A numerical analysis of large amplitude beam vibration under different boundary conditions and excitation patterns

Abstract

Large amplitude forced vibration behavior of thin beams under harmonic excitation is studied, incorporating the effect of geometric nonlinearity. Large amplitude free vibration analysis of the same system is also carried out to obtain the backbone curves. When a very small loading is considered, the system response becomes almost identical to the backbone curve, which confirms the validity of the large amplitude study. The sets of governing equations for both forced and free vibration problems are obtained using Hamilton’s principle, whereas the same for the static problem are obtained by applying the variational form of the energy principle. The solution of the static problem provides system stiffness as a function of deflection and it becomes an input to the corresponding free vibration problem. The results are presented in the nondimensional frequency–amplitude plane in the form of response curves, and the related backbone curves are shown to represent the complete system behavior. Results for different combinations of classical flexural boundary conditions and loading patterns are reported.

Nonlinear forced vibration analysis of higher order shear-deformable functionally graded microbeam resting on nonlinear elastic foundation based on modified couple stress theory

“…Hence, for a given value of excitation frequency, the peak load amplitude is employed to derive the governing equations using equation (5). 59 The work potential Vudl at peak load amplitude is thus derived using the following …”

“…The frequency-response curves are numerically generated using a combination of two different nonlinear equation solving algorithms 59 : iterative substitution method with successive relaxation 60 and Broyden’s method. 61 The Iterative substitution method is computationally efficient but it has a limited capability as it cannot provide the solution beyond the point of change in curvature in the in-phase region.…”

Non-linear forced vibration analysis of higher-order shear-deformable functionally graded material beam in thermal environment subjected to harmonic excitation and resting on non-linear elastic foundation

“…Now, for a given frequency of excitation, the peak load-amplitude is considered to derive the governing equation employing equation (6). 25 Hence, the work potential V at peak load-amplitude is given by…”

“…The frequency-response curves are generated by employing a combination of two different solution algorithms for non-linear equations: 25 one is the substitution method with successive relaxation 51 and the other one is Broyden’s algorithm. 52 The first one is computationally efficient but has limited capability.…”

“…Large amplitude forced vibration behaviors of homogeneous beams considering geometric nonlinearity are reported in the literature. [18][19][20][21][22][23][24][25] Based on Kelvin viscoelastic model and employing Galerkin method, Zhang et al 26 studied the steady-state periodic response of an axially moving viscoelastic beam. Forced vibration analysis of FGM beams under classical boundary conditions were reported in the literature.…”