Rotating Euler-Bernoulli Beam Research Articles

Vibration control of a pre-twisted rotating beam with nonlinear bi-stable attachments

The influence of the nonlinear bi-stable attachment on the vibration suppression of a pre-twisted rotating cantilever beam is investigated in this study. The kinetic and potential energies of the system, comprising a pre-twisted rotating Euler–Bernoulli beam and the nonlinear bi-stable attachments, are obtained, and the equations of motion are derived using the Rayleigh–Ritz method. The natural frequencies and mode shapes of the beam (without the attachments) obtained using the present formulation are validated with the results obtained from Ansys. Subsequently, the flapwise vibration control of the pre-twisted rotating beam subjected to a base excitation is studied. A parametric study is performed for various amplitudes of base excitation, rotational speeds, and the number and position of the bi-stable attachments. Results show significant vibration suppression capabilities of the nonlinear attachments, and hence the proposed system could potentially be used for rotating beam-type structures such as wind turbine blades.

Dynamic modeling and neural network compensation for rotating Euler-Bernoulli beam using a novel deformation description method

Dynamic models that consider multiple nonlinear factors including the two-dimensional deformation and higher-order modes are too computationally complex to be suitable for real-time control of the rotating Euler-Bernoulli beam. This article proposes a new dynamic modeling method according to the assumed mode method. The new modeling method not only has high modeling accuracy but also has a brief mathematical expression. In addition, based on the new modeling method, a neural network sliding mode control strategy using the saturation function is designed to improve the rotational accuracy of the Euler-Bernoulli beam. In the control law design, the saturation function is proposed instead of the unit step function to reduce the chattering phenomenon. Simulation and control experiments describe that the new dynamic modeling method has higher modeling accuracy than the modeling method based on the one-dimensional deformation; the proposed strategy can effectively decrease the rotation angle tracking error of the rotating Euler-Bernoulli beam and improve the control accuracy.

Selection of Radial Basis Functions for the Accuracy of Meshfree Galerkin Method in Rotating Euler-Bernoulli Beam Problem.

In this work, the radial basis function approximations are used to improve the accuracy of meshfree Galerkin method. The method is applied to the free vibration problems of non-rotating and rotating Euler–Bernoulli beams. The stiffness and mass matrices are derived by using conventional methods. In this meshfree method, only six nodes are considered within a single sub-domain. The parameters are varied for different approximations; the results are obtained with different approximations and found accurate. Two new basis function have been developed which are relatively accurate than conventional basis function: the first new basis function is obtained by multiplication of linear function to radial basis function and second new basis function is obtained by multiplying cubuic radial basis function to Gaussian radial basis function. The first few modes show same result that is available in literature using finite element method and higher modes are found very accurate as well. The result are found to be more accurate for first three modes of non-rotating and rotating Euler–Bernoulli beams where the cantilever beam boundary conditions are used; the first three modes do not change with the change in the parameter c of radial basis function.

Meshfree Galerkin Method for a Rotating Non-Uniform Rayleigh Beam with Refinement of Radial Basis Functions

The rotating Rayleigh beam problem is solved with meshfree method where the radial basis functions are explored. Numbers of basis functions have been used for meshfree methods which also include radial basis function. In this paper, the Gaussian radial basis function and multiquadrics radial basis functions are combined to get the new basis function which provides accuracy for higher natural frequencies. The radial basis functions satisfy the Kronecker delta property and it is easy to apply the essential boundary conditions. The Galerkin method is used for weak formulation. The matrices have been derived. The results are obtained for Gaussian radial basis function and new basis function. The results show more accurate values for fourth and fifth natural frequency with new basis function where only six nodes are used within the subdomain of trial and test function. The results are also obtained with conventional finite element method where forty, two node elements are considered. Also, the results are obtained for rotating Euler-Bernoulli beam to observe the difference in results with rotating Rayleigh beam.

Coriolis Force Sliding Mode Control Method for the Rotary Motion of the Central Rigid Body-Flexible Cantilever Beam System in TBM

Beam slab structure is often encountered in a complex tunnel boring machine. Beam slab structure is subject to dynamic load, which is easy to cause fatigue damage and affect its service life. Therefore, it is necessary to control the vibration of this kind of beam slab structure. In this study, the central rigid body-flexible beam model is established for the rotating beam and plate rotating around the y-axis. Based on the Hamilton variational principle, the dynamic equation of the central rigid body-flexible beam system is established, and the dynamic model of the central rigid body-flexible beam system considering the influence of Coriolis force and centrifugal force is given. The vibration control of the central rigid body-flexible beam system is studied. The vibration mode of the rotating Euler Bernoulli beam is determined by using the elastic wave and vibration mode theory. The influence of the rotating motion on the beam vibration is analyzed, and the variable structure control law is designed to suppress the beam vibration. Numerical simulation results show that the control method can effectively suppress the first-order and second-order vibration of the beam and verify the effectiveness of the control strategy.

Isogeometric analysis for in-plane free vibration of centrifugally stiffened beams including Coriolis effects

In this paper, in-plane free vibration of a rotating Euler-Bernoulli beam with uniform cross section is studied using a rotation-free isogeometric analysis formulation. Using Hamilton’s principle, the derived equations governing the in-plane bending and axial deformations are coupled by means the Coriolis terms. These two equations are transformed into a one linear gyroscopic eigenvalue problem via a weak formulation. The boundary conditions are enforced by means of Lagrange multipliers. Numerical tests of convergence are conducted and the obtained results are compared to the reference solutions to show the effectiveness and accuracy of the proposed method. Effects on natural frequencies, and mode shapes originated from Coriolis terms, angular speed, slenderness ration, hub radius and boundary conditions are numerically investigated.

General decay for a viscoelastic rotating Euler-Bernoulli beam

In this paper, we consider a viscoelastic rotating Euler-Bernoulli beam that has one end fixed to a rotated motor in a horizontal plane and to a tip mass at the other end. For a large class relaxation function \begin{document}$ q $\end{document} , namely, \begin{document}$ q^{\prime}(t) \leq -\zeta(t)H(q(t)) $\end{document} , where \begin{document}$ H $\end{document} is an increasing and convex function near the origin and \begin{document}$ \zeta $\end{document} is a nonincreasing function, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial decay.

Simplified modelling and analysis of a rotating Euler-Bernoulli beam with a single cracked edge

The natural frequencies and mode shapes of the flapwise and chordwise vibrations of a rotating cracked Euler-Bernoulli beam are investigated using a simplified method. This approach is based on obtaining the lateral deflection of the cracked rotating beam by subtracting the potential energy of a rotating massless spring, which represents the crack, from the total potential energy of the intact rotating beam. With this new method, it is assumed that the admissible function which satisfies the geometric boundary conditions of an intact beam is valid even in the presence of a crack. Furthermore, the centrifugal stiffness due to rotation is considered as an additional stiffness, which is obtained from the rotational speed and the geometry of the beam. Finally, the Rayleigh-Ritz method is utilised to solve the eigenvalue problem. The validity of the results is confirmed at different rotational speeds, crack depth and location by comparison with solid and beam finite element model simulations. Furthermore, the mode shapes are compared with those obtained from finite element models using a Modal Assurance Criterion (MAC).

Free Vibration Analysis of Rotating Euler–Bernoulli Beam with Exponentially Varying Cross-Section by Differential Transform Method

In this paper, the free vibration analysis of non-uniform rotating Euler–Bernoulli beam is carried out. It is assumed that the beam has exponentially decaying circular cross-section. In order to solve the problem, the differential transform method (DTM) is utilized. Based on our knowledge, we claim that the recurrence relation presented herein is an elaborate recurrence relation which has been obtained for ordinary differential equations. Non-dimensional natural frequencies of the beam are obtained and tabulated for different values of the beam parameters such as taper ratio and rotating speed. Furthermore, the finite element method (FEM) is employed to solve the problem. Comparison of the results obtained by DTM and FEM indicates the accuracy of proposed solutions.

Meshfree Galerkin method for a rotating Euler-Bernoulli beam

ABSTRACTIn this article, we solve the free vibration problem of a rotating Euler-Bernoulli beam using the meshfree Galerkin method. Radial basis functions are used for interpolation. An improved formulation of the rotating Euler-Bernoulli beam free vibration problem with the Galerkin method is explained for the first time, which results into a symmetric stiffness matrix and gives a significant computational advantage over the formulation given in the existing literature. A conventional hp-version of Galerkin finite element method is used for comparison. Results are obtained at different non-dimensional rotating speeds of a rotating beam. Results show excellent agreement with the existing literature.

Vibration analysis of a uniform pre-twisted rotating Euler–Bernoulli beam using the modified Adomian decomposition method

The governing equations for a pre-twisted rotating cantilever beam are derived and used for free vibration analysis of a pre-twisted rotating beam whose flexural displacements are coupled in two planes. First differential equations of motion of a rotating twisted beam, including terms due to centrifugal stiffening, are derived for an Euler–Bernoulli beam undergoing free natural vibrations. The general solutions of these equations are obtained on applying the Adomian modified decomposition method (AMDM). The AMDM allows the governing differential equations to become recursive algebraic equations and the boundary conditions to become simple algebraic frequency equations suitable for symbolic computation. With additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. Two main advantages of the application of the AMDM are, for the cases considered here, its fast convergence rate to the solution with the high degree of accuracy. As the AMDM technique is systematic, it is found straight-forward to modify boundary conditions from one case to the next. Comparison of results with published data showed the present calculations to be in reasonable agreement.

Output variance constrained bending control of rotating Euler–Bernoulli beam

In this article bending control of rotating Euler–Bernoulli beam is considered. It is assumed that the fixed-free elastic beam is attached to a servomotor using a variance constrained controller, specifically output variance constrained controller for vibration suppression. Equations of motion of the system obtained via Hamilton's principle and Galerkin method are used. The resulting linearized state-space models obtained considering just one or three modes are used for control system design. Output variance constrained controllers are designed in order to control bending at the beam tip and beam rotation angle with different variance constraint magnitudes. Closed-loop responses are analyzed when they experience white noise perturbations. Comparisons between system having tighter variance constraint and weaker variance constraint are also performed. Finally, robustness of Output variance constrained controllers with respect to the modeling uncertainty (i.e. variation of number of modes) is examined.

Characteristics of steady vibration in a rotating hub–beam system

A rotating beam features a puzzling character in which its frequencies and modal shapes may vary with the hub׳s inertia and its rotating speed. To highlight the essential nature behind the vibration phenomena, we analyze the steady vibration of a rotating Euler–Bernoulli beam with a quasi-steady-state stretch. Newton׳s law is used to derive the equations governing the beam׳s elastic motion and the hub׳s rotation. A combination of these equations results in a nonlinear partial differential equation (PDE) that fully reflects the mutual interaction between the two kinds of motion. Via the Fourier series expansion within a finite interval of time, we reduce the PDE into an infinite system of a nonlinear ordinary differential equation (ODE) in spatial domain. We further nondimensionalize the ODE and discretize it via a difference method. The frequencies and modal shapes of a general rotating beam are then determined numerically. For a low-speed beam where the ignorance of geometric stiffening is feasible, the beam׳s vibration characteristics are solved analytically. We validate our numerical method and the analytical solutions by comparing with either the past experiments or the past numerical findings reported in existing literature. Finally, systematic simulations are performed to demonstrate how the beam׳s eigenfrequencies vary with the hub׳s inertia and rotating speed.

Nonlinear Dynamics of a High-Dimensional Model of a Rotating Euler–Bernoulli Beam Under the Gravity Load

Nonlinear dynamic responses of an Euler–Bernoulli beam attached to a rotating rigid hub with a constant angular velocity under the gravity load are investigated. The slope angle of the centroid line of the beam is used to describe its motion, and the nonlinear integro-partial differential equation that governs the motion of the rotating hub-beam system is derived using Hamilton's principle. Spatially discretized governing equations are derived using Lagrange's equations based on discretized expressions of kinetic and potential energies of the system, yielding a set of second-order nonlinear ordinary differential equations with combined parametric and forced harmonic excitations due to the gravity load. The incremental harmonic balance (IHB) method is used to solve for periodic responses of a high-dimensional model of the system for which convergence is reached and its period-doubling bifurcations. The multivariable Floquet theory along with the precise Hsu's method is used to investigate the stability of the periodic responses. Phase portraits and bifurcation points obtained from the IHB method agree very well with those from numerical integration.

Modal tailoring and closed-form solutions for rotating non-uniform Euler–Bernoulli beams

In this paper, the free vibration of a rotating Euler–Bernoulli beam is studied using an inverse problem approach. We assume a polynomial mode shape function for a particular mode, which satisfies all the four boundary conditions of a rotating beam, along with the internal nodes. Using this assumed mode shape function, we determine the linear mass and fifth order stiffness variations of the beam which are typical of helicopter blades. Thus, it is found that an infinite number of such beams exist whose fourth order governing differential equation possess a closed form solution for certain polynomial variations of the mass and stiffness, for both cantilever and pinned-free boundary conditions corresponding to hingeless and articulated rotors, respectively. A detailed study is conducted for the first, second and third modes of a rotating cantilever beam and the first and second elastic modes of a rotating pinned-free beam, and on how to pre-select the internal nodes such that the closed-form solutions exist for these cases. The derived results can be used as benchmark solutions for the validation of rotating beam numerical methods and may also guide nodal tailoring.

Design and Development of a ?-Synthesis Controller for a Flexible Robotic Manipulator, Modelled as a Rotating Euler-Bernoulli Beam

In this paper, a robust control strategy for a robotic manipulator, modelled as a cantilever rotating Euler-Bernoulli beam, is developed. Imprecision in the payload mass, unknown properties of the manipulator link, and torque disturbance are included as the sources of uncertainty. The objective is to achieve a desired angular rotation while the vibration of the manipulator tip is suppressed and the control system remains in a stable region. The control input of the system is an external driving torque. For formulation of the continuous system, the mode summation technique is used and equations of motion are described in the Laplace domain. Then, unstructured uncertainties are included in the form of multiplicative input uncertainty. The μ-synthesis control approach is used and an H∞ optimal robust controller is developed based on the DK-iteration algorithm. Results show that the designed controller guarantees the robust stability and performance of the perturbed system against existing uncertainties. Consequently, stability of the closed-loop system, disturbance rejection, and trajectory tracking performance are achieved.

Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams

The Adomian modified decomposition method (AMDM) is employed in this paper for dynamic analysis of a rotating Euler-Bernoulli beam under various boundary conditions. Based on AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions as well as different offset length and rotational speeds are presented. The accuracy is assured from the convergence and comparison published results. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of rotating beams with arbitrary boundary conditions.

Dynamics of rotating composite beams: A comparative study between CNT reinforced polymer composite beams and laminated composite beams using spectral finite elements

This paper presents a spectral finite element formulation for uniform and tapered rotating CNT embedded polymer composite beams. The exact solution to the governing differential equation of a rotating Euler–Bernoulli beam with maximum centrifugal force is used as an interpolating function for the spectral element formulation. Free vibration and wave propagation analysis is carried out using the formulated spectral element. The present study shows the substantial effect of volume fraction and L/D ratio of CNTs in a beam on the natural frequency, impulse response and wave propagation characteristics of the rotating beam. It is found that the CNTs embedded in the matrix can make the rotating beam non-dispersive in nature at higher rotation speeds. Embedded CNTs can significantly alter the dynamics of polymer-nanocomposite beams. The results are also compared with those obtained for carbon fiber reinforced laminated composite rotating beams. It is observed that CNT reinforced rotating beams are superior in performance compared to laminated composite rotating beams.

A Time Domain Simulation Approach for Micro Milling Processes

Prediction of Process Machine Interactions (PMI) can be achieved using models for conventional milling processes in the time and frequency domain. However, instabilities such as regenerative chatter can also be observed in micro milling processes using filigree cemented carbide end mills. The dominant chatter frequencies are basically related to the end mills eigenfrequencies. Nevertheless, the dynamic characteristics of the machine tool structure and the work piece must not be neglected. In this paper a comprehensive time domain process model is presented. The machine tool dynamics are considered as non-coupled oscillators based on measured frequency response functions at the tool holder. The end mill is modeled as rotating Euler-Bernoulli beam with variable boundary conditions at the tool clamping. At first, the parameter identification for a geometric cutting force model is described. It contains the cutting edge radius as a time-depended parameter. Thereafter, the modeling and parameter identification of the structural parts are presented. A stability criterion is defined with special regard to the tool deflection at the TCP. Finally, simulation results at different stable and unstable operating points for full immersion cutting are discussed in detail and compared to experimental tests.

Vibration Control of Nonlinear Rotating Beam Using Piezoelectric Actuator and Sliding Mode Approach

In this research, we develop a general methodology for the vibration control of a nonlinear rotating beam. The dynamic model of a rotating Euler-Bernoulli beam integrated with a piezoelectric actuator is formulated. An integral sliding mode approach is proposed for the vibration control of the system with nonlinear coupling effect between the hub rotation and the beam transverse vibration. The vibration control is achieved by using the piezoelectric actuator only, whereas the motor torque is treated as a time-dependent external input. In the sliding mode control design, the sliding manifold is constructed using only the partial states of the system that are associated with the beam vibration. Particularly, utilizing the internal dynamics analysis, the nonlinear coupling effect of the rigid-body rotation and the beam transverse vibration is decoupled, and the robust stability of the system is guaranteed. This integral sliding mode control is continuous in nature, which can alleviate or avoid the chattering problem. A series of simulation studies demonstrate that the proposed control method can effectively suppress the beam vibration induced by the hub rotation and the external disturbance.