Nonlinear Coupled Transverse and Axial Vibration of a Compliant Structure: Free and Forced

2000 ◽  
Author(s):  
Seon M. Han ◽  
Haym Benaroya
Keyword(s):  
Equations Of Motion ◽  
Transverse Motion ◽  
Random Waves ◽  
Axial Motion ◽  
Axial Vibration ◽  
Compliant Structure ◽  
Coupled Equations ◽  
Compliant Tower ◽  
Ocean Environment ◽  
Forced Responses

Abstract A compliant tower in the ocean environment is modeled as a beam undergoing coupled transverse and axial motion. The beam is supported by a torsional spring and has a point mass at the other end. It is assumed that the strains are small but the rotation is moderate compared to the strain so that the equations of motion for the axial and transverse motion are nonlinearly coupled. The nonlinear coupled equations of motion are derived here using Hamilton’s principle. The forced responses due to random waves are observed here. The responses are obtained numerically using the finite difference approach.

Coupled Librational Motion of an Axi-Symmetric Satellite in a Circular Orbit

1969 ◽  
Vol 73 (704) ◽  
pp. 674-680 ◽  
Author(s):  
V. J. Modi ◽  
S. K. Shrivastava
Keyword(s):  
Circular Orbit ◽  
Equations Of Motion ◽  
Orbital Plane ◽  
Gradient Field ◽  
Gravity Gradient ◽  
Transverse Motion ◽  
Complex Nature ◽  
Coupled Equations ◽  
Librational Motion ◽  
Zero Velocity Curves

The review of the literature suggests that the planar motion of a rigid satellite in a gravity gradient field has been the subject of considerable investigation during the past ten years. In contrast the dynamic study of a satellite executing librational motion out of the orbital plane has received comparatively little attention. Such an investigation is important because as pointed out by Kane, for large amplitudes the transverse motion is strongly coupled with that in the plane. The lack of information may be due to the complex nature of the problem. The non-linear, non-autonomous, coupled equations of motion involving a large number of parameters are not amenable to simple, concise analysis. Some simplification may be achieved by restricting the satellite motion to a circular orbit. For this case, as indicated by Auelmann, closed zero-velocity curves exist under certain conditions which limit the amplitude of motion.

scholarly journals Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

2021 ◽  
Vol 9 (1) ◽  
pp. 76
Author(s):  
Duoc Nguyen ◽  
Niels Jacobsen ◽  
Dano Roelvink
Keyword(s):  
Surface Waves ◽  
Deep Water ◽  
Surf Zone ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Breaking Waves ◽  
Successful Implementation ◽  
Random Waves ◽  
Mean Motion ◽  
Generalized Lagrangian

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

Determination of Instability Boundaries for a Highly Flexible Prismatic-Jointed Robot Performing Repetitive Maneuvers

1991 ◽  
Author(s):  
Keith W. Buffinton
Keyword(s):  
Floquet Theory ◽  
Equations Of Motion ◽  
Perturbation Techniques ◽  
Axial Motion ◽  
The Third ◽  
Straightforward Application ◽  
Method Yield ◽  
Robotic Devices ◽  
Axially Moving

Abstract Presented in this work are the equations of motion governing the behavior of a simple, highly flexible, prismatic-jointed robotic manipulator performing repetitive maneuvers. The robot is modeled as a uniform cantilever beam that is subject to harmonic axial motions over a single bilateral support. To conveniently and accurately predict motions that lead to unstable behavior, three methods are investigated for determining the boundaries of unstable regions in the parameter space defined by the amplitude and frequency of axial motion. The first method is based on a straightforward application of Floquet theory; the second makes use of the results of a perturbation analysis; and the third employs Bolotin’s infinite determinate method. Results indicate that both perturbation techniques and Bolotin’s method yield acceptably accurate results for only very small amplitudes of axial motion and that a direct application of Floquet theory, while computational expensive, is the most reliable way to ensure that all instability boundaries are correctly represented. These results are particularly relevant to the study of prismatic-jointed robotic devices that experience amplitudes of periodic motion that are a significant percentage of the length of the axially moving member.

Dynamics of Spinning Disc Parametrically Excited by Noise

1999 ◽  
Author(s):  
Narayanan Ramakrishnan ◽  
N. Sri Namachchivaya
Keyword(s):  
Equations Of Motion ◽  
Planck Equation ◽  
Stochastic Averaging ◽  
Transverse Motion ◽  
Probability Density Distribution ◽  
Integral Of Motion ◽  
Parametrically Excited ◽  
One Dimensional ◽  
Small Intensity ◽  
Spinning Disc

Abstract The nonlinear dynamics of a circular spinning disc parametrically excited by noise of small intensity is investigated. The governing PDEs are reduced using a Galerkin reduction procedure to a two-DOF system of ODEs which, govern the transverse motion of the disc. The dynamics is simplified by exploiting the S1 invariance of the equations of motion of the reduced system and further, reduced by performing stochastic averaging. The resulting one-dimensional Markov diffusive process is studied in detail. The stationary probability density distribution is obtained by solving the Fokker-Planck equation along with the appropriate boundary conditions. The boundary behaviour is studied using an asymptotic approach. Some aspects of dynamical and phenomenological bifurcations of the stationary solution are also investigated. The scheme of things presented here can be applied in principle to a four-dimensional Hamiltonian system possessing one integral of motion in addition to the hamiltonian and having one fixed point.

Investigation of the Stability of Axially Moving Beams With Discrete Masses

2021 ◽  
Author(s):  
Konstantina Ntarladima ◽  
Michael Pieber ◽  
Johannes Gerstmayr
Keyword(s):  
Large Deformations ◽  
Equations Of Motion ◽  
Arbitrary Lagrangian Eulerian ◽  
Axial Motion ◽  
Multibody Modeling ◽  
Conveyor Belts ◽  
Axially Moving Beams ◽  
Concentrated Masses ◽  
Axially Moving ◽  
The Stability

Abstract The present paper addresses axially moving beams with co-moving concentrated masses while undergoing large deformations. For the numerical modeling, a novel beam finite element is introduced, which is based on the absolute nodal coordinate formulation extended with an additional Eulerian coordinate to represent the axial motion. The resulting formulation is well known as Arbitrary Lagrangian Eulerian (ALE) method, which is often used for axially moving beams and pipes conveying fluids. As compared to previous formulations, the present formulation allows us to introduce the Eulerian part by an independent coordinate, which fully incorporates the dynamics of the axial motion, while the shape functions remain independent of the beam coordinates and are thus constant. The proposed approach, which is derived from an extended version of Lagrange’s equations of motion, allows for the investigation of the stability of axially moving beams for a certain axial velocity and stationary state of large deformation. A multibody modeling approach allows us to extend the beam formulation for co-moving discrete masses, which represent concentrated masses attached to the beam, e.g., gondolas in ropeway systems, or transported masses in conveyor belts. Within numerical investigations we show that a larger number of discrete masses behaves similarly as the case of (continuously) distributed mass along the beam.

Natural frequency, damping and forced responses of sandwich plates with viscoelastic core and graphene nanoplatelets reinforced face sheets

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1165-1177 ◽  
Author(s):  
Ali Mohseni ◽  
Meisam Shakouri
Keyword(s):  
Sandwich Plate ◽  
Effective Properties ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Graphene Nanoplatelets ◽  
Complex Eigenvalue ◽  
Viscoelastic Core ◽  
Forced Vibration Analysis ◽  
Forced Responses

The free and forced vibration analysis of a sandwich plate with the viscoelastic core and face layers reinforced functionally with multilayered graphene nanoplatelets is presented. Different graphene nanoplatelet distributions are considered through the thickness, and the effective properties of the graphene reinforced nanocomposite are obtained by the rule of mixture. The equations of motion are extracted using Hamilton’s principle and assuming the classical thin plate theory for face layers and the first-order shear deformation theory for the thick viscoelastic core. Assuming the simply-supported boundary condition for all edges, the displacement components are proposed by Fourier series and the complex eigenvalue problem is solved to obtain the natural frequencies as well as the loss factors. The results are validated with available investigations, and effects of some important parameters on the free and forced responses of the sandwich plate are studied.

An Efficient Response Analysis Method for a Non-Linear/Parameter Changing System Using Sub-Structure Modes

1993 ◽  
Vol 207 (1) ◽  
pp. 41-52
Author(s):  
H K Kim ◽  
Y-S Park
Keyword(s):  
Equations Of Motion ◽  
Forced Response ◽  
Suggested Method ◽  
Response Analysis ◽  
Synthesis Methods ◽  
Lumped Mass ◽  
State Equations ◽  
Mass Model ◽  
Non Linear ◽  
Forced Responses

An efficient state-space method is presented to determine time domain forced responses of a structure using the Lagrange multiplier based sub-structure technique. Compared with the conventional mode synthesis methods, the suggested method can be particularly effective for the forced response analysis of a structure subjected to parameter changes with time, such as a missile launch system, and/or having localized non-linearities, because this method does not need to construct the governing equations of the combined whole structure. Both the loaded interface free-free modes and free interface modes can be employed as the modal bases of each sub-structure. The sub-structure equations of motion are derived using Lagrange multipliers and recurrence discrete-time state equations based upon the concept of the state transition matrix are formulated for transient response analysis. The suggested method is tested with two example structures, a simple lumped mass model with a non-linear joint and an abruptly parameter changing structure. The test results show that the suggested method is very accurate and efficient in calculating forced responses and in comparing it with the direct numerical integration method.

Inverse Identification of Non-Linear ODEs for the Dynamic Control of Material Testing Systems

2020 ◽  
Author(s):  
Evelyn M. Lunasin ◽  
Athanasios P. Iliopoulos ◽  
John G. Michopoulos ◽  
John C. Steuben
Keyword(s):  
Equations Of Motion ◽  
Dynamic Control ◽  
Material Testing ◽  
Naval Research ◽  
System Control ◽  
Numerical Verification ◽  
Inverse Identification ◽  
Coupled Equations ◽  
Actuation System ◽  
High Control

Abstract The development of advanced robotic systems for material testing by the U.S. Naval Research laboratory has expanded the set of requirements for mechatronic system control. This expansion lies beyond the limits of readily available control system capabilities because of the high control rate requirements. To establish this capability, a control mechanism based on the online identification of the ordinary differential equation governing the coupled equations of motion and deformation is proposed in the present work. The part of the proposed approach involving the inverse identification of the ODEs at hand is described in its general form first. Subsequently, the numerical verification is demonstrated via synthetic tests for a compliant actuation system with varying levels of noise injected in the system.

Axial Leakage Flow-Induced Vibration of Thin Cylindrical Shell With Respect to Circumferential Vibration

2002 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Atsuhiko Shintani ◽  
Masakazu Ono
Keyword(s):  
Cylindrical Shell ◽  
Equations Of Motion ◽  
Fluid Pressure ◽  
Navier Stokes ◽  
Leakage Flow ◽  
Thin Cylindrical Shell ◽  
Flow Induced Vibration ◽  
Navier Stokes Equation ◽  
Coupled Equations ◽  
Unstable Vibration

In this paper, the dynamic stability of a thin cylindrical shell subjected to axial leakage flow is discussed. In this paper, the third part of a study of the axial leakage flow-induced vibration of a thin cylindrical shell, we focus on circumferential vibration, that is, the ovaling vibration of a shell. The coupled equations of motion between shell and liquid are obtained by using Donnell’s shell theory and the Navier-Stokes equation. The added mass, added damping and added stiffness in the coupled equations of motion are described by utilizing the unsteady fluid pressure acting on the shell. The relations between axial velocity and the unstable vibration phenomena are clarified concerning the circumferential vibration of a shell. Numerical parametric studies are done for various dimensions of a shell and an axial leakage flow.

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