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This work presents a method to find the exact solutions for the free vibration analysis of a delaminated beam based on the Timoshenko type with different boundary conditions. The solutions are obtained by the method of Lagrange multipliers in which the free vibration problem is posed as a constrained variational problem. The Legendre orthogonal polynomials are used as the beam eigenfunctions. Natural frequencies and mode shapes of various Timoshenko beams are presented to demonstrate the efficiency of the methodology  

The large amplitude free vibration of an unsymmetrically laminated composite beam (LCB) on a nonlinear elastic foundation subjected to axial load has been studied. The equation of motion for the axial and transverse deformations of a geometrically nonlinear LCB is derived. Using the Ritz method, the governing equation is reduced to a time-dependent Duffing equation with quadratic and cubic nonlinearities. The homotopy analysis method (HAM) is used to obtain exact expressions for the dynamic response of the LCB. This study shows that the third-order approximation of the HAM leads to highly accurate solutions that are valid for a wide range of vibration amplitudes. The effects of different... 

A closed form solution is presented in this paper to study the dynamics of a composite beam with a single delamination under the action of a moving constant force. The delaminated beam is divided into four interconnected beams using the delamination limits as their boundaries. Governing motion equations are derived in which the differential stretching and the bending-extension coupling are considered. The method of modal analysis is adopted to derive analytically the dynamic response of each beam. The obtained results for the free vibrations of delaminated beam are verified against reported similar results in the literature. Moreover, the maximum dynamic response of such a beam is compared... 

The dynamic response of a delaminated composite beam under the motion of an oscillatory mass moving with a constant velocity has been studied. The delaminated composite beam is modeled as four interconnected sub-beams using the delamination limits as their boundaries. The constrained model is used to model the delamination region. The continuity and equilibrium conditions are forced to be satisfied between the adjoining beams. A set of derived governing differential equations along with those obtained by imposing boundary conditions are simultaneously solved in a closed form manner. The results for the response of the delaminated beam were compared with those of the intact beam. Furthermore,... 

This paper presents the dynamic response of a delaminated composite beam under the action of a moving oscillating mass. In this analysis the Poisson's effect is considered for the first time. Moreover, the effects of rotary inertia and shear deformation are incorporated. In our modeling linear springs are used between delaminated surfaces to simulate the dynamic interaction between sub-beams. To solve the governing differential equations of motion using modal expansion series, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes necessary for forced vibration analysis. The obtained results for the free and forced vibrations of beams are... 

This paper presents dynamic response of a delaminated composite beam under the action of moving oscillatory mass. The Poisson's effect, shear deformation and rotary inertia have been considered in this analysis. We have used the constrained mode model to simulate the behavior between the delaminated surfaces. Based on this model, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes for the delaminated beam. Then, the forced response is determined by employing the modal series expansion technique. The obtained results for the free and forced vibrations of beams are verified against reported similar results in the literature. Moreover, the... 

The dynamic response analysis of a delaminated composite beam with a general lay-up traversed under an arbitrary moving/non-moving force is presented. By employing the energy method and introducing a new finite element, the global mass and stiffness matrices for a Laminated Composite Beam (LCB) of Timoshenko type are derived in which the material couplings (bending-tension, bending-twist, and tension-twist couplings) with the Poisson's effect are considered. In deriving the governing equation the non-penetration condition is imposed by employing the method of Lagrange multipliers. Out of a self-developed finite element program, the natural frequencies and time response of such LCB are... 

This paper deals with the dynamic analysis of a delaminated composite beam under the action of moving oscillatory mass. The beam is analyzed as four interconnected sub-beams using the delamination limits as their boundaries. The constrained model is used to model the delamination region. The continuity and equilibrium conditions are satisfied between the adjoining beams. The beam response variation due to the delamination with respect to the intact beam has been investigated. Furthermore, the possible separation of the moving oscillator from the beam during the course of the motion is investigated by monitoring the contact force between the oscillator and the beam. The effect of the... 

In this paper, free vibration analysis of cross-ply layered composite beams (LCB) with finite length and rectangular cross-section rested on an elastic foundation is investigated by finite element method. Based on the Timoshenko beam theory which includes the shear deformation and rotary inertia, the stiffness and mass matrices of a LCB are obtained using the energy method. Then, the natural frequencies are calculated by employing eigenvalue technique. The obtained results are verified against existing data in the literatures for a LCB with no foundation and uniform cross-section. Good agreements are observed between these cases. In the same way, the natural frequencies of a specific case,... 

In this paper, the free vibration analysis of generally laminated composite beam (LCB) based on Timoshenko beam theory are presented using the method of Lagrange multipliers where in the free vibration problem is posed as a constrained variational problem. The effect of material couplings (bending-tension, bending-twist, and tension-twist couplings) with the effects of shear deformation, rotary inertia and Poisson's effect are taken into account. Analytical expression for the natural frequencies and mode shapes are presented. The calculated natural frequencies are verified against some available results in the literature and very good agreement is observed. Furthermore, the effects of some... 

A rather new semi-analytical method towards investigating the free vibration analysis of generally laminated composite beam (LCB) with a delamination is presented. For the first time the combined effects of material couplings (bending-tension, bending-twist, and tension-twist couplings) with the effects of shear deformation, rotary inertia and Poisson's effect are taken into account. The semi-analytical solution for the natural frequencies and mode shapes are presented by incorporating the constraint conditions using the method of Lagrange multipliers. To verify the validity and the accuracy of the obtained results, they were compared with the results from other available references. Very... 

The objective of this study is to present a brief survey on the geometrically nonlinear free vibrations of the Bernoulli-Euler, the Rayleigh, shear, and the Timoshenko beams with simple end conditions using the Homotopy Analysis Method (HAM). Expressions for the natural frequencies, the transverse deflection, postbuckling load-deflection relation to, and critical buckling load are presented. The results of nonlinear analysis are validated with the published results, and excellent agreement is observed. The effects of some parameters, such as slender ratio, the rotary inertia, and the shear deformation, are examined as other parameters are fixed. Copyright  

In this study, the dynamic response of the laminated composite beam with arbitrary lay-ups has been investigated within the framework of the third-order shear deformation theory using the finite element method. A new three-nodded finite element compliant with the theory is introduced next. To deal with the dynamic contact between the delaminated segments, unilateral contact constraints are employed in conjunction with Lagrange multiplier method. Furthermore, the Poisson's effect is incorporated in the formulation of the beam constitutive equation. Also, the higher-order inertia effects and material couplings (flexure-tensile, flexure-twist and tensile-twist couplings) are considered in the... 

The free vibration analysis of generally laminated composite beam (LCB) with a delamination is presented using the finite element method (FEM). The effect of material couplings (bending-tension, bending-twist, and tension-twist couplings) with the effects of shear deformation, rotary inertia, and Poisson's effect are taken into account. To verify the validity and the accuracy of this study, the numerical solutions are presented and compared with the results from available references and very good agreement observed. Furthermore, the effects of some parameters such as slenderness ratio, the rotary inertia, the shear deformation, material anisotropy, ply configuration, and delamination... 

This paper deals with the dynamic analysis of a delaminated Bernoulli-Euler beam with constrained model under the action of an external moving force. The beam is analyzed as four interconnected sub-beams using the delamination as their boundaries. The continuity and compatibility conditions are satisfied between adjoining beams. An analytical solution is presented in a basis of the series expansion of the unknown deflection. The delaminated beam response differences with the healthy beam have been investigated. Effect of the parameters like the moving speed of the force and the size, depth and spanwise location of the delamination on the dynamic response of the beam has been studied.... 

Human running can be stabilized in a wide range of speeds by automatically adjusting muscular properties of leg and torso. It is known that fast locomotion dynamics can be approximated by a spring loaded inverted pendulum (SLIP) system, in which leg is replaced by a single spring connecting body mass to ground. Taking advantage of the inherent stability of SLIP model, a hybrid control strategy is developed that guarantees a stable biped locomotion in sagittal plane. In the presented approach, nonlinear control methods are applied to synchronize the biped dynamics and the spring-mass dynamics. As the biped center of mass follows the mass of the mass-spring model, the whole biped performs a...