Dynamic Response of a Double Articulated Offshore Loading Structure to Noncollinear Waves and Current

1981 ◽  
Vol 103 (1) ◽  
pp. 41-47 ◽  
Author(s):  
R. K. Jain ◽  
C. L. Kirk
Keyword(s):  
Dynamic Response ◽  
Nonlinear Equations ◽  
Relative Motion ◽  
Integration Method ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Governing Equations ◽  
Steady Current ◽  
Highly Nonlinear ◽  
Lagrange’S Method

The three-dimensional dynamic response of a double articulated offshore loading structure to noncollinear waves and a steady current is studied for various waves and varying current directions. The governing equations of motion are derived by the Lagrange’s method where the wave and current forces are computed by a modified form of the Morison’s equation which takes account for the relative motion of the water particles with respect to the oscillating structure. The resulting highly nonlinear equations are solved by using a block integration method. The computed results predict complex whirling oscillations of the structure to noncollinear waves and current.

Response of Articulated Towers to Waves and Current

1978 ◽  
Vol 18 (05) ◽  
pp. 283-290 ◽  
Author(s):  
C.L. Kirk ◽  
R.K. Jain
Keyword(s):  
Dynamic Response ◽  
Equations Of Motion ◽  
Steady Current ◽  
Linear Waves ◽  
Sea Bed ◽  
Highly Nonlinear ◽  
Instantaneous Position ◽  
Combined Action ◽  
Interim Measure ◽  
Lagrange’S Method

Abstract The dynamic response of articulated towers to noncollinear Airy waves and steady current has been investigated, where the wave and current forces have been computed by a modified form of Morison's equation. The two equations of motion obtained by Lagrange's method describe the response in terms of meridional and circumferential angles. These equations are highly nonlinear and are solved numerically by the block integration method for various wave parameters. The predicted response is a complex whirling motion of the tower around a vertical axis. Introduction There has been an increasing use of mobile offshore systems in the North Sea for storing and loading oil into attendant tankers, particularly for fields that have a limited production capability or are too remote from refining or terminal facilities to warrant laying a pipeline. Mobile loading platforms are also used as an interim measure platforms are also used as an interim measure during pipeline laying for large fields and later as a backup system in case of pipeline failure.A typical type of mobile loading and storage system is the articulated buoyant loading tower, which may have either a single universal joint at the sea bed or a second joint nearer the surface. The tower is designed for a maximum tilt angle of 20 deg. under extreme tanker mooring conditions caused by wind, waves, and current. In assessing the performance characteristics and the strength of an performance characteristics and the strength of an articulated tower under severe environmental conditions, both with and without a tanker, it is essential to determine its dynamic response by theoretical methods, by model testing in a wind/ current/wave tank, and by measuring response of real structures on site.This paper analyzes the motion of a single articulated tower without a tanker under the combined action of forces resulting from current and a train of regular linear waves. The problem is of interest to both the operators and the designers of loading towers because it is important to estimate motion in moderate seas for the case of mooring an approaching tanker, as well as the extreme deflections that would occur under the 100-year design wave.The dynamic response of the tower is obtained by formulating the equations of motion by Lagrange's method. The wave forces are determined using a modified form of Morison's equation that accounts for the relative motion of the water particles with respect to the structure. The equations of motion are highly nonlinear and analytical solutions are not possible; thus, a numerical solution has been selected in which the block integration method has been used. The cases considered areorthogonal waves and current,collinear waves and current having the same or opposite directions of propagation, andthe directions of propagation inclined propagation, and (3) the directions of propagation inclined at an angle of 45 deg. DESCRIPTION OF PROBLEM In the schematic of the typical articulated tower shown in Fig. 1, the orthogonal fixed-axis reference system is chosen so that the X and Z axes are taken in a horizontal plane parallel to the sea bed. During motion of the tower the OZ'axis is perpendicular to the Y axis in the plane containing the perpendicular to the Y axis in the plane containing the OY axis and the tower axis OC, while OX'is normal to the YOZ'plane. The instantaneous position of the tower is completely determined by the coordinates psi, and theta, where psi is the angle between the planes psi, and theta, where psi is the angle between the planes YOC and YOZ, and theta is the meridional angle made by OC and OY in the instantaneous position of the plane YOZ'. plane YOZ'. The tower is subjected to the action of linear waves propagating in the direction of the X axis and a steady current of velocity v that may vary with depth. The direction of the current flow is at an angle a to the X axis. In the absence of waves, the tower will be in a static equilibrium position specified by coordinates (theta, pi/2-a). Under the combined action of waves and current the structure will oscillate around the OY axis. The purpose of this paper is to formulate and solve the equations of motion of the tower subjected to a variety of wave lengths, wave heights, and a current of constant velocity but with variable direction. SPEJ P. 283

Dynamic response of a pre-stressed bi-layered plate-strip subjected to an arbitrary inclined time-harmonic force

2017 ◽  
Vol 26 (3) ◽  
pp. 255-262
Author(s):  
AHMET DASDEMIR ◽  
Keyword(s):  
Dynamic Response ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Layered Plate ◽  
Field Problem ◽  
Time Harmonic ◽  
Governing System ◽  
Plate Strip ◽  
Linearized Theory ◽  
Complete Contact

Within the scope of the piecewise homogeneous body model with utilizing of the three dimensional linearized theory of elastic waves in initially stressed bodies the dynamical stress field problem in a bi-layered plate-strip with initial stress under the action of an arbitrary inclined timeharmonic force resting on a rigid foundation is investigated. The concrete materials such as a pair of Aluminum and Steel are selected. It is assumed that there exists a complete contact interaction between the layers. The mathematical modeling of the problem under consideration is carved out, and the governing system of the partial differential equations of motion is approximately solved by employing Finite Element Method. The numerical results related to the influence of certain parameters on the dynamic response of the plate-strip are presented.

scholarly journals New Analytical Model Used in Finite Element Analysis of Solids Mechanics

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1401 ◽  
Author(s):  
Sorin Vlase ◽  
Adrian Eracle Nicolescu ◽  
Marin Marin
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Elastic Solid ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Element Analysis ◽  
Elastic Multibody System

In classical mechanics, determining the governing equations of motion using finite element analysis (FEA) of an elastic multibody system (MBS) leads to a system of second order differential equations. To integrate this, it must be transformed into a system of first-order equations. However, this can also be achieved directly and naturally if Hamilton’s equations are used. The paper presents this useful alternative formalism used in conjunction with the finite element method for MBSs. The motion equations in the very general case of a three-dimensional motion of an elastic solid are obtained. To illustrate the method, two examples are presented. A comparison between the integration times in the two cases presents another possible advantage of applying this method.

A New Displacement Form for the Nonlinear Equations of Motion of Shells of Revolution

1985 ◽  
Vol 52 (3) ◽  
pp. 507-509 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Shells Of Revolution ◽  
Governing Equations ◽  
Hoop Strain ◽  
Axisymmetric Motion ◽  
Nonlinear Equations Of Motion ◽  
Surface Loads ◽  
Theory Of Shells

In the theory of shells of revolution undergoing torsionless, axisymmetric motion, an extensional and a bending hoop strain are introduced that are linear in the displacements, regardless of the magnitudes of the strains and the meridional rotation. The resulting equations of motion and boundary conditions are derived and some common conservative surface loads are listed along with their potentials. The governing equations appear to be the simplest possible in terms of displacements.

Nonlinear Wheelset Dynamic Response to Random Lateral Rail Irregularities

1974 ◽  
Vol 96 (4) ◽  
pp. 1168-1176 ◽  
Author(s):  
E. H. Law
Keyword(s):  
Dynamic Response ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Power Spectra ◽  
Railway Vehicle ◽  
Lateral Stiffness ◽  
Wheel Wear ◽  
Rail Irregularities ◽  
Nonlinear Equations Of Motion

The nonlinear equations of motion for a railway vehicle wheelset having profiled wheels and contact of the wheel flange with flexible rails are presented. The effects of spin creep and gyroscopic terms are included. The rails are considered to have random lateral irregularities which are described by prescribed power spectra. The equations of motion are integrated numerically and the effects on the dynamic response of quantities such as speed, track roughness, wheel wear, flange clearance, and lateral stiffness of the rails are investigated.

scholarly journals A Mathematical Model for Forced Vibration of Pre-Stressed Piezoelectric Plate-Strip Resting On Rigid Foundation

MATEMATIKA ◽  
2018 ◽  
Vol 34 (2) ◽  
pp. 419-431
Author(s):  
Ahmet Daşdemir
Keyword(s):  
Mathematical Model ◽  
Dynamic Response ◽  
Initial Stress ◽  
Piezoelectric Plate ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Rigid Foundation ◽  
Time Harmonic ◽  
Governing System ◽  
Plate Strip

A mathematical model to investigate the dynamic response of a piezoelectric plate-strip with initial stress under the action of a time-harmonic force resting on a rigid foundation is presented within the scope of the three-dimensional linearized theory of electro-elasticity waves in initially stressed bodies (TLTEEWISB). The governing system of equations of motion is solved by employing the Finite Element Method (FEM). The numerical results illustrating the dependencies of different problem parameters are investigated. In particular, the influence of a change in the value of the initial stress parameter on the dynamic response of the plate-strip is discussed.

scholarly journals A New Dynamic Model of a Two-Wheeled Two-Flexible-Beam Inverted Pendulum Robot

2020 ◽  
Author(s):  
Amin Mehrvarz ◽  
Mohammad Javad Khodaei ◽  
William Clark ◽  
Nader Jalili
Keyword(s):  
Differential Equations ◽  
High Performance ◽  
Inverted Pendulum ◽  
Equations Of Motion ◽  
Flexible Beam ◽  
Governing Equations ◽  
Natural Modes ◽  
Wheeled Inverted Pendulum ◽  
Highly Nonlinear ◽  
New Type

Abstract Inverted pendulums are traditional dynamic problems. If an inverted pendulum is used in a moving cart, a new type of exciting issues will appear. One of these problems is two-wheeled inverted pendulum systems. Because of their small size, high performance in quick driving, and their stability with controller, researchers and engineers are interested in them. In this paper, a new configuration of one specific robot is modeled, and its dynamic behavior is analyzed. The proposed model can move in two directions, and with a proper controller can keep its stability during the operation. In this robot, two cantilever beams are on the two-wheeled base, and they are excited by voltages to the attached piezoelectric actuators. The mathematical model of this system is obtained using the extended Hamilton’s Principle. The results show that the governing equations of motion are highly nonlinear and contain several coupled partial differential equations (PDEs). In order to extract the natural modes of the beams, the undamped, unforced equations of motion and boundary conditions of the beams are used. If a limited number of modes (N1 and N2) are selected for each beam, the coupled PDEs will be changed to N1 + N2 + 5 ordinary differential equations (ODEs). These complex equations are solved numerically, and the natural frequencies of the system are extracted. The system is then simulated in both lateral and horizontal plane movements. The simulation shows that the governing equations are correct, and the system is ready for designing a proper controller. It should be mentioned that in the future works, the derived equations will be validated experimentally, and a suitable control strategy will be applied to the system to make it automated and more applicable.

scholarly journals Direct integration method in three-dimensional elasticity and thermoelasticity problems for inhomogeneous transversely isotropic solids: governing equations in terms of stresses

2019 ◽  
pp. 102-105
Author(s):  
R. M. Kushnir ◽  
Y. V. Tokovyy ◽  
D. S. Boiko
Keyword(s):  
Stress Tensor ◽  
Integral Equations ◽  
Integration Method ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Solution Technique ◽  
Direct Integration ◽  
Governing Equations ◽  
Direct Integration Method

An efficient technique for thermoelastic analysis of inhomogeneous anisotropic solids is suggested within the framework of three-dimensional formulation. By making use of the direct integration method, a system of governing equations is derived in order to solve three-dimensional problems of elasticity and thermoelasticity for transversely isotropic inhomogeneous solids with elastic and thermo-physical properties represented by differentiable functions of the variable in the direction that is transversal to the plane of isotropy. By implementing the relevant separation of variables, the obtained equations can be uncoupled and reduced to second-kind integral equations for individual stress-tensor components and the total stress, which represents the trace of the stress tensor. The latter equations can be attempted by any of the numerical, analyticalnumerical, or analytical means available for the solution of the second-kind integral equations. In order to construct the solutions in an explicit form, an advanced solution technique can be developed on the basis of the resolvent-kernel method implying the series representation by the recurring kernels, computed iteratively by the original kernel of an integral equation.

scholarly journals Stress field formulation of linear electro-magneto-elastic materials

2019 ◽  
Vol 24 (12) ◽  
pp. 3806-3822
Author(s):  
A Amiri-Hezaveh ◽  
P Karimi ◽  
M Ostoja-Starzewski
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Governing Equations ◽  
Field Equations ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Sufficient Set

A stress-based approach to the analysis of linear electro-magneto-elastic materials is proposed. Firstly, field equations for linear electro-magneto-elastic solids are given in detail. Next, as a counterpart of coupled governing equations in terms of the displacement field, generalized stress equations of motion for the analysis of three-dimensional (3D) problems Are obtained – they supply a more convenient basis when mechanical boundary conditions are entirely tractions. Then, a sufficient set of conditions for the corresponding solution of generalized stress equations of motion to be unique are detailed in a uniqueness theorem. A numerical passage to obtain the solution of such equations is then given by generalizing a reciprocity theorem in terms of stress for such materials. Finally, as particular cases of the general 3D form, the stress equations of motion for planar problems (plane strain and Generalized plane stress) for transversely isotropic media are formulated.

Nonlinear Dynamic Response of Piezoelectric Cylindrical Shell with Delamination under Hygrothermal Conditions

2012 ◽  
Vol 204-208 ◽  
pp. 4698-4701
Author(s):  
Jin Hua Yang ◽  
De Liang Chen
Keyword(s):  
Cylindrical Shell ◽  
Finite Difference Method ◽  
Dynamic Response ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Transverse Motion ◽  
Governing Equations ◽  
Nonlinear Dynamic Response ◽  
The Finite Difference Method ◽  
Delamination Length

Abstract. On the basis of the nonlinear plate-shell and piezoelectric theory, the governing equations of motion for axisymmetrical piezoelectric delaminated cylindrical shell under hygrothermal conditions were derived. The governing equation of transverse motion was modified by contact force and thus the penetration between two delaminated layers could be avoided. The whole problem was resolved by using the finite difference method. In calculation examples, the effects of delamination length, depth and amplitude of load on the nonlinear dynamic response of the axisymmetrical piezoelectric delaminated shell under hygrothermal conditions were discussed in detail.

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