Nonconservative Follower Force Research Articles

The Sub and Super-Tangential Nonconservative Load in Stability Problem of Nanobeams with Sprung Masses

In this study, the critical load and natural vibration frequency of Euler–Bernoulli single nanobeams based on Eringen’s nonlocal elasticity theory are investigated. Cantilever nanobeams with attached sprung masses were subjected to compressed concentrated and distributed follower forces. The parameter that determines the direction of nonconservative follower forces was given the positive and negative values, therefore, sub-tangential and super-tangential load were analyzed. The stability analysis is based on dynamical stability criterion and was carried out using a numerical algorithm for solving segmental nanobeams with many boundary conditions. The presented algorithm is based on the exact solutions of motion equations which are derived from equilibrium conditions for each separated segment of the nanobeam. Two comparison studies are conducted to ensure the validity and accuracy of the presented algorithm. The excellent agreement of critical load for Beck’s nano-column on Winkler foundation observed was confirmed as reported by other researchers. The effect of different values of the nonlocality parameter, tangency coefficient, spring stiffness coefficient, location of sprung mass and the greater number of attached sprung masses on a critical load of nanobeams compressed by nonconservative load are discussed. One of the presented results shows that significant differences between local and nonlocal theory appear when the beam subjected to follower forces loses its stability by flutter.

Finding the strongest stable massless column with a follower load and relocatable concentrated masses

Summary We consider the problem of optimal placement of concentrated masses along a massless elastic column that is clamped at one end and loaded by a nonconservative follower force at the free end. The goal is to find the largest possible interval such that the variation in the loading parameter within this interval preserves stability of the structure. The stability constraint is nonconvex and nonsmooth, making the optimization problem quite challenging. We give a detailed analytical treatment for the case of two masses, arguing that the optimal parameter configuration approaches the flutter and divergence boundaries of the stability region simultaneously. Furthermore, we conjecture that this property holds for any number of masses, which in turn suggests a simple formula for the maximal load interval for $n$ masses. This conjecture is strongly supported by extensive computational results, obtained using the recently developed open-source software package granso (GRadient-based Algorithm for Non-Smooth Optimization) to maximize the load interval subject to an appropriate formulation of the nonsmooth stability constraint. We hope that our work will provide a foundation for new approaches to classical long-standing problems of stability optimization for nonconservative elastic systems arising in civil and mechanical engineering.

Three-dimensional nonlinear dynamics of prestressed active filaments: Flapping, swirling, and flipping.

Initially straight slender elastic filaments or rods with constrained ends buckle and form stable two-dimensional shapes when prestressed by bringing the ends together. Beyond a critical value of this prestress, rods can also deform off plane and form twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such deformed filaments subject to nonconservative active follower forces and fluid drag. We find that softly constrained filaments that are clamped at one end and pinned at the other exhibit stable two-dimensional planar flapping oscillations when active forces are directed toward the clamped end. Reversing the directionality of the forces quenches the instability. For strongly constrained filaments with both ends clamped, computations reveal an instability arising from the twist-bend-activity coupling. Planar oscillations are destabilized by off-planar perturbations resulting in twisted three-dimensional swirling patterns interspersed with periodic flipping or reversal of the swirling direction. These striking swirl-flip transitions are characterized by two distinct timescales: the time period for a swirl (rotation) and the time between flipping events. We interpret these reversals as relaxation oscillation events driven by accumulation of torsional energy. Each cycle is initiated by a fast jump in torsional deformation with a subsequent slow decrease in net torsion until the next cycle. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by nonconservative active forces. Taken together, our results suggest avenues by which prestress, elasticity, and activity may be used to design synthetic macroscale pumps or mixers.

Nonlinear Response of an Inextensible, Free–Free Beam Subjected to a Nonconservative Follower Force

Abstract A free–free beam with a compressive follower force applied to one end exhibits interesting flutter and limit cycle oscillation (LCO) responses. Here, the derivation from Lagrange's equations is given for the nonlinear inextensible beam with such a force applied. The inextensibility constraint is met with a Lagrange multiplier added to the Lagrangian, and the beam allowed three rigid body modes in planar motion in addition to its elastic deformation. The Rayleigh–Ritz modal expansion method and the Runge–Kutta method are used to calculate time histories of the forced beam response. This new model is validated against classical results for the stability boundary and new LCO bifurcation diagrams are computed.

Nonlinear Response of an Inextensible, Cantilevered Beam Subjected to a Nonconservative Follower Force

The dynamic stability of a cantilevered beam actuated by a nonconservative follower force has previously been studied for its interesting dynamical properties and its applications to engineering designs such as thrusters. However, most of the literature considers a linear model. A modest number of papers consider a nonlinear model. Here, a system of nonlinear equations is derived from a new energy approach for an inextensible cantilevered beam with a follower force acting upon it. The equations are solved in time, and the agreement is shown with published results for the critical force including the effects of damping (as determined by a linear model). This model readily allows the determination of both in-plane and out-of-plane deflections as well as the constraint force. With this novel transparency into the system dynamics, the nonlinear postcritical limit cycle oscillations (LCO) are studied including a concentration on the force which enforces the inextensibility constraint.

Nonlinear Oscillations Induced by Follower Forces in Prestressed Clamped Rods Subjected to Drag

Elastic-driven slender filaments subjected to compressive follower forces provide a synthetic way to mimic the oscillatory beating of biological flagella and cilia. Here, we use a continuum model to study the dynamical, nonlinear buckling instabilities that arise due to the action of nonconservative follower forces on a prestressed slender rod clamped at both ends and allowed to move in a fluid. Stable oscillatory responses are observed as a result of the interplay between the structural elastic instability of the inextensible slender rod, geometric constraints that control the onset of instability, energy pumped into the system by the active follower forces, and motion-driven fluid dissipation. Initial buckling instabilities are initiated by the effect of the follower forces and inertia; fluid drag subsequently allows for the active energy pumped into the system to be dissipated away and results in self-limiting amplitudes. By integrating the equations of equilibrium and compatibility conditions with linear constitutive laws, we compute the critical follower forces for the onset of oscillations, emergent frequencies of these solutions, and the postcritical nonlinear rod shapes for two forms of the drag force, namely linear Stokes drag and quadratic Morrison drag. For a rod with fixed inertia and drag parameters, the minimum (critical) force required to initiate stable oscillations depends on the initial slack and weakly on the nature of the drag force. Emergent frequencies and the amplitudes postonset are determined by the extent of prestress as well as the nature of the fluid drag. Far from onset, for large follower forces, the frequency of the oscillations can be predicted by evoking a power balance between the energy input by the active forces and the dissipation due to fluid drag.

On the nonlinear dynamics of shift gearbox models

When the so-called eek effect emerges while shifting a vehicle shift gearbox, car drivers perceive audible noise and vibrations of drivetrain components. Some physical approaches have been provided in order to explain the effect as well as to search for countermeasures. In Jehle and Fidlin (ZAMM J Appl Math Mech 94(11):911–916 (2014), the interaction of sliding clutch disk and gears is addressed in a rigid body model. The clutch disk’s non-conservative follower forces in combination with geometric coupling in gears imply the action of non-symmetric restoring- and velocity-proportional forces. The desired motion of gearbox components can therefore undergo a mode-coupling flutter instability, out of which vibrations emerge with exponentially growing amplitudes. As the contacts in clutch and gears hold non-smooth transitions such as the possibility of sticking and opening, dynamic solutions are both limited and complex. In this contribution, first of all model details concerning the gear formulation are revised: rigid and viscoelastic normal contact are compared, which imply differences in the nature of the solution and method of stability analysis. Nevertheless the two approaches are shown to converge. Finally, the dynamics behind the loss of stability of the desired motion is investigated.

Piezoelectric control of Hopf bifurcations: A non-linear discrete case study

Linear and non-linear analyses of a piezoelectric controlled non-linear Ziegler column are carried out in this paper. The aim is to evaluate the effects of a linear piezoelectric element on the Hopf bifurcations of the non-linear mechanical system, triggered by the non-conservative follower force. To this end a linear stability analysis, showing the performances of the controller in shifting forward the critical load of the uncontrolled system, is carried out and the role of the electro-mechanical coupling parameter and of the mechanical damping is investigated. The beneficial effects of the controller on the amplitude of the limit cycle occurring in the post-critical field are also discussed.

Risks of linear design of joined wings: a non-linear dynamic perspective in the presence of follower forces

Past work on Joined Wings pointed out the importance of including structural geometric non-linearities since the early stages of the design. However, the attention was mainly focused on conservative loading and several open questions needed an answer. In a effort to simulate aerodynamic-pressure-like loads, this research focuses on non-conservative follower forces. Several numerical evaluations demonstrate that follower loading exacerbates the risks of snap instability in Joined Wings. Snap type of instabilities introduce problems which do not pertain only to static responses. In fact, they are associated with the concept of bi-stable regions, with consequent risks of branch-jumping phenomena which are here shown to happen also for small vanishing perturbations and for load levels that are far from the limit one. The here studied phenomena suggest that the design process for Joined Wings has to be revisited and should focus, when stability is concerned, on the evaluation of the post-critical (if any) response: performing a non-linear analysis until the expected nominal limit load (set, for example, by the flight envelope) is reached is thus, a non-conservative approach and could lead to catastrophical consequences. Unfortunately, the above design directions may not be enough to guarantee a safe design: in this study, a case showing an isola-type of bifurcation is presented. Path-tracking techniques (i.e., arc-length family methods) could not detect such phenomena if not combined with ad hoc dynamic perturbation analyses. A further inherent property of the investigated cases is the sensitivity to parameters (e.g. the lower-to-upper-wing bending stiffness ratio): a small perturbation can produce a sudden appearance of a bi-stable region and isola type of bifurcation. This sensitiveness strongly interferes with the design process, being small deviation from the nominal properties (geometry, material, etc.) unavoidable at manufacturing level.

Mechanics of a mosquito bite with applications to microneedle design

The mechanics of a fascicle insertion into the skin by a mosquito of the type aedes aegypti has been studied experimentally using high-speed video (HSV) imaging, and analytically using a mathematical model. The fascicle is a polymeric microneedle composed of a ductile material, chitin. It has been proposed that the mosquito applies a non-conservative follower force component in addition to the Euler compressive load in order to prevent buckling and penetrate the skin. In addition, the protective sheath surrounding the fascicle (labium) provides lateral support during insertion. The mechanics model presented approximates the fascicle as a slender column supported on an elastic foundation (labium) subjected to non-conservative (Beck) and conservative Euler loads simultaneously at the end. Results show that the lateral support of the fascicle provided by the labium is essential for successful penetration by increasing the critical buckling load by a factor of 5. The non-conservative follower force application increases the buckling load by an additional 20% and may or may not be necessary for successful penetration. Experimental results showing the importance of the labium have been cited to validate the model predictions, in addition to the video observations presented in this work. This understanding may be useful in designing painless needle insertion systems as opposed to miniaturized hypodermic needles.

Dynamic instability characteristics of laminated composite plates subjected to partial follower edge load with damping

The study of vibration and dynamic instability behaviour of laminated composite plates subjected to partially distributed non-conservative follower forces is presented by using the finite element technique. The first-order shear deformation theory is used to model the plate, considering the effects of shear deformation and rotary inertia. The modal transformation technique is employed to the resulting equilibrium equation for subsequent analysis. Structural damping is introduced into the system in terms of equivalent viscous damping to study the significance of damping on stability characteristics. The effects of load width, boundary condition, aspect ratio, ply orientation, direction control of the load and damping parameters are considered for the stability behaviour of the plates. The results show that under follower loading, the system is susceptible to instability due to flutter alone or due to both flutter and divergence, depending on system parameters.

THE INFLUENCE OF COMBINED CONCENTRATED AND DISTRIBUTED FOLLOWER FORCES ON THE EIGENSOLUTIONS OF A SPINNING BEAM

The free vibrations of a spinning Timoshenko beam under the action of the non-conservative follower forces are solved analytically for the first time. Three types of follower-force system are considered: concentrated follower forces, distributed follower forces and, combined concentrated and distributed follower forces. The resulting characteristic equation for computing frequencies is very complicated and a novel solution algorithm involving symbolic algebraic operations with each term determined recursively, is proposed. Critical values of the concentrated follower forces that correspond to the onset of the flutter-type instability for the first four modes of vibration are presented.

Robust Active Vibration Control of a Bandsaw Blade

An analytical study of a vibrating bandsaw blade is presented. The blade is modeled as a plate translating over simply-supporting guides. Gyroscopic effects due to the blade’s axial motion as well as in-plane forces resulting from tensioning and the influence of the cutting force are included in the model. The latter is modeled as a nonconservative follower force on the cutting edge of the blade and shown to be destabilizing. A state-space model is developed which includes the effects of time-varying cutting forces and exogenous disturbances. Feedback control via a collocated force actuator/rate sensor is introduced and recent advances in robust control theory are used develop controllers which achieve robust stability and performance with respect to the time-varying model. [S0739-3717(00)01101-6]

Nonlinear Dynamics and Stability of a Two D.O.F. Elastic/Elasto-Plastic Model System

The local and global nonlinear dynamics of a two-degree-of-freedom model system is studied. The undeflected model consists of an inverted T formed by three rigid bars, with the tips of the two horizontal bars supported on springs. The springs exhibit an elasto-plastic response, including the Bauschinger effect. The vertical rigid bar is subjected to a conservative (dead) or non-conservative (follower) force having static and periodic components. First, the method of multiple scales is used for the analysis of the local dynamics of the system with elastic springs. The attention is focused at modal interaction phenomena in weak excitation at primary resonance and in hard sub-harmonic excitation. Three different asymptotic expansions are utilised to get a structural response for typical ranges of excitation parameters. Numerical integration of the governing equations is then performed to validate results of asymptotic analysis in each case. A full global nonlinear dynamics analysis of the elasto-plastic system is performed to reveal the role of plastic deformations in the stability of this system. Static 'force-displacement' curves are plotted and the role of plastic deformations in the destabilisation of the system is discussed. Large-amplitude non-linear oscillations of the elasto-plastic system are studied, including the influence of material hardening and of static and sinusoidal components of the applied force. A practical method is proposed for the study of a non-conservative elasto-plastic system as a non-conservative elastic system with an 'equivalent' viscous damping.

Effects of damping on the dynamic stability of a rod with an intermediate spring support subjected to follower forces

The equation of motion in matrix form of an Euler beam, including the effect of internal damping subjected to non-conservative follower forces, is formulated based on Lagrangian approach and the assumed mode method. The beam is assumed to rest on an intermediate spring support of large stiffness modeling a rigid support. The effect of slight damping on the jump phenomenon for the critical follower load with respect to variation in the support location is examined for a specific example of a simply supported-free rod. For some cases, the modes of instability in the form of flutter or divergence without damping are found to be unaffected by the presence of slight damping, although there may be a sharp decrease in the first critical load for instability by flutter. However, for specific locations of the intermediate spring support, the presence of a small amount of damping may change the mode of instability from divergence to flutter with a sharp decrease in the critical follower load. An interesting finding is that this mode of instability by flutter only occurs for an isolated and narrow range of follower loads.

Exact Stability Analysis Of Combined Flutter-Divergence Systems In The Presence Of Shear Deformation

The stability behaviour of a Timoshenko column with an intermediate support is examined, by solving the corresponding frequency equation. The structure is loaded by a non-conservative follower force, so that both static and dynamic instabilities can occur. It is suggested that the previously observed jump between divergence and flutter critical loads is in reality a jump between two different kinds of flutter instabilities.

Pseudo-follower reactive force as a means of improving the load carrying capacity of structural systems

This paper presents a method for improving the carrying capacity of structures due to the loss of overall structural stability. The proposed method is based on adding a displacement dependent force to a support according to a prescribed rule. Simple numerical examples are presented. These examples are confined to the application of the pseudo-follower reactive force and for in-plane framed structures. The results are obtained with the assumption that the change of the geometrical configuration due to loading is negligibly small before the instability occurs. The numerical results show that a significant improvement in the stability can be expected by the application of the proposed reaction system. In this paper only systems with non-conservative follower forces not explicitly dependent on time are considered.

Dynamic stability of a completely free plate subjected to a controlled non-conservative follower force

The dynamic instability of a rectangular plate with four free edges is investigated. One edge of the completely free plate is subjected to a uniformly distributed non-conservative follower force, the direction of which is controlled in proportion to the rotation angle of the loading edge. The plate may have both flutter and divergence type instabilities when the force is non-conservative, although it may have only divergence type instabilities when the force is conservative.

Vibration and stability of cracked thick rotating blades

A finite element model is applied to the vibration and stability of thick rotating blades with a single edge crack. The effects of transverse shear deformation and rotatory inertia are taken into account. The model can satisfy all the geometric and natural boundary conditions of a thick blade. The stability problems of a cracked rotating blade subjected to non-conservative follower forces are considered. The effects of crack locations and sizes on the dynamic behavior of a blade are studied. It is found that a crack can change the vibrational behavior and stability characteristics of the system.

Vibration and stability of non-conservative follower force systems

The equation which Hamilton called the “Law of Varying Action”, is applied to obtain direct analytical solutions to stationary, conservative and non-conservative follower force systems with various loadings and various constraints. The magnitude of the load required to produce buckling at zero frequency and/or to produce flutter, i.e., instability at non-zero frequency, is calculated. Frequency response curves and mode configurations are presented. Results from the direct analytical solution are compared to results from exact solutions for the few exact solutions which could be found in the literature. Several examples involving applied follower couples, both discrete and continuous, to which no published solutions could be found, are also presented. The paper clearly demonstrates the simplicity, generality, and accuracy available from Hamilton's law.