scholarly journals Jourdain’s variational equation and Appell’s equation of motion for nonholonomic dynamical systems

2003 ◽  
Vol 71 (1) ◽  
pp. 72-82 ◽  
Author(s):  
Li-Sheng Wang ◽  
Yih-Hsing Pao
Keyword(s):  
Dynamical Systems ◽  
Variational Equation ◽  
Equation Of Motion ◽  
Nonholonomic Dynamical Systems

scholarly journals Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

2016 ◽  
Vol 27 (6) ◽  
pp. 904-922 ◽  
Author(s):  
STEPHEN COOMBES ◽  
RÜDIGER THUL
Keyword(s):  
Dynamical Systems ◽  
Variational Equation ◽  
Period Doubling ◽  
Star Graph ◽  
Stability Function ◽  
Bifurcation Of Limit Cycles ◽  
Master Stability Function ◽  
Coupled Networks ◽  
Low Dimensional ◽  
Insight Into

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

Variational Equation of Motion for Coupled Flexure and Torsion of Bars of Thin-Walled Open Section Including Thermal Effect

1971 ◽  
Vol 38 (2) ◽  
pp. 502-506 ◽  
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Thermal Effect ◽  
Structural Dynamics ◽  
Variational Equation ◽  
Equation Of Motion ◽  
Bending Moments ◽  
Open Section ◽  
Thermal Bending ◽  
Axis Of Symmetry ◽  
Thin Walled ◽  
Special Case

Literature on flexure and torsion of bars of thin-walled open section is reviewed. The use of the variational equation of motion in solving problems of structural dynamics is further advocated. The variational equation of motion, together with the associated stress-displacement relations, is then derived for coupled flexure and torsion of the open section. Thermal effect is included, leading to a thermal twisting moment in addition to the usual thermal bending moments. For the special case of an open section with one axis of symmetry and with symmetrical heat input, only flexure is shown to be thermally inducible. The general result then reduces to the simple variational equation of flexural motion used in a separate study of the thermal flutter of a spacecraft boom.

Nonlinear Vibrations of Shallow Spherical Shells

1969 ◽  
Vol 36 (3) ◽  
pp. 451-458 ◽  
Author(s):  
P. L. Grossman ◽  
B. Koplik ◽  
Yi-Yuan Yu
Keyword(s):  
Nonlinear Vibrations ◽  
Variational Equation ◽  
Equation Of Motion ◽  
Dynamic Equations ◽  
Forced Oscillations ◽  
Finite Deflection ◽  
Spherical Shells ◽  
Nonlinear Dynamic Equations ◽  
Edge Conditions ◽  
Spherical Caps

Based on a system of nonlinear dynamic equations and the associated variational equation of motion derived for elastic spherical shells (deep or shallow), an investigation of the axisymmetric vibrations of spherical caps with various edge conditions is made by carrying out a consistent sequence of approximations with respect to space and time. Numerical results are obtained for both free and forced oscillations involving finite deflection. The effect of curvature is examined, with particular emphasis on the transition from a flat plate to a curved shell. In fact, in such a transition, the nonlinearity of the hardening type gradually reverses into one of softening.

scholarly journals A canonical form of the equation of motion of linear dynamical systems

2018 ◽  
Vol 474 (2211) ◽  
pp. 20170809
Author(s):  
Daniel T. Kawano ◽  
Rubens Goncalves Salsa ◽  
Fai Ma ◽  
Matthias Morzfeld
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Linear System ◽  
Canonical Form ◽  
Equation Of Motion ◽  
Positive Definite ◽  
Order Ordinary Differential Equation ◽  
Linear Dynamical Systems ◽  
Almost All

The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.

Application of Variational Equation of Motion to the Nonlinear Vibration Analysis of Homogeneous and Layered Plates and Shells

1963 ◽  
Vol 30 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Vibrations ◽  
Variational Equation ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Free Vibrations ◽  
Layered Plates ◽  
Variational Approximations ◽  
Elastic Systems ◽  
Plates And Shells

An integrated procedure is presented for applying the variational equation of motion to the approximate analysis of nonlinear vibrations of homogeneous and layered plates and shells involving large deflections. The procedure consists of a sequence of variational approximations. The first of these involves an approximation in the thickness direction and yields a system of equations of motion and boundary conditions for the plate or shell. Subsequent variational approximations with respect to the remaining space coordinates and time, wherever needed, lead to a solution to the nonlinear vibration problem. The procedure is illustrated by a study of the nonlinear free vibrations of homogeneous and sandwich cylindrical shells, and it appears to be applicable to still many other homogeneous and composite elastic systems.

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