Free Vibration Frequencies of Simply Supported Bars with Variable Cross Section

2021 ◽  
pp. 339-350
Author(s):  
Olga Szlachetka ◽  
Jacek Jaworski ◽  
Marek Chalecki
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Vibration Frequencies ◽  
Simply Supported

scholarly journals First natural frequency of multi-segment floor joists with variable cross section

2019 ◽  
Vol 28 (4) ◽  
pp. 526-538
Author(s):  
Marek Chalecki ◽  
Jacek Jaworski ◽  
Olga Szlachetka
Keyword(s):  
Natural Frequency ◽  
Cross Section ◽  
Constant Width ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Uniform Load ◽  
Simply Supported ◽  
Made In

The Rayleigh’s method can be used to determine the first natural frequency of beams with variable cross-section. The authors analyse multi-segment simply supported beams, symmetrical with respect to their midpoint, having a constant width and variable height. The beams consist generally of five segments. It has been assumed that the neutral bar axis deflected during vibrations has a shape of a beam deflected by a static uniform load. The calculations were made in Mathematica environment and their results are very close to those obtained with FEM.

Free Vibration of a Simply Supported Bar With a Linearly Variable Height of Cross Section

1962 ◽  
Vol 29 (3) ◽  
pp. 497-501 ◽  
Author(s):  
E. Krynicki ◽  
Z. Mazurkiewicz
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Exact Solutions ◽  
Cross Section ◽  
Approximate Solutions ◽  
Variable Cross ◽  
Simply Supported ◽  
Special Cases ◽  
Formal Integration ◽  
Simple Integration

The problem of vibration of nonhomogeneous bars or bars of variable cross section1 leads to differential equations, which are generally unsolvable by formal integration. It is known that functional coefficients occur in these equations, which make it difficult, if not impossible, to obtain exact solutions by simple integration. Several exact solutions obtained for a few special cases and also some interesting approximate solutions are mentioned in the paper.

scholarly journals Flexural Free Vibrations of Multistep Nonuniform Beams

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Guojin Tan ◽  
Wensheng Wang ◽  
Yubo Jiao
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Cross Section ◽  
Order Differential Equation ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
Exact Approach ◽  
Constant Coefficients ◽  
One Step

This paper presents an exact approach to investigate the flexural free vibrations of multistep nonuniform beams. Firstly, one-step beam with moment of inertia and mass per unit length varying as I(x)=α11+βxr+4 and m(x)=α21+βxr was studied. By using appropriate transformations, the differential equation for flexural free vibration of one-step beam with variable cross section is reduced to a four-order differential equation with constant coefficients. According to different types of roots for the characteristic equation of four-order differential equation with constant coefficients, two kinds of modal shape functions are obtained, and the general solutions for flexural free vibration of one-step beam with variable cross section are presented. An exact approach to solve the natural frequencies and modal shapes of multistep beam with variable cross section is presented by using transfer matrix method, the exact general solutions of one-step beam, and iterative method. Numerical examples reveal that the calculated frequencies and modal shapes are in good agreement with the finite element method (FEM), which demonstrates the solutions of present method are exact ones.

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