The existence of a steady state solution for a type of parabolic boundary value problem

1988 ◽  
Vol 19 (3) ◽  
pp. 413-419
Author(s):  
Enrique A. Gonzalez‐Velasco
Keyword(s):  
Steady State ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Steady State Solution ◽  
State Solution ◽  
Parabolic Boundary

Steady Motion of a Slack Belt Drive: Dynamics of a Beam in Frictional Contact With Rotating Pulleys

2020 ◽  
Vol 87 (12) ◽  
Author(s):  
Jakob Scheidl ◽  
Yury Vetyukov
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Boundary Value Problem ◽  
Contact Region ◽  
Boundary Value ◽  
Steady Motion ◽  
Steady State Solution ◽  
Belt Drive ◽  
Creep Theory ◽  
Lagrangian Finite Element

Abstract We seek the steady-state motion of a slack two-pulley belt drive with the belt modeled as an elastic, shear-deformable rod. Dynamic effects and gravity induce significant transverse deflections due to the low pre-tension. In analogy to the belt-creep theory, it is assumed that each contact region between the belt and one of the pulleys consists of a single sticking and a single sliding zone. Based on the governing equations of the rod theory, we for the first time derive the corresponding boundary value problem and integrate it numerically. Furthermore, a novel mixed Eulerian–Lagrangian finite element scheme is developed that iteratively seeks the steady-state solution. Finite element solutions are validated against semi-analytic results obtained by numerical integration of the boundary value problem. Parameter studies are conducted to examine solution dependence on the stiffness coefficients and the belt pre-tension.

BIFURCATION FROM AN EQUILIBRIUM OF THE STEADY STATE KURAMOTO–SIVASHINSKY EQUATION IN TWO SPATIAL DIMENSIONS

2002 ◽  
Vol 12 (01) ◽  
pp. 103-114 ◽  
Author(s):  
CHANGPIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Steady State ◽  
Perturbation Method ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Steady State Solution ◽  
State Solution ◽  
Asymptotic Expressions ◽  
Spatial Dimensions ◽  
Sivashinsky Equation

The paper deals with the steady state bifurcations of the Kuramoto–Sivashinsky (K–S) equation in two spatial dimensions with zero mean and periodic boundary value conditions. Applying the perturbation method, asymptotic expressions of the steady state solution branches that have bifurcated from the equilibrium are obtained. Furthermore, stability of the bifurcated solution branches is discussed.

A Nonlinear Theory of Sloshing in Rectangular Tanks

1974 ◽  
Vol 18 (04) ◽  
pp. 224-241 ◽  
Author(s):  
Odd M. Faltinsen
Keyword(s):  
Steady State ◽  
Power Series ◽  
Boundary Value Problem ◽  
Nonlinear Theory ◽  
Reasonable Agreement ◽  
Steady State Solution ◽  
State Solution ◽  
Two Dimensional ◽  
The Stability ◽  
Theory And Experiment

A two-dimensional, rigid, rectangular, open tank without baffles is forced to oscillate harmonically with small amplitudes of sway or roll oscillation in the vicinity of the lowest natural frequency for the fluid inside the tank. The breadth of the tank is 0(1) and the depth of the fluid is either 0(1) or in-finite. The excitation is 0(ε) and the response is 0(ε1/3). A nonlinear, inviscid boundary-value problem of potential flow is formulated and the steady-state solution is found as a power series in ε1/3 correctly to 0(ε). Comparison between theory and experiment shows reasonable agreement. The stability of the steady-state solution has been studied.

Response of a Submerged Cylindrical Shell to an Axially Propagating Step Wave

1965 ◽  
Vol 32 (4) ◽  
pp. 788-792 ◽  
Author(s):  
M. J. Forrestal ◽  
G. Herrmann
Keyword(s):  
Cylindrical Shell ◽  
Steady State ◽  
Wave Front ◽  
Transverse Shear ◽  
Shell Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Axial Coordinate

An infinitely long, circular, cylindrical shell is submerged in an acoustic medium and subjected to a plane, axially propagating step wave. The fluid-shell interaction is approximated by neglecting fluid motions in the axial direction, thereby assuming that cylindrical waves radiate away from the shell independently of the axial coordinate. Rotatory inertia and transverse shear deformations are included in the shell equations of motion, and a steady-state solution is obtained by combining the independent variables, time and the axial coordinate, through a transformation that measures the shell response from the advancing wave front. Results from the steady-state solution for the case of steel shells submerged in water are presented using both the Timoshenko-type shell theory and the bending shell theory. It is shown that previous solutions, which assumed plane waves radiated away from the vibrating shell, overestimated the dumping effect of the fluid, and that the inclusion of transverse shear deformations and rotatory inertia have an effect on the response ahead of the wave front.

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