Dynamic period equation: Derivation, relation to inhour equation, and precursor estimation

1999 ◽  
Vol 46 (3) ◽  
pp. 433-437 ◽  
Author(s):  
J.A. Bernard ◽  
L.W. Hu
Keyword(s):  
Period Equation

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.

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.

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.

scholarly journals Rayleigh Wave Dispersion for a Single Layer on an Elastic Half Space

1960 ◽  
Vol 13 (3) ◽  
pp. 498 ◽  
Author(s):  
BA Bolt ◽  
JC Butcher
Keyword(s):  
Numerical Solutions ◽  
Single Layer ◽  
Wave Dispersion ◽  
Elastic Half Space ◽  
Seismic Parameters ◽  
Period Equation ◽  
Phase And Group Velocities ◽  
The University ◽  
Rayleigh Wave Dispersion ◽  
Group Velocities

Numerical solutions of the period equation for Rayleigh waves in a single surface layer were calculated using the SILLIAC computer at the University of Sydney. Values of the phase and group velocities for both the fundamental and first higher mode are tabulated against period for eleven models. These related models allow a sensitivity analysis of the effect of variation in the seismic parameters.

Rayleigh waves in a half-space with bounded variation in density and rigidity

1974 ◽  
Vol 64 (4) ◽  
pp. 1263-1274
Author(s):  
C. R. A. Rao
Keyword(s):  
Power Series ◽  
Half Space ◽  
Bounded Variation ◽  
Rayleigh Waves ◽  
Body Wave ◽  
Equations Of Motion ◽  
Series Method ◽  
Power Series Method ◽  
Rigidity Modulus ◽  
Period Equation

abstract The stress equations of motion of elasticity are solved by the power series method for an inhomogeneous, isotropic elastic semi-space whose rigidity modulus μ and density ρ are defined by ρ = μ ∞ + ( μ 0 − μ ∞ ) exp ( − ε x 2 ) , ρ = ρ ∞ + ( ρ 0 − ρ ∞ ) exp ( − ε x 2 ) , ε > 0 , x 2 ∈ [ 0 , ∞ ]. The period equation for Rayleigh waves is derived and discussed numerically. The solutions may also be useful for body-wave studies.

The leaky-mode period equation—a plane-wave approach

1970 ◽  
Vol 60 (6) ◽  
pp. 1989-1998 ◽  
Author(s):  
L. E. Alsop
Keyword(s):  
Plane Wave ◽  
Phase Velocity ◽  
Half Space ◽  
Leaky Mode ◽  
Wave Number ◽  
Complex Frequency ◽  
The Real ◽  
Leaky Modes ◽  
Phase Velocity Dispersion ◽  
Period Equation

Abstract It is shown that the plane-wave picture of a leaky mode proposed by Burg, Ewing, Press and Stulkin (1951) yields the accepted period equation for leaky modes in a water layer a half-space. The resultant mode is formed by an inhomogeneous wave with real frequency and complex wave number and phase velocity. Another form of mode considered is that formed by a homogeneous wave in the guide with real phase velocity and complex frequency and wave number. The phase-velocity dispersion curve for this case is appropriate for determining shear-wave coupling to PL waves. The procedures of the article could be readily extended to the more complicated case of a solid layer over a half-space. It is also demonstrated that the derivative of the real part of angular frequency with respect to the real part of the wave number is a good approximation to the group velocity for leaky modes with low losses.

A fast, accurate method for computing group-velocity partial derivatives for Rayleigh and Love modes

1975 ◽  
Vol 65 (5) ◽  
pp. 1105-1114
Author(s):  
W. L. Rodi ◽  
P. Glover ◽  
T. M. C. Li ◽  
S. S. Alexander
Keyword(s):  
Group Velocity ◽  
Numerical Differentiation ◽  
Accurate Method ◽  
Model Parameters ◽  
Velocity Data ◽  
Partial Derivatives ◽  
Mode Group ◽  
Period Equation ◽  
Computer Codes ◽  
Force Calculation

Abstract A method for quickly and accurately calculating Rayleigh- and Love-mode group-velocity partial derivatives with respect to model parameters (m) is developed. The method requires computer codes that calculate C, U, and ∂C∂m|ω and employs numerical differentiation of ∂C∂m|ω to yield ∂U∂m|ω. The method is fast because ∂C∂m|ω and ∂U∂m|ω for all the model parameters can be obtained at a given frequency from only two solutions of the period equation. The accuracy of the method is established with two examples. For Love waves, the group-velocity partials computed by this method agree exactly with those obtained analytically by Novotny (1970). For Rayleigh waves, comparison with a “brute force” calculation of group-velocity partials showed agreement to the order of 0.00002. Systematic inversion of group-velocity data separately or in combination with phase-velocity data is computationally feasible using this method.

NUMERICAL SOLUTIONS FOR LOVE WAVE DISPERSION ON A HALF‐SPACE WITH DOUBLE SURFACE LAYER

Geophysics ◽  
1959 ◽  
Vol 24 (1) ◽  
pp. 12-29 ◽  
Author(s):  
James Dorman
Keyword(s):  
Surface Layer ◽  
Half Space ◽  
Crustal Structure ◽  
Love Wave ◽  
Numerical Solutions ◽  
Wave Dispersion ◽  
Columbia University ◽  
Period Equation ◽  
Double Surface ◽  
Love Wave Dispersion

The IBM 650 computer of the Watson Scientific Computing Laboratory, Columbia University, was programmed to obtain numerical solutions for the period equation for Love waves on a half‐space with a double surface layer. Solutions including higher modes for seven models of the continental crust‐mantle system are presented. This group of related cases shows that certain properties of the solutions are diagnostic of crustal structure. These relationships are illustrated graphically.

The period equation for primes p congruent to 1 (mod 5)

1971 ◽  
Vol 69 (1) ◽  
pp. 153-155
Author(s):  
A. R. Rajwade
Keyword(s):  
Primitive Root ◽  
Monic Polynomial ◽  
Symmetrical Form ◽  
Period Equation ◽  
Image Position

Let p = 1 + 5n be a rational prime congruent to 1 (mod 5). Let ζ = e2πi/p and let g be a primitive root mod p. Let the non-zero residues g, g2, …, gp-1 (mod) p be divided into five classes , ℬ, , , ℰ, where gν ∈ , ℬ, , , ℰ according as ν ≡ 0, 1, 2, 3, 4 (mod 5). Letbe the 5-nomial periods. Then it is well known (see (3)) that they are the roots of a monic polynomial with integral coefficients. Our object is to determine these coefficients in terms of the quantities A, B, C, D, E, Y, Z considered in a previous paper (2), p. 65. A large number of relations connecting these quantities have been obtained in the above-mentioned paper and we shall use these relations to simplify the coefficients and get them in a reasonably compact and symmetrical form.

A Forgotten Simple Pendulum Period Equation

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