Differential equations for librational motion of gravity-oriented rigid body

2015 ◽  
Vol 73 ◽  
pp. 51-57 ◽  
Author(s):  
E.A. Kosjakov ◽  
A.A. Tikhonov
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Librational Motion

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

scholarly journals New Aspects Concerning Dynamics of Rigid Solid Body in Plane-Parallel Motion Subjected to Real Constraints. Part One

2019 ◽  
Vol 24 (2) ◽  
pp. 175-180
Author(s):  
Vladimir Dragoş Tătaru ◽  
Mircea Bogdan Tătaru
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Analysis ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Original Method ◽  
Horizontal Force ◽  
Constraint Forces ◽  
Inclined Plane ◽  
Parallel Motion

Abstract The present paper approaches in an original manner the dynamic analysis of a wheel which climbs on an inclined plane under the action of a horizontal force. The wheel rolls and slides in the same time. The two movements, rolling and sliding are considered to be independent of each other. Therefore we are dealing with a solid rigid body with two degrees of freedom. The difficulty of approaching the problem lies in the fact that in the differential equations describing the motion of the solid rigid body are also present the constraint forces and these are unknown. For this reason they must be eliminated from the differential equations of motion. The paper presents as well an original method of the constraint forces elimination.

scholarly journals On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces

2021 ◽  
Vol 23 (2) ◽  
pp. 159-170
Author(s):  
Alexander A. Kosov ◽  
Eduard I. Semenov
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Exact Solutions ◽  
Rotational Motion ◽  
First Integral ◽  
System Of Differential Equations ◽  
Limit Values ◽  
Gyroscopic Forces ◽  
Parametric Families ◽  
Solutions Of Equations

Abstract. A nonlinear system of differential equations describing the rotational motion of a rigid body under the action of torque of potential and circular-gyroscopic forces is considered. For this torque, the system of differential equations has three classical first integrals: the energy integral, the area integral, and the geometric integral. For the analogue of the Lagrange case, when two moments of inertia coincide and the potential depends on one angle, an additional first integral is found and integration in quadratures is performed. A number of examples is considered where parametric families of exact solutions are considered. In these examples, polynomial or analytical functions were used as a potential. In particular, we construct families of periodic and almost periodic motions, as well as families of asymptotically uniaxial rotations. We also identified movements that have limit values of opposite signs for unlimited increase and decrease of time.

The temperature factor of an atom in a rigid vibrating molecule. II. Anisotropic thermal motion

1970 ◽  
Vol 26 (2) ◽  
pp. 260-262 ◽  
Author(s):  
G. S. Pawley ◽  
B. T. M. Willis
Keyword(s):  
Rigid Body ◽  
Temperature Factor ◽  
Thermal Motion ◽  
Mean Square ◽  
First Order ◽  
Translational Displacement ◽  
Librational Motion ◽  
The Mean ◽  
A Site ◽  
Body Theory

The analysis in part I is generalized to any crystal containing rigid molecules which undergo anisotropic translational and librational motion about a site fixed by symmetry. The treatment is correct to terms in (ui 2)2 and (ωi 2)2, where (ui 2) is the mean-square translational displacement of the molecule along the ith axis and (ωi 2) is the mean-square angular libration about the same axis. The first-order treatment to terms in (ui 2) and (ωi 2) is shown to be equivalent to the rigid-body theory in current use.

Three-Dimensional Frictional Rigid-Body Impact

1995 ◽  
Vol 62 (4) ◽  
pp. 893-898 ◽  
Author(s):  
V. Bhatt ◽  
Jeff Koechling
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Local Analysis ◽  
Integration Scheme ◽  
Numerical Integration Scheme ◽  
Rigid Body Model ◽  
Global Nature ◽  
The Impact

We present an analysis of the rigid-body model for frictional three-dimensional impacts, which was originally studied by Routh. Using Coulomb’s law for friction, a set of differential equations describing the progress and outcome of the impact process for general bodies can be obtained. The differential equations induce a flow in the tangent velocity space for which the trajectories cannot be solved for in a closed form, and a numerical integration scheme is required. At the point of sticking, the numerical problem becomes ill-conditioned and we have to analyze the flow at the singularity to determine the rest of the process. A local analysis at the point of sticking provides enough information about the global nature of the flow to let us enumerate all the possible dynamic scenarios for the sliding behavior during impact. The friction coefficient, and the mass parameters at the point of contact, determine the particular sliding behavior that would occur for a given problem. Once the initial conditions are specified, the possible outcome of the impact can then be easily determined.

Interaction Analysis of Simple form Objects with Fluid and Control Optimization

2021 ◽  
Author(s):  
◽  
Shravan Koundinya Vutukuru
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rigid Body ◽  
Simple Form ◽  
Body Fluid ◽  
Ansys Fluent ◽  
Structure Interaction ◽  
Alternative Approach ◽  
Fluid Interaction ◽  
Interaction Phenomenon

Fluid-rigid body interaction is an age-old phenomenon, but interestingly, a good approximated solution for the phenomenon pertaining to non-stationary body-fluid interaction is still non-existent. The solution is much more complicated due to huge system of simultaneous partial differential equations that are framed from multi-degrees of freedom, all elements in the spatial domain coupled together between all time steps. Additionally, when considering the spatial aspects of solving the system of partial differential equations, there arise a range of complexities from the type of solution technique (finite-differences, finite-volume, finite-element) and also from meshing techniques (moving, structured or unstructured). Even though advanced commercial fluid-structure interaction solvers are available, they are limited to simple objects and require frequent remeshing techniques that are time consuming and computationally expensive. The promotion work specifically focuses solely on rigid body-fluid (air) interaction and does not consider flow reattachment or flow separation phenomenon offering an alternative approach to study the interaction phenomenon and its advantages. The basic idea of the approximated theory in the current work is to have a simplified approach through a straightforward mathematical model without considering the viscous nature of fluid medium (air). Therefore, this is an approximate theory for non-stationary body and fluid interaction phenomenon considering inputs (post-processing results) from stationary rigid body-fluid interaction performed in ANSYS Fluent (2D and 3D) where the steady state RANS equation is solved with the help of turbulence model. The concept discussed in the work will offer an alternative approach for ‘space-time’ programming techniques and also help to solve the engineering tasks of optimization and synthesis for simple form objects without requiring huge computational efforts. A new world of science for autonomous robots (underwater robotic fish with single and dual tail actuator) is explored where in an on-board power pack technique (energy scavenging from surrounding medium) is proposed that purely based on the fluid and rigid body interaction phenomenon is analysed. Experiments on simple form objects were performed in ARMFIELD wind tunnel, available at Riga Technical University, at a constant speed of 10 m/s and validated with the computer program ANSYS Fluent (in 3D). All the latest techniques, advantages and importance related to fluid-structure interaction phenomenon are summarized in the literature review section through various databases available over internet.

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