A Forgotten Simple Pendulum Period Equation

2021 ◽  
Vol 59 (8) ◽  
pp. 646-647
Author(s):  
Peter F. Hinrichsen
Keyword(s):  
Simple Pendulum ◽  
Period Equation

Action-Minimizing Curves for Tonelli Lagrangians

2017 ◽  
Author(s):  
Alfonso Sorrentino
Keyword(s):  
Critical Value ◽  
The Other ◽  
Invariant Sets ◽  
Simple Pendulum ◽  
New Concepts ◽  
Peierls Barrier ◽  
Average Action

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

The Simple Pendulum

1913 ◽  
Vol 7 (108) ◽  
pp. 189
Author(s):  
G. Greenhill
Keyword(s):  
Simple Pendulum

The Nonlinear Simple Pendulum

1972 ◽  
Vol 79 (4) ◽  
pp. 348-355
Author(s):  
Fred Brauer
Keyword(s):  
Simple Pendulum

Measured natural periods of concrete shear wall buildings: insights for the design of Canadian buildings

2012 ◽  
Vol 39 (8) ◽  
pp. 867-877 ◽  
Author(s):  
Damien Gilles ◽  
Ghyslaine McClure
Keyword(s):  
Seismic Analysis ◽  
Dynamic Properties ◽  
Response Spectrum ◽  
Shear Wall ◽  
Mode Shapes ◽  
Design Stage ◽  
Base Shear ◽  
Period Equation ◽  
Input Load ◽  
Structural Engineers

Structural engineers routinely use rational dynamic analysis methods for the seismic analysis of buildings. In linear analysis based on modal superposition or response spectrum approaches, the overall response of a structure (for instance, base shear or inter-storey drift) is obtained by combining the responses in several vibration modes. These modal responses depend on the input load, but also on the dynamic characteristics of the building, such as its natural periods, mode shapes, and damping. At the design stage, engineers can only predict the natural periods using eigenvalue analysis of structural models or empirical equations provided in building codes. However, once a building is constructed, it is possible to measure more precisely its dynamic properties using a variety of in situ dynamic tests. In this paper, we use ambient motions recorded in 27 reinforced concrete shear wall (RCSW) buildings in Montréal to examine how various empirical models to predict the natural periods of RCSW buildings compare to the periods measured in actual buildings under ambient loading conditions. We show that a model in which the fundamental period of RCSW buildings varies linearly with building height would be a significant improvement over the period equation proposed in the 2010 National Building Code of Canada. Models to predict the natural periods of the first two torsion modes and second sway modes are also presented, along with their uncertainty.

The Simple Pendulum at Any Amplitude

1956 ◽  
Vol 40 (331) ◽  
pp. 34
Author(s):  
P. J. Bulman
Keyword(s):  
Simple Pendulum

Period equation for Rayleigh waves in a layer overlying a half space with a sinusoidal interface

1962 ◽  
Vol 52 (4) ◽  
pp. 807-822 ◽  
Author(s):  
John T. Kuo ◽  
John E. Nafe
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Half Space ◽  
Rayleigh Waves ◽  
Wave Length ◽  
Linear Equations ◽  
Wave Number ◽  
Interface Plane ◽  
Period Equation ◽  
Phase And Group Velocities

abstract The problem of the Rayleigh wave propagation in a solid layer overlying a solid half space separated by a sinusoidal interface is investigated. The amplitude of the interface is assumed to be small in comparison to the average thickness of the layer or the wave length of the interface. Either by applying Rayleigh's approximate method or by perturbating the boundary conditions at the sinusoidal interface, plane wave solutions for the equations which satisfy the given boundary conditions are found to form a system of linear equations. These equations may be expressed in a determinant form. The period (or characteristic) equations for the first and second approximation of the wave number k are obtained. The phase and group velocities of Rayleigh waves in the present case depend upon both frequency and distance. At a given point on the surface, there is a local phase and local group velocity of Rayleigh waves that is independent of the direction of wave propagation.

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