M.N. Hamdan

Bio: M.N. Hamdan is an academic researcher from King Fahd University of Petroleum and Minerals. The author has contributed to research in topics: Harmonic balance & Curvature. The author has an hindex of 8, co-authored 8 publications receiving 168 citations.

Geometrically Non-Linear Dynamic Model of a Rotating Flexible Arm

Abstract: In this paper, a mathematical model for a rotating flexible arm undergoing large planar flexural deformations is developed. The position of a typical material point along the span of the arm is described by using the inertial reference frame via a transformation matrix from the body co-ordinate system which is attached to the root of the rotating arm. The condition of inextensibility is employed to relate the axial and transverse deflections of the material point. The position and velocity vectors obtained, after imposing the inextensibility conditions, are used in the kinetic energy expression while the exact curvature is used in the potential energy. The Lagrangian dynamics in conjunction with the assumed modes method is utilized to derive directly the equivalent temporal equations of motion. The resulting non-linear model is discussed, simulated and the result of simulation are presented and compared to those obtained from the linear theory for different arm parameters.

Comparison of analytical techniques for nonlinear vibrations of a parametrically excited cantilever

Abstract: This paper is concerned with second-order approximations to the steady-state principal parametric resonance response of a vertically mounted flexible cantilever beam subjected to a vertical harmonic base motion. The unimodal form of the nonlinear equation describing the in-plane large amplitude parametric response of the beam, derived in Krishnamurthy (Ph.D. Thesis, Department of Mechanical Engineering, Washington State University, 1986) based on the previous analysis in Crespo da Silva and Glynn (Journal of Structural Mechanics 1978; 6:437–48), is analysed using the harmonic balance (HB) and the perturbation method of multiple time scales (MMS). Single term HB, two terms HB, and second-order MMS with reconstitution version I (Nayfeh and Sanchez, Journal of Sound and Vibration 1989; 24:483–97) and version II (Rahman and Burton, Journal of Sound and Vibration 1989; 133:369–79) approximations to the steady-sate frequency–amplitude curves of the principal parametric resonance for each of the first four natural modes of the cantilever beam are compared with each other and with those obtained by numerically integrating the unimodal equation of motion. The time transformation T= Ω t is used in obtaining these approximations; also detuning is used in obtaining the square of the forcing MMS approximations. The obtained results show that, for the problem under consideration, the MMS version II is, in comparison with MMS version I, simpler to apply and leads to qualitatively more accurate second-order results. These results, however, show that the MMS version II tends to produce appreciable over corrections to the first-order results and may breakdown at relatively low response amplitudes, whereas the two terms HB solutions tend to improve the first-order results and lead to fairly accurate results even for relatively large response amplitudes.

On the steady state response of oscillators with static and inertia non-linearities

Abstract: The concern of this work is the steady state periodic response having the same period as the excitation of strongly non-linear oscillatorsu+δu+mu+ϵ1u2u+ϵ1uu2+ϵ2u3=P cos Ωt, wherem=1, 0 or −1, ϵ1and ϵ1are positive parameters which may be arbitrarily large. Single-mode and two-mode harmonic balance (HB) approximations, and second order perturbation-multiple time scales (MMS) with reconstitution version I and version II approximations to the steady state amplitude frequency response curves are compared, for the casem=1 with each other, and with those obtained by numerically integrating the equation of motion. The transformation of timeT=Ωtand detuning in the square of forcing frequency are used in the MMS with reconstitution version I and version II. The objective here is to assess the accuracy of these approximate solutions in predicting the systems response over some range of system parameters by examining their ability or failure in establishing the correct qualitative behavior of the actual (numerical) solution. The casesm=0 andm=1, are studied for selected range of system parameter, using the single and two modes harmonic balance method and compared to those obtained numerically. It was shown that MMS version II, in addition to being appreciably simpler than MMS version I, leads to more accurate qualitative and quantitative results even when the non-linearity is not necessarily small.

Non-linear free vibrations of a rotating flexible arm

Abstract: The non-linear, moderately large amplitude flexural free vibrations of an arm clamped with a setting angle to a rigid rotating hub are studied. The shear deformation and rotary inertia effects are assumed to be negligible, but account is taken of axial inertia, non-linear curvature and the inextensibility condition. The Lagrangian approach in conjunction with the assumed modes method, assuming constant hub rotation speed, is used in a consistent manner to obtain the third order non-linear uni-modal temporal problem. Because of the strength of the non-linearities in the temporal problem, which includes elastic and inertial geometric stiffening as well as inertial softening terms, a time transformation method is employed to obtain an approximate solution to the frequency–amplitude relation of arm free oscillation. Results in non-dimensional form are presented graphically, for the effect of hub rotation speed, blade setting angle, and hub radius on the variation of the natural frequency with vibration amplitude.

Bifurcations and chaos of an immersed cantilever beam in a fluid and carrying an intermediate mass

Abstract: The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam–mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000 Shock and Vibration 7 , 179–194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration 244 , 453–479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.

A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic

Abstract: A vibration isolator consisting of a vertical linear spring and two nonlinear pre-stressed oblique springs is considered in this paper. The system has both geometrical and physical nonlinearity. Firstly, a static analysis is carried out. The softening parameter leading to quasi-zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry, pre-stress and the stiffness of the springs. The optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone. It also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position. For these values, a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities. It includes considering the possibilities of the appearance of period-doubling bifurcation and its development into chaotic motion. For this purpose, approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots, Poincare maps and Lyapunov exponents are used. Finally, the frequency at which the first period-doubling bifurcation appears is found and the effect of damping on this frequency determined.

Trajectory planning for residual vibration suppression of a two-link rigid-flexible manipulator considering large deformation

Abstract: In this paper an optimal trajectory planning technique for suppressing residual vibrations in two-link rigid-flexible manipulators is proposed. In order to obtain an accurate mathematical model, the flexible link is modeled by taking the axial displacement and nonlinear curvature arising from large bending deformation into consideration. The equations of motion of the manipulator are derived using the Lagrangian approach and the assumed modes method. For the trajectory planning, the joint angle of the flexible link is expressed as a cubic spline function, and then the particle swarm optimization algorithm is used to determine the optimal trajectory. The optimal trajectory thus obtained satisfies the minimum vibration condition. By performing numerical simulations, the effectiveness of the proposed trajectory planning technique is verified.

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

Abstract: This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.

Dynamics of a radially rotating beam with impact. Part 1. Theoretical and computational model

Abstract: The model uses a momentum balance method and a coefficient of restitution, and enables one to predict the rigid body motion as well as the elastic motion before and after impact

Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam

Abstract: The aim of this paper is to investigate the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. First, the extended Melnikov method for studying the Shilnikov-type multi-pulse homoclinic orbits and chaos in high-dimensional nonlinear systems is briefly introduced in the theoretical frame. Then, this method is utilized to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam. How to employ this method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example. Finally, the results of numerical simulation are given and also show that the Shilnikov-type multi-pulse chaotic motions can occur for the nonlinear non-planar oscillations of the cantilever beam, which verifies the analytical prediction.