Nonlinear Beam Theory Research Articles

Tiltrotor Wing-Root Vibratory Loads Reduction Through Higher Harmonic Control Actuation

This paper deals with the application of a via-swashplate higher harmonic control for the reduction of vibratory loads transmitted to tiltrotor fuselage by the proprotor-pyion-wing system. The control laws are derived through a quadratic optimal control methodology, based on the predictions given by an aeroelastic simulation tool developed in the past by the authors. It uses nonlinear beam theory to model the structural behavior ofproprotor blades and wing, and a potential aerodynamics boundary-element approach suitable to capture body-vortex interactions effects to evaluate wake inflow. Considering a XV-15-like configuration in airplane-and helicopter-mode flight, the numerical investigation examines different strategies in the identification of the optimal control law, analyzing their effectiveness and robustness in reducing the wing-root vibratory loads.

Variational asymptotic beam sectional analysis – An updated version

This paper discusses three recent updates to the variational asymptotic beam sectional analysis (VABS). The first update is a change to the warping constraints in terms of three-dimensional variables, so that one-dimensional beam variables are treated with more rigor. The second update, although its formulation has only been analytically derived but has not been implemented yet, is the incorporation of the effects due to applied loads. The third update is a more accurate energy transformation to generalized Timoshenko form, which is a crucial aspect in finding the stiffness constants to the generalized Timoshenko beam theory. Examples are presented to demonstrate that the updated energy transformation may yield significantly different stiffness predictions from previous versions of VABS and to show that the updated version is indeed more accurate. In addition to the updates, this paper includes a comprehensive derivation of the geometrically-exact nonlinear one-dimensional beam theory and the asymptotically-correct cross-sectional analysis that together form the basis of VABS.

On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange’s equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.

Investigation on Free Vibration of Buckled Beams

The objective of this study is to investigate the deformed configuration and free vibration around the deformed configuration of clamped buckled beams by co-rotational finite element formulation. The principle of virtual work, d'Alembert principle and the consistent second order linearization of the nonlinear beam theory are used to derive the element equations in current element coordinates. The governing equations for linear vibration are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position of the buckled beam. Numerical examples are studied to investigate the natural frequencies of clamped buckled beams with different slenderness ratios under different axial compression.

Hybrid state-space time integration of rotating beams

An efficient time integration algorithm for the dynamic equations of flexible beams in a rotating frame of reference is presented. The equations of motion are formulated in a hybrid state-space format in terms of local displacements and local components of the absolute velocity. With inspiration from Hamiltonian mechanics, where displacement and momentum have similar roles, both sets of state-space variables are interpolated between nodes by the same shape functions, leading to a general format where all inertia effects are represented via the classic constant mass matrix, while effects of the system rotation enter via global operations with the angular velocity vector. The algorithm is based on an integrated form of the equations of motion with energy and momentum conserving properties, if a kinematically consistent non-linear formulation is used. A consistent monotonic scheme for algorithmic energy dissipation in terms of local displacements and velocities, typical of structural vibrations, is developed and implemented in the form of forward weighting of appropriate mean value terms in the algorithm. The algorithm is implemented for a beam theory with consistent quadratic non-linear kinematics, valid for moderate finite rotations. The equations of this non-linear beam theory are generated in explicit form by extension of the constitutive stiffness and the geometric stiffness of a linear beam theory at the element level. The performance of the algorithm is illustrated by numerical examples.

Aerodynamic noise analysis of large horizontal axis wind turbines considering fluid–structure interaction

Aerodynamic noise is one of the most serious barriers in wind energy development. To develop technologies for wind turbine noise reduction and assessment, noise needs to be predicted precisely with special consideration given to blade flexibility. The numerical tool, WINFAS, which can simulate fluid–structure interaction, consists of three parts: the first part, the Unsteady Vortex Lattice Method, analyzes aerodynamics; the second part, the Nonlinear Composite Beam Theory, analyzes structure; and the third part uses a semi-empirical formula to analyze airfoil self-noise and the Lowson’s formula to analyze turbulence ingestion noise. In this study, using this numerical tool, the change in the noise strength due to blade flexibility was examined. This research showed that elastic blades decreased broadband noise because pitching motion reduced the angle of attack.

Investigation on steady state deformation and free vibration of a rotating inclined Euler beam

The steady state deformation and infinitesimal free vibration around the steady state deformation of a rotating inclined Euler beam at constant angular velocity are investigated by the corotational finite element method combined with floating frame method. The element nodal forces are derived using the consistent second order linearization of the nonlinear beam theory, the d'Alembert principle and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The equations of motion of the system are defined in terms of an inertia global coordinate system, which is coincident with a rotating global coordinate system rigidly tied to the rotating hub. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady state deformation and natural frequency of the rotating inclined beam.

Design and simulation of a carbon nanotube-based adjustable nano-electromechanical shock switch

In this paper, design and simulation of a carbon nanotube-based nano-electromechanical shock switch is reported. The switch is represented by a carbon nanotube placed over a ground electrode. Response of the nano-switch based on nonlinear beam theory is studied using Galerkin’s method. Up to five mode shapes have been utilized to capture the response of system and results are validated by comparing with Molecular Dynamics (MD) simulation. Due to their high stiffness, CNTs cannot be actuated to pull-in (switch ON) state by conventional mechanical shocks in range of 10–1000g under one-step voltage. Here, a multi-steps voltage modification is applied to enhance the actuation. Employing this method, a nano-switch with fixed geometry can be adjusted to be triggered by wide rang of mechanical shocks (10–1000g).

Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading

Abstract An exact, closed-form solution is obtained for the nonlinear static responses of beams made of functionally graded materials (FGM) subjected to a uniform in-plane thermal loading. The equations governing the axial and transverse deformations of FGM beams are derived based on the nonlinear first-order shear deformation beam theory and the physical neutral surface concept. The three equations are reduced to a single nonlinear fourth-order integral–differential equation governing the transverse deformations. For a fixed–fixed FGM beam, the equation and the corresponding boundary conditions lead to a differential eigenvalue problem, while for a hinged–hinged FGM beam, an eigenvalue problem does not arise due to the inhomogeneous boundary conditions, which result in quite different behavior between clamped and simply supported FGM beams. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for thermal post-buckling or bending deformation is obtained as a function of the applied thermal load. The exact solutions explicitly describe the nonlinear equilibrium paths of the deformed beam and thus are able to provide insight into deformation problems. To show the influence of the material gradients, transverse shear deformation, in-plane loading, and boundary conditions, numerical examples are given based on exact solutions, and some properties of the post-buckling and bending responses of FGM beams are discussed. The exact solutions obtained herein can serve as benchmarks to verify and improve various approximate theories and numerical methods.

Modeling of a Nonlinear Euler-Bernoulli Flexible Beam Actuated by Two Active Shape Memory Alloy Actuators

There are two different ways of using shape memory alloy (SMA) wire as an actuator for shape control of flexible structures: it can be either embedded within the composite laminate or externally attached to the structure. As the actuator can be placed at different offset distances from the beam, external actuators produce more bending moment and, consequently, considerabnle shape changes with the same magnitude of actuation force compared with the embedded type. Such a configuration also provides faster heat transfer rate owing to convection, which is very important in shape control applications that require a highfrequency response of SMA actuators. Although combination and physics-based modeling of externally attached SMA actuator wires and strips have been considered by many researchers, these studies have some drawbacks, which, if neglected, result in a number of errors in the theoretical results compared with experimental results. These shortcomings are considering the linear Euler-Bernoulli beam theory, not deriving the actuation force of the SMA from the constitutive equations, and taking into account the effect of only one SMA wire actuation force on, for example, the structure. These assumptions may lead to erroneous theoretical modeling results compared with experimental results. In this study, the aforementioned difficulties of attaching SMA actuators to the smart structures have been addressed. In other words, instead of linear beam theory, nonlinear beam theory is used in system modeling and, therefore, the proposed method and results are valid in large deflection and rotation behavior of beam. Also, in comparison to many other analyses that the effect of only one SMA wire is investigated, in the present research the effect of all active and inactive SMA wires is considered. Accordingly, the result of this paper can easily be generalized to the structure with several SMA wire actuations. Moreover, with the purpose of having practical applications in modeling and control, the heat transfer equations of all SMA wires are considered in the analysis and, as a result, the control inputs of the presented model are SMA wire electric currents rather than SMA temperatures. First, a flexible beam actuated by two active SMA actuators is modeled. Then, the Brinson constitutive equations and thermoelectric equations for SMA materials are coupled with the nonlinear beam behavior, and the coupled system of equations is numerically solved for some particular practical cases. Finally, the numerical results of the model simulation are verified against the experimental results using a test setup to validate the proposed model prediction. The implemented method used in this paper can be easily extended to the more complex smart structure with numerous externally attached SMA wires.

A continuum-mechanics interpretation of Reissner's non-linear shear-deformable beam theory

This article deals with the non-linear modelling of beams that are bent, sheared and stretched by external forces and moments. In the following, we restrict to plane-deformations and static conditions. Our task is to present a continuum mechanics-based interpretation of the celebrated large displacement finite deformation structural mechanics theory, which was presented by Eric Reissner [On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. The latter formulation was restricted to the notions of structural mechanics and thus did not use the notions of stress and strain, which are fundamental for continuum mechanics. Thus, the common continuum mechanics-based constitutive modelling at the stress–strain level cannot be utilized in connection with Reissner's original theory. Instead, Reissner suggested that constitutive relations between certain generalized strains (bending, shear and axial force strains) and generalized static entities (bending moments, shear and normal forces) should be evaluated from physical experiments. This means that the beam to be studied must be first built up, and the experiments must be performed for the real beam as a whole. Although such physical experiments are indeed to be performed in practice for safety reasons in sensible cases, for example, bridge decks or aircraft wings, it is nevertheless felt to be a drawback that the results of simple standardized stress–strain experiments concerning the constitutive behaviour of the materials, from which the beam is built up, cannot be used. Moreover, relying only on physical experiments on the whole beam means that computations (virtual experiments) can be made only after the beam has been built up. To overcome this problem, we subsequently present a continuum mechanics-based interpretation of Reissner's structural mechanics modelling, by attaching a proper continuum mechanics-based meaning to both the generalized static entities and the generalized strains in Reissner's theory [E. Reissner, On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. Consequently, these generalized static entities can be related to the generalized strains on the basis of a constitutive modelling on the stress–strain level. We show this in some detail in this contribution for a hyperelastic material proposed by Simo and Hughes [Computational Inelasticity, Springer, New York, 1998]. An illustrative numerical example is given which shows the results of large bending and axial deformation behaviour for different constitutive relations. This article represents an extended version of a preliminary work published in [H. Irschik and J. Gerstmayr, A hyperelastic Reissner-type model for non-linear shear deformable beams, Proceedings of the Mathmod 09 Vienna, I. Troch and F. Breitenecker, eds., 2009, pp. 1–7].

A Corotational Finite Element Method Combined with Floating Frame Method for Large Steady‐State Deformation and Free Vibration Analysis of a Rotating‐Inclined Beam

A corotational finite element method combined with floating frame method and a numerical procedure is proposed to investigate large steady‐state deformation and infinitesimal‐free vibrationaround the steady‐state deformation of a rotating‐inclined Euler beam at constant angular velocity. The element nodal forces are derived using the consistent second‐order linearization of the nonlinear beam theory, the d′Alembert principle, and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The governing equations for linear vibration are obtained by the first‐order Taylor series expansion of the equation of motion at the position of steady‐state deformation. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady‐state deformation and natural frequency of the rotating beam with different inclined angle, angular velocities, radius of the hub, and slenderness ratios.

Gust Response for Flexibly Suspended High-Aspect Ratio Wings

A theoretical aeroelastic study for a flexibly suspended high-aspect ratio wing aeroelastic model excited by gust loads is presented along with a companion wind-tunnel test. A flexible support system at the wing root has been constructed to simulate a rigid body mode. Structural equations of motion based on a nonlinear beam theory are combined with the ONERA aerodynamic model. Also, a dynamic perturbation analysis about a nonlinear static equilibrium is used to determine the small perturbation flutter boundary and structural natural frequencies. The effects of the flexible support system (bi-beam system) on the wing structural dynamics, the static aeroelastic displacement at root, nonlinear flutter, and gust response to a harmonic or a frequency sweep excitation are discussed. Also the effect of gust distribution along the span on the response is described based upon computations. The fair to good quantitative agreement between theory and experiment demonstrates that the present analysis method has reasonable accuracy.

Compliant folded beam suspension mechanism control for rotational dwell function generation using the state feedback linearization scheme

Abstract The control of a slider–crank mechanism driven by a compliant, folded beam mechanism is investigated in this study. The compliant folded beam mechanism, also called the compliant double arm mechanism, has application areas at both the micro and macro levels. The compliant folded beam mechanism control in micro dimensions, driven by a force actuator is investigated here. The kinematic synthesis of the mechanism has been studied using mathematically exact nonlinear Elastica theory. The equivalent stiffness of the large deflecting fixed-free flexible beam, the single arm beam and the complete folded beam mechanism (double arm mechanism) are computed and, the complete mechanism stiffness is represented by a polynomial function that is curve fitted to the nonlinear inextensible exact beam theory solution. The dynamic response of the mechanism is obtained by solving the nonlinear equation of motion numerically using Runge–Kutta methods. The emphasis of the paper is placed on the subject of control of compliant mechanisms incorporating nonlinear stiffness (having large deflections) for function generation. The trajectory control of double cranks is achieved by considering linearization by the state feedback since the compliant folded beam mechanism (the double parallel arm mechanism) has a nonlinear stiffness. A PD controller is then applied to the feedback linearized system. The controller coefficients are determined to satisfy desired specifications on the output of the system.

A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading

Due to the variation in material properties through the thickness, bifurcation buckling cannot generally occur for plates or beams made of functionally graded materials (FGM) with simply supported edges. Further investigation in this paper indicates that FGM beams subjected to an in-plane thermal loading do exhibit some unique and interesting characteristics in both static and dynamic behaviors, particularly when effects of transverse shear deformation and the temperature-dependent material properties are simultaneously taken into account. In the analysis, based on the nonlinear first-order shear deformation beam theory (FBT) and the physical neutral surface concept, governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform in-plane thermal loading are derived. Then, a shooting method is employed to numerically solve the resulting equations. The material properties of the beams are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and to be temperature-dependent. The effects of material constants, transverse shear deformation, temperature-dependent material properties, in-plane loading and boundary conditions on the nonlinear behavior of FGM beams are discussed in detail.

Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms

In this investigation, the effects of the knee-joint movements on the ligament tension and cross-section deformation are examined using large displacement nonlinear finite element/multibody system formulations. Two knee-joint models that employ different constitutive equations and significantly different deformation kinematics are developed and implemented to analyze the ligament dynamics in a computational solution procedure that integrates large displacement finite element and multibody system algorithms. The first model employs a lower fidelity large displacement cable element that does not capture the cross-section deformations and allows for using only nonlinear classical beam theory with a linear Hookean material law instead of a general continuum mechanics approach. In the second model, a higher fidelity large displacement beam model that captures more coupled deformation modes including Poisson modes as well the cross-section deformation is used. This higher fidelity model also allows for a straight forward implementation of general nonlinear constitutive models, such as Neo Hookean material laws, based on a general continuum mechanics approach. Cauchy stress tensor and Nanson’s formula are used to obtain an accurate expression for the ligament tension forces, which as shown in this investigation depend on the ligament cross section deformation. The two models are implemented in a general multibody system algorithm that allows introducing general constraint and force functions. The finite element/multibody system computational algorithm used in this investigation is based on an optimum sparse matrix structure and ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The results obtained in this investigation show that models that ignore coupled deformation modes including some Poisson modes and the cross-section deformations can lead to inaccurate prediction of the ligament forces. These simpler models, as demonstrated in this investigation, can be used to obtain only simplified expressions for the ligament tensions. A three-dimensional knee-joint model that consists of five bodies including two flexible bodies that represent the medial collateral ligament (MCL) and lateral collateral ligament (LCL) is used in the numerical comparative study presented in this paper. The large displacement procedure presented in this investigation can be applied to other types of Ligaments, Muscles, and Soft Tissues (LMST) in biomechanics applications.

Non-linear dynamics of a drill-string with uncertain model of the bit–rock interaction

The drill-string dynamics is difficult to predict due to the non-linearities and uncertainties involved in the problem. In this paper a stochastic computational model is proposed to model uncertainties in the bit–rock interaction model. To do so, a new strategy that uses the non-parametric probabilistic approach is developed to take into account model uncertainties in the bit–rock non-linear interaction model. The mean model considers the main forces applied to the column such as the bit–rock interaction, the fluid–structure interaction and the impact forces. The non-linear Timoshenko beam theory is used and the non-linear dynamical equations are discretized by means of the finite element method.

Flutter/LCO suppression for high-aspect ratio wings

AbstractAn experimental high-aspect ratio wing aeroelastic model with a device to provide a controllable slender body tip mass distribution has been constructed and the model response due to flutter and limit cycle oscillations has been measured in a wind tunnel test. A theoretical model has also been developed and calculations made to correlate with the experimental data. Structural equations of motion based on nonlinear beam theory are combined with the ONERA aerodynamic stall model (an empirical extension of Theodorsen aerodynamic theory that accounts for flow separation). A dynamic perturbation analysis about a nonlinear static equilibrium is used to determine the small perturbation flutter boundary which is compared to the experimentally determined flutter velocity and flutter frequency. Time simulation is used to compute the limit cycle oscillations response when the flutter/LCO control system is ON or OFF. Theory and experiment are in good agreement for predicting the flutter/LCO suppression that can be achieved with the control device.

Corotational Nonlinear Finite Element Formulation and Modeling of Electrostatically Actuated Microbeams

A complete nonlinear finite element model for coupled-domain microelectromechanical system devices with electrostatic actuation and squeeze film effect was developed. for this purpose, a corotational finite element formulation for the dynamic analysis of planer Euler-Bernoulli microbeams considering geometrical nonlinearities due to both large structural deformation and electrostatic actuation is developed. In this method, the internal forces due to deformation and residual stresses, the elemental inertias, and the damping effect of the squeeze-film are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory using d'Alembert and the virtual work principles. An incremental-iterative method based on the Newmark direct integration procedure and the Newton-Raphson algorithm is used to solve the nonlinear dynamic equilibrium equations. Several numerical examples are presented and their results are compared with those found experimentally, which indicate a very close agreement.

Effective coarse-grained simulations of super-thick multi-walled carbon nanotubes under torsion

Under torsion and beyond the buckling point, multi-walled carbon nanotubes (MWCNTs) develop a periodic wave-like rippling morphology. Here, we show that torsional rippling deformations can be accurately described by a simple sinusoidal shape function. Combining this observation with the geometry optimization, we develop an effective coarse-grained model that reproduces the complex nonlinear mechanical responses of thick MWCNTs under torsion predicted by large-scale atomistic simulations. Furthermore, the model allows us to simulate super-thick tubes, inaccessible by other coarse-grained methods. With this effective coarse-grained model, we show from an energetic analysis that the rippling deformation is a result of in-plane strain energy relaxation, penalized by the increase in the interlayer van der Waals interaction energy. Our simulations reveal that the torsional response of MWCNTs with up to 100 layers approximately follows a simple bilinear law, and the ratio of the torsional rigidities in the pre- and post-buckling regimes is nearly a constant, independent of the tube radius. In contrast, the bifurcation torsional strain powerly scales with the tube radius. We also find that the wave number in the circumferential direction linearly increases with tube radius, while the wavelength monotonically increases with tube radius, and approaches a constant in the limit of bulk graphite. The bilinear constitutive relation, together with the scaling law of the bifurcation torsional strain, furnishes a simple nonlinear beam theory, which facilitates the analysis of MWCNT bundles and networks.