Resonances for the beam equation

2016 ◽  
Author(s):  
Evgeny L. Korotyaev ◽  
Andrey Badanin
Keyword(s):  
Beam Equation

Microstructure simulation of early paper forming using immersed boundary methods

TAPPI Journal ◽  
2011 ◽  
Vol 11 (11) ◽  
pp. 23-30 ◽  
Author(s):  
ANDREAS MARK ◽  
ERIK SVENNING ◽  
ROBERT RUNDQVIST ◽  
FREDRIK EDELVIK ◽  
ERIK GLATT ◽  
...  
Keyword(s):  
Early Paper ◽  
Contact Model ◽  
Fiber Suspension ◽  
Beam Equation ◽  
Immersed Boundary ◽  
Paper Machine ◽  
Immersed Boundary Methods ◽  
Element Discretization ◽  
Microstructure Simulation ◽  
Boundary Methods

Paper forming is the first step in the paper machine where a fiber suspension leaves the headbox and flows through a forming fabric. Complex physical phenomena occur as the paper forms, during which fibers, fillers, fines, and chemicals added to the suspension interact. Understanding this process is important for the development of improved paper products because the configuration of the fibers during this step greatly influences the final paper quality. Because the effective paper properties depend on the microstructure of the fiber web, a continuum model is inadequate to explain the process and the properties of each fiber need to be accounted for in simulations. This study describes a new framework for microstructure simulation of early paper forming. The simulation framework includes a Navier-Stokes solver and immersed boundary methods to resolve the flow around the fibers. The fibers were modeled with a finite element discretization of the Euler-Bernoulli beam equation in a co-rotational formulation. The contact model is based on a penalty method and includes friction and elastic and inelastic collisions. We validated the fiber model and the contact model against demanding test cases from the literature, with excellent results. The fluid-structure interaction in the model was examined by simulating an elastic beam oscillating in a cross flow. We also simulated early paper formation to demonstrate the potential of the proposed framework.

Long-time dynamics of a class of nonlocal extensible beams with time delay

2021 ◽  
pp. 108128652110194
Author(s):  
Fengjuan Meng ◽  
Cuncai Liu ◽  
Chang Zhang
Keyword(s):  
Time Delay ◽  
Stability Property ◽  
Beam Equation ◽  
Exponential Attractors ◽  
Time Dynamics ◽  
Bounded Smooth Domain ◽  
Finite Dimensional ◽  
Long Time ◽  
Bounded Smooth ◽  
Long Time Dynamics

This work is devoted to the following nonlocal extensible beam equation with time delay: [Formula: see text] on a bounded smooth domain [Formula: see text]. The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.

Study on Sound Field of High Intensity Focused Ultrasound with Short Focal Length

2012 ◽  
Vol 195-196 ◽  
pp. 364-369 ◽  
Author(s):  
Jin Hua Zhao ◽  
Li Li Yu ◽  
Chun Hui ◽  
Bin Feng Huang ◽  
Chao Li ◽  
...  
Keyword(s):  
Nonlinear Effect ◽  
Sound Pressure ◽  
Focal Length ◽  
Sound Field ◽  
High Intensity Focused Ultrasound ◽  
Beam Equation ◽  
Focal Region ◽  
Interface Position ◽  
Short Focal Length ◽  
Planar Wave

In this paper, numerical simulation of sound field with short focal length is performed, which is based on spheroidal beam equation (SBE) in frequency-domain for transducer with a wide aperture angle. And we made some experiments on vitro bovine liver to explore the characteristic of sound pressure and-3dB sound focal region at different positions of incident interface. It is found that with a fixed curvature radius if the focal length is shorter under the skin, the amplitude of sound pressure will be higher on the focus and the shape of-3dB sound focal region will be smaller. When the incident interface is in the range of planar wave, nonlinear effect is strong and the focus will change with the interface position. Especially when the position is near to transition location between planar wave and spheroidal wave, the nonlinear effect is lowered. While the focus is closer to the sound source so as to burn the scarfskin easily. When the interface is in the range of spheroidal wave, the focus position changes little but the side lobe effect due to refraction is obvious. And the focusing performance of transducer will be affected. The experimental results validate the accuracy of theoretical results. It is concluded that the position of incident interface should be selected reasonably with short focal length in the treatment of superficial tissue.

scholarly journals Isospectrals of non-uniform Rayleigh beams with respect to their uniform counterparts

2018 ◽  
Vol 5 (2) ◽  
pp. 171717 ◽  
Author(s):  
Srivatsa Bhat K ◽  
Ranjan Ganguli
Keyword(s):  
Boundary Condition ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Beam Equation ◽  
The Finite Element Method ◽  
Rayleigh Beam ◽  
Uniform Beam ◽  
Governing Differential Equation ◽  
The Given

In this paper, we look for non-uniform Rayleigh beams isospectral to a given uniform Rayleigh beam. Isospectral systems are those that have the same spectral properties, i.e. the same free vibration natural frequencies for a given boundary condition. A transformation is proposed that converts the fourth-order governing differential equation of non-uniform Rayleigh beam into a uniform Rayleigh beam. If the coefficients of the transformed equation match with those of the uniform beam equation, then the non-uniform beam is isospectral to the given uniform beam. The boundary-condition configuration should be preserved under this transformation. We present the constraints under which the boundary configurations will remain unchanged. Frequency equivalence of the non-uniform beams and the uniform beam is confirmed by the finite-element method. For the considered cases, examples of beams having a rectangular cross section are presented to show the application of our analysis.

On the Effects of a Flexible Structure on Boundary Layer Stability and Transition

2011 ◽  
Vol 133 (7) ◽  
Author(s):  
Ashraf Al Musleh ◽  
Abdelkader Frendi
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Young Modulus ◽  
Boundary Layer Transition ◽  
Beam Equation ◽  
Flexible Structure ◽  
Boundary Layer Stability ◽  
Wall Shear ◽  
Fluid Wall Shear Stress

Delaying the onset of boundary layer transition has become a major research area in the last few years. This delay can be achieved by either active or passive control techniques. In the present paper, the effects of flexible or compliant structures on boundary layer stability and transition is studied. The Orr-Sommerfeld equation coupled to a beam equation representing the flexible structure is solved for a Blasius type boundary layer. A parametric study consisting of the beam thickness and material properties is carried out. In addition, the effect of fluid wall shear stress on boundary layer stability is analyzed. It is found that high density and high Young modulus materials behave like rigid structures and therefore do not alter the stability characteristic of the boundary layer. Whereas low density and low Young modulus materials are found to stabilize the boundary layer. High values of fluid wall shear stress are found to destabilize the boundary layer. Our results are in good agreement with those published in the literature.

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