scholarly journals Thermal post-buckling & large amplitude free vibration analysis of Timoshenko beams: Simple closed-form solutions

2014 ◽  
Vol 38 (17-18) ◽  
pp. 4548-4558 ◽  
Author(s):  
Jagadish Babu Gunda
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Large Amplitude ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Timoshenko Beams ◽  
Closed Form Solutions ◽  
Post Buckling

Closed-Form Vibration Analysis of Sagged Cable/Mass Suspensions

1992 ◽  
Vol 59 (4) ◽  
pp. 923-928 ◽  
Author(s):  
S.-P. Cheng ◽  
N. C. Perkins
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Vibration Analysis ◽  
Asymptotic Form ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Asymptotic Model ◽  
Form Analysis ◽  
Closed Form Solutions ◽  
Nonlinear Motion

A theoretical model is presented which describes the three-dimensional nonlinear motion of a sagged cable that supports an array of discrete masses. An asymptotic form of this general model is derived for the linear response of a cable/mass suspension having small equilibrium curvature and horizontal supports. While the asymptotic model remains rich enough to capture dominant sagged cable effects, it is simple enough to permit closed-form analysis. A free vibration analysis is pursued that leads to closed-form solutions for problems which, heretofore, were analyzed using purely numerical methods. Among the advantages of the present method is its ability to pro vide results for: (1) complex mass arrays, (2) high-order modes, and (3) dynamic cable tension. Examples highlight the key role played by mass array symmetry and lead to new conclusions regarding free vibration.

scholarly journals An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Winkler Foundation ◽  
Closed Form Solutions ◽  
Stiffness Matrices

This study presents an analytical solution to the free vibration analysis of a uniform Timoshenko beam. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. A material law combining bending, shear, curvature, and natural frequency is presented. This complete study is based on this material law and closed-form solutions are found. The free vibration response of single-span systems, as well as that of spring-mass systems, is analyzed. Closed-form formulations of matrices expressing the boundary conditions are presented; the natural frequencies are determined by solving the eigenvalue problem. First-order dynamic stiffness matrices in local coordinates are determined. Finally, second-order analysis of beams resting on an elastic Winkler foundation is conducted.

Large Amplitude Forced Vibration Analysis of Stiffened Plates under Harmonic Excitation

2012 ◽  
pp. 26-61
Author(s):  
Anirban Mitra ◽  
Prasanta Sahoo ◽  
Kashinath Saha
Keyword(s):  
Free Vibration ◽  
Forced Vibration ◽  
Large Amplitude ◽  
Vibration Analysis ◽  
Mathematical Formulation ◽  
Free Vibration Analysis ◽  
Harmonic Excitation ◽  
Stiffened Plates ◽  
Backbone Curve ◽  
Forced Vibration Analysis

Large amplitude forced vibration behaviour of stiffened plates under harmonic excitation is studied numerically incorporating the effect of geometric non-linearity. The forced vibration analysis is carried out in an indirect way in which the dynamic system is assumed to satisfy the force equilibrium condition at peak excitation amplitude. Large amplitude free vibration analysis of the same system is carried out separately to determine the backbone curves. The mathematical formulation is based on energy principles and the set of governing equations for both forced and free vibration problems derived using Hamilton’s principle. Appropriate sets of coordinate functions are formed by following the two dimensional Gram-Schmidt orthogonalization procedure to satisfy the corresponding boundary conditions of the plate. The problem is solved by employing an iterative direct substitution method with an appropriate relaxation technique and when the system becomes computationally stiff, Broyden’s method is used. The results are furnished as frequency response curves along with the backbone curve in the dimensionless amplitude-frequency plane. Three dimensional operational deflection shape (ODS) plots and contour plots are provided in a few cases.

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