Dynamics of Geometrically Exact Sandwich Beams/1-D Plates: Computational Aspects

1995 ◽  
Author(s):  
L. Vu-Quoc ◽  
H. Deng
Keyword(s):  
Cross Section ◽  
Piecewise Linear ◽  
Equation Of Motion ◽  
Initial Position ◽  
Galerkin Projection ◽  
Multilayer Structures ◽  
Sandwich Beams ◽  
Finite Rotations ◽  
Numerical Examples ◽  
Computational Aspects

Abstract A Galerkin projection of the equation of motion of geometrically-exact sandwich beams, applicable to 1-D plates, formulated in Vu-Quoc & Ebcioğlu [1995] is presented here. The beam could take any arbitrary initial position in the 2-D space. Each layer of the beam could have different material constants with no restriction on the mass distribution and layer thickness; further, each layer could take on different length, thus allows the modeling of multilayer structures with ply drop-off. Finite rotations and shear deformation are accommodated for in each layer. The deformed cross section of the beam is continuous, piecewise linear. The continuity of displacement across the interlayer boundaries is exactly enforced. Extensive numerical examples including sandwich structures with ply drop-off, free flying and spin up maneuver of sandwich beams are presented to illustrate the applicability and versatility of the proposed formulation.

Galerkin Projection for Geometrically Exact Sandwich Beams Allowing for Ply Drop-off

1995 ◽  
Vol 62 (2) ◽  
pp. 479-488 ◽  
Author(s):  
Loc Vu-Quoc ◽  
H. Deng
Keyword(s):  
Finite Element ◽  
Large Deformation ◽  
Shear Deformation ◽  
Galerkin Projection ◽  
Element Formulation ◽  
Recent Theory ◽  
Sandwich Beams ◽  
Finite Rotations ◽  
Numerical Examples ◽  
Number Of Layers

A Galerkin projection of the equations of equilibrium for a recent theory of geometrically exact sandwich beams that allow finite rotations and shear deformation in each layer is presented. The continuity of the displacement across the layers is exactly satisfied. The resulting finite element formulation can accommodate large deformation. The number of layers is variable, with layer lengths and thicknesses not required to be the same, thus allowing the modeling of sandwich structures with ply drop-off. Numerical examples are presented which underline the salient features of the formulation. Saint-Venant principle is demonstrated for very short sandwich beams.

Dynamic Formulation for Geometrically Exact Sandwich Beams and One-Dimensional Plates

1995 ◽  
Vol 62 (3) ◽  
pp. 756-763 ◽  
Author(s):  
Loc Vu-Quoc ◽  
I. K. Ebcioglu
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Inertial Frame ◽  
Multibody System ◽  
Beam Dynamics ◽  
Sandwich Beams ◽  
Finite Rotations ◽  
One Dimensional ◽  
Dynamic Formulation ◽  
Consistent Linearization

A new theory of sandwich beams/one-dimensional plates is presented with finite rotations and shear allowed in each layer. The layers, variable in number from one to three, need not have the same thickness and the same length, thus allowing for ply drop-off. Restricting to planar deformation, the cross section has a motion identical to that of a multibody system that consists of rigid links connected by hinges. Large deformation and large overall motion are accommodated, with the beam dynamics referred directly to an inertial frame. An important approximated theory is developed from the general nonlinear equations. The classical linear theory is recovered by consistent linearization.

Computational Aspects of the Analysis of Piecewise Linear Systems

1998 ◽  
Vol 31 (17) ◽  
pp. 85-90
Author(s):  
T. Manavis ◽  
C. Yfoulis ◽  
A. Muir ◽  
N.B.O.L. Pettit ◽  
P.E. Wellstead
Keyword(s):  
Linear Systems ◽  
Piecewise Linear ◽  
Piecewise Linear Systems ◽  
Computational Aspects

scholarly journals Multi-moving Loads Induced Vibration of FG Sandwich Beams Resting on Pasternak Elastic Foundation

2021 ◽  
Vol 18 (9) ◽  
Author(s):  
Wachirawit SONGSUWAN ◽  
Monsak PIMSARN ◽  
Nuttawit WATTANASAKULPONG
Keyword(s):  
Dynamic Response ◽  
Elastic Foundation ◽  
Excitation Frequency ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Sandwich Beams ◽  
Moving Loads ◽  
Layer Thickness Ratio ◽  
Pasternak Elastic Foundation

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

Dynamics of a Continuous System With Clearance and Motion Limiting Stops

1999 ◽  
Author(s):  
P. Metallidis ◽  
S. Natsiavas
Keyword(s):  
Piecewise Linear ◽  
Research Work ◽  
Continuous System ◽  
Continuous Model ◽  
Equation Of Motion ◽  
Model System ◽  
Periodic Motions ◽  
System Parameters ◽  
Rigid Rods ◽  
The Stability

Abstract The present study generalises previous research work on the dynamics of discrete oscillators with piecewise linear characteristics and investigates the response of a continuous model system with clearance and motion-limiting constraints. More specifically, in the first part of this work, an analysis is presented for determining exact periodic response of a periodically excited deformable rod, whose motion is constrained by a flexible obstacle. This methodology is based on the exact solution form obtained within response intervals where the system parameters remain constant and its behavior is governed by a linear equation of motion. The unknowns of the problem are subsequently determined by imposing an appropriate set of periodicity and matching conditions. The analytical part is complemented by a suitable method for determining the stability properties of the located periodic motions. In the second part of the study, the analysis is applied to several cases in order to investigate the effect of the system parameters on its dynamics. Special emphasis is placed on comparing these results with results obtained for similar but rigid rods. Finally, direct integration of the equation of motion in selected areas reveals the existence of motions, which are more complicated than the periodic motions determined analytically.

Fast Solution of Transient Elastohydrodynamic Line Contact Problems Using the Trajectory Piecewise Linear Approach

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Daniel Maier ◽  
Corinna Hager ◽  
Hartmut Hetzler ◽  
Nicolas Fillot ◽  
Philippe Vergne ◽  
...  
Keyword(s):  
Piecewise Linear ◽  
Contact Problems ◽  
Galerkin Projection ◽  
Line Contact ◽  
Reduced Order ◽  
Fast Solution ◽  
Speed Up ◽  
Newton Raphson ◽  
First Time ◽  
Piecewise Linear Approach

In order to obtain a fast solution scheme, the trajectory piecewise linear (TPWL) method is applied to the transient elastohydrodynamic (EHD) line contact problem for the first time. TPWL approximates the nonlinearity of a dynamical system by a weighted superposition of reduced linearized systems along specified trajectories. The method is compared to another reduced order model (ROM), based on Galerkin projection, Newton–Raphson scheme and an approximation of the nonlinear reduced system functions. The TPWL model provides further speed-up compared to the Newton–Raphson based method at a high accuracy.

Study on Micropump Using Boiling Bubbles

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Yasuo Koizumi ◽  
Hiroyasu Ohtake
Keyword(s):  
Cross Section ◽  
Flow Rate ◽  
Equation Of Motion ◽  
Flow Channel ◽  
Stationary Condition ◽  
Short Channel ◽  
Short Side ◽  
Liquid Plug ◽  
Average Flow Velocity ◽  
Long Channel

A micropump was developed using boiling and condensation in a microchannel. The length and hydraulic diameter of the semi-half-circle cross-section microchannel having two open tanks at both ends were 26mm and 0.465mm, respectively. A 0.5×0.5mm2 electrically heated patch was located at the offset location from the center between both ends of the microchannel, at a distance of 8.5mm from one end and at a distance of 17mm from the other end. The microchannel and the two open tanks were filled with distilled water. The heating patch was heated periodically to cause cyclic formation of a boiling bubble and its condensation. By this procedure, flow from the short side (8.5mm side) to the long side was created. The flow rate increased as the heating rate was increased. The obtained maximum average flow velocity and flow rate were 10.4mm∕s and 2.16mm3∕s, respectively. The velocity of an interface between the bubble and the liquid plug during the condensing period was much faster than that during the boiling period. During the condensing period, the velocity of the interface at the short channel side (8.5mm side) was faster than that at the long channel side (17mm side). The equation of motion of liquid in the flow channel was solved in order to calculate the travel of liquid in the flow channel. The predicted velocities agreed well with the experimental results. The velocity differences between the short side and the long side, as well as those between the boiling period and the condensing period, were expressed well by the calculation. Liquid began to move from the stationary condition during both the boiling and the condensing periods. The liquid in the inlet side (short side) moved faster than that in the outlet side (long side) during the condensing period because the inertia in the short side was lower than that in the long side. Since the condensation was much faster than boiling, this effect was more prominent during the condensing period. By iterating these procedures, the net flow from the short side to the long side was created.

The Scattering Waves by Two Spheres in Solid

2013 ◽  
Vol 423-426 ◽  
pp. 1640-1643
Author(s):  
Yan Ru Zhang ◽  
Pei Jun Wei
Keyword(s):  
Wave Function ◽  
Cross Section ◽  
Spherical Wave ◽  
Addition Theorem ◽  
Wave Functions ◽  
Interface Condition ◽  
Elastic Sphere ◽  
Numerical Examples ◽  
Transmitted Waves ◽  
Expansion Coefficients

The scattering waves by two elastic spheres in solid are studied. The incident wave, the scattering waves in the host and the transmitted waves in the elastic spheres are all expanded in the series form of spherical wave functions. The total waves are obtained by addition of all scattered waves from individual elastic sphere. The addition theorem of spherical wave function is used to perform the coordinates transform for the scattering waves from different spheres. The expansion coefficients of scattering waves are determined by the interface condition between the elastic spheres and the solid host. The scattering cross section is computed as numerical examples.

Galerkin Projections for Delay Differential Equations

2003 ◽  
Author(s):  
Pankaj Wahi ◽  
Anindya Chatterjee
Keyword(s):  
Delay Differential Equations ◽  
Galerkin Projection ◽  
Bifurcation Diagrams ◽  
Shape Functions ◽  
Projection Technique ◽  
Numerical Examples ◽  
Finite Dimensional ◽  
Qualitative Dynamics ◽  
Delay Differential ◽  
Galerkin Projections

We present a Galerkin projection technique by which finite-dimensional ODE approximations for DDE’s can be obtained in a straightforward fashion. The technique requires neither the system to be near a bifurcation point, nor the delayed terms to have any specific restrictive form, nor even the delay, nonlinearities and/or forcing to be small. We show through several numerical examples that the systems of ODE’s obtained using this procedure can accurately capture the dynamics of the DDE’s under study, and that the accuracy of solutions increases with increasing numbers of shape functions used in the Galerkin projection. Examples studied here include a linear constant coefficient DDE as well as forced nonlinear DDE’s with one or more delays and possibly nonlinear delayed terms. Parameter studies, with associated bifurcation diagrams, show that the qualitative dynamics of the DDE’s can be captured satisfactorily with a modest number of shape functions in the Galerkin projection.

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