On Computation of the Motion of Elastic Rods

1974 ◽  
Vol 41 (3) ◽  
pp. 777-780 ◽  
Author(s):  
R. P. Nordgren

A computational method is developed for the finite-amplitude three-dimensional motion of inextensible elastic rods with equal principal stiffnesses. The method also applies to the two-dimensional motion of such rods with unequal principal stiffnesses. For these two classes of problems the equations of the classical theory of rods are reduced to a non-linear vector equation of motion together with the inextensibility condition and appropriate boundary and initial conditions. Consistent finite-difference approximations are introduced and a semi-explicit method of solution is devised. The approximate limitation for numerical stability of the method is shown to be the same as for the usual explicit method in linear beam dynamics. By way of example the method is applied to the free fall of a circular pipe through water onto a rigid plane from a suspended initial configuration.

2021 ◽  
Vol 263 (5) ◽  
pp. 1041-1052
Author(s):  
Martin Richter ◽  
Gregor Tanner ◽  
Bruno Carpentieri ◽  
David J. Chappell

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.


2011 ◽  
Vol 121-126 ◽  
pp. 3657-3661
Author(s):  
Dun Zhang ◽  
Yuan Zheng ◽  
Ying Zhao ◽  
Jian Jun Huang

Numerical simulation of three-dimensional transient turbulent flow in the whole flow passage of a Francis turbine were based upon the large eddy simulation(LES) technique on Smargorinsky model and sliding mesh technology. The steady flow data simulated with the standard k-εmodel was used as the initial conditions for the unsteady simulation. The results show that LES can do well transient turbulent flow simulation in a Francis turbine with complex geometry. The computational method provides some reference for exploring the mechanism of eddy formation in a complex turbulent of hydraulic machinery.


2013 ◽  
Vol 732 ◽  
pp. 77-104 ◽  
Author(s):  
Cunbiao Lee ◽  
Zhuang Su ◽  
Hongjie Zhong ◽  
Shiyi Chen ◽  
Mingde Zhou ◽  
...  

AbstractThe free-fall motion of a thin disk with small dimensionless moments of inertia (${I}^{\ast } \lt 1{0}^{- 3} $) was investigated experimentally. The transition from two-dimensional zigzag motion to three-dimensional spiral motion occurs due to the growth of three-dimensional disturbances. Oscillations in the direction normal to the zigzag plane increase with the development of this instability. At the same time, the oscillation of the nutation angle decreases to zero and the angle remains constant. The effects of initial conditions (release angle) were investigated. Two kinds of transition modes, zigzag–spiral transition and zigzag–spiral–zigzag intermittence transition, were observed to be separated by a critical Reynolds number. In addition, the solution of the generalized Kirchhoff equations shows that the small ${I}^{\ast } $ is responsible for the growth of disturbances in the third dimension (perpendicular to the planar motion).


Author(s):  
John Wiechel ◽  
Sandra Metzler ◽  
Dawn Freyder ◽  
Nick Kloppenborg

Reconstructing the mechanics and determining the cause of a person falling from a height in the absence of witness observations or a statement from the victim can be quite challenging. Often there is little information available beyond the final resting position of the victim and the injuries they sustained. The mechanics of a fall must follow the physics of falling bodies and this physics provides an additional source of information about how the fall occurred. Computational, physics-based simulations can be utilized to model the free-fall portion of the fall kinematics and to analyze biomechanical injury mechanisms. However, an accurate determination of the overall fall kinematics, including the initial conditions and any specific contributions of the person(s) involved, must include the correct position and posture of the individual prior to the fall. Frequently this phase of the analysis includes voluntary movement on the part of the fall victim, which cannot be modeled with simulations using anthropomorphic test devices (ATDs). One approach that has been utilized in the past to overcome this limitation is to run the simulations utilizing a number of different initial conditions for the fall victim. While fall simulations allow the initial conditions of the fall to be varied, they are unable to include the active movement of the subject, and the resulting interaction with other objects in the environment immediately prior to or during the fall. Furthermore, accurate contact interactions between the fall victim and multiple objects in their environment can be difficult to model within the simulation, as they are dependent on the knowledge of material properties of these objects and the environment such as elasticity and damping. Motion capture technology, however, allows active subject movement and behaviors to be captured in a quantitative, three-dimensional manner. This information can then be utilized within the fall simulation to more accurately model the initial fall conditions. This paper presents a methodology for reconstructing fall mechanics using a combination of motion capture, human body simulation, and injury biomechanics. This methodology uses as an example a fall situation where interaction between the fall victim and specific objects in the environment, as well as voluntary movements by the fall victim immediately prior to the accident, provided information that could not be otherwise obtained. Motion capture was first used to record the possible motions of a person in the early stages of the fall. The initial position of the fall victim within the physics based simulation of the body in free fall was determined utilizing the individual body segment and joint angles from the motion capture analysis. The methodology is applied to a real world case example and compared with the actual outcome.


2019 ◽  
Vol 880 ◽  
Author(s):  
Stefan Zammert ◽  
Bruno Eckhardt

Plane Poiseuille flow, the pressure-driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien–Schlichting (TS) waves, and another route, the bypass transition, that can be triggered with finite-amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance $2H$ apart, and in a domain of width $2\unicode[STIX]{x03C0}H$ and length $2\unicode[STIX]{x03C0}H$, the subcritical instability to TS waves sets in at $Re_{c}=5815$ and extends down to $Re_{TS}\approx 4884$. The bypass route becomes available above $Re_{E}=459$ with the appearance of three-dimensional, finite-amplitude travelling waves. Below $Re_{c}$, TS transition appears for a tiny region of initial conditions that grows with increasing Reynolds number. Above $Re_{c}$, the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent state in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.


Fluids ◽  
2021 ◽  
Vol 6 (10) ◽  
pp. 348
Author(s):  
Thomas Meunier ◽  
J. H. LaCasce

The finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here, we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second-order velocity structure function.


1979 ◽  
Vol 46 (1) ◽  
pp. 132-138 ◽  
Author(s):  
J. W. Klahs ◽  
J. H. Ginsberg

The equations of motion governing the three-dimensional finite-amplitude response of a plate in arbitrary space motion are derived and shown to lead to dynamic coupling between the transverse and in-plane displacement. A general method of solution for such problems is demonstrated in an example involving a simply supported rectangular plate spinning about an axis parallel to an edge and nutating through a small angle. The method involves an asymptotic expansion using the derivative expansion version of the method of multiple time scales, in conjunction with the Galerkin method. A critical spin rate leading to the loss of stability in divergence is determined. Then, a numerical example of resonant excitation of one principal coordinate demonstrates that the nonlinear response resembling the one obtained from linear theory may lose stability in favor of a second response in which several principal coordinates are mutually excited. Consideration of the interaction between in-plane and transverse displacements is shown to be crucial to the prediction of this “unusual” response.


Author(s):  
Thomas Meunier ◽  
J.H. LaCasce

The Finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990’s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second order velocity structure function.


2009 ◽  
Vol 635 ◽  
pp. 1-26 ◽  
Author(s):  
NICOLAS PÉRINET ◽  
DAMIR JURIC ◽  
LAURETTE S. TUCKERMAN

We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, p. 49). The finite-amplitude results are compared with the experiments of Kityk et al (Phys. Rev. E, vol. 72, 2005, p. 036209). In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.


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