Related papers: Small oscillations of the pendulum, Euler's method…

We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichlet boundary conditions.…

A geometric form of Euler-Lagrange equations is developed for a chain pendulum, a serial connection of $n$ rigid links connected by spherical joints, that is attached to a rigid cart. The cart can translate in a horizontal plane acted on by…

This paper is based on MacColl's [1] solution of the equation of motion for a linear (harmonic) oscillator subject to the laws of special relativity in the rest frame of the center of attraction. MacColl's result can be extended to the…

We present a mathematical and numerical analysis on a control model for the time evolution of a multi-layered piezoelectric cantilever with tip mass and moment of inertia, as developed by Kugi and Thull [31]. This closed-loop control system…

We describe a 8085 microprocessor interface developed to make reliable time period measurements. The time period of each oscillation of a simple pendulum was measured using this interface. The variation of the time period with increasing…

The aim of this paper is to study and classify the multiplicity of distinguished limits and asymptotic solutions for the advection equation with a general oscillating velocity field with the systematic use of the two-timing method. Our…

We derive nonlinear stability results for numerical integrators on Riemannian manifolds, by imposing conditions on the ODE vector field and the step size that makes the numerical solution non-expansive whenever the exact solution is…

For the system of an inverted spherical pendulum with friction and a periodically moving pivot point we prove the existence of at least one periodic solution with the additional property of being falling-free. The last means that the…

Allowing for space- and time-dependence of mass in Klein--Gordon equations resolves the problem of negative probability density and of violation of Lorenz covariance of interaction in quantum mechanics. Moreover it extends their…

This paper proposes and analyzes a finite difference method based on compact schemes for the Euler-Bernoulli beam equation with damping terms. The method achieves fourth-order accuracy in space and second-order accuracy in time, while…

Recent equations of motion for the large deflections of a cantilevered elastic beam are analyzed. In the traditional theory of beam (and plate) large deflections, nonlinear restoring forces are due to the effect of stretching on bending;…

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method…

A new nonlinear equation governing asymptotic dynamics of ripples is derived by using a short wave perturbative expansion on a generalized version of the Green-Naghdi system. It admits peakon solutions with amplitude, velocity and width in…

For the study of highly nonlinear, conservative dynamic systems, finding special periodic solutions which can be seen as generalization of the well-known normal modes of linear systems is very attractive. However, the study of…

The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order…

The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order…

We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations.

A novel derivation of non-stationary solutions of 3D Euler equations for incompressible inviscid flow is considered here. Such a solution is the product of 2 separated parts: - one consisting of the spatial component and the other being…

The humble pendulum is often invoked as the archetype of a simple, gravity driven, oscillator. Under ideal circumstances, the oscillation frequency of the pendulum is independent of its mass and swing amplitude. However, in most real-world…

Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such…