Seher Durmaz

High-Order Energy Balance Method to Nonlinear Oscillators

Energy balance method (EBM) is extended for high-order nonlinear oscillators. To illustrate the effectiveness of the method, a cubic-quintic Duffing oscillator was chosen. The maximum relative errors of the frequencies of the oscillator read 1.25% and 0.6% for the first- and second-order approximation, respectively. The third-order approximation has an accuracy as high as 0.008%. Excellent agreement of the approximated frequencies and periodic solutions with the exact ones is demonstrated for several values of parameters of the oscillator.

High Order Hamiltonian Approach to Nonlinear Oscillators

This paper suggests a high order Hamiltonian approach to nonlinear oscillators. Two examples are given to illustrate the effectiveness of the method. The maximum relative error of Duffing oscillator obtained by Hes Hamiltonian approach is 2.22% ; while the present approach leads to 0.19% and 0.009% accuracy, respectively, for the second and third approximations.

Approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire

In this paper, the approximate solutions of the mathematical model of a mass attached to a stretched elastic wire are presented. At the beginning of the study, the equation of motion is derived in a detailed way. Hes max-min approach, Hes frequency-amplitude method and the parameter-expansion method are implemented to solve the established model. The numerical results are further compared with the approximate analytical solutions for both a small and large amplitude of oscillations, and a very good agreement is observed. The relative errors are computed to illustrate the strength of agreement between the numerical and approximate analytical results.

He's Variational Approach to Multiple Coupled Nonlinear Oscillators

In this paper, Hes variational approach is extended to investigate coupled nonlinear multi-degree-offreedom (mdof) oscillatory systems. A mass-spring system with nonlinear stiffness is considered as an example and approximate amplitude-frequency relations for 2-DOF and 3-DOF nonlinear systems are obtained.

Approximate solutions for nonlinear transverse vibrations of elastically restrained tapered beams

This paper introduces the approximate solutions of the mathematical model of an elastically restrained tapered beam. At the beginning of the study, the equation of motion is derived in a detailed way. The frequency–amplitude relation is deduced and solved numerically. The nonlinear natural frequencies for the transverse vibrations of an elastically restrained tapered beam are provided using Mathematica software. The max–min approach, the frequency–amplitude method and the parameter-expansion method are applied in order to obtain an approximated solution. The approximate analytical results are further compared with the numerical results for both small and large amplitude oscillations, and very good agreement is observed.

Aeroelastic Response of an Aircraft Wing Modeled as a Thin-Walled Composite Beam

This study reports static and dynamic aeroelastic analyses of an aircraft wing in an incompressible flow. A swept thin-walled composite beam with a biconvex cross-section is used as the structural model that incorporates a number of non-classical effects such as material anisotropy, transverse shear deformation and warping restraint. A symmetric lay-up configuration i.e. circumferentially asymmetric stiffness (CAS) is further adapted to this model to generate the coupled motion of flap-wise bending-torsion-transverse shear. For this beam model, the unsteady aerodynamic loads are expressed using Wagners function in the time-domain as well as using Theodorsen function in the frequency-domain. The divergence and the flutter speeds are evaluated for several ply angles and the aeroelastic response of an aircraft wing exposed to gust load is examined. The effects of transverse shear, fiber-orientation and sweep angle on the aeroelastic instabilities and the aeroelastic response of the beam are further discussed.Copyright

Geometrically Nonlinear Vibration Analysis of Thin-Walled Composite Beams

In this study, accounting for large displacements a geometrically nonlinear theory, which is valid for laminated thin-walled composite beams of open and closed cross sections, is developed. The beam model incorporates a number of non-classical effects such as material anisotropy, transverse shear deformation and warping restraint. Moreover, the directionality property of thin-walled composite beams produces a wide range of elastic couplings. In this respect, symmetric lay-up configuration i.e. Circumferentially Asymmetric Stiffness (CAS) is adapted to this model to generate coupled motion of flapwise bending-torsion-flapwise transverse shear. Initially, free vibration analyses are carried out for the linear model of the shearable and the non-shearable thin-walled composite beams. Similar to the linear model, the displacement-based nonlinear equations are derived by the variational formulation, considering the geometric non-linearity in the von Karman sense. Finally, the static and the dynamic analyses for the nonlinear beam model are carried out addressing the effects of transverse shear, fiber-orientation and sweep angle on the nonlinear frequencies and the static response of the beam.Copyright

Limit Cycle Oscillations of Swept-back Wings In an Incompressible Flow

This study focuses on the limit cycle oscillations (LCO) of cantilever swept-back wings containing a cubic nonlinearity in an incompressible flow. The governing aeroelastic equations of two degrees-of-freedom swept wings are derived through applying the strip theory and unsteady aerodynamics. In order to apply strip theory, mode shapes of the cantilever beam are used. The harmonic balance method is used to calculate the frequencies of LCOs. Linear flutter analyses are conducted for several values of sweep angles to obtain the flutter boundaries. Nomenclature Λ a = non-dimensional distance from wing section mid-chord to elastic axis Λ b = wing section semi-chord normal to the elastic axis h C , α C = damping coefficients in plunge and pitch EI = bending stiffness α I = mass moment of inertia about elastic axis α ξ F F , = structural nonlinearity in plunge and plunge GJ = torsional stiffness l = span length of wing a L = lift force about elastic axis LCO = limit cycle oscillations m = wing mass per unit length a M = pitching moment about elastic axis α r = radius of gyration about elastic axis U = free-stream velocity Λ U = non-dimensional velocity L U Λ = non-dimensional linear flutter speed α x = non-dimensional distance from elastic axis to center of mass α ξ β β , = constants in nonlinear terms α ξ γ γ , = constants in nonlinear terms α ξ ζ ζ , = viscous damping ratios in plunge and pitch ( ) τ φ = Wagner’s function h φ = first plunge mode shape function α φ = first pitch mode shape function h ω , α ω = natural frequencies in plunge and pitch

The nonlinear transverse vibrations of narrow microbeams by energy balance method based on collocation method

In this paper, the microbeam, which incorporates with the mid-plane stretching effect and distributed electrostatic force, is considered. After the beam model is introduced briefly, the governing equation of motion of the microbeam is solved by the energy balance method (EBM). The first and the second order approximate periodic solutions are obtained and plotted for various values of the parameters of the equation. The numerical solutions (Runge-Kutta) are also computed to illustrate good agreement between both results.

Free vibration analysis of rotating thin-walled composite beams

In this paper, both analytical and finite element formulations of a rotating thin-walled composite beam are derived for the flapwise bending, chordwise bending and torsional displacements. In the analytical part, derivation of both strain and kinetic energy expressions are made and the equations of motion are obtained by applying the Hamiltons principle. The equations of motion are solved by applying the Differential Transform Method (DTM). In the finite element part, structural matrices are obtained by using the derived energy expressions and a finite element code is written to calculate the natural frequencies. The analysis is carried out for anti-symmetric lay-up configuration also referred as Circumferentially Uniform Stiffness (CUS). Effect of the ply orientation on the natural frequencies is investigated. The natural frequencies are validated by making comparisons with the results in literature. Consequently, it is observed that there is a good agreement between the results.