Tangential Follower Force Research Articles

Vibration and Stability of a Non-Uniform L-Shaped Beam Subjected to a Follower Force (In-Plane Vibration)

An analysis is presented for the vibration and stability of a non-uniform L-shaped beam subjected to a tangential follower force distributed over the centerline by use of the transfer matrix approach. For this purpose, the governing equations of the beam are written as a coupled set of first order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly varying depths and breadths, subjected to a concentrated follower force, and the natural frequencies and flutter loads are calculated numerically, to provide information about the effects on them of varying cross-section, opening angle, shape and the stiffness of the supports.

Non-conservative Instability Of A Timoshenko Beam Resting On Winkler Elastic Foundation

The transfer matrix method is used to investigate the influence of a Winkler elastic foundation on the non-conservative instability of uniform Timoshenko beams. It is found that the critical flutter load for a cantilever Timoshenko beam subjected to an end-concentrated or linearly distributed tangential follower force will first decrease as the elastic foundation modulus is increased and, as the elastic foundation modulus increases beyond the corresponding critical value, which corresponds to the lowest critical flutter load, it will increase instead. It is also observed that when the elastic foundation modulus is large enough, the critical flutter load for a cantilever Timoshenko beam can be greater than that of a Bernoulli-Euler beam, and there exists a corresponding critical turning point for the critical flutter load. At this critical turning point, the instability mechanism changes. The instability mechanisms for a beam subjected to end concentrated and linearly distributed tangential follower force are different.

Vibration Mode of a Free-Free Column Subjected to Follower Forces.

The dynamic instability of a free-free column subjected to a tangential follower force has been investigated. A suspended column, in which one end is supported by a rotational spring and the other end is free and thrusted, is proposed to simulate the dynamic instability phenomena of a portion from the nodal point to the trailing end of an accelerated free-free column in space. These two systems are formulated using FEM, and the vibration modes are compared to discuss the equivalency of the two systems. The rotational spring constants satisfying the equivalency are obtained for the cases with and without structural damping.

Effect of structural damping on flutter of plates with a follower force

A plate-like large space structure may undergo dynamic instabilities when it is thrusted by a nonconservative compressive force. A flexible rectangular plate with four free edges, one of which is subjected to a tangential follower force, is considered. The effects of structural damping are studied here because small damping may destabilize the nonconservative system. The calculation shows that the thrusted free-edged plate with structural damping also has both divergence and flutter types of instability, and the flutter thrust load with small structural damping is drastically lower than that without damping. The destabilizing effect depends on the slenderness ratio of the rectangular plate.

On the dynamic response of certain separable and non-selfadjoint systems

In this paper the dynamic response of certain separable and non-selfadjoint systems has been investigated. Specifically, the dynamic characteristics of isotropic, homogeneous, linearly elastic and spatially one-dimensional systems such as clamped-free rods under constant uniformly distributed tangential follower forces have been determined and presented. The dynamic response of such a system when perturbed by a time varying driving force has also been determined and numerical examples are presented.

Vibration and Stability of a Clamped-Elastically Restrained Timoshenko Column under Nonconservative Load.

The influence of shear deformation and rotatory inertia on the vibration and stability of a clamped-elastically restrained column subjected to a nonconservative force is studied in the present paper. The direction of an external force following the deflection of the column at its elastically restrained end is generally expressed in terms of the arbitrary angle dependent on the slope at the column end. In the analysis, the governing equations derived according to Hamilton's principle are used. The depen-dence of the eigenfrequency, and flutter and divergence loads on the slenderness ratio is numerically examined for various spring stiffnesses and follower force directions. It is shown that although the effect of shear deformation and rotatory inertia is almost to destabilize the system, there may be a case in which it is to stabilize the system in connection with the spring stiffness and follower force direction. Detailed discussion is also presented for the special case of a clamped-free column subjected to a fully tangential follower force in comparison with the other study.

Dynamic stability of a rectangular plate with four free edges subjected to a follower force

One of the four free edges of the plate is subjected to a tangential follower force. The flutter load and divergence load are obtained for various slenderness ratios of the rectangular plate by means of modal analysis. In the calculation, many weak instabilities were found

Dynamic response of elastic rods subjected to uniformly distributed, tangential follower forces

This paper treats the solution of separable non-selfadjoint initial-boundary-value problems using Green's function approach. Specifically, the dynamic response of an isotropic, homogeneous and linearly elastic clamped-free rod when perturbed by a time varying driving force and under constant uniformly distributed tangential follower forces is determined.

A dynamic method for measuring the critical load of a non-conservative composite column

Based on the authors previous investigation on nonconservative problems, a simple experimental model simulating a composite cantilever with a tangential follower force is developed. In the experiments the difficult task of producing follower end force is avoided by solving a conservative problem with related eigensolution, and the experimental verification of the non-conservative problem is accomplished indirectly

A Simply Supported Column Under a Tangential Follower Force as a Self-Adjoint System

A Simply Supported Column Under a Tangential Follower Force as a Self-Adjoint System

Stability of a rectangular plate under nonconservative and conservative forces

A study is made of the flutter and divergence instabilities of a rectangular plate with two independent loading parameters. The plate is subjected to the combined action of a tangential follower force and a unidirectional axial force along one edge. Two opposite sides of the plate are simply supported, one side being clamped and the other being a free edge where the in-plane forces act. Depending on the relative magnitudes of the follower and axial forces, the plate may lose its stability by flutter or divergence. The flutter problem is solved by maximizing the flutter load over the frequency and thereby obtaining the maximum point of an eigencurve. The stability boundaries are given for plates with different aspect ratios.The two-dimensional nature of the problem reveals some interesting results not observed in the one-dimensional counterpart of the problem, which is a cantilevered column under vertical and follower forces.The effect of an elastic foundation on the stability boundaries are determined. The influence of Poisson’s ratio on the flutter load and frequency is investigated. It is shown that no general rule can be formulated as to the effect of Poisson’s ratio on the flutter load, and that this effect will vary according to the aspect ratio and axial load.

Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force

An analysis is presented for the vibration and stability of a non-uniform Timoshenko beam subjected to a tangential follower force distributed over the center line by use of the transfer matrix approach. For this purpose, the governing equations of a beam are written in a coupled set of first-order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly, parabolically and exponentially varying depths, subjected to a concentrated, uniformly distributed or linearly distributed follower force, and the natural frequencies and flutter loads are calculated numerically, from which the effects of the varying cross-section, slenderness ration, follower force and the stiffness of the supports on them are studied.

Stability of a Rectangular Simply Supported Plate Subjected to Nonincreasing Tangential Follower Forces

Stability of a Rectangular Simply Supported Plate Subjected to Nonincreasing Tangential Follower Forces

On a variational principle for the clamped-free rod subjected to tangential follower forces

On a variational principle for the clamped-free rod subjected to tangential follower forces

On the solution of the stability problem of elastic rods subjected to triangularly distributed, tangential follower forces

In order to complete the existing theory of beams subjected to follower forces, an exact solution for the case of triangularly distributed loads is presented. The results obtained may be compared with those reported in the literature and which were based on approximate calculations by means of the Method of Galerkin and the Finite Difference Method. The conclusion is that there is good agreement between all these results, thus confirming the dependability of the approximate methods.

On an unduly simplified model in the non-conservative problems of elastic stability

The peculiar features of instability of Dzhanelidze's model, which is composed of a massless flexible bar and a concentrated mass at the tip and is subjected to a tangential follower force at the tip, are considered. By discussing the eigenvalue curves of Pflüger's model, which has distributed mass in addition to the case of Dzhanelidze's model, and of the corresponding discrete model, it is shown that the critical value, which one must use the dynamic method to obtain, can be identical with the classical Euler value obtainable by the static method only as a result of ignoring a crucial mass. The mathematical meaning of the infinitely large critical frequency is pertinently interpreted.

On the Buckling of Thin Plates Subjected to Nonconservative Follower Forces

It is shown that under appropriate circumstances (proper edge-lengths’ ratio and boundary conditions) a plate subjected to nonconservative follower forces may remain a pseudo-divergence or divergence type system. In such a case, the designer may apply a lower bound theorem which allows him to replace the nonconservative follower forces by the corresponding conservative forces and to design the plate with respect to the easily available first conservative buckling load. Since the calculation is usually carried out by means of Galerkin’s method, the reliability of this method is checked. Although the method turns our to be non-uniformly convergent, the first branch of the characteristic curves is obviously accurately reproduced by means of Galerkin’s method, so that the decision as to whether a problem belongs to the pseudo-divergence or divergence type or not is fairly safe. One of the examples reveals that from the theoretical point of view buckling is also possible for a tensile tangential follower force...

On the solution of the stability problem of elastic rods subjected to uniformly distributed, tangential follower forces

On the solution of the stability problem of elastic rods subjected to uniformly distributed, tangential follower forces

An extremum principle for the buckling problem of the clamped-clamped rod subjected to tangential follower forces

An extremum principle for the buckling problem of the clamped-clamped rod subjected to tangential follower forces

Stability of elastic rods via Liapunov's second method

Stability of elastic rods subjected to nonconservative tangential follower forces is rigorously investigated by means of Liapunov's Second Method in the version of Movchan, Knops, and Wilkes, i.e., adapted to the specific nature of continuous systems. The result is that for such, rods internal viscous damping should have a sufficiently large magnitude in order to ensure stability. This is in agreement with, the well known fact that small damping destabilizes in the presence of nonconservative forces.