Subtangential Forces Research Articles

Electromechanical stability analysis of smart double-nanobeam systems

In the present article, the electromechanical stability analysis of double-nanobeam-systems (DNBS) under subtangential forces is studied based on Eringen’s nonlocal elasticity theory. The two cantilever nanobeams are assumed parallel and coupled by an enclosing visco-Pasternak medium. Furthermore, each nanobeam is surrounded by two piezoelectric layers and the system resting on Winkler elastic medium. The governing equations of motion and associated boundary conditions are derived by the extended Hamilton’s principle. Applying the extended Galerkin’s approach, the partial differential equations are transformed to ordinary differential equations. The effects of the nonlocal parameter, piezoelectric voltage, aspect ratios and surface effects on the vibration and stability characteristics of the DNBS are explained in detail. It is observed that the critical flutter and divergence loads predicted by the classical continuum theory have more value than the nonlocal continuum theory. In order to validate the accuracy and applicability of the proposed model, the numerical results are confirmed by comparing the results with those obtained in the literature.

Interaction effect of cracks on flutter and divergence instabilities of cracked beams under subtangential forces

The dynamic stability of cracked beams under conservative and non-conservative forces and for various boundary conditions is investigated. In order to extend the formulation of a previous paper by the first two authors, here doubly cracked Euler–Bernoulli beams subjected to triangularly distributed subtangential forces are considered. Cracked sections are modelled through the theory of fracture mechanics and involve a line-spring stiffness matrix. The finite element method (FEM) is used to perform numerical computations. The stability maps, obtained from the eigenvalue analysis, define the divergence and flutter domains. The proposed procedure can also tackle general multi-cracked beams.

Divergence and Flutter Instability of Damped Laminated Beams Subjected to a Triangular Distribution of Nonconservative Forces

The divergence and flutter instability behavior of the damped laminated beams subjected to a triangular distribution of subtangential forces are investigated based on the finite element model. The formal engineering approach of the mechanics for the laminated beam theory is presented and the extended Hamilton's principle is employed to obtain the mass, damping, elastic stiffness, geometric stiffness matrices, and the load correction stiffness matrix due to the nonconservative forces, respectively. The methods for the evaluation of the divergence and flutter loads of the nonconservative systems are briefly introduced in case of considering and neglecting external and internal damping effects. Throughout numerical examples, the influence of various parameters on the dynamic instability behavior of the laminated beams is newly investigated: 1) the variation of the divergence and flutter loads due to the nonconservativeness with respect to the fiber angle change, 2) the influence the boundary condition of beams on the instability region of the divergence-flutter system, 3) the influence of external and internal damping on the flutter load by analysis of complex natural frequencies.

Cracks Interaction Effect on the Dynamic Stability of Beams under Conservative and Nonconservative Forces

This study is an extension of the paper by E. Viola and A. Marzani [1] where the eigenfrequencies and critical loads of a single cracked beam subjected to conservative and nonconservative forceshave been investigated. Here the aim is to analyze the dynamic stability of T cross section beams withmultiple cracks. A doubly cracked Euler-Bernoulli beam subjected to triangularly distributed subtangential forces, which are the combination of axial and tangential forces, is considered. The governingequation of the system is derived via the extended Hamilton’s principle in which the kinetic energy, theelastic potential energy, the conservative work and the nonconservative work are taken into account. Thelocal flexibility matrix for a beam with T cross-section is used to model the cracked section. The resultsshow that for given boundary conditions cracked beams become unstable in the form of either flutter ordivergence depending on the crack parameters, the nonconservativeness of the applied load as well as theinteraction of the two cracks.

Dynamic stability behavior of damped laminated beam subjected to uniformly distributed subtangential forces

Abstract The dynamic stability behavior of damped laminated beam with various boundary conditions subjected to the uniformly distributed subtangential forces is investigated using the finite element formulation. The formal engineering approach of the mechanics for the thin-walled laminated beam based on kinematic assumptions consistent with Vlasov beam theory is used. An extended Hamilton’s principle is employed to obtain the mass-, damping-, elastic stiffness-, geometric stiffness matrices, and the load correction stiffness matrix due to the subtangential forces, respectively. The method for the evaluation of critical values for divergence and flutter of the nonconservative systems is briefly introduced in case of considering and neglecting damping effects. Throughout numerical examples, the influence of various parameters on the dynamic stability behavior of the nonconservative laminated beam is newly investigated: (1) the variation of the divergence and flutter loads due to the nonconservativeness with respect to the fiber orientation, (2) the effect of boundary condition on the instability region of the divergence-flutter system, and (3) the influence of external and internal damping on the flutter load.

Stability Maps of a Cracked Timoshenko Beam Resting on Elastic Soils under Sub-Tangential Forces

The goal of this paper is to generate the stability maps for an elastic cracked beam resting on elastic soils and loaded by a constant distribution of sub-tangential forces. The soil behavior is simulated by a two-parameter Pasternak model. Firstly, the extend version of the Hamilton principle is used to formulate the weak form of the governing equation for the undamaged beam problem. Secondly, a finite element procedure, in which the effect of an open surface crack is computed via the Line-spring model, leads to the discrete governing equation for the cracked beam. Finally, the effect of several parameters on divergence or flutter instability, such as the crack depth and location, the non-conservativeness of the applied load as well as the stiffness of the two soil parameters, is highlighted.

Two approaches to the dynamic analysis of foundation beams subjected to subtangential forces

In this paper the dynamic analysis of a foundation beam on a two parameter elastic soil is illustrated in the presence of subtangential follower force. The differential quadrature method and the integral variational formulation Rayleigh–Ritz method are employed. The limiting subtangential coefficient is calculated, which gives the boundary between static and dynamic instability. The paper ends with some numerical examples which confirm the usefulness of the proposed methods.

Effect of boundary conditions on the stability of beams under conservative and non-conservative forces

This paper, which is an extension of a previous work by Viola et al. (2002), deals with the dynamic stability of beams under a triangularly distributed sub-tangential forces when the effect of an elastically restrained end is taken into account. The sub-tangential forces can be realised by a combination of axial and tangential follower forces, that are conservative and non-conservative forces, respectively. The studied beams become unstable in the form of either flutter or divergence, depending on the degree of non-conservativeness of the distributed sub-tangential forces and the stiffness of the elastically restrained end. A non-conservative parameter is introduced to provide all possible combinations of these forces. Problems of this kind are usually, at least in the first approximation, reduced to the analysis of beams according to the Bernoulli-Euler theory if shear deformability and rotational inertia are negligible. The equation governing the system may be derived from the extended form of Hamilton`s principle. The stability maps will be obtained from the eigenvalue analysis in order to define the divergence and flutter domain. The passage from divergence to flutter is associated with a noticeable lowering of the critical load. A number of particular cases can be immediately recovered.

Crack effect on dynamic stability of beams under conservative and nonconservative forces

The purpose of this paper is to investigate the dynamic stability of beams containing a single crack subjected to conservative and nonconservative forces. The governing equation of the system can be derived from the extended Hamilton’s principle in which the kinetic energy, the elastic potential energy, the conservative work and the nonconservative work must be taken into account. The local flexibility matrix of a beam of a rectangular cross-section with a single edge crack is employed in order to perform numerical analysis. The investigated cracked beams are subjected to triangularly distributed subtangential forces, which are the combination of axial and tangential forces. The studied cracked beams become unstable in the form of either flutter or divergence, depending on crack parameters and on the degree of nonconservativeness of the load, when boundary conditions are fixed.

Dynamic stability of an elastically restrained column subjected to triangulary distributed subtangential forces

The dynamic stability of a column subjected to triangularly distributed subtangential forces is investigated. The column is clamped at one end, and is supported rigidly at an arbitrary location. The column studied becomes unstable in the form of either flutter or divergence, depending on the degree of nonconservativeness of distributed subtangential forces and the location of support. The stability maps, which are produced from eigenvalue analyses, present two different types of instability domains. Through the numerical simulation, the following results are obtained: (i) When the nonconservativeness parameter, α, is less than 0.5, divergence appears only regardless of the existence of the rigid support. (ii) For α>0.5, the instability type changes with the position of a rigid support. (iii) When a rigid support exists, the transition position from flutter to divergence increases as nonconservativeness parameter increases. However, the transition position from divergence to flutter is decreased with increasing nonconservativeness. (iv) Boundary conditions of columns affect greatly both the instability type and the critical distributed subtangential force.

Dynamic stability of Timoshenko columns subjected to subtangential forces

This paper deals with the dynamic stability of cantilevered Timoshenko vertical columns having a tip rigid body and subjected to the action of subtangential forces. Here the tip rigid body assumes a rocket motor. The subtangential force can be realized by a combined action of the dead load of the tip rigid body and the pure follower thrust induced by the rocket motor. It is assumed the rocket motor is a rigid body, not a mass point as it has been assumed so far. It is shown that the tip rigid body does not affect the critical force for α≤0.5, whereas for α>0.5, the tip rigid body parameters M∗, J∗ and L∗ R have a great effect on the flutter force. The effect of the shear deformation parameter for S≥10 4 on the critical force is very small. The rotary inertia parameter for R<10 −4 has negligible effect on the critical force.

Post-flutter analysis of beams subjected to subtangential forces

This paper presents an analytical study of the post-flutter behavior of beams subjected to both distributed and concentrated subtangential forces. First, Hamilton's principle is used to derive the variational equation of motion. Next, the Ritz-Galerkin method is applied to yield a discretized system of equations. The linear terms in the system of equations are then uncoupled by transforming the system into its quasi-canonical form before the method of multiple scales is utilized to solve for analytical solutions. It is observed that for ideal beams without damping, the post-flutter behavior depends on the initial conditions of beams.

Vibration and stability of elastic columns subjected to triangularly distributed sub-tangential forces

The dynamic behaviour of elastic columns under the action of triangularly distributed sub-tangential forces is investigated to fill the gap between the classical conservative and typical non-conservative cases. The stability maps which are obtained from the eigenvalue analysis determine the divergence and flutter domains. It is shown that either flutter or divergence instability occurs depending on the non-conservativeness of the applied force.