Hakan Boyacı

Non-linear vibrations and stability of an axially moving beam with time-dependent velocity

Non-linear vibrations of an axially moving beam are investigated. The non-linearity is introduced by including stretching effect of the beam. The beam is moving with a time-dependent velocity, namely a harmonically varying velocity about a constant mean velocity. Approximate solutions are sought using the method of multiple scales. Depending on the variation of velocity, three distinct cases arise: (i) frequency away from zero or two times the natural frequency, (ii) frequency close to zero, (iii) frequency close to two times the natural frequency. Amplitude-dependent non-linear frequencies are derived. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.

NON-LINEAR VIBRATIONS OF A SIMPLE–SIMPLE BEAM WITH A NON-IDEAL SUPPORT IN BETWEEN

Abstract A simply supported Euler–Bernoulli beam with an intermediate support is considered. Non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. An approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Ideal and non-ideal frequencies as well as frequency-response curves are contrasted.

Generation of root finding algorithms via perturbation theory and some formulas

Abstract Perturbation theory is systematically used to generate root finding algorithms. Depending on the number of correction terms in the perturbation expansion and the number of Taylor expansion terms, different root finding formulas can be generated. The way of separating the resulting equations after the perturbation expansion alters the root-finding formulas also. Well known cases such as Newton–Raphson and its second correction, namely the Householder’s iteration, are derived as examples. Moreover, higher order algorithms which may or may not be the corrections of well known formulas are derived. The formulas are contrasted with each other as well as with some new algorithms obtained by modified Adomian Decomposition Method proposed in Ref. [S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation 145 (2003) 887–893].

Effects of Non-ideal Boundary Conditions on Vibrations of Microbeams

Effects of non-ideal boundary conditions on the vibrations of microbeams are investigated. Stretching effect as well as axial force is included along with the non-ideal boundary conditions. The Method of Multiple Time Scales (a perturbation technique) is employed to solve the non-dimensional equation of motion for subharmonic and superharmonic resonance cases. The frequencies and mode shapes obtained are compared with the ideal boundary conditions case and the differences between them are contrasted on frequency response curves.

Nonlinear vibrations and stability analysis of axially moving strings having nonideal mid-support conditions

In this study, nonlinear vibrations of an axially moving string are investigated. The main difference of this study from other studies is that there is a nonideal support between the opposite sides, which allows small displacements. Nonlinear equations of motion and boundary conditions are derived using Hamilton’s principle. Equations of motion and boundary conditions are converted to nondimensional form. Thus, the equations become independent from geometry and material properties. The method of multiple scales, a perturbation technique, is used. A harmonically varying velocity function is chosen for modeling the axial movement. String as a continuous medium is investigated in two regions. Vibrations are investigated for three different cases of the excitation frequency Ω. Stability analysis is carried out for these three cases, and stability boundaries are determined for the principle parametric resonance case. Thus, differences between ideal and nonideal boundary conditions are investigated.

Vibrations of stretched damped beams under non-ideal boundary conditions

A simply supported damped Euler-Bernoulli beam with immovable end conditions are considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections and moments. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.

A new perturbation technique in solution of nonlinear differential equations by using variable transformation

A perturbation algorithm using a new variable transformation is introduced. This transformation enables control of the independent variable of the problem. The problems are solved with new transformation: Classical Duffing equation with cubic nonlinear term and a singular perturbation problem. Results of multiple scales, Lindstedt Poincare method, new method and numerical solutions are contrasted.

BEAM VIBRATIONS WITH NON-IDEAL BOUNDARY CONDITIONS

A simply supported damped Euler-Bernoulli beam with immovable end conditions is considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections and moments. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.