Torsional Vibration Modes of Tapered Bars

1952 ◽  
Vol 19 (2) ◽  
pp. 220-222
Author(s):  
H. E. Fettis
Keyword(s):  
Differential Equation ◽  
Torsional Vibration ◽  
Harmonic Functions ◽  
Vibration Modes

Abstract The author offers a solution for the differential equation which governs the torsional vibration modes of a shaft or bar, these modes being simple harmonic functions of time.

Torsional Vibrations of Shells of Revolution

1961 ◽  
Vol 28 (4) ◽  
pp. 571-573 ◽  
Author(s):  
H. Garnet ◽  
M. A. Goldberg ◽  
V. L. Salerno
Keyword(s):  
Differential Equation ◽  
Governing Equation ◽  
Torsional Vibration ◽  
Conical Shell ◽  
Order Differential Equation ◽  
Annular Plate ◽  
Thin Shells ◽  
Torsional Vibrations ◽  
Vibration Modes ◽  
Shells Of Revolution

Torsional-vibration modes are uncoupled from the bending and extensional modes in thin shells of revolution. The solution for the uncoupled torsional modes then depends upon a linear second-order differential equation. The governing equation is subsequently solved for the frequencies of a conical shell. A tabulation of the first five frequencies for varying ratios of the terminal radii is presented. These frequencies are identical to those of an annular plate which has the same supports as the conical shell.

scholarly journals Boundary Isomorphism between Dirichlet Finite Solutions of Δu = Pu and Harmonic Functions

1973 ◽  
Vol 50 ◽  
pp. 7-20 ◽  
Author(s):  
Ivan J. Singer
Keyword(s):  
Differential Equation ◽  
Riemann Surface ◽  
Laplace Operator ◽  
Nontrivial Solution ◽  
Harmonic Functions ◽  
Green’S Functions ◽  
Hölder Continuous ◽  
Open Riemann Surface ◽  
Continuous Density ◽  
The Laplace Operator

Consider an open Riemann surface R and a density P(z)dxdy (z = x + iy), well defined on R. As was shown by Myrberg in [3], if P ≢ 0 is a nonnegative α-Hölder continuous density on R (0 < α ≤ 1) then there exists the Green’s functions of the differential equationp>on R, where Δ means the Laplace operator. As a consequence, there always exists a nontrivial solution on R.

scholarly journals The Solution of Mathteu's Differential Equation

1915 ◽  
Vol 34 ◽  
pp. 176-196 ◽  
Author(s):  
John Dougall
Keyword(s):  
Differential Equation ◽  
Harmonic Functions ◽  
Hypergeometric Equation ◽  
Successful Theory ◽  
The One ◽  
Hyperbolic Cylinders ◽  
Degenerate Cases

The determination of the harmonic functions of elliptic and hyperbolic cylinders depends on the solution of Mathieu's differential equation. This equation, it has been remarked by Professor Whittaker, is the one which naturally comes up for study after the hypergeometric equation has been disposed of. Its solution presents difficulties which do not arise in connection with the hypergeometric equation or its degenerate cases, and it cannot, I think, be said that any discussion of the equation has yet been given with which the student of analysis can rest content. The treatment given below, though certainly incomplete at some points, seetns to follow the lines along which a thoroughly successful theory may be hoped for.

scholarly journals On electrostatics in a gravitational field

1928 ◽  
Vol 118 (779) ◽  
pp. 184-194 ◽  
Keyword(s):  
Differential Equation ◽  
Harmonic Functions ◽  
Gravitational Force ◽  
Algebraic Expression ◽  
Uniform Field ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Gravitational Fields ◽  
Limiting Case ◽  
General Method

In a recent paper, Prof. Whittaker has discussed the effect, according to the general theory of relativity, of gravitation on electromagnetic phenomena. In particular, he has discussed electrostatics in gravitational fields of two kinds, namely (i) the field due to a single gravitating mass, in which case space-time has the metric discovered by Schwarzschild, and (ii) a limiting case of this, called a quasi-uniform field, in which the gravitational force is, in the neighbourhood of the origin, uniform. Whittaker’s general method, so far as electrostatical problems were concerned, was to solve the partial differential equation satisfied by the electrostatic potential in terms of generalised harmonic functions, and then, from these, to build up other solutions. In this way, he succeeded in finding an algebraic expression which represents the potential of a single electron in the quasi-uniform field; he did not, however, obtain a corresponding algebraic expression for the potential of an electron in the Schwarzschild field.

scholarly journals Numerical Investigation of Flapwise-Torsional Vibration Model of a Smart Section Blade with Microtab

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Nailu Li ◽  
Mark J. Balas ◽  
Hua Yang ◽  
Wei Jiang ◽  
Kaman T. Magar
Keyword(s):  
Torsional Vibration ◽  
Numerical Data ◽  
Vibration Modes ◽  
Aerodynamic Model ◽  
Aeroelastic Model ◽  
Vibration Model ◽  
Divergence Instability ◽  
Blade Section ◽  
Control Capability ◽  
Control Study

This study presents a method to develop an aeroelastic model of a smart section blade equipped with microtab. The model is suitable for potential passive vibration control study of the blade section in classic flutter. Equations of the model are described by the nondimensional flapwise and torsional vibration modes coupled with the aerodynamic model based on the Theodorsen theory and aerodynamic effects of the microtab based on the wind tunnel experimental data. The aeroelastic model is validated using numerical data available in the literature and then utilized to analyze the microtab control capability on flutter instability case and divergence instability case. The effectiveness of the microtab is investigated with the scenarios of different output controllers and actuation deployments for both instability cases. The numerical results show that the microtab can effectively suppress both vibration modes with the appropriate choice of the output feedback controller.

On discrete harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 194-206 ◽  
Author(s):  
H. A. Heilbronn
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Difference Equation ◽  
Harmonic Functions ◽  
The Difference ◽  
Discrete Harmonic Functions ◽  
Image Position

A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equationThis equation can be considered as the direct analogue either of the differential equationor of the integral equationin the notation normally employed to harmonic functions.

Stability of Cracked Rotors in the Coupled Vibration Mode

1988 ◽  
Vol 110 (3) ◽  
pp. 356-359 ◽  
Author(s):  
C. A. Papadopoulos ◽  
A. D. Dimarogonas
Keyword(s):  
Harmonic Functions ◽  
Vibration Mode ◽  
Coupling Effect ◽  
Equations Of Motion ◽  
Vibration Modes ◽  
Coupled Vibration ◽  
Compliance Matrix ◽  
Stiffness Coefficients ◽  
Local Flexibility

A transverse surface crack is known to add to a shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented, in the general case by way of a 6 × 6 compliance matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by the off-diagonal terms. In addition, when the shaft rotates the crack opens and closes. Then the differential equations of motion have periodically varying stiffness coefficients and the solution can be expressed as a sum of harmonic functions of time. A method for the determination of the intervals of instability of the first and of second kind is developed. The results have been presented in stability charts in the frequency vs. depth of the crack domain. The coupling effect due to the crack leads to very interesting results such as new frequencies and vibration modes.

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