Mehmet Pakdemirli

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.

Approximate analytical solutions for the flow of a third-grade fluid in a pipe

Abstract The flow of a third-grade fluid in a pipe with heat transfer is considered. Constant viscosity, Reynolds model viscosity and Vogels model viscosity cases are treated separately. Approximate analytical solutions are presented for each case using perturbations. The criteria for which the solutions are valid are determined for the dimensionless parameters involved. The analytical solutions are contrasted with the finite difference solutions given in Massoudi and Christie (Int. J. Non-Linear Mech. 30 (1995) 687–699) and within admissible parameter range, a close match is achieved.

Lie group analysis of creeping flow of a second grade fluid

The two-dimensional equations of motion for the slowly flowing second grade fluid are written in cartesian coordinates neglecting the inertial terms. By employing Lie group analysis, the symmetries of the equations are calculated. The Lie algebra consists of four finite parameter Lie group transformations, one being the scaling symmetry and the others being translations. Two different types of solutions are found using the symmetries. Using the translations in x and y coordinates, an exponential type of exact solution is constructed. For the scaling symmetry, the outcoming ordinary differential equations are more involved and only a series type of approximate solution is presented. Finally, some boundary value problems are discussed.

New perturbation-iteration solutions for Bratu-type equations

Perturbation-iteration theory is systematically generated for both linear and nonlinear second-order differential equations and applied to Bratu-type equations. Different perturbation-iteration algorithms depending upon the number of Taylor expansion terms are proposed. Using the iteration formulas derived using different perturbation-iteration algorithms, new solutions of Bratu-type equations are obtained. Solutions constructed using different perturbation-iteration algorithms are contrasted with each other as well as with numerical solutions. It is found that algorithms with more Taylor series expansion terms yield more accurate results.

Similarity Transformations for Partial Differential Equations

The importance of similarity transformations and their applications to partial differential equations is discussed. The theory has been presented in a simple manner so that it would be beneficial at the undergraduate level. Special group transformations useful for producing similarity solutions are investigated. Scaling, translation, and the spiral group of transformations are applied to well-known problems in mathematical physics, such as the boundary layer equations, the wave equation, and the heat conduction equation. Finally, a new transformation including the mentioned transformations as its special cases is also proposed.

Analysis of One-to-One Autoparametric Resonances in Cables — Discretization vs. Direct Treatment

We discuss solution methods for nonlinear vibrations of cables having small initial sag-to-span ratios. One-to-one internal resonances between the in-plane and out-of-plane modes as well as primary resonances of the in-plane mode are considered. Approximate solutions are obtained by two different approaches. In the first approach, the method of multiple scales is applied directly to the governing partial-differential equations and boundary conditions. In the second approach, the equations are first discretized, and then the method of multiple scales is applied to the resulting ordinary-differential equations. It is shown that treatment of the discretized system is inaccurate compared to direct treatment of the partial-differential system. Discrepancies between the two solutions appear even at the first level of approximation. Stability analyses of the amplitude and phase modulation equations for both methods are also performed.

A comparison of two perturbation methods for vibrations of systems with quadratic and cubic nonlinearities

Discreatizing the nonlinear partial differential equations of vibration problems and then solving the resulting ordinary differential equations by perturbation techniques is a quite common method. The disereatization process simplifies the equations much and the ordinary differential equations are easier to handle. However the process may lead to inaccurate results in the case of quadratic and cubic nonlinearities. The issue was first discussed by Nayfeh, Nayfeh and Mook [1]. They considered the nonlinear dynamic response of a relief valve and showed the discreapancies between the two methods for this specific problem. Pakdemirli, Nayfeh and Nayfeh [2] presented the discreapancies between the methods for a nonlinear cable vibration problem. A summary of the related work done up to now has been presented by Nayfeh, Nayfeh and Pakdemirli [3].

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.

The boundary layer equations of third-grade fluids

Abstract Steady, two-dimensional, incompressible boundary layer equations for a fluid of grade three are derived using a special coordinate system. For the inviscid flow around an arbitrary object, the streamlines are the ψ-coordinates and velocity potential lines are the ψ-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, are then shown to be independent of the body shape immersed into the flow. In deriving the boundary layer equations, method of matched asymptotic expansion is used. Then, it is shown that the equations do not have similarity solutions. Finally, the shear stress on the boundary for the coordinate system is also calculated.

Nonlinear Vibrations of a Beam-Spring-Mass System

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.