scholarly journals Location of eigenmodes of Euler–Bernoulli beam model under fully non-dissipative boundary conditions

2019 ◽  
Vol 475 (2231) ◽  
pp. 20190544
Author(s):  
Marianna A. Shubov
Keyword(s):  
Boundary Conditions ◽  
Half Plane ◽  
Adjoint Problem ◽  
Beam Model ◽  
Control Parameters ◽  
Bernoulli Beam ◽  
Left Half ◽  
The Right ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

The Euler–Bernoulli beam model with non-conservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and off-diagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped–free self-adjoint problem have been established.

scholarly journals Numerical Solution of Bending of the Beam with Given Friction

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 898
Author(s):  
Michaela Bobková ◽  
Lukáš Pospíšil
Keyword(s):  
Sliding Friction ◽  
Quadratic Function ◽  
Building Construction ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Sustainable Building ◽  
Construction Design ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model ◽  
Fixed Beam

We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks.

Vibration Analysis of Drug Delivery CNTs Using Transfer Matrix Method

2011 ◽  
Author(s):  
Anooshiravan Farshidianfar ◽  
Ali A. Ghassabi ◽  
Mohammad H. Farshidianfar ◽  
Mohammad Hoseinzadeh
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Equation Of Motion ◽  
Beam Model ◽  
Multi Wall Carbon Nanotubes ◽  
Resonance Frequencies ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

The free vibration and instability of fluid-conveying multi-wall carbon nanotubes (MWCNTs) are studied based on an Euler-Bernoulli beam model. A theory based on the transfer matrix method (TMM) is presented. The validity of the theory was confirmed for MWCNTs with different boundary conditions. The effects of the fluid flow velocity were studied on MWCNTs with simply-supported and clamped boundary conditions. Furthermore, the effects of the CNTs’ thickness, radius and length were investigated on resonance frequencies. The CNT was found to posses certain frequency behaviors at different geometries. The effect of the damping corriolis term was studied in the equation of motion. Finally, a useful simplification is introduced in the equation of motion.

Spectral analysis of Euler–Bernoulli beam model with distributed damping and fully non-conservative boundary feedback matrix

2021 ◽  
pp. 1-38
Author(s):  
Marianna A. Shubov
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Damping Coefficient ◽  
Control Input ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Boundary Feedback ◽  
Feedback Matrix ◽  
The Right ◽  
Euler Bernoulli Beam

The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms.

scholarly journals Critical Velocity of High-Performance Yarn Transversely Impacted by Razor Blade

Fibers ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 95 ◽  
Author(s):  
Boon Lim ◽  
Jou-Mei Chu ◽  
Benjamin Claus ◽  
Yizhou Nie ◽  
Wayne Chen
Keyword(s):  
Critical Velocity ◽  
High Speed ◽  
High Performance ◽  
Hertzian Contact ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Razor Blade ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model ◽  
Hertzian Contact Model

A ballistic parameter that influences the ballistic performances of a high-performance yarn is the critical velocity. The critical velocity is defined as the projectile striking velocity that causes instantaneous rupture of the yarn upon impact. In this study, we performed ballistic experiments to determine the critical velocity of a Twaron® yarn transversely impacted by a razor blade. A high-speed camera was integrated into the experimental apparatus to capture the in-situ deformation of the yarn. The experimental critical velocity demonstrated a reduction compared to the critical velocity predicted by the classical theory. The high-speed images revealed the yarn specimen failed from the projectile side toward the free end when impacted by the razor blade. To improve the prediction capability, the Euler–Bernoulli beam and Hertzian contact models were used to predict the critical velocity. For the Euler–Bernoulli beam model, the critical velocity was obtained by assuming the specimen ruptured instantaneously when the maximum flexural strain reached the ultimate tensile strain of the yarn upon impact. On the other hand, for the Hertzian contact model, the yarn was assumed to fail when the indentation depth was equivalent to the diameter of the yarn. The errors between the average critical velocities determined from experiments and the predicted critical velocities were around 19% and 48% for the Euler–Bernoulli beam model and Hertzian contact model, respectively.

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