Tangential Follower Force Research Articles

Nonlinear vibrations of fractional viscoelastic PET membranes subjected to tangential follower force

Nonlinear forced vibrations of simply supported viscoelastic Polyethylene Terephthalate (PET) membranes subjected to a harmonic excitation and tangential follower force are investigated in this paper. The consideration of vibration systems under follower force requires special attention to the modeling of non-conservative force and its influence on nonlinear analysis. Herein, a fractional Kelvin-Voigt Hamilton-based framework accounting for the viscoelastic PET membrane is developed. Based on the Hamilton principle for the non-conservative system, the mathematical modeling of viscoelastic PET membrane taking into account non-conservative force and geometric nonlinearity is formulated, and the derived equations are discretized via the Galerkin schemes. A technique of multiple scales is employed to numerically acquire the frequency–response curves with different system configurations. Numerical investigations are conducted to analyze the effects of tangential follower force, viscous coefficient, as well as fractional order on the dynamic stability of the considered structure. Further ramifications of the present study point to the paramount importance of tangential follower force and an accurate viscoelasticity model on the dynamic behaviors of the viscoelastic structures.

Dynamics of functionally graded porous rotating Rayleigh microbeams with longitudinal movement in complex fields

As a first attempt, the vibration of rotating functionally graded porous Rayleigh beams under longitudinal motion in hygro-thermo-magnetic fields is scrutinized numerically and analytically based on the scale-dependent strain gradient theory. Also, parametric studies are performed to diagnose the impacts of critical factors such as the functionally graded index, rotary inertia parameter, porosity distribution, foundation characteristics, environmental loads, boundary conditions, axial and tangential follower forces on the microbeam stability. It is assumed that the material properties of the microbeam are graded across the thickness according to a power-law function with six different porosity models. The dynamical equations of the functionally graded microbeam are derived based on the extended Hamilton’s principle. Then, backward and forward vibrational frequencies are computed with the aid of the Galerkin discretization scheme and eigenvalue analysis. For validation purposes, the results are compared with published reports. Furthermore, the instability border of the microbeam is determined via an analytical approach. Critical rotational and longitudinal speeds as well as Campbell and stability diagrams are obtained to identify the stability evolution. The results declared that as the functionally graded index and rotary inertia parameter ascend, divergence and flutter speeds of the microbeam descend. The presented outcomes could be helpful in the optimum design of inhomogeneous gyroscopic microsystems.

Nonconservative Stability Analysis of Columns with Various Loads and Boundary Conditions

In this paper, the stability analysis of the elastic columns subjected to seven different types of the nonconservative force is investigated on the basis of fully intrinsic beam equations. The generalized differential quadrature method is used for the discretization of the first-order intrinsic equations and corresponding boundary conditions. Altogether, four important boundary conditions—simply supported, clamped-simply supported, clamped-free, and clamped-clamped conditions—are considered. Furthermore, the effect of the combined action of an end-concentrated force and a distributed tangential follower force is investigated. To confirm the validity of the proposed intrinsic formulations, the present results are compared with those obtained from classical formulations. Our results reveal that the fully intrinsic formulation is a suitable framework to model nonconservative problems.

Flutter and vibration of elastically restrained nanowires under a nonconservative force

AbstractThis paper investigates vibration and flutter behavior of a cantilevered nanowire under the action of a subtangential follower force. Taking the surface effects into account, we apply Hamilton principle to derive a governing equation for an elastically restrained nanobeam. A characteristic equation is obtained. Force‐frequency interaction curves are displayed to show the influence of the surface elasticity, surface stress, restraint stiffness on divergence and flutter force. Natural frequencies and divergence loads are calculated for conservative forces at the free end. For a generalized follower force with various tangency coefficients, dynamic instability of nanocantilevers is presented and flutter loads are calculated for subtangential, supertangential and tangential follower forces. Flutter loads are sensitive to translational and rotational spring stiffness and surface elasticity. Some special cases with some vanishing parameters reduce to the well‐known results.

Analysis of dynamic instabilities in bridges under wind action through a simple friction-based mechanical model

In the field of stability of structures under nonconservative loads, the concept of follower force has long been debated by scientists due to the lack of actual experimental evidence. Bigoni and Noselli’s work [2] aimed to investigate flutter and divergence instability phenomena through a purely mechanical model with Coulomb friction represents a praiseworthy attempt to shed light on this issue. A two-degree-of-freedom (DOF) system, conceived as a variant of the Ziegler column, was set up experimentally. The follower load was induced by a frictional force acting on a wheel mounted at the column end, so that the rolling friction vanishes and the sliding frictional force keeps always coaxial to the column, thus representing a tangential follower force. Along this research line, in this contribution a model is elaborated that stems from the analysis of an elastically supported rigid plate that represents the behaviour of a bridge deck suspended on springs and subjected to a wind-induced force. The wind force has been simulated by a Coulomb friction force acting on a wheel mounted on the plate aerodynamic centre, so that the sliding friction force keeps perpendicular to the plate axis throughout the system motion, thus representing a follower force. To properly reproduce the wind force, the friction force is applied to the wheel by a lever mechanism wherein one of the two lever arms involves the plate rotation via a particular circular guide. The corresponding equations of motion of the bridge deck are derived in a completely dimensionless form. Depending on the mechanical characteristics of the plate and the magnitude of the friction force, stability, flutter or divergence phenomena may occur. The occurrence of these phenomena is numerically investigated by integration of the equations of motion. The development of an experimental framework of the model to corroborate these intuitions is the object of an ongoing research.

General descriptions of follower forces derived via a geometrically exact inverse contact algorithm

SummaryThe main feature of the geometrically exact theory of contact is that all objects, which are necessary for computation, weak form and residual, linearized weak form, and tangent matrices are given in a covariant closed form in the local coordinate system corresponding to the geometry of contact pairs. This allows easily to construct computational algorithms for the normal and tangential follower forces as an inverse contact algorithm. In this case, following the definition of the follower forces as given and not changing in the local coordinate system, we have to modify all objects for the contact taking into account the definition of follower forces instead of constitutive relationships for the contact interfaces. The main feature is that the tangent matrices for both normal and tangential part being split into the rotational and the curvature parts are symmetric for any order of approximation. The following numerical examples are selected in the current article to illustrate the effectiveness of implementation: (1) the modeling of a pure bending with a moment applied either as a pair of single forces or a distributed follower forces (pressure) in both 2D and 3D cases; (2) modeling of inflation of a plate as application of the distributed follower normal forces (pressure); and (3) modeling of twisting of a beam with the rectangular cross‐section as application of the tangential follower forces algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

The destabilizing effect of external damping: Singular flutter boundary for the Pflüger column with vanishing external dissipation

Elastic structures loaded by non-conservative positional forces are prone to instabilities induced by dissipation: it is well-known that internal viscous damping destabilizes the marginally stable Ziegler's pendulum and Pflüger column (of which the Beck's column is a special case), two structures loaded by a tangential follower force. The result is the so-called ‘destabilization paradox’, where the critical force for flutter instability decreases by an order of magnitude when the coefficient of internal damping becomes infinitesimally small. Until now external damping, such as that related to air drag, is believed to provide only a stabilizing effect, as one would intuitively expect. Contrary to this belief, it will be shown that the effect of external damping is qualitatively the same as the effect of internal damping, yielding a pronounced destabilization paradox. Previous results relative to destabilization by external damping of the Ziegler's and Pflüger's elastic structures are corrected in a definitive way leading to a new understanding of the destabilizating role played by viscous terms.

Effect of Uniformly Distributed Tangential Follower Force on the Stability of Rotating Cantilever Tube Conveying Fluid

In this paper, the Euler-Bernoulli beam model is used to predict the structural instability of rotating cantilever tubes conveying fluid and subjected to uniform distributed tangential compressive load. The governing equation of motion and boundary conditions of the system are derived using the Hamilton's principle. Then, Galerkin method is applied in order to transform the resulting equation into a general eigenvalue problem. The present analysis is validated by comparing the results with those available in literature. Furthermore, the model is utilized to elucidate the stability characteristics of the system for different conditions.

Stability of Axially Moving Piezolaminated Viscoelastic Plate Subjected to Follower Force

The stability of the moving viscoelastic plate with the piezoelectric layer subjected to uniformly distributed tangential follower force is investigated. The force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force. The differential equation of the axially moving viscoelastic rectangular plate with piezoelectric layer subjected to uniformly distributed tangential follower force is formulated on the basis of the Kirchhoff thin plate theory and the two-dimensional viscoelastic differential constitutive relation. The complex eigenvalue equations are established by the differential quadrature method. Via numerical calculation, the curves of real parts and imaginary parts of dimensionless complex frequencies versus uniformly distributed tangential follower force and dimensionless moving speed are obtained. The effects of nonconservative force, dimensionless axially moving speed, and dimensionless applied voltages on the stability of axially moving nonconservative viscoelastic plate with piezoelectric layer are analyzed.

Exact solutions for the stability of viscoelastic rectangular plate subjected to tangential follower force

An exact solution procedure is formulated for the stability analysis of viscoelastic rectangular plate with two opposite edges simply supported and other two edges clamped as well as the viscoelastic rectangular plate with one edge clamped and other three edges simply supported under the action of tangential follower force. Firstly, by assuming the transverse displacement (W) as independent functions which automatically satisfies the simply supported boundary conditions, the governing partial differential equation is reduced to an ordinary differential equation with variable coefficients. Then, by the normalized power series method and applying the boundary conditions yield the eigenvalue problem of finding the roots of a fourth-order characteristic determinant. The results show that the aspect ratio λ and the dimensionless delay time H have great effects on the types of instability and the critical loads of the viscoelastic plates.

Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force

The paper addresses the issue of effectively using the direct numerical method for static analysis of the flexible curved uniform cantilever beam under a tip-concentrated follower force. The angle of inclination of the follower force with respect to the deformed axis of the beam remains unchanged during deformation. After changing the variables, the original non-linear boundary value problem transforms into the initial-value problem for pendulum equation. The resulting initial value problem is solved numerically using a modified Numerov's method. In contrast to the usually used iteration methods (e.g. shooting technique), the problem is solved without iterations by direct numerical method. Some qualitative conclusions were made using Kirchhoff's kinetic analogy. It is shown that there are no critical loads in the Euler sense (divergence) for any values of the initial curvature and angle of inclination of the follower force. An extension of direct numerical method to curved spring-hinged cantilever subjected to follower force is also proposed. The paper presents some equilibrium configurations of the uniform curved fixed and spring-hinged cantilevers under normal and tangential follower force obtained by direct method.

Pre-Buckling Transverse Free Vibration of an Asymmetrically Supported Euler Beam Subjected to Distributed Follower Forces

Based on the accurate geometrical theory for the extensible elastic beams, an exact mathematical model of post-buckling transverse free vibration of Euler beams subjected to a distributed tangential follower force along the central axis are established. By using shooting method, pre-buckling free vibrations of both simply supported and fixed Euler beam are solved and the responses of small amplitude vibration are obtained. The numerical results show that all the frequencies of unbuckled beams decrease continuously with the increment of the load parameters.

Dynamic stability of functionally graded materials skew plates subjected to uniformly distributed tangential follower forces

The dynamic stability of functionally graded material (FGM) skew thin plate subjected to uniformly distributed tangential follower force is investigated. The material properties are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents based on the Voigt model. In skew coordinate system, the differential equations of motion of the FGM skew plate subjected to uniformly distributed tangential follower force are derived by the Kirchhoff thin plate theory, and the different boundary conditions are obtained of the plate for arbitrary curve edges. By eliminating the in-plane displacement components on the neutral plane, the differential equations of motion can be expressed in terms of deflection only. Then the equations are discretized by the differential quadrature method, and the curves of real parts and imaginary parts of the first second-order dimensionless complex frequencies vs. uniformly distributed tangential follower force are obtained. The effects of the gradient index, skew angle and aspect ratio on the instability type and the corresponding critical load of the non-conservation FGM skew plate are analyzed.

Buckling and post-buckling behaviour of semicircular functionally graded material arches: a theoretical study

In this study, behaviour of a thin semicircular arch made of functionally graded material subjected to radial and tangential follower forces, as well as thermal loading, based on the theory of large deformation of the arches and critical buckling load and post-buckling of structures is investigated. Assuming thin arch, governing equations of the arch behaviour are derived using kinematics and static equilibrium. Resulting equations including different support conditions form a set of highly coupled and non-linear boundary value differential equations. To solve the problem, the well-known numerical boundary value problems of shooting method are employed. By gradual increase of loading, the buckling and post-buckling behaviour is closely monitored. Several examples corresponding to different combinations of support conditions/loadings/non-homogeneity/slenderness ratio to illustrate the performance of the proposed algorithm are presented.

Vibration Characteristics of the Non-Conservative Thermoelastic Coupling Plate

The vibration characteristics of the thermoelastic coupling rectangular plate under the action of uniformly distributed tangential follower force are investigated. The coupled thermoelastic differential equation of the plate under the action of uniformly distributed tangential follower force was derived. Dimensionless complex frequencies of the thermoelastic coupling rectangular plate with one edge clamped and other three edges simply supported, two opposite edges simply supported and other two edges clamped were calculated by the differential quadrature method. The effects of the dimensionless thermoelastic coupling factor on the stability and critical load of the thin plate were analyzed. The results show that the flutter loads of the coupled modes increase with the increase of the dimensionless coupled thermoelastic factor and the aspect ratio.

Experimental evidence of flutter and divergence instabilities induced by dry friction

Flutter and divergence instabilities have been advocated to be possible in elastic structures with Coulomb friction, but no direct experimental evidence has ever been provided. Moreover, the same types of instability can be induced by tangential follower forces, but these are commonly thought to be of extremely difficult, if not impossible, practical realization. Therefore, a clear experimental basis for flutter and divergence induced by friction or follower-loading is still lacking. This is provided for the first time in the present article, showing how a follower force of tangential type can be realized via Coulomb friction and how this, in full agreement with the theory, can induce a blowing-up vibrational motion of increasing amplitude (flutter) or an exponentially growing motion (divergence). In addition, our results show the limits of a treatment based on the linearized equations, so that nonlinearities yield the initial blowing-up vibration of flutter to reach eventually a steady state. The presented results give full evidence to potential problems in the design of mechanical systems subject to friction, open a new perspective in the realization of follower-loading systems and of innovative structures exhibiting ‘unusual’ dynamical behaviors.

Stability of a Cracked Viscoelastic Plate of Varying Thickness Subjected to Follower Force

The stability of the cracked visoelastic rectangular plate of varying thickness subjected to uniformly distributed tangential follower force is investigated. Two models describing thickness variation are parabolic and linear variations along one edge of the plate. This is done in order to show how critical loads for cracked visoelastic thin plate subjected to follower force can be obtained using the two-dimensional viscoelastic differential constitutive relation and the thin-plate theory. The differential equations in the Laplace domain are established. The complex eigen-value equations are derived by the differential quadrature method. The generalized eigen-value under different boundary conditions is calculated. The effects of the plate parameters, the crack parameters and the dimensionless delay time on the stability of the viscoelastic plates are analyzed. The crack is found to change the type of instability and reduce the stability of varying thickness viscoelastic plate.

Dynamic stability of thermoelastic coupling moving plate subjected to follower force

The dynamic characteristics and stability of the moving thermoelastic coupling rectangular plate subjected to uniformly distributed tangential follower force are investigated. Based on the heat conduction equation containing the thermoelastic coupling term and the thin plate theory, the thermoelastic coupling differential equation of motion of the rectangular plate under the action of uniformly distributed tangential follower force is established. Dimensionless complex frequencies of the moving thermoelastic coupling rectangular plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method. The effects of the dimensionless thermoelastic coupling factor and dimensionless moving speed on the stability and critical load of the moving plate are analyzed. The results show that the divergence loads of the first order mode increase with the increase of the dimensionless thermoelastic coupling factor, and decrease with increasing the dimensionless moving speed.

О задаче устойчивости составного консольного стержня при одновремeнном действии “следящей” и неизменного направления сил, приложенных в пролёте.

О задаче устойчивости составного консольного стержня при одновремeнном действии “следящей” и неизменного направления сил, приложенных в пролёте.

경사 종동력과 끝질량을 갖는 크랙 보의 안정성 해석

In this paper, the purpose is to investigate the stability and variation of natural frequency of a cracked cantilever beams subjected to follower force and tip mass. In addition, an analysis of the flutter instability(flutter critical follower force) of a cracked cantilever beam as slenderness ratio and crack severity is investigated. The governing differential equations of a Timoshenko beam subjected to an end tangential follower force is derived via Hamilton's principle. The two coupled governing differential equations are reduced to one fourth order ordinary differential equation in terms of the flexural displacement. Finally, the influence of the slenderness ratio and crack severity on the critical follower force, stability and the natural frequency of a beam are investigated.