Dynamic model of large amplitude vibration of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

Abstract

In this paper, a mathematical model of large amplitude vibration of a uniform cantilever beam arising in the structural engineering is proposed. Two efficient and easy mathematical techniques called variational iteration method and He's variational approach are used to solve the governing differential equation of motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th order. An excellent agreement of the approximate frequencies and periodic solutions with the numerical results and published results has been demonstrated. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering and physics. The Structural engineering theory is based upon physical laws and empirical knowledge of the structural performance of different landscapes and materials. Many engineering structures can be modelled as a slender, flexible cantilever beam carrying a lumped mass with rotary inertia at an intermediate point along its span; hence they experience large-amplitude vibration (Wu, 2003;Herisanu and Marinca, 2010;Cveticanin and Kovacic, 2007;Hamdan and Shabaneh, 1997;Hamdan and Dado, 1997). Key wordsIn general, such problems are not amenable to exact treatment and approximate techniques must be resorted to. So, many new techniques have appeared in the open literature such as: variational iteration method (He, 2007a;Barari et al., 2011), homotopy perturbation method Saravi et al., 2013), homotopy analysis method (Liao, 2003;Khan et al., 2012)and some other methods Ghasempoor et al., 2102;Akbarzade and Khan, 2012;Alinia et al., 2011;Ganji, 2012;Torabi et al., 2012;

A multimode approach to geometrically non-linear forced vibration of beams carrying point masses

“…Nikkar et al tackled the large vibration amplitudes of a uniform cantilever beam with an intermediate attached mass using the variational iteration method and He's variational approach to solve the equation of motion. The results showed a good agreement with those obtained by the Range-Kutta fourth order method [41]. Souayeh and Kacem developed a model integrating both geometric and electrostatic nonlinearities for the computation of nonlinear vibrations of electrostatically actuated carbon nanotube-based mass sensors.…”

Nonlinear Vibration of an Elastic Soft String: Large Amplitude and Large Curvature

“…Equation (17) is strong nonlinear system due to the nonlinear terms q 3 and q 5 which are far more large than the linear term . Although He's variational method [6,12,15,19], harmonic balance method [18], and Homotopy analysis method [16] are used widely in the strong nonlinear system, the modified CNFM method [29,[34][35][36] gives more accurate results based on the experience of the authors. Next, we use the modified CNFM approach to solve (17).…”

“…In their model, the von Karman type nonlinear strain-displacement relationship is employed, and the effects of transverse shear deformation are included based upon the Timoshenko beam theory. Similarly, to further seek the nonlinear frequencies, Gunda and Gupta [11] investigated the vibration of a composite beam, Nikkar and Bagheri [12] explored the cantilever beam with an intermediate lumped mass, and Yu and Wu et al [13] studied the beam with immovable spring-hinged ends. More related works include that Raju and Rao [14] formulated the nonlinear vibration of the beam using multiterm admissible 2 Mathematical Problems in Engineering functions for the first mode.…”