The Transverse Vibration of a Doubly Tapered Beam

1977 ◽  
Vol 44 (1) ◽  
pp. 123-126 ◽  
Author(s):  
D. O. Banks ◽  
G. J. Kurowski
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Cross Section ◽  
Transverse Vibration ◽  
Eigenvalues And Eigenfunctions ◽  
Free Boundary Conditions ◽  
Tapered Beam ◽  
Transverse Vibrations ◽  
The Cross ◽  
Homogeneous Beam

We analyze the transverse vibrations of a thin homogeneous beam which is symmetric with respect to the x-y and x-z planes. The cross section of the beam at x is assumed to have the form D(x)={(x,y,z)|x∈[0,1],y=xαy1,z=xβz1,(y1,z1)∈D1} where D1 is the cross section at x = 1. Expressions are obtained from which the eigenvalues and eigenfunctions can be easily found for 0 ≤ α < 2 and all combinations of clamped, hinged, guided, and free boundary conditions at both ends of the beam.

Exact Solutions for Transverse Vibrations of Beams and Plates of Parabolic Thickness Variation

2004 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Orthogonal Polynomials ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Thickness Variation ◽  
Singular Problems ◽  
Free Boundary Conditions ◽  
Transverse Vibrations

This paper presents the class of nonuniform beams and nonuniform axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations, respectively, have been identified to be eigenvalue singular problems of orthogonal polynomials. Recent published results regarding a fourth order differential equation and eigenvalue singular problem of classical orthogonal polynomials allowed this study, which extends the class of nonuniform beams and circular nonuniform plates having exact solutions for the problem of free transverse vibrations. The geometry of the elements belonging to the class presented in this paper consists of beams convex parabolic thickness variation and polynomial width variation with the axial coordinate, and plates of convex parabolic thickness variation with the radius. Two boundary value problems of transverse vibrations of beams are reported: 1) complete beam (sharp at either end) with free-free boundary conditions, and 2) half-beam, i.e. a half of the symmetric complete beam, with the large end hinged and sharp end free. The boundary value problem of circular complete plate (zero thickness at zero and outer radii) with free-free boundary conditions has been also reported. For all these boundary value problems the exact mode shapes were Jacobi polynomials and the exact dimensionless natural frequencies were found from the eigenvalues of the eigenvalue singular problems of orthogonal polynomials.

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

scholarly journals Identification of Corrosion on a Hollow Tube Using Vibration

2014 ◽  
Vol 564 ◽  
pp. 176-181
Author(s):  
S.T. Cheng ◽  
Nawal Aswan Abdul Jalil ◽  
Zamir A. Zulkefli
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Impact Loading ◽  
Structural Damage ◽  
Natural Frequencies ◽  
Localized Corrosion ◽  
Free Boundary Conditions ◽  
The Impact ◽  
Impact Vibration

Vibration based technique have so far been focused on the identification of structural damage. However, not many studies have been conducted on the corrosion identification on pipes. The objective of this paper is to identify corrosion on pipes from vibration measurements. A hollow pipe, 500 mm in length with 63.5 mm in diameter was subjected to impact loading using an impact hammer to identify the natural frequency of the tube in two conditions i) without any corrosion and ii) with an induced localized 40 mm by 40 mm corrosion at the middle of the pipe. The shift of natural frequencies of the structures under free boundary conditions was examined for each node of excitation. The results showed that there is a shift in natural frequency of the pipe, between 3 and 4 Hz near to the corrosion area. It can suggested that that the impact vibration is capable of identifying of localized corrosion on a hollow tube.

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