On the Lagrange’s Method for Floating Bodies

2006 ◽  
Author(s):  
Keyvan Sadeghi
Keyword(s):  
Kinetic Energy ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
The Body ◽  
Floating Body ◽  
Floating Bodies ◽  
Lagrange’S Equation ◽  
Lagrange’S Method ◽  
Linear Radiation

It is shown that, in the context of a linear theory, all fluid radiation actions on a floating body can be solely represented by a part of the fluid mechanical energy corresponding to the wetted surface of the body. In this regard, it is indicated that the linear radiation damping can be expressed by a fluid kinetic energy which has a bilinear form. Then from the Lagrange’s equations of motion, an equation of motion is derived that is called the conjugate Larange’s equation of motion. A variant of Hamilton’s principle is also introduced as the variational generator of the conjugate Lagrange’s equation of motion.

scholarly journals Generalized eigenfunction method for floating bodies

2011 ◽  
Vol 667 ◽  
pp. 544-554 ◽  
Author(s):  
COLM J. FITZGERALD ◽  
MICHAEL H. MEYLAN
Keyword(s):  
Time Domain ◽  
Water Waves ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Two Dimensions ◽  
Floating Body ◽  
Floating Bodies ◽  
Abstract Wave Equation ◽  
Generalized Eigenfunction ◽  
The Time Domain

We consider the time domain problem of a floating body in two dimensions, constrained to move in heave and pitch only, subject to the linear equations of water waves. We show that using the acceleration potential, we can write the equations of motion as an abstract wave equation. From this we derive a generalized eigenfunction solution in which the time domain problem is solved using the frequency-domain solutions. We present numerical results for two simple cases and compare our results with an alternative time domain method.

The energy distribution resulting from an impact on a floating body

2000 ◽  
Vol 417 ◽  
pp. 157-181 ◽  
Author(s):  
A. A. KOROBKIN ◽  
D. H. PEREGRINE
Keyword(s):  
Kinetic Energy ◽  
Energy Distribution ◽  
Water Flow ◽  
Compression Wave ◽  
The Body ◽  
The Other ◽  
Body Motion ◽  
Floating Body ◽  
Pressure Impulse ◽  
Work Done

The initial stage of the water flow caused by an impact on a floating body is considered. The vertical velocity of the body is prescribed and kept constant after a short acceleration stage. The present study demonstrates that impact on a floating and non-flared body gives acoustic effects that are localized in time behind the front of the compression wave generated at the moment of impact and are of major significance for explaining the energy distribution throughout the water, but their contribution to the flow pattern near the body decays with time. We analyse the dependence on the body acceleration of both the water flow and the energy distribution – temporal and spatial. Calculations are performed for a half-submerged sphere within the framework of the acoustic approximation. It is shown that the pressure impulse and the total impulse of the flow are independent of the history of the body motion and are readily found from pressure-impulse theory. On the other hand, the work done to oppose the pressure force, the internal energy of the water and its kinetic energy are essentially dependent on details of the body motion during the acceleration stage. The main parameter is the ratio of the time scale for the acoustic effects and the duration of the acceleration stage. When this parameter is small the work done to accelerate the body is minimal and is spent mostly on the kinetic energy of the flow. When the sphere is impulsively started to a constant velocity (the parameter is infinitely large), the work takes its maximum value: Longhorn (1952) discovered that half of this work goes to the kinetic energy of the flow near the body and the other half is taken away with the compression wave. However, the work required to accelerate the body decreases rapidly as the duration of the acceleration stage increases. The optimal acceleration of the sphere, which minimizes the acoustic energy, is determined for a given duration of the acceleration stage. Roughly speaking, the optimal acceleration is a combination of both sudden changes of the sphere velocity and uniform acceleration.If only the initial velocity of the body is prescribed and it then moves freely under the influence of the pressure, the fraction of the energy lost in acoustic waves depends only on the ratio of the body's mass to the mass of water displaced by the hemisphere.

Nonlinear impulse of ocean waves on floating bodies

2012 ◽  
Vol 697 ◽  
pp. 316-335 ◽  
Author(s):  
Paul D. Sclavounos
Keyword(s):  
Time Derivative ◽  
Ocean Waves ◽  
The Body ◽  
Offshore Wind ◽  
Floating Body ◽  
Offshore Platforms ◽  
Wave Loads ◽  
Floating Bodies ◽  
Bernoulli’S Equation ◽  
Hydrostatic Force

AbstractA new formulation is presented of the nonlinear loads exerted on floating bodies by steep irregular surface waves. The forces and moments are expressed in terms of the time derivative of the fluid impulse which circumvents the time-consuming computation of the temporal and spatial derivatives in Bernoulli’s equation. The nonlinear hydrostatic force on a floating body is shown to point vertically upwards and the nonlinear Froude–Krylov force and moment are derived as the time derivative of an impulse that involves the time derivative of a simple integral of the ambient velocity potential over the time-dependent body wetted surface. The nonlinear radiation and diffraction forces and moments are expressed as time derivatives of two impulses, a body impulse and a free surface impulse that represents higher-order wave loads acting along the body waterline. Numerical results are presented illustrating the accuracy of the new force expressions. Applications discussed include the nonlinear seakeeping of ships and offshore platforms and the extreme wave loads and responses of offshore wind turbines.

Three-dimensional kinematics and limb kinetic energy of running cockroaches.

1997 ◽  
Vol 200 (13) ◽  
pp. 1919-1929 ◽  
Author(s):  
R Kram ◽  
B Wong ◽  
R J Full
Keyword(s):  
Kinetic Energy ◽  
High Speed ◽  
Mechanical Energy ◽  
Center Of Mass ◽  
Three Dimensional ◽  
The Body ◽  
Morphological Data ◽  
Fast Running ◽  
Total Mechanical Energy ◽  
Reaction Forces

We tested the hypothesis that fast-running hexapeds must generate high levels of kinetic energy to cycle their limbs rapidly compared with bipeds and quadrupeds. We used high-speed video analysis to determine the three-dimensional movements of the limbs and bodies of cockroaches (Blaberus discoidalis) running on a motorized treadmill at 21 cm s-1 using an alternating tripod gait. We combined these kinematic data with morphological data to calculate the mechanical energy produced to move the limbs relative to the overall center of mass and the mechanical energy generated to rotate the body (head + thorax + abdomen) about the overall center of mass. The kinetic energy involved in moving the limbs was 8 microJ stride-1 (a power output of 21 mW kg-1, which was only approximately 13% of the external mechanical energy generated to lift and accelerate the overall center of mass at this speed. Pitch, yaw and roll rotational movements of the body were modest (less than +/- 7 degrees), and the mechanical energy required for these rotations was surprisingly small (1.7 microJ stride-1 for pitch, 0.5 microJ stride-1 for yaw and 0.4 microJ stride-1 for roll) as was the power (4.2, 1.2 and 1.1 mW kg-1, respectively). Compared at the same absolute forward speed, the mass-specific kinetic energy generated by the trotting hexaped to swing its limbs was approximately half of that predicted from data on much larger two- and four-legged animals. Compared at an equivalent speed (mid-trotting speed), limb kinetic energy was a smaller fraction of total mechanical energy for cockroaches than for large bipedal runners and hoppers and for quadrupedal trotters. Cockroaches operate at relatively high stride frequencies, but distribute ground reaction forces over a greater number of relatively small legs. The relatively small leg mass and inertia of hexapeds may allow relatively high leg cycling frequencies without exceptionally high internal mechanical energy generation.

Drift Motion of Floating Bodies Under the Action of Green Water

2021 ◽  
Author(s):  
Sasan Tavakoli ◽  
Luofeng Huang ◽  
Alexander V. Babanin
Keyword(s):  
Numerical Simulations ◽  
Water Waves ◽  
The Body ◽  
Green Water ◽  
Floating Body ◽  
Floating Bodies ◽  
Drift Motion ◽  
Upper Deck ◽  
Steep Waves ◽  
The Impact

Abstract Numerical simulations are peformed to model the dynamic motions of a free floating body exposed to water waves. The solid body has low freeboard and draft, and its upper deck can be washed by the steep waves. Thus, the green water phenomenon occurs as large waves interact with the floating body. The aim of the research is to improve the understanding of the green water emerging above the upper deck of a floating plate. A thin floating body with barriers is also modeled. For the case of the body equipped with barriers, no green water occurs. Green water has been seen to affect the wave field and the dynamic motions of the plate. It is observed that when water can wash the upper surface of the floating object, drift speed is slightly decreased as a proportion of the energy of waves is dissipated above the body. Water waves are seen to impact the upper surface of the thin floating body as the green water flows over its upper deck. Furthermore, water is seen to impact the plate as its front edge re-enters the water. The first water impact only occurs when the floating body is not equipped with any barrier. By sampling the numerical simulations, it is observed that the non-dimensional value of the impact pressure, resulting from the green water, is larger for the case of smaller wavelength.

scholarly journals Treating the Solid Pendulum Motion by the Large Parameter Procedure

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
A. I. Ismail
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
The Body ◽  
Large Parameter ◽  
Lagrange’S Equation ◽  
External Moment ◽  
Dynamical Description ◽  
Quasi Linear System

In this paper, we consider the dynamical description of a pendulum model consists of a heavy solid connection to a nonelastic string which suspended on an elliptic path in a vertical plane. We suppose that the dimensions of the solid are large enough to the length of the suspended string, in contrast to previous works which considered that the dimensions of the body are sufficiently small to the length of the string. According to this new assumption, we define a large parameter ε and apply Lagrange’s equation to construct the equations of motion for this case in terms of this large parameter. These equations give a quasi-linear system of second order with two degrees of freedom. The obtained system will be solved in terms of the generalized coordinates θ and φ using the large parameter procedure. This procedure has an advantage over the other methods because it solves the problem in a new domain when fails all other methods for solving the problem in such a domain under these conditions. It is one of the most important applications, when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment or the rotational motion of a heavy solid in a uniform gravity field or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems. The results of this paper also are useful in moving bridges and the swings. For satisfying the validation of the obtained solutions, we consider numerical considerations by one of the numerical methods and compare the obtained analytical and numerical solutions.

Nonlinear vibration analysis of a ┴-shaped mass attached to a clamped–clamped microbeam under electrostatic actuation

2016 ◽  
Vol 231 (11) ◽  
pp. 2147-2158 ◽  
Author(s):  
M Zamanian ◽  
A Karimiyan ◽  
SAA Hosseini ◽  
H Tourajizadeh
Keyword(s):  
Nonlinear Vibration ◽  
Electrostatic Force ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Electrostatic Actuation ◽  
Concentrated Force ◽  
Assumed Mode Method ◽  
Lagrange’S Equation ◽  
Comparison Functions ◽  
Horizontal Part

This article studies the nonlinear vibration of a ┴-shaped mass attached to a clamped–clamped microbeam under electrostatic actuation considering the effect of stretching. The DC and AC electrostatic force is applied to the horizontal part of ┴-shaped mass. The dynamic solution is studied using two methods of modeling. In the first model, the ┴-shaped mass is considered as a rigid body between two flexible microbeams. Then, the discretized equation of motion is derived using Lagrange’s equation combined with assumed mode method. The vibration mode shape of linear system is used as the comparison functions. In the second model, the dynamical effect of ┴-shaped mass is modeled as a concentrated force and moment, and it is introduced in the equation of motion by the Dirac function. Afterwards, the equation of motion is discretized using Galerkin method. In both methods of modeling, the equations of motion are solved using two methods. The first method is approximate analytical perturbation and the other one is Runge–Kutta numerical method. The effect of geometrical dimension of ┴-shaped mass on the nonlinear shift of resonance frequency and dynamic pull-in voltage is studied. The efficiency and accuracy of the presented formulations is verified by comparing the obtained results by two methods of modeling and two methods of solution.

Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions: Part II—System Equations

1990 ◽  
Vol 112 (2) ◽  
pp. 215-224 ◽  
Author(s):  
S. Nagarajan ◽  
D. A. Turcic
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Elastic Response ◽  
Mutual Dependence ◽  
Generalized Coordinates ◽  
Lagrange’S Equation ◽  
Elastic Mechanism

The first step in the derivation of the equations of motion for general elastic mechanism systems was described in Part I of this work. The equations were derived at the elemental level using Lagrange’s equation and the generalized coordinates were both the rigid body degrees of freedom, and the elastic degrees of freedom of element ‘e’. Each rigid body degree of freedom gave rise to a scalar equation of motion, and the elastic degrees of freedom of element e gave rise to a vector equation of motion. Since both the rigid body degrees of freedom and elastic degrees of freedom are considered as generalized coordinates, the equations derived take into account the mutual dependence between the rigid body and elastic motions. This is important for mechanisms that are built using lightweight and flexible members and which operate at high speeds. A schematic diagram of how the equations of motion are obtained in this work is shown in Fig. 1 in Part I. The transformation step in the figure refers to the rotational transformation of the nodal elastic displacements (which were measured in the element coordinate system), so that they are measured in terms of the reference coordinate system. This transformation is necessary in order to ensure compatibility of the displacement, velocity and acceleration of the degrees of freedom that are common to two or more links during the assembly of the equations of motion. This final set of equations after assembly are obtained in closed form, and, given external torques and forces, can be solved for the rigid body and elastic response simultaneously taking into account the mutual dependence between the two responses.

scholarly journals Kinetic Energy Harvesting from Human Hand Movement by Mounting micro Electromagnetic Generator

2019 ◽  
Vol 115 ◽  
pp. 02005
Author(s):  
F. R. Pathan
Keyword(s):  
Energy Harvesting ◽  
Kinetic Energy ◽  
Thermal Energy ◽  
Human Body ◽  
Mechanical Energy ◽  
Hand Movement ◽  
The Body ◽  
Human Hand ◽  
Electromagnetic Generator ◽  
Static Energy

A comprehensive review of design and experimentation is presented in this research paper on sustainable renewable energy scavenging from Human body movement using Micro electromagnetic kinetic energy harvester to powering wearable, portable electronics, implantable medical devices etc. The body location which is chosen as the harvester is human hand between elbow and shoulder. Human body harvest energy in two ways i,e, mechanical energy and thermal energy. Mechanical energy is of two kinds one is static energy and the other one is kinetic energy. Due to motion or displacement or enforcement excitation the kinetic energy is extracted. The electric charges which remains imbalance on the surface or within a material is static energy. Thermal energy is extracted from the dissipation of heat from human body. Human body parts and organs generate energy through two types of activities are voluntary and involuntary. The energy which are produced by voluntary activities are high as people intentionally does work by body motion, walk, run. The generated energy by involuntary organs like heart, breathing, artery are smaller compare to voluntary energy harvesting. One process of energy harvesting is by use of micro electromagnetic generator, flexible and stretchable piezoelectric, triboelectric, electromagnetic induction, PVDF cantilever mounting on human body. The harvester prototype is cylindrical magnet L40xD10 mm size which is mounted on human hand for energy harvesting. While in movement of hand the produced wave forms by magnetic generator are measured and recorded for calculation. Analyzing the received data it has been found that the generated power by micro electromagnetic vibration generator from movement of human hand are 319 RMS μW and 2.48 RMS mV with a frequency of 0.25 Hz and power density of about 2.48μW/cm³.

Mechanical Energy Expenditures and Movement Efficiency in Full Body Reaching Movements

2010 ◽  
Vol 26 (1) ◽  
pp. 32-44 ◽  
Author(s):  
Daohang Sha ◽  
Christopher R. France ◽  
James S. Thomas
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Target Location ◽  
Mechanical Energy ◽  
The Body ◽  
Reaching Movements ◽  
Whole Body ◽  
Joint Power ◽  
Total Mechanical Energy ◽  
Full Body

The effect of target location, speed, and handedness on the average total mechanical energy and movement efficiency is studied in 15 healthy subjects (7 males and 8 females with age 22.9 ± 1.79 years old) performing full body reaching movements. The average total mechanical energy is measured as the time average of integration of joint power, potential energy, and kinetic energy respectively. Movement efficiency is calculated as the ratio of total kinetic energy to the total joint power and potential energy. Results show that speed and target location have significant effects on total mechanical energy and movement efficiency, but reaching hand only effects kinetic energy. From our findings we conclude that (1) efficiency in whole body reaching is dependent on whether the height of the body center of mass is raised or lowered during the task; (2) efficiency is increased as movement speed is increased, in part because of greater changes in potential energy; and (3) the CNS does not appear to use movement efficiency as a primary planning variable in full body reaching. It may be dependent on a combination of other factors or constraints.

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