Flexible Euler-Bernoulli Beam Research Articles

Influence of control parameters on the stabilization of an Euler-Bernoulli flexible beam

In this work, we numerically study the influence of control parameters on the stabilization of a flexible Euler-Bernoulli beam fixed at one end and subjected at the other end to a force control and a punctual moment control proportional respectively to velocity and rotation velocity. First, we analyze the displacement stabilization and the asymptotic behavior of the beam energy using a stable numerical scheme, resulting from the Crank-Nicholson algorithm for time discretization and the finite element method based on the approximation by Hermite's cubic polynomial functions, for discretization in space. Then, by means of the finite element method, we represent the spectrum of the operator associated with this beam problem and we carry out a qualitative study of thelocus of the eigenvalues according to the positive control parameters. From these studies we conclude that rotation velocity control has more effect on the stabilization of the beam compared to velocity control. Finally, this result is confirmed by a sensitivity study on the control parameters involved in the stabilization of the beam.

A Numerical Study for a Flexible Euler-Bernoulli Beam with a Force Control in Velocity and a Moment Control in Rotating Velocity

In this paper, we numerically study a flexible Euler-Bernoulli beam with a force control in velocity and a moment control in rotating velocity. First, we show the existence and uniqueness of the weak solution using Faedo-Galerkin’s method with the intermediate spaces. Then, we use the finite elements method with the cubic Hermite polynomials for the approximation of (1.1)–(1.5) in space such that the semi-discrete scheme obtained is stable and convergent. In addition, an a-priori error estimate is obtained. Finally, we perform numerical simulations in order to validate this method.

Existence and uniqueness of a hybrid system with variable coefficients

In this paper we consider a long flexible Euler-Bernoulli beam with boundary conditions imposed at the two ends, the resulting model being called hybrid system. The beam is hybrid in the sense that it holds both rigid and elastic motions. Our main result is to show the existence and uniqueness of the weak solution of close-loop system. The closed-loop system stability is shown through Lyapunov-based analysis.

Cubic static stall model for nonlinear aeroelastic behavior of high-aspect-ratio flexible composite wings

In this paper, by defining a new paradigm for nonlinear aerodynamic equations of flow separation and static stall, a new form of nonlinear aeroelastic equations for a highly flexible composite wing with torsional and bending movement is presented. Structural modeling is carried out using nonlinear general flexible Euler–Bernoulli beam equations in the third-order. Combining the unsteady Wagner model and the nonlinear lift coefficient-angle of attack lead to the aerodynamic equations for simulating stall using a cubic approximation. The aeroelastic equations are obtained utilizing Hamilton’s principle and Lagrange equations. A time-history integration method is used to solve the integro-differential nonlinear aeroelastic equations. The obtained results are compared with previous studies for validation, and there is good agreement between the results. The results show that the use of the cubic curve instead of the piecewise linear curves which is commonly used in other references, although, causes an apparent complication of the equations but reduces the errors. It is also observed that the limit cycle oscillations speed changes by changing the angle of fibers, and the maximum of this instability speed occurs at an angle of about twenty degree.

Numerical approximation of the spectrum for a flexible Euler-Bernoulli beams with a force control in rotation and velocity rotation

In this paper, we use asymptotic techniques and the finite differences method to study the spectrum of the differential operator arising in the exponential stabilization of the following differential equation $y_{tt}+y_{xxxx}=0,$ $0<x<1,$ $t\geq 0$ with boundary conditions $y(0,t)=y_{x}(0,t)=0,t\geq 0,$\ $y_{xxx}(1,t)=0,t\geq 0,$\ $y_{xx}(1,t)=-\alpha y_{xt}(1,t)-\beta y_{x}(1,t),t\geq 0.$

Control of Flexible Euler-Bernoulli Beam with Input/Output Delay and Stochastic Disturbances

We consider the problem of stability enhancement of an undamped flexible beam with a tip mass in presence of input delay and random disturbances. In absence of delay this problem is classically solved through output feedback based on a suitable approximation of an infinite-dimensional Kalman filter. To cope with the presence of input or output delays we derive and compare two solutions, one based on a predictor from estimates in the past and the other one based on a filter with delayed measurements. An identical delay bound in closed form is derived for both solutions and we show that by an appropriate choice of the control gain it is possible to stabilize the system in presence of arbitrarily large delays. A modular structure is proposed for the case of arbitrary gain and delay bound. Finally, we consider the problem of deriving a finite-dimensional approximation of the predictor.

Bio-inspired gust mitigation system for a flapping wing UAV: modeling and simulation

The growing use of drones today necessitates improved UAVs performance and capabilities. These requirements include flights in operational environments full of obstacles which present navigation and stability problems due to large-scale turbulence. To address this major concern, turbulence alleviation capabilities of natural counterparts have been studied in depth. Birds use inherent active and passive flow mechanism in their flapping wings and also deflection of covert feathers to stabilize themselves in turbulent airflows. This paper presents a novel bio-inspired gust mitigation system (GMS) for flapping wing UAVs mimicking covert feathers of birds. GMS senses the forces experienced from turbulent airflows and actuates to perform local airflow manipulations to alleviate them. GMS comprises of electromechanical feathers capable of deflecting out of the airfoil once they encounter turbulence. Modeling of single electromechanical feather assists in development of a complete GMS model that is further integrated in wing, modeled as flexible Euler–Bernoulli beam, and a complete dynamic model of flapping wing is obtained using bond graph modeling approach. We perform digital simulations and compute state-space equations to analyze model internal dynamics and responses. Comparison of the simulation results of wing without GMS and GMS-integrated wing, in response to vertical gust, confirms the efficacy of offered design. Furthermore, a good agreement between the present simulation results and experimental results from the literature validates the proposed model. As a result, the offered design successfully marks an initial step toward research into bio-inspired active gust mitigation systems for flapping wing UAVs.

Vibration analysis of rotating fully-bonded and delaminated sandwich beam with CNTRC face sheets and AL-foam flexible core in thermal and moisture environments

In the present study, the high-order free vibration analysis of rotating fully-bonded and delaminated sandwich beams; with and without vertical contact; containing AL-foam flexible core and carbon nanotubes reinforced composite (CNTRC) face sheets subjected to thermal and moisture field are investigated by using generalized differential quadrature method (GDQM). The compressible core and face sheets of sandwich beam, respectively, are composed of Aluminum alloy foam with variable mechanical properties in the thickness direction and CNTRC with temperature dependent material properties. In this study, the high-order sandwich panel theory (HSAPT) for AL-foam flexible core and Euler-Bernoulli beam theory for CNTRC face sheets are considered. By employing Hamilton’s principle, the governing partial differential equations of motion and associated boundary and continuity conditions for various types of regions (fully-bonded, delaminated with contact, delaminated without contact) are derived and then discretized by using GDQM. The final formulations lead to 14 partial differential equations for the entire structure including five equations for fully-bonded two-headed parts of AL-foam cored sandwich beam (AL-FCSB) and four equations for delaminated middle part of AL-FCSB beam which are combined in axial and transverse deformations. A parametric study is performed to investigate the influence of some important parameters such as existence of delaminated region, type of delaminated region (with or without contact), longitudinal position of delaminated region, slenderness ratio, face sheet thickness ratio, CNT volume fraction, temperature rise, moisture concentration, rotating speed, and hub radius. The obtained results reveal that the 1st frequency of delaminated AL-FCSB beam, whether with or without vertical contact, is less remarkably than ones of fully-bonded AL-FCSB beam which its value for the case of delaminated ‘with contact’ is larger than that of ‘without contact’. Moreover, the 1st frequency variation of the delaminated AL-FCSB beam is symmetrical with regard to the longitudinal position of the debonded region such that the 1st natural frequency declines with moving the debonded region toward the center of the beam. The study of vibration behavior of rotating sandwich beams is very important in design of rotating structural systems, specially damaged ones, such as airplanes, helicopter rotor blades, and robot arms. One of the most important types of damage encountered in mentioned cases is the decomposition of two layers or delamination. Working these rotating structures in the media, are always along with variations of temperature and humidity and hence their mechanical properties may be changed due to the environment conditions.Communicated by S. Velinsky

Adaptive actuator fault-tolerant control for a three-dimensional Euler–Bernoulli beam with output constraints and uncertain end load

This paper studied an adaptive actuator fault-tolerant control scheme for the flexible Euler–Bernoulli beam in the three-dimensional space with output constraints and uncertain end load. The dynamic models are represented by partial differential equations (PDEs) and ordinary differential equations (ODEs). When part of the actuator fails, an adaptive control scheme is designed to regulate the vibration and stabilize the flexible three-dimensional Euler–Bernoulli beam. Barrier Lyapunov Function (BLF) is adopted to realize output constraints of the system. Adaptive control law with projection mapping operator is designed to compensate for the end load which is uncertain and bounded. The goal of this paper is to suppress the displacement of the flexible three-dimensional Euler–Bernoulli beam which can be constrained in given bounds under actuator fault and uncertain, bounded end load. It is confirmed that the proposed control scheme can deal with the vibration, adaptive actuator fault-tolerant control, uncertain and bounded end load and output constraints of the system simultaneously. Finally, numerical simulations illustrate the effectiveness and feasibility of the method.

Vibration Control for Two Kinds of Flexible Mechanical Systems With Full State Output Constraints

This paper concentrates on the vibration control problem of a flexible string system and a flexible Euler-Bernoulli beam system for considering full state output constraints, i.e., deflection and speed constraints. For either system, a boundary control law is developed based on a barrier Lyapunov function. By employing the control scheme, the deflection and speed are limited into desirable regions. With the help of a rigorous unique solution proof for the time derivative of the barrier Lyapunov function, the asymptotic stability of the considered system is achieved via LaSalle's Invariance Principle. Finally, numerical simulations are provided to verify the control performances of the proposed controls.

Quantifying Chaos by Various Computational Methods. Part 2: Vibrations of the Bernoulli-Euler Beam Subjected to Periodic and Colored Noise.

In this part of the paper, the theory of nonlinear dynamics of flexible Euler–Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied.

Dynamic analysis of vehicle–track interaction due to wheel flat using bond graph

The dynamic response of a railway track under a moving train in the presence of a wheel with flat has been studied over many years. The force at the wheel–rail interface is mainly responsible for vehicle and track components deterioration and adds to the maintenance cost. So, reliable predictions of wheel–rail interaction forces are of prime concern to get the key factors responsible for damage of vehicle and track components. In most of the studies, a symmetrical vehicle–track model with linearity in track components behavior is assumed for simplification. This may lead to incorrect results in some situation. In this paper, wheel–rail impact dynamics is investigated by considering an asymmetrical vehicle–track model with due consideration to nonlinear behavior of track. Some nonlinear factors such as loss of wheel–rail contact, nonlinearity in pad, and ballast behavior are taken into consideration. A combined vehicle–track bond graph model is developed to study the wheel–track interaction dynamics. The rail is modeled as a flexible Euler Bernoulli beam resting on discrete support. The nonlinear Hertzian contact theory is used to accomplish the dynamic interactions between the vehicle and the track. Time response of forces, displacements, velocities, and accelerations of the related components of the vehicle and the track are obtained. It has been found that, though the wheel flat exists on leading right wheels, its effect has also been transferred to other components of the vehicle. The obtained results further lead to provide a better understanding of the interaction dynamics at the wheel–track interface with attention to the nonlinear behavior of pad and ballast.

Dissipative Numerical Method for a Flexible Euler-Bernoulli Beam with a Force Control in Rotation and Velocity Rotation

In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.

Non-symmetric and chaotic vibrations of Euler-Bernoulli beams under harmonic and noisy excitations

In this paper we study non-linear dynamics of flexible Euler-Bernoulli beams subjected to harmonic load and white noise. We report that in the case of continuous mechanical systems like the studied Euler-Bernoulli beams the action of white noise yieldes novel and unexpected phenomena, i.e. symmetric transversal loads and symmetric boundary conditions imply non-symmetric beam vibration. In addition, regions of non-symmetric beam vibrations for two values of white noise intensity are given and the charts of regular and chaotic vibrations of the studied beams are presented.

Vibration Control of a Flexible Beam With Output Constraint

In this paper, we present the vibration control design for a Euler–Bernoulli beam with the boundary output constraint. To prevent the constraint violation, a novel barrier Lyapunov function is employed for the control design and stability analysis. This paper represents an important step in extending the barrier Lyapunov function theory to distributed parameter systems. Model-based barrier control is proposed to suppress the vibration of a flexible Euler–Bernoulli beam under the boundary output constraint. Then, adaptive control is designed to handle the system parametric uncertainties. The vibration suppression is well achieved without violation of the constraint. Numerical simulations are provided to illustrate the performance of the control system.

Chaotic and synchronized dynamics of non-linear Euler–Bernoulli beams

In this work the mathematical modeling and analysis of the chaotic dynamics of flexible Euler–Bernoulli beams is carried out. Algorithms reducing the studied objects associated with the boundary value problems are reduced to the Cauchy problem through both the Finite Difference Method (FDM) with approximation of O(c2) and the Finite Element Method (FEM). The constructed Cauchy problem is solved via the fourth- and sixth-order Runge–Kutta methods. The validity and reliability of the obtained results is rigorously discussed. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincarè and pseudo-Poincarè maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. In particular, we study a transition from symmetric to asymmetric vibrations and we explain this phenomenon. Vibration-type charts are reported regarding two control parameters: amplitude q0 and frequency ωp of the uniformly distributed periodic excitation. Furthermore, we have detected and illustrated chaotic vibrations of the Euler–Bernoulli beams for different boundary conditions and different beams thickness. In addition, we study chaotic dynamics and synchronization of multi-layer beams coupled only via boundary conditions. Computational examples of the theoretical investigations are given, where geometric, physical and design non-linearities are taken into account.

Chaotic dynamics of flexible Euler-Bernoulli beams

Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c(2)) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q(0) and frequency ω(p) of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.

Static Stall Model in Aeroelastic Analysis of a Flexible Wing with Geometrical Nonlinearity

This paper presents the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with a nonlinear unsteady aerodynamic model. A high-aspect ratio and flexibility are the specific characteristics of this type of wing. Wing flexibility, coupled with a long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear flexible Euler-Bernoulli beam equations is used for the structural modeling. An unsteady nonlinear aerodynamic considering static stall based on the Wagner function is used for determination of aerodynamic loading of the wing. Combining these two types of formulations yields fully nonlinear integro/differential aeroelastic equations. Using Galerkin’s method, a mode summation technique, and a numerical method, the governing equations are solved to predict the nonlinear aeroelastic response of a wing in the stall region. The results obtained are compared with the results given in the literature. In addition, the effects of the static stall characteristics of some typical airfoils on the aeroelastic behavior of a flexible wing are investigated. The results indicate that in a range of static stall angles of attacks there is a specific angle of attack in which the limit cycle oscillation boundary speed has a maximum value.

Response spectrum of a coupled flexible shaft-flexible beam system for cycloidal input motion

In this study, flexible Euler-Bernoulli type beam attached to a flexible shaft-rigid disk system which is driven by a servomotor is considered. A cycloidal rise motion is assumed for disk rotation. Equation of motion for the flexible shaft-rigid disk system is developed and the disk motion for the cycloidal input motion is obtained. In the second step, elastic beam which is attached to the rigid disk is modeled with the boundary conditions of flexible root-free end. Mode summation technique is assumed for the solution of the equations. Analytical solutions are obtained and spectrum of beam tip vibration amplitudes with respect to frequency ratio is obtained. Frequency ratio is defined as the elastic beam first natural frequency divided by the cycloidal rise frequency. Results show that at frequency ratios of 2, 3, 4, … beam tip vibration amplitudes become zero. With the proper selection of either first natural frequency of elastic beam or the cycloidal rise frequency, it is possible to drive elastic shaft-rigid disk-elastic beam system without exciting vibrational motion.

An efficient method for nonlinear aeroelasticy of slender wings

This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler–Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulation yields nonlinear integro-differentials aeroelastic equations. Using the Galerkin’s method and a mode summation technique, the governing equations will be solved by introducing a numerical method without the need to adding any aerodynamic state space variables and the corresponding equations related to these variables of the problem. The obtained equations are solved to predict the aeroelastic response of the problem. The obtained results for a test case are compared with those of some other works and show a good agreement between results.