Derivation of First-Order Difference Equations for Dynamical Systems by Direct Application of Hamilton’s Principle

1970 ◽  
Vol 37 (2) ◽  
pp. 276-278 ◽  
Author(s):  
J. M. Vance ◽  
A. Sitchin
Keyword(s):  
Difference Equations ◽  
Equations Of Motion ◽  
Lagrangian Multiplier ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Order Difference ◽  
First Order ◽  
Finite Difference Equations ◽  
Lagrangian Multiplier Method ◽  
Dynamics Problems

In dynamics problems where the equations of motion are eventually reduced to finite-difference equations for numerical integration on a digital computer, an auxiliary condition exists that permits the application of the Lagrangian multiplier method to Hamilton’s principle in order to obtain directly a set of first-order difference equations. These equations are equivalent to Hamilton’s canonical equations and are derived without the necessity to obtain the Hamiltonian or take time derivatives.

Free vibration analysis of a porous rotor integrated with regular patterns of circumferentially distributed functionally graded piezoelectric patches on inner and outer surfaces

2020 ◽  
Vol 32 (1) ◽  
pp. 82-103
Author(s):  
Yaser Heidari ◽  
Mohsen Irani Rahaghi ◽  
Mohammad Arefi
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
External Work ◽  
First Order ◽  
Functionally Graded Piezoelectric

This article studies dynamic characteristics of a novel porous cylindrical hollow rotor based on the first-order shear deformation theory and Hamilton’s principle. The proposed model is made from a core including aluminum with porosity integrated with an arrangement of functionally graded piezoelectric patches placed on its inner and outer surfaces with a customized circumferential orientation. The piezoelectric patches are subjected to applied electric potential as sensor and actuator. The kinematic relations are developed based on the first-order shear deformation theory. Hamilton’s principle is used to derive governing equations of motion with calculation of strain and kinetic energies and external work. Solution procedure of the partial differential equations of motion is developed using Galerkin technique for simple boundary conditions. The accuracy and trueness of this work is justified using a comprehensive comparison with previous valid references. A large parametric study is presented to show influence of significant parameters such as dimensionless geometric parameters, porosity coefficient, angular speed, inhomogeneous index, and characteristics of patches on the mode shapes, natural frequencies, and critical speeds of the structure.

A Two-Dimensional Numerical Solution for Elastic Waves in Variously Configured Rods

1971 ◽  
Vol 38 (1) ◽  
pp. 62-70 ◽  
Author(s):  
J. L. Habberstad
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Difference Equations ◽  
Pressure Pulse ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
First Order ◽  
Finite Difference Equations ◽  
Viscosity Term

The exact equations of motion governing elastic, axisymmetric wave propagation in a cylindrical rod are approximated by a first-order finite-difference scheme. This difference scheme is based on a displacement rather than a velocity formulation, thereby making it unnecessary to explicitly introduce an artificial viscosity term into the finite-difference equations. The resulting difference equations are used in conjunction with the boundary and initial conditions 10 study: (a) a pressure pulse applied to the end of a semi-infinite bar, (b) a bar composed of two materials joined together at some point along its length, and (c) a bar containing a discontinuity in cross section. The numerical results so obtained are compared to available experimental data and other analytical-numerical solutions.

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