Dynamics of a Plate in Large Overall Motion

1989 ◽  
Vol 56 (4) ◽  
pp. 887-892 ◽  
Author(s):  
A. K. Banerjee ◽  
T. R. Kane
Keyword(s):  
Rigid Body ◽  
Reference Frame ◽  
Elastic Plate ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Present Theory ◽  
Thin Elastic Plate ◽  
Vibration Modes ◽  
Simply Supported ◽  
Rigid Body Motions

Equations of motion are formulated for a thin elastic plate that is executing small motions relative to a reference frame undergoing large rigid body motions (three-dimensional rotation and translation) in a Newtonian reference frame. As an illustrative example, a spin-up maneuver for a simply-supported rectangular plate is examined, and the vibration modes of such a plate are used to show that the present theory captures the phenomenon of dynamic stiffening.

Vibration of Beams and Plate Strips With Three-Dimensional Flexibility

1989 ◽  
Vol 56 (1) ◽  
pp. 228-231 ◽  
Author(s):  
Manuel Stein
Keyword(s):  
Cross Sections ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Beam Theory ◽  
Vibration Modes ◽  
Natural Vibrations ◽  
Accurate Representation ◽  
Simply Supported

Studies making use of higher vibration modes and frequencies have indicated a need for a more accurate beam theory. Equations of motion are developed here that give a more accurate representation of the dynamic behaivor of a beam than conventional beam theory. Results are obtained using these equations for the natural vibrations of simply-supported aluminum beams of rectangular cross-sections. These results are compared to results from conventional beam theory, and they are examined to identify where various effects are important.

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.

On Saint-Venant's Problem for Helicoidal Beams

2015 ◽  
Vol 83 (2) ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Fem Analysis ◽  
Original System ◽  
Form Solution ◽  
Hamiltonian Matrix ◽  
Jordan Structure ◽  
Rigid Body Motions

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

The Three-Dimensional Dynamics for Toroidal Drive

2012 ◽  
Vol 562-564 ◽  
pp. 536-539
Author(s):  
Li Zhong Xu ◽  
Jin Liang Li ◽  
Ya Jun Li
Keyword(s):  
Natural Frequencies ◽  
Cone Angle ◽  
Three Dimensional ◽  
Drive System ◽  
Vibration Modes ◽  
Mesh Stiffness ◽  
Coupled Modes ◽  
Bearing Stiffness ◽  
Rigid Body Motions ◽  
Half Cone

In this paper, a model to simulate the dynamic behavior of the toroidal drive is developed. The three-dimensional dynamic model includes all six rigid body motions of the stator, worm, rotor and the planets. Using the model, the natural frequencies and vibration modes of the drive system are investigated. The vibration modes are classified into single modes and coupled modes. The single modes include planet mode, worm mode and stator mode. The vibration and frequency characteristics of different modes are obtained. The relation between modes and half cone angle of the planet tooth is discussed. The relation between vibrations and bearing stiffness is also discussed. When the bearing stiffness is about 10 times of the mesh stiffness, some vibration displacements of the drive system are quite small and can be neglected. Meanwhile, the dynamic equations for the drive system can be simplified.

scholarly journals Nonlinear interactions in deformable container cranes

2015 ◽  
Vol 230 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Matthew P Cartmell
Keyword(s):  
Rigid Body ◽  
Internal Resonance ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Higher Order ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Internal Resonances ◽  
Container Cranes ◽  
Bending Mode

Nonlinear dynamic interactions in harbour quayside cranes due to a two-to-one internal resonance between the lowest bending mode of the deformable boom and the in-plane pendular mode of the container are investigated. To this end, a three-dimensional model of container cranes accounting for the elastic interaction between the crane boom and the container dynamics is proposed. The container is modelled as a three-dimensional rigid body elastically suspended through hoisting cables from the trolley moving along the crane boom modelled as an Euler-Bernoulli beam. The reduced governing equations of motion are obtained through the Euler-Lagrange equations employing the boom kinetic and stored energies, derived via a Galerkin discretisation based on the mode shapes of the two-span crane boom used as trial functions, and the kinetic and stored energies of the rigid body container and the elastic hoisting cables. First, conditions for the onset of internal resonances between the boom and the container are found. A higher order perturbation treatment of the Taylor expanded equations of motion in the neighbourhood of a two-to-one internal resonance between the lowest boom bending mode and the lowest pendular mode of the container is carried out. Continuation of the fixed points of the modulation equations together with stability analysis yields a rich bifurcation behaviour, which features Hopf bifurcations. It is shown that consideration of higher order terms (cubic nonlinearities) beyond the quadratic geometric and inertia nonlinearities breaks the symmetry of the bifurcation equations, shifts the bifurcation points and the stability ranges, and leads to bifurcations not predicted by the low order analysis.

The Use of Vector Techniques in Variational Problems

1986 ◽  
Vol 108 (2) ◽  
pp. 141-145 ◽  
Author(s):  
L. J. Everett ◽  
M. McDermott
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Double Pendulum ◽  
Coordinate Transformations ◽  
Elastic Deformations ◽  
Variational Techniques ◽  
Convenient Means ◽  
Moving Coordinate ◽  
Rigid Body Motions

A convenient means for applying vector mathematics to variational problems is presented. The total and relative variations of a vector are defined and results which follow from these definitions are developed and proved. These results are then used to express the variation of a functional using vector techniques rather than the classical scalar or matrix techniques. The simple problems of deriving equations of motion for a rigid body and for a rigid double pendulum are presented as examples of the technique. The key advantages of the method are that (1) it allows the investigator who is familiar and proficient with vector techniques to apply these skills to variational problems and (2) it greatly simplifies the application of variational techniques to problems which include both rigid body motions and elastic deformations. This is accomplished by providing the techniques necessary for computing the variation of a vector defined in a moving coordinate system without using coordinate transformations.

scholarly journals Equivalence of Euler equations and torque-angular momentum relation

2021 ◽  
Vol 18 (1) ◽  
pp. 136
Author(s):  
V. Tanriverdi
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Reference Frame ◽  
Euler Equations ◽  
Equations Of Motion ◽  
The Body ◽  
Moments Of Inertia ◽  
Body Reference ◽  
Stationary Reference Frame ◽  
Momentum Relation

Euler derived equations for rigid body rotations in the body reference frame and in the stationary reference frame by considering an infinitesimal part of the rigid body.Another derivation is possible, and it is widely used: transforming torque-angular momentum relation to the body reference frame.However, their equivalence is not shown explicitly.In this work, for a rigid body with different moments of inertia, we calculated Euler equations explicitly in the body reference frame and in the stationary reference frame and torque-angular momentum relation.We also calculated equations of motion from Lagrangian.These calculations show that all four of them are equivalent.

scholarly journals Periodic orbits in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Awadhesh Kumar Poddar ◽  
Divyanshi Sharma
Keyword(s):  
Perturbation Theory ◽  
Rigid Body ◽  
Coordinate System ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
General Perturbation ◽  
Triaxial Rigid Body ◽  
Three Dimensional Coordinate

AbstractIn this paper, we have studied the equations of motion for the problem, which are regularised in the neighbourhood of one of the finite masses and the existence of periodic orbits in a three-dimensional coordinate system when μ = 0. Finally, it establishes the canonical set (l, L, g, G, h, H) and forms the basic general perturbation theory for the problem.

Sign in / Sign up

Export Citation Format

Share Document