Various methods of determining the natural frequencies and damping of composite cantilever plates. 2. Approximate solution by Galerkin's method for the trinomial model of damping

1997 ◽  
Vol 33 (1) ◽  
pp. 15-22
Author(s):  
V. S. Ekel'chik ◽  
V. M. Ryabov
Keyword(s):  
Approximate Solution ◽  
Natural Frequencies ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Trinomial Model

Stability of a Beam on an Elastic Foundation Subjected to a Nonconservative Load

1980 ◽  
Vol 47 (1) ◽  
pp. 116-120 ◽  
Author(s):  
Z. Celep
Keyword(s):  
Approximate Solution ◽  
Numerical Calculation ◽  
Elastic Foundation ◽  
Cantilever Beam ◽  
Numerical Study ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
The Common ◽  
The Stability

In this investigation, the influence of a Winkler type of elastic foundation on the stability of the cantilever beam subjected to a nonconservative load which consists of a vertical and a follower components is studied. In addition to the common transverse foundation modulus, a rotatory foundation modulus is considered. Approximate solution is obtained by using Galerkin’s method. Numerical calculation are reported and displayed for various combinations of the nonconservativeness parameter, transverse and rotatory modulus of the foundation, distance of the point of application of the load and that of the transverse spring. As a result of the numerical study unexpected feature of stability of the cantilever beam in contrast to the behavior of the column is identified.

Free Dry and Wet Vibration of 2-Way Tapered Hollow Marine Rudder With Non-Classical Pivot: Theoretical Study

2015 ◽  
Author(s):  
Rahul Jindal ◽  
Nabanita Datta
Keyword(s):  
Rigid Body ◽  
Natural Frequencies ◽  
Added Mass ◽  
Chord Length ◽  
Mode Shapes ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Uniform Beam ◽  
Aspect Ratios ◽  
Beam Mode

A theoretical analysis of free dry and wet vibration of a trapezoidal, 2-way tapered, marine spade rudder, is presented. The rudder is considered as a hollow Kirchhoff’s plate, with the chord section as a NACA profile. The chord length and the thickness taper from the top to the bottom, over the vertical span. The rudder is pivoted at the top, with the pivot behind the leading edge. The pivot is modeled as a combination of a translational and a rotational spring, in order to include the rigid body modes of the rudder vibration. The span-wise and chord-wise non-uniform beam vibration is first analyzed by the Rayleigh-Ritz method, in order to establish the non-uniform beam mode shapes. The span-wise beam is a linearly tapered vertical cantilever, with non-classical edge at the top and free at the bottom. The chord-wise section is a 2-span beam with the ends free, and four continuity conditions at the pivot. The non-uniform mode shapes, in either direction, are a weighted summation of the uniform beam mode shapes, which also satisfy the boundary/continuity conditions. They now act as admissible spatial functions to the plate vibration, which is analyzed by the Galerkin’s method. Eigenvalue analysis generates the plate natural frequencies. A weighted superposition, of the product of the beam mode shapes, in either direction, generates the plate mode shapes. Alternately, uniform beam mode shapes are used as admissible functions into the Galerkin’s method for the plate natural frequencies and mode shapes. The natural frequencies are generated for various positions of the rudder stock along the chord length. The pivot conditions (in both translational and rotational rigid body degree of freedom) influence the prominence of the rigid body mode shapes. The natural frequencies are analyzed for various pivot fixities, taper ratios, and aspect ratios of the plate. This is followed by the wet vibration analysis of the rudder. First, 2D strip theory is used to generate the added mass of each chord section. Constant strength source distribution technique is used to generate the added mass in sway and yaw of a 2D aerofoil. Each flexural and torsional mode is associated with its own added mass. Various empirical corrections are done to account for the 3D flow. Finally, 3D panel method is used to generate the modal added masses, and hence the wet natural frequencies. The added mass coefficient is generated for various aerofoil fineness ratios, pivot fixities, taper ratios, and aspect ratios of the plate.

scholarly journals The Approximated Solution for The Nonlinear Second Order Delay Multi-Value Problems

2011 ◽  
Vol 8 (1) ◽  
pp. 175-178
Author(s):  
Baghdad Science Journal
Keyword(s):  
Approximate Solution ◽  
Second Order ◽  
Galerkin’S Method ◽  
Galerkin's Method

This paper is attempt to study the nonlinear second order delay multi-value problems. We want to say that the properties of such kind of problems are the same as the properties of those with out delay just more technically involved. Our results discuss several known properties, introduce some notations and definitions. We also give an approximate solution to the coined problems using the Galerkin's method.

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