Forced Motions of Timoshenko Beams

1955 ◽  
Vol 22 (1) ◽  
pp. 53-56
Author(s):  
G. Herrmann
Keyword(s):  
Free Vibration ◽  
Elastic Beam ◽  
Time Dependent ◽  
Transverse Shear Deformation ◽  
Transverse Displacement ◽  
Timoshenko Beams ◽  
Rotatory Inertia ◽  
Forced Motion ◽  
Dependent Variables ◽  
The One

Abstract Timoshenko’s theory of flexural motions in an elastic beam takes into account both rotatory inertia and transverse-shear deformation and, accordingly, contains two dependent variables instead of the one transverse displacement of classical theory of flexure. For the case of forced motions, the solution involves complications not usually encountered. The difficulties may be surmounted in several ways, one of which is presented in this paper. The method described makes use of the property of orthogonality of the principal modes of free vibration and uses the procedure of R. D. Mindlin and L. E. Goodman in dealing with time-dependent boundary conditions. Thus the most general problem of forced motion is reduced to a free-vibration problem and a quadrature.

Analytical Treatment of In-Plane Parametrically Excited Undamped Vibrations of Simply Supported Parabolic Arches

2003 ◽  
Vol 125 (1) ◽  
pp. 73-79 ◽  
Author(s):  
Dimitris S. Sophianopoulos ◽  
George T. Michaltsos
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Dynamic Stability ◽  
Parametric Excitation ◽  
Stability Problem ◽  
Axial Loading ◽  
Time Dependent ◽  
Analytical Treatment ◽  
Simply Supported ◽  
Forced Motion

The present work offers a simple and efficient analytical treatment of the in-plane undamped vibrations of simply supported parabolic arches under parametric excitation. After thoroughly dealing with the free vibration characteristics of the structure dealt with, the differential equations of the forced motion caused by a time dependent axial loading of the form P=P0+Pt cos θt are reduced to a set of Mathieu-Hill type equations. These may be thereafter tackled and the dynamic stability problem comprehensively discussed. An illustrative example based on Bolotin’s approach produces results validating the proposed method.

A New Theory of Elastic Sandwich Plates—One-Dimensional Case

1959 ◽  
Vol 26 (3) ◽  
pp. 415-421
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Infinite Plate ◽  
Theory Of Elasticity ◽  
Dimensional Case ◽  
Transverse Shear Deformation ◽  
Sandwich Plates ◽  
Rotatory Inertia ◽  
One Dimensional ◽  
The Core ◽  
The One ◽  
Final Equation

Abstract A new flexural theory of isotropic elastic sandwich plates is deduced from the theory of elasticity. The one-dimensional case is presented in this paper. The theory includes the effects of transverse-shear deformation and rotatory inertia in both the core and faces of the sandwich, and no limitation is imposed upon the magnitudes of the ratios between the thicknesses, material densities, and elastic constants of the core and faces of the sandwich. The method used is an extension of one due to Mindlin [1], and the results reduce to those of his for the corresponding homogeneous plates as special limiting cases. A final equation also may be simplified and reduced to the corresponding results of Reissner [2], Hoff [3], and Eringen [4] for the bending of ordinary sandwich plates that have thin faces. Results of the theory are applied to the problem of bending of a cantilever plate subjected to load at the unsupported end and to the problem of propagation of straight-crested waves in an infinite plate.

Forced Motions of Elastic Rods

1954 ◽  
Vol 21 (3) ◽  
pp. 221-224
Author(s):  
G. Herrmann
Keyword(s):  
Equation Of Motion ◽  
Elastic Rod ◽  
Dimensional Theory ◽  
Compressional Waves ◽  
One Dimensional ◽  
Forced Motion ◽  
Lagrange’S Equation ◽  
Rod Theory ◽  
Dependent Variables ◽  
The One

Abstract In a recent paper by Mindlin and Herrmann, a one-dimensional theory of compressional waves in an elastic rod was described. This theory takes into account both radial inertia and radial shear stress and, accordingly, contains two dependent variables instead of the one axial displacement of classical rod theory. The solution of the equations for the case of forced motions thus involves complications not usually encountered. The difficulties may be surmounted in several ways, one of which is presented in this paper. The method described makes use of Lagrange’s equation of motion and reduces the most general problem of forced motion to a free vibration problem and a quadrature.

Vibration and Stability of an Elastic Beam Subjected to a Periodic Axial Load With Time-Dependent Displacement Excitation at Both Ends

1991 ◽  
Author(s):  
Xiao-Feng Wu ◽  
Adnan Akay
Keyword(s):  
Axial Load ◽  
Perturbation Technique ◽  
Harmonic Balance Method ◽  
Elastic Beam ◽  
Time Dependent ◽  
Transverse Load ◽  
Order Partial Differential Equation ◽  
Weighted Residual Method ◽  
Hill’S Equation ◽  
Hill's Equation

Abstract This paper concerns the transverse vibrations and stabilities of an elastic beam simultaneously subjected to a periodic axial load, a distributed transverse load, and time-dependent displacement excitations at both ends. The equation of motion derived from Bernoulli-Euler beam theory is a fourth-order partial differential equation with periodic coefficients. To obtain approximate solutions, the method of assumed-modes is used. The unknown time-dependent function in the assumed-modes method is determined by a generalized inhomogeneous Hill’s equation. The instability regions possessed by this generalized Hill’s equation are obtained by both the perturbation technique up to the second order and the harmonic balance method. The dynamic response and the corresponding spectrum of the transversely oscillating elastic beam are calculated by the weighted-residual method.

Dynamic Behavior of Aircraft Wings Modeled as Doubly-Tapered Composite Thin-Walled Beams

1999 ◽  
Author(s):  
Sungsoo Na ◽  
Liviu Librescu
Keyword(s):  
Composite Materials ◽  
Free Vibration ◽  
Dynamic Response ◽  
Dynamic Behavior ◽  
Transverse Shear ◽  
Dynamical Behavior ◽  
Time Dependent ◽  
Aircraft Wings ◽  
Helicopter Blades ◽  
Thin Walled

Abstract A study of the dynamical behavior of aircraft wings modeled as doubly-tapered thin-walled beams, made from advanced anisotropic composite materials, and incorporating a number of non-classical effects such as transverse shear, and warping inhibition is presented. The supplied numerical results illustrate the effects played by the taper ratio, anisotropy of constituent materials, transverse shear flexibility, and warping inhibition on free vibration and dynamic response to time-dependent external excitations. Although considered for aircraft wings, this analysis and results can be also applied to a large number of structures such as helicopter blades, robotic manipulator arms, space booms, tall cantilever chimneys, etc.

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