Critical Follower Force Research Articles

On large deformation and stability of microcantilevers under follower load

In the present study, a novel geometrically exact nonlinear beam model based on the nonlinear displacement field, centerline-inextensibility condition, Euler-Bernoulli beam hypothesis, and modified couple stress theory is established to investigate the extremely large deformations and stability characteristics of size-dependent microcantilevers subjected to a tip-concentrated non-conservative (follower type) load with arbitrary inclination angle. The newly developed mathematical formulation explains the size effect phenomenon in the microcantilevers with only one material length scale parameter. The nonlinear shooting method for two-point boundary value problems is utilized to assess the deformed configuration of microsystem based on the quasistatic form of nonlinear mathematical model. Afterwards, the linearized dynamical mathematical model around the nonlinear deformed configurations of microsystem is numerically solved using the Galerkin technique to examine the stability of obtained responses. A comprehensive investigation is then conducted to highlight the role of size-dependency in the critical follower force at which the microsystem undergoes a flutter instability, as well as the nonlinear large deformation of microsystem in the pre-flutter region.

Dynamic Stability and Response of Inclined Beams Under Moving Mass and Follower Force

This paper is concerned with the dynamic stability and response of an inclined Euler–Bernoulli beam under a moving mass and a moving follower force. The extended Hamilton’s principle is used to derive the governing equation of motion and the boundary conditions for this general moving load/force problem. Considering a simply supported beam, one can solve the problem analytically by approximating the spatial part of the deflection with a Fourier sine series. Based on the formulation and method of solution, sample dynamic responses are determined for a beam that is inclined at 30[Formula: see text] with respect to the horizontal. It is shown that the dynamic response of the beam under a moving mass is rather different from an equivalent moving follower force. Also investigated herein are the dynamic stability of inclined beams under moving load/follower force which are described by four key variables, viz. the speed of the moving mass/follower force, concentrated mass to the beam distributed mass, vibration frequency and the magnitude of the moving mass/follower force. The critical axial load and the critical follower force are different when they are located at different positions in the beam; except for the special case when they are at the end of the beam.

Geometrically nonlinear vibration of anisotropic composite beams using isogeometric third-order shear deformation theory

Due to broad usage of anisotropic composite beams in modern engineering structures, the main goal of the present work is to examine their geometrically nonlinear vibration. To this end, a third-order shear deformation theory with a nonlinear von-Kármán strain field is used for anisotropic beams and combined with the advantages of the isogeometric framework. The layup properties are assumed to be anisotropic in the depth direction and a transient tip follower force is considered. The governing nonlinear equations of vibration are integrated by means of the Newmark approach and solved with the Newton–Raphson method. Flutter loads as well as natural frequencies are obtained by eigenvalue analysis. The effects of various important factors such as non-prismatic shape, orientation of the composite fibers, critical follower force and bifurcation point are studied, using both h- and p-refinements. The results show that the nonlinear vibration and flutter characteristics of the anisotropic composite beams are completely different from those for orthotropic and isotropic ones. Thick beams with anisotropic layups are more sensitive to the shear parameter than conventional ones and deform primarily in shear mode rather than in bending. Anisotropic beams reveal a higher flutter instability force than other cases for a given shear parameter value. Another important phenomenon is that the stress distribution in anisotropic layups shows irregular patterns both in depth and time. Anisotropic layups with ply angles between 15° and 45° seem to present enhanced nonlinear performance with respect to other layup choices. A coarse mesh of quartic C3 B-splines is observed to provide high accuracy for nonlinear deflections in anisotropic cases even for rather low shear parameters.

Nonlinear time domain and stability analysis of beams under partially distributed follower force

• A geometrically exact beam formulation is used to obtain the stability and nonlinear time domain analysis. • A wide range of beams with different flexibility ratios are utilized in this investigation. • The behavior of the eigenvalues at pre- and post-instability as well as the frequency and time domain results are obtained. • The stability and nonlinear behavior are highly affected by partially follower forces length and position. This paper aims to investigate linear and nonlinear behavior of beams subjected to externally applied partially distributed follower forces. In this investigation, the nonlinear composite beam theory of Hodges is used. The system of nonlinear equations is linearized about the equilibrium, or rest structure state, and the linear system is solved numerically. The effects of follower force position on the behavior of eigenvalues at pre- and post-instability are reported. Additionally, the contours of critical follower force are obtained by changing the position of follower force in span-wise and chord-wise directions. The effects of different parameters such as the length, and position of follower force and the ratios of stiffnesses on the critical follower force as well as the nonlinear limit cycle oscillation (LCO) are reported. The obtained results indicate that the length and the position of the partially distributed follower forces considerably affect the stability of the beam.

Reduction of the actuator oscillations in the flying vehicle under a follower force

Flexible behaviors in new aerospace structures can lead to a degradation of their control and guidance system and undesired performance. The objectives of the current work are to analyze the vibration resulting from the propulsion force on a Single Stage to Orbit (SSTO) launch vehicle (LV). This is modeled as a follower force on a free-free Euler-Bernoulli beam consisting of two concentrated masses at the two free ends. Once the effects on the oscillation of the actuators are studied, a solution to reduce these oscillations will also be developed. To pursue this goal, the stability of the beam model is studied using Ritz method. It is determined that the transverse and rotary inertia of the concentrated masses cause a change in the critical follower force. A new dynamic model and an adaptive control system for an SSTO LV have been developed that allow the aerospace structure to run on its maximum bearable propulsion force with the optimum effects on the oscillation of its actuators. Simulation results show that such a control model provides an effective way to reduce the undesirable oscillations of the actuators.

Instability Characteristics of a Free-Free Multi-Stepped Timoshenko Beam with Concentrated Masses Subjected to Follower Force

Thispaperpresentsdynamicandstaticinstabilitycharacteristicsofafree-freemulti-steppedTimoshenkobeamsub-jected to a follower force, where each step has a point mass with transverse and rotary inertia. A finite element code isdeveloped to model two-dimensional multi-stepped Timoshenko beams using the extended Hamilton’s principle. Theeffects of each section parameters on the type of instability and critical follower force have been studied for differentmodelsofmulti-steppedbeams.Itisconcludedthattheassumptionofauniformbeamtostudymostaerospacestructuressuch as launch vehicles could lead to an unrealistic instability analysis, where any small change in the characteristicsparameters for each step could change the instability behavior significantly. Furthermore, this paper introduces a newparametric approach concept for analyzing the characteristics effects of a multi-step aerospace structure.Key Words: Multi-Stepped Timoshenko Beam, Follower Force, Finite Element, Static and Dynamic Instability

The dependence of the jump in the critical follower force for a viscoelastic bar on the form of the non-linear internal viscosity

The transverse oscillations of a viscoelastic bar under the action of a follower force are considered. The constitutive relation is the non-linear relation between the stress, strain and strain rate, where the non-linear part is described by a homogeneous form of the fifth degree. A numerical-analytical investigation of the stability showed that the three values of the critical follower force, corresponding to each of the three coefficients of viscosity available in the model, are independent of these coefficients and change in comparison with the case of a cubic viscosity.

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.

Effects of a flexible joint on instability of a free-free jointed bipartite beam under the follower and transversal forces

This paper deals with the problem of the instability regions of a free-free flexible jointed bipartite beam under the follower and transversal forces as a realistic simulation of a two-stage aerospace structure. The aim of this study is to analyze the effects of the characteristics of a flexible joint on the beam instability to use maximum bearable propulsion force. A parametric study is conducted to investigate the effects of the stiffness and the location of the joint on the critical follower force by the Ritz method and the Newmark method, then to research the vibrational properties of the structure. It has been shown that the nature of instability is quite unpredictable and dependent on the stiffness and the location of the joint. The increase of the follower force or the transversal force will increase the vibration of the model and consequently cause a destructive phenomenon in the control system of the aerospace structure. Furthermore, this paper introduces a new concept of the parametric approach to analyze the characteristics effects of a flexible two-stage aerospace structure joint design.

경사 종동력을 받는 티모센코 보의 안정성에 미치는 크랙의 영향

In this paper, the purpose is to investigate the stability of cracked Timoshenko cantilever beams subjected to subtangential follower force. In addition, an analysis of the instability(critical follower force of flutter and divergence) of a cracked beam as slenderness ratio and subtangential coefficient is investigated. The governing differential equations of a Timoshenko beam subjected to an end tangential follower force is derived via Hamilton's principle. The crack is assumed to be in the first mode of fracture and to be always opened during the vibrations. The results of this study will contribute to the safety test and stability estimation of structures of a cracked beam subjected to subtangential follower force.

끝단질량과 종동력을 가진 크랙 외팔 보의 안정성 해석

In this paper a dynamic behavior(natural frequency) of a cracked cantilever beam subjected to follower force is presented. In addition, an analysis of the flutter and buckling instability of a cracked cantilever beam subjected to a follower compressive load is presented. Based on the Euler-Bernoulli beam theory, the equation of motion can be constructed by using the Lagrange's equation. The vibration analysis on such cracked beam is conducted to identify the critical follower force for flutter instability based on the variation of the first two resonant frequencies of the beam. Besides, the effect of the crack's intensity and location on the flutter follower force is studied. The crack section is represented by a local flexibility matrix connecting two undamaged beam segments. The crack is assumed to be in the first mode of fracture and to be always opened during the vibrations.

Complex Analysis of Flutter and Buckling of Beams under Rotational and Transverse Spring Constraints

The flutter and buckling analysis of a beam, subjected to a follower force, is studied theoretically. The beam is constrained by a pair of transverse and rotational springs at one end and fixed at the other end. The effects of the transverse and rotational spring stiffnesses are studied to obtain the transition condition differentiating two distinct types of instability that co-existed in the beam. The critical follower force is derived for both flutter and buckling of the beam. In addition, the variations of the parametric curves for the critical follower force are analyzed, and the abrupt drop or rise points in the curve are taken as the indicators of the transition between different types of instability. The research discussed in this paper may be used as a benchmark for the flutter and buckling analysis of beams. Moreover, this work may provide an accurate model in dealing with the flutter and buckling analysis and design of general beams.

Dynamic stability of cantilevered timoshenko columns subjected to a rocket thrust

The dynamic stability of cantilevered Timoshenko columns subjected to a rocket thrust is described. It is assumed that the thrust force is produced by the installation of a solid rocket motor to the tip end of the cantilevered columns. The rocket motor is assumed to be a rigid body having finite sizes, but not a mass point as it has been assumed so far. A finite element model of the columns is formulated through the extended Hamilton's principle. The dynamic stability of the columns is investigated with respect to (i) the magnitude, rotary inertia and size of the rocket motor and (ii) the shear deformation and rotary inertia parameter of the columns. The theoretical analysis predicts a proper combination of the magnitude and the position of the rocket motor to stabilize the columns. It is shown that the size and rotary inertia of the rocket motor have a great effect on the critical follower force, on the other hand the rotary inertia parameter of the column hardly affects the critical follower force as long as the rotary inertia parameter of the column is small. It is found that when shear deformation parameter is large enough, the shear deformation parameter has a slight effect on the critical follower force.