scholarly journals STABILIZATION OF THE GENERALIZED RAO-NAKRA BEAM BY PARTIAL VISCOUS DAMPING

2021 ◽  
Author(s):  
Mohammad AKIL ◽  
Zhuangyi Liu
Keyword(s):  
Wave Equations ◽  
Sufficient Conditions ◽  
Beam Equation ◽  
Viscous Damping ◽  
Bernoulli Beam ◽  
Polynomial Stability ◽  
Transversal Displacement ◽  
Dissipative Mechanism ◽  
Necessary And Sufficient ◽  
Euler Bernoulli Beam

In this paper, we consider the stabilization of the generalized Rao-Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler-Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group two to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group one are left for future study.

The inverse problem for the Euler-Bernoulli beam

1986 ◽  
Vol 407 (1832) ◽  
pp. 199-218 ◽  
Keyword(s):  
Inverse Problem ◽  
Spectral Data ◽  
Sufficient Conditions ◽  
Sectional Area ◽  
Bernoulli Beam ◽  
Scaling Factors ◽  
Cross Sectional ◽  
End Conditions ◽  
Necessary And Sufficient ◽  
Euler Bernoulli Beam

It has long been known that two scaling factors and three spectra, corresponding to three different end-conditions, are required to determine the cross-sectional area A(x) and second moment of area I(x) of an Euler-Bernoulli beam. What has not been known are the necessary and sufficient conditions on the spectral data which will yield positive functions A(x), I(x) . Such a set of conditions is derived in this paper.

scholarly journals Stabilization of Variable Coefficients Euler-Bernoulli Beam with Viscous Damping under a Force Control in Rotation and Velocity Rotation

2017 ◽  
Vol 9 (6) ◽  
pp. 1
Author(s):  
Bomisso G. Jean Marc ◽  
Tour\'{e} K. Augustin ◽  
Yoro Gozo
Keyword(s):  
Exponential Stability ◽  
Force Control ◽  
Riesz Basis ◽  
Closed Loop ◽  
Variable Coefficients ◽  
Viscous Damping ◽  
Closed Loop System ◽  
Bernoulli Beam ◽  
Spectral System ◽  
Euler Bernoulli Beam

This paper investigates the problem of exponential stability for a damped Euler-Bernoulli beam with variable coefficients clamped at one end and subjected to a force control in rotation and velocity rotation. We adopt the Riesz basis approach for show that the closed-loop system is a Riesz spectral system. Therefore, the exponential stability and the spectrum-determined growth condition are obtained.

scholarly journals Application of ADM Using Laplace Transform to Approximate Solutions of Nonlinear Deformation for Cantilever Beam

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Ratchata Theinchai ◽  
Siriwan Chankan ◽  
Weera Yukunthorn
Keyword(s):  
Laplace Transform ◽  
Cantilever Beam ◽  
Adomian Decomposition Method ◽  
Small Deformation ◽  
Approximate Solutions ◽  
Beam Equation ◽  
Bernoulli Beam ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Euler Bernoulli Beam

We investigate semianalytical solutions of Euler-Bernoulli beam equation by using Laplace transform and Adomian decomposition method (LADM). The deformation of a uniform flexible cantilever beam is formulated to initial value problems. We separate the problems into 2 cases: integer order for small deformation and fractional order for large deformation. The numerical results show the approximated solutions of deflection curve, moment diagram, and shear diagram of the presented method.

Comparison of Methods for Modeling a Hydraulic Loader Crane With Flexible Translational Links

2015 ◽  
Vol 137 (10) ◽  
Author(s):  
Henrik C. Pedersen ◽  
Torben O. Andersen ◽  
Brian K. Nielsen
Keyword(s):  
Order Approximation ◽  
Mode Shapes ◽  
Beam Equation ◽  
Flexible Structure ◽  
Bernoulli Beam ◽  
Finite Segment ◽  
Single Beam ◽  
Beam Elements ◽  
Euler Bernoulli Beam ◽  
Mode Order

When modeling flexible robots and structures for control purposes, most often the assumed modes (AMs) method is used to describe the deformation in combination with a floating reference frame formulation. This typically has the benefit of obtaining a low-order, but accurate model of the flexible structure, if the number of modes and AMs are properly chosen. The basis for using this method is, however, that the vibrations (deflections) are time and position independent, i.e., the expression is separable in space and time. This holds for the classic Euler–Bernoulli beam equation, but essentially does not hold for translational links. Hence, special care has to be taken when including flexible translational links. In the current paper, different methods for modeling a hydraulic loader crane with a telescopic arm are investigated and compared using both the finite segment (FS) and AMs method. The translational links are approximated by a single beam, respectively, multiple beam elements, with both one and two modes and using different mode shapes. The models are all validated against experimental data and the comparison is made for different operating scenarios. Based on the results, it is found that in most cases a single beam, low mode order approximation is sufficient to accurately model the mechanical structure and this yields similar results as the FS method.

Noether Symmetry Analysis of the Dynamic Euler-Bernoulli Beam Equation

2016 ◽  
Vol 71 (5) ◽  
pp. 447-456 ◽  
Author(s):  
A.G. Johnpillai ◽  
K.S. Mahomed ◽  
C. Harley ◽  
F.M. Mahomed
Keyword(s):  
Conservation Laws ◽  
Power Law ◽  
Numerical Solutions ◽  
Noether Symmetry ◽  
Numerical Code ◽  
Beam Equation ◽  
Noether Symmetries ◽  
Bernoulli Beam ◽  
Symmetry Classification ◽  
Euler Bernoulli Beam

AbstractWe study the fourth-order dynamic Euler-Bernoulli beam equation from the Noether symmetry viewpoint. This was earlier considered for the Lie symmetry classification. We obtain the Noether symmetry classification of the equation with respect to the applied load, which is a function of the dependent variable of the underlying equation. We find that the principal Noether symmetry algebra is two-dimensional when the load function is arbitrary and extends for linear and power law cases. For all cases, for each of the Noether symmetries associated with the usual Lagrangian, we construct conservation laws for the equation via the Noether theorem. We also provide a basis of conservation laws by using the adjoint algebra. The Noether symmetries pick out the special value of the power law, which is –7. We consider the Noether symmetry reduction for this special case, which gives rise to a first integral that is used for our numerical code. For this, we then find numerical solutions using an in-built function in MATLAB called bvp4c, which is a boundary value solver for differential equations that are depicted in five figures. The physical solutions obtained are for the deflection of the beam with an increase in displacement. These are given in four figures and discussed.

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