial chimney with flexibility of soil. Eng Struct 27(1):25–34
Wang A-P, Lin Y-H (2007) Vibration control of a tall building subjected to earthquake excitation. J Sound Vib 299(4):757–773
Taranath BS (2011) Structural analysis and design of tall buildings: steel and composite construction. CRC Press, Boca Raton
Wang G, Wereley NM (2004) Free vibration analysis of rotating blades with uniform tapers. AIAA J 42(12):2429–2437
Yan S-X, Zhang Z-P, Wei D-J, Li X-F (2011) Bending vibration of rotating tapered cantilevers by integral equation method. AIAA J 49(4):872–876
Wei D, Liu Y, Xiang Z (2012) An analytical method for free vibration analysis of functionally graded beams with edge cracks. J Sound Vib 331(7):1686–1700
Kambampati S, Ganguli R, Mani V (2013) Rotating beams isospectral to axially loaded nonrotating uniform beams. AIAA J 51(5):1189–1202
Kim H, Yoo H Hee, Chung J (2013) Dynamic model for free vibration and response analysis of rotating beams. J Sound Vib 332(22):5917–5928
Behera L, Chakraverty S (2014) Free vibration of nonhomogeneous Timoshenko nanobeams. Meccanica 49(1):51–67
Vo TP, Thai H-T, Nguyen T-K, Inam F (2014) Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica 49(1):155–168
Elishakoff I, Zaza N, Curtin J, Hashemi J (2014) Apparently first closed-form solution for vibration of functionally graded rotating beams. AIAA J 52(11):2587–2593
Rajasekaran S, Tochaei EN (2014) Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order. Meccanica 49(4):995–1009
Sarkar K, Ganguli R (2014) Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams. Meccanica 49(6):1469–1477
Bambill D, Rossit C, Felix D (2015) Free vibrations of stepped axially functionally graded Timoshenko beams. Meccanica 50(4):1073–1087
Wattanasakulpong N, Charoensuk J (2015) Vibration characteristics of stepped beams made of FGM using differential transformation method. Meccanica 50(4):1089–1101
Paidoussis MP, Des Trois Maisons PE (1971) Free vibration of a heavy, damped, vertical cantilever. J Appl Mech 38:524
Caddemi S, Caliò I (2009) Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks. J Sound Vib 327(3):473–489
Liu M-F, Chang T-P (2010) Closed form expression for the vibration problem of a transversely isotropic magneto-electro-elastic plate. J Appl Mech 77(2):024502
Stojanovic V, Kozic P, Janevski G (2013) Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory. J Sound Vib 332(3):563–576
Sarkar K, Ganguli R (2013) Closed-form solutions for non-uniform Euler–Bernoulli free-free beams. J Sound Vib 332(23):6078–6092
Yokoyama T (1990) Vibrations of a hanging Timoshenko beam under gravity. J Sound Vib 141(2):245–258
Abramovich H (1993) Free vibrations of gravity loaded composite beams. Compos Struct 23(1):17–26
Naguleswaran S (1991) Vibration of a vertical cantilever with and without axial freedom at clamped end. J Sound Vib 146(2):191–198
Naguleswaran S (2004) Transverse vibration of an uniform Euler–Bernoulli beam under linearly varying axial force. J Sound Vib 275(1):47–57
Virgin LN, Santillan ST, Holland DB (2007) Effect of gravity on the vibration of vertical cantilevers. Mech Res Commun 34(3):312–317
Hijmissen JW, Van Horssen WT (2007) On aspects of damping for a vertical beam with a tuned mass damper at the top. Nonlinear Dyn 50(1–2):169–190
Xi LY, Li XF, Tang GJ (2013) Free vibration of standing and hanging gravity-loaded Rayleigh cantilevers. Int J Mech Sci 66:233–238
Gladwell GM (2005) Inverse problems in vibration. Kluwer, New York
Becquet R, Elishakoff I (2001) Class of analytical closed-form polynomial solutions for guided-pinned inhomogeneous beams. Chaos Solitons Fractals 12(8):1509–1534
Elishakoff I, Becquet R (2000) Closed-form solutions for natural frequency for inhomogeneous beams with one sliding support and the other pinned. J Sound Vib 238(3):529–539
Elishakoff I, Candan S (2001) Apparently first closed-form solution for vibrating: inhomogeneous beams. Int J Solids Struct 38(19):3411–3441
Yamazaki F, Member A, Shinozuka M, Dasgupta G (1988) Neumann expansion for stochastic finite element analysis. J Eng Mech 114(8):1335–1354
Shinozuka M, Astill CJ (1972) Random eigenvalue problems in structural analysis. AIAA J 10(4):456–462
Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainty in applied mechanics. Elsevier, Amsterdam
Choi S-K, Grandhi RV, Canfield RA, Pettit CL (2004) Polynomial chaos expansion with latin hypercube sampling for estimating response variability. AIAA J 42(6):1191–1198
Huang S, Mahadevan S, Rebba R (2007) Collocation-based stochastic finite element analysis for random field problems. Probab Eng Mech 22(2):194–205
Vom Scheidt J, Purkert W (1983) Random eigenvalue problems. North Holland, New York
Ghanem R, Ghosh D (2007) Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition. Int J Numer Methods Eng 72(4):486–504
Adhikari S (2007) Joint statistics of natural frequencies of stochastic dynamic systems. Comput Mech 40(4):739–752
Wolfram S (1999) The Mathematica Book. Cambridge University Press, Cambridge
Udupa KM, Varadan TK (1990) Hierarchical finite element method for rotating beams. J Sound Vib 138(3):447–456
Hodges DY, Rutkowski MY (1981) Free-vibration analysis of rotating beams by a variable-order finite-element method. AIAA J 19(11):1459–1466
Gunda JB, Singh AP, Chhabra PS, Ganguli R (2007) Free vibration analysis of rotating tapered blades using Fourier-\(p\) super element. Struct Eng Mech 27(2):243–257
Vinod KG, Gopalakrishnan S, Ganguli R (2007) Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements. Int J Solids Struct 44(18):5875–5893
Sarkar K, Ganguli R (2013) Rotating beams and non-rotating beams with shared eigenpair for pinned-free boundary condition. Meccanica 48(7):1661–1676
Bartle RG, Sherbert DR (2000) Introduction to real analysis. Wiley, New York