Response and Discretization Methods for Axially Moving Materials

1991 ◽  
Vol 44 (11S) ◽  
pp. S279-S284 ◽  
Author(s):  
J. A. Wickert ◽  
C. D. Mote
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
The State ◽  
Space Formulation ◽  
Axially Moving ◽  
Effective Use ◽  
Gyroscopic Systems ◽  
Acceleration Component

Through a convective acceleration component, the equations of motion for axially-moving materials are skew-symmetric in the state space formulation, so that the response problem is best analyzed within the broader context of continuous gyroscopic systems. With particular application to the prototypical traveling string and beam models, a modal analysis that associates degrees of freedom with the complex state eigenfunctions and their conjugates is presented. This procedure is well-suited for harmonic excitation sources, and in some instances, it is more convenient than previous methods which decompose the modal coordinates, eigenfunctions, and generalized forces into real and imaginary components. Also from the state space perspective, Rayleigh’s quotient for gyroscopic systems provides a variational method for determining the eigensolutions of axially-moving materials. Ritz discretization of the quotient can make effective use of the speed-adapting modes of the traveling string and beam models as they are rich in phase, as well as amplitude, content.

Cylindrically Anisotropic Tubes Containing a Line Dislocation

2006 ◽  
Author(s):  
Chung-Hao Wang
Keyword(s):  
State Space ◽  
Dislocation Line ◽  
Longitudinal Axis ◽  
The State ◽  
Fourier Series Expansion ◽  
General Solutions ◽  
Mixed Dislocation ◽  
Line Dislocation ◽  
Space Formulation ◽  
Cylindrically Anisotropic

An analytical solution of the problem of a cylindrically anisotropic tube which contains a line dislocation is presented in this study. The state space formulation in conjunction with the eigenstrain theory is proved to be a feasible and systematic methodology to analyze a tube with the existence of dislocations. The state space formulation which expediently groups the displacements and the cylindrical surface traction can construct a governing differential matrix equation. By using Fourier series expansion and the well developed theory of matrix algebra, the asymmetrical solutions are not only explicit but also compact in form. The dislocation considered in this study is a kind of mixed dislocation which is the combination of edge dislocations and a screw dislocation and the dislocation line is parallel to the longitudinal axis of the tube. The degeneracy of the eigen relation and the technique to determine the inverse of a singular matrix are thoroughly discussed, so that the general solutions can be applied to the case of isotropic tubes, which is one of the novel features of this research. The results of isotropic problems, which are belong to the general solutions, are compared with the well-established expressions in the literature. The satisfied correspondences of these comparisons indicate the validness of this study. A cylindrically orthotropic tube is also investigated as an example and the numerical results for the displacements and tangential stress on the outer surface are displayed. The effects on surface stresses due to the existence of a dislocation appear to have a characteristic of localized phenomenon.

Transient Vibration Analysis of Open Circular Cylindrical Shells

2005 ◽  
Vol 128 (3) ◽  
pp. 366-374 ◽  
Author(s):  
Selvakumar Kandasamy ◽  
Anand V. Singh
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Grid Point ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Middle Surface ◽  
The State ◽  
Displacement Fields ◽  
Transient Vibration ◽  
Rotational Components

A numerical method based on the Rayleigh-Ritz method has been presented for the forced vibration of open cylindrical shells. The equations are derived from the three-dimensional strain-displacement relations in the cylindrical coordinate system. The middle surface of the shell represents the geometry, which is defined by an angle that subtends the curved edges, the length, and the thickness. The displacement fields are generated with a predefined set of grid points on the middle surface using considerably high-order polynomials. Each grid point has five degrees of freedom, viz., three translational components along the cylindrical coordinates and two rotational components of the normal to the middle surface. Then the strain and kinetic energy expressions are obtained in terms of these displacement fields. The differential equation governing the vibration characteristics of the shell is expressed in terms of the mass, stiffness, and the load consistent with the prescribed displacement fields. The transient response of the shell with and without damping is sought by transforming the equation of motion to the state-space model and then the state-space differential equations are solved using the Runge-Kutta algorithm.

Modeling and Simulation of Motorcycle Ride Comfort Based on Bump Road

2010 ◽  
Vol 139-141 ◽  
pp. 2643-2647 ◽  
Author(s):  
Dong Mei Yuan ◽  
Xiao Mei Zheng ◽  
Ying Yang
Keyword(s):  
Degrees Of Freedom ◽  
Ride Comfort ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Dynamics Model ◽  
Body Dynamics ◽  
Space Formulation ◽  
Simulation Results ◽  
Multi Body

Through analyzing the motion when motorcycle runs on the bump road, the 5-DOF multi-body dynamics model of motorcycle is developed, the degrees of freedom include vertical displacement of sprung mass, rotation of sprung mass, vertical displacement of driver, and vertical displacement of front and rear suspension under sprung mass. According to Lagrange Equation, the differential equations of motion and state-space formulation are derived. Then bump road is simulated by triangle bump, and input displacement is programmed by MATLAB. With the input of bump road, motorcycle ride comfort is simulated, and the simulation results are verified by experiment results combined with two channels tire-coupling road simulator. It indicates that the simulation results and experiment results match well; the 5-DOF model has guidance for development of motorcycle ride comfort.

scholarly journals Thermal Effects on Nonlinear Vibrations of an Axially Moving Beam with an Intermediate Spring-Mass Support

2013 ◽  
Vol 20 (3) ◽  
pp. 385-399 ◽  
Author(s):  
Siavash Kazemirad ◽  
Mergen H. Ghayesh ◽  
Marco Amabili
Keyword(s):  
Thermal Loading ◽  
Nonlinear Vibrations ◽  
Equations Of Motion ◽  
Time Integration ◽  
Harmonic Excitation ◽  
Axially Moving Beam ◽  
Moving Beam ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Continuation Technique

The thermo-mechanical nonlinear vibrations and stability of a hinged-hinged axially moving beam, additionally supported by a nonlinear spring-mass support are examined via two numerical techniques. The system is subjected to a transverse harmonic excitation force as well as a thermal loading. Hamilton's principle is employed to derive the equations of motion; it is discretized into a multi-degree-freedom system by means of the Galerkin method. The steady state resonant response of the system for both cases with and without an internal resonance between the first two modes is examined via the pseudo-arclength continuation technique. In the second method, direct time integration is employed to construct bifurcation diagrams of Poincaré maps of the system.

Analytical Investigation of Aero-Elastic Behavior of Aircraft Composite Box Wings in Unsteady Compressible Flow

2011 ◽  
Author(s):  
T. Farsadi ◽  
H. Haddadpour ◽  
U. Yuceoglu
Keyword(s):  
State Space ◽  
Compressible Flow ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Analytical Investigation ◽  
The State ◽  
Elastic Behavior ◽  
Aeroelastic System ◽  
The Time Domain ◽  
Flight Regimes

In this study, an anisotropic thin-walled “Composite Box Beam” as the “Wing System” is used to consider the effects of the fiber orientation and the lay-up configuration on the aeroelastic stability and the dynamic response of an aircraft wing. The present “Circumferentially Asymmetric Stiffness Model (CAS)” takes into account a group of non-classical effects; such as the transverse shear, the material anisotropy, warping inhibition, etc. The “Aerodynamic Strip Method” based on “Wagner Functions” in unsteady compressible flow are used to simulate compressible unsteady aerodynamic effects in the “state space” form. In addition, the mass, the stiffness and the damping matrices of the present non-conservative aeroelastic system are formed so that, the “Extended Galerkin’s Method (EGM)” and the “Separation of Variables Method” can be employed. As a result, the coupled and linear “Governing System of Dynamic Equations” are obtained. After then, by transforming matrices into the “state space” and “state vector” forms, the problem under study is finally converted into an “Eigenvalue Problem and Analysis”. Hence, the “flutter” and the “divergence” speeds for various layer configurations with different geometric and material properties and fiber orientations were obtained. Furthermore, by solving the aforementioned equations of motion in the time domain, the aeroelastic responses of the “Composite Box Wing System” for different flight regimes are computed. The present numerical results were compared and are verified with some existing experimental results in the literature. Based on these, some brief but important conclusions are presented.

Geometric description of time-dependent finite-dimensional mechanical systems

2020 ◽  
Vol 25 (11) ◽  
pp. 2050-2075
Author(s):  
Simon R. Eugster ◽  
Giuseppe Capobianco ◽  
Tom Winandy
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Mechanical Systems ◽  
Second Order ◽  
The State ◽  
Time Dependent ◽  
Finite Dimensional ◽  
Affine Bundle ◽  
Time Position

Using the non-standard geometric structure proposed by Loos, we present a coordinate-free formulation of the theory for time-dependent finite-dimensional mechanical systems with n degrees of freedom. The state space containing the system’s information on time, position and velocity is defined as a (2 n+1)-dimensional affine bundle over an ( n+1)-dimensional generalized space-time. The main goal is to present a geometric postulate that characterizes a second-order vector field whose integral curves describe the motions of a time-dependent finite-dimensional mechanical system. The core objects of the postulate are differential two-forms on the state space, called action forms, which are in a bijective relation with second-order vector fields. The requirements for a differential two-form to be an action form allow for a coordinate-free definition of non-potential forces, which may depend on time, position and velocity. Finally, we show that not only Lagrange’s equations but also Hamilton’s equations follow directly as mere coordinate representations of the same coordinate-free postulate.

A method for deriving a non-singular state-space formulation based on Rubin's model for Pogo analysis

2018 ◽  
Vol 233 (8) ◽  
pp. 2717-2730
Author(s):  
Shujun Tan ◽  
Qingwei Wang ◽  
Zhigang Wu ◽  
Yunfei Yang ◽  
Ziwen Yu
Keyword(s):  
State Space ◽  
State Space Model ◽  
The State ◽  
Dynamic Equations ◽  
Time Varying ◽  
The Finite Element Method ◽  
Space Model ◽  
Assembly Method ◽  
Standard Manner ◽  
Space Formulation

A method for deriving a non-singular state-space formulation based on Rubin's model for Pogo analysis is presented in this study. Because of the non-singularity, the state-space model can be directly used in frequency-domain analyses and time-domain simulations. To describe the assembly method concisely, the dynamic equations of the nine types of independent elements are described in a standard manner. The nine types of elements are divided into two classes according to characteristics of the dynamic equations. The mapping relationship between the local and global numbers of elements and nodes is obtained by numbering all of the elements and nodes. By integrating the element stiffness matrixes to obtain the total stiffness matrix used in the finite element method, the coefficient matrixes of the improved Rubin's model are assembled from the coefficient matrixes of all of the elements according to the mapping relationship. Based on the non-singular model, the time-varying simulation with the nonlinear property of the accumulator can be done conveniently by revising the state-space model. The successful application of this method to a Pogo analysis of a certain type of CZ rocket used in China verifies the correctness and efficiency of the method of this study.

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