Solution of Nonlinear Equilibrium Equations

2005 ◽  
pp. 429-473
Author(s):  
Niels Saabye Ottosen ◽  
Matti Ristinmaa
Keyword(s):  
Equilibrium Equations ◽  
Nonlinear Equilibrium

Instability of Discrete Pseudo-Conservative Structural Systems

1991 ◽  
Vol 44 (11S) ◽  
pp. S194-S198 ◽  
Author(s):  
Anibal E. Mirasso ◽  
Luis A. Godoy
Keyword(s):  
Classical System ◽  
Potential Energy Function ◽  
Structural Systems ◽  
Equilibrium Path ◽  
Equilibrium Equations ◽  
Total Potential ◽  
Approximate Form ◽  
Nonlinear Equilibrium ◽  
Mechanical Models ◽  
New Formulation

Critical and postcritical states of pseudo-conservative discrete structural systems are studied by means of a new formulation leading to a classification of critical states and to an approximate form of the postcritical equilibrium path. The nonlinear equilibrium equations are derived from the total potential energy function of a classical system, but with the addition of at least one control parameter. The follower force effect is thus included by nonlinear constraints to the equilibrium equation. The nonlinear equations are solved by perturbation techniques. Finally the theory is applied to investigate the instability of some simple mechanical models.

NONLINEAR GENERALIZED BEAM THEORY FOR COLD-FORMED STEEL MEMBERS

2003 ◽  
Vol 03 (04) ◽  
pp. 461-490 ◽  
Author(s):  
N. SILVESTRE ◽  
D. CAMOTIM
Keyword(s):  
Numerical Technique ◽  
Beam Theory ◽  
Equilibrium Equations ◽  
Steel Members ◽  
Mode Decomposition ◽  
Nonlinear Equilibrium ◽  
Geometrical Imperfections ◽  
Post Buckling ◽  
Cold Formed Steel ◽  
Generalized Beam Theory

A geometrically nonlinear Generalized Beam Theory (GBT) is formulated and its application leads to a system of equilibrium equations which are valid in the large deformation range but still retain and take advantage of the unique GBT mode decomposition feature. The proposed GBT formulation, for the elastic post-buckling analysis of isotropic thin-walled members, is able to handle various types of loading and arbitrary initial geometrical imperfections and, in particular, it can be used to perform "exact" or "approximate" (i.e., including only a few deformation modes) analyses. Concerning the solution of the system of GBT nonlinear equilibrium equations, the finite element method (FEM) constitutes the most efficient and versatile numerical technique and, thus, a beam FE is specifically developed for this purpose. The FEM implementation of the GBT post-buckling formulation is reported in some detail and then employed to obtain numerical results, which validate and illustrate the application and capabilities of the theory.

scholarly journals Optimal Shape and First Integrals for Inverted Compressed Column

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 334
Author(s):  
Enes Kacapor ◽  
Teodor M. Atanackovic ◽  
Cemal Dolicanin
Keyword(s):  
Variational Principle ◽  
First Integrals ◽  
Arbitrary Cross Section ◽  
Optimal Shape ◽  
Equilibrium Equations ◽  
Concentrated Force ◽  
Cross Sectional ◽  
Nonlinear Equilibrium ◽  
Elastic Column ◽  
Special Case

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.

NUMERICAL ANALYSIS OF CIRCULAR CONCRETE-FILLED STEEL TUBULAR SLENDER BEAM-COLUMNS WITH PRELOAD EFFECTS

2013 ◽  
Vol 13 (03) ◽  
pp. 1250065 ◽  
Author(s):  
VIPULKUMAR ISHVARBHAI PATEL ◽  
QING QUAN LIANG ◽  
MUHAMMAD N. S. HADI
Keyword(s):  
Numerical Model ◽  
Order Effects ◽  
Computational Algorithms ◽  
Equilibrium Equations ◽  
Steel Tubes ◽  
Beam Columns ◽  
Concrete Confinement ◽  
Nonlinear Equilibrium ◽  
Initial Geometric Imperfections ◽  
Loading Eccentricity

This paper presents a new numerical model for the nonlinear analysis of circular concrete-filled steel tubular (CFST) slender beam-columns with preload effects, in which the initial geometric imperfections, deflections caused by preloads, concrete confinement and second order effects are incorporated. Computational algorithms are developed to solve the nonlinear equilibrium equations. Comparative studies are undertaken to validate the accuracy of computational algorithms developed. Also included is a parametric study for examining the effects of the preloads, column slenderness, diameter-to-thickness ratio, loading eccentricity, steel yield stress and concrete confinement on the behavior of circular CFST slender beam-columns under eccentric loadings. The numerical model is demonstrated to be capable of predicting accurately the behavior of circular CFST slender beam-columns with preloads. The preloads on the steel tubes can affect significantly the behavior of CFST slender beam-columns and must be taken into account in the design.

Effects of rotor blade-tip geometry on helicopter trim and control response

2017 ◽  
Vol 121 (1239) ◽  
pp. 637-659 ◽  
Author(s):  
M. Rohin Kumar ◽  
C. Venkatesan
Keyword(s):  
Equations Of Motion ◽  
Rotor Blades ◽  
Equilibrium Equations ◽  
Aerodynamic Model ◽  
Nonlinear Equilibrium ◽  
Control Response ◽  
Blade Tip ◽  
Nonlinear Equations Of Motion ◽  
The Cost ◽  
And Control

ABSTRACTFor performance improvement and noise reduction, swept and anhedral tips have been incorporated in advanced-geometry rotor blades. While there are aerodynamic benefits to these advanced tip geometries, they come at the cost of complicated structural design and weight penalties. The effect of these tip shapes on loads, vibration and aeroelastic response are also unclear. In this study, a comprehensive helicopter aeroelastic analysis which includes rotor-fuselage coupling shall be described and the analysis results for rotor blades with straight tip, tip sweep and tip anhedral shall be presented and compared. The helicopter modelled is a conventional one with a hingeless single main rotor and single tail rotor. The blade undergoes flap, lag, torsion and axial deformations. Tip sweep, pretwist, precone, predroop, torque offset and root offset are included in the model. Aerodynamic model includes Peters-He dynamic wake theory for inflow and the modified ONERA dynamic stall theory for airloads calculations. The complete 6-dof nonlinear equilibrium equations of the fuselage are solved for analysing any general flight condition. Response to pilot control inputs is determined by integrating the full set of nonlinear equations of motion with respect to time. The effects of tip sweep and tip anhedral on structural dynamics, trim characteristics and vehicle response to pilot inputs are presented. It is shown that for blades with tip sweep and tip anhedral/dihedral, the 1/rev harmonics of the root loads reduce while the 4/rev harmonics of the hub loads increase in magnitude. Tip dihedral is shown to induce a reversal of yaw rate for lateral and longitudinal cyclic input.

Fundamental Characteristics of a New Variable Torque Clutch With Skewed Rollers

1999 ◽  
Vol 123 (3) ◽  
pp. 436-446 ◽  
Author(s):  
Ming Feng ◽  
Kyosuke Ono ◽  
Kenji Mimura
Keyword(s):  
Design Parameters ◽  
Equilibrium Equations ◽  
Linear Distribution ◽  
Outer Race ◽  
Geometrical Properties ◽  
Torque Transmission ◽  
Nonlinear Equilibrium ◽  
Torque Capacity ◽  
New Type ◽  
Theoretical Results

In this paper, a new type of a clutch by the name of the variable torque clutch with skewed rollers is first introduced and second investigated both theoretically and experimentally. It is comprised of an inner and an outer race that are each in spatial line contact with the crossed axis cylindrical rollers. Torque transmission is delivered by a slipping induced between the rollers and the races due to skewing the rollers. The equations of the race surfaces are derived and the geometrical properties are analyzed. Based on the kinematic analysis, a roller-wedge model is proposed for this clutch in order to visualize the motion at the tangency of the rollers and the races. By assuming the linear distribution of the contact force along the spatial contact line, the transmitted torque capacity and kinematic characteristics can be evaluated properly from the solution of a set of nonlinear equilibrium equations. Several prototypes of this clutch are manufactured and measured to show the validity of this design idea and the theoretical results. The computational results are found to coincide with the experimental data. In addition, the influences of the design parameters on the fundamental characteristics are discussed in detail.

Buckling of Shear-Deformable Multi-Layered Rings due to Fluid-Pressure Loading

2001 ◽  
Author(s):  
Jack McNamara ◽  
Li Liu ◽  
Anthony M. Waas
Keyword(s):  
Fluid Pressure ◽  
Stacking Sequence ◽  
Finite Element Method Analysis ◽  
Pressure Loading ◽  
Equilibrium Equations ◽  
Nonlinear Equilibrium ◽  
Method Analysis ◽  
Shear Deformable ◽  
Theoretical Results ◽  
Thickness Shear

Abstract This paper is concerned with the analysis of composite rings subjected to external fluid pressure loading. Nonlinear equilibrium equations, linear stability equations, and critical fluid-pressure loads are found for thin multi-layered shear deformable rings. The extensions presented here can be shown to be generalizations of the theory given in [1]. The theory shows that introduction of multiple layers of material introduces coupling between bending and extension. The results are used to show that shear deformation is important when R h < 10 , as well as when the ratio of through thickness shear modulus to Young’s modulus becomes small. The latter has consequences when composite materials are used for the ring layers. The results are also used to show that for coupling between bending and extension the critical fluid-pressure will increase or decrease depending on the stacking sequence. For the example presented in this paper, the predicted critical fluid-pressure loading was higher for the stiffer material located on the inside of a two-layer ring. In all cases, the theoretical results are compared to a finite element method analysis.

Method of integral equations in the polytropic theory of stars with axial rotation. II. Polytropes with indices $n>1$

2021 ◽  
Vol 8 (3) ◽  
pp. 474-485
Author(s):  
M. V. Vavrukh ◽  
◽  
D. V. Dzikovskyi ◽  
Keyword(s):  
Integral Form ◽  
Axial Rotation ◽  
Geometrical Parameters ◽  
Surface Form ◽  
Equilibrium Equations ◽  
Method Of Integral Equations ◽  
Mass Moment ◽  
Self Consistent ◽  
Nonlinear Equilibrium ◽  
Mass Moment Of Inertia

A new method for finding solutions of the nonlinear equilibrium equations for rotational polytropes was proposed, which is based on a self-consistent description of internal region and periphery using the integral form of equations. Dependencies of geometrical parameters, surface form, mass, moment of inertia and integration constants on angular velocity were calculated for indices $n=2.5$ and $n=3$.

Effects of symmetric imperfections on the behavior of bistable struts

2016 ◽  
Vol 22 (12) ◽  
pp. 2240-2252 ◽  
Author(s):  
Jianguo Cai ◽  
Xiaowei Deng ◽  
Jian Feng
Keyword(s):  
Virtual Work ◽  
Variable Geometry ◽  
Equilibrium Equations ◽  
Buckling Mode ◽  
Concentrated Load ◽  
Virtual Work Principle ◽  
Shallow Arches ◽  
Buckling Behavior ◽  
Nonlinear Equilibrium ◽  
Post Buckling

The behavior of a bistable strut for variable geometry structures was investigated in this paper. A three-hinged arch subjected to a central concentrated load was used to study the effect of symmetric imperfections on the behavior of the bistable strut. Based on a nonlinear strain–displacement relationship, the virtual work principle was adopted to establish both the pre-buckling and buckling nonlinear equilibrium equations for the symmetric snap-through buckling mode. Then the critical load for symmetric snap-through buckling was obtained. The results show that the axial force is in compression before the arch is buckled, but it becomes in tension after buckling. Thus, the previous formulas cannot be used for the analysis of post-buckling behavior of three-hinged shallow arches. Then, the principle of virtual work was also used to establish the post-buckling equilibrium equations of the arch in the horizontal and vertical directions as well as the static boundary conditions, which are very important for bistable struts.

scholarly journals A numerical survey of nonlinear dynamical responses of discrete pantographic beams

2021 ◽  
Author(s):  
Emilio Turco
Keyword(s):  
Dynamical Behavior ◽  
Integration Scheme ◽  
Dynamical Response ◽  
Static Case ◽  
Equilibrium Equations ◽  
Response Curves ◽  
Nonlinear Dynamical ◽  
Nonlinear Equilibrium ◽  
Standard Integration ◽  
Inertia Forces

AbstractMaterials and structures based on pantographic cells exhibit interesting mechanical peculiarities. They have been studied prevalently in the static case, both in linear and nonlinear regime. When the dynamical behavior is considered, available literature is scarce probably for the intrinsic difficulties in the solution of this kind of problems. The aim of this work is to contribute to filling of this gap by addressing the dynamical response of pantographic beams. Starting from a simple spring mechanical model for pantographic beams, the nonlinear equilibrium problem is formulated directly for such a discrete system also considering inertia forces. Successively, the solution of the system of equilibrium equations is sought by means of a stepwise strategy based on a non-standard integration scheme. Here, only harmonic excitations are considered and, for large displacements, frequency-response curves are thoroughly discussed for some significant cases.

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