Nonlinear Transverse Vibrations of a Spinning Disk

1964 ◽  
Vol 31 (1) ◽  
pp. 72-78 ◽  
Author(s):  
Jerzy L. Nowinski
Keyword(s):  
Large Amplitude ◽  
Close Agreement ◽  
Vibration Mode ◽  
Linear Case ◽  
Field Equations ◽  
Spinning Disk ◽  
Transverse Vibrations ◽  
Von Karman ◽  
Large Amplitude Vibrations ◽  
Von Kármán

Using the von Karman field equations, large amplitude vibrations of a spinning disk are analysed. For definiteness, the disk is assumed to be free, and its deflection is represented by a two-term polynomial. A vibration mode associated with two nodal diameters is studied in more detail. A familiar phenomenon of a decreasing period of vibration with an increasing amplitude is corroborated. The results specialized to the linear case show a close agreement with the classical results of Lamb and Southwell. The dependence of the membrane stresses on the amplitude of vibration and the velocity of spin is discussed.

Large Amplitude Vibrations of Clamped Circular Plate of Variable Thickness

1984 ◽  
Vol 51 (1) ◽  
pp. 207-210 ◽  
Author(s):  
S. K. Chaudhuri
Keyword(s):  
Circular Plate ◽  
Variable Thickness ◽  
Large Amplitude ◽  
Nonlinear Oscillations ◽  
Numerical Results ◽  
Varying Thickness ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Plate Of Variable Thickness ◽  
Large Amplitude Vibrations

In this paper nonlinear oscillations of a clamped circular plate of linearly varying thickness have been investigated using von Karman equations expressed in terms of displacement components. Numerical results obtained have been compared and discussed.

A theoretical study of in-plane bending of a single unreinforced mitred-bend

1966 ◽  
Vol 1 (3) ◽  
pp. 264-276 ◽  
Author(s):  
N Jones ◽  
R Kitching
Keyword(s):  
Strain Energy ◽  
Close Agreement ◽  
Theoretical Study ◽  
Compatibility Conditions ◽  
Curved Pipe ◽  
Exact Analysis ◽  
Von Karman ◽  
Plane Bending ◽  
Experimental Values ◽  
Von Kármán

It can be shown that an exact analysis of the deformations or stresses in a single mitred-bend would involve some terms which rapidly decay along the pipe axes and others which would have a relatively long damping length. The overall flexibility and radial deformations of a single unreinforced mitred-bend have been estimated by minimizing the strain energy contributed by the long damping length terms in a manner similar to that used by von Kármán for curved pipe-bends. Finally, the short damping length terms are introduced in order to satisfy the equilibrium and compatibility conditions across the oblique intersection of the two pipes forming the mitred-bend. The theoretical values of overall flexibility, diameter changes, and stresses were found to be in close agreement with experimental values for a single right-angled unreinforced mitred-bend.

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.

scholarly journals A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 714
Author(s):  
Jiujiang Wang ◽  
Xin Liu ◽  
Yuanyu Yu ◽  
Yao Li ◽  
Ching-Hsiang Cheng ◽  
...  
Keyword(s):  
Analytical Modeling ◽  
Ultrasonic Transducer ◽  
Governing Equations ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Collapse Mode ◽  
Capacitive Micromachined Ultrasonic Transducer ◽  
Gap Height ◽  
Von Kármán ◽  
Analytical Expressions

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

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