Semi-Discrete Analytical Solution of Lumped Mass Finite Element Method for the Longitudinal Vibrations of an Elastic Bar

2013 ◽  
Vol 273 ◽  
pp. 234-239
Author(s):  
Jing Song Pan ◽  
Ji Chan Wang
Keyword(s):  
Finite Element Method ◽  
Exact Solution ◽  
Finite Element ◽  
Analytical Solution ◽  
Longitudinal Vibration ◽  
Analytical Approximation ◽  
Longitudinal Vibrations ◽  
Lumped Mass ◽  
Elastic Bar ◽  
Element Method

The objective of this paper is to present semi-discrete analytical method for the longitudinal vibration of an elastic bar. Using lumped mass finite element method, we first obtain a system of second order ordinary differential equations. In terms of some transform technique we obtain the exact solution to the system, i.e. excellently semi-discrete analytical approximation to the longitudinal vibration. An example is given to illustrate the effectiveness of the proposed method.

scholarly journals Evaluation of Pile’s Buckling Under Axial Load by B-Spline Method and Comparison With Finite Element Method and Exact Solution

2018 ◽  
Vol 8 (2) ◽  
pp. 29-34
Author(s):  
A. Moghaddam ◽  
A. Nayeri ◽  
S.M. Mirhosseini
Keyword(s):  
Finite Element Method ◽  
Exact Solution ◽  
Finite Element ◽  
Numerical Methods ◽  
High Volume ◽  
Spline Method ◽  
B Spline ◽  
Volume Calculations ◽  
Reaction Coefficient ◽  
Element Method

Abstract Although various analytical and numerical methods have been proposed by researchers to solve equations, but use of numerical tools with low volume calculations and high accuracy instead of other numerical methods with high volume calculations is inevitable in the analysis of engineering equations. In this paper, B-Spline spectral method was used to study buckling equations of the piles. Results were compared with the calculated amounts of the exact solution and finite element method. Uniform horizontal reaction coefficient has been used in most of proposed methods for analyzing buckling of the pile on elastic base. In reality, soil horizontal reaction coefficient is nonlinear along the pile. So, in this research by using B-Spline method, buckling equation of the pile with nonlinear horizontal reaction coefficient of the soil was investigated. It is worth mentioning that B-Spline method had not been used for buckling of the pile.

scholarly journals Analysis and Research on Longitudinal Vibration Characteristics of Deep Sea Mining Pipe Based on Finite Element Method

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Qiang Liu ◽  
Linjing Xiao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Frequency ◽  
Vibration Amplitude ◽  
Longitudinal Vibration ◽  
Excitation Frequency ◽  
Vibration Characteristics ◽  
Wind Condition ◽  
Vibration Intensity ◽  
Element Method

This paper aimed to study the longitudinal vibration characteristics of the 5000 m mining pipe in the ocean under different working wind conditions, offset angle, damping, and ore bin weight. Based on the finite element method, the mining pipe is simplified into beam element and discretized, and the physical and mathematical models of the mining pipe system are established. The Wilson-θ direct integral method is adopted for numerical calculation. The results show that the longitudinal vibration of the mining pipe is irregular, which presents the phenomenon of oscillation. The vibration amplitude decreases first and then increases from top to bottom, the minimum vibration amplitude appears at 1000 m, and the maximum vibration amplitude appears at the top of the mining pipe. Under the same working wind condition, the overall longitudinal vibration amplitude of the mining pipe can be increased by increasing the ore bin weight and the offset angle, but neither of them can change the frequency of the longitudinal vibration. The closer the excitation frequency generated by different working wind conditions is to the natural frequency, the larger the mining pipe longitudinal vibration amplitude is. The closer the vibration frequency generated by the same excitation frequency is to the natural frequency, the stronger the vibration intensity is, and when damping is added, the vibration intensity decreases faster.

Sign in / Sign up

Export Citation Format

Share Document