Non‐Linear Vibration of a Cantilever Beam of Variable Cross‐Section

Abstract

Es werden freie Schwingungen grober Amplitude eines diinnen elastischen Balkens fester Lange mit variablem Querschnitt untersucht. Obwohl groJe Auslenkungen und Drehungen betrachtet werden, sind die Dehnungen klein. Es wird gezeigt, wie man mittels des Galerkinverfahrens Naherungslosungen erhalt, und es wird der Effekt des variablen Querschnitts untersucht. Fur den Balken mit konstantem Querschnitt werden die Ergebnisse mit denen aus der Arbeit 131 verglichen, und es wird. die exakte Losung fur den Spezialfall einer in dieser Arbeit gewonnenen gewohnlichen Differentialgleichung betrachtet. Zusatzlich wird eine einfache Methode fur statische Auslenkungen vorgeschlagen.Large amplitude free vibration of an inextensible thin elastic beam of a variable cross-section is analysed. Although large deflections and rotations are considered the strains are smdl. It is shown how approximate solutions can be obtained by using Galerkin's method, and the effect of the variation of cross-section is investigated. For the beam of constant cross-section the results are compared with those of paper [3] and the exact solution of the special case of an ordinary differential equation obtained in the paper is considered. Additionally, a simple method for static deflections of the beam is proposed. MeHHoro cesetrm. X o~a paccMaTpusamca 60nbmue oTnnonemn .EI BpaweHm, ne@opMauuu ocTamTcfi ManbiMu. I I o K~~~I B~~T c~, nan iionywTb npu6nulrce~~~1e pemeHm nocpencmoM MeToaa FanepHma, II usysae~ca B@@BKT nepeMemoro cesemm. Peaynb~a~br paccseTa B cnysae 6anm ~O C T O R H H O~O ceseHm cpasmBamcR c p e a y n b~a~a~u pa6om f3] u pacciwaqmsaews TosHoe pememe B sacTHoM cnyqae nony-Yemoro B pa6o~e O~~I~H O B~H H O~O nu@@epeHuuamHoro ypamemrr. AononmTenbHo npeAnaraeTcR npo-C T O~~ MeToa ~n r r CTaTmecKux oTnnoHeH&i. H3y' IaIOTCR CB060AHbIe ~o n e 6 a~u r r 60nbmux aMnnllTyA TOHHOa YnpJTOii 6anm IIOCTOFIIHHOfi AJIIlHbI II nepe-14, 280-282 (1976). 303-306 (1981).

An Improved Analytical Method for Vibration Analysis of Variable Section Beam

“…Taylor expansion cannot be completed. Obviously, it is almost impossible to substitute the fitting function T 1 (x) with k � 200 into the equations (8), (9), and (12) for iterative computations. erefore, in practical application, ADM is limited by the expansion order k of the fitting function, so it cannot be fully applied to solve the arbitrary section beam.…”

“…Firstly, divide the whole structure into p segments according to the section function of the whole structure, as shown in Figure 4. en, the modal functions of each section could be established by using equations (8), (9), and (12), respectively. erefore, they could be rewritten as follows:…”

“…In order to solve this problem and analyze its vibration characteristics, various analytical methods are proposed. Basing on the Galerkin method, Pielorz and Nadolski [9] developed a modal analysis method for a thin beam with a variable section. Laura et al [10] optimized the Rayleigh-Ritz method to obtain natural frequency of the beam with constant width and hyperbolic thickness.…”

Cylindrical Motion of Infinite Cylindrical Shells (Beamshells)

“…Equations for the large, classical, flexural motion of straight, cantilevered beams of variable cross section have been presented by Pielorz & Nadolski (1986) and solved approximately by Galerkin's method for a linearly tapered beam, initially deformed by a constant end load and then released from rest.…”

Nonlinear free vibration of thin functionally graded beams using the finite element method

“…Descriptions of the concept and the fabrication methods of FGMs can be found in textbooks by Miyamoto et al (1999) and Sobczak and Drenchev (2009). Nonlinear free vibrations of homogeneous beams have been analyzed using the approximate continuum approach (Woinowsky-Krieger, 1950;Burgreen, 1951;Srinivasan, 1965;Wagner, 1965;Evensen, 1968;Raju et al, 1976;Prathap and Varadan, 1978;Pielorz and Nadolski, 1986;Lewandowski, 1987;Singh et al, 1990b;Pillai and Rao, 1992) and by using the finite element method (FEM) (Mei, 1973a(Mei, ,b, 1986Rao et al, 1976a,b;Bhashyam and Prathap, 1980;Reddy and Singh, 1981;Sarma and Varadan, 1982, 1983, 1985Dumir and Bhaskar, 1988;Sarma et al, 1988;Singh et al, 1990a;Marur and Prathap, 2005;Gupta et al, 2009). In addition, Ray and Bert (1969) conducted an experiment to verify some analytical methods for nonlinear free vibrations of homogeneous beams, and Marur (2001) presented a review of the work published on this topic from the 1950s to the 1990s.…”